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CLARKSON UNIVERSITY

Detection and Parameter Extraction of Low Probability of Intercept Radar Signals Using the Reassignment Method and the Hough Transform

A Dissertation

By

Daniel Lee Stevens

Department of Electrical and Computer Engineering

Submitted in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy, Electrical and Computer Engineering

November 22, 2010

Accepted by the Graduate School

_________, _______________________

Date, Dean of the Graduate School

UMI Number: 3437800

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The undersigned have examined the dissertation entitled Detection and Parameter

Extraction of Low Probability of Intercept Radar Signals Using the Reassignment

Method and the Hough Transform presented by Daniel Lee Stevens, a candidate for

the degree of Doctor of Philosophy, Electrical and Computer Engineering and hereby

certify that it is worthy of acceptance.

______________________ ___________________________________

Date Dr. Stephanie A. Schuckers

______________________ ___________________________________

Date Dr. Robert J. Schilling

______________________ ___________________________________

Date Dr. Abdul J. Jerri

______________________ ___________________________________

Date Dr. Jeremiah J. Remus

______________________ ___________________________________

Date Dr. Andrew J. Noga

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Abstract

Current radar systems employ new types of signals which are difficult to detect and

characterize. These signals are known as low probability of intercept radar signals,

which have as their goal to see and not be seen by intercept receivers, devices which are

designed to detect and extract information from radar emissions. Digital intercept

receivers are currently moving away from Fourier-based analysis and towards classical

time-frequency analysis techniques, based on the Wigner-Ville distribution, Choi-

Williams distribution, spectrogram, and scalogram, for the purpose of analyzing low

probability of intercept radar signals (e.g. triangular modulated frequency modulated

continuous wave and frequency shift keying). Although these classical time-frequency

techniques are an improvement over the Fourier-based analysis, they still suffer from a

lack of readability, due to poor time-frequency localization, cross-term interference, and a

mediocre performance in low signal-to-noise ratio environments. This lack of readability

may lead to inaccurate detection and parameter extraction of these radar signals, making

for a less informed (and therefore less safe) intercept receiver environment. In this

dissertation, the reassignment method, because of its ability to smooth out cross-term

interference and improve time-frequency localization, is proposed as an improved signal

analysis technique. In addition, the Hough transform, due to its ability to suppress cross-

term interference, separate signals from cross-term interference, and perform well in a

low signal-to-noise ratio environment, is also proposed as an improved signal analysis

technique. With these qualities, both the reassignment method and the Hough transform

have the potential to produce better readability and consequently, more accurate signal

detection and parameter extraction metrics. Simulations are presented that compare time-

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frequency representations of the classical time-frequency techniques with those of the

reassignment method and the Hough transform. Two different triangular modulated

frequency modulated continuous wave low probability of intercept radar signals and two

frequency shift keying low probability of intercept radar signals (4-component and 8-

component) were analyzed. The following metrics were used for evaluation of the

analysis: percent error of: carrier frequency, modulation bandwidth, modulation period,

time-frequency localization, and chirp rate. Also used were: percent detection, number of

cross-term false positives, lowest signal-to-noise ratio for signal detection, and plot

(processing) time. Experimental results demonstrate that the squeezing and

smoothing qualities of the reassignment method, along with the cross-term interference

suppressing/separating and low signal-to-noise ratio performance qualities of the Hough

transform, do in fact lead to improved readability over the classical time-frequency

analysis techniques, and consequently, provide more accurate signal detection and

parameter extraction metrics than the classical time-frequency analysis techniques. In

addition, the Hough transform was utilized to detect, extract parameters, and properly

identify a real-world low probability of intercept radar signal in a low signal-to-noise

ratio environment, where classical time-frequency analysis failed. In summary, this

dissertation provides evidence that the reassignment method and the Hough transform

have the potential to outperform the classical time-frequency analysis techniques, which

are the current state-of-the-art, cutting edge techniques for this arena. An improvement

in performance can easily translate into saved equipment and lives. Future work will

include automation of metrics extraction process, analysis of additional low probability of

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intercept radar waveforms of interest, and analysis of other real-world low probability of

intercept radar signals utilizing more powerful computing platforms.

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Acknowledgements

Id like to thank the following people, who all played a part in this major

accomplishment:

My better-half, Donna, my loving wife, who lives the life of a Proverbs 31 woman.

Abraham, Solomon, Joy, Judah, Christian, and Glory, my six blessings from the Lord,

who continually radiate life, and who are always an inspiration to me.

My Mom, Judy, whose prayers through the years have kept me on the straight and

narrow.

My sister Cathy (Snorque), and my brothers Mike and Jake, who are undoubtedly smarter

than I am, but whose heads arent quite as large.

My Father, Roger, who was himself a brilliant man, and who is currently part of that

great cloud of witnesses that we will once again see one day.

Prof. Stephanie Schuckers, who has, in both a personal and professional way, guided me

through this process, all the while believing in this non-traditional (i.e. old ) student.

She has been as much a friend as an advisor, as we are at similar stages in our lives.

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Dr. Dick Wiley, Dr. Phil Pace, Dr. Steven Kay, and Dr. G. Faye Boudreaux-Bartels: for

allowing me to bounce ideas and questions off of their sharp minds.

My committee members: Prof. Bob Schilling (who helped me get the rust out early on),

Prof. Abdul Jerri (who always has a smile on his face), Prof. Jeremiah Remus (one of the

nicest people around), and Dr. Andy Noga (aka Dr. Demodulation - who pushed all the

right buttons to get the banana!) Thank-you for your feedback and your helpful

suggestions.

Prof. Tom Ortmeyer: for introducing me to the program.

John Grieco, Steve Johns, Ed Starczewski, and Dr. John Maher: for their in-house

support.

Charlie Estrella: for his MATLAB wisdom (Go Raiders!).

Finally, I would like to thank Jesus, because apart from Him I can do nothing (John

15:5), but with Him I can do all things (Phil 4:13).

