QR SIGNAL DETECTION IN THE PRESENCE OF AM NOISE

The Pennsylvania State University

The Graduate School

Department of Electrical Engineering

QR SIGNAL DETECTION IN THE

PRESENCE OF AM NOISE

A Dissertation in

Electrical Engineering

by

Abdullah G. Almahri

c© 2013 Abdullah G. Almahri

Submitted in Partial Fulfillmentof the Requirements

for the Degree of

Doctor of Philosophy

December 2013

The dissertation of Abdullah G. Almahri was reviewed and approved* by the following:

Constantino C. LagoaProfessor of Electrical EngineeringDissertation AdviserChair of Committee

Jeffrey SchianoProfessor of Electrical Engineering

David MillerProfessor of Electrical Engineering

Patrick M. LenahanProfessor of Engineering Science and Mechanics

Kultegin AydinProfessor of Electrical EngineeringHead of the Department of Electrical Engineering

*Signatures are on file in the Graduate School.

iii

Abstract

The thesis proposes a matched filter approach to detect quadrupole resonance (QR)

signals in the presence of disturbance from AM stations. Detecting QR signals is a

challenge due to several reasons. One is the amplitude of a QR signal is typically on

the same level of thermal noise, which makes it very susceptible to noise interferences.

External Radio Frequency (RF) interferences, such as AM signals, and internal RF

interferences, ones from inside the search volume, pose another challenge and contribute

to the low SNR values observed. AM stations broadcast within the same frequency band

of QR signals, which is a problem for QR detection. A third important challenge we face

is the uncertainty in the QR signal characteristics.

To motivate the use of a matched filter approach, a matched filter (under the assumption

that the QR signal is known) was compared to the generic energy detector in theory and it

resulted in a performance improvement. The work proposes a detector referred to as the

batch matched filter, which uses a gridding technique to search for unknown QR signal

parameters and attempts to match the filter to the shape of the QR signal present. This

approach resulted in a performance gain when compared to the generic energy detector

using simulation and experimental data, where the QR signal is unknown. To further

improve performance we introduced an approach that would also match the filter to the

noise present in addition to the QR signal. This approach is referred to as the batch

whitened matched filter and when properly matched to the noise outperforms both the

batch matched filter and energy detector.

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Table of Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Chapter 2. QR Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 QR Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Quadrupole Interaction . . . . . . . . . . . . . . . . . . . . . 132.1.2 Relaxation Mechanisms . . . . . . . . . . . . . . . . . . . . . 19

2.2 Observed Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 RF Excitation Pulse Sequences . . . . . . . . . . . . . . . . . . . . . 202.4 QR Detection Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter 3. Challenges, Signal Characteristics and Data Generation . . . . . . . 253.1 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Challenges Due to Uncertainty QR Signal . . . . . . . . . . . 253.1.2 Challenges Due to External and Internal RFI Signals . . . . . 28

3.2 RFI Mitigation Methods . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 QR Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4 Noise Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.1 Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4.2 AM Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5 Averaged Nm Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6 Aggregating Nm Trials . . . . . . . . . . . . . . . . . . . . . . . . . . 393.7 Experimental Data Versus Simulation Data . . . . . . . . . . . . . . 41

3.7.1 Experimental Data Collected . . . . . . . . . . . . . . . . . . 413.7.2 Simulation Data Generated . . . . . . . . . . . . . . . . . . . 42

Chapter 4. Motivation for Using Matched Filter . . . . . . . . . . . . . . . . . . 444.1 Energy Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Detection Algorithm Comparison Under Noise Assumptions . . . . . 50

4.3.1 Energy Detector in the Presence of Thermal Noise . . . . . . 534.3.2 Energy Detector in the Presence of AM and Thermal Noise . 564.3.3 Matched Filter in the Presence of Thermal Noise . . . . . . . 60

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4.3.4 Matched Filter in the Presence of AM and Thermal Noise . . 634.3.5 Thermal Noise Comparison . . . . . . . . . . . . . . . . . . . 674.3.6 AM and Thermal Noise Comparison . . . . . . . . . . . . . . 68

4.4 Algorithm Comparison, No Noise Assumptions . . . . . . . . . . . . 694.4.1 Band and Low pass Filtered Thermal Noise . . . . . . . . . . 704.4.2 Band and Low pass Filtered AM and Thermal Noise . . . . . 73

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Chapter 5. Batch Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.1 Error in the QR signal Description . . . . . . . . . . . . . . . . . . . 815.2 Batch Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.3 Batch Matched Filter versus Energy Detector, Unknown QR signal . 90

5.3.1 Simulation Data . . . . . . . . . . . . . . . . . . . . . . . . . 915.3.2 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . 91

5.4 The Effect of Finer Gridding on the Performance of the Batch MatchedFilter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.5 Adaptive Grid Batch Matched Filter . . . . . . . . . . . . . . . . . . 985.5.1 Simulation Data . . . . . . . . . . . . . . . . . . . . . . . . . 1015.5.2 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . 102

5.6 Alternative Detection Decisions . . . . . . . . . . . . . . . . . . . . . 107

Chapter 6. Batch Whitened Matched Filter . . . . . . . . . . . . . . . . . . . . . 1166.1 Whitened Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . 117

6.1.1 Estimating Whitening Matrix using the Autocorrelation Method 1216.1.2 Estimating Whitening Matrix using the Covariance Method . 1236.1.3 The Effect of All Zero Filters On a QR Signal . . . . . . . . . 125

6.2 Batch Whitened Matched Filter . . . . . . . . . . . . . . . . . . . . . 1296.3 Batch Whitened Matched Filter versus Energy Detector, Unknown

QR signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.3.1 Simulation Data . . . . . . . . . . . . . . . . . . . . . . . . . 1356.3.2 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . 152

6.4 Batch Adaptive Whitened Matched Filter . . . . . . . . . . . . . . . 1686.4.1 Whitening Filter Order that Least Effects the QR Signal . . . 1746.4.2 Whitening Filter Order, Minimum Description Length Algo-

rithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.4.3 Simulation Data . . . . . . . . . . . . . . . . . . . . . . . . . 1776.4.4 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . 186

Chapter 7. Batch Whitened Robust Matched Filter . . . . . . . . . . . . . . . . 1937.1 Robust Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 1967.2 Analytical Solutions For Robust Matched Filters Over Particular Un-

certainty Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2007.2.1 Spherical Signal Set and Noise Uncertainty Bounded by a Ma-

trix Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

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7.2.2 Elliptic Signal Set and Noise Uncertainty Bounded by theFrobenius Matrix Norm or the 2-Norm . . . . . . . . . . . . . 202

7.3 The Scenario Approach . . . . . . . . . . . . . . . . . . . . . . . . . 2077.4 Characterizing a Set of QR Signals Through Sampling . . . . . . . . 208

7.4.1 Smallest Sphere Containing the Set of QR Signals . . . . . . 2097.4.1.1 Spherical Set Central Signal Examples . . . . . . . . 211

7.5 Robust Matched Filter by Maximizing SNR Using Sampling . . . . . 2167.5.1 Robust Matched Filter Examples in the presence of Thermal

Noise, Maximizing SNR . . . . . . . . . . . . . . . . . . . . . 2187.6 Batch Whitened Robust Matched Filter . . . . . . . . . . . . . . . . 2227.7 Batch Robust Matched Filter in Presence of Thermal Noise . . . . . 227

7.7.1 Frequency Robust Batch Matched Filters in the Presence ofThermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 228

7.7.2 Robust Batch Matched Filters in the Presence of Thermal Noise 2307.8 Batch Whitened Robust Matched Filter in the Presence of AM and

Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2337.8.1 Simulation Data in the Presence of AM and Thermal Noise . 2337.8.2 Experimental Data in the Presence of AM and Thermal Noise 237

Chapter 8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2418.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Appendix A. Mean and Variance of Energy Detector Test Statistic, in the Presenceof AM and Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . 246

Appendix B. Mean and Variance of Matched Filter Test Statistic, in the Presenceof AM and Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . 256

Appendix C. MATLAB Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259C.1 Data Generation Function . . . . . . . . . . . . . . . . . . . . . . . . 259

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

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List of Tables

6.1 Simulation, SNR = -22 dB, Performance Comparison on SimulationData, Band-passed White Gaussian AM, with the QR and AM at 2.5kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.2 Simulation, SNR = -22 dB, Performance Comparison on SimulationData, Band-passed White Gaussian AM, with the QR and AM at 5kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139





6.7 Performance Comparison on Experimental Data A, BP White GaussianAM, with the QR and AM at 6.25 kHz. . . . . . . . . . . . . . . . . . . 165

6.8 Performance Comparison on Experimental Data B, BP White GaussianAM, with the QR and AM at 10 kHz. . . . . . . . . . . . . . . . . . . . 166

6.9 Performance Comparison on Experimental Data C, BP White GaussianAM, with the QR and AM at -8 kHz. . . . . . . . . . . . . . . . . . . . 167

6.10 Performance Comparison on Experimental Data D, BP White GaussianAM, with the QR and AM at 12.5 kHz. . . . . . . . . . . . . . . . . . . 167

6.11 Simulation, SNR = -22 dB, Performance Comparison on SimulationData, Band-passed White Gaussian AM, with the QR and AM at 2.5kHz. Adaptive Whitening Filter Order Selection. . . . . . . . . . . . . . 179

6.12 Simulation, SNR = -22 dB, Performance Comparison on SimulationData, Band-passed White Gaussian AM, with the QR and AM at 5kHz. Adaptive Whitening Filter Order Selection. . . . . . . . . . . . . . 179




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6.17 Performance Comparison on Experimental Data A, BP White GaussianAM, with the QR and AM at 6.25 kHz. Adaptive Whitening Filter OrderSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.18 Performance Comparison on Experimental Data B, BP White GaussianAM, with the QR and AM at 10 kHz. Adaptive Whitening Filter OrderSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

6.19 Performance Comparison on Experimental Data C, BP White GaussianAM, with the QR and AM at -8 kHz. Adaptive Whitening Filter OrderSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.20 Performance Comparison on Experimental Data D, BP White GaussianAM, with the QR and AM at 12.5 kHz. Adaptive Whitening Filter OrderSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

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List of Figures

2.1 Simplified block diagram of QR detection system . . . . . . . . . . . . . 132.2 QR energy levels and transition frequencies for nitrogen-14, an I=1 nu-

cleus with η 6= 0 [17]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 QR frequencies of different explosives/chemicals [3] . . . . . . . . . . . . 172.4 Lorentzian distribution of transition frequencies [17]. . . . . . . . . . . . 182.5 Block diagram of QR spectrometer . . . . . . . . . . . . . . . . . . . . . 24

3.1 QR Signal with a Frequency of 3.016 MHz and T2∗ of 500e-6 . . . . . . 26

3.2 Comparison of QR+AM and QR: (Top) QR Signal with a Frequency of3.016 MHz and T2

∗ of 500e-6 (Center) AM Signal with Carrier Frequencyof 3.016 MHz (Bottom) QR plus AM Signal, SNR = -12 dB . . . . . . . 30

3.3 Illustration of Phase Cycling on SLSE sequences. . . . . . . . . . . . . . 32

4.1 Energy Detector’s Theoretical PDF plots vs Simulation PDF plots forThermal Noise, SNR = 20 log10( A

σtn) = -50 dB . . . . . . . . . . . . . . 55

4.2 Energy Detector’s, Theoretical ROC plot vs Simulation ROC plot forThermal Noise, SNR = 20 log10( A

σtn) = -50 dB . . . . . . . . . . . . . . 57

4.3 Energy Detector’s Theoretical PDF plots vs Simulation PDF plots forAM and Thermal Noise, SNR = 20 log10( A

Aη) = -66 dB . . . . . . . . . . 59

4.4 Energy Detector’s, Theoretical ROC plot vs Simulation ROC plot forAM and Thermal Noise, SNR = 20 log10( A

Aη) = -66 dB . . . . . . . . . . 61

4.5 Matched Filter’s Theoretical PDF plots vs Simulation PDF plots forThermal Noise, SNR = 20 log10( A

σtn) =-50 dB . . . . . . . . . . . . . . . 63

4.6 Theoretical ROC plot vs Simulation ROC plot for Thermal Noise, SNR= 20 log10( A

σtn) = -50 dB . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.7 Matched Filter’s Theoretical PDF plots vs Simulation PDF plots for AMand Thermal Noise, SNR = 20 log10( A

Aη) = -66 dB . . . . . . . . . . . . 66

4.8 Theoretical ROC plot vs Simulation ROC plot for AM & Thermal Noise,SNR = 20 log10( A

Aη) = -66 dB . . . . . . . . . . . . . . . . . . . . . . . . 68

4.9 Energy of noise signals vs number of averages, Log scale. . . . . . . . . . 704.10 Energy Detector PDF plots in the presence of Thermal Noise, SNR =

20 log10( Aσtn

) =-12 dB, Theoretical PDF plots vs Simulation and Exper-iment without Noise Assumptions . . . . . . . . . . . . . . . . . . . . . . 72

4.11 Matched Filter PDF plots in the presence of Thermal Noise, SNR =20 log10( A

σtn) = -12 dB, Theoretical PDF plots vs Simulation and Exper-

iment without Noise Assumptions . . . . . . . . . . . . . . . . . . . . . . 744.12 Energy Detector PDF plots in the presence of band-passed white gaussian

AM and Thermal Noise, SNR = 20 log10( AAη

) = -30 dB, Theoretical PDFplots vs Simulation and Experiment without Noise Assumptions. Nm =10, Number of experiments = 50. . . . . . . . . . . . . . . . . . . . . . . 75

x

4.13 Energy Detector ROC plots in the presence of band-passed white gaus-sian AM Thermal Noise, SNR = 20 log10( A

Aη)= -30 dB, Theoretical PDF

plots vs Simulation and Experiment without Noise Assumptions. Nm =10, Number of experiments = 50. . . . . . . . . . . . . . . . . . . . . . . 76

4.14 Matched Filter PDF plots in the presence of band-passed white gaussianAM and Thermal Noise, SNR = 20 log10( A

Aη) = -30 dB, Theoretical PDF

plots vs Simulation and Experiment without Noise Assumptions. . . . . 774.15 Matched Filter PDF plots in the presence of band-passed white gaussian

AM and Thermal Noise, SNR = 20 log10( AAη

) = -30 dB, Theoretical PDFplots vs Simulation and Experiment without Noise Assumptions. . . . . 77

5.1 Inner Product versus Frequency Error of the Filter . . . . . . . . . . . . 835.2 Inner Product versus Phase Error of the Filter . . . . . . . . . . . . . . 845.3 Inner Product versus Decaying Parameter Error of the Filter . . . . . . 855.4 Uncertain Frequency Matched Filter ROC plots in the presence of Band-

passed White Gaussian AM, SNR = -30 dB . . . . . . . . . . . . . . . . 865.5 Batch Matched Filter with 1 kHz Frequency gridding, ROC plots in the

presence of Different QR Frequencies and Band-passed White GaussianAM, Simulation SNR = -22 dB . . . . . . . . . . . . . . . . . . . . . . . 87

5.6 Simulation SNR = -22 dB, Performance Comparison, BP White GaussianAM, with the QR and AM at 2.5 kHz . . . . . . . . . . . . . . . . . . . 92

5.7 Simulation SNR = -22 dB, Performance Comparison, BP White GaussianAM, with the QR and AM at 5 kHz . . . . . . . . . . . . . . . . . . . . 92





5.12 Experiment A, Performance Comparison on Experiment Data, BP WhiteGaussian AM, with the QR and AM at 6.25 kHz . . . . . . . . . . . . . 96

5.13 Experiment B, Performance Comparison on Experiment Data, BP WhiteGaussian AM, with the QR and AM at 10 kHz . . . . . . . . . . . . . . 96

5.14 Experiment C, Performance Comparison on Experiment Data, BP WhiteGaussian AM, with the QR and AM at -8 kHz . . . . . . . . . . . . . . 97

5.15 Experiment D, Performance Comparison on Experiment Data, BP WhiteGaussian AM, with the QR and AM at 12.5 kHz . . . . . . . . . . . . . 97

5.16 Simulation SNR = -22 dB, Adaptive versus “Brute Force” Gridding Per-formance Comparison, BP White Gaussian AM, with the QR and AMat 2.5 kHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.17 Simulation SNR = -22 dB, Adaptive versus “Brute Force” Gridding Per-formance Comparison, BP White Gaussian AM, with the QR and AMat 5 kHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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5.18 Simulation SNR = -22 dB, Adaptive versus “Brute Force” Gridding Per-formance Comparison, BP White Gaussian AM, with the QR and AMat 7.5 kHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.19 Simulation SNR = -22 dB, Adaptive versus “Brute Force” Gridding Per-formance Comparison, BP White Gaussian AM, with the QR and AMat 10 kHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.20 Simulation SNR = -22 dB, Adaptive versus “Brute Force” Gridding Per-formance Comparison, BP White Gaussian AM, with the QR and AMat 12.5 kHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.21 Simulation SNR = -22 dB, Adaptive versus “Brute Force” Gridding Per-formance Comparison, BP White Gaussian AM, with the QR and AMat 15 kHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.22 Experiment A, Adaptive versus “Brute Force” Gridding PerformanceComparison on Experiment Data, BP White Gaussian AM, with the QRand AM at 6.25 kHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.23 Experiment B, Adaptive versus “Brute Force” Gridding PerformanceComparison on Experiment Data, BP White Gaussian AM, with theQR and AM at 10 kHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.24 Experiment C, Adaptive versus “Brute Force” Gridding PerformanceComparison on Experiment Data, BP White Gaussian AM, with theQR and AM at -8 kHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.25 Experiment D, Adaptive versus “Brute Force” Gridding PerformanceComparison on Experiment Data, BP White Gaussian AM, with the QRand AM at 12.5 kHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.26 Simulation SNR = -22 dB, Comparing Detection Decision Methods, BPWhite Gaussian AM, with the QR and AM at 2.5 kHz . . . . . . . . . . 110

5.27 Simulation SNR = -22 dB, Comparing Detection Decision Methods, BPWhite Gaussian AM, with the QR and AM at 5 kHz . . . . . . . . . . . 111

5.28 Simulation SNR = -22 dB, Comparing Detection Decision Methods, BPWhite Gaussian AM, with the QR and AM at 7.5 kHz . . . . . . . . . . 112

5.29 Simulation SNR = -22 dB, Comparing Detection Decision Methods, BPWhite Gaussian AM, with the QR and AM at 10 kHz . . . . . . . . . . 112

5.30 Simulation SNR = -22 dB, Comparing Detection Decision Methods, BPWhite Gaussian AM, with the QR and AM at 12.5 kHz . . . . . . . . . 113

5.31 Simulation SNR = -22 dB, Comparing Detection Decision Methods, BPWhite Gaussian AM, with the QR and AM at 15 kHz . . . . . . . . . . 113

5.32 Experiment A, Comparing Detection Decision Methods on ExperimentData, BP White Gaussian AM, with the QR and AM at 6.25 kHz. . . . 114

5.33 Experiment B, Comparing Detection Decision Methods on ExperimentData, BP White Gaussian AM, with the QR and AM at 10 kHz. . . . . 114

5.34 Experiment C, Comparing Detection Decision Methods on ExperimentData, BP White Gaussian AM, with the QR and AM at -8 kHz. . . . . 115

5.35 Experiment D, Comparing Detection Decision Methods on ExperimentData, BP White Gaussian AM, with the QR and AM at 12.5 kHz. . . . 115

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6.1 The output of a 6th order FIR filter applied to a simulated 12.5 kHz QRsignal is compared to the input signal. . . . . . . . . . . . . . . . . . . . 129

6.2 Magnitude of the Frequency Response of an 8 pole Butterworth filterwith a 20 kHz cutoff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.3 Magnitude (dB) of the Frequency Response of an 8 pole Butterworthfilter with a 20 kHz cutoff. . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.4 Simulation, SNR = -22 dB, Performance Comparison on SimulationData, Band-passed White Gaussian AM, with the QR and AM at 2.5kHz. Autocorrelation Whitening Method. . . . . . . . . . . . . . . . . . 142

6.5 Simulation, SNR = -22 dB, Performance Comparison on SimulationData, Band-passed White Gaussian AM, with the QR and AM at 2.5kHz. Covariance Whitening Method. . . . . . . . . . . . . . . . . . . . . 142

6.6 Simulation, SNR = -22 dB, Performance Comparison on SimulationData, Band-passed White Gaussian AM, with the QR and AM at 5kHz. Autocorrelation Whitening Method. . . . . . . . . . . . . . . . . . 143

6.7 Simulation, SNR = -22 dB, Performance Comparison on SimulationData, Band-passed White Gaussian AM, with the QR and AM at 5kHz. Covariance Whitening Method. . . . . . . . . . . . . . . . . . . . . 143









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6.16 Top: Frequency Response of Modulating Signal (Bandpassed GaussianNoise). Bottom: Frequency Response of an AM Signal with a CarrierFrequency of fc < 10 kHz. 0 = 0 Hz, 1 = fc- 40 Hz, 2 = fc+40 Hz, 3 =10 kHz - fc and 4 = fc+10 kHz. . . . . . . . . . . . . . . . . . . . . . . 152

6.17 Frequency Response of Whitening Filter When the AM Signal’s CarrierFrequency is fc < 10 kHz. 1 = fc- 40 Hz, 2 = fc+40 Hz, 3 = 10 kHz -fc and 4 = fc+10 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.18 Fast Fourier Transform of a 2.5 kHz QR signal. . . . . . . . . . . . . . . 1536.19 Frequency Response of the Ne = 5 Whitening Filters of Order 3, De-

signed Using the Covariance Method. Simulation Data, SNR = -22 dB,Experiment 1 of Band-passed White Gaussian AM, with the QR and AMat 2.5 kHz Data Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.20 Frequency Response of the Ne = 5 Whitening Filters of Order 4, De-signed Using the Covariance Method. Simulation Data, SNR = -22 dB,Experiment 1 of Band-passed White Gaussian AM, with the QR and AMat 2.5 kHz Data Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154




6.24 Filter Lock Frequency for Different Experiments, When Using A Covari-ance Whitening Filter of Order 3. Simulation Data, SNR = -22 dB,Band-passed White Gaussian AM, with the QR and AM at 2.5 kHz. . . 156





xiv

6.29 Frequency Response of the Ne = 5 Whitening Filters of Order 3, De-signed Using the Autocorrelation Method. Simulation Data, SNR = -22dB, Experiment 1 of Band-passed White Gaussian AM, with the QR andAM at 2.5 kHz Data Set. . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.30 Frequency Response of the Ne = 5 Whitening Filters of Order 4, De-signed Using the Autcorrelation Method. Simulation Data, SNR = -22dB, Experiment 1 of Band-passed White Gaussian AM, with the QR andAM at 2.5 kHz Data Set. . . . . . . . . . . . . . . . . . . . . . . . . . . 159




6.34 Filter Lock Frequency for Different Experiments, When Using An Au-tocorrelation Whitening Filter of Order 3. Simulation Data, SNR = -22dB, Band-passed White Gaussian AM, with the QR and AM at 2.5 kHz. 161





6.39 Performance Comparison on Experiment Data, BP White Gaussian AM,with the QR and AM at 6.25 kHz. Autocorrelation Method. . . . . . . . 168

6.40 Performance Comparison on Experiment Data, BP White Gaussian AM,Covariance Method, with the QR and AM at 6.25 kHz. . . . . . . . . . 169

6.41 Performance Comparison on Experiment Data, BP White Gaussian AM,with the QR and AM at 10 kHz. Autocorrelation Method. . . . . . . . . 169

6.42 Performance Comparison on Experiment Data, BP White Gaussian AM,with the QR and AM at 10 kHz. Covariance Method. . . . . . . . . . . 170

6.43 Performance Comparison on Experiment Data, BP White Gaussian AM,with the QR and AM at -8 kHz. Autocorrelation Method. . . . . . . . . 170

xv

6.44 Performance Comparison on Experiment Data, BP White Gaussian AM,with the QR and AM at -8 kHz. Covariance Method. . . . . . . . . . . . 171

6.45 Performance Comparison on Experiment Data, BP White Gaussian AM,with the QR and AM at 12.5 kHz. Autocorrelation Method. . . . . . . . 171

6.46 Performance Comparison on Experiment Data, BP White Gaussian AM,with the QR and AM at 12.5 kHz. Covariance Method. . . . . . . . . . 172

6.47 Simulation, SNR = -22 dB, Performance Comparison on SimulationData, Band-passed White Gaussian AM, with the QR and AM at 2.5kHz. Whitened with either the Autocorrelation Method or the Covari-ance Method using an Adaptive Filter Order that either Minimizes theEffect on the QR Signal or based on the MDL algorithm. . . . . . . . . 183

6.48 Simulation, SNR = -22 dB, Performance Comparison on SimulationData, Band-passed White Gaussian AM, with the QR and AM at 5kHz. Whitened with either the Autocorrelation Method or the Covari-ance Method using an Adaptive Filter Order that either Minimizes theEffect on the QR Signal or based on the MDL algorithm. . . . . . . . . 183





6.53 Performance Comparison on Experiment Data A, BP White GaussianAM, with the QR and AM at 6.25 kHz. Whitened with either the Auto-correlation Method or the Covariance Method using an Adaptive FilterOrder that either Minimizes the Effect on the QR Signal or based on theMDL algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

xvi

6.54 Performance Comparison on Experiment Data B, BP White GaussianAM, with the QR and AM at 10 kHz. Whitened with either the Auto-correlation Method or the Covariance Method using an Adaptive FilterOrder that either Minimizes the Effect on the QR Signal or based on theMDL algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

6.55 Performance Comparison on Experiment Data C, BP White GaussianAM, with the QR and AM at -8 kHz. Whitened with either the Auto-correlation Method or the Covariance Method using an Adaptive FilterOrder that either Minimizes the Effect on the QR Signal or based on theMDL algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

6.56 Performance Comparison on Experiment Data D, BP White GaussianAM, with the QR and AM at 12.5 kHz. Whitened with either the Auto-correlation Method or the Covariance Method using an Adaptive FilterOrder that either Minimizes the Effect on the QR Signal or based on theMDL algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

7.1 Central Signal, For the Set of QR Signals with Fixed Phase and T ∗2

andFrequency Values Between 12.25 kHz and 12.75 kHz. . . . . . . . . . . . 212


andFrequency Values Between 11.5 kHz and 12.5 kHz. . . . . . . . . . . . . 212


andFrequency Values Between 12 kHz and 14 kHz. . . . . . . . . . . . . . . 213

7.4 Central Signal, For the Set of QR Signals with Fixed Frequency andPhase and T ∗

2Values Between 400e-6 and 800e-6. . . . . . . . . . . . . . 214

7.5 Central Signal, For the Set of QR Signals with Fixed Frequency and T ∗2

and Phase Values Between -0.4 and 0.4 Radians. . . . . . . . . . . . . . 2147.6 Central Signal, of the Smallest Sphere Containing the Set of QR Signals

with Frequency Values Between 11. 5 kHz and 12.5 kHz, T ∗2

Values Be-tween 400e-6 and 800e-6, and Phase Values Between -0.4 and 0.4 Radians. 215

7.7 Robust Matched Filter, For Thermal Noise and the Samples from the Setof QR Signals with Fixed Phase and T ∗

2and Frequency Values Between

12.25 kHz and 12.75 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . 2197.8 Robust Matched Filter, For Thermal Noise and the Samples from the Set

of QR Signals with Fixed Phase and T ∗2

and Frequency Values Between11.5 kHz and 12.5 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

7.9 Robust Matched Filter, For Thermal Noise and the Samples from the Setof QR Signals with Fixed Phase and T ∗

2and Frequency Values Between

12 kHz and 14 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2207.10 Robust Matched Filter, For Thermal Noise and the Samples from the Set

of QR Signals with Fixed Frequency and Phase and T ∗2

Values Between400e-6 and 800e-6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

7.11 Robust Matched Filter, For Thermal Noise and the Samples from the Setof QR Signals with Fixed Frequency and T ∗

2and Phase Values Between

-0.4 and 0.4 Radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

xvii

7.12 Robust Matched Filter, For Thermal Noise and the Samples from theSet of QR Signals with Frequency Between 11.5 kHz and 12.5 kHz, T ∗

2Between 400e-6 and 800e-6 and Phase Between -0.4 and 0.4 Radians. . . 223

7.13 Robust Matched Filter, For Thermal Noise and the Set of QR SignalSamples Versus the Robust Matched Filter, For Thermal Noise and theSmallestSphere Containing the QR Signal Samples. . . . . . . . . . . . . 223

7.14 Comparing the Robust Matched Filter for the set covering the frequencies11.5 kHz to 12.5 kHz, to the sample signals with the largest and smallestfrequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

7.15 Simulation, SNR = -16 dB, Thermal Noise, Batch of Robust MatchedFilters versus using a Batch Matched Filter, The QR signal parameters,decaying parameter, T ∗

2, and phase ,φ, remained constant while the fre-

quency varied between 5 kHz to 15 kHz. . . . . . . . . . . . . . . . . . . 2297.16 Simulation, SNR = -12 dB, Thermal Noise, Batch of Robust Matched

Filters versus using a Batch Matched Filter, The QR signal parametersvaried as follows, frequency between 5 kHz to 15 kHz, decaying parameterT ∗

2between 400e-6 and 800e-6, and phase φ between −π/2 to π/2. . . . 232

7.17 Simulation, SNR = -18 dB, BPWGAM Noise, Batch of Robust MatchedFilters versus using a Batch Matched Filter, The QR signal parametersvaried as follows, frequency between 5 kHz to 15 kHz, decaying parameterT ∗

2between 400e-6 and 800e-6, and phase φ between −π/2 to π/2. . . . 235

7.18 Simulation, SNR = -18 dB, BPWGAM Noise, Batch of Whitened RobustMatched Filters versus Batch Whitened Matched Filters, QR signal pa-rameters varied as follows, frequency between 5 kHz to 15 kHz, decayingparameter T ∗

2between 400e-6 and 800e-6, and phase φ between −π/2 to

π/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

xviii

Acknowledgments

I would like to thank everybody who influenced the completion of this thesis in one way

or another. First and foremost, I would like to thank the source of all knowledge and

reason, the Almighty Allah, for making things work out for me.

On the personal side, this thesis is the end product of unwavering support from my

loving family. My parents, my father who I look to as a role model and my mother who

never stops praying for me, constantly pushed me towards the pursuit of knowledge and

wisdom. My brother, Faisal, and two sisters, Ghalia and Sara, have always inspired me

and supported in my strive for success.

I would also like to express immeasurable gratitude to my adviser, Dr. Constantino

M. Lagoa, for his invaluable, scholarly insights, guidance, encouragement and unfailing

support. In addition to being an outstanding teacher and a seasoned scholar, Dr. Lagoa

was a caring coach, a morale booster, and a supporter at times when I was about to falter.

I would also like to thank each one of my committee members, Dr. Jeffrey Schiano, Dr.

David Miller and Dr. Patrick M. Lenahan for their valuable input and comments that

helped fine tune this document.

The many students, teachers, and social supervisors who gave me their trust and their

time and shared their deeply-felt beliefs and attitudes with me, made a qualitative contri-

bution to the development of this thesis. I have learned a lot from them, and understand

xix

them to be sources of knowledge in their own right. Furthermore, I’m grateful to the

many friends who have supported me and encouraged me throughout the years.

I also acknowledge, for the record and from the heart, my debt to the Abu-Dhabi National

Oil Company for sponsoring my graduate studies at the Pennsylvania State University.

I also thank in particular the dedicated staff of the Scholarship Department at ADNOC,

for their assistance and support throughout my academic journey.

1

Chapter 1

Introduction

1.1 Motivation

The idea of using quadrupole resonance (QR) spectroscopy as an explosive detection

technology started more than 30 years ago in an attempt to detect improvised explosive

devices used against American soldiers during the Vietnam war [36; 29]. The North Viet-

namese forces would recycle American munitions as satchel charges and seed roadways

with them. Metal detectors were unable to detect these satchels loaded with explosives.

Hirschfeld proposed that NQR might provide a means for directly detecting the explo-

sive material, and therefore provide a means for discriminating between the explosive

satchels and decoys [30]. Marino [36] was the first to detect NQR signals in RDX (which

makes up approximately 91% of C4 [1]), and he later presented a review paper on NQR

spectroscopy of explosive materials that included TNT, PETN (which has been used

by both the underwear and shoe bombers [33; 2]), RDX, and HMX. Research funding

diminished after the withdrawal of American forces from Vietnam in 1973. Fifteen years

later, the destruction of a Pan AM Flight 103 over Scotland restored interest in explosive

detection.

2

1.2 Literature Review

Researchers at the Naval Research Laboratory (NRL) noted that x-ray detection sys-

tems and magnetometers used at aviation security points are unable to detect plastic

explosives. This led to the development of NQR technology for civil aviation security.

Buess showed that a pulsed NQR spectrometer can detect sub-kilogram quantities of

explosives [34; 35]. At least two commercial NQR detection systems have been devel-

oped. Quantum Magnetics, now a subsidiary of Morpho, in San Diego, California, and

British Technology Group (BTG) in conjunction with Smith et al., at King’s College in

London, have produced NQR detection systems for narcotics and explosives detection

in airline baggage. Recently, the SEE Corporation in Perth, Australia, has also started

work on NQR detection systems for aviation, landmine, and postal applications. With

funding from DARPA, Quantum Magnetics also conducted field trials of an NQR system

for detection of mines containing RDX.

Researchers in the former Soviet Union began investigating NQR as a means to detect

AT landmines during the war in Afghanistan. Grechishkin, at the Kaliningrad State

University in Russia, developed an NQR detection system that could sweep a one-square

meter area in ten seconds with a detection rate over ninety percent for mines buried

within 10 cm of the surface [63]. His group also demonstrated that the NQR system

could detect 2.5 kg of RDX buried 35 cm underground using a RF power level of 1 kW

[25]. Recently, Grechishkin described a method for determining the burial depth based

on finding the optimal frequency offset in a RF pulse sequence [26].

3

In addition to the mentioned systems, there are several other QR explosive detection

prototypes such as the chemical sniffers and others that even combine x-ray and QR

detection technology. While the technology has progressed significantly in the last three

decades, present day detectors still suffer from high false alarm rates [9]. The physical

basis for QR detection is the electrical properties of atomic nuclei and their surround-

ing electronic environment [52]. Atomic nuclei with spin angular momentum greater

than one-half possess both an electric quadrupole moment and a magnetic dipole mo-

ment, and are referred to as quadrupolar nuclei. If a quadrupolar nucleus experiences

an electric-field gradient tensor due to the surrounding electric charges, the resulting

electrostatic interaction energy produces preferred orientations of the nucleus. It is pos-

sible to perturb the orientation of quadrupolar nuclei by subjecting them to an external

radio frequency (RF) magnetic field, at a resonant frequency that is material dependent.

As the resonant frequency is strongly dependent on the electric field gradient tensor,

different chemical compounds containing the same quadrupolar nuclei will have distinct

resonant frequencies [19].

As of now, no QR system has been approved for civil aviation security by the Transporta-

tion Security Administration.The low success rate of these explosive detection machines

is due to the several challenges presented next. Threat quantities of explosives are not

easy to detect, due to four main obstacles. The first, is that the amplitude of a QR signal

is typically on the same level if not smaller than the amplitude of thermal noise, which

makes it very susceptible to noise interference [9]. External RF interferences such as AM

signals pose another challenge, and they are another source of noise that contributes to

4

the low SNR values. AM stations broadcast within the same frequency band of QR sig-

nals, which is a problem for QR detection. Internal noise sources (ones from within the

search volume such as RF interferences) which include ringing produced by the search

coil [9; 45] and piezoelectric responses, pose another challenge. The excitation of a QR

response requires the application of a pulsed RF magnetic field within the search volume.

Currents induced within conductive materials located in the search volume cause decay-

ing magnetic fields that lead to unacceptable false alarm rates. The fourth challenge we

face is uncertainty in the QR signal characteristics. The signal may contain more than

a single frequency, depending on the temperature and strains of the explosive material.

Explosive material within a bomb is required to have a uniform temperature to obtain

a signal with a very narrow bandwidth. We also face uncertainty in the decaying shape

of the QR signal. The envelope of the QR signal is often thought of as a Lorentzian

distribution, though at times it may look more Gaussian.

To overcome these challenges, several attempts at increasing the SNR ratio have been

made. Some sought to increase the SNR ratio by increasing the amplitude of the QR

signal. Smith et al. attempted this through interweaving of different pulse sequences [54].

Schiano et al. used feedback optimization to optimize the pulse parameters and gain an

increase in the QR signal [31]. Schiano also used narrowband superconducting HTS coils

to gain an increase in SNR [49]. The HTS coils managed to amplify the QR signal by

orders of magnitude and at the same time suppress noise due to their narrow passband.

Unfortunately, the HTS coil also amplifies any noise that falls within its passband as

much as it would amplify the QR signal. Another approach to increasing the SNR,

5

is to try to decrease the noise. Ernst, [21], showed that, for the case of uncorrelated

and stationary noise, signal averaging is an efficient and simple method to decreasing

the noise. Signal averaging decreases the standard deviation of the noise by the square

root of the number of averages. Another way to decrease the noise is to shield from

external RF interferences, which is impractical when attempting to detect explosives

in land mines and on humans at aviation security check points due to claustrophobic

experiences. Suits proposed using a gradiometer [56], which is sensitive only to spatial

gradients of the magnetic fields, as another approach to limiting the level of interference

which enters the receiver.

Others have used signal processing methods to improve detection and false alarm rates.

Since this work focuses on signal processing algorithms, only literature with a common

focus will be reviewed. The most widely used signal detection method is the energy

detector, due to its simplicity. The detector transforms the collected QR signal into the

frequency domain and the power at the frequency bin of interest is calculated. Then,

using a preset threshold, the presence of the target of interest is determined. According

to [45] this method works well when the signal-to-interference-plus-noise ratio (SNR) is

high. Although in the more practical scenario of land mine detection, where the SNR

ratio is usually low [4; 5], it becomes difficult to obtain a good performance rate using

this method alone.

Other signal processing algorithms focused on RFI mitigation. Tantum et al. [46] used

an adaptive noise cancellation method to reduce the RFI’s for QR. This method is used

in a similar fashion in QM’s active approach for RFI reduction [9]. By using a 1-tap

6

least mean squares algorithm, it has been reported that the adaptive noise cancellation

method [64; 60] can reduce the RFI’s by almost 40 dB [46]. The drawback however, is

that this method may amplify noise from signal cancellation, [64]. RFI mitigation for

landmine detection by QR was also investigated in Liu et al. [22]. They exploited both

the spatial and temporal correlation of the RFI’s and proposed a combined approach to

mitigate the RFI’s efficiently and effectively improve the TNT detection performance.

They first considered only exploiting the spatial correlation of the RFI’s and proposed

a maximum likelihood (ML) estimator for signal amplitude estimation and a constant

false alarm rate (CFAR) detector for TNT detection. Then, they used a multi-channel

autoregressive (MAR) model to take into account the temporal correlation of the RFI’s.

Third, they made use of the spatial and temporal correlations of the RFI’s using a (2-

D) robust Capon beamformer (RCB) followed by the ML method for improved RFI

mitigation. Finally, they combined the merits of all the three methods and applied it

to TNT detection. Using experimental results they showed that the combined method

outperforms all the three proposed methods but still does not provide enough of an SNR

improvement to robustly detect a QR signal.