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Table of Contents Chapter Page

Abstract .............................................................................................................................. iii

Acknowledgements ............................................................................................................ vi

Table of Contents ............................................................................................................. viii

List of Figures .................................................................................................................... xi

List of Tables ................................................................................................................. xxiv

1. Introduction ......................................................................................................................1

1.1. Intercept Receiver Signal Analysis Techniques ........................................................3

1.2. Classical Time-Frequency Analysis Techniques .......................................................4

1.3. The Reassignment Method ........................................................................................9

1.4. The Hough Transform..............................................................................................11

2. Background ....................................................................................................................14

2.1. LPI Radar Overview ................................................................................................14

2.2. LPI Radar Characteristics ........................................................................................15

2.3. LPI Radar Waveforms .............................................................................................16

2.4. LPI Radar Applications and Examples ....................................................................18

2.5. Detection of LPI Radars: Intercept Receiver Overview ..........................................20

2.6. Intercept Receiver Signal Analysis Techniques ......................................................21

2.6.1. Time-Frequency Analysis ..................................................................................21

2.6.2. Wigner-Ville Distribution (WVD) .....................................................................22

2.6.3. Choi-Williams Distribution (CWD) ..................................................................31

2.6.4. Spectrogram (Short-Time Fourier Transform (STFT)) .....................................32

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2.6.5. Scalogram (Wavelet Transform) ........................................................................34

2.6.6. The Reassignment Method ................................................................................38

2.6.7. The Hough Transform........................................................................................49

2.7. Previous Related Studies..........................................................................................58

2.7.1. Detection of Low Probability of Intercept Radar Signals ...............................58

2.7.2. Extraction of Polyphase Radar Modulation Parameters using a Wigner-Ville

Distribution Radon Transform..........................................................................................58

2.7.3. Periodic Wigner-Ville Hough Transform .......................................................60

2.7.4. Detection and Parameter Estimation of Chirped Radar Signals .....................61

3. Methodology ..................................................................................................................64

4. Detection and Parameter Extraction of Low Probability of Intercept Radar Signals

Using the Reassignment Method .......................................................................................85

4.1. Abstract ....................................................................................................................85

4.2. Introduction ..............................................................................................................86

4.3. Methodology ............................................................................................................91

4.4. Results ....................................................................................................................102

4.5. Discussion ..............................................................................................................109

4.6. Conclusions ............................................................................................................119

5. Detection and Parameter Extraction of Low Probability of Intercept Radar Signals

Using the Hough Transform ............................................................................................122

5.1. Abstract ..................................................................................................................122

5.2. Introduction ............................................................................................................123

5.3. Methodology ..........................................................................................................128

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5.4. Results ....................................................................................................................138

5.5. Discussion ..............................................................................................................148

5.6. Conclusions ............................................................................................................156

6. Joint Sequential Use of the Reassignment Method and the Hough Transform ...........158

7. Overall Conclusions and Future Work ........................................................................161

References ........................................................................................................................165

Appendix A ......................................................................................................................176

Appendix B ......................................................................................................................184

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List of Figures

Figure Page

Figure 1: WVD of a triangular modulated FMCW signal at an SNR of 0 dB (256

samples) .............................................................................................................................27

Figure 2: Same as Figure 1, but with the SNR level increased to 10 dB. Though some of

the noise has disappeared, the cross-terms remain ............................................................27

Figure 3: Same as Figure 1, but with the SNR level increased to 20 dB. Most of the noise

has disappeared, but the cross-terms remain, demonstrating that cross-terms are virtually

unaffected by the SNR level ..............................................................................................28

Figure 4: Example of cross-term false positives (XFPs). There are 4 true signals

(fc1=1KHZ, fc2=1.75KHz, fc3=0.75KHz, fc4=1.25KHz) and 6 XFPs (cross-terms that

were wrongly declared as signal detections) (ct1=0.875KHz, ct2=1KHz, ct3=1.125KHz,

ct4=1.25KHz, ct5=1.375KHz, ct6=1.5KHz). WVD of a 4-component FSK signal at

SNR=10dB (512 samples) .................................................................................................29

Figure 5: RSPWVD of triangular modulated FMCW, 256 samples, SNR=10dB.

Threshold of 5% of the maximum intensity (upper left) and 1% threshold (upper right).

The SNR has been increased to 25dB (lower-left) and to 50dB (lower-right). The plot on

the upper left gives the appearance of perfectly localized distributions for the chirp

signals. However, the plot on the upper right (1% threshold) reveals artifacts that are

clearly not perfectly localized. In order to determine if these artifacts are due to noise,

the SNR is raised to 25dB (lower left). Many of the artifacts disappear, but some remain.

The SNR is then raised to 50dB (lower right) to see if the remaining artifacts are also

noise related, but most of the artifacts in the 25dB plot are also in the 50dB plot. For

time-frequency reassignment, it is known that the localization is slightly degraded at

those points where the IF cannot be considered as quasi-linear within a TF smoothing

window (i.e. the curved areas of the signal in Figure 5) causing artifacts. In addition,

when more than one component is within a TF smoothing window (i.e. the curves where

the chirp signals meet in Figure 5), a beating effect occurs, resulting in interference

fringes which may manifest as artifacts .............................................................................45

Figure 6: Time-frequency plot on the left and Hough transform plot on the right. A point

in the TF plot maps to a sinusoidal curve in the HT plot. A line (signal) in the TF plot

maps to a point in the HT plot. The rho and theta values of the point in the HT plot can

be used to back-map to the TF plot in order to find the location of the line (signal) (good

if time-frequency plot is cluttered with noise and/or cross-term interference and signal is

not visible) .........................................................................................................................51

Figure 7: Hough transform of the WVD of a triangular modulated FMCW signal at an

SNR of 10 dB (512 samples). Each point has a unique theta and rho value which can be

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used to back-map to the time-frequency (WVD) representation in order to locate the 4

signals, as depicted in Figure 7 ..........................................................................................53

Figure 8: The unique theta and rho values extracted from Figure 6 are used to back-map

to the time-frequency representation (WVD of a triangular modulated FMCW signal

(SNR=10dB, #samples=512)) in order to locate the 4 chirp signals that make up the 4

legs of the triangular modulated FMCW signal .................................................................55

Figure 9: Threshold percentage determination. This plot is an amplitude vs. time (x-z

view) of CWD of FSK 4-component signal (512 samples, SNR= -2dB). For visually

detected low SNR plots (like this one), the percent of max intensity for the peak z-value

of each of the signal components was noted (here 82%, 98%, 87%, 72%), and the lowest

of these 4 values was recorded (72%). Ten test runs were performed for each time-

frequency analysis tool, for each of the 4 waveforms. The average of these recorded low

values was determined and then assigned as the threshold for that particular time-

frequency analysis tool. Note - the threshold for CWD is 70%........................................68

Figure 10: Percent detection (time-frequency). CWD of 4-component FSK (512 samples,

SNR=10dB) with threshold value automatically set to 70%. From this threshold plot, the

signal was declared a (visual) detection because at least a portion of each of the 4 FSK

signal components was visible ...........................................................................................69

Figure 11: Percent detection (Hough transform). This plot is a theta-intensity (x-z view)

of the HT of an RSPWVD (512 samples, SNR=10dB). Signal declared a (visual)

detection because at least a portion of each of the signal components exceeded the noise

floor threshold ....................................................................................................................70

Figure 12: Example of cross-term false positives (XFPs). WVD of a 4-component FSK

signal at SNR=10dB (512 samples). There are 4 true signals (fc1=1KHz, fc2=1.75KHz,

fc3=0.75KHz, fc4=1.25KHz) and 6 XFPs (cross-terms that were wrongly declared as

signal detections because they passed the signal detection criteria listed in the percent

detection section above) (ct1=0.875KHz, ct2=1KHz, ct3=1.125KHz, ct4=1.25KHz,

ct5=1.375KHz, ct6=1.5KHz) .............................................................................................71

Figure 13: Determination of carrier frequency. CWD of a triangular modulated FMCW

(256 samples, SNR=10dB). From the frequency-intensity (y-z) view, the maximum

intensity value is manually determined. The frequency corresponding to the max

intensity value is the carrier frequency (here fc=1062.4 Hz).............................................72

Figure 14: Determination of carrier frequency. CWD of a 4-component FSK (512

samples, SNR=10dB). From the frequency-intensity (y-z) view, the 4 maximum

intensity values (1 for each carrier frequency) are manually determined. The frequencies

corresponding to those 4 max intensity values are the 4 carrier frequency (here

fc1=10001 Hz, fc2=1753Hz, fc3=757Hz, fc4=1255Hz) ...................................................73

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Figure 15: Modulation bandwidth determination. CWD of a triangular modulated

FMCW (512 samples, SNR=10dB) with threshold value automatically set to 20%. From

this threshold plot, the modulation bandwidth was measured manually from the highest

frequency value of the signal (top red arrow) to the lowest frequency value of the signal

(bottom red arrow) in the y-direction (frequency) .............................................................74

Figure 16: Modulation bandwidth determination. CWD of a 4-component FSK (512

samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold

plot, the modulation bandwidth was measured manually from the highest frequency value

of the signal (top red arrow) to the lowest frequency value of the signal (bottom red

arrow) in the y-direction (frequency) .................................................................................75

Figure 17: Modulation period determination. CWD of a triangular modulated FMCW

(512 samples, SNR=10dB) with threshold value automatically set to 20%. From this

threshold plot, the modulation period was measured manually from the highest frequency

value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red

arrow) in the x-direction (time) ..........................................................................................77

Figure 18: Modulation period determination. CWD of a 4-component FSK (512 samples,


modulation period was measured manually from the left side of the signal (left red arrow)

to the right side of the signal (right red arrow) in the x-direction (time). This was done

for all 4 signal components, and the average value was determined .................................78

Figure 19: Time-frequency localization determination. CWD of a triangular modulated


this threshold plot, the time-frequency localization was measured manually from the left

side of the signal (left red arrow) to the right side of the signal (right red arrow) in both

the x-direction (time) and the y-direction (frequency). Measurements were made at the

center of each of the 4 legs, and the average values were determined. Average time and frequency thickness values were then converted to: % of entire x-axis and % of entire y-axis .....................................................................................................................................80

Figure 20: Time-frequency localization determination for the CWD of a 4-component

FSK (512 samples, SNR=10dB) with threshold value automatically set to 20%. From

this threshold plot, the time-frequency localization was measured manually from the top

of the signal (top red arrow) to the bottom of the signal (bottom red arrow) in the y-

direction (frequency). This frequency thickness value was then converted to: % of entire y-axis. .......................................................................................................................81

Figure 21: Lowest detectable SNR (time-frequency). CWD of 4-component FSK (512

samples, SNR=-2dB) with threshold value automatically set to 70%. From this threshold

plot, the signal was declared a (visual) detection because at least a portion of each of the

4 FSK signal components was visible. For this case, any lower SNR would have been a

non-detect. Compare to Figure 10, which is the same plot, except that it has an SNR

level equal to 10dB ............................................................................................................82

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Figure 22: Lowest detectable SNR (Hough transform). This plot is a theta-intensity (x-z

view) of the HT of an RSPWVD (512 samples, SNR=-5dB). Signal declared a (visual)


floor threshold. For this case, any lower SNR would have been a non-detect. Compare

to Figure 11, which is the same plot, except that it has an SNR level equal to 10dB .......83








frequency analysis tool. Note - the threshold for CWD is 70%......................................95




signal components was visible ...........................................................................................96

Figure 25: Modulation bandwidth determination. CWD of a triangular modulated


this threshold plot, the modulation bandwidth was measured manually from the highest

frequency value of the signal (top red arrow) to the lowest frequency value of the signal

(bottom red arrow) in the y-direction (frequency) .............................................................98

Figure 26: Modulation bandwidth determination. CWD of a 4-component FSK (512

samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold

plot, the modulation bandwidth was measured manually from the highest frequency value

of the signal (top red arrow) to the lowest frequency value of the signal (bottom red

arrow) in the y-direction (frequency) .................................................................................99






level equal to 10dB ..........................................................................................................101

Figure 28: Time-frequency localization comparison between the classical time-frequency

analysis tools (left) and the reassignment method (right). The uppper two plots are the

spectrogram (left) and the reassigned spectrogram (right) for a triangular modulated

FMCW signal (256 samples, SNR=10dB), and the lower two plots are the scalogram

(left) and the reassigned scalogram (right) for a 4-component FSK signal (512 samples,

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SNR=10dB). The reassignment method gives a much more concentrated time-frequency

localization than does its classical time-frequency analysis counterparts .......................104