Another group focused on estimation as an approach for QR detection. One example

is the average power detector based on a power spectral estimation algorithm, which

has been proposed by Tan et al. in [69]. It has been reported in [69] that this detector

outperforms the non-adaptive Bayesian detector by using distinguishable features of the

QR signal and RFI in the frequency domain [59]. However, just like the energy detector,

the average power detector suffers from low SNR and, therefore it is preferably used after

7

RFI mitigation. Tan et al. [70] have derived a Cramer-Rao lower bound by considering

the RFI as a colored non-Gaussian process. A two-step adaptive Kalman filter to estimate

and detect a QR signal in the post-mitigation signal [59; 71] has been proposed by Tan

et al. It has been shown in [71] that this method can provide robust landmine detection

performance. However, to obtain the coefficient and covariance matrices, this method

requires training data, which might not be available.

Signal amplitude estimation, with known signal waveform and phase delays, is another

method that has been used for landmine detection by Jiang et al. [68; 67]. In [68] they

proposed a maximum likelihood (ML) estimator and a Capon estimator and derived

closed-form expressions for the bias and mean-squared errors of both estimators in the

the presence of spatially colored but temporally white interference and noise [68]. Both

of these estimators have also been shown to be asymptotically statistically efficient for

large data snapshots. To consider the more general case where the interference and

noise are both temporally spatially colored, an alternative least square (ALS) method

has also been proposed. Using numerical simulations, in [68], Jiang et al. showed that in

most cases the ALS approach outperforms the model-mismatched maximum likelihood

(M3L) method, which ignores the temporal correlation of the interference noise. On the

other hand the M3L is slightly better in worst cases, when the desired signal and the

interference are closely spaced in the temporal frequency domain. Both these methods

work well in the particular situations mentioned, though neither of them can robustly

detect a QR signal.

8

Other QR signal detection algorithms have been studied by Stegna [55]. These include

the Bayesian method and the maximum entropy (ME) method. The Bayesian method

has been reported to be the most robust method against noise. However, it requires a pri-

ori information which may not be available. The ME method has been shown to degrade

rapidly as the SNR decreases and has been reported to be the most computationally

intensive among the three methods.

Jakobsson et al. used the characteristic of temperature dependency of the QR frequencies

to develop several methods for QR signal detection. Among these is a non-linear least

squares method, an approximate maximum likelihood detector (AML), and a frequency

selective AML detector [7; 8; 6].

In regards to using the matched filter as a possible detector, some only mentioned it is

a possible detector, if the exact QR signal was known, and did not evaluate its perfor-

mance. Others proposed using filters matched to signals other than QR signal. Tan [59]

and Garroway et al. [11] have proposed using the matched filter as a possible filter to

maximize SNR when the signal s is a deterministic one and the noise is white, though

they have not evaluated its performance. Tan has also developed a filter, the complex-

valued quadrature matched filter, using the generalized likelihood ratio test under the

assumption that the RFI noise is white after RFI mitigation and averaging [59]. This

filter is referred to as a matched filter, since it uses estimates of the QR signal’s param-

eter but is actually different than what is generally referred to as a matched filter. The

filter assumes perfect demodulation of the QR signal s, i.e. that the resonant frequency

9

is exactly known, and the magnitude and phase of the complex envelope of the QR sig-

nal are found using maximum likelihood estimates and knowledge of their distributions.

The resulting filter, after demodulation and RFI mitigation, demodulates the complex

envelope of the QR signals by assuming its frequency is where the FFT of the envelope

peaks. The signal is then segmented and the square magnitude is averaged and compared

to a threshold.

Others have proposed or used filters that are matched to signals other than the QR

signal to detect explosives. Goldman et al.[24] developed a method that sends an elec-

tromagnetic signal into the ground and receives a response. The response is processed to

generate an image and determine whether a mine is present. He states that the SNR ratio

can be improved by using a matched filter and then continues to state that the matched

filter response is unknown and the step will be skipped. Barrall et. al [23] worked on

a method to cancel extraneous signals by irradiating the target with a specific sequence

of electromagnetic pulses referred to as SLIME. QR signals of one phase are subtracted

from QR signals having the opposite phase, resulting in a cumulative echo signal and

simultaneously subtracting out the same-phase extraneous signal. He proposed using

a weighing factor when averaging the echoes to increase the SNR. That is due to the

fact that the SNR decreases with time after transmission of the excitation pulse. The

weighting factors are chosen so that the weighting assigned to the echoes corresponds

to the decay envelope of the echo signal. He refers to this as matched filter exponential

weighting; i.e. matched to an exponential function with a decay constant. Bulsara et.

al [10] designed a stochastic resonator signal detector to detect the presence of a QR

10

signal. A stochastic resonator comprises of a multistable nonlinear device for coupling

to an input signal and a control signal coupled to the multi-stable nonlinear device for

varying asymmetry among stable states of the multi-stable nonlinear device. The in-

teraction of the input signal with the control signal in the multistable nonlinear device

generates an output signal having an amplitude responsive to the input signal amplitude

and a frequency range that comprises harmonics from the product of the control signal

and the input signal. The matched filter is used to detect the presence of harmonics in

these frequencies.

1.3 Approach

Although all the previously mentioned methods contributed to the problem of robustly

detecting a QR signal in the presence of AM RFI’s, none were capable of completely

solving the problem. These methods have also not exploited the shape of the QR signal.

Even though the exact description of the QR signal is unknown, the general shape is.

This leads to the idea of matched filters and robust matched filters, which is the approach

proposed in this thesis.

The thesis first motivates the use of matched filters in QR detection by showing the im-

provements in detection rates when more information about the signal is used. Though,

to be able to evaluate and compare the performance of the matched filter to the generic

energy detector we assume that exact knowledge of the QR signal is attainable, an

assumption placed just for the sake of an elementary comparison. Receiver operating

characteristic (ROC) plots, a graphical plot of sensitivity vs specificity, are used as a

11

measure of comparison between the two detection algorithms. Theoretical calculations

of the distribution of these plots will be compared to simulation and experimental values

for both the energy detector and matched filter algorithms. Under the given assump-

tions, the matched filter outperforms the energy detector. The method of estimating

the QR signal and noise characteristics is then presented. These estimates are used to

obtain a filter closely matched to the QR signal, had one been present, and the noise

statistics. The method of designing matched filters robust over signal and noise sets is

then presented, whose ultimate aim is to gain robustness to variances in the QR signal.

The following chapter presents QR spectroscopy and the experimental procedure used.

This is followed by a chapter that will present the observed QR and noise signals. The

motivation behind using the matched filter as a detector is then presented in Chapter

4. This chapter is followed by an introduction of our approach of estimating a filter

matched to the QR signal present. This is followed by a chapter, that combines the

work from the former chapter with an approach to estimating the statistics of the noise

signal present, to design a filter matched to both the QR signal and the noise present.

The thesis then ends with a chapter that introduces a matched filter robust to a signal

and noise sets pair. This chapter only uses simulation data to analyze the performance

of robust filters for sets of QR signals with varying parameters. Though, the ultimate

goal is to design filters robust to uncertainties in the general shape of the QR signal

instead of uncertainties in the QR signal parameters.

12

Chapter 2

QR Spectroscopy

Although the focus of this thesis is on the signal processing algorithms of QR detection,

it is important to provide a brief overview of QR spectroscopy. This chapter starts

by introducing the physics behind the QR detection process. The chapter then moves

on to introducing the signals observed in QR. This is followed by an introduction of

the excitations pulses used in QR detection. Lastly, the QR detection procedure is

introduced. Figure 2.1 is a simplified block diagram of the QR detection system.

The idea behind QR detection is straightforward. Certain material, such as explosives,

contain nuclei that form a resonant system because of their interaction between their

electrical quadrupole moment and electrical field gradient. By applying a radio frequency

pulse, whose frequency matches this interaction energy, it is possible to perturb the sys-

tem. Once a system is perturbed, it prodcasts a signal at the same frequency whose

presence would reveal the presence of the explosive material. The QR spectrometer

shown sends an RF pulse or a series of RF pulses, of frequency matched to the material

of interest. The probe then receives a response that is passed to the QR spectrome-

ter. Briefly, the spectrometer digitizes and demodulates the received signal. A signal

processing algorithm is then applied to determine the presence of a QR signal.

13

Fig. 2.1 Simplified block diagram of QR detection system

2.1 QR Physics

This section introduces the QR physics, first by describing the quadrupole interaction

between the electrical quadrupole moment of a nucleus and the electric field gradient

(EFG) generated by surrounding electrons. The section then discusses the prince relax-

ation constants for quadrupole resonance.

2.1.1 Quadrupole Interaction

QR spectroscopy is the study of the quadrupole interaction between the electrical quadrupole

moment of a nucleus and the electric field gradient generated by surrounding electrons

[20; 19], mentioned above. This can be used not only to determine the presence or ab-

sence of nuclei with electro quadrupole moments but also the signature of the chemical

bonds that these nuclei form.

The quadrupole moment is a quantitative description of the spatial distribution of the

charge of the nucleus [20]. The spatial distribution of the charge density can be expressed

as a multipole expansion, which is the sum of an infinite number of charge distributions

[27]. The first, is a monopole which is a spherical distribution, while the second term is

a dipole, which has the shape of a dumbbell. The third term, the quadrupole moment,

14

can be viewed as two anti-parallel electric dipoles, hence the name quadrupole. Though

the only charge distribution of interest to us is the quadrupole moment. The electric

quadrupole moment is a tensor that can be described with a single parameter eQ, where

e is the magnitude of an electron’s charge and Q is a scalar parameter that measures

the departure of the electric charge distribution from the spherical symmetry. When the

nuclear charge density is spherical, then the electric quadrupole parameter eQ is zero.

This value is positive, when the nuclear charge density is elongated along an axis of

symmetry and the charge density is shaped like an ellipse. On the other hand when eQ

is negative, the charge distribution is flattened like a frisbee.

The other component needed for quadrupole interaction, the electric field gradient (EFG)

is determined by the charge distribution within the bonds that the nucleus forms with

other atoms. The components of the tensor EFG can be reduced to two, which are the

maximal electric field gradient (EFG), eq and the asymmetry parameter, η. A non-zero

EFG’s interaction with the monopole, dipole, or any higher odd moment of the multipole

expansion of the nuclear charge distribution results in a zero torque acting on the nuclei

[19]. On the other hand, the EFG’s interaction with the quadrupole moment and higher

order even moments result in a non-zero torque, but only the one with the quadrupole

moment results in a torque large enough to be observed. Therefore, the interaction of

the EFG gradient with the quadrupole moment is the only one of interest. This torque,

a result of the interaction, is proportional to e2Qq, the quadrupole coupling constant

[20]. Therefore, the nucleus has certain preferred orientations, and each orientation

corresponds to a separate electrostatic interaction energy [52].

15

The quadrupole nuclei also posses angular momentum, referred to as spin, S. This

angular momentum is a vector quantity that is quantized by the axioms of quantum

mechanics. The magnitude of the vector is related to the spin quantum number I, which

can be an integer or a half integer value greater than zero. The possible spin directions m,

are functions of I, the spin quantum number. The value of m can be any value between

-I to +I in increments of one [18]. For example an I = 1, leads to three possible values

of m [52], (-1, 0 and 1). Quantization of the nuclear spin also leads to the quantization

of the electrostatic interaction energy [19]. Therefore a nitrogen-14 nuclei with a spin I

equal to one, has three preferred orientations of the nucleus denoted by x, y, and z, and

the interaction energy associated with these orientations are:

Ez =−e2qQR

2(2.1)

Ex =e2qQ

4(1− η) (2.2)

Ey =e2qQ

4(1 + η) (2.3)

Since eQ is constant for a given nucleus, the above energy levels are dependent on the

largest electric field gradient (EFG), eq, and the asymmetry parameter of the EFG, η.

These two parameters are a character of the bonds the quadrupole atom forms within

its molecule. In QR spectroscopy (the study of quadrupole interactions) these values are

determined by observing the transition between orientations and using these observations

the values of η and eq are calculated. These values are determined by the structure of

16

the chemical bond formed by the nuclei. Figure (2.2) uses a level diagram to show the

three electrostatic energies.

Fig. 2.2 QR energy levels and transition frequencies for nitrogen-14, an I=1 nucleus withη 6= 0 [17].

The diagram shows the lowest energy level is Ez, and the highest is Ey. The difference

between any two energy levels scaled by Planck’s constant, h, defines the QR transition

frequencies. The 14N transition frequencies are:

νd =Ey − Ex

h=

12e2qQ

hη (2.4)

ν− =Ex − Ez

h=

34e2qQ

h(1− η

3) (2.5)

ν+ =Ey − Ez

h=

34e2qQ

h(1 +

η

3) (2.6)

17

As stated earlier, when the asymmetry parameter, η is zero, the energy levels Ex and

Ey are degenerate since they represent the same energy [52]. Figure (2.2) displays the

three transition frequencies with respect to their energy levels.

The values eq and η (the EFG parameters) for a given nucleus are sensitive to factors

such as strains and impurities that determine the local chemical environment. Therefore

nuclei within a material can have slight differences in transition frequencies which results

in difference in the energy levels [20]. The different molecular orientations and impuri-

ties combined create a distribution of transition frequencies centered at the transition

frequency, ν∗ = ω∗/(2π). This frequency is unique for each type of explosive. Figure

(2.3) shows the QR frequencies of some explosives and chemicals.

Fig. 2.3 QR frequencies of different explosives/chemicals [3]

Torsional motion of the molecules from thermal agitation distorts the local EFG and

further alters the transition frequency. For two unsharp energy levels, the distribution

of transition frequencies is typically Gaussian, but is often modeled as a Lorentzian

distribution to ease the calculations [19]. The Lorentzian distribution is

18

L(ω) =1π

T ∗2

1 + (T ∗2

)2(ω − ω∗)2 (2.7)

where ω∗ is the transition frequency, T ∗2

is the inverse linewidth parameter (decaying

parameter). The above distribution has a full-width at half-maximum (FWHM) of 2/T ∗2

rad/sec [40]. A plot of a Lorentzian distribution is shown in Figure (2.4). A larger number

of imperfections in a crystal, leads to a smaller T ∗2

, which results in a larger FWHM [20].

The mass of the sample (the number of QR nuclei) is directly proportional to the area

under the Lorenztian. Therefore, as the line broadens, it decreases in amplitude making

it more difficult to identify the QR signals [39].

Fig. 2.4 Lorentzian distribution of transition frequencies [17].

19

The following subsection discusses the relaxation mechanisms of QR and time constants

associated with QR spectroscopy.

2.1.2 Relaxation Mechanisms

The principle relaxation parameters for QR are the spin-lattice and spin-spin relaxation

constants [38]. The spin-lattice relaxation time constant, T1, describes the length of

time needed for the RF energy absorbed by the nuclei (from pulsing) to be dissipated

by the nuclei to the surrounding lattice. The lattice is the general name for all other

degrees of freedom of the system besides spin orientation such as translational motion of

the molecules [50]. This relaxation time constant, T1, is physically proportional to the

time it takes the nuclei returning to their thermal equilibrium orientation. The spin-spin

relaxation time constant, T2eff , describes how the energy is exchanged among nuclei

through the interaction of their magnetic dipole moments. This interaction produces a

perturbation in the transition frequencies that causes the magnetic moments of precessing

nuclei to interfere destructively. Unlike the T ∗2

decay constant, which is the result of a

time-independent disturbance in transition frequencies, the T2eff relaxation is caused by

random fluctuations. Since the latter loss can not be recovered, it is termed a relaxation

process [50]. It can be shown that T1 > T2eff > T ∗2

.

For example if a sample is in thermal equilibrium and is pulsed the largest possible QR

signals is produced and after a few T2eff time constants the observed signal will vanish

due to destructive interference among the processing nuclei. Therefore, to obtain the

maximum response from a sequence of RF pulses one must wait a minimum of three

T1 constants for thermal equilibrium magnetization to be restored. Reducing the pulse

20

spacing causes the amplitude of QR signals from successive pulses to decrease. Therefore,

the pulse sequences used will wait at least three T1 between pulses. The following section

introduces the observed signals in QR detection.

2.2 Observed Signals

Two commonly observed signals in QR detection are the free induction decays (FIDs)

and the spin echoes. The two kinds of signals are described as follows [45; 52; 53; 51; 44].

• Free Induction Decays: The FID is a decaying signal caused by the interaction

between the oscillating magnetic field of the applied RF pulse with the magnetic

moment of the quadrupolar nucleus. This signal is observed immediately following

the applied RF pulse. Due to the fast decay of this signal and the ringing after

applying the RF pulse, this signal is hardly used for QR detection.

• Spin Echoes: After applying an RF pulse, spins become dephased causing the

QR signal to diminish [45]. Using appropriate pulse sequences, these spins can be

momentarily placed back in phase to generate useful spin echoes. These echoes can

be observed for a longer period of time than the FIDs, which makes them useful

for QR detection.

2.3 RF Excitation Pulse Sequences

The design of the RF pulse excitation sequence is a vital step in the process of obtaining

a useful QR signal. Though the QR signal is small in amplitude, the SNR ratio can be

improved by coherently adding the individual echoes acquired from each pulse. This is

21

led to the study and development of multi-pulse sequences in QR applications, which

can drastically improve the detection capability of the QR signal. The most commonly

used multi-pulse sequences represent some form of the spin-lock spin echo (SLSE) or the

strong off-resonant comb (SORC).

The spin-lock spin echo sequence, introduced in the 1970’s, generates a sequence of de-

caying spin-echoes. These echoes appear in between the rephasing pulses of the sequences

and decay in an approximately exponential manner. The strong off-resonant comb pulse

sequence is a steady state free procession, which was first introduced in the early 1980s

[47], composed of a sequence of equally spaced off-resonant pulses. This SORC sequence

generates a sequence of stable non-decaying spin-echoes [15], located at the rephasing

pulses. The spin echoes from both these sequences can be coherently added to improve

the SNR, [15]. One of the advantages of using the SORC sequence is that the sequences

last for as long as the rephasing pulse is applied, unlike SLSE. Unfortunately the SORC

sequence may not be used with all explosives, an example of such an explosive is TNT.

Another advantage of using the SORC sequence is that the amplitude of the generated

echoes are comparable to the free induction decay signal. The signal processing algo-

rithms discussed in this thesis are applicable independently of which pulse sequence is

used, though the type of pulse sequence used will affect the decay factor of the echoes.

The energy or equivalently the product of the amplitude and width of the RF pulses

determine the angle in which the nucleus is rotated. The frequency of the RF pulses

must match the energy difference between any two interaction energy levels. The next

section provides a brief overview of the QR detection procedure.

22

2.4 QR Detection Procedure

The process of detecting a QR signal can be summarized in the following steps:

• A series of RF magnetic pulses, known as excitation pulses, are generated and

emitted by a transmitter.

• Each pulse, from the series of pulses, perturbs the alignment of the nitrogen nuclei

within the material.

• During the inter-pulse interval, the precessing nuclei relax back to the their original

state. This motion of magnetic moment of the nuclei induces a voltage across the

probe coil, which is the QR response

• The QR response is received by the receiver and after passing through an amplifier

and digital converter, the received signal is sent to a computer based processor.

• The processor then analyzes the received signal, based on a specific algorithm

(such the one developed in this thesis), and a decision on whether the sought after

explosive is present is made.

To better understand the digitized received signals at the output of the QR spectrometer

(Schiano [48]), we need to provide a more detailed description of the spectrometer. A

block diagram describing the QR spectrometer is shown in Figure 2.5. During the receiv-

ing mode, the spectrometer uses a double heterodyne system that includes a quadrature

phase detector to demodulate the QR response to a lower frequency so that it can be

digitized using an A/D converter. The frequency of the PTS D310 is set to ωr+10 MHz,

23

where ωr = ω∗+ω0, ω∗ is the QR transition frequency (or the spectrometer frequency we

are searching at ωs) and ω0 is the desired receiver offset frequency. The signal induced

across the probe is first amplified and then heterodyned with the PTS D310 output

signal. The output of the first heterodyne is then ran through a bandpass filter at 10

MHz resulting in signal whose spectra is centered at 10 MHz +ω0. This signal is het-

erodyned a second time in a quadrature phase detector at 10 MHz to produce in-phase

and quadrature signals whose spectra are approximately centered at ω0.

The signal spectra are not exactly centered at the ω0, since the frequency of the PTS

D310 is hardly ever set exactly to ωr + 10 MHz due to uncertainty in the QR transition

frequency, ω∗. The QR transition frequency of a material is temperature sensitive and

in some cases more sensitive than others. For example, the explosive RDX varies by

approximately 0.45 kHz/◦C [32], while piperazine is highly insensitive to changes in

temperature which is why it was used in collecting experimental data. The two signals,

the in-phase and quadrature, are then low passed by a filter with a cutoff frequency

of 20 kHz. They are then digitized and combined within the computer to produce the

complex-valued signal.

24

LP

F

Cu

toff =

5M

Hz

RF

Gate

LP

F

Mix

erP

hase S

hifter

{0, 9

0, 1

80, 2

70}

20 d

B

10

MH

z

\4 N

etwo

rk

35 d

B

LP

F

Cu

toff=

5M

Hz

RF

Gate

20 d

B

Mix

er

BP

F

BW

= 1

00

KH

z

Cen

ter= 1

0M

Hz

LN

A

Match

ing

Netw

ork

Pro

be

L, R

Mix

er

Mix

er

LP

F

Cu

toff=

20

KH

z

LP

F

Cu

toff=

20

KH

z

SI (t)

90°

Freq

uen

cy

Synth

esizerR

PS

GC

om

pu

ter

Pow

er

Am

p

Clo

ck R

eference

RP

SG

SQ(t)

sp (t)

wr +

10

MH

z

wr +

10

MH

z

10

MH

z

Fig. 2.5 Block diagram of QR spectrometer

25

Chapter 3

Challenges, Signal Characteristics and Data Generation

This chapter discusses the challenges faced in QR detection and provides a description

of the QR and corrupting noise signals.

3.1 Challenges

There are several challenges one faces in QR detection. Some are the result of uncer-

tainty in the QR signal, while others are due to external and internal radio frequency

interferences. Some of these challenges are discussed in the following subsections.

3.1.1 Challenges Due to Uncertainty QR Signal

In an ideal case, with no interferences, a spin echo received in between pulses by the coil

would resemble the signal shown in the top part of Figure (3.1), which is a sinusoidal

signal that exponentially decays about the center.

Depending on the spatial distribution of the magnetic field across your sample, this

signal can at times be more gaussian shaped than exponentially decaying about the

center, which brings in uncertainty in the general shape of the signal. It may also

contain more than a single frequency, depending on the temperature and strains of the

explosive material. Explosive material within a bomb is required to have a uniform

temperature to obtain a signal with a very narrow bandwidth. All these reasons cause

26

0 50 100 150 200 250 300 350 400 450−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Am

plit

ude, V

Number of Samples

Fig. 3.1 QR Signal with a Frequency of 3.016 MHz and T2∗ of 500e-6

uncertainty in the general shape, frequency and phase of the QR echo. The signal can

be approximately represented mathematically with the following equation.

s(t) = e(−|t|/T2∗) cos(ω∗t+ φ), −T/2 ≤ t ≤ T/2 (3.1)

where ω∗ is the QR transition frequency, φ is the phase, T2∗ is the decaying parameter,

and T is the length of the data acquisition window. Unfortunately achieving acceptable

correct detection and false alarm rates for quantities of explosives that would pose a

threat is not easy, partly due to the low SNR of QR measurements [9].

Another challenge one faces is the uncertainty in the QR transition frequency, ω∗, the

phase φ, and time constant (decaying rate) T2∗. The surrounding temperature and

mechanical strains on the explosive significantly affect the QR transition frequencies. As

a result, some QR detection systems will require the temperature of scanned materials

27

to lie within a specified range. Due to this, the transition frequency, ω∗, in Equation

(3.1) is only known to fall within an upper and lower bound as follows,

ωlb ≤ ω∗ ≤ ωub (3.2)

The values ωlb = ωs − RBW /2 and ωub = ωs + RBW /2 are the lower and upper bound

frequency values respectively, and are dependent upon the known values ωs, spectrometer

frequency, and RBW , the bandwidth of the receiver, which is 20 kHz in our experimental

setup. The bandwidth of the receiver, RBW , varies with the type of coil used.

Uncertainty in the phase of the QR signal poses yet another challenge in QR detection.

The phase in Equation (3.1) is only known to fall within upper and lower bound as

follows,

−π ≤ ω∗ ≤ π (3.3)

Yet another challenge is uncertainty in the explosives time constant, T2∗. The quality

of the explosive highly affects the time constant, T2∗, of the QR response, for example

a fixed quantity of plastic explosive manufactured in an explosive manufacturing plant

will have a different time constant than the same quantity of explosive manufactured in

less artisanal methods. As a result the time constant, T2∗, is only known to fall within

an upper and lower bound as follows

T2∗lb ≤ T2

∗ ≤ T2∗ub. (3.4)

28

Presented next, are the challenges imposed from noise signals.

3.1.2 Challenges Due to External and Internal RFI Signals

Interferences from external RF sources pose another challenge when trying to detect a

QR signal. Since the frequency of the QR signal is low, between 0.5 and 6 MHz [44],

they are unavoidably corrupted by external radio frequency interferences, RFIs, located

in their frequency band. AM stations and ignition noises can trigger false alarms when

attempting to detect a QR signal. To overcome the challenge posed by external RFIs

it is essential for one to understand the source of these interferences. Of main interest

to us are RFIs from AM broadcast stations, since they are a major contributor to the

external interferences.

AM station frequency bands can be classified into three categories: long wave, medium

wave and short wave. Two of which, the medium and short wave, overlap with the

frequencies of QR signals from certain materials. The medium band is the one used

for commercial broadcasting and falls between 520 kHz and 1, 610 kHz, with channel

spacing of 10 kHz in the U.S. and 9 kHz elsewhere. The third one, is the short wave

band and is used for audio services intended to be heard at great distances from the

transmitting station, and falls in the range of 1.711 MHz −30 MHz with channel spacing

of 5 kHz. Among those three bands fall clear channel stations, which can be heard across

the country due to their large transmitting power.

One particular clear channel station to be concerned about is the Chicago based WLS

channel, which broadcasts at 890 kHz, the exact QR frequency of PETN. This is a

29

problem for QR detection. Another example where the QR frequency of the material

falls within the AM broadcast band is the frequency of the explosive TNT, which is

at 842 kHz at normal room temperature [9]. This frequency falls within the AM radio

frequency band and as a result the performance QR signal detection at that frequency

is seriously degraded.

Therefore to detect these weak QR signals, it becomes essential to mitigate the RFIs

using methods similar to [9; 45; 68; 59]. A single corrupting AM signal can be described

using the following mathematical equation.

AM i(t) = A(1 +m(t)) cos(ωct+ φi) 0 ≤ t ≤ T (3.5)

where φi is a random phase shift between π and −π that is variable and T is the period

of the QR signal. The frequency of the carrier is ωc and the message being modulated

is m(t).

The center of Figure (3.2) shows an AM signal, while bottom of Figure (3.2) shows a QR

signal corrupted by an AM signal with an SNR ratio of 0.25 or -12 dB. The resemblance

between the AM signal and the corrupted QR signal is obvious, even with this high SNR,

when compared to SNR values observed without noise shielding. Shielding can minimize

external RF interferences, though there exist situations where shielding is not applicable.

One particular example is the detection of anti-tank land mines. This makes the detection

of these mines a particular difficult task in the evening, when worldwide broadcasts

interfere with the detection process. Other sources of RF interference are internal ones,

from sources within the search volume. The excitation of a QR response requires the

30

application of a pulsed RF magnetic field within the search volume. Currents induced

within conductive materials located in the search volume, cause decaying magnetic fields

that lead to unacceptable false alarm rates.

0 50 100 150 200 250 300 350 400 450−1

0

1

Am

plit

ude, V

0 50 100 150 200 250 300 350 400 450−5

0

5

Am

plit

ude, V

0 50 100 150 200 250 300 350 400 450−5

0

5

Am

plit

ude, V

Number of Samples

Fig. 3.2 Comparison of QR+AM and QR: (Top) QR Signal with a Frequency of 3.016MHz and T2

∗ of 500e-6 (Center) AM Signal with Carrier Frequency of 3.016 MHz (Bot-tom) QR plus AM Signal, SNR = -12 dB

Another concern is the ringing produced by the search coil and the remaining hardware

[9; 45]. The coil can not immediately obtain the QR signal although the exciting pulse

has ended [45]. Performing phase cycling on the pulse sequence is an efficient way to

reduce the ringing problem [9; 58; 45]. Another method is using an active Q-spoiling

system which allows the probes natural response to relax rapidly following the excitation

pulse to permit detection of the QR response, [42]. Other RFI mitigation methods are

discussed in the following section.

31

3.2 RFI Mitigation Methods

Signal averaging, which is discussed in the following section, is another efficient method

used to tackle the major challenge of low SNR ratio [9; 15]. For uncorrelated noise, the

SNR increases by the square root of the number of averaged QR signals [15]. However

this method is ineffective in the presence of correlated noise or interferences such as AM

noise, [15]. Optimizing the search coil design can also improve the SNR ratio and hence

improve the probability of detecting a QR signal. Several design issues and ways to

improve the design have been addressed by Suits et al. [13; 57; 12; 14].

The SNR can also be enhanced by reducing the RFIs. Both active and passive strategies

in the hardware design have been considered [9]. The Naval Research Laboratory (NRL)

designed gradiometer coil, which measures the difference in magnetic flux [13]. This is

considered a passive approach and they claim that the gradiometer coil can reduce the

far field magnetic interference by 30 dB [9]. However, the coil may reduce the QR signal

at the same time [13], which in turn would reduce the probability of detection. In [45],

asymmetric gradiometers have been proposed to partly solve the QR signal loss problem

[45].

An efficient QR coil (used as the main antenna) and a set of external remote antennas

(used as reference antennas) have been developed by QM to reduce RFIs [9]. The goal of

the main antenna is to receive both RFIs and a QR signal, while the reference antennas

only receive RFIs [9]. The RFI”s from the reference antennas are used to mitigate the

RFIs at the main antenna. This method’s main disadvantage is that it requires high

dynamic range and an accurate balance of antennas. The approach proposed in this

32

thesis addresses these challenges using signal processing methods. The following section

discusses phase cycling, which improves the SNR ratio and eliminates the free induction

decay.

3.3 QR Signals

In the absence of external RFI signals, the signals observed at the receiving probe are the

Spin Echo, the Free Induction Decay (FID), and coherent spurious noise from glitches

and RF gates. The Spin Echo is usually used for QR detection for two main reasons.

For one, it occurs away from the falling edge of the pulse, where parasitic transients

appear, and second its twice as long as the FID. Therefore, it is of interest to eliminate

the presence of the FID, spurious noise and DC offset. Fortunately, phase cycling, which

is illustrated in Figure (3.3), is an efficient method to do so.

τ/2 τ

θα

θα+180

φα+90

φα+90

φα+90

φα+90

Fig. 3.3 Illustration of Phase Cycling on SLSE sequences.

33

When the phase of the first lock pulse is changed by 180 degrees, the phase of the echoes

are changed accordingly while the phase of the FID and noise are not. Performing

this procedure on successive pulse sequences is called phase cycling. The sequence of

echoes, resulting from the difference of the two sequences, are then averaged to obtain a

single echo. In the case of uncorrelated noise, the root-mean-squared amplitude of the

noise component in the composite signal decreases with the square-root of the number

of averages, at the expense of increasing detection time [21].

As was discussed in Section 2.3, there are several different excitation pulses used for QR

detection. The two most commonly used multi-pulse sequences are the spin-lock spin

echo (SLSE) and the strong off-resonanct comb(SORC). Other pulse sequences represent

some form of the SLSE or SORC sequences. The type of pulse sequence used affects the

single averaged echo resulting from a pair of phase cycled pulse sequences, as will be

shown in the mathematical description of the averaged echo.

As was mentioned in the previous section, in an ideal case, with no interferences, an echo

received at the coil can be approximated as,

sp(t) = e(−|t|/T2∗) cos(ω∗t+ φ), (3.6)

which is a single frequency spin-echo with a Lorentzian distribution in the absence of

noise. The parameter T2∗ defines the decaying rate of QR signal, while ω∗ is the QR

transition frequency, and both are material specific. To use the above mathematical

34

form of the QR signal, we place the following assumption on the received QR signal

throughout this thesis.

Assumption 1. The full width at half maximum, FWHM1, of sp(t) in equation (3.6) is

sufficiently small compared to the 3 dB bandwidth of the probe, the bandpass filter and

the low pass filter in the quadrature phase detector.

When the signal in Equation (3.6) is passed through the QR spectrometer, if Assumption

1 holds, the output signals are the following discrete time signals

sI [k] = Ae(−|−T/2+k∆|/T2∗) cos (ωbb(−T/2 + k∆) + φbb) (3.7)

sQ[k] = Ae(−|−T/2+k∆|/T2∗) sin (ωbb(−T/2 + k∆) + φbb) (3.8)

for k = 0, 1, . . . , Ns−1, where ωbb is the base-band frequency, φbb is the base-band phase,

T is the length of the data acquisition window, Ns is the number of discrete samples,

and ∆ = T/(Ns − 1) is the time increment in between sample points. We will assume

in this thesis that the base-band frequency, phase and decaying parameters are the only

unknowns in the QR’s signal description. The above two signals when combined form

the complex-valued digital signal,

sb[k] = sI [k] + jsQ[k] k = 0, 1, . . . , Ns − 1 (3.9)

= Ae(−|−T/2+k∆|/T2∗)e(jωbb(−T/2+k∆)+φbb) k = 0, 1, . . . , Ns − 1

1The full width at half maximum of a signal is the width of the frequency range where lessthan the half the signal’s power is attenuated, i.e. the power is at least half the maximum.

35

Although the output of the spectrometer is complex, the algorithms discussed in this

thesis will be compared while only exploiting the real part (3.7). Phase cycling and

averaging the real part from a pair of pulse sequences would be the next step for the

detection algorithms discussed in this work. Depending on the pulse sequence used in

generating the QR signals, one can experience decay across pulse echoes (SLSE). We will

assume that the averaged QR signal is a result of the following assumption.

Assumption 2. The ith QR echo of pulse sequence j decays with a factor of e−(i−1)τ/T2eff ,

where τ is the width of an echo and T2eff is a constant that is material dependent. This

constant describes how energy is exchanged among nuclei through the interaction of their

magnetic dipole moments. The interaction causes the magnetic moment of processing

nuclei to interfere destructively, which leads to a decay in the echoes [38]. When using a

SORC pulse sequence the value of T2eff is considered infinite (i.e. no decay). To simplify

the data simulation and calculations (Chapter 4) we assume no decay of echoes across

sequences for both SLSE and SORC sequences (the time in-between pulse sequences is

greater than 3T1, nuclei almost at thermal equilibrium), although the results can be ex-

tended to incorporate any decay. In other words the ith echo from the jth sequence will

have the same amplitude as the ith echo from the kth sequence.

Under Assumption 2, the real part of the averaged phase cycled QR echo, from a pair of

pulse sequences of Ne echoes each, can be represented as

spc[k] = β ∗Ae(−|−T/2+k∆|/T2∗) cos (ωbb(−T/2 + k∆) + φbb). (3.10)

36

for k = 0, 1, . . . , Ns − 1, where β is defined as

β =

Ne∑i=1

e−(i−1)τ/T2eff

Ne. (3.11)

When using the SORC sequence, which results in non-decaying echoes, the decaying

factor β is one. The thesis first evaluates and compares the performance of the energy

detector and matched filter under the assumption that the QR signal is exactly known in

Chapter 4. The thesis then moves on to evaluating and comparing the proposed batch

matched filter algorithm to the energy detector without this assumption in Chapter

5. Before comparing the two detection algorithms, a description of the observed noise

signals is presented next.

3.4 Noise Characteristics

The noise characteristic discussed in this thesis is a superposition of thermal and external

RF interference from AM signals. Due to its simplicity, thermal noise is discussed first

followed by AM signals.

3.4.1 Thermal Noise

Thermal noise received at the probe is processed by the QR spectrometer in the same

way a QR signal would be. A thermal noise signal at the output of the QR spectrometer

may be represented as,

nbi [k] = nIi [k] + jnQi [k] k = 0, 1, . . . , Ns − 1, (3.12)

37

where the nIi [k] and nQi [k] are filtered white gaussian random variables with a dis-

tribution, N(0, σ2tn

). Filtering the thermal noise correlates the noise samples, and the

samples can no longer be considered as white gaussian random variables. Phase cycling

and averaging the real part from a pair of pulse sequences leads to the following signal,

npc[k] =1

2Ne

Ne∑i=1

[nI [ζ(k, i)]− nI [ζ(k, i) + τd]] k = 0, 1, . . . , Ns − 1, (3.13)

where ζ(k, i) = k∆ + (i− 1)τ and τd is the time delay between the pulse sequences used

in phase cycling. The averaged phase cycled signal npc[k] for each k is the average of

2Ne noise samples.

3.4.2 AM Signals

Interferences from external RF sources pose a bigger threat, compared to internal RF

interferences, when trying to detect a QR signal. Since the frequency of the QR signal is

low (0.5 - 6 MHz [44]), they are unavoidably corrupted by external RFI sources located

in their frequency band. Of main interest to us are RFIs from AM broadcast stations,

since they are a major contributor to the external interferences.

A corrupting AM signal is represented in Equation (3.5). This AM signal is processed

by the QR spectrometer in the same way a QR signal is. Therefore, an AM signal at

the output of the spectrometer can be represented as,

ηIi [k] = Aηi(1 +mi[k∆]) cos (∆ωAMk∆ + φ) (3.14)

38

ηQi [k] = Aηi(1 +mi[k∆]) sin (∆ωAMk∆ + φ) (3.15)

for k = 0, 1, . . . , Ns− 1, where ∆ωAM is the frequency of the AM signal after demodula-

tion, m[.] is the broadcasted message, and φ is an unknown random phase shift between

π and −π. Equations (3.14) and (3.15) are combined to form the following composite

complex signal,

ηbi[k] = Aηi(1 +mi[k])e(j∆ωAM (k∆)+φ) k = 0, 1, . . . , Ns − 1. (3.16)

Again, even though the output of the QR spectrometer is complex, we will only be

utilizing the real part, the output of one channel. Phase cycling and averaging the real

part from a pair of pulse sequences, of Ne echoes each, would result in the following AM

signal,

ηpc[k] =Aη2Ne

Ne∑i=1

[(1 +m[ζ(k, i)]) cos (∆ωAM (ζ(k, i)) + φ)− (3.17)

(1 +m[ζ(k, i) + τd]) cos (∆ωAM (ζ(k, i) + τd) + φ)]

for k = 0, 1, . . . , Ns − 1, where ζ(k, i) = k∆ + (i− 1)τ and τd is the time delay between

the pulse sequences used in phase cycling.

3.5 Averaged Nm Trials

After Nm phase cycled trials are repeated, the Nm signals can be be averaged into

one signal, which is the approach taken when using the energy detector to detect the

39

presence of a QR signal. The averaged, uncorrupted QR signal, corrupting AM signal

and corrupting thermal noise signal have the form shown in Equations (3.18), (3.19) and

(3.20) respectively.

savg[k] =Nm−1∑i=0

spci [k]/Nm (3.18)

ηavg[k] =Nm−1∑i=0

ηpci [k]/Nm (3.19)

navg[k] =Nm−1∑i=0

npci [k]/Nm (3.20)

for k = 0, 1, . . . , Ns − 1, where Ns is the number of samples per trial and the index i

represents trial i + 1. The signals spci [k], ηpci [k] and npci [k] represent the phase cycled

signals and have the mathematical form shown in Equations (3.10), (3.17) and (3.13).