Figure 29: Cross-term interference of the WVD (left) and ability of the RSPWVD to

reduce cross-term interference (right). The uppper two plots are the WVD (left) and the

RSPWVD (right) for a triangular modulated FMCW signal (512 samples, SNR=10dB),

and the lower two plots are the WVD (left) and the RSPWVD (right) for a 8-component

FSK signal (512 samples, SNR=10dB). The WVD displays a lot of cross-term

interference. There appears to be one additional triangle signal in the middle of the two-

triangle signal in the upper-left plot, and there appears to be 9 additional signals to go

along with the 8-component signal in the lower-left plot. The RSPWVD has drastically

reduced the cross-term interference found in the WVD plots, making for more readable

presentations. Notice also that the RSPWVD is more concentrated than its WVD

counterpart (as per Table 1) .............................................................................................105

Figure 30: Readability degradation due to reduction in SNR. Spectrogram, triangular

modulated FMCW, modulation bandwidth=500Hz, 512 samples. SNR=10dB (left), 0dB

(center), -4dB (left). Readability degrades as SNR decreases, negatively affecting the

accuracy of the metrics extracted, as per Table 2. ...........................................................107

Figure 31: Comparison between Task 1 (left) and Task 2 (right). The uppper two plots

are both spectrogram plots of a triangular modulated FMCW signal (512 samples,

SNR=10dB), Task 1 (modulation bandwidth=500Hz) is on the left and Task 2

(modulation bandwidth=2400Hz) is on the right. The lower two plots are CWD plots of

the same signals. The Task 2 plots (right) have a larger modulation bandwidth than Task

1 plots, therefore the signals appear taller and more upright than the Task 1 signals .....109

Figure 32: Percent detection (Hough transform). This plot is a theta-intensity (x-z view)

of the HT of an RSPWVD (512 samples, SNR=10dB). Signal declared a (visual)


floor threshold ..................................................................................................................132








frequency analysis tool. Note - the threshold for CWD is 70%....................................133




signal components was visible .........................................................................................134

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Figure 35: S Example of cross-term false positives (XFPs). WVD of a 4-component FSK

signal at SNR=10dB (512 samples). There are 4 true signals (fc1=1KHz, fc2=1.75KHz,

fc3=0.75KHz, fc4=1.25KHz) and 6 XFPs (cross-terms that were wrongly declared as

signal detections because they passed the signal detection criteria listed in the percent

detection section above) (ct1=0.875KHz, ct2=1KHz, ct3=1.125KHz, ct4=1.25KHz,

ct5=1.375KHz, ct6=1.5KHz) ...........................................................................................135

Figure 36: Lowest detectable SNR (Hough transform). This plot is a theta-intensity (x-z

view) of the HT of an RSPWVD (512 samples, SNR=-5dB). Signal declared a (visual)


floor threshold. For this case, any lower SNR would have been a non-detect. Compare

to Figure 32, which is the same plot, except that it has an SNR level equal to 10dB .....136






level equal to 10dB ..........................................................................................................137

Figure 38: Cross-term comparison between classical time-frequency analysis techniques

(left) and the Hough transform (right). Top left: WVD of a triangular modulated FMCW

signal (512 samples, SNR=10dB) (left). Top right: the Hough transform of the WVD of

a triangular modulated FMCW signal (512 samples, SNR=10dB). Bottom left: WVD of

an FSK (8-component) signal (512 samples, SNR=10dB). Bottom right: the Hough

transform of the WVD of an FSK (8-component) signal (512 samples, SNR=10dB).

Upper 2 plots the Hough transform (right) has eliminated the cross-term interference that the WVD (left) displays, making it easier to see the signal (better readability) in the

Hough transform plot (the four bright spots which represent the four legs of the triangular

modulated FMCW signal). The WVD appears to have another triangle signal between

the outer two triangle signals. Lower 2 plots - In the WVD plot (left) there are 8 signal

components and 9 cross-term components that appear to be signal components, all

melded in together with one another. The 8 signal components are located at 5 distinct

frequencies (one at 0.75KHz, two at 1KHz, two at 1.25KHz, two at 1.5KHz, and one at

1.75KHz). In the Hough transform plot (right) there are 8 signal components (located at

5 different frequencies (3 on the left-hand side of the plot and 2 in the middle of the plot))

plus 2 cross-term components, clearly separated from the signal components. The Hough

transform plot makes it easier to see the signal components (better readability) as

compared to the WVD plot ..............................................................................................141

Figure 39: Low SNR comparison between classical time-frequency analysis techniques

(left) and the Hough transform (right). Top left: CWD of a triangular modulated FMCW

signal (512 samples, SNR=-6dB) (left). Top right: the Hough transform of the CWD of a

triangular modulated FMCW signal (512 samples, SNR=-6dB). Bottom left: CWD of an

FSK (8-component) signal (512 samples, SNR=-3dB). Bottom right: the Hough

transform of the CWD of an FSK (8-component) signal (512 samples, SNR=-3dB).

xvii

Upper 2 plots - Though the signal is not visible in the CWD plot (left) (due to the low

SNR (-6dB)), the four bright spots that represent the four legs of the triangular modulated

FMCW signal are clearly seen in the Hough transform of the CWD plot (right). Each

bright spot has a unique rho and theta value that can be used to back-map to the time-

frequency representation (here CWD) and find the location of the 4 (non-visible) chirps

that make up the triangular modulated FMCW signal. Lower 2 plots - Though the signal

components are not visible in the CWD plot (due to the low SNR (-3dB)), the 5 bright

spots (3 on the left and 2 in the middle) corresponding to the 5 different frequencies of

the 8 FSK components are clearly visible in the Hough transform of the CWD plot (as are

2 cross-term components). The Hough transform does a good job of detecting the signal

components and separating the signal components from the cross-term components, all in

a low SNR environment ...................................................................................................142

Figure 40: Readability degradation due to reduction in SNR. Spectrogram, triangular

modulated FMCW, modulation bandwidth=500Hz, 512 samples. SNR=10dB (left), 0dB

(center), -4dB (left). Readability degrades as SNR decreases, negatively affecting the

accuracy of the metrics extracted, as per Table 5 ............................................................144