3.6 Aggregating Nm Trials

Instead of averaging the repeated Nm phase cycled trials, we can form an aggregated

discretized signal by stacking the Nm signals in time. This approach is used when using a

matched filter based algorithm to detect the presence of a QR signal. A stacked discrete,

40

uncorrupted QR signal, corrupting AM signal and a corrupting thermal noise signal has

the form shown in Equation (3.21), (3.22) and (3.23) respectively,

s[k] =Nm−1∑i=0

spci [k − iNs] (3.21)

η[k] =Nm−1∑i=0

ηpci [k − iNs] (3.22)

n[k] =Nm−1∑i=0

npci [k − iNs] (3.23)

for k = 0, 1, . . . , Nm × Ns − 1, where Ns is the number of samples per trial and the

index i represents trial i+ 1. The signals spci [k], ηpci [k] and npci [k] represent the phase

cycled signals and have the mathematical form shown in Equations (3.10), (3.17) and

(3.13), where spci [k] = 0, ηpci [k] = 0 and npci [k] = 0 for k < 0 and k > Ns − 1. Note

that the random phase φ in Equation (3.17) does not vary across echoes, Ne, but varies

across trials (experiments), Nm, therefore φ remains constant throughout ηpc[k] but not

throughout η[k].

41

3.7 Experimental Data Versus Simulation Data

This section discusses the collection of experimental data in the laboratory and the

simulation data generated in the following two sections respectively.

3.7.1 Experimental Data Collected

The general procedure of data collection was briefly discussed in Section 2.4 and shown in

Figure (2.1). The data used in this thesis was collected using a shielded RF probe, which

collects the signals and passes them on to QR spectrometer introduced in Section 2.4 and

presented using a simplified block diagram in Figure (2.5). This section focuses on the

stage of data collection prior to the processing block of the the QR spectrometer in Figure

(2.1). The sample/chemical used in generating the QR signal was piperazine, which has a

a signature frequency of 3.016 MHz. Piperazine has a small spin-lattice relaxation time,

T1, which is related to the wait time in between pulse sequences [40]. The insensitivity

of its transition frequency to temperature variations is another attractive feature of this

material [40]. These characteristics make it a suitable material to conduct experiments

with. Since the RF probe used is shielded the only signals picked up by the probe other

than the QR signal is thermal noise and RF interferences present within the shielded

box if any, and therefore the AM noise had to be fed into the shielded probe.

A typical AM signal has the form shown in Equation (3.14), where AM is the modulated

signal and is typically a music or speech signal, which is bandpassed between 40 Hz and

10 kHz. To generate consistent data, we chose to use bandpassed thermal noise instead

of music or speech in generating the AM signal. Filtering thermal noise introduces

42

correlation between the noise samples, increasing the signals resemblance to speech/music

signals. This increase in resemblance reduces the effect of using noise as the message

signal m[k], versus speech or music, on the detection algorithm.

The signal was generated by playing the modulating signal, thermal noise band passed

between 40 Hz and 10 kHz, using a CD player and feeding it into a function generator,

which in turn modulates the signal at a frequency of 3.016 MHz, the same frequency

as the QR signal. In between the CD player and the function generator is a step up

ratio transformer, whose goal is to step up the voltage to achieve a hundred percent

modulation. Through a 40 dB amplifier and a coaxial cable, the modulated signal at the

output of the function generator is then coupled into a 50 ohm resistor in the shielded RF

probe. The audio level at the CD player can be adjusted to ensure a modulation level of

approximately 100 percent. The RF output level at the function generator may also be

adjusted to achieve the desired SNR level. The AM signal is then picked up by the RF

probe along with any other noise and QR signal present. These signals are processed by

the QR spectrometer, before it is processed by the signal processing detection algorithm

used.

3.7.2 Simulation Data Generated

Using MATLAB code, we attempt to closely simulate the data generated using the QR

spectrometer in the lab. To simulate the signals received by the RF probe we used

the models in Equations (3.7) and (3.14) to represent the QR signal and the AM noise

respectively. The modulated signal, m, in Equation (3.14), was represented with thermal

noise bandpassed by an 8-pole Butterworth filter with cutoff frequencies of 40 Hz and

43

10 kHz. The next step is to simulate the passing of the signals at the probe through the

QR spectrometer.

Simulating a QR spectrometer even as simplified as the one in Figure (2.5) is difficult,

due to the inaccuracy in representing the various filters. Briefly, the QR spectrometer

demodulates the signal at the probe to the baseband and passes it through a low pass

filter at the output. In simulation the process of demodulation was eliminated by just

generating the QR signal and the AM signal at the baseband frequency. We simulated

the low pass filter at the output with an 8-pole Butterworth filter with a cutoff frequency

of 20 kHz. Due to the simplicity of our simulation discrepancies between simulation and

experimental data would not be unusual. The simulation was kept simple, since the

algorithms developed in this thesis will be tested on both experimental and simulation

data and therefore using a sophisticated simulation will not further contribute to the

results. The main goal of the simulation procedure was to efficiently test algorithms

prior to the availability of laboratory experimental data. The simulation code used is

shown in Appendix C.1.

44

Chapter 4

Motivation for Using Matched Filter

Several signal processing QR detection methods have been discussed and compared in

[59]. The one that resulted in the most uniform performance levels, which is also the

simplest one to implement, is the energy detector [59]. These characteristics make it

a suitable algorithm to compare our matched filter based approach against. The goal

of this chapter is to motivate the use of a matched filter for QR signal detection by

comparing its performance, under the assumption that the QR signal description is

known, to the generic energy detector. The two detectors will be compared in theory

and using simulation and experimental data. Since the matched filter requires exact

knowledge of the QR signal, the two detectors are compared under the assumption that

the QR signal description is attainable, which is an unrealistic scenario, though the

improvement in performance demonstrated by the matched filter provides a basis for

pursuing a detection algorithm based on the matched filter.

The two detectors are compared using the QR signal and noise characteristics presented

in the previous chapter. It starts by introducing both detectors and their individual test

statistics. It then moves on to comparing both detectors under particular assumptions

placed on the noise. Under these noise assumptions, approximate expressions for the

theoretical mean and variance of the test statistics are then provided for both the pres-

ence of thermal noise only and in the presence of both thermal and AM noise. These

45

expressions are then used to generate theoretical ROC plots, which are compared to

ROC plots generated from simulations under the same conditions.

The chapter then moves on to comparing the two detectors when no assumptions are

placed on the noise. Without these noise assumptions, the derived moments of the

two detectors test statistics, which lead to the theoretical ROC plots are not valid and

therefore the detectors are compared using ROC plots generated using simulation and

experimental data only.

Before introducing the two detection algorithms, some parameters and assumptions that

will be used by both detectors, are defined and discussed. The first is the total energy

of a QR signal from a single phase cycled experiment denoted as, εs, and from a string

of Nm phase cycled trials (experiments) denoted as, ε, as follows respectively.

εs =Ns−1∑k=0

[Ae(−|−T/2+k∆|/T2∗) cos (ωbb(−T/2 + k∆) + φbb)]

2. (4.1)

ε = Nm ∗ εs (4.2)

The measure of SNR used throughout this thesis to describe simulation data is dependent

on the amplitude of the QR signal (prior to any decay), the standard deviation of thermal

noise and the amplitude of the AM signal. We define the measure of signal-to-noise ratio

46

(SNR) in the presence of thermal noise, SNRtn, and the signal-to-noise ratio in the

presence of AM and thermal noise, SNRamtn, as

SNRtn = 20× log10

(A

σtn

)(4.3)

SNRamtn = 20× log10

(A

Aη

)(4.4)

where, A is the amplitude of a received QR echo (3.7), Aη is the amplitude of a received

AM signal (3.14), and σtn is the standard deviation of the thermal noise, prior to passing

through the QR spectrometer and prior to averaging and phase cycling. Note, that the

SNR in the presence of AM and thermal noise, is calculated with respect to the amplitude

of the AM signal only. This is due to the fact that in all of the experimental setups used

in this thesis, the standard deviation of the thermal noise is more or less constant, and

is typically on the same level as the amplitude of the QR signal [9]. Therefore, in

the presence of AM and thermal noise, the standard deviation of the thermal noise is

held constant and equal to the amplitude of the QR signal, σtn = A. Several different

definitions of signal to noise ratio (SNR) have been used in the literature. One of which,

is the ratio of the QR signal’s energy over the noise’s energy. This measure of SNR

leads to values lower than the SNR definition used in this thesis. The energy detector is

presented next before comparing its performance to the matched filter.

47

4.1 Energy Detector

In an allocated time slot, all collected signals are averaged into a single signal and the

energy of averaged signal is calculated as follows,

Energy =Ns−1∑k=0

xavg[k]2. (4.5)

where xavg is the total averaged received signal, the sum of signals in Equations (3.18),

(3.19) and (3.20). If the energy is above a preassigned threshold value, γ, then it is

assumed that a QR signal is present and not otherwise. Therefore, the test statistic is

equal to the energy of the signal.

T (x) =Ns−1∑k=0

xavg[k]2.

4.2 Matched Filter

This section introduces the matched filter and its test statistic. Let the input to the

discrete-time linear filter with impulse response {h[k], k = 0, 1, ...., Ns − 1} be given by

x[k] = s[k] + n[k] k = 0, 1, ..., Ns − 1 (4.6)

where {n[k], k = 0, ..., Ns − 1} is a zero-mean stochastic process with covariance matrix

48

Σ =

E[n[0], n[0]] E[n[0], n[1]] · · · E[n[0], n[Ns − 1]]

E[n[1], n[0]] E[n[1], n[1]] · · · E[n[1], n[Ns − 1]]

......

. . ....

E[n[Ns − 1], n[0]] E[n[Ns − 1], n[1]] · · · E[n[Ns − 1], n[Ns − 1]]

(4.7)

and {s[k]; k = 0, ..., Ns−1} is a deterministic signal. The ratio between the output power

due to the signal and the expected output power due to the noise according to [37] is

the signal-to-noise ratio and is denoted by ρ.

ρ(h; s,Σ) =< h, s >2

< h,Σh >≤ C (4.8)

The notation, <,>, represents the standard inner product and C is the maximum value

of SNR for the optimal pseudo-signal. The linear system, h[k], that maximizes the output

signal-to-noise ratio, ρ, when the QR signal is a deterministic one embedded in additive

stochastic (random) noise is known as the Matched Filter [37]. Since the second-order

statistics of the input noise determines the power of the noise at the output of the linear

filter, a complete description of the QR signal and the second order statistics of the noise

is necessary in order to derive the corresponding matched filter. The matched filter is

derived as follows. Cross-multipication of Equation (4.8) yields,

< h, s >2 −C < h,Σh >≤ 0

49

This can be maximized by setting the gradient (the derivative w.r.t. h) equal to zero,

i.e.

2 < h, s > s− 2CΣh = 0

By rearranging and noting that C/ < h, s > is a constant and can be, without loss of

generality, set equal to unity, the result is the well known Matched Filter equation [37]:

s = (C/ < h, s >)Σh = Σh (4.9)

The solution of equation (4.9) maximizes the SNR and is the Matched Filter:

h∗ = Σ−1s (4.10)

with output SNR given by

ρ∗(h∗; s,Σ) =< h, s >=< Σ−1s, s > (4.11)

Throughout this chapter, we will assume no knowledge of the present noise statistics and

instead design the matched filter based on the assumption that the corrupting noise is

white with unit variance hence, in the presence of AM noise, this filter is matched only

to the QR signal and not the noise.

Such a filter has the following form, when it is matched exactly to the received QR signal,

50

h[k] =Nm−1∑i=0

spc[k − iNs], (4.12)

where spc[k] is shown in Equation (3.10), is applied. In this case the test statistic at the

output of the filter is

T (x) =N−1∑k=0

x[k] ∗ h[k]

the inner product of x[k] and h[k], where N = Nm × Ns. If the value of T (x, h), the

test statistic, is above a threshold then its assumed that a QR signal is present, and not

otherwise. This test uses the LTI filter that maximizes the SNR in Equation (4.8), in the

presence of white gaussian noise. The filter can only be used when an exact description

of the received QR signal is attainable, subsequent chapters will address the problem

when the QR description is not exactly known but the general description of the signal

is.

4.3 Detection Algorithm Comparison Under Noise Assumptions

The statistics of noise signals passing through the QR spectrometer are changed as they

pass through its linear time invariant (LTI) filters. These LTI filters correlate the noise

samples making the theoretical analysis very difficult, due to the inapplicability of the

classical central limit theorem (CLT) for independent variables. This theorem states that

the sum of a sufficiently large number of independent variables, each with a well defined

51

mean and well defined variance, will be approximately normally distributed. Deriving

an explicit expression for the mean and variance of the test statistic without being able

to apply the CLT, is not simple, especially for the energy detector where the received

signal is squared and cross terms are introduced. Therefore, to reduce the complexity

of the derivation of explicit expressions for the mean and variance of the test statistics

for the two detection algorithms, we must place some assumptions on the noise. These

assumptions are placed for the sole purpose of a theoretical performance comparison. In

the following section the test statistics of the two algorithms are compared without these

assumptions using both simulation and experimental data. For now the following is the

assumption placed on the thermal noise picked up by the coil.

Assumption 3. Assume that the filters in the quadrature phase detector do not alter the

noise signal’s statistics.

If the approximation in Assumption 3 holds, the real part of the white noise signal at the

output of the QR spectrometer is a white gaussian random variable with a distribution,

N(0, σ2tn

). Therefore, the averaged phase cycled signal npc[k], Equation 3.13, under this

assumption for each k is the sum of 2Ne white gaussian random variables, which results

in a white gaussian random variable with a variance of σ2tn

defined as,

σ2tn

= σ2tn/2Ne. (4.13)

Further since navg[k] is the average of Nm phase cycled signals, npc[k], for each k it is a

gaussian random variable with the following variance,

52

σ2avg,tn

= σ2tn/Nm = σ2

tn/(2NmNe). (4.14)

Computing the statistical parameters of the test statistics in the presence of AM noise

is a more difficult task, even if Assumption 3 holds, due to the correlation between

sample points in the broadcasted message m[k], Equation (3.17). To simplify these

calculations, we will assume the broadcasted message samples are uncorrelated, again the

following section will compare the test statistics of the two algorithms without placing

any assumptions on the noise. For now, we make the following assumption on the

broadcasted message m[k]:

Assumption 4. The broadcasted message in an AM signal at a certain time k, m[k], is a

normal random variable with the distribution m[k] ∼ N (0, σ2m

), where σ2m

is small in an

attempt to confine 1 +m[k] between 0 and 2 with high probability. The random variables

m[k] and m[k + h] are identically and independently distributed for all h = ±1,±2, . . ..

As in Assumption 3, we will assume that the quadrature phase detector does not alter

the AM signal’s statistics.

In reference to the above assumption, a value of σm that worked in restricting, 1 +m[k]

to 0− 2 with high probability was, σm = .25. Further the above assumption leads to an

AM signal that is a random variable at time k, with the same distribution as 1 +m[k],

N (1, σ2m

).

53

Before presenting the comparison of the two detectors, we point out that all simulation

plots in this section were generated with the following values. The number of echoes

per pulse sequence, Ne, and the number of phase cycled trials, Nm, were fixed to 2 and

2500, respectively. For each of the different noise cases, the above values were used to

generate 500 experiments with a QR signal present and 500 experiments without. In the

presence of thermal noise the signal to noise ratio was, SNRtn = −50 dB, while in the

presence of both AM and thermal noise the signal to noise ratio was SNRamtn = −66

dB, the standard deviation of the broadcasted message m[k], σm = .25, and the standard

deviation of the thermal noise is set equal to the amplitude of the QR signal, σtn = A.

This is due to the fact that the QR signal amplitude is approximately on the same level

as thermal noise [9].

4.3.1 Energy Detector in the Presence of Thermal Noise

When using the energy detector the received Nm phase cycled trials are not stacked in

time but averaged in time resulting in a signal of length Ns, the number of samples in a

single echo. If xavg[k] is the processed received signal,

H0 : xavg[k] = navg[k] (4.15)

H1 : xavg[k] = savg[k] + navg[k] (4.16)

54

where savg[k] is the QR signal described in Equation (3.18), and navg[k] is the white

gaussian thermal noise signal described in Equation (3.20), which has a variance of

σ2avg,tn

(defined in Equation (4.14)) for each k, when Assumption 3 holds. These signals

are obtained from averaging the Nm phase-cycled trials, as is explained in Section 3.5.

By Assumption 3, xavg[k] is a normal random variable for each k and hence x2avg

[k] is a

chi-squared random variable. Therefore, both by definition and Assumption 3, the test

statistic is the sum of Ns independent and identically distributed chi-squared random

variables. Each of theseNs random variables, has the following first and second moments.

H0 :µ(xavg[k]) = σ2avg,tn

σ2(xavg[k]) = 2σ4avg,tn

H1 :µ(xavg[k]) = σ2avg,tn

+ savg[k]2 σ2(xavg[k]) = 2σ4avg,tn

+ 4savg[k]2σ2avg,tn

Since Ns is large, the sum of the Ns independent random variables, by the central limit

theorem, can be approximated as a gaussian random variable with a mean and variance

equal to the sum of means and the sum of variances respectively. Therefore the test

statistic, in the presence of thermal noise, has the following approximate distributions,

for the cases shown in Equations (4.15) and (4.16) respectively.

55

H0 : T (x) ∼ N (Nsσ2avg,tn

, Ns2σ4avg,tn

) (4.17)

H1 : T (x) ∼ N (Nsσ2avg,tn

+ β2εs, Ns2σ4avg,tn

+ 4β2εsσ2avg,tn

) (4.18)

The variables β, σ2avg,tn

and εs are defined in Equations (3.11), (4.14) and (4.1) re-

spectively. These approximate probability density functions (PDFs), of the test statistic

when a QR signal is present, (4.18), and when one is not, (4.17), were compared to PDFs

obtained from simulation in Figure (4.1). The simulation data was generated using the

parameters described at the beginning of section 4.3. This data was generated to match

the assumptions placed on the noise. As is apparent from the figure, the theoretical plots

closely matched the simulation plots.

8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 130

0.1

0.2

0.3

0.4

0.5

0.6

0.7

False Positive Distribution Theoretical

False Positive Distribution Simulation

True Positive Distribution Theoretical

True Postive Distribution Simulation

Fig. 4.1 Energy Detector’s Theoretical PDF plots vs Simulation PDF plots for ThermalNoise, SNR = 20 log10( A

σtn) = -50 dB

56

The approximated theoretical PDFs lead to a probability of false alarm (false positive),

PFA,

PFA = P (T (x) > γ, | H0) (4.19)

= P

Z >(γ −Nsσ

2avg,tn

)√Ns2σ4

avg,tn

where Z is the standard normal random variable and γ is a fixed threshold value. On

the other hand the probability of correct detection (true positive), PD, is approximately

PD = P (T (x) > γ, | H1) (4.20)

= P

Z >(γ − β2εs −Nsσ

2avg,tn

)√Ns2σ4

avg,tn+ 4β2εsσ

2avg,tn

The approximated theoretical probability of false alarm as a function of γ, PFA(γ), was

plotted versus the probability of correct detection as a function of γ, PD(γ) to generate

a ROC plot which was compared to one from simulation data, generated using the

parameters described in top of this section, in Figure (4.2). Again as was expected from

the previous figure, the simulation and theoretical plots matched.

4.3.2 Energy Detector in the Presence of AM and Thermal Noise

A more realistic case, is when the QR signal is received within AM and thermal noise,

as is shown below. If xavg[k] is the processed received signal,

57

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PFA

, Probability of False Alarm

PD

, P

robabili

ty o

f C

orr

ect D

ete

ction

ROC in the presence of Thermal Noise, SNR −50 dB = 20log10

(A/σtn

)

Theoretical ROC, Energy Detector

Simulation ROC, Energy Detector

Guess

Fig. 4.2 Energy Detector’s, Theoretical ROC plot vs Simulation ROC plot for ThermalNoise, SNR = 20 log10( A

σtn) = -50 dB

H0 : xavg[k] = ηavg[k] + navg[k] (4.21)

H1 : xavg[k] = savg[k] + ηavg[k] + navg[k] (4.22)

where savg[k], ηavg[k] and navg[k] are the averaged QR signal, AM signal and the white

gaussian thermal noise signal described in Equations (3.18), (3.19) and (3.20). The

gaussian thermal noise signal has a variance of σ2avg,tn

, defined in Equation (4.14). By

definition, the noise Assumptions 3 and 4, and the fact that the AM and thermal noise

sources are independent, the test statistic is the sum of Ns independent random variables.

Since Ns is large, the test statistic can be approximated as a gaussian random variable

58

by the central limit theorem. This random variable, in the presence of AM and thermal

noise, has the following approximate distributions

H0 : T (x) ∼ N (µE,FA, σ2E,FA

) (4.23)

H1 : T (x) ∼ N (µE,D, σ2E,D

) (4.24)

for the cases shown in Equations (4.21) and (4.22) respectively. The values of µE,FA,

σ2E,FA

, µE,D and σ2E,D

are defined as follows,

µE,FA = Ns

(σ2avg,tn

+A2η

2NeNm

σ2m

2

)

σ2E,FA

= Ns

(A4η

(2NeNm)2

3σ4m

4+ 2σ4

avg,tn+ 4

(σ2avg,tn

) A2η

2NeNm

σ2m

2

)

µE,D = µE,FA + β2εs

σ2E,D

= σ2E,FA

+ 4β2εs

(A2η

2NeNm

σ2m

2+ σ2

avg,tn

)

59

where β, εs, and σ2avg,tn

are shown in Equations (3.11), (4.1) and (4.14) respectively. The

variables Aη and σm represent the AM signal’s amplitude and standard deviation of the

modulating signal m[k], respectively. A detailed calculation of the mean and variance of

the energy detector test statistic is provided in Appendix A.

The PDFs of the test statistic in Equations (4.23) and (4.24) were compared to PDFs

obtained from simulation in Figure (4.3). The simulation data was generated using the

parameters discussed in the beginning of the section. As is apparent in the figure, the

theoretical plots closely matched the simulation plots.

9 10 11 12 13 14 15 16 17 18 19 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

False Positive Distribution Theoretical

False Positive Distribution Simulation

True Positive Distribution Theoretical

True Postive Distribution Simulation

Fig. 4.3 Energy Detector’s Theoretical PDF plots vs Simulation PDF plots for AM andThermal Noise, SNR = 20 log10( A

Aη) = -66 dB

The distributions in Equations (4.23) and (4.24) lead to the following probability of false

alarm, PFA, and probability of correct detection, PD,

60

PFA = P (T (x) > γ, | H0) (4.25)

= P

(Z >

(γ − µE,FA)σE,FA

)

PD = P (T (x) > γ, | H1) (4.26)

= P

(Z >

(γ − µE,D)σE,D

),

where Z is the standard normal random variable and γ is a fixed threshold value. Plotting

the probability of false alarm as a function of γ, PFA(γ), versus the probability of correct

detection as a function of γ, PD(γ), leads to a theoretical ROC plot. This theoretical

ROC plot was plotted against a simulation ROC plot in Figure (4.4). The simulation

data used in this plot was generated using the simulation parameters discussed at the

beginning of the section.

4.3.3 Matched Filter in the Presence of Thermal Noise

Let the processed received signal, x[k], be

H0 : x[k] = n[k] (4.27)

61

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PFA


PD

, P

robabili

ty o

f C

orr

ect D

ete

ction



Guess

Fig. 4.4 Energy Detector’s, Theoretical ROC plot vs Simulation ROC plot for AM andThermal Noise, SNR = 20 log10( A

Aη) = -66 dB

H1 : x[k] = s[k] + n[k] (4.28)

where s[k] is the QR signal described in Equation (3.21), and n[k] is the white gaussian

thermal noise signal in Equation (3.23), which has a variance of σ2tn

= σ2tn/(2Ne) when

Assumption 3 holds. These signals are obtained from stacking the Nm phase-cycled

trials in time, as is described in Section 3.6. In this case the test statistic T (x, h) in the

presence of thermal noise, which corresponds to the Equations (4.27) and (4.28), has the

following distributions,

H0 : T (x, h) ∼ N (0, εσ2tn

) (4.29)

62

H1 : T (x, h) ∼ N (βε, εσ2tn

). (4.30)

The values of β (the decaying factor of the QR signal), σ2tn

(the variance of the phase

cycled thermal noise), and ε (the energy of the Nm QR signals without a decaying factor)

are defined in Equations (3.11), (4.13) and (4.2) respectively. The theoretical PDFs in

Equations (4.29) and (4.30) are plotted versus simulation PDFs in Figure (4.5). The

simulation PDFs were generated from data that matches the assumptions placed on the

noise with the parameters discussed at the beginning of the section. The theoretical

PDFs plotted closely match the simulation PDFs plotted and the slight discrepancy

is due to using a finite number of experiments to estimate the simulation PDFs. The

theoretical PDFs in Equations (4.29) and (4.30) lead to a probability of false alarm, PFA,

and a probability of correct detection, PD, defined as

PFA = P (T (x, s) > γ, | H0) = P

(Z > γ/

√εσ2tn

), (4.31)

PD = P (T (x, s) > γ, | H1) = P

(Z > (γ − βε)/

√εσ2tn

), (4.32)


the probability of false alarm as a function of γ, PFA(γ), versus the probability of the

correct detection as a function of γ, PD(γ), results in a theoretical ROC plot. This

plot and an ROC plot generated through simulation, using the simulation parameters

mentioned at the beginning of the section, are plotted in Figure (4.6) and the match is

63

apparent. The theoretical and simulation ROC plots for the energy detecter are shown

on the same figure for comparison purposes. The improvement in performance when

using the matched filter becomes apparent.

−600 −400 −200 0 200 400 600 800 10000

0.5

1

1.5

2

2.5

3x 10

−3

PFA

Theoretical PDF, −50 dB

PD


PFA

Simulation PDF, −50 dB, 500 Experiments

PD


Fig. 4.5 Matched Filter’s Theoretical PDF plots vs Simulation PDF plots for ThermalNoise, SNR = 20 log10( A

σtn) =-50 dB

4.3.4 Matched Filter in the Presence of AM and Thermal Noise

Now considering the matched filter case, where the processed received signal, x[k], is

H0 : x[k] = η[k] + n[k], (4.33)

H1 : x[k] = s[k] + η[k] + n[k], (4.34)

64

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PFA


PD

, P

robabili

ty o

f C

orr

ect D

ete

ction



Theoretical ROC, Matched Filter

Simulation ROC, Matched Filter

Guess

Fig. 4.6 Theoretical ROC plot vs Simulation ROC plot for Thermal Noise, SNR =20 log10( A

σtn) = -50 dB

where s[k] is the QR signal described in Equation (3.21), η[k] is the AM signal in Equation

(3.22), and n[k] is the white gaussian thermal noise signal, described in Equation (3.23).

Applying a filter matched to the QR signal of the form in Equation (4.12), which is

non-optimal since it’s only matched to the QR signal and not the noise, leads to a

non-optimal test statistic. The test statistic, T (x, h), by definition and under noise

Assumptions 3 and 4, is a sum of independent gaussian random variables and therefore is

also a gaussian random variable, with a mean and variance equal to the sum of means and

variances respectively. The approximate distributions of the test statistics corresponding

to Equations (4.33) and (4.34) are

H0 : T (x, h) ∼ N (µMF,FA, σ2MF,FA

) (4.35)

65

H1 : T (x, h) ∼ N (µMF,D, σ2MF,D

) (4.36)

The values µMF,FA, σ2MF,FA

, µMF,D and σ2MF,D

are defined as,

µMF,FA = 0

σ2MF,FA

= ε(σ2AM

+ σ2tn

)

µMF,D = βε

σ2MF,D

= ε(σ2AM

+ σ2tn

)

where β, ε and σ2tn

are defined in Equations (3.11), (4.2) and (4.13) and σ2AM

is defined

as

σ2AM

= (A2η/(4Ne))σ

2m. (4.37)

A detailed calculation of the mean and variance of the matched filter test statistic, in the

presence of AM and thermal noise, is in Appendix B. The PDFs of the test statistic in

Equations (4.35) and (4.36) were compared to PDFs obtained from simulation in Figure

(4.7). The simulation plot was generated using data with the parameters discussed in

the beginning of the section. The plot shows a very close match, which validates our

66

calculations. The non exact match is due to using a finite number of experiments to

estimate the simulation plots.

−600 −400 −200 0 200 400 600 800 10000

0.5

1

1.5

2

2.5x 10

−3

PFA


PD


PFA


PD


Fig. 4.7 Matched Filter’s Theoretical PDF plots vs Simulation PDF plots for AM andThermal Noise, SNR = 20 log10( A

Aη) = -66 dB

Based on the theoretical test statistic approximate PDFs, the explicit expression for the

probability of false alarm, PFA, is

PFA = P (T (x, h) > γ, |H0) (4.38)

= P

(Z >

γ

σMF,FA

),

and for the probability of correct detection, PD, is

67

PD = P (T (x, h) > γ, |H1) (4.39)

= P

(Z >

(γ − µMF,D)σMF,D

),


the theoretical probability of false alarm as a function of γ, PFA(γ), versus the probability

of correct detection as a function of γ, PD(γ), generates a theoretical ROC plot. This plot

is compared to an ROC plot obtained from simulation in Figure (4.8) and the match is

apparent. As mentioned earlier, the simulation data was generated using the simulation

parameters discussed in the beginning of the section, and with the assumptions placed

on the noise when calculating the theoretical expressions. The same figure displays the

performance of the energy detector, and the improvement in performance when utilizing

information about the signal is apparent.

4.3.5 Thermal Noise Comparison

When comparing the performance of the matched filter to the energy detector in Figure

(4.6), the improvement is significant. In the case of gaussian noise, not utilizing the shape

of the QR signal and squaring the received signal, as in the energy detector, makes the

first moment of the test statistic a function of the second moment of the noise, which is

nonzero. This brings the PDF of the test statistic in the case of no QR signal closer to

the PDF of the test statistic when a QR signal is present. On the other hand the first

moment of the test statistic of the matched filter is zero when a QR signal is not present,

68

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PFA


PD

, P

robabili

ty o

f C

orr

ect D

ete

ction



Theoretical ROC, Matched Filter

Simulation ROC, Matched Filter

Guess

Fig. 4.8 Theoretical ROC plot vs Simulation ROC plot for AM & Thermal Noise, SNR= 20 log10( A

Aη) = -66 dB

since it depends on the first moment of the noise which is zero. This is noticeable when

comparing the PDFs of the test statistic from the energy detector (Figure (4.1)) to the

PDFs of the test statistic from the matched filter (Figure (4.5)) and it leads to worst

correct detection and false alarm rates .

4.3.6 AM and Thermal Noise Comparison

We noticed a significant decrease in performance for both detectors when the noise

present is AM and thermal. For the same reasons mentioned in Section 4.3.5, the matched

filter again outperforms the energy detector when comparing their performance using

Figure (4.8). Not utilizing the shape of the QR signal and squaring the received signal,

as in the energy detector, makes the first moment of the test statistic a function of

the second moment of the noise, which is nonzero. This brings the PDF of the energy

detector’s test statistic in the case of no QR signal closer to the PDF of the energy

69

detector’s test statistic when a QR signal is present. This is apparent when comparing

the PDFs of the matched filter’s test statistics (Figure (4.7) to the PDFs of the energy

detector’s test statistics (Figure (4.3)).

4.4 Algorithm Comparison, No Noise Assumptions

The distributions of the AM and thermal noise signals change when Assumptions 3 and

4 are not in place. In reality noise is filtered as it passes through the quadrature phase

detector, which correlates the noise samples. The correlation in the noise causes the

performance of both detectors to deteriorate, since the standard deviation of the noise

does not fall by the square root of the number of averages anymore. To depict this

drop in performance, when the noise samples are correlated from filtering, we plotted

the energy of the noise versus the number of averages in Figure (4.9). The green and

magenta lines in the figure, which correspond to modulated white gaussian noise and

white gaussian noise respectively, closely resemble a straight line, i.e the energy of the

noise drops in a straight line in the log scale. On the other hand, the energy of the noise

when the noise signals are filtered do not drop in a straight line in the log scale and

create a ripple effect.

Without the assumptions on the noise, the probability of correct detection and the

probability of false alarm in the presence of thermal noise for both detectors, shown in

Equations (4.20), (4.19), (4.32) and (4.31) respectively, can not be applied. Abandoning

Assumption 4, leads to a nonwhite broadcasted message m[k]. The AM signal is also

filtered when passing through the quadrature phase detector. In this case the central

limit theorem can not be applied and therefore the averaged AM signal can not be

70

100

101

102

103

10−6

10−5

10−4

10−3

10−2

10−1

Number of Averages

Energ

y

Experiment, Low passed Thermal

Experiment, Low passed BPWGAM

Simulation, Low passed Thermal

Simulation, Low passed BPWGAM

Simulation, Thermal

Simulation, WGAM

Fig. 4.9 Energy of noise signals vs number of averages, Log scale.

approximated with a normal distribution. Without Assumptions 3 and 4, the calculations

for the probability of correct detection and the probability of false alarm in the presence

of AM and thermal noise for both detectors, shown in Equations (4.26), (4.25), (4.39)

and (4.38) respectively, can not be applied.

The only assumption placed when comparing the performance of the matched filter to the

energy detector in this section is that an exact description of the QR signal is attainable.

4.4.1 Band and Low pass Filtered Thermal Noise

In the case when the thermal noise is band and low passed, the noise samples become

random variables correlated in time. Therefore, calculating the mean and variance of the

test statistic for both detectors is a difficult task. The test statistic for both detectors is

not necessarily a normal random variable neither. This section, will compare the perfor-

mance of both detectors in the ideal case, when the noise is white, to the performance of

both detectors using experimental and simulation data when no assumptions are placed

71

on the noise. The noise in the experimental data has been band and low pass filtered by

the QR spectrometer, while in simulation data thermal was band and low passed using

a simulated filters.

In the case of the Energy Detector, the test statistic may not be approximated as a normal

random variable, without the noise assumptions, since it is the sum of a large number of

random variables that are not necessarily independent. Though to depict the increase in

the standard deviation of the test statistic, which leads to a drop in performance, when

the random variables are correlated, we will assume that the test statistic when the noise

is filtered is a normal random variable. This allows us to compare the PDFs of the test

statistic in the ideal case to the PDFs from experimental and simulation data.

Figure (4.10), compares the PDFs of the test statistic of the energy detector, when a

QR signal is present and when one is not. The PDFs for the ideal thermal noise case

were generated using Equations (4.17) and (4.18), while the experimental and simulation

PDFs were generated using the mean and variance calculated from 100 experiments. The

signal to noise ratio used was SNRtn = 20 log 10(A/σtn) = −12 dB, where σtn is the

thermal noise’s standard deviation before filtering and A is the amplitude of the QR

signal. From the figure it is apparent that the standard deviation of the test statistics

increases from band pass filtering the noise, while the mean stays fairly unchanged. This

would lead to a drop in the performance of the detector. The figure also shows that the

results from simulation data closely resemble the results from experimental data, which

further validates the simulation data.

72

From looking at the PDFs in Figure (4.10), it would be pointless to plot to the corre-

sponding ROC curves, since it will not depict a drop in performance from filtering. In all

three cases, a threshold can be chosen, where the sum of the areas under the red curve

and the blue curve to the right of it, which represent the probability of correct detection

and the probability of false alarm respectively, are approximately 1 and 0.

−0.4 −0.2 0 0.2 0.4 0.6 0.80

5

10

15

20

25

30

35

40

45

FP pdf Experiment, mean = 0.13028, std. dev = 0.02529

FP pdf Calculation, mean = 0.13202, std. dev = 0.0089623

FP pdf Simulation, mean = 0.13378, std. dev = 0.022683

TP pdf Experiment, mean = 0.40927, std. dev = 0.061445

TP pdf Calculation, mean = 0.39377, std. dev = 0.01997

TP pdf Simulation, mean = 0.37125, std. dev = 0.045273

Fig. 4.10 Energy Detector PDF plots in the presence of Thermal Noise, SNR =20 log10( A

σtn) =-12 dB, Theoretical PDF plots vs Simulation and Experiment without

Noise Assumptions

Figure (4.11) compares the PDFs of the matched filter, assuming an exact description

of the QR signal is attainable, when the corrupting noise is thermal to the case when

the noise is band and low passed thermal noise. When the thermal noise is filtered the

test statistic is not a sum of independent normal random variables anymore. The inner

product of the filter with the received signal, h[k]x[k], at a time k is correlated to the

inner product at time k + 1. This due to the fact that the filtered thermal noise in x[k]

for each k is not an independent normal random variable. Though, unlike in the case

73

of the energy detector, the test statistic is still a normal random variable since it is the

sum of dependent but jointly normal random variables.

To depict the decrease in performance resulting from band pass filtering the noise, as we

did with the energy detector, we plot the distributions of the test statistic in both cases

in Figure (4.11). When comparing the distributions of the test statistic when the noise

is filtered to the ideal case case, it is apparent that band and low pass filtering the noise

causes the standard deviation of the test statistic to increase while the mean remains

fairly equal. This leads to worse detection performance, since it makes the test statistics

less separable.

Plotting the corresponding ROC curves is pointless, since they will not show the drop in

performance caused by filtering. This is due to the fact that in all three cases a threshold

can be chosen, where the sum of the areas under the red curve and the blue curve to

the right of it, which represent the probability of correct detection and the probability

of false alarm respectively, are approximately 1 and 0. Comparing the PDFs of the test

statistics of the energy detector, Figure (4.10), to the matched filter, Figure (4.11), shows

that the test statistics of the energy detector are less separable.

4.4.2 Band and Low pass Filtered AM and Thermal Noise

In the presence of thermal noise and AM noise, both noise sources are actually band

and low pass filtered. In experiment this was performed by the quadrature detector,

while in simulation 8-pole Butterworth filters were used. The modulated signal, m[k],

is band passed with cutoff frequencies of 40 Hz and 10 kHz, and therefore not normal

74

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.5

1

1.5

2

2.5


FP pdf Simulation, mean = −0.0037341, std. dev = 0.38943

FP pdf Calculation, mean = 0, std. dev = 0.17093




Fig. 4.11 Matched Filter PDF plots in the presence of Thermal Noise, SNR =20 log10( A

σtn) = -12 dB, Theoretical PDF plots vs Simulation and Experiment without

Noise Assumptions

and independent for each k. As was stated in the previous section, calculating the

test statistic under these circumstances is very difficult. The test statistics is again not

necessarily a normal random variable when the AM and thermal noise are filtered for

both detectors. This section, will compare the performance of both detectors in the ideal

case, when the noise is white thermal noise plus amplitude modulated thermal noise, to

the performance of both detectors on experimental and simulation data. The noise in the

experimental data has been band and low pass filtered by the QR spectrometer, while

in simulation data it was band and low passed using simulated filters.

As was mentioned earlier the energy detector’s test statistic is not a normal random

variable in the case when the noise is filtered. Though to compare the distribution of

the energy detector test statistic in experiment to the case of ideal noise, we will assume

the test statistic is a normal random variable even when the noise is filtered. Figure

75

(4.12) compares the distributions of the test statistic from calculation, Equations (4.23)

and (4.24), to the test statistics from experimental and simulation data when no noise

is filtered. The mean and variance of the test statistics for experimental and simulation

data, where calculated based on 50 experiments. As is in the previous section these plots

depict the increase in the standard deviation of the test statistic in the case of filtered

noise, which is due to the correlation in the noise samples after filtering. The plot also

shows that the test statistic from simulation data is very close to the test statistic from

experimental data. This increase in standard deviation decreases the performance level

of the energy detector. The corresponding ROC plots are shown in Figure (4.13). As

expected, the energy detector in the ideal uncorrelated noise case, outperforms the energy

detector when applied to experimental and simulation data with filtered noise.

−1 0 1 2 3 4 50

1

2

3

4

5

6

7

8


FP pdf Calculation, mean = 0.67632, std. dev = 0.054462

FP pdf Simulation, mean = 0.51913, std. dev = 0.090475TP pdf Experiment, mean = 0.72802, std. dev = 0.13476



Fig. 4.12 Energy Detector PDF plots in the presence of band-passed white gaussian AMand Thermal Noise, SNR = 20 log10( A

Aη) = -30 dB, Theoretical PDF plots vs Simulation

and Experiment without Noise Assumptions. Nm = 10, Number of experiments = 50.

76

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

False Positive

Tru

e P

osiv

tive

Experiment

Calculations

Simulations

Fig. 4.13 Energy Detector ROC plots in the presence of band-passed white gaussian AMThermal Noise, SNR = 20 log10( A

Aη)= -30 dB, Theoretical PDF plots vs Simulation and

Experiment without Noise Assumptions. Nm = 10, Number of experiments = 50.

The same experimental and simulation data used to compare the performance of the en-

ergy detector where used to compare the performance of the matched filter. To visually

depict the difference in the test statistic of the matched filter in the ideal non-filtered

independent noise case, Equations (4.33) and (4.34), to the filtered noise case, the dis-

tributions of the test statistics are plotted in Figure (4.14). Its apparent that the test

statistic’s standard deviation increases, when the noise is filtered in the experimental and

simulation data. It is also apparent that the difference in the mean of the test statistic

when a a QR signal is present to when one is not remain fairly constant, therefore we

can conclude that the performance of the matched filter decreases when the noise is

correlated. The corresponding ROC plots are shown in Figure (4.15). The plot does not

show a difference in the ROC curves since the signal to noise ratio, which was chosen to

match the experimental data, is high and the matched filter was able to perfectly detect

the presence of a QR signal in all cases.

77

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4


FP pdf Simulation, mean = 0.060147, std. dev = 0.49506

FP pdf Calculation, mean = 0, std. dev = 0.38691




Fig. 4.14 Matched Filter PDF plots in the presence of band-passed white gaussian AMand Thermal Noise, SNR = 20 log10( A


and Experiment without Noise Assumptions.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Filter Matched to QR, Experimental

Filter Matched to QR, Simulation

Filter Matched to QR, Calculation

Fig. 4.15 Matched Filter PDF plots in the presence of band-passed white gaussian AMand Thermal Noise, SNR = 20 log10( A


and Experiment without Noise Assumptions.

78

4.5 Conclusion

In conclusion, the matched filter, designed for the presence of thermal noise, outperforms

the energy detector in the two cases, first in the presence of thermal noise only and second

in the presence of both thermal and band-passed white gaussian AM noise. The major

drawback to the matched filter is that it requires knowledge of the QR signal, which

is addressed in subsequent chapters. The improvement in performance of the matched

filter is due to the larger separation between the distribution of the test statistic when

a QR signal is present to the distribution of the test statistic when one is not. This

improvement in the separation of the test statistic is apparent when comparing the

distributions shown in Figure (4.12) to the distributions shown in Figure (4.14). This

increase in separation of the test statistic, is due to the fact that the variances of the

matched filter’s test statistic are fairly equal, whether a QR signal is present or not and

is a function of the second moment of the noise. On the other hand the variance of the

energy detector’s test statistic is much larger than that of the matched filter, when a

QR signal present, and decreases when one is not. The difference between the mean of

the test statistic when a QR signal is present to when one is not, is constant and equal

to the energy of the QR signal.

The following chapter introduces an approach to designing a filter matched to the QR

signal present and compares its performance to the energy detector using ROC plots.

79

Chapter 5

Batch Matched Filter

As a first attempt to solving the QR detection problem, we propose a gridding approach

to determine the unknown QR parameters and design a filter matched to each value on

the grid, creating a batch of matched filters. Using the output of the batch of filters a

decision whether a QR signal is present is made.

The received corrupted version of the QR signal after Nm experiments is the sum of

the phase cycled QR, AM and thermal noise signals shown in Equations (3.21), (3.22)

and (3.23) respectively. Since the observed QR signal’s base band frequency ωbb =

ω∗ − ωs + ω0, (note that ωs was denoted as ω∗ in Section 2.4), base-band phase φbb,

and the decaying response parameter T2∗ are unknowns, we are unable to design a

periodic filter matched to the QR signal. However, we can utilize the knowledge that the

observed base band frequency ωbb, base band phase φbb and decaying parameter T2∗ fall

within known upper and lower bounds, as is shown in Equations (5.1), (5.2) and (5.3)

respectively, to design a batch of matched filters to use for QR detection.

ωlb ≤ ωbb ≤ ωub (5.1)

−π ≤ φbb ≤ π (5.2)

80

T ∗2 lb≤ T ∗

2≤ T ∗

2 ub(5.3)

Designing a matched filter for each possible frequency, phase and decaying parameter

within the bounded ranges above is not feasible. We therefore chose to discretize the

range of values, creating a grid of possible values, and design a filter matched to each

value on the grid. Out of this batch of filters, the one that results in the maximum

output is the chosen filter and its output is compared to a threshold to determine the

presence of a QR signal. Since this approach does not design a filter for each possible QR

signal that might be present, the problem where any one of the QR signal’s parameters

(frequency, phase and decaying parameter), does not match any of the values on the grid

should be addressed.

The following section discusses this problem and shows that small uncertainties in the

QR signal parameters have a small effect on the inner product of the QR signal with the

filter, which is the value that enables one to detect the presence of the QR signal. This

section is followed by a more detailed description of the “brute force” batch matched

filter algorithm, which is then followed by a section that compares the performance of

the “brute force” approach to the batch matched filter to the generic energy detector

algorithm. The improvement in performance gained from finer gridding of the parameters

is then discussed in a following section. An adaptive gridding approach to the batch

matched filter is introduced in Section 5.5. The idea behind the adaptive grid approach,

is to perform a coarse global grid and once a grid point is selected, perform a finer local

search around that point. The performance of the batch filter using this approach is then

81

compared to the “brute force” approach to the batch matched filter. Unless specified

otherwise, when referring to the batch matched filter in the subsequent chapters, the

reference is to the “brute force” approach described in Section 5.2 and not the smart

adaptive gridding approach. The chapter ends with a section that discusses possible

thresholding methods that can be used.

5.1 Error in the QR signal Description

This section addresses the issue when the frequency, phase or decaying parameter of

the matched filter are not exactly matched to the QR signal. It shows that for small

uncertainties in the QR signal parameters the performance of the matched filter is not

significantly affected, but as the error increases the performance of the filter quickly starts

to deteriorate. The drop in performance of the matched filter, is due to the drop in the

value of Equation (4.2), the inner product of the filter and the QR signal present. As

this value decreases, the mean of the test statistic when a QR signal is present becomes

closer to the mean of the test statistic when a QR signal is not. If the QR signal within

the received signal can be described as,

s[k] =A

αse(−|−T/2+k∆|/T2

∗) cos (ωbb(−T/2 + k∆) + φbb). (5.4)

for k = 0, 1, . . . , Ns− 1, where αs normalizes the signal in the l2 norm. If we assume the

error variables in frequency, phase and decaying parameter are denoted by ∆ωbb, ∆φbb,

and ∆T2∗ respectively, then the mismatched filter can be described as

82

hu[k] =B

αBe(−|−T/2+k∆|/(T2

∗+∆T2∗) cos ((ωbb + ∆ωbb)(−T/2 + k∆) + φbb + ∆φbb), (5.5)

for k = 0, 1, . . . , Ns − 1, where αB normalizes the signal in the l2 norm. With this

error, the inner product of the QR signal present and the filter, which was represented

in Equation (4.2), for Nm = 1, changes to,

εu =Ns−1∑k=0

AB

αsαh[e(−|−T/2+k∆|/T2

∗)e(−|−T/2+k∆|/(T2∗+∆T2

∗)] . . . (5.6)

[cos ((ωbb + ∆ωbb)(−T/2 + k∆) + φbb + ∆φbb) cos(ωbb(−T/2 + k∆) + φbb)]

=Ns−1∑k=0

AB

αsαh[e(−|−T/2+k∆|/T2

∗)e(−|−T/2+k∆|/(T2∗+∆T2

∗)] . . .

[cos (∆ωbb(−T/2 + k∆) + ∆φbb)− . . .

sin(ωbb(−T/2 + k∆) + φbb) sin((ωbb + ∆ωbb)(−T/2 + k∆) + φbb + ∆φbb)]

.

Next we discuss how this value changes with respect to sole changes in the error of the

individual parameters, starting with frequency. A plot of this inner product, εu versus

the error in frequency is shown in Figure (5.1), where the received QR signal had a

frequency of 12.1 kHz and the filter’s frequency error varied from 0 to 3 kHz. The plot

shows that the energy rapidly drops as the frequency offset increases causing a drop

in performance of the matched filter. Based on this plot, the gridding in frequency in

83

the following section was set to 50 Hz, which corresponds to a value of 100π for ω∆.

This value was chosen since it leads to small discrepancies between εu and the energy

of the QR signal present.This value is varied in other sections of the chapter, when the

improvement in performance from finer gridding is discussed.

0 500 1000 1500 2000 2500 30000

0.2

0.4

0.6

0.8

1

1.2

1.4E

nerg

y, ε

u

Frequency (Hz) Offset

Fig. 5.1 Inner Product versus Frequency Error of the Filter

The inner product, εu, is also plotted versus the error in phase in Figure (5.2). In this

case the received QR signal had a frequency of 12.1 kHz and a phase of 0 radians. It

is apparent from the plot that the inner product again sharply drops for small shifts in

phase, as is shown, a phase shift of π/2 leads to an εu of approximately zero. Based on

this plot, the value chosen as the interval in between phase grid points in the following

section is 0.1 radians, since it leads to small differences between εu and the energy of the

QR signal present.

Lastly the inner product, εu, is plotted versus error in the decaying parameter in Figure

(5.3). The plot was generated using a QR signal with a decaying parameter T2∗ = .9e−3,

and a filter where the error in the decaying parameter swept the interval [−.9e−3, 2e−3].

The plot shows that inner product, drops at a much slower rate when the decaying

84

parameter of the filter is larger than that of the QR signal present, compared to the

rate of decrease when the decaying parameter of the filter is lower than that of the QR

signal. This is because a larger decaying parameter implies less decay (i.e. as this value

increases the filter more closely resembles a sinusoid versus a sinusoid decaying about the

center). The plot shown in the figure lead to the decision of using 100e-6 as the gridding

interval when designing the filters in the following section. This is due to the fact that,

an error of 100e-6 led to a small difference between the energy of the QR signal and the

value εu.

0 1 2 3 4 5 6 7−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Energ

y, ε

u

Phase (Radians) Offset

Fig. 5.2 Inner Product versus Phase Error of the Filter

Figure (5.4) uses simulation data, with a QR signal at 12.1 kHz and an SNR of −30 dB, to

demonstrate the drop in the performance of the matched filter when the frequency is not

matched. Since the batch matched filter approach is discussed in the following section,

the simulation results are presented without discussing the filter design process itself.

85

−5 0 5 10 15 20

x 10−4

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Energ

y, ε

u

Decaying Parameter Offset

Fig. 5.3 Inner Product versus Decaying Parameter Error of the Filter

The simulation assumed knowledge of the phase and decaying parameter to demonstrate

the effects of error in frequency only. While a mismatch of 100 Hz and 500 Hz showed a

slight drop in performance, a mismatch of 1 kHz dropped the performance to almost the

guessing level. This problem of error in the QR signal parameters can be overcome by

fine gridding. Coarse gridding leads to a drop in the performance of the filter when the

QR signal present falls in between frequency grid values, this phenomena is demonstrated

in Figure (5.5). The figure shows ROC curves for several simulation data setups, using a

batch matched filter, with a frequency gridding interval of 1 kHz, ω∆ = 2000π. When the

QR signal present is at 12.5 kHz (a value centered between the possible filter frequency

values of 12 kHz and 13 kHz) the performance of the filter is poor and increases as the

QR signal gets closer to 13 kHz, one of the possible filter frequency values. The following

section throughly describes the batch matched filter algorithm.

86

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Prob of False Alarm

Pro

b o

f C

orr

ect

De

tectio

n

Exact Freq

QR Freq at 12.1 kHz, Filter at 12 kHz

QR Freq at 12.1 kHz, Filter at 11.6 kHz

QR Freq at 12.1 kHz, Filter at 11.1 kHz

Guess

Fig. 5.4 Uncertain Frequency Matched Filter ROC plots in the presence of Band-passedWhite Gaussian AM, SNR = -30 dB

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm

QR @ 12.5 kHz, Closest Possible Filter Frequencies around QR, 12 & 13 kHz





QR @ 13 kHz, Closest Possible Filter Frequencies around QR, 12 & 13 kHz

Fig. 5.5 Batch Matched Filter with 1 kHz Frequency gridding, ROC plots in the presenceof Different QR Frequencies and Band-passed White Gaussian AM, Simulation SNR =-22 dB

87

5.2 Batch Matched Filter

Let the intervals of admissible values for frequency ωbb, phase φbb and decaying parameter

T2∗ in equations (5.1), (5.2), and (5.3) respectively, be discretized and converted into

sets of admissible values represented in the following vectors:

ωa = [ωlb, ωlb + ω∆, ωlb + 2ω∆, . . . , ωub] (5.7)

φa = [−π,−π + φ∆,−π + 2φ∆, . . . , π] (5.8)

Ta = [T2∗lb, T2

∗lb + T2

∗∆, T2

∗lb + 2T2

∗∆, . . . , T2

∗ub] (5.9)

where Ndω , Ndφ , and NdT are the number of discrete frequency values, discrete phase

values and discrete decaying values respectively, and ω∆ = (ωub − ωlb)/(Ndω − 1), φ∆ =

(2π)/(Ndφ − 1) and T2∗

∆ = (T2∗ub − T2

∗lb)/(NdT − 1) are the discretization periods for

the frequency, phase and decaying parameter respectively.

The next step is to choose a triplet of values from the vectors ωa, φa and Ta and use them

to design a corresponding periodic matched filter of period T . If there are Ndω , Ndφ and

NdT values in the vectors ωa, φa and Ta respectively, then there are Ndω × Ndφ × NdT

possible pair combinations. If the lth value of vector ωa, the mth value of vector φa

and the nth value of vector Ta are the chosen triplets then the corresponding normalized

estimate of the QR signal would be

88

sl,m,n[k] =1

α(l,m, n)e(−|−T/2+k∆|/Ta(n))cos(ωa(l)(−T/2 + k∆) + φa(m)) (5.10)

for k = 0, 1, . . . , Ns − 1, where T is the length of the data acquisition window, Ns is the

number of discrete samples, ∆ = T/(Ns − 1) and the constant α(l,m, n) is the second

norm of the QR signal defined as,

α(l,m, n) =Ns−1∑k=0

[e(−|−T/2+k∆|/Ta(n))cos(ωa(l)(−T/2 + k∆) + φa(m))]2. (5.11)

Normalizing the filter by, α(l,m,m), the l2 norm is a necessary step to ensure that each

filter’s contribution to its inner product with the QR signal, had one been present, is

dependent on the match between them. Therefore, from Equation (4.12), the matched

filter, designed with white noise of unit variance in mind, for the above chosen QR signal

parameter values is the following periodic filter:

h[k] =Nm−1∑i=0

sl,m,n[k − iNs], (5.12)

for k = 0, 1, . . . , Nm × Ns − 1. After passing the received corrupted signal through the

filter with the above impulse response, the value of the Nm ×Nsth sample, which is

equivalent to the inner product of the received signal and the filter, is recorded and the

process is repeated for all different combinations of frequency ωbb, phase φbb and decaying

89

parameter T2∗ values. Repeating this procedure Ndω ×Ndφ×NdT times, for the different

possible triplets, leads to a vector Θ of values of length Ndω ×Ndφ ×NdT .

Θ = [θ1, . . . , θNdω×Ndφ×NdT] (5.13)

Finally the maximum value of the vector Θ is compared to a threshold value θthreshold

and the filter corresponding to this maximum is the chosen filter out of the batch.

Normalizing all the filters in the batch by their l2 norms increases the likelihood of the

chosen filter being the closest description of the QR signal present. If maximum value

of the vector Θ, is greater than the threshold then we conclude that a QR signal exists

within the corrupted signal, otherwise we conclude that the received signal is just noise.

Determining the value of θthreshold is still a matter that requires further work. There are

several possible ways of determining the value of θthreshold, one of which, is to collect

a measurement of the noise and use it to determine the value of θthreshold. The above

process of choosing the filter is equivalent to selecting the filter, hn[k], that maximizes

yn, the inner product of the filter with the received signal x[k], as follows

y = maxn

(yn) = maxn

(hn[k]Tx[k]

)= max

n

(|hn[k]||x[k]| cos γ

)

where γ is the angle between the received signal, x[k], and the filter hn[k]. The above

value is maximized when the value of the angle, γ, is zero, which corresponds to a value

of hn[k], equal to a scalar multiple of x[k]. For the selected filter, hn[k], to be x[k], the

received noise would have to zero and the QR signal present would have to be one of the

90

filters in the batch. If one assumes that the chosen filter out of the batch is equal to the

QR signal present, s[k], then the resulting signal to noise ratio defined in Equation (4.8)

is reduced to

ρ(s; s,Σ) =(s[k]T s[k])2

s[k]TΣs[k].

Had Σ been equal to the identity matrix, i.e. the noise present is white with unit variance,

the above signal to noise ratio would reduce to s[k]T s[k], which is the optimal signal to

noise ratio when the noise is white.

5.3 Batch Matched Filter versus Energy Detector, Unknown QR signal

This section compares the performance of the batch matched filter to the generic energy

detector, when the QR signal is unknown and no assumptions are placed on the noise,

which is the case in reality. The following subsection compares the performance of the

two detectors using simulation data. This subsection is followed by one that compares

their performance using experimental data collected in the laboratory.

For consistency, when comparing the batch matched filter to the energy detector (using

simulation and experimental data in the following two subsections) the batch matched

filter’s design parameters were fixed. The design of the filter assumes the presence of

thermal noise and therefore uses a Σ = I. The method gridded using the following

values, ωlb = 0, ωub = 40000π, ω∆ = 100π, φ∆ = .1, T ∗2 lb

= 300e − 6, T ∗2 ub

= 900e − 6

and T ∗2 ∆

= 100e−6. The above values lead to Ndω = 401, Ndφ = 63 and NdT = 7 in each

of the three dimensions, which corresponds to Ndω ×Ndφ ×NdT = 176841 filters in the

91

batch. The computational time of each one of those filters, which is an inner product, is

cheap and the process can be parallelized. The following sections will discuss the effect of

finer gridding on performance and will introduce a smarter adaptive gridding approach,

which reduces computational time.

5.3.1 Simulation Data

This section uses simulation data to compare the performance of the batch matched filter,

designed using the parameters at the beginning of the section, to the energy detector.

Six sets of simulation data of different frequencies, where the QR signal frequency and

the AM signal frequency are equal, were generated. The selected frequencies were 2.5, 5,

7.5, 10, 12.5 and 15 kHz. Each data set consisted of 100 experiments, where the received

signal is the sum of QR and AM, and 100 experiments, where the received signal consists

of only AM noise. Each experiment consisted of Nm = 5 trials strung in time, where

each is the average of 20 phase cycled echoes.

ROC plots were used to compare the performance of the batch matched filter to the

energy detector. Figures (5.6-5.11) compare the performance of the two detectors for

the different frequencies. The figures show that the batch matched filter outperforms

the energy detector for all cases. An outperformance by the batch matched filter was

expected since it was demonstrated in Chapter 4, when the QR signal was assumed to

be known.

92

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm

Energy DetectorBatch Matched Filter Detector, L

2 Normalization

Fig. 5.6 Simulation SNR = -22 dB, Performance Comparison, BP White Gaussian AM,with the QR and AM at 2.5 kHz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm


2 Normalization

Fig. 5.7 Simulation SNR = -22 dB, Performance Comparison, BP White Gaussian AM,with the QR and AM at 5 kHz

93

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm


2 Normalization


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm


2 Normalization


94

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

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0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm

Energy Detector, Area Under the Curve 0.81505Batch Matched Filter, L

2 Normalization, Area Under the Curve 0.8464


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

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1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm


2 Normalization


95

5.3.2 Experimental Data

This section uses experimental data to compare the performance of the batch matched

filter, designed using the parameters at the beginning of the section, to the energy

detector. Four sets of experiment data of different frequencies, where the QR signal

baseband frequency and AM signal baseband frequency are equal, were collected. The

four sets, which will be referred to as A, B, C and D have frequencies 6.25, 10, 8 and 12.5

kHz respectively. Each set consisted of 100 experiments, where the received signal is the

sum of a QR signal an AM signal and thermal noise, and the another 100 experiments,

where the received signal is the sum of an AM signal and thermal noise. Each experiment

consists of Nm = 5 trials strung in time, where each is the average of 20 phase-cycled

echoes.

ROC plots were used to compare the performance of the two detectors for the different

experimental sets. These ROC plots are shown in Figures (5.12)-(5.15) for experiments

A-D respectively. The batch matched filter with the gridding parameters mentioned at

the beginning of the section, is represented by the blue curve in the figures. The green

represents a batch matched filter with different gridding parameters, than the ones above,

and will be discussed in the following section. The figures show that the batch matched

filter outperforms the energy detector in all cases.

96

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm

Energy Detector, Area Under the Curve 0.9455

Batch Matched Filter, Area Under the Curve 0.97745

Batch Matched Filter, Finest Grid, Area Under the Curve 0.97745

Fig. 5.12 Experiment A, Performance Comparison on Experiment Data, BP White Gaus-sian AM, with the QR and AM at 6.25 kHz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm



Batch Matched Filter, L2 Normalization, Finest Grid, Area Under the Curve 0.9987

Fig. 5.13 Experiment B, Performance Comparison on Experiment Data, BP White Gaus-sian AM, with the QR and AM at 10 kHz

97

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm

Exp C, −8 kHz Freq, BP White Gaussian AM




Fig. 5.14 Experiment C, Performance Comparison on Experiment Data, BP White Gaus-sian AM, with the QR and AM at -8 kHz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm

Exp D, 12.5 kHz Freq, BP White Gaussian AM




Fig. 5.15 Experiment D, Performance Comparison on Experiment Data, BP White Gaus-sian AM, with the QR and AM at 12.5 kHz

98

5.4 The Effect of Finer Gridding on the Performance of the Batch

Matched Filter

This section discusses the effect of finer gridding on the performance of the batch matched

filter. In Section 5.1, it has been shown that finer gridding leads to an improvement in the

performance of the batch matched filter when compared to coarser gridding. Though

it is important to finely grid the parameters of the QR signal, there is a limit to the

increase in performance gained from finer gridding.

Figures (5.12)-(5.15) demonstrate the saturation in the performance gained, from finer

gridding. The green curve represents the performance of a batch matched filter with

finer gridding parameters, compared to the batch matched filter discussed in the previous

section, the blue curve. For the green curve, the distance in between frequency (ω∆),

phase (φ∆) and decaying constant (T ∗2 ∆

) values was cut by a half, leading to 23 = 8 times

the number of filters used to generate the blue curve. It is apparent from the figures that

the performance of the two batches is almost identical. An adaptive gridding approach

to batch matched filters is discussed in the following section.

5.5 Adaptive Grid Batch Matched Filter

This section introduces a smarter adaptive gridding approach to batch matched filters

to reduce the number of filters one grids through. The idea is to coarsely grid through

the QR signal parameters, initially, and follow it with a finer local grid around the QR

signal parameters selected by the coarser grid. This approach is described in more detail

next.

99

To start, the intervals of admissible values for frequency ωbb, phase φbb, and decaying

parameter T2∗, are discretized into a set of admissible values as shown in equations

(5.1), (5.2), and (5.3) respectively. The number of discrete frequency Ndω , phase Ndφ

and decaying parameter NdT values in this initial coarse grid are small when compared

to the number of values used in the “brute force” approach. The next step is to select

the point on the grid that results in the maximum output. This process is discussed

in detail in Section 5.2, and therefore will not be repeated here. The objective of the

coarser grid is to locate a filter with parameters close to the parameters of the QR signal,

had one been present. The QR parameters selected from the coarse grid are then used

to define a local search region and locate a filter that matches the QR signal present as

follows.

Let the frequency, phase and decaying parameter selected as a result of the global coarse

search be denoted as ωG, φG and T ∗2 G

respectively. The local search region is then defined

by the two values of frequency, phase and decaying parameter on the global grid closest

to each of the selected QR parameter values. More specifically, each of the QR signal

parameters in the local search is bounded from above by either the closest value on the

coarse grid larger than the selected value, or by the selected value itself, in the case it is

the largest value on that grid. Similarly, each of the QR signal parameters is bounded

from below by either the closest value on the coarse grid smaller than the selected value,

or the selected value itself, in the case it is the smallest value on that grid. The upper

and lower bounds for each of the QR signal parameters in the local search can be defined

as follows,

100

ωlbl = max{(ωG − ω∆), ωlb} ωubl = min{(ωG + ω∆), ωub} (5.14)

φlbl = max{(φG − φ∆),−π} φubl = min{(φG + φ∆), π} (5.15)

T ∗2 lbl

= max{(T ∗2 G− T ∗

2 ∆), T ∗

2 lb} T ∗

2 ubl= min{(T ∗

2 G+ T ∗

2 ∆), T ∗

2 ub} (5.16)

where ω∆, φ∆ and T ∗2 ∆

are the distances between grid points in the coarse grid, and

(ωlb, ωub), (φlb, φub), and (T ∗2 lb

, T ∗2 ub

) are the lower and upper bound values that define

the QR signal parameters in the same coarse grid. Therefore, the intervals that define

the local search region can be finely discretized and converted into a set of admissible

values represented in the following vectors:

ωal = [ωlbl , ωlbl + ω∆l, ωlbl + 2ω∆l

, . . . , ωubl ] (5.17)

φal = [φlbl , φlbl + φ∆l, φlbl + 2φ∆l

, . . . , φubl ] (5.18)

Tal = [T2∗lbl, T2∗lbl

+ T2∗

∆l, T2∗lbl

+ 2T2∗

∆l, . . . , T2

∗ubl

] (5.19)

where Ndωl, Ndφl

, and NdTlare the number of discrete frequency values, discrete phase

values and discrete decaying values respectively in the local search. The distance in

between values of frequency, phase and decaying parameter in the local search are defined

as ω∆l= (ωubl − ωlbl)/(Ndωl

− 1), φ∆l= (φubl − φlbl)/(Ndφl

− 1) and T2∗

∆l= (T2

∗ubl−

T2∗lbl

)/(NdTl− 1), respectively. The filter in the local batch of filters that results in the

maximum output is chosen, and by comparing that value to a threshold the decision

101

whether a QR signal is present is made. This process is discussed in detail in Section

5.2, and will not be repeated here. The total number of filters used in this adaptive grid

approach to the batch matched filter is equal to the sum of the filters in the initial coarse

batch plus the number of filters in the later local fine batch, Ndω ×Ndφ ×NdT +Ndωl×

Ndφl× NdTl

. The performance of this adaptive gridding approach to batch matched

filters is compared to the “brute force” approach, discussed in Section 5.2, using both

simulation and experimental data next.

When comparing the “brute force” batch matched filter to the adaptive grid batch

matched filter (using simulation and experimental data in the following two subsections)

the “brute force” batch matched filter’s design parameters were fixed and are the same

as the ones used in Section 5.3. The “brute force” gridding parameters used resulted

in 176, 841. The adaptive grid batch matched filter it is being compared to used the

following values, ωlb = 0, ωub = 40000π, ω∆ = 1000π, ω∆l= 50π, φ∆ = .2, φ∆l

= .05,

T ∗2 lb

= 300e− 6, T ∗2 ub

= 900e− 6, T ∗2 ∆

= 200e− 6 and T ∗2 ∆l

= 50e− 6. The above values

lead to Ndω = 41 different frequencies, Ndφ = 32 different phases and NdT = 4 different

decaying parameters in the initial batch of filters. On the other hand within the local

search, the above grid values lead to Ndωl= 41 different frequencies, Ndφl

= 9 different

phases and NdTl= 9 different decaying parameters in the local batch of filters. This

corresponds to a total of Ndω ×Ndφ ×NdT +Ndωl×Ndφl

×NdTl= 5248 + 3321 = 8569

different filters between both batches. The adaptive grid approach used a significantly

smaller batch of filters, when compared to the “brute force” batch of filters, 176841,

which leads to significantly lower computational time. The two approaches to batch

102

matched filters are compared using simulation and experimental data in the following

subsections.


This section uses simulation data to compare the performance of the adaptive gridding

approach to batch matched filters to the “brute force” approach of batch matched filters.

Six sets of simulation data of different frequencies, where the QR signal frequency and

the AM signal frequency are equal, were used. The selected frequencies were 2.5, 5, 7.5,

10, 12.5 and 15 kHz. Each data set consisted of 100 experiments, where the received

signal is the sum of QR and AM, and 100 experiments, where the received signal only

AM noise. Each experiment consisted of Nm = 5 trials strung in time, where each is

the average of 20 phase cycled echoes. These data sets are used to compare the two

approaches to the batch matched filter in Figures (5.16)-(5.21).

The adaptive gridding approach used a significantly smaller batch of filters, when com-

pared to the batch of filters used in the “brute force” approach in Section (5.3) (176841),

which leads to significantly lower computational time. In all simulation data sets, the

adaptive gridding approach to batch matched filters achieves approximately equal per-

formance to that achieved by “brute force” approach with less computational time.

Even though, the adaptive grid’s local grid was finer than the grid of the “brute force”

approach, nowhere did the adaptive grid significantly outperform the “brute force” ap-

proach.

103

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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0.3

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0.5

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0.9

1

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b C

orr

ect D

ete

ction

Prob False Alarm



Batch Matched Filter, L2 Normalization, Adaptive Grid, Area Under the Curve 0.88825

Fig. 5.16 Simulation SNR = -22 dB, Adaptive versus “Brute Force” Gridding Perfor-mance Comparison, BP White Gaussian AM, with the QR and AM at 2.5 kHz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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Fig. 5.17 Simulation SNR = -22 dB, Adaptive versus “Brute Force” Gridding Perfor-mance Comparison, BP White Gaussian AM, with the QR and AM at 5 kHz

104

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Batch Matched Filter, L2 Normalization, Adpative Grid, Area Under the Curve 0.8889


106


This section uses experimental data to compare the adaptive gridding approach to the

brute force approach to batch matched filters. Four sets of experiment data of different

frequencies, where the QR signal baseband frequency and AM signal baseband frequency

are equal, were used. The four sets, which will be referred to as A, B, C and D have

frequencies 6.25, 10, 8 and 12.5 kHz respectively. Each set consisted of 100 experiments,

where the received signal is the sum of a QR signal an AM signal and thermal noise, and

the another 100 experiments, where the received signal is the sum of an AM signal and

thermal noise. Each experiment consists of Nm = 5 trials strung in time, where each is

the average of 20 phase-cycled echoes.

The “brute force” approach to batch matched filters, used the same parameters discussed

in Section 5.3. The adaptive grid used the parameters listed at the beginning of this

section. The two approaches are compared in Figures (5.22)-(5.25) using experimental

data.

The adaptive gridding approach to batch matched filters achieves performance levels

equal to that of the “brute force” method, with less computational time. As was observed

with the simulation, even though the adaptive grid’s local grid was finer than the grid of

the “brute force” approach, nowhere did the adaptive grid outperform the “brute force”

approach.

In conclusion, the adaptive approach to batch matched filters was able to achieve the

same performance as the “brute force” approach using less computational time. The

107

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Fig. 5.22 Experiment A, Adaptive versus “Brute Force” Gridding Performance Com-parison on Experiment Data, BP White Gaussian AM, with the QR and AM at 6.25kHz

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Batch Matched Filter, L2 Normalization, Adpative Grid, Area Under the Curve 0.9989

Fig. 5.23 Experiment B, Adaptive versus “Brute Force” Gridding Performance Com-parison on Experiment Data, BP White Gaussian AM, with the QR and AM at 10kHz

108

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Fig. 5.24 Experiment C, Adaptive versus “Brute Force” Gridding Performance Compar-ison on Experiment Data, BP White Gaussian AM, with the QR and AM at -8 kHz

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Fig. 5.25 Experiment D, Adaptive versus “Brute Force” Gridding Performance Com-parison on Experiment Data, BP White Gaussian AM, with the QR and AM at 12.5kHz

109

following chapters will use the “brute force” approach to matched filters with a fine grid,

unless specified otherwise. The following section discusses an alternative approach to

the maximum output used in making a QR detection decision.

5.6 Alternative Detection Decisions

This section investigates the effect of using several filter outputs, to determine the pres-

ence of a QR signal, on performance. The idea is to base the QR detection decision not

only on the maximum output (out of the batch of filters) but also the output of a group

of surrounding filters, with QR parameters close to the parameters of the filter with the

maximum output. The goal is to decrease the false alarm rate for a fixed probability of

correct detection by using several output values versus one.

We chose to average, using equal weights, the maximum output value of the batch

matched filter and the output of a group of six filters, that surround the filter resulting

in the maximum output. The average output value is then compared to a threshold to

determine the presence of a QR signal. The group of filters used are chosen as follows.

Let ωmax, φmax, and T2∗max denote the frequency, phase and decaying parameter re-

spectively of the filter resulting in the maximum output and ω∆, φ∆ and T2∗

∆ denote

the distance between grid points in the frequency, phase and decaying parameter dimen-

sions respectively. In which case the group of filters would have the following QR signal

parameters:

1. φ = φmax, T2∗ = T2

∗max, ω∗ = ωmax − αω∆

2. φ = φmax, T2∗ = T2

∗max, ω∗ = ωmax + αω∆

110

3. ω∗ = ωmax, T2∗ = T2

∗max, φ = φmax − αφ∆

4. ω∗ = ωmax, T2∗ = T2

∗max, φ = φmax + αφ∆

5. ω∗ = ωmax, φ = φmax T2∗ = T2

∗max − αT2

∗∆

6. ω∗ = ωmax, φ = φmax T2∗ = T2

∗max + αT2

∗∆

where α is a multiple that can be varied to vary the distance between the QR signal

parameters of the filters. Using the gridding parameters described in Section 5.3, the

above average thresholding method was applied for the following three different values

of α, 0.5, 1, and 2. Both simulation and experimental data sets were used to compare

the performance of this method to the simple maximum output decision method used

in Section 5.3. The ROC curves generated using the different data sets are shown in

Figures (5.26)-(5.35), starting with the simulation data.

From comparing the ROC plots in the figures, it is apparent that using the average

output of a group of filters, does not improve the performance resulting from using the

maximum output only as a decision criteria. This means that the correlation between an

AM signal and the filter (resulting in the maximum output) decreases almost in the same

amount as the correlation between a QR signal and the filter (resulting in the maximum

output) as the filter parameters are varied within a local region surround the maximum

output filter. Therefore, due to the lack of improvement in performance, when using

the average output, and the simplicity of the maximum output approach, the decision

criteria that will be used in the rest of the thesis will be the maximum output value.

111

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Batch Matched Filter, L2 Normalization, Threshold on Average of Points 0.5 Delta Away from Center, Area Under the Curve 0.8856

Batch Matched Filter, L2 Normalization, Threshold on Average of Points 1 Delta Away from Center, Area Under the Curve 0.8858


Fig. 5.26 Simulation SNR = -22 dB, Comparing Detection Decision Methods, BP WhiteGaussian AM, with the QR and AM at 2.5 kHz

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Fig. 5.27 Simulation SNR = -22 dB, Comparing Detection Decision Methods, BP WhiteGaussian AM, with the QR and AM at 5 kHz

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Fig. 5.32 Experiment A, Comparing Detection Decision Methods on Experiment Data,BP White Gaussian AM, with the QR and AM at 6.25 kHz.

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Fig. 5.33 Experiment B, Comparing Detection Decision Methods on Experiment Data,BP White Gaussian AM, with the QR and AM at 10 kHz.

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Fig. 5.34 Experiment C, Comparing Detection Decision Methods on Experiment Data,BP White Gaussian AM, with the QR and AM at -8 kHz.

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Fig. 5.35 Experiment D, Comparing Detection Decision Methods on Experiment Data,BP White Gaussian AM, with the QR and AM at 12.5 kHz.

116

The following chapter introduces an approach that utilizes a noise estimate to better

match the filter to the noise present, instead of designing the matched filter on the

assumption that the noise present is white.

117

Chapter 6

Batch Whitened Matched Filter

The previous chapter introduced the batch matched filter and its performance was com-

pared to the generic energy detector, where it was shown to outperform it. That filter

attempts to match to the QR signal, had one been present in the received signal, though

it does not utilize any information about the noise to better improve the detection per-

formance. The selected filter out of the batch of matched filters would be the linear filter

that maximizes SNR (Equation 4.8), if it exactly matched the QR signal and the present

noise is white. Unfortunately, AM noise is not white, therefore the batch matched filter

is not optimal even if it locks to the right QR signal.

To be able to design the matched filter that maximizes SNR, not only is an exact descrip-

tion of the QR signal required but so is the second moment of the noise present. Since

AM noise is not stationary, the noise statistics change, we are unable to design a filter

matched to both the QR signal and the noise. In Section (5.3), we used a “brute force”

gridding technique to match to the QR signal present. This is combined with a method

introduced in this chapter that provides an estimate of the noise statistics and leads to

a filter that is better matched to the noise present. Before introducing the method that

will provide an estimate of the noise statistics, we need to overcome the non-stationarity

challenge of the AM noise. To do that, we will assume the following,

118

Assumption 5. Assume that throughout the length of a single echo (less than 2 ms), an

averaged AM noise signal is wide sense stationary, i.e. the first and second moments do

not vary with respect to time.

The above assumption is a reasonable one, since the length of an echo window is very

short. Based on the above assumption the received AM signal in Equation (3.22), consists

of Nm wide sense stationary, WSS, AM signals strung in time. So each ηpci is a WSS

signal with its own noise statistics. This enables us to design a string of matched filters,

where each is matched to the QR signal present and a different set of noise statistics.

The chapter starts by introducing a batch whitened matched filter with a fixed order

whitening filter. The chapter then moves on to introducing methods that adaptively

select the whitening filter order and compares their performance.

6.1 Whitened Matched Filter

In a typical QR detection applications such as landmine detection and luggage scanning

the SNR is very low due to external RF interferences [9; 31]. Therefore, if the weak

stationarity condition holds (Assumption 5) and the noise statistics are unknown, an

estimate of the noise statistics can be obtained from the received signal if the following

assumption, which is reasonable in QR detection applications, holds.