Figure 41: Comparison between Task 1 (left) and Task 2 (right). The uppper two plots

are both spectrogram plots of a triangular modulated FMCW signal (512 samples,

SNR=10dB), Task 1 (modulation bandwidth=500Hz) is on the left and Task 2

(modulation bandwidth=2400Hz) is on the right. The lower two plots are CWD plots of

the same signals. The Task 2 plots (right) have a larger modulation bandwidth than Task

1 plots, therefore the signals appear taller and more upright than the Task 1 signals .....145

Figure 42: Spectrogram (left) and Hough transform of Spectrogram (right) of area where

Task 5 real-world signal was located. Signal was not visible in the spectrogram, but was

visible in the Hough transform, due to the Hough transforms ability to extract signals from low SNR environments. ..........................................................................................146

Figure 43: Hough transform of Spectrogram (zoomed-in on receiver IF bandwidth

(~750MHz to 1250MHz)) of area where Task 5 real-world signal was located. Signal is

clearly visible at theta=2.872 and rho=228.8 ...................................................................147

Figure 44: Spectrogram (zoomed-in on receiver IF bandwidth (~750MHz to 1250MHz)).

Shows how the Hough transform theta and rho values back-map to the time-frequency

representation for signal location .....................................................................................148

TASK 1

Figure 45: Test 1 Plot WVD, 256, 10 ...........................................................................184

Figure 46: Test 2 Plot WVD, 256, 0 .............................................................................185

Figure 47: Test 3 Plot WVD, 256, -2 ............................................................................186

xviii

Figure 48: Test 4 Plot WVD, 512, 10 ...........................................................................187

Figure 49: Test 7 Plot CWD, 256, 10 ...........................................................................189

Figure 50: Test 8 Plot CWD, 256, 0 .............................................................................190

Figure 51: Test 9 Plot CWD, 256, -3 ............................................................................191

Figure 52: Test 10 Plot CWD, 512, 10 .........................................................................192

Figure 53: Test 11 Plot CWD, 512, 0 ...........................................................................193

Figure 54: Test 12 Plot CWD, 512, -3 ..........................................................................194

Figure 55: Test 13 Plot Spectrogram, 256, 10 ..............................................................195

Figure 56: Test 14 Plot Spectrogram, 256, 0 ................................................................196 Figure 57: Test 15 Plot Spectrogram, 256, -3 ...............................................................197 Figure 58: Test 16 Plot Spectrogram, 512, 10 ..............................................................198

Figure 59: Test 17 Plot Spectrogram, 512, 0 ................................................................199 Figure 60: Test 18 Plot Spectrogram, 512, -4 ...............................................................200

Figure 61: Test 19 Plot Scalogram, 256, 10..................................................................201 Figure 62: Test 20 Plot Scalogram, 256, 0....................................................................202 Figure 63: Test 21 Plot Scalogram, 256, -2 ..................................................................203 Figure 64: Test 22 Plot Scalogram, 512, 10..................................................................204 Figure 65: Test 23 Plot Scalogram, 512, 0....................................................................205 Figure 66: Test 24 Plot Scalogram, 512, -3 ..................................................................206 Figure 67: Test 25 Plot Reassigned Spectrogram, 256, 10 ...........................................207 Figure 68: Test 26 Plot Reassigned Spectrogram, 256, 0 .............................................208 Figure 69: Test 27 Plot Reassigned Spectrogram, 256, -2 ............................................209 Figure 70: Test 28 Plot Reassigned Spectrogram, 512, 10 ...........................................210

xix

Figure 71: Test 29 Plot Reassigned Spectrogram, 512, 0 .............................................211 Figure 72: Test 30 Plot Reassigned Spectrogram, 512, -3 ............................................212 Figure 73: Test 31 Plot Reassigned Scalogram, 256, 10 ..............................................213 Figure 74: Test 32 Plot Reassigned Scalogram, 256, 0 ................................................214 Figure 75: Test 33 Plot Reassigned Scalogram, 256, -2 ...............................................215 Figure 76: Test 34 Plot Reassigned Scalogram, 512, 10 ..............................................216 Figure 77: Test 35 Plot Reassigned Scalogram, 512, 0 ................................................217 Figure 78: Test 36 Plot Reassigned Scalogram, 512, -3 ...............................................218 Figure 79: Test 37 Plot RSPWVD, 256, 10 ..................................................................219 Figure 80: Test 38 Plot RSPWVD, 256, 0 ....................................................................220 Figure 81: Test 39 Plot RSPWVD, 256, -3 ...................................................................221 Figure 82: Test 40 Plot RSPWVD, 512, 10 ..................................................................222 Figure 83: Test 41 Plot RSPWVD, 512, 0 ....................................................................223 Figure 84: Test 42 Plot RSPWVD, 512, -3 ...................................................................224 Figure 85: Test 43 Plot HT of WVD, 256, 10 ..............................................................225

Figure 86: Test 44 Plot HT of WVD, 256, 0 ................................................................226

Figure 87: Test 45 Plot HT of WVD, 256, -3 ...............................................................227 Figure 88: Test 46 Plot HT of WVD, 512, 10 ..............................................................228 Figure 89: Test 49 Plot HT of CWD, 256, 10 ...............................................................230

Figure 90: Test 50 Plot HT of CWD, 256, 0 .................................................................231

Figure 91: Test 51 Plot HT of CWD, 256, -4 ...............................................................232 Figure 92: Test 52 Plot HT of CWD, 512, 10 ...............................................................233

Figure 93: Test 53 Plot HT of CWD, 512, 0 .................................................................234

xx

Figure 94: Test 54 Plot HT of CWD, 512, -6 ...............................................................235 Figure 95: Test 55 Plot HT of RSPWVD, 256, 10 .......................................................236 Figure 96: Test 56 Plot HT of RSPWVD, 256, 0 .........................................................237 Figure 97: Test 57 Plot HT of RSPWVD, 256, -4 ........................................................238 Figure 98: Test 58 Plot HT of RSPWVD, 512, 10 .......................................................239 Figure 99: Test 59 Plot HT of RSPWVD, 512, 0 .........................................................240 Figure 100: Test 60 Plot HT of RSPWVD, 512, -5 ......................................................241 TASK 2