Assumption 6. Assume that phase-cycled echo, xpc, has low SNR ratio, i.e. the value

of the QR signal is very small when compared to the present noise.

If the received phase cycled signal, xpc[k], and present noise, χpc[k], are

119

xpc[k] = spc[k] + ηpc[k] + npc[k]

χpc[k] = ηpc[k] + npc[k]

we are unable to obtain the noise statistics due to the unavailability of χpc[k]. Though,

a good estimate of the received noise, χpc[k], would be the received signal, xpc[k],

χpc[k] = ηpc[k] + npc[k] = xpc[k]. (6.1)

This would lead to an error equal to the value of the QR signal, small when compared

to the value of χpc[k]

χpc[k]− χpc[k] = spc[k]

This estimate of the received noise, χpc, can be used to estimate the covariance of the

noise, Σ, and design a matched filter that uses both information about the QR signal

and the noise as follows,

h = Σ−1spc

Since Σ−1 is a real symmetric positive definite matrix, the above equation can be rewrit-

ten as

120

h = W TWspc

where W is a real nonsingular matrix. The inner product of this filter with the received

signal xpc is

y = hTxpc = (W TWspc)Txpc = (Wspc)

T (Wxpc) = (sw)Txw.

From the above equation, it becomes apparent that applying a filter, h (matched to

both the QR signal and the present noise) to the received signal, xpc is equivalent to

multiplying both the received signal, xpc, and the filter that is only matched to the

QR signal, spc, by a filtering matrix, W , before taking their inner product. Therefore,

estimating the inverse covariance matrix Σ−1 is equivalent to estimating the matrix W .

The term (sw)Txw in the above equation resembles the matched filter in the presence

of white noise, when the optimal filter is the QR signal. Therefore, we can refer to the

matrix W as the whitening matrix.

There exist several methods to estimate the whitening matrix. Some of which are based

on Prony’s all-pole modeling method, which is an approach used to model a particular

signal. Applying the inverse model to the same signal should result in white noise, hence

applying the inverse model of the received noise to the received echo, xpc, should whiten

the received noise and result in xw. Prony’s method owes its popularity to its accurate

representation of many different types of signals in many different applications. In speech

processing for example, an acoustic tube model for speech production leads to an all-pole

121

model [28]. Other reasons we chose to focus on all-pole modeling algorithms, is one their

special structure and two the small effect the inverse model (an all-zero model) has on

the QR signal, which is discussed in Section 6.1.3.

We used two all-pole modeling methods, which are based on Prony’s all-pole normal

equations, to model the signal. These two methods are introduced next and their perfor-

mance is compared later in this chapter. The first is the autocorrelation method which

is discussed in the following subsection. The second method is the covariance method,

discussed in the subsection following that. They both have their advantages and disad-

vantages as will be discussed in the following subsections, though the most important

advantage the autocorrelation method has over the covariance method is that it guaran-

tees stability of the estimated model. In our case, even if the covariance method results

in an unstable model estimate of the noise, it can still be used, since its inverse (all-zero

model) is what is used to whiten to the signal. One advantage the covariance method

has over the autocorrelation, is that it has been known to provide more accurate signal

models [28].

In our particular case, we are attempting to model an AM signal, which has a sinusoidal

component, and therefore the signal model estimate should have poles close to the unit

circle. The autocorrelation method forces these poles to fall within the unit circle, which

leads to some inaccuracy when attempting to model a sinusoidal signal. On the other

hand, since the covariance method does not place this restriction on the poles of the

model, it can lead to more accurate sinusoidal signal models.

122

6.1.1 Estimating Whitening Matrix using the Autocorrelation Method

The autocorrelation all-pole modeling method, based on Prony’s method, is introduced in

this subsection. The special structure found in the all pole Prony normal equations, lead

to fast and efficient algorithms for finding the all-pole parameters. The autocorrelation

method, also guarantees that the resulting noise model estimate is a stable one.

Assume x[n] for n = 0, 1, . . . , N −1, is the signal we would like to model the z-transform

of, using a pth order all pole model of the form

H[z] =b[0]

1 +∑p

k=1 ap[k]z−k,

then using the autocorrelation method, the all pole coefficients, ap[k], are found by

minimizing the following error

εp =∞∑n=0

|e[n]|2

where,

e[n] = x[n] +p∑

k=1

ap[k]x[n− k].

Since x[n] is only known for 0 ≤ n ≤ N − 1, the error e[n] can only be evaluated for

p ≤ n ≤ N − 1. To enable us to calculate e[n], the autocorrelation method assumes that

x[n] = 0 outside the interval [0, N − 1], which is a drawback since the signal is usually

non-zero outside the known interval. It has been shown that solving the following normal

equations for the coefficients ap[k] minimizes εp [28].

123

p∑l=1

ap[l]rx[k − l] = −rx[k]; k = 1, 2, . . . , p

rx[k] =N−1∑n=k

x[n]x∗[n− k]; k ≥ 0

These coefficients lead to a minimum error value of,

{εp}min = rx[0] +p∑

k=1

ap[k]r∗x[k]. (6.2)

Note that in our particular case, where the signals are real, the conjugates of the signals

above are equal to the signals themselves. The effect of the model order on the per-

formance of the batch whitened matched filter is discussed in Section 6.3 and different

methods of choosing the filter order are introduced in Section 6.4.

Now that an estimate model of the noise signal has been obtained, we can apply the

inverse model, 1/H[z], to the received signal, which would lead to an approximately

white signal. Therefore, this filter can be referred to as a whitening filter.

The inverse of the model is an all zero model and can be represented as follows

Hw[z] = Hinv[z] =1

H[z]=

1 +∑p

k=1 ap[k]z−k

b0,

where, b0 is a scaling factor and can be set to one without affecting the resulting whitened

signal. Applying the above whitening filter to a signal is equivalent to multiplying

124

the signals by the matrix below of size N × N , where N is length of the vector x[n].

Throughout the rest of this thesis, this matrix, W , be referred to as the whitening matrix.

W =

1 0 . . . . . . . . . . . . 0

a1 1 0 . . . . . . . . . 0

......

. . . . . .... . . . 0

ap ap−1 . . . 1 0 . . . 0

0 ap ap−1 . . . 1 . . . 0

.... . . . . . . . .

.... . .

...

0 . . . 0 ap ap−1 . . . 1

(6.3)

6.1.2 Estimating Whitening Matrix using the Covariance Method

Using the autocorrelation method, the signal being modeled is set to zero outside it’s

known interval of values. On the other hand, with the covariance method the filter

coefficients are found by minimizing an error that does not depend upon the values

outside that known interval. Not placing the assumption that the signal is zero outside

the known interval, usually leads to a more accurate model of the signal. On the contrary,

unlike the autocorrelation method the covariance method does not guarantee a stable

noise model estimate. Although, since the inverse model (an all-zero model) is what is

used to whiten the signal, an unstable covariance model can still be used as an estimate

model of the noise signal.

As earlier, if we assume x[n] for n = 0, 1, . . . , N − 1, is the signal we would like to model

the z-transform of, using an all pole model of the form

125

H[z] =b[0]

1 +∑p

k=1 ap[k]z−k,

then using the covariance method, the all pole coefficients, ap[k], are found by minimizing

the following error

εpC =

N−1∑n=p

|e[n]|2

where,

e[n] = x[n] +p∑

k=1

ap[k]x[n− k].

Minimizing the sum of the error over the interval p ≤ n ≤ N − 1 instead of [0,∞], as

in the autocorrelation method, enables one to solve for the coefficients without placing

assumptions on the signal. It has been shown that solving the following normal equations

for the coefficients ap[k], minimizes the error εpC [28].

p∑l=1

ap[l]rx[k, l] = −rx[k, 0]; k = 1, 2, . . . , p

rx[k, l] =N−1∑n=p

x[n− l]x∗[n− k]; k, l ≥ 0

These coefficients lead to a minimum error value of,

{εpC}min

= rx[0, 0] +p∑

k=1

ap[k]rx[0, k]. (6.4)

126

Note that in our particular case, where the signals are real, the conjugates of the signals

above are equal to the signals themselves. The effect of the model order on the perfor-

mance of the batch whitened matched filter using this method and the autocorrelation

method is discussed in Section 6.3, and different ways to choosing the filter order are

introduced in Section 6.4. As was done in the previous subsection, the inverse of the

above estimated model is considered the whitening filter model, Hw[z]. This filter can

be represented as a N ×N matrix, W , with the form shown in equation (6.3), which will

be referred to as the whitening matrix. The following subsection describes the effect of

these all zero whitening filters on the shape of the QR signal.

6.1.3 The Effect of All Zero Filters On a QR Signal

When one of the whitening filters, described in the two subsections above, Hw[z], is

applied to the following QR signal, spc,

spc[k] = Ae(−|−T/2+k∆|/T2∗) cos (ωbb(−T/2 + k∆) + φbb)

=Ae(−|−T/2+k∆|/T2

∗)

2

[ej(ωbb(−T/2+k∆)+φbb) + e−j(ωbb(−T/2+k∆)+φbb)

].

the resulting signal is sw,

127

sw[k] =p∑

n=0

anspc[k − n]

=p∑

n=0

anAe(−|−T/2+(k−n)∆|/T2

∗)

2

[ej(ωbb(−T/2+(k−n)∆)+φbb) + . . .

e−j(ωbb(−T/2+(k−n)∆)+φbb)]

=Aej(ωbb(−T/2+k∆)+φbb)

2

p∑n=0

ane(−|−T/2+(k−n)∆|/T2

∗)e−j(ωbbn∆) + . . .

Ae−j(ωbb(−T/2+k∆)+φbb)

2

p∑n=0

ane(−|−T/2+(k−n)∆|/T2

∗)ej(ωbbn∆) (6.5)

for the case when, (k− p)∆ ≥ T/2 or k ≥ (Ns− 1)/2 + p, the signal sw can be simplified

to

sw[k] =Aej(ωbb(−T/2+k∆)+φbb)e(T/2−k∆)/T2

∗

2

p∑n=0

ane(n∆/T2

∗)e−j(ωbbn∆) + . . .

Ae−j(ωbb(−T/2+k∆)+φbb)e(T/2−k∆)/T2∗

2

p∑n=0

ane(n∆/T2

∗)ej(ωbbn∆) (6.6)

=Aej(ωbb(−T/2+k∆)+φbb)e(T/2−k∆)/T2

∗

2

p∑n=0

ane−j(ωbbn∆) + . . .


2

p∑n=0

anej(ωbbn∆) (6.7)

=Aej(ωbb(−T/2+k∆)+φbb)e(T/2−k∆)/T2

∗

2Hw(ej(ωbb∆)) + . . .


2Hw(e−j(ωbb∆)) (6.8)

where ane(n∆/T2

∗) in Equation (6.6) has been replaced with an in Equation (6.7), and

the value Hw represents the Fourier transform of the filter with coefficients an at the

128

frequency ωbb∆. Since, FIR filters with real coefficients have conjugate symmetry, i.e

Hw(e−j(ωbb∆)) = H∗w

(ej(ωbb∆)), as follows

Hw(e−j(ωbb∆)) = Re{H(ej(ωbb∆))} − jIm{H(ej(ωbb∆))}

= |H(ej(ωbb∆))|{−∠H(ej(ωbb∆))}

the Equation (6.8) can be rewritten as

sw[k] =A

2

∣∣∣Hw

(ej(ωbb∆)

)∣∣∣ e(T/2−k∆)/T2∗ [ej(ωbb(−T/2+k∆)+φbb+〈Hw(ej(ωbb∆))) + . . .

e−j(ωbb(−T/2+k∆)+φbb+〈Hw(ej(ωbb∆)))]

= A∣∣∣Hw

(ej(ωbb∆)

)∣∣∣ e(T/2−k∆)/T2∗

cos(ωbb(−T/2 + k∆) + φbb + 〈Hw

(ej(ωbb∆)

)). (6.9)

The above equation shows that for k ≥ (Ns − 1)/2 + p the whitening filter only affects

the magnitude and phase of the QR signal and keeps the shape and frequency of the

signal intact. Using the same procedure above it can be shown that for the case when

k ≤ (Ns − 1)/2 the whitening filter again only affects the magnitude and phase of the

signal, leaving the shape and frequency intact. Due to the similarity of the derivation

with the previous step, only a general overview of the derivation will be presented. For

k ≤ (Ns− 1)/2, the processed QR signal at the output of the QR signal, Equation (6.5),

is reduced to

129

sw[k] =Aej(ωbb(−T/2+k∆)+φbb)e(−T/2+k∆)/T2

∗

2

p∑n=0

ane(−n∆/T2

∗)e−j(ωbbn∆) + . . .

Ae−j(ωbb(−T/2+k∆)+φbb)e(−T/2+k∆)/T2∗

2

p∑n=0

ane(−n∆/T2

∗)ej(ωbbn∆)

=Aej(ωbb(−T/2+k∆)+φbb)e(−T/2+k∆)/T2

∗

2

p∑n=0

ane−j(ωbbn∆) + . . .


2

p∑n=0

anej(ωbbn∆)

=Aej(ωbb(−T/2+k∆)+φbb)e(−T/2+k∆)/T2

∗

2Hw(ej(ωbb∆)) + . . .


2Hw(e−j(ωbb∆))

=A

2

∣∣∣Hw

(ej(ωbb∆)

)∣∣∣ e(−T/2+k∆)/T2∗ [ej(ωbb(−T/2+k∆)+φbb+〈Hw(ej(ωbb∆))) + . . .

e−j(ωbb(−T/2+k∆)+φbb+〈Hw(ej(ωbb∆)))]

= A∣∣∣Hw

(ej(ωbb∆)

)∣∣∣ e(−T/2+k∆)/T2∗

cos(ωbb(−T/2 + k∆) + φbb + 〈Hw

(ej(ωbb∆)

)). (6.10)

Equations (6.10), for k ≤ (Ns − 1)/2, and (6.9), for k ≥ (Ns − 1)/2 + p, show that

the shape and frequency of the whitened QR signal remain unchanged and only the

magnitude and phase are affected. For the p−1 values, (Ns−1)/2 < k < (Ns−1)/2+p,

the shape of the QR signal may be altered depending on the filter, and the result is

described using Equation (6.5). To demonstrate how small the effect of the FIR filter is

on the general shape of the QR signal, a simulated QR signal of frequency 12.5 is passed

through a random 6th order FIR filter and the output is compared to the input in Figure

6.1. The plot shows that the effect of the FIR filter is small.

130

0 50 100 150 200 250 300 350 400 450−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

QR signal

QR signal passed through a 6th

order FIR filter

Fig. 6.1 The output of a 6th order FIR filter applied to a simulated 12.5 kHz QR signalis compared to the input signal.

6.2 Batch Whitened Matched Filter

In Section 6.1 we have shown that, the inner product of the whitened QR signal with the

whitened received signal is equivalent to the inner product of the linear filter, matched

to both the QR signal and the noise, with the received signal. We then moved to

estimating the whitening filter, when the noise statistics are unknown in Section 6.1.1.

Now we will combine the gridding approach used in Chapter 5, with our approach to

estimating the whitening filter discussed in the previous section, to design a filter that

is not only matched to the QR signal but also the noise statistics. This filter should

show an improvement in performance over the batch matched filter designed in Chapter

5, which did not use any information about the noise.

Estimating the whitening filter of order p for each of the Nm received phase cycled

echoes, xpci for i = 0, 1, . . . , Nm − 1, results in Nm different whitening matrices Wi for

131

i = 0, 1, . . . , Nm−1. The next step would be to whiten each of the received phase cycled

echoes xpci , with its corresponding whitening matrix. The whitening matrix attempts

to equally spread the energy of the signal into the frequency spectrum. Since we know

that the QR signal frequency is less than ωub, we chose to low pass the whitened received

signal with a low pass filter with a cuttoff frequency of ωub. The low pass filter chosen

was a Butterworth filter due to its maximally flat passband. Though other filters such

as the Chebyshev Type I / Type II and the Elliptic have faster roll offs, the slow roll off

of the Butterworth filter can be compensated for by increasing the order of the filter. An

increase in the filter’s order leads to an increase in the filter’s delay, though this effect is

not as important to us as the flat pass band and fast roll off.

The frequency response of a low pass Butterworth filter is

HBW [ω] =

√1

1 + (ω/ωc)2n

where ω is the frequency in radians, ωc = ωub is the cutoff frequency in radians, and n

is the low pass filter order, which we set to 8 since it demonstrated a fast enough roll

off. From the frequency response of the filter, the amplitude of the filter at the cutoff

frequency, ωub, is 1/√

2, which is −3 dB or half the power. The magnitude of the filter

beyond the cut-off frequency rolls off at a -6n dB per octave, where an octave is two

times the frequency. The magnitude of the frequency response and the magnitude in dB

of the frequency response of an 8 pole Butterworth filter with a cutoff frequency of 20

kHz are shown in Figures (6.2) and (6.3) below.

132

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency (Hz)

Magnitude

Transfer Function Design

Fig. 6.2 Magnitude of the Frequency Response of an 8 pole Butterworth filter with a 20kHz cutoff.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

−60

−50

−40

−30

−20

−10

0

10

Frequency (Hz)

Magnitude (

dB

)

Transfer Function Design

Fig. 6.3 Magnitude (dB) of the Frequency Response of an 8 pole Butterworth filter witha 20 kHz cutoff.

133

The low pass filter can be be easily represented by a matrix, M . Whitening and low

passing the Nm received signal leads to xwi signals of the following form,

xwi = βiMWixpci ,

where,

βi =

∑Ns−1k=0

x2pci

[k]∑Ns−1k=0 (MWixpci [k])2

.

Stringing these Nm signals in time leads to, xw

xw[k] = [xw0[k], xw1

[k −Ns], . . . .xwNm−1[k − (Nm − 1)Ns], (6.11)

for k = 0, 1, . . . , Nm ×Ns − 1. Now that the received phase cycled echoes are whitened,

our goal is to design a filter matched to the whitened and low passed QR signal present

in the received signal using the gridding approach discussed in Chapter 5. If we assume

that the discretization of the interval of admissible values for frequency, phase and de-

caying parameter, in Equations (5.1), (5.2) and (5.3) respectively, is fine enough then

the frequency, phase and decaying parameter of the pre-whitened QR signal must fall in

the vector of values in Equations (5.7), (5.8), and (5.9) respectively.

Therefore a triplet of values from the vectors ωa, φa and Ta are chosen and used to

design a corresponding whitened matched filter of period T . There are Ndf ×Ndφ ×NdT

different triplets to choose from, where Ndf , Ndφ and NdT are the number of values in

vectors ωa, φa and Ta respectively. If the lth value of vector ωa, the mth value of vector

134

φa and the nth value of vector Ta are the chosen values then the corresponding estimate

of the whitened low passed matched filter for phase cycled echo, i, is

swi(l,m, n)[k] =1

αwi(l,m, n)MWie

(−|−T/2+k∆|/Ta(n))cos(ωa(l)(−T/2 + k∆) + φa(m))

for k = 0, 1, . . . , Ns − 1. The value T is the length of the data acquisition window, Ns is

the number of discrete samples, ∆ = T/(Ns − 1) and the constant αwi(l,m, n) is

αwi(l,m, n) =

√√√√Ns−1∑k=0

[MWie(−|−T/2+k∆|/Ta(n))cos(ωa(l)(−T/2 + k∆) + φa(m))]2.

Therefore, the estimated whitened low passed matched filter for the above QR signal

parameter values chosen, comprises of the Nm whitened low passed matched filters,

swi(l,m, n), strung in time,

h[k] = [sw0(l,m, n)[k], sw1

(l,m, n)[k −Ns], . . . , swNm−1(l,m, n)[k − (Nm− 1)Ns],

for k = 0, 1, . . . , Nm×Ns−1. After sending our received whitened signal through the filter

with the above impulse response, the value of the Nm ×Nsth sample, which is equivalent

to the inner product of the received whitened signal and the estimated whitened matched

filter, is recorded. This process is repeated for a different combination of values for the

135

frequency ωbb, φbb and decaying parameter T2∗. Repeating this experiment Ndf ×Ndφ ×

NdT times leads to a vector Θ of values of length Ndf ×Ndφ ×NdT .

Θ = [θ1, . . . , θNdf×Ndφ×NdT] (6.12)

Finally the maximum value of the vector Θ is compared to a threshold value θthreshold and

the filter corresponding to this maximum is the chosen filter out of the whitened batch of

filters. If the maximum value is greater than the threshold then we would conclude that

a QR signal exists within the corrupted signal, otherwise we would conclude that the

received signal is just noise. The process of choosing the filter is equivalent to selecting

the filter, hn[k], that maximizes yn, the inner product of the filter with the whitened

received signal xw[k] in Equation (6.11), over the set of possible filters as follows

y = maxn

(yn) = maxn

(hn[k]Txw[k]

)= max

n

(|hn[k]||xw[k]| cos γ

)

where γ is the angle between the received signal, xw[k], and the filter hn[k]. If the

chosen filter out of the batch is equal to the whitened QR signal present, sw[k], then the

resulting signal to noise ratio defined in Equation (4.8) is reduced to

ρ(sw; sw, I) =(sw[k]T sw[k])2

sw[k]T Isw[k]= (sw[k]T sw[k]) = (s[k]TΣ−1s[k])

where I is the identity matrix and corresponds the covariance of the noise after whitening,

assuming the whitening filter is exact. The matrix, Σ, on the other hand is the covariance

of the noise pre-whitening. The above signal to noise ratio is the same as the signal

136

noise ratio resulting from the optimal matched filter, shown in Equation (4.11). This

is expected since we are assuming that the estimate of the whitening filter is exact and

the chosen whitened filter out of the whitened batch of filters is the whitened QR signal

present, which would lead to an optimal matched filter.

6.3 Batch Whitened Matched Filter versus Energy Detector, Unknown

QR signal

Simulation data was first used, to test the performance of several batch whitened matched

filters for a range of whitening filter orders, where the whitening filter was designed

using either the autocorrelation or the covariance method. The performance of these

filters was then compared it to the performance of the batch matched filter, described in

Chapter 5, and the energy detector. These filters were the compared using experimental

data. For both the autocorrelation and covariance methods, the problem of choosing the

order of the all zero whitening filter has not been addressed and the order was varied

between 3 and 7, to evaluate the performance of the batch whitened matched filter

for different whitening filter orders. Section 6.4 introduces two different methods for

choosing the whitening filter order and compares their performance. The order of the

low pass butterworth filter on the other hand, was fixed to 8 with a cutoff frequency of

20 kHz.


Six sets of simulation data, the same ones used to compare the batch matched filter to

the energy detector, where the QR and the AM baseband frequencies were equal. The

137

first set had a frequency of 2.5 kHz, and the frequency was increased by 2.5 kHz for each

following set, leading to a frequency of 15 kHz for the sixth data set. The simulation

data setup and batch matched filter gridding parameters are provided in section 5.3.1,

therefore they will not be repeated here.

The batch whitened matched filters, for both the autocorrelation and the covariance

whitening methods with the different whitening filter orders, used the following values

to grid, ωlb = 0, ωub = 40000π, ω∆ = 200π, φ∆ = .1, T ∗2 lb

= 800e− 6, T ∗2 ub

= 1000e− 6

and T ∗2 ∆

= 100e − 6. The above values lead to Ndf = 201, Ndφ = 63 and NdT = 3 in

each of the three dimensions, which corresponds to Ndf × Ndφ × NdT = 34, 989 filters

in the batch. The number of filters in the batch was reduced for the batch whitened

matched filter when compared to the batch matched filter due to the increase in each

filter’s computation time. The increase in computation time is a result of having to

whiten each filter in the batch with Nm different whitening filters, Wi, for i = 1, . . . , Nm.

The performance of the batch whitened matched filter might be slightly improved had

the gridding been finer, i.e. a larger batch of filters are used, though this leads to an

increase in computation time. The effect of finer gridding on performance was discussed

in Section 6.3.2 and therefore will not be discussed here.

ROC plots were used to compare the performance of the batch whitened matched filter

to the batch matched filter and the energy detector. Figures (6.4-6.15) compare the

performance of the different detectors for the different frequencies. Each figure compares

the energy detector and the batch matched filter to the batch whitened matched filter,

using either the autocorrelation or the covariance method, as the whitening filter order

138

is varied. Figures (6.4) and (6.5) compare the performance of the different detectors

using the 2.5 kHz data set. These results are summarized in Table (6.1). In Figure (6.4),

among the batch whitened matched filters (whitened using the autocorrelation method),

the one that used a 7th order all zero whitening filter performed best among the rest

of the batch whitened matched filters. By comparing the areas under the curve in the

figure, this filter had performance approximately equal to that of the batch matched

filter and therefore did not lead to an improvement over the performance of the batch

matched filter, though both of these filters outperformed the energy detector. On the

other hand, in the case of the covariance method, the best performing batch whitened

matched filter that used a 4th order whitening filter performed the best among the batch

whitened match filters, with an area under the curve of 0.854. This filter performed

worse than the batch matched filter, which had an area under the curve of 0.886, but

better than the energy detector, which had an area under the curve of 0.74. Therefore,

the batch whitened matched filters, whitened using the autocorrelation method and the

one whitened using the covariance method, did not outperform the batch matched filter,

though very comparable performance was achieved.

Whitening Filter Order Autocorrelation Method Covariance Method No Whitening0 0.8863 0.685 0.5204 0.661 0.8545 0.753 0.5786 0.866 0.8117 0.878 0.706

Table 6.1 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data,Band-passed White Gaussian AM, with the QR and AM at 2.5 kHz.

139

Figure (6.6) & (6.7) used the 5 kHz data set to compare the performance of the batch

whitened matched filters for different whitening filter orders, when estimated using the

autocorrelation method and the covariance method respectively. These results are sum-

marized in Table (6.2). Among the batch whitened matched filters, whitened with the

autocorrelation method, the best performance was achieved by the batch that used a 6th

order whitening filter. The performance of this filter, with an area under the curve of

0.894, was lower than the performance of the batch matched filter, with an area under

the curve of 0.92. Though, both these filters outperformed the energy detector, which

had an area under the curve of 0.8. In Figure (6.7), the same data set was used to

compare the performance of the batch whitened match filters for different whitening

filter orders, when estimated using the covariance method. The best filter among the

different batch whitened matched filters was achieved with a 7th order whitening filter

estimated using the covariance method, with an area under the curve of 0.97. This area

significantly outperformed both the batch matched filter and the energy detector, which

achieved areas under the curve of 0.92 and 0.8 respectively. For this particular data set,

the batch whitened matched filter, whitened using the covariance method, was able to

outperform the batch matched filter, but the batch whitened matched filter, whitened

using the autocorrelation method, did not.

The 7.5 kHz data set was used to compare the batch whitened matched filters for dif-

ferent whitening filter orders, when estimated using the autocorrelation method and the

covariance method in Figures (6.8) and (6.9) respectively. These results are summa-

rized in Table (6.3). The best performing detector among the batch whitened matched

140


Table 6.2 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data,Band-passed White Gaussian AM, with the QR and AM at 5 kHz.

filters, when the autocorrelation method was used was the one that used a 4th order

filter. This filter, with an area under the curve of 0.915, outperformed both the batch

matched filter, with an area under the curve of 0.889, and the energy detector, with an

area under the curve of 0.796. For the covariance method, the best performing filter

was the one that used a 5th order whitening filter, which resulted in an area under the

curve of 0.954, which outperformed the best whitening filer in the autocorrelation case,

the batch matched filter, and the energy detector, with areas under the curve of 0.915,

0.889 and 0.796 respectively. Again, this shows that the covariance method provided the

better whitening filter estimate, which resulted in the best performance.



141

The performance of the batch whitened matched filters for different whitening filter

orders, when estimated using the autocorrelation and the covariance methods were com-

pared using the 10 kHz data set, in Figures (6.10) and (6.11) respectively. These results

are summarized in Table (6.4). In the case of the autocorrelation method, the batch

whitened matched filter which used a 5th order whitening filter performed best with an

area under the cube of 0.910. This filter outperformed both the batch matched filter and

the energy detector, with areas under the curve of 0.873 and 0.778 respectively. When

the whitening filters were estimated using the covariance method, the batch whitened

matched filter with a 3rd order whitening filter outperformed all the others, with an

area under the curve of 0.944, which is larger than the and 0.910 achieved by the batch

whitened matched filter, with a 5th order whitening filter estimated using the autocorre-

lation method. This again shows that the covariance method with the right filter order

can result in a better estimate of the whitening filter.



The same comparison of the filters discussed above is shown in Figures (6.12) and (6.13)

for the 12.5 kHz data set. These results are summarized in Table (6.5). In Figure

(6.12), for the case of the autocorrelation method, the best batch whitened matched

142

filter was achieved with a 6 whitening filter, resulting in an area under the curve of

0.921. This filter outperformed both the batch matched filter and the energy detector,

which achieved areas under the curve of 0.846 and 0.815 respectively. For the case of

the covariance method, the best performing filter was achieved by the batch whitened

matched filter with a 4th order whitening filter, which resulted in an area under the curve

of 0.908. This filter did not outperform the best filter when the autocorrelation method

was used, but the performance is close, 0.908 versus 0.921.



The 15 kHz data set was used to compare these same filters using the autocorrelation

and the covariance methods in Figures (6.14) and (6.15) respectively. The results from

these figures are summarized in Table (6.6). The best performing method from the

autocorrelation method, was achieved with a 7th order whitening filter, with an area

under the curve 0.852. This filter resulted in worst performance when compared to the

batch matched filter but still outperformed the energy detector, with areas under the

curve 0.884 and 0.804 respectively. On the other hand, the best filter from using the

covariance method, which also used a 7th order filter, resulted in an area under the curve

of 0.972, which outperforms both the batch matched filter and the energy detector.

143

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm

Energy Detector, AUC 0.7396Batch Matched Filter Detector, L

2 Normalization, AUC 0.88555

Batch Whitened Matched Filter Detector, Whitening Filter Order 3, L2 Normalization, AUC 0.68505





Fig. 6.4 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data, Band-passed White Gaussian AM, with the QR and AM at 2.5 kHz. Autocorrelation Whiten-ing Method.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm



Batch Whitened Matched Filter Detector, Covariance Method, Whitening Filter Order 3, AUC 0.52035





Fig. 6.5 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data, Band-passed White Gaussian AM, with the QR and AM at 2.5 kHz. Covariance WhiteningMethod.

144

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.6 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data, Band-passed White Gaussian AM, with the QR and AM at 5 kHz. Autocorrelation WhiteningMethod.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.7 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data, Band-passed White Gaussian AM, with the QR and AM at 5 kHz. Covariance WhiteningMethod.

145

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.8 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data, Band-passed White Gaussian AM, with the QR and AM at 7.5 kHz. Autocorrelation Whiten-ing Method.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.9 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data, Band-passed White Gaussian AM, with the QR and AM at 7.5 kHz. Covariance WhiteningMethod.

146

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.10 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data,Band-passed White Gaussian AM, with the QR and AM at 10 kHz. AutocorrelationWhitening Method.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.11 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data,Band-passed White Gaussian AM, with the QR and AM at 10 kHz. Covariance Whiten-ing Method.

147

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.12 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data,Band-passed White Gaussian AM, with the QR and AM at 12.5 kHz. AutocorrelationWhitening Method.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm



Batch Whitened Matched Filter Detector, Covariance Method, Whitening Filter Order 3, # of Unstable Filters 0, AUC 0.74655





Fig. 6.13 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data,Band-passed White Gaussian AM, with the QR and AM at 12.5 kHz. CovarianceWhitening Method.

148

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.14 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data,Band-passed White Gaussian AM, with the QR and AM at 15 kHz. AutocorrelationWhitening Method.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.15 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data,Band-passed White Gaussian AM, with the QR and AM at 15 kHz. Covariance Whiten-ing Method.

149



In conclusion, based on the results from simulation data above, the order of the whitening

filter significantly affects performance. The drops in performance of the batch whitened

matched filters for some whitening filter orders, is due to a couple of reasons which will be

discussed next. The first and most important is that for a particular whitening method

certain filter orders lead to inaccurate whitening filter estimates. These inaccurate esti-

mates can lead to amplifications of non-QR frequencies and suppression of the QR signal

frequency. This leads to locking to the incorrect QR signal frequency when gridding in

frequency, which is demonstrated in the next paragraph using the 2.5 kHz simulation

data set. Another reason, which is not as significant as was shown in Section 6.1.3,

is that the whitening filter partially destroys the shape of the QR signal present. The

variance in performance observed as the whitening filter order is varied is more apparent

when the covariance method was used to estimate the whitening filter.

In an attempt to understand the variance in the results, we plotted the frequency re-

sponse of the 5 whitening filters used to whiten a QR plus AM experiment of 5 echoes

(Ne = 5), for different whitening filter orders. The first experiment of QR in the presence

of AM from the 2.5 kHz simulation data set was used for this particular analysis. Prior

150

to plotting the whitening filters for different filter orders, we plotted what we expect

the frequency response in theory of a whitening filter to look like. To whiten an AM

signal, the whitening filter’s frequency response should be the inverse of the AM signal’s

frequency response. A bandpassed white gaussian AM signal with a carrier at 2.5 kHz,

which is less than the upper cutoff frequency of the bandpass filter (10 kHz) has over-

lapping sidebands, which lead to an uneven frequency spectrum. Figure (6.16) displays

the frequency response of the bandpassed white gaussian noise, while the bottom part

displays the frequency response of the bandpassed white gaussian AM signal. Therefore

in theory we would expect the whitening filter’s frequency spectrum to be the inverse of

the frequency spectrum of the AM signal, shown in the bottom of Figure (6.17). The

spectrum of the whitening filter in theory should try to equally amplify the frequency

bands 2.46 kHz-2.54 kHz and 7.5 kHz-12.5 kHz. The filter should also provide a larger

amplification of the frequencies greater than 12.5 kHz, while suppressing the remaining

frequencies.

To compare the frequency response of the whitening filter we expect in theory to the

frequency responses observed in experiment, the responses of the whitening filters for

filter orders 3-7 for both design methods were plotted. The frequency response of the

5 whitening filters (one for each echo of the experiment) designed using the covariance

method for filter orders 3-7 are shown in Figures (6.19)-(6.23) respectively. The spec-

trums of the 5 whitening filters (all-zero filters) of a particular order are overlaid in a

single figure along with the frequencies the filter’s zeros correspond to. The angle be-

tween the location of the zero and the positive real axis is what determines this frequency.

151

All filters attempt to mimic the expected whitening filter but some are limited by the

number of zeros (filter order).

Comparing these figures shows that the closest match between the expected whitening

filter and the whitening filters observed occurs when the filter order used is of 7th or-

der. The spectrum of the whitening filters of 4th and 6th order in Figure (6.20) also

closely resembles the spectrum of the whitening filter in theory, with the exception of

not suppressing the frequencies below 2.46 kHz. Therefore one expects these filter orders

to lead to good performance. Figures (6.19) and (6.21) clearly show that the whitening

filters of orders 3 and 5 suppress the QR signal’s frequency (2.5 kHz) when compared

to other frequencies. These filter estimates attempt to suppress the 0-2.46 kHz band

by placing one of the filter’s zeros on the positive real axis, which corresponds to the

zero frequency. The magnitude of frequency response of the filters should then rise at

2.46 kHz to follow the whitening filter in theory but does not rise fast enough to prevent

a suppression of the 2.5 kHz QR signal frequency, whose spectrum is shown in Figure

(6.18). This phenomena leads to the filter often locking to incorrect frequencies when

gridding. This does not occur with the 7th order filter due to the increase in the model

order, which increases the maneuverability of the frequency response.

The filter estimates of even order, 4 and 6, place the zeros in complex conjugates to

suppress higher frequencies since placing a zero on the positive real axis to suppress

the 0-2.46 kHz frequency band limits another zero to a non-complex value. Since these

whitening filter estimates do not attempt to suppress the 0-2.46 kHz frequency band, the

QR signal’s frequency of 2.5 kHz is not suppressed in the same manner as when a 3rd

152

or 5th order filter is used. The frequencies the filter locks to when whitening, using the

covariance method, for the 2.5 kHz data set with filter orders 3-7 are shown in Figures

(6.24) to (6.28) respectively. These figures show that the suppression of the QR signal’s

frequency leads to the matched filter locking to the incorrect frequency resulting in poor

performance, shown in Table 6.1.

For comparison, the frequency responses of the whitening filters designed using the au-

tocorrelation method for filter orders 3-7 are shown in Figures (6.29)-(6.33) respectively.

When using the autocorrelation method to design the whitening filters we observe more

variance in the shape the of the spectrum across echoes. We also notice that the spec-

trum of the whitening filters of 5th and 6th order closely resemble the whitening filter

expected in theory with the exception of suppressing the frequency band 0-2.46 kHz.

This helps the matched filter lock to the right frequency when gridding, which is appar-

ent when comparing the plots of the filter’s lock frequencies for different filter orders in

Figures (6.34)-(6.38).

These results stress the importance of selecting the right filter order when whitening

to gain an improvement in performance. Overall the covariance method provided the

better estimate of the whitening filters, with the exception of the 2.5 kHz and 12.5 kHz

data sets, where the performance was worst but not significantly worse. These drops in

performance are outweighed by the gains achieved by the covariance method in other

data sets. When the filter order is properly chosen, the batch whitened matched filter

lead to improvements in performance, when compared to the batch matched filter and

153

the energy detector. The next section compares the performance of the batch whitened

matched filter to other detectors using experimental data.

6

-�

6

-�

6

M(f)

AM(f)

6

40 Hz 10 kHz

fc 4321

f

f

Fig. 6.16 Top: Frequency Response of Modulating Signal (Bandpassed Gaussian Noise).Bottom: Frequency Response of an AM Signal with a Carrier Frequency of fc < 10 kHz.0 = 0 Hz, 1 = fc- 40 Hz, 2 = fc+40 Hz, 3 = 10 kHz - fc and 4 = fc+10 kHz.


This section uses experimental data to compare the performance of the batch whitened

matched filter to the batch matched filter and the energy detector. Four sets of ex-

perimental data of different frequencies, where the QR signal frequency and AM signal

frequency are equal, were collected. The four sets, which will be referred to as A, B,

C and D have frequencies 6.25, 10, -8 and 12.5 kHz respectively. Each set consisted of

100 experiments, where the received signal is the sum of a QR signal an AM signal and

thermal noise, and another 100 experiments, where the received signal is the sum of an

AM signal and thermal noise. Each experiment consists of Nm = 5 echoes strung in

time, where each is the average of 20 phase-cycled echoes.

154

-�

6

f

W(f)

0 1 2 3 4

Fig. 6.17 Frequency Response of Whitening Filter When the AM Signal’s Carrier Fre-quency is fc < 10 kHz. 1 = fc- 40 Hz, 2 = fc+40 Hz, 3 = 10 kHz - fc and 4 = fc+10kHz.

102

103

104

105

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency, Hz

Norm

aliz

ed M

agnitude

Fig. 6.18 Fast Fourier Transform of a 2.5 kHz QR signal.

155

10−1

100

101

102

103

104

105

−20

−15

−10

−5

0

5

10

15

20

Frequency (Hz)

Magnitude (

dB

)

15940.776 0

16052.9356 0

15588.996 0

15790.9769 0

16852.9618 0

Fig. 6.19 Frequency Response of the Ne = 5 Whitening Filters of Order 3, Designed Usingthe Covariance Method. Simulation Data, SNR = -22 dB, Experiment 1 of Band-passedWhite Gaussian AM, with the QR and AM at 2.5 kHz Data Set.