Figure 101: Test 61 Plot WVD, 512, 10 .......................................................................242 Figure 102: Test 64 Plot CWD, 512, 10 .......................................................................244

Figure 103: Test 65 Plot CWD, 512, 0 .........................................................................245

Figure 104: Test 66 Plot CWD, 512, -3 ........................................................................246 Figure 105: Test 67 Plot Spectrogram, 512, 10 ............................................................247

Figure 106: Test 68 Plot Spectrogram, 512, 0 ..............................................................248 Figure 107: Test 69 Plot Spectrogram, 512, -4 .............................................................249 Figure 108: Test 70 Plot Scalogram, 512, 10................................................................250 Figure 109: Test 71 Plot Scalogram, 512, 0..................................................................251 Figure 110: Test 72 Plot Scalogram, 512, -3 ................................................................252 Figure 111: Test 73 Plot Reassigned Spectrogram, 512, 10 .........................................253 Figure 112: Test 74 Plot Reassigned Spectrogram, 512, 0 ...........................................254 Figure 113: Test 75 Plot Reassigned Spectrogram, 512, -3 ..........................................255 Figure 114: Test 76 Plot Reassigned Scalogram, 512, 10 ............................................256 Figure 115: Test 77 Plot Reassigned Scalogram, 512, 0 ..............................................257

xxi

Figure 116: Test 78 Plot Reassigned Scalogram, 512, -3 .............................................258 Figure 117: Test 79 Plot RSPWVD, 512, 10 ................................................................259 Figure 118: Test 80 Plot RSPWVD, 512, 0 ..................................................................260 Figure 119: Test 81 Plot RSPWVD, 512, -3 .................................................................261 Figure 120: Test 82 Plot HT of WVD, 512, 10 ............................................................263 Figure 121: Test 85 Plot HT of CWD, 512, 10 .............................................................264

Figure 122: Test 86 Plot HT of CWD, 512, 0 ...............................................................265

Figure 123: Test 87 Plot HT of CWD, 512, -6 .............................................................266 Figure 124: Test 88 Plot HT of RSPWVD, 512, 10 .....................................................267 Figure 125: Test 89 Plot HT of RSPWVD, 512, 0 .......................................................268 Figure 126: Test 90 Plot HT of RSPWVD, 512, -6 ......................................................269 TASK 3

Figure 127: Test 91 Plot WVD, 512, 10 .......................................................................270 Figure 128: Test 94 Plot CWD, 512, 10 .......................................................................272 Figure 129: Test 95 Plot CWD, 512, 0 .........................................................................273 Figure 130: Test 96 Plot CWD, 512, -2 ........................................................................274 Figure 131: Test 97 Plot Spectrogram, 512, 10 ............................................................275

Figure 132: Test 98 Plot Spectrogram, 512, 0 ..............................................................276 Figure 133: Test 99 Plot Spectrogram, 512, -3 .............................................................277 Figure 134: Test 100 Plot Scalogram, 512, 10..............................................................278 Figure 135: Test 101 Plot Scalogram, 512, 0................................................................279 Figure 136: Test 102 Plot Scalogram, 512, -3 ..............................................................280 Figure 137: Test 103 Plot Reassigned Spectrogram, 512, 10 .......................................281

xxii

Figure 138: Test 104 Plot Reassigned Spectrogram, 512, 0 .........................................282 Figure 139: Test 105 Plot Reassigned Spectrogram, 512, -3 ........................................283 Figure 140: Test 106 Plot Reassigned Scalogram, 512, 10 ..........................................284 Figure 141: Test 107 Plot Reassigned Scalogram, 512, 0 ............................................285 Figure 142: Test 108 Plot Reassigned Scalogram, 512, -4 ...........................................286 Figure 143: Test 109 Plot RSPWVD, 512, 10 ..............................................................287 Figure 144: Test 110 Plot RSPWVD, 512, 0 ................................................................288 Figure 145: Test 111 Plot RSPWVD, 512, -3 ...............................................................289 Figure 146: Test 112 Plot HT of WVD, 512, 10 ..........................................................290 Figure 147: Test 112 Plot X-Y View - HT of WVD, 512, 10 ......................................290

Figure 148: Test 113 Plot HT of CWD, 512, -3 ...........................................................292 Figure 149: Test 113 Plot X-Y View - HT of CWD, 512, -3 .......................................292

Figure 150: Test 114 Plot HT of RSPWVD, 512, -4 ....................................................294 Figure 151: Test 114 Plot X-Y View - HT of RSPWVD, 512, -4 ................................294

TASK 4

Figure 152: Test 115 Plot WVD, 512, 10 .....................................................................296 Figure 153: Test 118 Plot CWD, 512, 10 .....................................................................298 Figure 154: Test 119 Plot CWD, 512, 0 .......................................................................299 Figure 155: Test 120 Plot CWD, 512, -1 ......................................................................300 Figure 156: Test 121 Plot Spectrogram, 512, 10 ..........................................................301

Figure 157: Test 122 Plot Spectrogram, 512, 0 ............................................................302 Figure 158: Test 123 Plot Spectrogram, 512, -1 ...........................................................303 Figure 159: Test 124 Plot Scalogram, 512, 10..............................................................304

xxiii

Figure 160: Test 125 Plot Scalogram, 512, 0................................................................305 Figure 161: Test 126 Plot Scalogram, 512, -3 ..............................................................306 Figure 162: Test 127 Plot Reassigned Spectrogram, 512, 10 .......................................307 Figure 163: Test 128 Plot Reassigned Spectrogram, 512, 0 .........................................308 Figure 164: Test 129 Plot Reassigned Spectrogram, 512, -2 ........................................309 Figure 165: Test 130 Plot Reassigned Scalogram, 512, 10 ..........................................310 Figure 166: Test 131 Plot Reassigned Scalogram, 512, 0 ............................................311 Figure 167: Test 132 Plot Reassigned Scalogram, 512, -2 ...........................................312 Figure 168: Test 133 Plot RSPWVD, 512, 10 ..............................................................313 Figure 169: Test 134 Plot RSPWVD, 512, 0 ................................................................314 Figure 170: Test 135 Plot RSPWVD, 512, -3 ...............................................................315 Figure 171: Test 136 Plot HT of WVD, 512, 10 ..........................................................316 Figure 172: Test 136 Plot X-Y View - HT of WVD, 512, 10 ......................................316