10−1

100

101

102

103

104

105

−10

−5

0

5

10

15

20

25

30

Frequency (Hz)

Magnitude (

dB

)

18312.1716 7948.42976

18546.3477 7521.10352

18317.2419 7300.70097

18942.616 7699.4048

18831.1747 6767.84707


156

10−1

100

101

102

103

104

105

0

5

10

15

20

25

30

35

Frequency (Hz)

Magnitude (

dB

)

19967.7218 0 12113.7013

20765.0191 0 12269.6128

19703.1101 0 12661.3224

20163.391 0 12685.898

20509.4906 0 13226.3461


10−1

100

101

102

103

104

105

5

10

15

20

25

30

Frequency (Hz)

Magnitude (

dB

)

22366.4942 15792.7695 5795.1099

22811.8713 15506.5736 5592.1293

22236.6989 15436.6406 6340.65634

22383.8017 15364.4763 5454.28908

22745.3083 16353.2561 5547.37598


157

10−1

100

101

102

103

104

105

15

20

25

30

35

40

45

Frequency (Hz)

Magnitude (

dB

)

25687.0526 18087.6676 0 10048.4705

27492.1822 19071.2527 0 10912.4108

26548.0555 17897.4936 0 11100.8156

26301.9618 18174.9786 0 11103.9658

26519.6586 18713.7305 0 11153.2078


0 10 20 30 40 50 60 70 80 90 1000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Experiment Number

Fre

quency

QR+AM

AM

Fig. 6.24 Filter Lock Frequency for Different Experiments, When Using A CovarianceWhitening Filter of Order 3. Simulation Data, SNR = -22 dB, Band-passed WhiteGaussian AM, with the QR and AM at 2.5 kHz.

158

0 10 20 30 40 50 60 70 80 90 1000

2000

4000

6000

8000

10000

12000

14000

16000

18000

Experiment Number

Fre

quency

QR+AM

AM


0 10 20 30 40 50 60 70 80 90 1000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Experiment Number

Fre

quency

QR+AM

AM


159

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Experiment Number

Fre

quency

QR+AM

AM


0 10 20 30 40 50 60 70 80 90 1000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

4

Experiment Number

Fre

quency

QR+AM

AM


160

10−1

100

101

102

103

104

105

−20

−15

−10

−5

0

5

10

15

Frequency (Hz)

Magnitude (

dB

)

15956.7229 0

16081.6208 0

14301.4517 0

15799.9945 0

16868.2305 0

Fig. 6.29 Frequency Response of the Ne = 5 Whitening Filters of Order 3, DesignedUsing the Autocorrelation Method. Simulation Data, SNR = -22 dB, Experiment 1 ofBand-passed White Gaussian AM, with the QR and AM at 2.5 kHz Data Set.

10−1

100

101

102

103

104

105

−20

−15

−10

−5

0

5

10

15

20

Frequency (Hz)

Magnitude (

dB

)

17399.8302 6097.92043

16609.3191 0

125000 15541.38191

17940.4648 7526.31278

17777.8309 3787.75707

Fig. 6.30 Frequency Response of the Ne = 5 Whitening Filters of Order 4, DesignedUsing the Autcorrelation Method. Simulation Data, SNR = -22 dB, Experiment 1 ofBand-passed White Gaussian AM, with the QR and AM at 2.5 kHz Data Set.

161

10−1

100

101

102

103

104

105

−20

−15

−10

−5

0

5

10

15

20

Frequency (Hz)

Magnitude (

dB

)

125000 18123.51393 7257.324685

125000 18465.2566 7246.797043

15542.3624 0 125000

17982.936 8490.77132 0

125000 18585.17551 7353.330738


10−1

100

101

102

103

104

105

−20

−15

−10

−5

0

5

10

15

Frequency (Hz)

Magnitude (

dB

)

125000 18156.9937 7333.681651

18491.2032 7252.594601 125000

98172.3295 15921.5328 0

18276.3431 11372.1635 0

125000 18582.30647 7354.599543


162

10−1

100

101

102

103

104

105

−20

−15

−10

−5

0

5

10

15

Frequency (Hz)

Magnitude (

dB

)

100916.0819 19074.94026 11193.81573 0

99867.5428 18901.6683 8295.95793 0

125000 82241.98819 16991.11989 8069.693501

76764.2871 19875.1087 0 12653.0729

100804.4583 19446.67553 0 12773.50619


0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Experiment Number

Fre

quency

QR+AM

AM

Fig. 6.34 Filter Lock Frequency for Different Experiments, When Using An Autocorrela-tion Whitening Filter of Order 3. Simulation Data, SNR = -22 dB, Band-passed WhiteGaussian AM, with the QR and AM at 2.5 kHz.

163

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Experiment Number

Fre

quency

QR+AM

AM


0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Experiment Number

Fre

quency

QR+AM

AM


164

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Experiment Number

Fre

quency

QR+AM

AM


0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Experiment Number

Fre

quency

QR+AM

AM


165

The batch matched filter grid parameters are provided in section 5.3.2 and therefore are

not repeated here. The batch whitened match filter on the other hand, for both the

autocorrelation and the covariance whitening methods with the different whitening filter

orders, used the following values to grid, ωlb = 0, ωub = 40000π, ω∆ = 200π, φ∆ = .1,

T ∗2 lb

= 300e − 6, T ∗2 ub

= 500e − 6 and T ∗2 ∆

= 100e − 6. The above values lead to

Ndf = 201, Ndφ = 63 and NdT = 3 in each of the three dimensions, which corresponds

to Ndf × Ndφ × NdT = 37989 filters in the batch. Again the number of filters in the

batch was reduced, when compared to the batch matched filter, due to the increase in

computation time per filter. The increase in computation time is due to having to whiten

each filter in the batch, with the Nm filters. A finer grid, leading to an increase in the

number of filters in the batch, may lead to a slight increase in performance. Though the

effect of finer gridding on performance was discussed in Section and therefore will not

be discussed here.

ROC plots were used to compare the performance of the batch whitened matched filter

to the batch matched filter and the energy detector using four different experimental

setups. Figures (6.39-6.46) compare the performance of the different detectors for the

different experimental setups. As with the simulation data, each figure compares the

energy detector and the batch matched filter to the batch whitened matched filter, using

either the autocorrelation or the covariance method, as the whitening filter order is

varied.

166

The experimental set A was used to compare the performance of the batch whitened

matched filters for different whitening filter orders, when the whitening filters were es-

timated using the autocorrelation method and the covariance method in Figures (6.39)

and (6.40) respectively. The results from these figures are summarized in Table (6.7).

For both modeling methods, the batch whitened matched filter that performed best was

the one that used a 7th order whitening filter. The best filter that used autocorrela-

tion method, lead to an area under the curve (AUC) of 0.997, while the best filter that

used the covariance method lead to an AUC of 0.9949, both which outperformed the

batch matched filter and the energy detector, which lead to AUCs of 0.977 and 0.946,

respectively.


Table 6.7 Performance Comparison on Experimental Data A, BP White Gaussian AM,with the QR and AM at 6.25 kHz.

The experimental setup B was used to compare the batch whitened match filters, which

used estimates using the autocorrelation and covariance methods in Figures (6.41) and

(6.42), respectively. These results are summarized in Table (6.8). The best performance

using the autocorrelation method, used a filter order of 6 and achieved an AUC of 1,

while the one from the covariance method, used a filter order of 5 and achieved an AUC

of 0.998. In the case of the autocorrelation method, the batch whitened matched filter

167

outperformed the batch matched filter, which lead to an AUC of 0.999, but that was not

the case with the covariance method. Although, the difference in performance among

these three filters is negligible. The three filters on the other hand, all significantly

outperformed the energy detector, which had an AUC of 0.984.

Whitening Filter Order Autocorrelation Method Covariance Method No Whitening0 0.9993 0.799 0.9944 0.977 0.9955 0.995 0.9986 1 0.99617 0.9991 0.9963

Table 6.8 Performance Comparison on Experimental Data B, BP White Gaussian AM,with the QR and AM at 10 kHz.

For the case of experimental setup C, the batch whitened matched filters which used

the autocorrelation and the covariance methods are shown in Figures (6.43) and (6.44),

respectively. The results from these figures are summarized in Table (6.9). The best

filter using the autocorrelation method was achieved using a 7th order filter and lead to

an AUC of 0.9994, while the best one using the covariance method was achieved using

a 3rd order filter and lead to an AUC of 0.9992. When comparing the performance of

these filters, the difference is negligible, but both outperformed the batch match filter

and the energy detector, which lead to AUCs of 0.988 and 0.923 respectively.

The fourth experimental setup D was used to compare the batch whitened matched fil-

ters for different whitening filter order, which used the autocorrelation and covariance

methods to estimate the whitening filters in Figures (6.45) and (6.46), respectively. The

168


Table 6.9 Performance Comparison on Experimental Data C, BP White Gaussian AM,with the QR and AM at -8 kHz.

results from these figures are summarized in Table (6.10). In the case of the autocorrela-

tion method, the best performance was achieved using a 7th order filter, and achieved an

AUC of 0.9935. This filter did not outperform the batch matched filter, which achieved

an AUC of 0.9943, but the difference in performance is negligible. On the other hand,

in the case of the covariance method the best filter, achieved using 7th, which lead to an

AUC of 0.9997 did outperform the batch matched filter. All these filters outperformed

the energy detector, which has an AUC of 0.923. To reiterate, the covariance method

batch whitened matched filter outperformed the batch matched filter, which outper-

formed the autocorrelation method batch matched filter, but the different between the

last two is negligible.


Table 6.10 Performance Comparison on Experimental Data D, BP White Gaussian AM,with the QR and AM at 12.5 kHz.

169

In the following cases: experiment B (covariance method), experiment D (autocorre-

lation), the best performing batch whitened matched filter performs worse than batch

matched filter. Though, the difference in performance is negligible, especially when com-

pared to the gains in performance achieved by the whitening approach in other cases.

In conclusion, the best performing batch whitened matched filter, for both the autocor-

relation and the covariance method, either outperforms or achieves approximately equal

performance, when compared to the batch matched filter.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.39 Performance Comparison on Experiment Data, BP White Gaussian AM, withthe QR and AM at 6.25 kHz. Autocorrelation Method.

6.4 Batch Adaptive Whitened Matched Filter

This section introduces two adaptive ways to choosing the whitening filter order that can

be applied when modeling with both the autocorrelation and covariance methods. The

first is a method that selects the whitening filter order that least affects the estimated QR

signal intended to match the QR signal present, had one been present. The second selects

170

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.40 Performance Comparison on Experiment Data, BP White Gaussian AM, Co-variance Method, with the QR and AM at 6.25 kHz.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm






Batch Whitened Matched Filter Detector, Whitening Filter Order 6, L2 Normalization, AUC 1


Fig. 6.41 Performance Comparison on Experiment Data, BP White Gaussian AM, withthe QR and AM at 10 kHz. Autocorrelation Method.

171

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.42 Performance Comparison on Experiment Data, BP White Gaussian AM, withthe QR and AM at 10 kHz. Covariance Method.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.43 Performance Comparison on Experiment Data, BP White Gaussian AM, withthe QR and AM at -8 kHz. Autocorrelation Method.

172

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.44 Performance Comparison on Experiment Data, BP White Gaussian AM, withthe QR and AM at -8 kHz. Covariance Method.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.45 Performance Comparison on Experiment Data, BP White Gaussian AM, withthe QR and AM at 12.5 kHz. Autocorrelation Method.

173

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm








Fig. 6.46 Performance Comparison on Experiment Data, BP White Gaussian AM, withthe QR and AM at 12.5 kHz. Covariance Method.

the whitening filter order based on the minimum description length (MDL) algorithm

[28], which attempts to minimize the noise model error incorporating a penalty function

that increases as the filter order does. The goals of the two adaptive methods of selecting

the whitening filter order are “orthogonal”. One focuses on reducing the effect of the

whitening filter on the QR signal while the other focuses on reducing the noise modeling

error. The performance of the batch whitened matched filter when adaptively selecting

the whitening filter order, for both the autocorrelation and the covariance, is compared

and discussed in this section. The performance of the batch whitened match filters, when

the whitening filter order is chosen adaptively, are compared using simulation data in

Subsection 6.4.3 and experimental data in Subsection 6.4.4.

Previously, in Section 6.2, the Nm phase cycled received echoes are whitened using one

of modeling methods (either the covariance or the autocorrelation method) with a fixed

order and then low passed. Then, each filter in a batch of filters, matched to a set of

174

QR signals, is whitened and low passed using the same whitening and low pass filters

applied to the received signal. A decision on the presence of a QR signal is made based

on the maximum output of the batch of whitened match filters.

Unlike then, the gridding to match the QR signal parameters is performed pre-whitening

when designing the filters discussed in this section. The filters discussed in this section

use the approach in Section 5.2 to grid through the QR signal parameters and obtain

a periodic filter of Nm signals matched to the QR signal present. Since this part of

the process was discussed earlier it will not be repeated here. The next step would

be to whiten each of the Nm echoes of the filter (matched to the QR signal) using

a whitening filter (estimated by either the autocorrelation or covariance method) of

adaptively selected order. Selecting the whitening filter order is done according to one of

the methods introduced in this section. Each of these Nm whitened QR signals is then

low passed using the same 8-pole butterworth filter used in Section 6.2, denoted by M .

The Nm received echoes , xpci for i = 0, 1, . . . Nm− 1, are then whitened and low passed

using the same filters used to whiten and low pass the Nm echoes of the filter (matched

to the QR signal). Let the periodic filter matched to the QR signal, chosen as a result

of the process described in Section 5.2, be denoted as

h[k] =Nm−1∑i=0

sl,m,n[k − iNs], (6.13)

for k = 0, 1, . . . , Nm × Ns − 1, where sl,m,n[k] is a filter normalized in the l2 norm and

is matched to the lth, mth and nth frequency, phase and decaying parameter values

175

respectively. Each of these individual echoes is whitened using an estimated whitening

model (autocorrelation in Section 6.1.1 or covariance in Section 6.1.2) of order chosen

according to the least affect on the QR signal or the MDL algorithm. The following

two subsections describe how to adaptively choose the whitening filter order. These

methods can be applied whether the modeling method used is the autocorrelation or the

covariance method.

6.4.1 Whitening Filter Order that Least Effects the QR Signal

For a particular modeling method (auctorrelation or covariance method), the ith received

echo xpci is used to estimate a whitening model Wip of order p. This is repeated for a

number of model orders, Np, represented as a vector [plb, plb + 1, . . . , pub]. This process

leads to Np whitening matrices (Wi1, . . . ,Wi

Np). The whitening matrix that maximizes

maxjβijMWi

j sl,m,n[k]sTl,m,n

[k], (6.14)

is chosen, Wimax, where sl,m,n[k] is the normalized filter matched to the chosen QR

signal parameters, M is the 8-pole butterworth low pass filter, and βij is a constant that

normalizes the whitened low passed QR signal estimate MWij sl,m,n[k]. Maximizing the

above value ensures choosing the whitening filter order, between plb and pub that leads

to a whitening model that least deviates the whitened low passed QR signal from the

QR signal. This process is repeated for each of the Nm received signals resulting in Nm,

whitening matrices (W1max, . . . ,W1

max), where each can have a different model order.

176

The chosen whitening matrix for echo i, Wimax, along with the low pass filter M is

applied to the corresponding received echo i, as follows

xwi = βimaxMWi

maxxpci ,

where,

βimax =

∑Ns−1k=0

x2pci

[k]∑Ns−1k=0 (MWi

maxxpci [k])2.

Stringing these Nm signals in time leads to, xwmax

xwmax[k] = [xw0

[k], xw1[k −Ns], . . . .xwNm−1

[k − (Nm − 1)Ns], (6.15)

for k = 0, 1, . . . , Nm ×Ns − 1. Now that the received Nm echoes are whitened, the next

step is to whiten and low pass each of the Nm echoes of the periodic filter matched to

the QR signal as follows

hmax[k] = [sw0(l,m, n)[k], sw1

(l,m, n)[k −Ns], . . . , swNm−1(l,m, n)[k − (Nm− 1)Ns]

where,

swi(l,m, n)[k] =1

αwimax(l,m, n)

MWimaxe(−|−T/2+k∆|/Ta(n))cos(ωa(l)(−T/2 + k∆) + φa(m))

177

for k = 0, 1, . . . , Ns − 1. The value T is the length of the data acquisition window, Ns

is the number of discrete samples, ∆ = T/(Ns − 1) and αwimax(l,m, n) is a normalizing

constant.

The inner product of xwmax and hmax is then compared to threshold to determine the

presence of a QR signal. The following subsection describes the method of adaptively

choosing the whitening filter order according to the minimum description length (MDL)

algorithm [28].

6.4.2 Whitening Filter Order, Minimum Description Length Algorithm

This subsection explains how to use the minimum description length algorithm to adap-

tively determine the whitening filter order. For a particular modeling method (auctorre-

lation or covariance method), the ith received echo xpci is used to estimate a whitening

model Wip of order p. This is done for a number of model orders, Np, represented as a

vector [plb, plb+1, . . . , pub]. This process leads to Np whitening matrices (Wi1, . . . ,Wi

Np).

According to the minimum description algorithm, the whitening filter order p that min-

imizes the following function is chosen [28].

minp

[Ns log εp + (logNs)p] (6.16)

In the above equation, p represents the whitening filter order, Ns represents the number

of samples in an echo, and εp represents the minimized error shown in Equations (6.2)

and (6.4) when modeling using the autocorrelation and covariance methods respectively.

The idea behind the MDL algorithm is to increase the model order p until the modeling

178

error εp is minimized. However, since the error is a monotonically nonincreasing function

of the model order p, the MDL algorithm incorporates a penalty function that increases

with the model order p. This process is repeated with each of the Nm received signals

resulting in Nm, whitening matrices (W1max, . . . ,W1

max), where each can have a different

model order.

The rest of the procedure is identical to the procedure described in the above subsection

and therefore will not be repeated here. The following subsections use both simulation

and experimental data to compare the performance of the batch whitened matched filter,

where the order is selected adaptively.


Figures (6.47)-(6.52) use simulation data to discuss the performance of the batch adap-

tive whitened matched filter. The whitening filters were estimated according to either the

autocorrelation or the covariance method, and the whitening filter order is chosen adap-

tively to either minimize the effect on the QR signal or according to the MDL algorithm.

The method gridded using the following values, ωlb = 0, ωub = 40000π, ω∆ = 100π,

φ∆ = .1, T ∗2 lb

= 300e−6, T ∗2 ub

= 900e−6 and T ∗2 ∆

= 100e−6. The above values lead to

Ndω = 401, Ndφ = 63 and NdT = 7 in each of the three dimensions, which corresponds to

Ndω ×Ndφ ×NdT = 176841 filters in the batch. After selecting the filter that resulted in

the maximum output, the adaptive whitening filters were applied. The filter order was

varied between plb = 3 and pub = 7 when adaptively selecting the whitening filter order.

179

The ROC plots in the figures compare the performance of the following four batch

adaptive whitened matched filters to the batch matched filter and the energy detector.

1. Batch Adaptive Whitened Matched Filter, whitened using autocorrelation method

with an adaptive order that least affects the QR signal.

2. Batch Adaptive Whitened Matched Filter, whitened using covariance method with

an adaptive order that least affects the QR signal.


with an adaptive order selected according to the MDL algorithm.


an adaptive order selected according to the MDL algorithm.

The ROC plots for the 2.5 kHz data set are shown in Figure (6.47). These results are

summarized in Table (6.11). The best batch adaptive whitened matched filter was the

one that used the autocorrelation method to model the noise with a filter order chosen

according to the MDL algorithm. This filter achieved an area under the curve (AUC)

of 0.875, which is lower but comparable to the performance of the batch matched filter

that achieved an AUC of 0.886. The other three batch adaptive whitened matched filters

achieved AUCs lower than that of the energy detector.

The best performing batch adaptive whitened matched filter when using the 5 kHz data

set, was the one that used the covariance method to model the noise with a filter of order

chosen according to the MDL algorithm. This is shown in Figure (6.48), where the ROC

curves of the different filters are compared. The results from this figure are summarized

180

Adaptive Whitening Autocorrelation Covariance No WhiteningFilter Order Method Method

None 0.886Least Effect on QR 0.706 0.660

MDL Algorithm 0.875 0.698

Table 6.11 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data,Band-passed White Gaussian AM, with the QR and AM at 2.5 kHz. Adaptive WhiteningFilter Order Selection.

in Table (6.12). The best filter achieved an AUC of 0.935, which is larger than the

AUC of the batch matched filter of 0.920. The three remaining batch adaptive whitened

matched filter achieved AUCs that are lower than the AUC of the batch matched filter,

but larger than the AUC of the energy detector of 0.8.




Table 6.12 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data,Band-passed White Gaussian AM, with the QR and AM at 5 kHz. Adaptive WhiteningFilter Order Selection.

According to the 7.5 kHz simulation data set, the best performing batch adaptive

matched filter was the one that used the autocorrelation method to model the noise

with a filter of order that least affects the QR signal. The ROC plots of this filter and

others are shown in Figure (6.49) and the results are summarized in Table (6.13). The

best filter achieved an AUC of 0.936, which is larger than the AUC of the batch matched

filter of 0.889. The other three batch adaptive whitened matched filters achieved AUCs

181

lower than that of the batch matched filter, but larger than the AUC of the energy de-

tector of 0.79. Although, one of those three, the one that used the covariance method to

model the noise with a filter of order that least affects the QR signal, achieved an AUC

of 0.87, which is close to that of the batch matched filter.




Table 6.13 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data,Band-passed White Gaussian AM, with the QR and AM at 7.5 kHz. Adaptive WhiteningFilter Order Selection.

The performance of the batch adaptive whitened matched filters when applied to the

10 kHz data set is shown in Figure (6.50) and summarized in Table (6.14). The best

performing one was the one that used the covariance method to model the noise with a

filter order that least affects the QR signal. This filter achieved an AUC of 0.907, which

is larger than the AUC of the batch matched filter of 0.873. The two adaptive filters

that used the autocorrelation method to model the noise achieved AUCs lower than the

batch matched filter, but higher than the energy detector, which achieved an AUC of

0.77. The filter that used the covariance method to model the noise with a filter of order

chosen according to the MDL algorithm performed very poorly.

The batch adaptive whitened matched filters when applied to the 12.5 kHz data set

were compared in Figure (6.51) and summarized in Table (6.15). The one that used the

autocorrelation method to model the noise with a filter of order chosen according to the

182





MDL algorithm performed best. This filter achieved an AUC of 0.898, which is larger

than the AUC achieved by the batch matched filter of 0.874. The one that used the

covariance method to model the noise with a filter order that least affects the QR signal

achieved an AUC of 0.840, which is lower than the AUC of the batch matched filter,

but higher than the 0.815 achieved by the energy detector. The remaining two achieved

AUCs lower than that achieved by the energy detector.




Table 6.15 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data,Band-passed White Gaussian AM, with the QR and AM at 12.5 kHz. Adaptive Whiten-ing Filter Order Selection.

The performance of the four filters when used on the 15 kHz data set was compared in

Figure (6.52) and are summarized in Table (6.16). The best performing filter in this case

was the one that used the covariance method to model the noise with a filter of order

chosen according to the MDL algorithm, which achieved an AUC of 0.953. The second

best performing filter was the once that used the covariance method to model the noise

183

with a filter order that least affects the QR signal and achieved an AUC of 0.942. Both

these filters outperformed the batch matched filter, which achieved an AUC of 0.884.

The two filters that used the autocorrelation method achieved AUCs lower than that of

the batch matched filter but higher than that of the energy detector, AUC 0.8.





In conclusion, throughout all the simulation data at least one of the batch adaptive

whitened matched filters among the four compared in this section was able to achieve

equal or better performance than the batch matched filter. Although, in some cases

the batch adaptive whitened match filter achieved significantly lower performance when

compared to the batch matched filter. Further analysis of these filters needs to be done

to find the reason behind this drop in performance, especially since we have seen, in

Section 6.1.3, that the effect an FIR filter has on the QR signal is small. One can

attempt to combine the approaches used in designing these filters to design a filter that

would outperform the batch matched filter consistently.

184

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

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0.8

0.9

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rob C

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ction

Prob False Alarm



Batch Whitened Matched Filter Detector, Autocorrelation Method, Adaptive Whitening Order, AUC 0.7058

Batch Whitened Matched Filter Detector, Autocorrelation Method, Minimum Description Length Order, AUC 0.87515

Batch Whitened Matched Filter Detector, Covariance Method, Adaptive Whitening Order, AUC 0.6599

Batch Whitened Matched Filter Detector, Covariance Method, Minimum Description Length Order, AUC 0.69775

Fig. 6.47 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data,Band-passed White Gaussian AM, with the QR and AM at 2.5 kHz. Whitened witheither the Autocorrelation Method or the Covariance Method using an Adaptive FilterOrder that either Minimizes the Effect on the QR Signal or based on the MDL algorithm.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm







Fig. 6.48 Simulation, SNR = -22 dB, Performance Comparison on Simulation Data,Band-passed White Gaussian AM, with the QR and AM at 5 kHz. Whitened witheither the Autocorrelation Method or the Covariance Method using an Adaptive FilterOrder that either Minimizes the Effect on the QR Signal or based on the MDL algorithm.

185

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1P

rob C

orr

ect D

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ction

Prob False Alarm








0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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186

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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0.6

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

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0.5

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187


Figures (6.53)-(6.56) use experimental data to discuss the performance of the batch

adaptive whitened matched filter. The whitening filters were estimated according to

either the autocorrelation or the covariance method, and the whitening filter order is

chosen adaptively to either minimize the effect on the QR signal or according to the

MDL algorithm. The method gridded using the following values, ωlb = 0, ωub = 40000π,

ω∆ = 100π, φ∆ = .1, T ∗2 lb

= 300e − 6, T ∗2 ub

= 900e − 6 and T ∗2 ∆

= 100e − 6. The

above values lead to Ndω = 401, Ndφ = 63 and NdT = 7 in each of the three dimensions,

which corresponds to Ndω ×Ndφ ×NdT = 176841 filters in the batch. After selecting the

filter that resulted in the maximum output, the adaptive whitening filters were applied.

The range of filters to choose from is lower bounded by plb = 3 and upper bounded by

pub = 7, when adaptively selecting the whitening filter order.

The ROC plots in the figures below, compare the performance of the following four batch

adaptive whitened matched filters to the batch matched filter and the energy detector.


with an adaptive order that least affects the QR signal.


an adaptive order that least affects the QR signal.


with an adaptive order selected according to the MDL algorithm.

188


an adaptive order selected according to the MDL algorithm.

The four filters above were compared using the experimental data set A (6.25 kHz) in

Figure (6.53) and the results are summarized in Table (6.17). In this case all four batch

adaptive whitened matched filters outperformed the batch matched filter, which achieved

an AUC of 0.977, and the energy detector, which achieved an AUC of 0.945. The best

performing filter was the one that used the autocorrelation method to model the noise

with a filter of order that least affects the QR signal. The second best was the one that

used the covariance method to model the noise with a filter of order that least affects

the QR signal. The third best filter was the one that used the autcorrelation method

to model the noise with a filter of order selected according to the MDL algorithm. The

fourth filter is the one that used the autocorrelation method to model the noise with

a filter of order chosen according to the MDL algorithm and achieved an AUC slightly

higher than that of the batch matched filter.




Table 6.17 Performance Comparison on Experimental Data A, BP White Gaussian AM,with the QR and AM at 6.25 kHz. Adaptive Whitening Filter Order Selection

The same four filters were compared using the experimental data set B (10 kHz) in

Figure (6.54) and the results from this figure are summarized in Table (6.18). Two of

189

the four filters outperformed the batch matched filter, which achieved an AUC of 0.9987,

and other two did not outperform the batch matched filter but outperformed the energy

detector which achieved an AUC of 0.984. The best filter used a combination of the

covariance method to model the noise and selecting the filter order that minimizes the

effect on the QR signal, while the second best used a combination of the autocorrelation

method to model the noise and the MDL algorithm to select the filter order.


None 0.9987Least Effect on QR 0.961 1


Table 6.18 Performance Comparison on Experimental Data B, BP White Gaussian AM,with the QR and AM at 10 kHz. Adaptive Whitening Filter Order Selection

The experimental data set C (8 kHz) was used to compare the four adaptive filters in

Figure (6.55). The results are summarized in Table (6.19). The best performance was

achieved by using the covariance model to whiten the noise and selecting the filter order

that minimizes the effect on the QR signal and resulted in an AUC of 0.9987. Using either

covariance method to model the noise or the autocorrelation method, while selecting the

filter order according to the MDL algorithm did not change the performance and resulted

in an AUC of 0.99865. The three filters mentioned above all outperformed the batch

matched filter, which achieved an AUC of 0.987. Using the autocorrelation method to

model the noise, while selecting the filter order that minimizes the affect on the QR

signal resulted in an AUC of 0.979, which is lower than that of the batch matched filter

but higher than that of the energy detector with an AUC of 0.923.

190




Table 6.19 Performance Comparison on Experimental Data C, BP White Gaussian AM,with the QR and AM at -8 kHz. Adaptive Whitening Filter Order Selection

The fourth experimental data set D (12.5 kHz) was used to compare the four filters in

Figure (6.56) and the results were then summarized in Table (6.20). The best perfor-

mance was achieved by using the covariance method to model the noise with a filter of

selected by the MDL algorithm. This filter achieved an AUC of 0.9997, which is higher

than the 0.994 achieved by the batch matched filter. The second best achieved an AUC

of 0.9964, which also outperformed the batch matched filter and used the covariance

method to model the noise with a filter order that least affects the QR signal. The other

two that used the autocorrelation method to model the noise did not outperform the

batch matched filter, but outperformed the energy detector which had an AUC of 0.956.




Table 6.20 Performance Comparison on Experimental Data D, BP White Gaussian AM,with the QR and AM at 12.5 kHz. Adaptive Whitening Filter Order Selection

In conclusion the batch adaptive whitened matched filter performed better overall on the

experimental data versus the simulation data. Throughout the four experimental data

sets, at least two of the batch adaptive whitened matched filters outperformed the batch

191

matched filter. On the other hand, the batch adaptive whitened match filter that used

the autocorrelation method to model the noise using a filter of order that least affects the

QR signal performed poorly on three of the four experimental sets (B, C and D). Further

analysis needs to be done to find the reason behind this drop in performance, especially

since we have seen, in Section 6.1.3, that the effect an FIR filter has on the QR signal is

small and the adaptive method for selecting the whitening filter order attempts further

to minimize the effect on the QR signal. As was mentioned earlier, one can combine the

different approaches used by these filters to design a filter that consistently outperforms

the batch matched filter.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

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Fig. 6.53 Performance Comparison on Experiment Data A, BP White Gaussian AM,with the QR and AM at 6.25 kHz. Whitened with either the Autocorrelation Method orthe Covariance Method using an Adaptive Filter Order that either Minimizes the Effecton the QR Signal or based on the MDL algorithm.

192

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Batch Whitened Matched Filter Detector, Covariance Method, Adaptive Whitening Order, AUC 1


Fig. 6.54 Performance Comparison on Experiment Data B, BP White Gaussian AM,with the QR and AM at 10 kHz. Whitened with either the Autocorrelation Method orthe Covariance Method using an Adaptive Filter Order that either Minimizes the Effecton the QR Signal or based on the MDL algorithm.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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Fig. 6.55 Performance Comparison on Experiment Data C, BP White Gaussian AM,with the QR and AM at -8 kHz. Whitened with either the Autocorrelation Method orthe Covariance Method using an Adaptive Filter Order that either Minimizes the Effecton the QR Signal or based on the MDL algorithm.

193

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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Fig. 6.56 Performance Comparison on Experiment Data D, BP White Gaussian AM,with the QR and AM at 12.5 kHz. Whitened with either the Autocorrelation Method orthe Covariance Method using an Adaptive Filter Order that either Minimizes the Effecton the QR Signal or based on the MDL algorithm.

194

Chapter 7

Batch Whitened Robust Matched Filter

In Chapter 6, a gridding technique was combined with a whitening filter to design a

filter closely matched to both the QR signal and noise, when the QR signal and second

order statistics of the input noise are unknown. The performance of this filter drops,

when the parameters (frequency, phase and decaying value) of the QR signal present,

spc, fall in-between the values on the grid. For example, if the QR signal present, spc,

had a frequency of 12.1 kHz, a phase of .15 radians and a decaying parameter of 450e−6

and we grid in intervals of 200 Hz, .1 radians and 100e − 6 for frequency, phase and

decaying parameter respectively, none of the filters in the batch would exactly match

the QR signal present. This effect can be significantly reduced by using finer grids in

frequency, phase and decaying parameter, i.e. reducing each of the gridding intervals.

Another factor that can degrade performance is uncertainty in the general shape of the

QR signal. The ultimate goal of the robust matched filter approach in this chapter is

to address this uncertainty in the general shape of the QR signal, caused by narrow

bandwidth filters and receiver coils (HTS coil).

To address this uncertainty in the general shape of the signal, we propose designing a set

of filters, where each is robust to a set of QR signals and select the filter that results in

the maximum output. Each of the robust filters would be robust to a set of QR signals

with a fixed frequency, phase and decaying parameter but varies in the general shape.

195

Unfortunately, due to the lack of data with variances in the general shape of the QR

signal, we can not yet address the problem of robustness to these variances. Instead

this chapter will develop the tools to be used in designing the robust matched filters

and these methods will be applied to address robustness to variances in the QR signal

parameters and assume the general shape is known. These tools can later be applied

to design filters robust to uncertainties in the QR signal’s general shape for fixed signal

parameters.

Although uncertainties in the whitening filter (noise covariance matrix) are inevitable,

due to inaccuracies in the modeling estimates, this chapter will not address them but

will introduce the notion of filters that optimize performance for a pair of QR signal and

covariance matrix sets. The work will address the problem of designing a filter optimized

for a set of QR signals, while assuming our whitening approach proposed in Section 6.1

results in an exact match.

The chapter combines the gridding technique, the whitening filter, and the robust matched

filter introduced in Chapter 5, Section 6.1 and Section 7.5 respectively to form a batch

of robust whitened matched filters. As was mentioned earlier, the ultimate goal is to

achieve robustness to uncertainties in the QR signal’s shape but for now the uncertainty

will be restricted to the QR signal parameters that could be caused by unfine gridding.

Briefly the idea is to design a batch of whitened robust matched filters, where each filter

is whitened according to the noise present and is robust to a set of QR signals. Each

one of these filters is robust to a different set of QR signals, with frequency, phase and

196

decaying parameter that fall within intervals, which represents uncertainty in the signal’s

parameters. This batch of filters is used to determine the presence of a QR signal.

The following section introduces the idea of filters optimized for a pair of QR signal

and covariance matrix sets. This is followed by a section that introduces analytical

solutions for filters robust to two different pairs of QR signal and noise covariance sets,

one of which was derived by Verdu and Poor and the other was derived in this thesis. A

sampling approach to solving optimization problems, referred to as the scenario approach

is then introduced in Section 7.3. This approach to solving optimization problems will

be used in Section 7.4 in characterizing the set of QR signals using sampling, which is

a necessary step to use any of the analytical solutions introduced Section 7.2. Since

this chapter assumes that the whitening approach to estimating the covariance filter

leads to an exact value for the covariance matrix, the problem of characterizing a set

of covariance matrices is not addressed. The chapter moves on to directly using the

scenario approach to design an optimized matched filter for a set of QR signal samples

by solving an optimization problem, without having to first characterize the set of QR

signals. This is the most feasible approach for designing filters robust to a set of the

QR signals. This approach is combined with the gridding technique and whitening filter

introduced in Chapter 5 and Section 6.1 to create a batch robust whitened matched filter

in Section 7.6. The performance of this filter in the presence of noise is discussed using

simulation data in the last section of the chapter.

197

7.1 Robust Matched Filter

In the case when the signal and noise are not exactly known but are known to fall

within a particular uncertainty set, since no filter will be optimum for every member

of the uncertainty set, it is of interest to design a robust filter. The robust filter would

guarantee a certain level of performance for the set of possible input signal and noise

pairs, independently of which member of the uncertainty set is actually present.

There are several different approaches to designing this robust filter; one well established

approach is the minimax strategy, used in this thesis. The goal here is to optimize the

performance of the filter for the worst-case signal-noise pair. There are several studies

on the minimax strategy to robust matched filtering [16; 62; 61], but this section will

focus on robust finite-length discrete time matched filters.

Consider the following problem formulation [65]: Let H be a Hilbert space with inner

product < ·, · > and P a set of bounded, linear, self-adjoint, positive operators mapping

H to itself. Now, if the input signal and noise quantities are known only to belong to

some uncertainty classes S ⊂ H, N ⊂ P , then a possible optimal filter design strategy is

to choose the one that exhibits the best performance for the worst-case signal and noise

pair in Q = S×N (the cartesian product of the uncertainty sets) i.e., to choose hR such

that

hR = arg maxh∈H

inf(s,Σ)∈S×N

ρ(h; s,Σ) (7.1)

198

Another way of explaining Equation (7.1), is that of a least favorable signal and noise

pair, (sL,ΣL), defined by the relationship

(sL,ΣL) = arg min(s,Σ)∈S×N

ρ(h∗(s,Σ); s,Σ), (7.2)

where, h∗(s,Σ), is the optimal matched filter for the signal and noise pair (s,Σ). So

(sL,ΣL) is the pair in the uncertainty class with minimum optimal signal-to-noise ratio.

This design problem can be thought of as a game (H,S×N, ρ), where the designer tries

to maximize the function ρ by selecting a filter from the set H and the opponent (nature)

tries to minimize it by choosing a signal and noise pair from S×N . The following shows

that the filter, hL, optimal to the signal noise pair (sL,ΣL), referred to as a saddle point,

is in fact the minimax robust filter.

If a saddle point exists for this game, which means if there is a point [hL, (sL,ΣL)] that

satisfies

ρ(h; sL,ΣL) ≤ ρ(hL; sL,ΣL) ≤ ρ(hL; s,Σ) (7.3)

for all h ∈ H, s ∈ S, and Σ ∈ N , then hL has its worst performance at (sL,ΣL), and any

other filter has worst behavior at (sL,ΣL), which implies that hL is the minimax robust

filter. Furthermore, from the definition of h∗(s,Σ), we have

ρ(hL; s,Σ) ≤ ρ(h∗(s,Σ); s,Σ) (7.4)

which, together with Equation (7.3) implies that (sL,ΣL) is a least favorable pair.

199

The existence and characterization of a saddle point for the robust matched filtering

problem in (7.1) can be straightforwardly derived from (4.8) and (7.3) and are summa-

rized in the following lemma. A proof of this lemma can be found in [65].

Lemma 7.1. (hL, (sL,ΣL)) is a saddle point for (H,S ×N, ρ) if and only if

1. ΣLhL = sL

2. | < sL, hL > | ≤ | < s, hL > | ∀s ∈ S

3. 0 ≤ < hL, (ΣL − Σ)hL > ∀Σ ∈ N

Solving the equations of Lemma (7.1) would result in a saddle point of (H,S×N, ρ). The

above lemma is an important result and can be used to derive analytical filter solutions

for specific uncertainty models. Another important result presented in [65] is Theorem

(7.2), which requires defining the term regular pair. The following theorem, [65] defines

the term regular pair.

Theorem 7.1. Denote hL = h∗(sL,ΣL), Q = S × N (the cartesian product of the

uncertainty sets) and define the functional f : H ×H × P × P × [0, 1] → C where C is

the complex scalar field of H by

f(a, b,A,B, α) ≡< b−Bh∗(a,A),h∗(a− α(a− b),A− α(A−B)) > (7.5)

200

If for every (s,Σ) ∈ Q such that (sα,Σα) = (1−α)(sL,ΣL)+α(s,Σ) ∈ Q for all α ∈ [0, 1],

we have that f(sL, s,ΣL,Σ, .) is right continuos at the origin, then (hL, (sL,ΣL)) is a

regular pair for (H,Q, ρ).

The regularity condition can also be seen as requiring the difference of performance

achieved by the optimal filter, hα, at (sα,Σα) (a point close to (sL,ΣL)) and by hL at

(sα,Σα), divided by α, to go to zero when α goes to zero. This condition is used in

the following theorem, which simplifies the robust matched filtering problem for the case

when the uncertainty set is convex.

Theorem 7.2. Suppose that Q = S ×N , the cartesian product of the uncertainty sets,

is a convex set and that ρ(h; , ) is convex on Q for every h ∈ H.

Then if the pair of filter and signal/noise (hL; sL,ΣL) is a regular pair of (H,S ×N, ρ),

the following statements are equivalent,

1. (sL,ΣL) is a least favorable operating point of (H,S ×N, ρ)

2. (hL; sL,ΣL) is a saddle point point solution of (H,S ×N, ρ)

It has been shown in [41] that the signal to noise ratio defined in (4.8) is convex in (s,Σ)

for every h ∈ H. Theorem 7.1 stated that under a mild continuity condition on the

behavior of the matched filter around a given operating point (sL,ΣL), this point and

its optimal filter form a regular pair. Furthermore, it has been proved in [65], that the

invertibility of ΣL is a sufficient condition for the continuity condition to hold. Therefore,

if the uncertainty class Q is convex and the regularity condition holds (i.e. the conditions

of Theorem 7.2 are satisfied) then the problem of finding a saddle point (minimax robust

201

filter), whenever one exists, is reduced to that of finding a least favorable pair of signal

and noise. This need not be easier than the original problem of finding a minimax robust

matched filter. However, the existence of an explicit expression for ρ∗(.) (Equation 4.11)

reduces the search for least favorable operating points to a convex minimization problem,

at which point, a directional derivative approach can be used.

The results of Lemma 7.1 and Theorems 7.1 and 7.2 were used by Verdu and Poor in [66]

to derive analytical solutions for several types of uncertainty sets. As examples, one of

these cases is presented in the next section as along with the solution to another signal

and noise pair (derived in this thesis using the results of Lemma 7.1).

7.2 Analytical Solutions For Robust Matched Filters Over Particular

Uncertainty Sets

This section provides analytic solutions to the matched filter robust to specific signal and

noise uncertainty sets. Considering the time-domain formulation, where H = <k, P =

{Σ ∈ <k×k,Σ > 0}, s = [s[0], . . . , s[k−1]]T and h = [h0, . . . , h[k−1]]T , h[i] = h[k−1− i],

and s[i] and h[i] are the values of the signal and of the filter response, respectively, at

the ith sample. The following subsection presents the analytic solution for a filter robust

to a spherical signal set and matrix norm bounded noise set pair, derived by Verdu and

Poor [66]. This is followed by a another subsection that derives the analytical solution

of the filter robust to an elliptical signal set and matrix norm bounded noise set.

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7.2.1 Spherical Signal Set and Noise Uncertainty Bounded by a Matrix

Norm

If the signal uncertainty is modeled by a bound on the mean-square distortion S1 as

follows,

S1 = {s ∈ <k,k−1∑i=0

|s[i]− s0[i]|2 ≤ 42} (7.6)

and the noise uncertainty is modeled by a bound on some matrix norm of the deviation

from a nominal, N1

N1 = {Σ ∈ <k×k, ||Σ− Σ0|| ≤ ε,Σ > 0} (7.7)

where the norm in (7.7) is a unit matrix norm (if ||I|| = 1, then ||.|| is a unit matrix

norm). It has been proven by Verdu and Poor in [66] that the optimal robust filter is

of the form

hL = (Σ0 + (ε+ σs2)I)−1s0 (7.8)

where σs2 is defined by σs

2||hL||2 = 4. The corresponding least favorable signal and

noise pair has been shown to have the following form,

sL = s0 − σs2hL (7.9)

ΣL = Σ0 + εI (7.10)

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Verdu and Poor in [66] have also considered other signal uncertainty sets, where the

distortion measure is bounded in the l1 and l∞ norms, and other noise covariance un-

certainty sets, where the perturbations from the nominal covariance matrix are bounded

in the l2 norm. Unfortunately in [66] the case when the uncertainty in the QR signal

is unevenly spaced is not discussed. Uneven weights can be placed on the uncertainty

of the QR signal if an ellipse is used to to describe the uncertainty set. Fortunately

enough, Lemma 7.1 can be used to derive analytical solutions for particular signal and

noise uncertainty sets.

7.2.2 Elliptic Signal Set and Noise Uncertainty Bounded by the Frobenius

Matrix Norm or the 2-Norm

We used the lemma to derive an analytical solution for a signal uncertainty set bounded

by an ellipse and a noise uncertainty set modeled by a bound in either the Frobenius

norm or the l2 norm (the results is unchanged for both these norms). If the signal

uncertainty is modeled by an ellipse S2 as follows,

S2 = {s ∈ <k, (s− s0)TW (s− s0) ≤ 42} (7.11)

which is a convex set. The noise uncertainty is modeled by a bound on the deviation

from the nominal. This bound can be in any matrix norm that satisfies the property

||xxT || = xTx, where x ∈ <k×1. Both the Frobenius and the 2-norm satisfy this property,

so without loss of generality we will define the noise uncertainty set in the l2 norm, N2,

as follows,

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N2 = {Σ ∈ <k×k, ||Σ− Σ0||2 ≤ ε,Σ > 0}. (7.12)

The minimax solution for the uncertainty sets S2 and N2, can characterized by the

following result.

Proposition 1. (hL, (sL,ΣL)) is a saddle point for (H,S2 ×N2, ρ) if and only if

sL = s0 − λW−1hL and λ = 4||W−1/2hL||2

with

ΣL = Σ0 + σn2hLhL

T and σn2||hL||22 = ε

and

ΣLhL = sL

Proof: Using the above proposition we can solve for the robust minimax filter hL as

follows

205

ΣLhL = sL

(Σ0 + σn2hLhL

T )hL = s0 − λW−1hL

Σ0hL + σn2||hL||

22hL + λW−1hL = s0

Σ0hL + εhL + λW−1hL = s0

(Σ0 + ε+ λW−1)hL = s0

hL = (Σ0 + ε+ λW−1)−1s0

and

λ||W−1/2hL|| = 4

λ||(Σ0W1/2 + εW 1/2 + λW−1/2)−1s0|| = 4

where the function of λ, f(λ), to the left of the equality sign has a solution, since f(0) = 0

and the limλ→∞ f(λ) = ||W 1/2s0||. Next, we show that Lemma 7.1.2 is true, which is

equivalent to 〈s− sL, hL〉 ≥ 0, ∀s ∈ S2 since S2 is a convex set.

206

< s− sL, hL > = < W 1/2(s− sL),W−1/2hL >

= < W 1/2((s− s0)− (sL − s0)),W−1/2hL >

= < W 1/2(s− s0),W−1/2hL > − < W 1/2(sL − s0),W−1/2hL >

= < W 1/2(s− s0),W−1/2hL > +λ < W−1/2hL,W−1/2hL >

≥ −| < W 1/2(s− s0),W−1/2hL > |+4||W−1/2hL||

≥ −||W 1/2(s− s0)||||W−1/2hL||+4||W−1/2hL|| ≥ 0

Lastly, we show that Lemma 7.1.3 is true as follows.

< hL, (ΣL − Σ)hL > = < hL, (Σ0 + σn2hLhL

T − Σ)hL >

= σn2(hL

ThL)2+ < hL, (Σ0 − Σ)hL >

= ε||hL||22+ < hL, (Σ0 − Σ)hL >

207

where

< hL, (Σ0 − Σ)hL > ≥ −| < hL, (Σ0 − Σ)hL > |

≥ −||hL||||(Σ0 − Σ)hL||

≥ −||hL||2||(Σ0 − Σ)hL||

therefore,

< hL, (ΣL − Σ)hL > ≥ ε||hL||22− ||hL||

2||(Σ0 − Σ)hL||

≥ ||hL||22(ε− ||Σ0 − Σ||)

≥ 0.

To utilize the analytical solutions for the robust matched filters derived in the section

above, a characterization of the set of QR signals one would like to be robust to is needed.

This problem is addressed in Section 7.4, though before that the scenario approach to

probabilistic robust design is introduced in the next section. This approach is used to

characterize signal sets through sampling in Section 7.4, and this same approach can be

used to characterize an ellipse or any other shaped signal set. The problem of character-

izing noise uncertainty sets will not be discussed, since throughout the remainder of this

chapter we will be assuming that our whitening approach to estimating the covariance

matrix leads to accurate values. The scenario approach to robust design is also used in

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solving for matched filters robust, maximizes SNR, to a set of QR signals where the pa-

rameters vary within lower and upper bounds in the presence of thermal noise in Section

7.5. The following section uses a sampling approach to characterize a set of signals.

7.3 The Scenario Approach

This section introduces a probabilistic approach to robustness referred to as the scenario

approach. The scenario approach can be used to modify a robust optimization problem

with an infinite number of constraints into a one with a finite number of constraints.

If we consider our optimization problem to be

minimize cT [h; γ]

subject to J(s, h) ≤ γ ∀s ∈ S.

where, S ⊂ <k, h ∈ H ⊂ <k, J(s, h) : S ×H → < is convex in h, for all s ∈ S and H

is a convex and closed subset of <k. Then an alternative approach to solving the above

difficult problem is to redefine it using the scenario approach [43], as follows. If s is a

random variable variable with an assigned probability distribution over S, then s(1,...,N)

denotes a multi-sample s1, . . . , sN of independent samples of S extracted according to

some probability distribution. These samples, s(1,...,N), represent the randomly selected

scenarios and used in defining the following scenario design problem [43].

minimize cT [h; γ]

subject to J(si, h) ≤ γ i = 1, . . . , N.

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This is a standard convex program, with a finite number of constraints, and as such

it is usually computationally tractable. If the number of samples N is properly chosen

then the optimal solution returned by the scenario approach is with high probability

robust to the original design problem. Tempo [43], discusses the relationship between

N and the probability of robustness of the original problem but since in reality the

computational solver used is what places the limitations on N , this relationship is not

discussed here. This approach is used in the following section to characterize signal sets

by solving robust design problems. The scenario approach is also used in Section 7.5 to

design robust matched filters, without characterizing the uncertainty set, by solving an

optimization problem.

7.4 Characterizing a Set of QR Signals Through Sampling

Since this chapter will address robustness to uncertainties in the QR signal parameters

and not uncertainties in the general shape, examples of characterizing QR sets with

parametric uncertainty are presented in this section. This approach to characterizing

sets can be applied to characterizing any QR signal set, such as a set with uncertainty

in the general shape of the signal. The characterization of signal sets is required to use

any of the results in the section above.

The section uses a sampling approach to characterize the set of QR signals, S,

s[k] = Ae(−|−T/2+k∆|/T2∗) cos (ωbb(−T/2 + k∆) + φbb).

210

bounded in frequency, phase and decaying parameter as follows,

ωi ≤ ωbb ≤ ωi+1

φi ≤ φbb ≤ φi+1

T ∗2 i≤ T ∗

2 bb≤ T ∗

2 i+1.

Unfortunately the set that contains the set of QR signal signals with the listed bounds

in frequency, phase and decaying parameter is not necessarily a sphere, an ellipse or any

other particular shape. Though we can attempt to find the smallest sphere or ellipse

that contains this set of signals. The following subsection describes the approach taken

to find the smallest sphere that contains the set of signals.

7.4.1 Smallest Sphere Containing the Set of QR Signals

As shown in Equation (7.6), a sphere can be characterized by two values the center signal

s0 and the radius of the sphere ∆. Let S be the set containing all the QR signals with

frequency ωbb, phase φbb, and decaying parameter T ∗2 bb

, bounded as follows

ω1 ≤ ωbb ≤ ω2,

φ1 ≤ φbb ≤ φ2,

211

T ∗2 1≤ T ∗

2 bb≤ T ∗

2 2,

respectively, then solving following convex optimization problem with quadratic con-

straints leads to the smallest sphere containing the set of QR signals.

minimize ∆

subject to (s− s0)T (s− s0) ≤ ∆2, ∀s ∈ S.

The above optimization problem has an uncountable number of constraints, so to simplify

the problem, the scenario approach to convex optimization is used. The result is the

following convex optimization problem, whose solution leads to the central signal s0, and

the radius, ∆.

minimize ∆

subject to (si − s0)T (si − s0) ≤ ∆2, i = 1, . . . ,m.

The number of variables in the above problem is, Ns+1 equal to the number of samples,

Ns in the signal s0 plus the radius, ∆. The number of quadratic constraints is equal to

the number of samples, m, taken from the set S. Increasing the number of samples, m,

leads to a better estimate of s0, though this would result in a larger number of constraints

leading to longer computational times. Since the solution to the robust matched filter

is computed offline, computational time is not a restriction and the number of samples

used should be only limited by the computational power of the solver used. Most efficient

solvers, can solve a quadratic convex problems with thousands of variables and thousands

212

of constraints. The length of the QR signal window is less than 2ms, with a sampling rate

of 250 kHz the number of samples Ns is less than 500. Therefore the above problem can

be solved with thousands of constraint, i.e. samples from the signal set. What follows

are a number of central signal examples of the smallest spheres containing QR signals

bounded in in frequency, phase and decaying parameter.

7.4.1.1 Spherical Set Central Signal Examples

Figures (7.1), (7.2) and (7.3) display the central signal estimate, s0, of the smallest

sphere containing different sets of QR signals. The estimates were generated from a 1000

samples of signals with a decaying parameter of 600e − 6, a phase of 0, and bounded

in the following frequency ranges respectively, 12.25 kHz and 12.75 kHz, 11.5 kHz and

12.5 kHz, and 12 kHz and 14 kHz. Each of these estimates was generated by solving an

optimization problem with a 1000 quadratic constraints, corresponding to a 1000 signal

samples within the frequency bounds, and 435 variables. Comparing the central signal

to the signal samples with the largest and smallest frequencies shows that the central

signal has a frequency approximately equal to the central frequencies 12.5 kHz, 12 kHz

and 13 kHz respectively. In Figure (7.1) when the frequency interval defining the set

was small, the central signal is a QR signal with a frequency equal to the midpoint of

the range, 12.5 kHz, but decays at a faster rate when compared to the individual signal

samples.

Figure (7.4) displays the central signal estimate, s0, for the smallest sphere that contains

another set of QR signals. The QR signals in this set have a fixed frequency of 12 kHz, a

fixed phase of 0 radians, and T ∗2

values that varied between 400e− 6 and 800e− 6. One

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0

0.2

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0.8

1

Central Signal, Frequency Approximately equal to center frequency 12.5 kHz

Min Frequency QR Signal 12250.3293

Max Frequency QR Signal 12749.8963

Fig. 7.1 Central Signal, For the Set of QR Signals with Fixed Phase and T ∗2

and FrequencyValues Between 12.25 kHz and 12.75 kHz.

0 50 100 150 200 250 300 350 400 450−1

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Central Signal, Frequency Approximately equal to center frequency 12e3




and FrequencyValues Between 11.5 kHz and 12.5 kHz.

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0

0.2

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1

Central Signal, Frequency Approximately equal to center frequency 13 kHz




and FrequencyValues Between 12 kHz and 14 kHz.

thousand QR signal samples were used to estimate the central signal. When comparing

the central signal to the signal sample with the smallest and largest T ∗2

value, its obvious

that the central QR signal has the same frequency and phase as the QR signal samples

but has a T ∗2

that is halfway in between 400e-6 and 800e-6.

Figure (7.5) displays yet another central signal estimate, s0, for the smallest sphere

containing the set of QR signals with a frequency of 12 kHz, a T ∗2

value of 600e-6

and a phase value between -0.4 and 0.4 radians. A 1000 QR signal samples from the

set were used to estimate the central signal. The figure shows that the central signal

approximately has a frequency and decaying parameter equal to the QR signal samples

but a phase in between the maximum and minimum phase in the set of signals.

The above figures demonstrate how the signal central to the smallest sphere containing

the signal samples, generated from varying one of the three parameters, compares to the

signal samples with the varied parameter’s extreme range values.

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0

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1

Central Signal of the Smallest Sphere Containing the Signal SamplesMin T

2

* QR Signal 0.00040059

Max T2

* QR Signal 0.00079991

Fig. 7.4 Central Signal, For the Set of QR Signals with Fixed Frequency and Phase andT ∗

2Values Between 400e-6 and 800e-6.

0 50 100 150 200 250 300 350 400 450−1

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0

0.2

0.4

0.6

0.8

1

Central Signal of the Smallest Sphere Containing the Signal Samples

Min Phase QR Signal −0.39894

Max Phase QR Signal 0.39888

Fig. 7.5 Central Signal, For the Set of QR Signals with Fixed Frequency and T ∗2

andPhase Values Between -0.4 and 0.4 Radians.

216

Next we solve an optimization problem to find an estimate of the central signal of the

smallest sphere containing a set of QR signals, where all three parameters are varied.

Two thousand QR signal samples were generated, where the frequency varied between

11.5 kHz and 12.5 kHz, the decaying parameter, T ∗2

varied between 400e-6 and 800e-6,

and the phase varied between -0.4 and 0.4 radians. Figure (7.6) displays the central

signal estimate, s0, for the smallest sphere containing these signal samples.

0 50 100 150 200 250 300 350 400 450−1

−0.8

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0

0.2

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1


Fig. 7.6 Central Signal, of the Smallest Sphere Containing the Set of QR Signals withFrequency Values Between 11. 5 kHz and 12.5 kHz, T ∗

2Values Between 400e-6 and

800e-6, and Phase Values Between -0.4 and 0.4 Radians.

Using the results from Section 7.2.1, in the case of known thermal noise statistics, i.e.

σ0 = I and ε = 0, scaled versions of the central signals of spherical sets are the filters

robust to these sets of signals. Setting Σ0 = 0, in Equation (7.8) demonstrates this

result. Therefore, the central signals shown in Figures (7.1)-(7.6) are the filters in the

217

presence of thermal noise and no noise uncertainty, robust to the smallest spherical set

of signals that contains the set of QR signals described in each figure.

An important point to bring up is that characterizing the smallest set that contains

the QR signal samples and then applying a filter robust to the characterized set is

a conservative approach since it results in a filter robust to a set larger than the set

of QR signals. We therefore present another approach to designing minimax robust

matched filters, which does not require characterizing the smallest set containing the

signal samples. This approach maximizes performance for the worst-case QR signal

sample.

7.5 Robust Matched Filter by Maximizing SNR Using Sampling

If we assume that the whitening matched filter presented in Section (6.1) provides an

accurate estimate of the signal statistics, in other words the covariance matrix estimate

is exact, another approach to solving the robust matched filter for a set of QR signals

is to solve an optimization problem that maximizes the SNR for the a set of QR signals

and not the smallest sphere or ellipse that contains it. The SNR is defined as,

ρ(h, s,Σ) =(hT s)2

hTΣh.

Since we are assuming Σ is known, without loss of generality we can equate it to the iden-

tity matrix, when demonstrating this approach. In this case the problem of maximizing

the SNR over a set of QR signals, S, can be defined as

218

maximize ε

subject to hTh ≤ 1

hT s ≥ ε, ∀s ∈ S

The above optimization problem maximizes the numerator of the SNR, defined above,

over the set of QR signals, while restricting the l2 norm of the filter to be less then or

equal to one. The above optimization problem has an uncountable number of constraints.

Earlier we proposed defining the smallest sphere that contains the set, S, and applied

the analytic solution to the filter robust to the spherical set, h. The approach proposed

in this section is to use the scenario approach to solving the above optimization problem,

which converts the optimization problem to

maximize ε

subject to hTh ≤ 1

hT si ≥ ε, i = 1, . . . ,m.

Doing this eliminates the need to characterize the smallest set of a particular shape that

includes the QR signals and allows us to design a filter robust to the QR signals and

not the smallest set contain the signals. Increasing the number of signal samples to the

limit of the optimization solver used, increases the computational time but also increases

the robustness of the filter. The following are examples of the robust filter discussed for

different sets of the QR signal. This approach can be applied to any uncertainty signal

set, as long as samples of the signal set are obtainable.

219

7.5.1 Robust Matched Filter Examples in the presence of Thermal Noise,

Maximizing SNR

Figures (7.7), (7.8) and (7.9) display the filters that maximize the SNR for the signal

samples used, in the presence of thermal noise with a covariance matrix equal to the

identity matrix. A thousand samples of signals with a decaying parameter of 600e− 6, a

phase of 0 were used, which were bounded in the following frequency ranges respectively,

12.25 kHz and 12.75 kHz, 11.5 kHz and 12.5 kHz, and 12 kHz and 14 kHz. Each of

the three optimization problems had one quadratic constraint, corresponding to the l2

norm bound of the filter, and 1000 linear constraints, corresponding to the 1000 signal

samples within the frequency bounds, and 435 variables equal to the filter’s sample points

plus ε. Comparing the robust filter to the signal samples with the largest and smallest

frequencies shows that center portion of the central signal has a frequency equal to the

central frequencies 12.5 kHz, 12 kHz and 13 kHz. Note that the robust matched filters,

that maximize SNR of over the signal samples with variances in frequency only, shown

in Figures (7.7), (7.8) and (7.9), discussed in this section are almost the same as the

matched filters robust to the smallest spherical sets containing the samples mentioned

in the presence of thermal noise, shown in Figures (7.1), (7.2) and (7.3).

Figure (7.10) shows the filter that maximizes the SNR, in the presence of thermal noise,

of QR signal samples taken from the set of signals with a frequency of 12 kHz, a phase

of 0 radians, and T ∗2

values between 400e-6 and 800e-6. The figure shows the filter along

with the QR signal samples with the maximum and minimum T ∗2

values used. A 1000

QR signals samples were used to solve for the robust filter, which resembles the QR

220

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Robust Filter, Maximizing over signal samples, Approx equal to center frequency 12.5 kHz



Fig. 7.7 Robust Matched Filter, For Thermal Noise and the Samples from the Set of QRSignals with Fixed Phase and T ∗

2and Frequency Values Between 12.25 kHz and 12.75

kHz.

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Robust Filter, Maximizing over signal samples, Approx equal to center frequency 12e3




2and Frequency Values Between 11.5 kHz and 12.5 kHz.

221

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Robust Filter, Maximizing over signal samples, Approx equal to center frequency 13 kHz




2and Frequency Values Between 12 kHz and 14 kHz.

signal sample with the lowest T ∗2

value. In this case we see a difference between the filter

discussed above and the filter that would maximize the SNR for the smallest sphere

containing the discussed signal samples, which is shown in Figure (7.4). In that figure,

the central signal to the spherical set (also the robust filter for the spherical set that

contains the QR signal set, in the presence of thermal noise) had a T2∗ value equal to

the central value.

Figure (7.11) shows the filter that maximizes the SNR, in the presence of thermal noise,

of QR signal samples taken from the set of signals with a frequency of 12 kHz, a decaying

parameter, T ∗2

, of 600e-6 and phase values between -0.4 and 0.4 radians. The figure plots

the filter along with the QR signal samples with the maximum and minimum phase

values used. A 1000 QR signal samples were used to solve for the filter, which has a

frequency and decaying parameter equal to the QR signal samples, but a phase value

central to the minimum and maximum phase values that define the set.

222

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Robust Filter, Maximizing over Signal SamplesMin T

2

* QR Signal 0.00040059

Max T2

* QR Signal 0.00079991

Fig. 7.10 Robust Matched Filter, For Thermal Noise and the Samples from the Set ofQR Signals with Fixed Frequency and Phase and T ∗

2Values Between 400e-6 and 800e-6.

0 50 100 150 200 250 300 350 400 450−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Robust Filter, Maximizing over Signal Samples

Min Phase QR Signal −0.39894

Max Phase QR Signal 0.39888

Fig. 7.11 Robust Matched Filter, For Thermal Noise and the Samples from the Set of QRSignals with Fixed Frequency and T ∗

2and Phase Values Between -0.4 and 0.4 Radians.

223

The filters that maximize the SNR in the presence of thermal noise for the QR signal

samples as the frequency, decaying parameter, and phase are individually varied are

shown in Figures (7.8), (7.10) and (7.11) respectively. The next figure shows the robust

filter, maximizing the SNR, in the presence of thermal noise as all three parameters,

frequency, decaying parameter and phase are varied concurrently. Two thousand QR

signal samples were generated as the frequency, decaying parameter, and phase was var-

ied between 11.5 and 12.5 kHz, 400e-6 and 800e-6, and -0.4 and 0.4 radians respectively.

The robust matched filter for the 2000 QR signal samples generated, which maximizes

SNR in the presence of thermal noise, is shown in Figure (7.12). This filter was compared

to the central signal of the smallest sphere containing the 2000 QR signal samples (the

robust matched filter for the smallest spherical set of signals, that contains the QR signal

set, in the presence of thermal noise) in Figure (7.13).

In conclusion, the difference between the matched filters robust to QR signal samples

and the matched filter robust to the smallest spherical set containing the QR signals is

apparent. This reinforces the idea that using the robust matched filter for the smallest

spherical set containing the set of QR signals proposed in Section 7.2.1 and 7.4.1 is a

conservative approach.

7.6 Batch Whitened Robust Matched Filter

Now that the robust matched filter has been introduced in Section 7.5, it can be combined

with the gridding technique, which was first introduced in Chapter 5, and the whitening

filter introduced in Section 6.1 to create a batch whitened robust matched filter. The

224

0 50 100 150 200 250 300 350 400 450−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Robust Filter, Maximizing over Signal Samples,

Fig. 7.12 Robust Matched Filter, For Thermal Noise and the Samples from the Set ofQR Signals with Frequency Between 11.5 kHz and 12.5 kHz, T ∗

2Between 400e-6 and

800e-6 and Phase Between -0.4 and 0.4 Radians.

0 50 100 150 200 250 300 350 400 450−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Robust Filter, Maximizing over Signal Samples


Fig. 7.13 Robust Matched Filter, For Thermal Noise and the Set of QR Signal SamplesVersus the Robust Matched Filter, For Thermal Noise and the SmallestSphere Contain-ing the QR Signal Samples.

225

ultimate goal is to apply this filter to uncertainty in the general shape of the QR signal.

For demonstration purposes it was applied to uncertainty in the QR signal parameters

caused by coarser gridding.

The approach discussed in the above section is used to create a batch of robust matched

filters, where each is optimized for intervals of the gridding parameters, frequency, phase

and decaying parameter. If we define the following the following intervals

ωi ≤ ωbb ≤ ωi+1

φi ≤ φbb ≤ φi+1

T ∗2 i≤ T ∗

2 bb≤ T ∗

2 i+1.

as, ωinti , φinti and T ∗2 inti

respectively, then the larger intervals of possible values

ωlb ≤ ωbb ≤ ωub

−π ≤ φbb ≤ π

T ∗2 lb≤ T ∗

2 bb≤ T ∗

2 ub

can be represented as a number of smaller intervals strung together as follows,

226

[ωint1 , ωint2 , . . . , ωintNintω], (7.13)

[φint1 , φint2 , . . . , φintNφT], (7.14)

[T ∗2 int1

, T ∗2 int2

, . . . , T ∗2 intNintT

]. (7.15)

Designing a robust matched filter, optimized for each possible combination of smaller

intervals leads to Nintω ×Nintφ ×Nintφ robust matched filters.

The next step is to whiten each of the robust matched filter in the batch the same way

the phase cycled received echoes are whitened, which is discussed in Section 6.1. The

Nm whitening filters, obtained from the Nm received phase cycled echoes, xpc, using

the approach discussed in Section 6.1, are then applied to each of the robust matched

filters. As mentioned in Section 6.1, the whitening filter attempts to equally spread the

energy of the signal into the frequency spectrum. Therefore we chose to low pass the

whitened received signal with a low pass filter with a cutoff frequency of ωub, since we

know that the QR signal frequency is less than that cutoff. The type of low pass filter

used, is discussed in Section 6.1, and therefore will not be discussed here. If we denote

the matched filter robust to the lth, mth and nth intervals in Equations (7.13) (7.14) and

(7.15) as sr(l,m, n), then the whitened robust matched filter of the ith phase-cycled echo

is

227

swri(l,m, n)[k] =1

αwi(l,m, n)MWisr(l,m, n)[k]

where M is the low pass butter worth filter defined in section 6.2, Wi is the whitening

filter for the phase-cycled echo i, and αwi(l,m, n) is defined as

αwi(l,m, n) = ||MWisr(l,m, n)[k]||2.

Therefore, the whitened matched filter robust to the QR signals within the lth, mth, and

nth, frequency, phase and decay parameter intervals respectively, is the Nm whitened

low passed robust matched filters, swri(l,m, n), strung in time as follows

hr[k] = [swr0(l,m, n)[k], sw1(l,m, n)[k −Ns], . . . , swNm−1

(l,m, n)[k − (Nm− 1)Ns],

for k = 0, 1, . . . , Nm×Ns− 1. The inner product of the whitened, phase-cycled received

echo in Equation (6.11), with each of the Nintω ×Nintφ ×NintT possible whitened robust

matched filters is recorded in the vector Θ of the following form,

Θ = [θ1, . . . , θNintω×Nintφ×NintT]. (7.16)

Finally the maximum value of the vector Θ is compared to a threshold value θthreshold

and the filter corresponding to this maximum is the chosen filter out of the whitened

batch of robust filters. If the maximum value is greater than the threshold then we

would conclude that a QR Signal exists within the corrupted signal, otherwise we would

228

conclude that the received signal is just noise. If using one of the adaptive methods

to whitening, proposed in the previous chapter, gridding through the robust filters is

performed pre whitening and the selected filter is whitened according to the adaptive

whitening algorithm. As was mentioned in the previous chapter determining the value of

θthreshold is still a matter that requires further work. There are several possible ways of

determining the value of θthreshold, one of which, is to collect a measurement of the noise

and use it to determine the value of θthreshold. The next section compares the performance

of the batch robust matched filter, without whitening, to the batch matched filter in the

presence of thermal noise using simulation data. The section following that compares the

performance of the batch whitened robust matched filter to the batch whitened matched

filter in the presence of thermal and AM noise using both simulation data. Both these

filters were whitened with a 6th order whitening filter estimated using the autocorrelation

method.

7.7 Batch Robust Matched Filter in Presence of Thermal Noise

In this section, the performance of the robust matched filter, designed using the approach

above, is compared to the performance of the batch matched filter, discussed in Chapter

5, in the presence of the thermal noise using simulation data. The section starts by

analyzing filters robust to uncertainties in frequencies first, in the following subsection.

The section then moves on to analyzing filters robust to uncertainties in all three QR

signal parameters (frequency, phase and decaying parameter).

229

7.7.1 Frequency Robust Batch Matched Filters in the Presence of Thermal

Noise

0 50 100 150 200 250 300 350 400 450−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Robust Filter, Maximizing over signal samples, Approx equal to center frequency 12e3



Fig. 7.14 Comparing the Robust Matched Filter for the set covering the frequencies 11.5kHz to 12.5 kHz, to the sample signals with the largest and smallest frequencies.

At first, 1000 experiments of QR plus thermal noise with an SNR of −16 dB and a 1000

experiments of only thermal noise were generated, where the frequency of the QR signal

present in an experiment is a random value between 5 kHz and 15 kHz and the decaying

parameter and the phase remained unchanged as 600e-6 and 0 respectively. To analyze

the performance of the robust matched filter it was compared to three batch matched

filters, using the above data. One batch matched filter had a frequency gridding interval

of 100 Hz, another used a frequency gridding interval of 1 kHz, and the third used a

frequency gridding interval of 2 kHz. All filters assumed that the decaying parameter

and the phase are known. The performance of these filters were compared to two batches

of robust matched filters, where the first batch of filters composed of 21 matched filters

230

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm

Energy DetectorBatch Matched Filter, 1 kHz Grid, Fixed T

2

* & Phase

Batch Matched Filter, 100 Hz Grid, Fixed T2

* & Phase

Gridding a batch of Robust Matched Filters 1 kHz appartBatch Matched Filter, 2 kHz Grid, Fixed T

2

* & Phase

Gridding a batch of Robust Matched Filters 2 kHz appart

Fig. 7.15 Simulation, SNR = -16 dB, Thermal Noise, Batch of Robust Matched Filtersversus using a Batch Matched Filter, The QR signal parameters, decaying parameter,T ∗

2, and phase ,φ, remained constant while the frequency varied between 5 kHz to 15

kHz.

robust to the following frequency ranges [-0.5 kHz, 0.5 kHz], [0.5 kHz, 1.5 kHz], . . . , [19.5

kHz, 20.5 kHz]. As an example, one of these robust matched filters, the one robust to

the set covering the frequency interval [11.5 kHz, 12.5 kHz], is shown in Figure (7.14).

The filter is compared to the samples with the largest and smallest frequencies in the set.

The robust matched filters maximizes the performance for the worst case signal among

the signals sampled out of the set. The second batch of filters composed of 10 matched

filters robust to the following ranges [0 kHz, 2 kHz], [2 kHz, 4 kHz], . . . , [18 kHz, 20

kHz]. The performance of these filters were compared using ROC plots in Figure (7.15).

As was expected, fine gridding using a batch of matched filters with frequencies 100 Hz

apart performed the best. When comparing the performance of the batch of 21 matched

filters, with frequencies 0 kHz, 1 kHz, . . . , 20 kHz, to the performance of the batch of

21 robust match filters, robust to the ranges [-0.5 kHz, 0.5 kHz], [0.5 kHz, 1.5 kHz], . . . ,

231

[19.5, 20.5 kHz], their performance is more or less equal. The performance of these two

detectors are shown in the green and blue curves. As the frequency intervals between

filters in the batch of matched filters increased the performance of the batch of matched

filters decreased. This is apparent when comparing the black curve, where the batch of

matched filters are 2 kHz apart, to the green curve, where the batch of filters are only 1

kHz apart. Though as the interval of frequencies the robust matched filter is robust to is

increased from 1 kHz (the blue curve) to 2 kHz (the cyan curve), the performance drop

is insignificant when compared to the drop seen when using batch matched filters. Since

the robust matched filters that are robust to 2 kHz frequency intervals cover twice the

range of frequencies as the robust matched filters robust to 1 kHz intervals, the number

of filters required in the batch to cover the complete range of frequencies is reduced by a

factor of two leading to half the computation time required. Therefore, if computation

time is a factor and fine gridding is not an option then using a batch of robust matched

filter is superior to using a batch of matched filters, which is apparent when comparing

the cyan curve to the black curve.

7.7.2 Robust Batch Matched Filters in the Presence of Thermal Noise

Another set of data was generated was generated, where the QR signals varied in fre-

quency, phase and decaying parameter and not only frequency. A 1000 experiments of

QR plus thermal noise and 1000 experiments of only thermal noise were generated. The

QR signal present at a given experiment, had a random frequency varied between 5 kHz

and 15 kHz, a random phase between −π/2 and π/2 and a random decaying parameter

232

between 400e-6 and 800e-6. This data was used to compare the performance of two batch

robust matched filters and three batch matched filters in Figure (7.16).

As expected, the best performing filter was the batch matched filter with a fine grid

of, 100 Hz, 0.1 radians and 100e-6 in between grid ding values for frequency, phase and

decaying parameter respectively. Such a batch of filters consists of Ndω ×Ndφ ×NdT =

201× 63× 5 = 63, 315 filters, which is the largest batch among filters compared in this

figure. The in-between grid values for frequency, phase, and decaying parameter for the

other two filters were 1 kHz, 0.2 radians, and 200e-6 respectively for one, and 2 kHz,

0.2 radians, and 200e-6 for the other. The number of filters in the above two batches of

matched filters were Ndω × Ndφ × NdT = 21 × 32 × 3 = 2016 and Ndω × Ndφ × NdT =

11× 32× 3 = 1056 respectively. Figure (7.16) shows that as the values in-between grid

parameters of the batch matched filters increased the performance dropped, which is

apparent from comparing the magenta, green and black curves. The first batch robust

matched filter used consisted of a group of filters robust to QR signals with frequency,

phase and decaying parameter that fall within the intervals of size 1 kHz, 0.2 radians and

200e-6 respectively. This batch robust matched filter consists of Nintω ×Nintφ ×NintT =

20× 32× 3 = 1920, individual filters, which is comparable to the number of filters in the

batch matched filter with 2 kHz spacing in-between frequency values. Comparing the

green curve to the blue curve shows that in this case the batch matched filter and the

batch robust matched filter have approximately equal performance. The second batch

robust matched filter consisted of a group of filters robust to QR signals with frequency,

phase and decaying parameter that fall within the intervals size 2 kHz, 0.2 radians and

233

200e-6 respectively. This leads to Nintω ×Nintφ ×NintT = 10 × 32 × 3 = 960, filters in

the batch, which is comparable to the number of filters in the batch matched filter with

1 kHz frequency spacing, the black curve. When comparing the batch robust matched

filter to the batch matched filter in this case, the batch robust matched filter, the cyan

curve, outperforms the batch matched filter, the black curve.

In conclusion, Figure (7.16) shows that batch robust matched filter and the batch

matched filter perform almost equally, when the intervals the filters are robust to and

the in-between grid values are small, the green and blue curve. On the other hand when

these values are increased and the number of filters in the batches are reduced, the batch

robust matched filter outperforms the batch matched filter, comparing the cyan curve

to the black curve.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm

Energy Detector

Batch Matched Filter, Grid Intervals, 100Hz, 100e−6 Decaying Parameter, 0.1 Radians

Batch Matched Filter, Grid Intervals, 1 kHz, 200e−6 Decaying Parameter, 0.2 Radians

Batch Robust Matched Filter, Grid Intervals, 1 kHz, 200e−6 Decaying Parameter, 0.2 Radians



Fig. 7.16 Simulation, SNR = -12 dB, Thermal Noise, Batch of Robust Matched Filtersversus using a Batch Matched Filter, The QR signal parameters varied as follows, fre-quency between 5 kHz to 15 kHz, decaying parameter T ∗

2between 400e-6 and 800e-6,

and phase φ between −π/2 to π/2.

234

7.8 Batch Whitened Robust Matched Filter in the Presence of AM

and Thermal Noise

The purpose of this section is to compare gridding using filters robust to sets of QR

signals with varying QR parameters versus gridding using filters matched to specific

QR signals, in the presence of AM and thermal noise. The section compares the two

detectors, the batch whitened matched filter and the batch whitened robust matched

filter using simulation data. The section then moves on to demonstrating why comparing

the filters using the available experimental data is aimless.

7.8.1 Simulation Data in the Presence of AM and Thermal Noise

The simulation data used in this section was a 1000 experiments of QR plus AM and

thermal noise and a 1000 experiments of only AM and thermal noise. The QR signal

present in an experiment had a frequency that varied between 5 kHz and 15 kHz, a

phase between −π/2 and π/2, and a decaying parameter between 400e-6 and 800e-6.