Figure 173: Test 137 Plot HT of CWD, 512, -3 ...........................................................318

Figure 174: Test 137 Plot X-Y View - HT of CWD, 512, -3 .......................................318 Figure 175: Test 138 Plot HT of RSPWVD, 512, -4 ....................................................320

Figure 176: Test 138 Plot X-Y View - HT of RSPWVD, 512, -4 ................................320

xxiv

List of Tables

Table 1: Signal Processing Tool viewpoint of the overall test metrics (average percent

error) for the 4 classical time-frequency analysis techniques (WVD, CWD, spectrogram,

scalogram), along with their combined average (TF) and for the 3 reassignment methods

(reassigned spectrogram, reassigned scalogram, RSPWVD), along with their combined

average (RM). The parameters extracted are listed in the left-hand column: carrier

frequency (fc), modulation bandwidth (modbw), modulation period (modper), time-

frequency localization in the x-direction (tf res-x), time-frequency localization in the y-

direction (tf res-y), chirp rate (cr), percent detection (% det), number of cross-term false

positives (#XFP), lowest detectable SNR (low snr), plot time (plottime) .......................103

Table 2: SNR viewpoint of overall test metrics (average percent error) for the classical

time-frequency analysis techniques (TF) and for the reassignment method (RM) for

SNR=10dB, 0dB, and lowest detectable SNR (low snr). The parameters extracted are

listed in the left-hand column: carrier frequency (fc), modulation bandwidth (modbw),

modulation period (modper), time-frequency localization in the x-direction (tf res-x),

time-frequency localization in the y-direction (tf res-y), chirp rate (cr), percent detection

(% det), number of cross-term false positives (#XFP), plot time (plottime) ...................106

Table 3: Task 1, 2, 3, and 4 viewpoint of overall test metrics (average percent error) for

the classical time-frequency analysis techniques (TF) and for the reassignment method

(RM). Task1=triangular modulated FMCW signal (modulation bandwidth=500Hz),

Task2=triangular modulated FMCW signal (modulation bandwidth=2400Hz), Task

3=FSK (4-component) signal, Task 4=FSK (8-component) signal. The parameters

extracted are listed in the left-hand column: carrier frequency (fc), modulation bandwidth

(modbw), modulation period (modper), time-frequency localization in the x-direction (tf

res-x), time-frequency localization in the y-direction (tf res-y), chirp rate (cr), percent

detection (% det), number of cross-term false positives (#XFP), lowest detectable SNR

(low snr), plot time (plottime) ..........................................................................................108

Table 4: Signal Processing Tool viewpoint of the overall test metrics (average percent

error) for the 4 classical time-frequency analysis techniques (WVD, CWD, spectrogram,

scalogram) along with their combined average (TF) and for the 2 Hough transform

methods (WVD + HT, CWD + HT), along with their combined average (HT). The

parameters extracted are listed in the left-hand column: chirp rate (cr), percent detection

(% det), # of cross-term false positives (#XFP), lowest detectable SNR (low snr) .........139

Table 5: SNR viewpoint of overall test metrics (average percent error) for the classical

time-frequency analysis techniques (TF) and for the Hough transform methods (HT) for

SNR=10dB, 0dB, and lowest detectable SNR (low SNR). The parameters extracted are

listed in the left-hand column: chirp rate (cr), percent detection (% det), number of cross-

term false positives (#XFP). ............................................................................................143

Table 6: Task 1, 2, 3, and 4 viewpoint of overall test metrics (average percent error) for

the classical time-frequency analysis techniques (TF) and for the Hough transform (HT).

xxv

Task1=triangular modulated FMCW signal (modulation bandwidth=500Hz),

Task2=triangular modulated FMCW signal (modulation bandwidth=2400Hz), Task

3=FSK (4-component) signal, Task 4=FSK (8-component) signal. The parameters

extracted are listed in the left-hand column: chirp rate (cr), percent detection (% det),

number of cross-term false positives (#XFP), lowest detectable SNR (low snr) ............144

Table 7: Overall test metrics for the classical time-frequency analysis techniques

(Classical TF), reassignment method (RM), Hough transform (HT), and reassignment

method plus the Hough transform (RM + HT). The parameters extracted are listed in the

left-hand column: chirp rate, lowest detectable SNR (Low SNR), and percent detection

(% Detection) ...................................................................................................................159

Table 8: Test Matrix for Task 1 through Task 5 ..............................................................176

TASK 1

Table 9: Test 1 Metrics WVD, 256, 10 .........................................................................184

Table 10: Test 2 Metrics WVD, 256, 0 .........................................................................185

Table 11: Test 3 Metrics WVD, 256, -2 .......................................................................186

Table 12: Test 4 Metrics WVD, 512, 10 .......................................................................187

Table 13: Test 7 Metrics CWD, 256, 10 .......................................................................189

Table 14: Test 8 Metrics CWD, 256, 0 .........................................................................190

Table 15: Test 9 Metrics CWD, 256, -3........................................................................191

Table 16: Test 10 Metrics CWD, 512, 10 .....................................................................192


Table 18: Test 12 Metrics CWD, 512, -3......................................................................194

Table 19: Test 13 Metrics Spectrogram, 256, 10 ..........................................................195

Table 20: Test 14 Metrics Spectrogram, 256, 0 ............................................................196 Table 21: Test 15 Metrics Spectrogram, 256, -3 ..........................................................197 Table 22: Test 16 Metrics Spectrogram, 512, 10 ..........................................................198

Table 23: Test 17 Metrics Spectrogram, 512, 0 ............................................................199

xxvi

Table 24: Test 18 Metrics Spectrogram, 512, -4 ..........................................................200