The carrier of the AM signal present in an experiment had a fixed frequency of 10 kHz.

To start, this data was used to compare the performance of two batch robust matched

filters and three batch matched filters in Figure (7.17). The same data is then used

to compare the performance of the batch whitened robust matched filter to the batch

whitened matched filter in Figure (7.18).

In Figure (7.17) the best performing filter, was the batch matched filter with a fine grid

of, 100 Hz, 0.1 radians and 100e-6 in between gridding values for frequency, phase and

decaying parameter respectively. As was stated above, this batch of filters consists of

235

Ndω ×Ndφ ×NdT = 201× 63× 5 = 63, 315 filters and therefore this filter has the largest

computational time. The in-between grid values for frequency, phase, and decaying

parameter for the other two filters were 1 kHz, 0.2 radians, and 200e-6 respectively for

one, and 2 kHz, 0.2 radians, and 200e-6 for the other. The number of filters in the

above two batches of matched filters were Ndω ×Ndφ ×NdT = 21 × 32 × 3 = 2016 and

Ndω ×Ndφ ×NdT = 11× 32× 3 = 1056 respectively.

The first batch robust matched filter used consisted of a group of filters robust to QR

signals with frequency, phase and decaying parameter that fall within the intervals of size

1 kHz, 0.2 radians and 200e-6 respectively. This batch robust matched filter consists of

Nintω ×Nintφ×NintT = 20×32×3 = 1920, individual filters, which is comparable to the

number of filters in the batch matched filter with 1 kHz spacing in-between frequency

values. Comparing the green curve to the blue curve shows that in this case the batch

matched filter and the batch robust matched filter have more or less equal performance.

The second batch robust matched filter consisted of a group of filters robust to QR signals

with frequency, phase and decaying parameter that fall within intervals of size 2 kHz, 0.2

radians and 200e-6 respectively. This leads to Nintω ×Nintφ×NintT = 10×32×3 = 960,

filters in the batch, which is comparable to the number of filters in the batch matched

filter with 2 kHz frequency spacing, the black curve. When comparing the batch robust

matched filter to the batch matched filter in this case, the batch robust matched filter,

the cyan curve, outperforms the batch matched filter, the black curve. Next these filters

will integrate the whitening approach and their performance will be compared using the

same set of data.

236

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm

Energy Detector

Batch Matched Filter, Grid Intervals, 100 Hz, 100e−6 Decaying Parameter, 0.1 Radians





Fig. 7.17 Simulation, SNR = -18 dB, BPWGAM Noise, Batch of Robust Matched Fil-ters versus using a Batch Matched Filter, The QR signal parameters varied as follows,frequency between 5 kHz to 15 kHz, decaying parameter T ∗

2between 400e-6 and 800e-6,

and phase φ between −π/2 to π/2.

As expected in Figure (7.18) the best performing filter, was the batch whitened matched

filter with a fine grid of, 100 Hz, 0.1 radians and 100e-6 in between gridding values for

frequency, phase and decaying parameter respectively. As was stated above, this batch

of filters consisted of 63,315 filters, where each one had to be whitened with the same

whitening filter used to whiten the received phase cycled echo. For the other two batch

whitened matched filters, the in between grid values for frequency, phase, and decaying

parameter were 1 kHz, 0.2 radians, and 200e-6 respectively for one, and 2 kHz, 0.2

radians, and 200e-6 for the other. The number of filters in the above two batches of

matched filters were 2016 and 1056 respectively.

In reference to the batch whitened robust matched filters, the first batch used consisted

of a group of filters robust to QR signals with frequency, phase and decaying parameter

that fall within the intervals of size 1 kHz, 0.2 radians and 200e-6 respectively. This

237

batch of robust matched filter consisted of 1920, individual filters, which is comparable

to the number of filters in the batch matched filter with 1 kHz spacing in-between

frequency values. Comparing the green curve to the blue curve, shows that in this

case the batch whitened matched filter and the batch whitened robust matched filter

have more or less equal performance. The second batch whitened robust matched filter

consisted of a group of filters robust to QR signals with frequency, phase and decaying

parameter that fall within intervals of size 2 kHz, 0.2 radians and 200e-6 respectively.

This batch contains 960 filters, which is comparable to the number of filters in the batch

whitened matched filter with 2 kHz frequency spacing, the black curve. When comparing

the batch whitened robust matched filter to the batch whitened matched filter in this

case, the batch whitened robust matched filter, the cyan curve, outperforms the batch

whitened matched filter, the black curve.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b C

orr

ect D

ete

ction

Prob False Alarm

Energy Detector

Batch Whitened Matched Filter, Grid Intervals, 100 Hz, 100e−6 Decaying Parameter, 0.1 Radians

Batch Whitened Robust Matched Filter, Grid Intervals, 1 kHz, 200e−6 Decaying Parameter, 0.2 Radians

Batch Whitened Robust Matched Filter, Grid Intervals, 2 kHz, 200e−6 Decaying Parameter, 0.2 Radians

Batch Whitened Matched Filter, Grid Intervals, 1 kHz, 200e−6 Decaying Parameter, 0.2 Radians

Batch Whitened Matched Filter, Grid Intervals, 2 kHz, 200e−6 Decaying Parameter, 0.2 Radians

Fig. 7.18 Simulation, SNR = -18 dB, BPWGAM Noise, Batch of Whitened RobustMatched Filters versus Batch Whitened Matched Filters, QR signal parameters variedas follows, frequency between 5 kHz to 15 kHz, decaying parameter T ∗

2between 400e-6

and 800e-6, and phase φ between −π/2 to π/2.

238

Simulation results, have shown that in cases of fine gridding, a batch whitened robust

matched filter and a batch whitened matched filter, with a comparable number of filters

in the batch, perform more or less equally and therefore robustness in the QR signal

parameters is unnecessary. Though the ultimate aim of robust filters is to be applied to

uncertainty in the QR signal shape, while fine gridding in the QR signal parameters. In

cases of unfine gridding in the QR signal parameters the batch whitened robust (robust

to QR signal parameters) matched filter outperforms a batch whitened matched filter,

with a comparable number of filters in the batch. Therefore, the batch whitened robust

matched filter (robust to uncertainties in the QR signal parameters) should only replace

the batch whitened matched filter, when fine gridding of the QR parameters is not an

option. In this case decreasing the number of filters in the batch of whitened robust

matched filters, does not degrade performance the performance as it does for the batch

whitened matched filter.

Unfortunately due to the lack of experimental data with varying QR signal parameters,

this phenomena can not be tested on experimental data and the following section explains

why.

7.8.2 Experimental Data in the Presence of AM and Thermal Noise

The goal of the robust matched filter is to optimize performance for a set of QR signals.

The set represents uncertainty in the signal and can be in the general shape of the QR

signal or in the QR parameters. The experimental data collected lacks this uncertainty

or variance in the QR parameter or the signal’s general shape so it is pointless to use to

analyze the performance of robust filters. Using simulation data above we generated QR

239

signals with varying QR signal parameter and we have shown that on average a batch

of robust filters, robust to uncertainties in QR signal parameters, should only be used

when fine grading can not be used.

The experimental data collected for this thesis consisted of four different sets of data,

where each set’s QR signal had a fixed frequency, phase and decaying parameter. The

lack of variance in the QR signal parameters of the experimental data, prevents us from

properly comparing the performance of the batch whitened robust matched to the batch

whitened matched filter. The following example illustrates this point.

Let the experimental set contain a QR signal with frequency, phase and decaying pa-

rameter of 12.1 kHz, 0 radians and 500e-6 respectively. To illustrate, let us assume he

following five batches of whitened filters are applied, where the first three are matched

filters and the last two are robust matched filters.

1. In between grid values: frequency ω∆ = 2π×100, phase φ∆ = 0.1 radians, decaying

parameter T ∗2 ∆

= 100e− 6.

2. In between grid values: frequency ω∆ = 2π × 1000, phase φ∆ = 0.2 radians,

decaying parameter T ∗2 ∆

= 200e− 6.

3. In between grid values: frequency ω∆ = 2π × 2000, phase φ∆ = 0.2 radians,

decaying parameter T ∗2 ∆

= 200e− 6.

4. Filters robust to intervals of length: frequency 2π × 1000, phase 0.2 radians, de-

caying parameter 200e-6.

240

5. Filters robust to intervals of length: frequency 2π × 2000, phase 0.2 radians, de-

caying parameter 200e-6.

From the discussion of batch robust matched filters in Chapter 6, we would expect the

performance of the batch of filters on average to drop as the in between grid values

increase. Therefore, the first batch is expected perform better than the second and the

second is expected to perform better than the third, as the QR signal parameters varied.

Though, since the frequency, phase and decaying parameter of the QR signal present in

the example above are fixed to 12 kHz, 0 radians, and 500e-6, its parameters are always

100 Hz, 0.1 radians, and 100e-6 away from one of the filters in the batch, for both the

second and third batch. Therefore the performance of the second batch should not be

significantly better than the third one.

On the other hand when comparing the last two batches of whitened robust matched

filters to each other, the first batch will always outperform the second batch since the

second batch of filters are robust to larger sets of signals. The second and fourth batch of

filters contain almost the same number of filters, while the third and fifth batch contain

almost the same number of filters.

In other words, for QR signals with random signal parameters, increasing the in between

grid values when using a batch whitened matched filter, and increasing the length of the

intervals the filters are robust to when using a batch whitened robust matched filter,

causes a drop in performance. On the other hand, when the QR signal parameters of the

signal present are fixed (the case with experimental data) this is not always the case. For

specific cases, where the distance of the signal’s parameters to the parameters of one of

241

the filters remain unchanged, increasing the in between grid values does significantly drop

performance. Therefore, to properly compare the performance of the batch whitened

matched filter to the batch of whitened robust matched filter a set of QR signals with

random parameters is needed.

The batch whitened robust matched filter discussed in this chapter can be applied to any

uncertainty in the QR signal parameters. As mentioned earlier, the ultimate goal is to

apply this filter to uncertainty the general shape of the QR signal and use fine gridding

to minimize uncertainty in the QR signal parameters.

242

Chapter 8

Conclusion

In conclusion, the matched filter approaches to QR signal detection, proposed in this

thesis, have shown significant improvements over the generic energy detector. Among

the different matched filter approaches, the batch whitened matched filter (discussed in

Chapter 6) with a properly chosen filter order performed best.

The motivation behind using a matched filter approach was presented in Chapter 4

by comparing the test statistic of the matched filter, under the assumption that the

QR signal is known, to that of the energy detector. This is an unrealistic case, but it

demonstrated an advantage to using a matched filter approach in QR detection. The

distributions of the test statistics in the presence of a QR signal and in the absence of

one, are more separable in the case of the matched filter when compared to the energy

detector.

This was followed by Chapter 5, which presented a gridding approach to the matched

filter referred to as the batch matched filter, which does not assume any knowledge

about the QR signal present. The approach utilizes the knowledge that the unknown

QR signal parameters, frequency, phase and decaying parameter, fall within lower and

upper bounds and discretizes the interval of admissible values to create a finite set of

possible values, for each of the QR signal parameters. These sets, one for each of the three

QR signal parameters, form a grid, where each point refers to a possible combination of

243

values for the QR signal parameters. The number of points on grid is dependent on the

number of discrete values in each of the three QR signal parameters. A set of filters are

then designed, where each is matched to one of the grid points, which are used to detect

the presence of the QR signal. The performance of this filter is dependent on how finely

discretized the unknown QR signal parameters are. A finely gridded batch matched

filter significantly outperforms the energy detector, though it has been shown that the

improvement gained from fine gridding reaches a point of saturation. The computational

time increases as the grid becomes finer. This chapter also introduces an adaptive way

to gridding that can significantly reduce the number of filters in a batch and achieve the

same performance as the brute force method. The adaptive method performs a global

coarser grid search, which is followed by a local finer grid around the grid point selected

by the global grid. Another important point to make, is the fact that the batch matched

filter is ignorant to the type of noise present and does not utilize any information about

the noise in detecting the QR signal.

The filter discussed above is then combined with a whitening filter in Chapter 6, resulting

in a filter referred to as the batch whitened matched. The whitening filter is the inverse

of the noise model estimate, estimated under the assumption that the present noise is

wide sense stationary across the length of an echo, and uses it to whiten the received

signal and the batch of filters. The WSS assumption is a reasonable one, since the length

of an echo is less than 2 ms. As was shown in Chapter 6, estimating a whitening filter

is equivalent to estimating the covariance matrix of the noise present, since one can

be derived from the other. With the properly chosen whitening filter order, this filter

244

outperformed both the batch matched filter and the energy detector, which was expected

since it not only attempts to match to the QR signal present but also the noise statistics.

In estimating the whitening filter the chapter proposed two methods, the autocorrelation

and the covariance methods. The chapter also proposed an adaptive method to selecting

the whitening filter, the first would select the whitening filter order that would least

affect the QR signal while the second used the minimum description length algorithm

to select the filter order. The chapter compared the adaptive methods to selecting the

whitening filter, by comparing the batch whitened matched filters, whitened with a filter

of order chosen by one of the adaptive order selecting methods and estimated using one

of the modeling estimates. This results in four different filters, and according to the

experimental data, the filter that used the covariance method to model the filter with

an order that least affects the QR signal consistently outperformed the batch matched

filter. The results from the simulation data did not point to one filter that consistently

outperformed the remaining filters. Integrating the whitening filter with the gridding

approach further increases the computational time required for the filter to make a

decision, since not only is the received signal whited but so is each of the filters in the

batch. Though this can be significantly reduced by using the adaptive gridding approach

introduced in Chapter 5.

Chapter 7 presents the batch whitened robust matched filter, which combines the whiten-

ing filter with a batch of robust matched filter, where each optimizes SNR for a set of

QR signals instead of a single signal. The ultimate goal is to design a batch of filters,

each matched to particular QR signal parameters, robust to uncertainty in the general

245

shape of the signal. Finely gridding the QR signal parameters reduces the effect of un-

certainty in the QR signal parameters, and therefore robustness to QR signal parameters

is unnecessary. Due to the lack of experimental data with variances in the general shape

of the QR signal, designing filters robust to uncertainty in the general shape QR signal

was not possible. The chapter lays out the tools needed to design a batch of whitened

robust (robust to uncertainty in general QR signal shape) matched filters. For demon-

stration, the chapter compared the performance of the batch whitened matched filter to

the batch whitened robust (robust to uncertainty in QR parameters) matched filter, and

has shown that when fine gridding can be performed robustness to the QR parameters

is unnecessary.

The scenario approach, used in the designing filters robust to variances in the QR pa-

rameters can be used in designing filters robust to any type of uncertainty in the QR

signal, as long as signal samples are obtainable. Variances in the general shape in the

QR signal can be caused by narrow bandwidth filters and coils.

8.1 Future Work

There are several areas one can further address, which will be discussed in this section.

One of which is to further analyze the modeling methods used to estimate the noise

models. Another, is to further analyze the adaptive methods used to choosing the

whitening filter order. These two points can help one develop a method, where the

output of the four different batch adaptive whitened matched filters compared in Section

(6.4) can be used to determine the presence of a QR signal. This idea stems from the

246

fact that for any of the data sets (simulation and experimental) used to compare the

four filters, at least one outperformed the batch matched filter.

Further work also needs to be done with the regards to the robust filters. First, experi-

mental data with variances in the general shape of the QR signal needs to be collected.

This data can be used as signal samples to design filters robust to variances in the general

shape of the QR signal. The performance of these robust filters should then be compared

to the performance of the matched filters using data with variances in the general QR

signal shape.

247

Appendix A

Mean and Variance of Energy Detector Test Statistic,

in the Presence of AM and Thermal Noise

From section (4.1) the test statistic of the energy detector is of the following form

T (x) =Ns−1∑k=0

x[k]2, (A.1)

where x[k] is the received signal. The test statistic is the sum of Ns independent random

variables and since Ns is large, it can be approximated as a gaussian random variable.

To find the mean and variance of the test statistic, we first have to find the mean and

variance of the received signal squared for a certain value of k, x[k]2.

Under the assumption that received signal is the sum of averaged QR, AM and thermal

signals, the received signal squared for a certain value of k is equal to,

x2[k] =s2avg

[k] + η2avg

[k] + n2avg

[k] + 2savg[k]ηavg[k]

+ 2savg[k]navg[k] + 2ηavg[k]navg[k]

248

where navg[k] is a random variable and has the distribution N(0, σ2avg,tn

). The variable

ηavg[k] is also a normal random variable that is also a function of the random phase of

the AM signal, φ. The first moment of the random variable ηavg[k], µη[k], is

µη[k] =Aη

2NeNm

Nm−1∑i=0

Ne∑j=1

[cos (∆ωAM (D(i, j, k) + φ)− (A.2)

cos (∆ωAM (D(i, j, k) + τd) + φ)]

while the second moment, σ2η[k], is

σ2η[k] =E{η2

avg[k]} − µ2

η[k] (A.3)

=

(A2η

(2NeNm)2

)E

Nm−1∑

i=0

Ne∑j=1

(1 +m[D(i, j, k)]) cos (∆ωAM (D(i, j, k)) + φ)−

(1 +m[D(i, j, k)]) cos (∆ωAM (D(i, j, k) + τd) + φ)

)2−(

A2η

(2NeNm)2

)Nm−1∑i=0

Ne∑j=1

cos (∆ωAM (D(i, j, k)) + φ)−

cos (∆ωAM (D(i, j, k) + τd) + φ)

)2

=

(A2η

(2NeNm)2

)Nm−1∑i=0

Ne∑j=1

σ2m

[cos2 (∆ωAM (D(i, j, k) + φ)+

cos2 (∆ωAM (D(i, j, k) + τd) + φ)]

249

Since the first and second moments of a normal random variable completely describe the

distribution, we can move on to calculating the mean and variance of the received signal

squared for a certain value k, x2[k]. The mean value of the received signal squared,

µx2 [k], as a function of k, would be

µx2 [k] = E{x2[k]} (A.4)

= s2avg

[k] + (σ2η[k] + µ2

η[k]) + σ2

avg,tn+ 2savg[k]µη[k],

where σ2avg,tn

= σ2tn/(2NeNm) and µη[k] and σ2

η[k] are defined in Equations (A.2) and

(A.3) respectively. The variance of the received signal squared at a specific value k,

250

σ2x2 [k] =V ar{x2[k]} (A.5)

=V ar{η2avg

[k]}+ V ar{n2avg

[k]}+ 4s2avg

[k]V ar{ηavg[k]}+

4s2avg

[k]V ar{navg[k]}+ 4V ar{ηavg[k]navg[k]}+

2Cov{η2avg

[k], n2avg

[k]}+ 2Cov{η2avg

[k], 2savg[k]ηavg[k]}+

2Cov{η2avg

[k], 2savg[k]navg[k]}+

2Cov{η2avg

[k], 2ηavg[k]navg[k]}+

2Cov{n2avg

[k], 2savg[k]ηavg[k]}+

2Cov{n2avg

[k], 2savg[k]navg[k]}+

2Cov{n2avg

[k], 2ηavg[k]navg[k]}+

2Cov{2savg[k]ηavg[k], 2savg[k]navg[k]}+

2Cov{2savg[k]ηavg[k], 2ηavg[k]navg[k]}+

2Cov{2savg[k]navg[k], 2ηavg[k]navg[k]}

Some of the values in Equation (A.5) can be further simplified as follows. The vari-

ables ηavg[k] and navg[k] are normal random variables with the following distributions

N(µη[k], σ2η[k]) and N(0, σ2

avg,tn[k]), we therefore can use their first and second moments

to find all their higher moments. The following covariance terms are equal to zero due

to the random variables being independent.

251

Cov{η2avg

[k], n2avg

[k]}, Cov{η2avg

[k], 2savg[k]navg[k]}

Cov{n2avg

[k], 2savg[k]ηavg[k]}, Cov{2savg[k]ηavg[k], 2savg[k]navg[k]}

The other variances and covariances in Equation (A.5) will be further simplified sepa-

rately, before simplifying the whole expression, as follows.

V ar{η2avg

[k]} = E{η4avg

[k]} − (E{η2avg

[k]})2

= (µ4η[k] + 6µ2

η[k]σ2

η[k] + 3σ4

η[k])− (µ2

η[k] + σ2

η[k])2

= 2σ4η[k] + 4µ2

η[k]σ2

η[k]

V ar{n2avg

[k]} = E{n4avg

[k]} − (E{n2avg

[k]})2

= (3σ4avg,tn

[k])− (σ2avg,tn

[k])2

= 2σ4avg,tn

[k]

252

V ar{ηavg[k]navg[k]} = E{η2avg

[k]n2avg

[k]} − (E{ηavg[k]navg[k]})2

= E{η2avg

[k]}E{n2avg

[k]}

= (σ2η[k] + µ2

η[k])σ2

avg,tn

Cov{η2avg

[k], 2savg[k]ηavg[k]} =2savg[k](E{η3avg

[k]}−

E{η2avg

[k]}E{ηavg[k]})

=2savg[k]((µ3η[k] + 3µη[k]σ2

η[k])−

(µ3η[k] + µη[k]σ2

η[k]))

=4savg[k]µη[k]σ2η[k]

Cov{η2avg

[k], 2ηavg[k]navg[k]} =2E{navg[k]}[E{η3avg

[k]}−

E{η2avg

[k]}E{ηavg[k]}]

=0

253

Cov{n2avg

[k], 2savg[k]navg[k]} =2savg[k][E{n3avg

[k]}−

E{n2avg

[k]}E{navg[k]}]

=2savg[k]E{n3avg

[k]}

=0

Cov{n2avg

[k], 2ηavg[k]navg[k]} =2E{ηavg[k]}[E{n3avg

[k]}−

E{n2avg

[k]}E{navg[k]}]

=2E{ηavg[k]}E{n3avg

[k]}

=0

Cov{2savg[k]ηavg[k], 2ηavg[k]navg[k]} =4savg[k][E{η2avg

[k]}E{navg[k]}

− E{ηavg[k]}2E{navg[k]}]

=0

254

Cov{2savg[k]navg[k], 2ηavg[k]navg[k]} =4savg[k][E{n2avg

[k]}E{ηavg[k]}

− E{navg[k]}2E{ηavg[k]}]

=4savg[k]σ2avg,tn

[k]µη[k]

Hence the Equation (A.5) can be simplified into the following,

σ2x2 [k] =(2σ4

η[k] + 4µ2

η[k]σ2

η[k]) + (2σ4

avg,tn[k]) + (4s2

avg[k]σ2

η[k])+ (A.6)

4s2avg

[k]σ2avg,tn

+ (4(σ2η[k] + µ2

η[k])σ2

avg,tn)+

8savg[k]µη[k]σ2η[k] + 8savg[k]σ2

avg,tn[k]µη[k]

The two expressions in Equations (A.6) and (A.4) represent the variance and the mean

respectively of the received signal squared for a certain value k. These two expressions

are functions of the first and second moments, µη[k] and σ2η[k] shown in Equations (A.2)

and (A.3), of ηavg[k]. Since µη[k] and σ2η[k] are functions of the random variable φ, so

is the mean and variance of squared received signal at k. The random variable φ is the

phase of the AM signal’s carrier which varies from experiment to experiment, therefore

the expected value of Equations (A.6) and (A.4) with respect to φ must be taken. Some

useful expected values with respect to φ follow,

255

E{µη[k]}φ = 0 E{µ2η[k]}φ = 0

E{σ2η[k]}φ =

A2η

2NeNm

σ2m

2E{σ4

η[k]}φ =

A4η

(2NeNm)2

3σ4m

8

E{µ2η[k]σ2

η[k]}φ = 0 E{µη[k]σ2

η[k]}φ = 0

Therefore the expected values of µx2 [k] and σ2x2 [k] with respect to φ are

E{µx2 [k]}φ = s2avg

[k] +A2η

2NeNm

σ2m

2+

σ2tn

2NeNm(A.7)

E{σ2x2 [k]}φ =

A4η

(2NeNm)2

3σ4m

4+ 2

σ4tn

(2NeNm)2 + (A.8)

4s2avg

[k]

(A2η

2NeNm

σ2m

2+

σ2tn

2NeNm

)+

4

(A2η

2NeNm

σ2m

2

)σ2tn

2NeNm.

Summing the expected values of the mean and variance, in Equations (A.7) and (A.8),

of the squared received signal over k = 0, 1, . . . , Ns−1, leads to the mean and variance of

the energy detector’s test statistic. The mean and variance of the distribution of the false

alarm is achieved by setting the QR signal in Equations (A.7) and (A.8) respectively to

zero.

256

µE,FA = Ns

(A2η

2NeNm

σ2m

2+

σ2tn

2NeNm

)

σ2E,FA

= Ns

(A4η

(2NeNm)2

3σ4m

4+

2σ4tn

(2NeNm)2

)

While the mean and variance of the distribution of the true positive is achieved by

summing the values in Equations (A.7) and (A.8) respectively over k = 0, 1, . . . , Ns − 1.

This leads to the following values,

µE,D = µE,FA + β2εs

σ2E,D

= σ2E,FA

+ 4β2εs

(A2η

2NeNm

σ2m

2+

σ2tn

2NeNm

)

257

Appendix B

Mean and Variance of Matched Filter Test Statistic,

in the Presence of AM and Thermal Noise

From section (4.2) the test statistic of the matched filter detector is of the following form

T (x, h) =N−1∑k=0

x[k] ∗ h[k],

the inner product of x[k] and h[k], where N = Nm ×Ns. This is a sum of independent

gaussian random variables, which is a gaussian random variable with a specific mean

and variance. To find the mean and variance of the test statistic, we first have to find

the mean and variance of the product x[k] ∗ h[k] for a specific value of k, under the

assumption that a QR signal exists within the received signal x[k], has the following

statistics,

E{x[k] ∗ h[k]} =β ∗ [Ae(−|−T/2+k∆|/T2∗) cos (∆ωr(−T/2 + k∆))]2+ (B.1)

h[k] ∗ (Aη2Ne

Ne∑i=1

[cos (∆ωAM (k∆ + (i− 1)τ) + φ)+

cos (∆ωAM (k∆ + (i− 1)τ + τd) + φ)])

258

V ar{x[k] ∗ h[k]} =h[k]2 ∗ (A2η

4N2e

Ne∑i=1

σ2m∗ (B.2)

[cos2 (∆ωAM (k∆ + (i− 1)τ) + φ)+

cos2 (∆ωAM (k∆ + (i− 1)τ + τd) + φ)] + σ2tn

)

where φ is a random variable that is uniformly distributed between −π and π. Taking

the expectation of the mean and variance of x[k]∗h[k] in Equations (B.1) and (B.2) with

respect to φ leads to the following statistics,

E{E{x[k] ∗ h[k]}}φ = β ∗ [Ae(−|−T/2+k∆|/T2∗) cos (∆ωr(−T/2 + k∆))]2 (B.3)

E{V ar{x[k] ∗ h[k]}}φ = h[k]2 ∗ (A2η

4Neσ2m

+ σ2tn

) (B.4)

Summing the expected values of the mean and variance, in Equations (B.3) and (B.4),

of the squared received signal over k = 0, 1, . . . , Nm × Ns − 1, leads to the mean and

variance of the matched filter’s test statistic. The mean and variance of the distribution

of the false alarm, achieved by setting the QR signal in Equations (B.3) and (B.4) to

zero and summing over k = 0, 1, . . . , Nm ×Ns − 1, are

µMF,FA = 0

259

σ2MF,FA

= ε ∗ (σ2AM

+ σ2tn

)

where σ2tn

and ε are defined in Equations (4.13) and (4.2) and σ2AM

is defined as

σ2AM

= (A2η/(4 ∗Ne)) ∗ σ

2m. (B.5)

On the other hand the mean and variance of the distribution of correct detection,

achieved by summing Equations (B.3) and (B.4) over k = 0, 1, . . . , Nm ×Ns − 1, are

µMF,D = β ∗ ε

σ2MF,D

= ε ∗ (σ2AM

+ σ2tn

)

where σ2tn

, σ2AM

and ε are defined in Equations (4.13) (B.5) and (4.2).

260

Appendix C

MATLAB Code

C.1 Data Generation Function

The following Matlab code was used to generate the simulated data used to test the

various algorithms.

%% m-file to simulate generation Sevaral QR and Noise Signals in SLSE Exp.

%% 11 April 2012

function [qr,qrtn,tn,qramtn,amtn]=sim_data_SD_BPWGAM(wstar,fo,T2star,T2eff,QR_amp,w_AM,AM_mag,AM_st,therm_std,Nm,Ne,Ns,fs,T_daq,b);

% Sample parameters

% wstar % QR frequency

% T2star % The decaying parameter

% T2eff % Decay of echoes across sequence

% QR_amp % 3.5 g Piperazine producses a 25 mV avg signal

% w_AM % location of AM carrier

% fo % Spectrometer offset frequency

261

% Pulse sequence and spectrometer parameters

% Nm % Length of modulation sequence

% Ne % Collect Ne echoes following each lock pulse (10)

% Ns % Number of samples per acquired spin echo

% fs % AD sample rateke

% Ts =1/fs % AD sample period

% T_daq = 1.738e-3; % Acutal Data Acquision window time

td = 100e-3; % time delay before lock pulse

t_save = 0; % time required to save a waveforem in seconds

%td = 100e-3; % time delay before lock pulse

%t_save = 11; % time required to save a waveforem in seconds

tau = 2e-3; % time between echoes

% Generating modulation sequence

b_cmp = bitcmp(b,1);

262

b_mod = reshape([b b_cmp]’,2*Nm,1);

ws = wstar;

wo = 2*pi*fo; % Frequency offset

% Centroid of baseband QR Signal

w_bb_QR = ws + wo - wstar;

Ns_am = round(Ns*(tau/T_daq)); % Number of samples used up by data

% acquisition window

Ns_d = round(Ns*(td/T_daq)); % Number of samples used by the

% time delay between sequences

Ns_save = round(Ns*(t_save/T_daq)); % Number of samples used by the

% save time between sequences

w_bb_AM = ws + wo - w_AM; % Centroid of AM spectra in baseband

263

% AM_mag*(1+m[k]), where m varies between -1 and 1, therefor max = 2*AM_mag

SNR = 20*log10(QR_amp/(AM_mag));

% time vector for each acquisition window

Ts = 1/fs;

ns = 1:Ns;

t_echo = (ns - Ns/2)*Ts;

t_am = linspace(0,T_daq,Ns);

% Initialize storage vectors for signal averaging

Nm = 2*Nm; % Generate twice the number of sequences for phase cycling

zv = zeros(Nm,Ns);

QR_seq = zv;

AM_tn1 = zv;

264

AM_tn2 = zv;

tn_001 = zv;

tn_002 = zv;

%% Design low pass Butterworth Filter

fc = 20e3; % 3 dB Cuttoff Frequency

%% Normalized, 1 corresponds to fs/2

wn = fc/(.5*fs); % Normalized Cuttoff Frequency

% Butter worth filter design

Ord = 8;

ftype = ’low’;

% Butter worth filter, Zero-Pole-Gain Design

[z, p, k] = butter(Ord,wn,ftype);

[sos,g] = zp2sos(z,p,k);

H = dfilt.df2sos(sos,g);

265

% % Plot and compare the results (As the order increases the transfer

% % function design can go unstable due to rounding errors)

%

% [H_mag,w_mag]=freqz(H,fs);

%

% % unwarp frequency

%

% w_mag = w_mag/(2*pi);

% w_mag = w_mag*fs;

%

% figure(1)

% semilogx(w_mag,20*log10(abs(H_mag)),’g’)

% ylim([-150 50])

% xlabel(’Frequency (Hz)’)

% ylabel(’Magnitude (dB)’)

% legend(’Pole Zero Gain Design’)

% title(’8th Order Butterworth Lowpass Filter Frequency Response’)

%% Design Bandpass Butterworth Filter

fs_CD = 44.1e3; % Sampling Frequency

fc1 = 40; % 3 dB Cuttoff Frequency left

266

fc2 = 1e4; % 3 dB Cuttoff Frequency right

%% Normalized, 1 corresponds to fs/2

wn1 = fc1/(.5*fs_CD); % Normalized Cuttoff Frequency left

wn2 = fc2/(.5*fs_CD); % NOrmailized Cutoff Frequency right

fn_bp = [fc1 fc2];

wn_bp = [wn1 wn2];

% Butterworth filter design

Ord_bp = 4;

ftype = ’bandpass’;

% Butterworth filter, Zero-Pole-Gain Design

[z_bp, p_bp, k_bp] = butter(Ord_bp,wn_bp,ftype);

[sos_bp,g_bp] = zp2sos(z_bp,p_bp,k_bp);

H_bp = dfilt.df2sos(sos_bp,g_bp);

% % Plot and compare the results (As the order increases the transfer

% % function design can go unstable due to rounding errors)

267

%

% [H_bp_mag,w_bp_mag]=freqz(H_bp,fs_CD);

%

% Unwarp frequency

%

% w_bp_mag = w_bp_mag/(2*pi);

% w_bp_mag = w_bp_mag*fs_CD;

%

% figure(1)

% semilogx(w_bp_mag,20*log10(abs(H_bp_mag)),’g’)

% ylim([-150 50])

% xlabel(’Frequency (Hz)’)

% ylabel(’Magnitude (dB)’)

% legend(’Pole Zero Gain Design’)

% title(’8th Order Butterworth Bandpass Filter Frequency Response’)

% Generating average vectors for QR Signal and RFI sources

% Initial time delays - used in calculating the AM carrier phase

time_delay_exp_1 = 0;

268

start_phase1 = 2*pi*rand-pi;

start_phase2 = 2*pi*rand-pi;

k_001 = ns;

t_sin = [-T_daq/2:Ts:T_daq/2];

t_sin = t_sin(1:Ns);

for km=1:Nm

for ke=1:Ne

% QR Signals, add the Ne decaying echoes into one row

QR_mag = exp(-abs(t_echo)/T2star);

QR_sig = QR_mag.*exp(i*(w_bb_QR*t_echo + b_mod(km)*pi));

269

QR_sig = filter(H,QR_sig);

QR_sig = ((QR_amp.*exp(-(ke-1)*tau/T2eff))/max(abs(real(QR_sig))))*QR_sig;

QR_seq(km,:) = QR_seq(km,:) + QR_sig;

% Amplitude Modulated Thermal noise of standard deviation

m1 = randn(1,Ns);

m1 = filter(H_bp,m1);

m1 = (AM_st/(((m1*m1’)/size(m1,2))^.5))*m1;

m2 = randn(1,Ns);

m2 = filter(H_bp,m2);

m2 = (AM_st/(((m2*m2’)/size(m2,2))^.5))*m2;

AMtn1 = (5+ m1).*cos(w_bb_AM*(t_am+time_delay_exp_1) - start_phase1);

270

AMtn1 = filter(H,AMtn1);

AMtn1 = (AM_mag/(((AMtn1*AMtn1’)/size(AMtn1,2))^.5))*AMtn1;

AMtn2 = (5+ m2).*cos(w_bb_AM*(t_am+time_delay_exp_1) - start_phase2);

AMtn2 = filter(H,AMtn2);

AMtn2 = (AM_mag/(((AMtn2*AMtn2’)/size(AMtn2,2))^.5))*AMtn2;

AM_tn1(km,:) = AM_tn1(km,:) + AMtn1;

AM_tn2(km,:) = AM_tn2(km,:) + AMtn2;

% Form complex thermal noise

therm_amp=therm_std;

tn_real1 = randn(1,Ns);

tn_real1 = filter(H,tn_real1);

tn_real1 = (therm_amp/(((tn_real1*tn_real1’)/size(tn_real1,2))^.5))*tn_real1;

271

tn_cmpl1 = randn(1,Ns);

tn_cmpl1 = filter(H,tn_cmpl1);

tn_cmpl1 = (therm_amp/(((tn_cmpl1*tn_cmpl1’)/size(tn_cmpl1,2))^.5))*tn_cmpl1;

tn1 = (tn_real1+i*tn_cmpl1);

tn_real2 = randn(1,Ns);

tn_real2 = filter(H,tn_real2);

tn_real2 = (therm_amp/(((tn_real2*tn_real2’)/size(tn_real2,2))^.5))*tn_real2;

tn_cmpl2 = randn(1,Ns);

tn_cmpl2 = filter(H,tn_cmpl2);

tn_cmpl2 = (therm_amp/(((tn_cmpl2*tn_cmpl2’)/size(tn_cmpl2,2))^.5))*tn_cmpl2;

tn2 = (tn_real2+i*tn_cmpl2);

272

% Thermal Noise different for sping echo

tn_001(km,:) = tn_001(km,:) + tn1;

tn_002(km,:) = tn_002(km,:) + tn2;

% Updating the time delay for the next echo

k_001 = k_001 + Ns_am;

time_delay_exp_1 = time_delay_exp_1 + tau;

end % End of one pulse train that produces Ne echoes

%time_delay_exp_1 = time_delay_exp_1 + td + ps_timing_jitter(kr+(km-1)*Nr,1);

time_delay_exp_1 = time_delay_exp_1 + td;

k_001 = k_001 + Ns_d;

273

% Adding the computer saving time + time_jitter random between 0 &

% 1 s if km is even, (i.e. after the an even number of pulses)

if(mod(km,2)==0)

% time jitter is a random between 0 and 1s when saving

%t_rand = rand;

% jitter is zero for no save time

% t_rand = tau*rand;

t_rand = 0;

%Even

time_delay_exp_1 = time_delay_exp_1 + t_save + t_rand;

k_001 = k_001 + Ns_save + ceil(Ns*(t_rand/T_daq));

end

end

274

%Perform phase cycling using the modulation sequence and its complement

qr_seq(:,:) = QR_seq(:,:);

qrtn_seq(:,:)= QR_seq(:,:) + tn_001(:,:);

tn_seq(:,:) = tn_002(:,:);

qramtn_seq(:,:)= QR_seq(:,:) + AM_tn1(:,:) + tn_001(:,:);

amtn_seq(:,:) = AM_tn2(:,:) + tn_002(:,:);

qr(:,:) = (qr_seq(1:2:Nm-1,:) - qr_seq(2:2:Nm,:))./(2*Ne);

qrtn(:,:) = (qrtn_seq(1:2:Nm-1,:) - qrtn_seq(2:2:Nm,:))./(2*Ne);

tn(:,:) = (tn_seq(1:2:Nm-1,:) - tn_seq(2:2:Nm,:))./(2*Ne);

qramtn(:,:) = (qramtn_seq(1:2:Nm-1,:) - qramtn_seq(2:2:Nm,:))./(2*Ne);

amtn(:,:) = (amtn_seq(1:2:Nm-1,:) - amtn_seq(2:2:Nm,:))./(2*Ne);

275

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Vita: Abdullah G. Almahri

Abdullah Ghazi Almahri, born on the 22nd of Janauary 1983, is the son of Ghazi Ab-

dulla Almahri and Khitam Yousef Maslamani. After completing his work at George C.

Marshall High School in Ankara, Turkey, he was sponsored by the UAE’s Ministry of

Higher Education to complete a Bachelor of Science in Electrical Engineering at The

Penn State University, University Park and completed his degree in December of 2004.

In 2005, Abdullah joined the Abu Dhabi National Oil Company. Shortly after that he

was granted a study leave to rejoin Penn State University to complete a Masters of

Science in Electrical Engineering, which he received in August of 2007. Abdullah then

resumed his studies at Penn State to pursue a Ph.D in Electrical Engineering.

Documents

QR SIGNAL DETECTION IN THE PRESENCE OF AM NOISE