Table 25: Test 19 Metrics Scalogram, 256, 10 .............................................................201 Table 26: Test 20 Metrics Scalogram, 256, 0 ...............................................................202 Table 27: Test 21 Metrics Scalogram, 256, -2 ..............................................................203 Table 28: Test 22 Metrics Scalogram, 512, 10 .............................................................204 Table 29: Test 23 Metrics Scalogram, 512, 0 ...............................................................205 Table 30: Test 24 Metrics Scalogram, 512, -3 ..............................................................206 Table 31: Test 25 Metrics Reassigned Spectrogram, 256, 10 .......................................207 Table 32: Test 26 Metrics Reassigned Spectrogram, 256, 0 .........................................208 Table 33: Test 27 Metrics Reassigned Spectrogram, 256, -2 .......................................209 Table 34: Test 28 Metrics Reassigned Spectrogram, 512, 10 .......................................210 Table 35: Test 29 Metrics Reassigned Spectrogram, 512, 0 .........................................211 Table 36: Test 30 Metrics Reassigned Spectrogram, 512, -3 .......................................212 Table 37: Test 31 Metrics Reassigned Scalogram, 256, 10 ..........................................213 Table 38: Test 32 Metrics Reassigned Scalogram, 256, 0 ............................................214 Table 39: Test 33 Metrics Reassigned Scalogram, 256, -2 ...........................................215 Table 40: Test 34 Metrics Reassigned Scalogram, 512, 10 ..........................................216 Table 41: Test 35 Metrics Reassigned Scalogram, 512, 0 ............................................217 Table 42: Test 36 Metrics Reassigned Scalogram, 512, -3 ...........................................218 Table 43: Test 37 Metrics RSPWVD, 256, 10..............................................................219 Table 44: Test 38 Metrics RSPWVD, 256, 0................................................................220 Table 45: Test 39 Metrics RSPWVD, 256, -3 ..............................................................221 Table 46: Test 40 Metrics RSPWVD, 512, 10..............................................................222

xxvii

Table 47: Test 41 Metrics RSPWVD, 512, 0................................................................223 Table 48: Test 42 Metrics RSPWVD, 512, -3 ..............................................................224 Table 49: Test 43 Metrics HT of WVD, 256, 10 ..........................................................225

Table 50: Test 44 Metrics HT of WVD, 256, 0 ............................................................226

Table 51: Test 45 Metrics HT of WVD, 256, -3 ...........................................................227 Table 52: Test 46 Metrics HT of WVD, 512, 10 ..........................................................228 Table 53: Test 49 Metrics HT of CWD, 256, 10 ..........................................................230

Table 54: Test 50 Metrics HT of CWD, 256, 0 ............................................................231

Table 55: Test 51 Metrics HT of CWD, 256, -4 ...........................................................232 Table 56: Test 52 Metrics HT of CWD, 512, 10 ..........................................................233


Table 58: Test 54 Metrics HT of CWD, 512, -6 ...........................................................235 Table 59: Test 55 Metrics HT of RSPWVD, 256, 10 ...................................................236 Table 60: Test 56 Metrics HT of RSPWVD, 256, 0 .....................................................237 Table 61: Test 57 Metrics HT of RSPWVD, 256, -4 ....................................................238 Table 62: Test 58 Metrics HT of RSPWVD, 512, 10 ...................................................239 Table 63: Test 59 Metrics HT of RSPWVD, 512, 0 .....................................................240 Table 64: Test 60 Metrics HT of RSPWVD, 512, -5 ....................................................241 TASK 2

Table 65: Test 61 Metrics WVD, 512, 10 .....................................................................242 Table 66: Test 64 Metrics CWD, 512, 10 .....................................................................244


Table 68: Test 66 Metrics CWD, 512, -3......................................................................246

xxviii

Table 69: Test 67 Metrics Spectrogram, 512, 10 ..........................................................247

Table 70: Test 68 Metrics Spectrogram, 512, 0 ............................................................248 Table 71: Test 69 Metrics Spectrogram, 512, -4 ..........................................................249 Table 72: Test 70 Metrics Scalogram, 512, 10 .............................................................250 Table 73: Test 71 Metrics Scalogram, 512, 0 ...............................................................251 Table 74: Test 72 Metrics Scalogram, 512, -3 ..............................................................252 Table 75: Test 73 Metrics Reassigned Spectrogram, 512, 10 .......................................253 Table 76: Test 74 Metrics Reassigned Spectrogram, 512, 0 .........................................254 Table 77: Test 75 Metrics Reassigned Spectrogram, 512, -3 .......................................255 Table 78: Test 76 Metrics Reassigned Scalogram, 512, 10 ..........................................256 Table 79: Test 77 Metrics Reassigned Scalogram, 512, 0 ............................................257 Table 80: Test 78 Metrics Reassigned Scalogram, 512, -3 ...........................................258 Table 81: Test 79 Metrics RSPWVD, 512, 10..............................................................259 Table 82: Test 80 Metrics RSPWVD, 512, 0................................................................260 Table 83: Test 81 Metrics RSPWVD, 512, -3 ..............................................................261 Table 84: Test 82 Metrics HT of WVD, 512, 10 ..........................................................262 Table 85: Test 85 Metrics HT of CWD, 512, 10 ..........................................................264


Table 87: Test 87 Metrics HT of CWD, 512, -6 ...........................................................266 Table 88: Test 88 Metrics HT of RSPWVD, 512, 10 ...................................................267 Table 89: Test 89 Metrics HT of RSPWVD, 512, 0 .....................................................268 Table 90: Test 90 Metrics HT of RSPWVD, 512, -6 ....................................................269 TASK 3

xxix

Table 91: Test 91 Metrics WVD, 512, 10 .....................................................................270 Table 92: Test 94 Metrics CWD, 512, 10 .....................................................................272 Table 93: Test 95 Metrics CWD, 512, 0 .......................................................................273 Table 94: Test 96 Metrics CWD, 512, -2......................................................................274 Table 95: Test 97 Metrics Spectrogram, 512, 10 ..........................................................275

Table 96: Test 98 Metrics Spectrogram, 512, 0 ............................................................276 Table 97: Test 99 Metrics Spectrogram, 512, -3 ..........................................................277 Table 98: Test 100 Metrics Scalogram, 512, 10 ...........................................................278 Table 99: Test 101 Metrics Scalogram, 512, 0 .............................................................279 Table 100: Test 102 Metrics Scalogram, 512, -3 ..........................................................280 Table 101: Test 103 Metrics Reassigned Spectrogram, 512, 10...................................281 Table 102: Test 104 Metrics Reassigned Spectrogram, 512, 0.....................................282 Table 103: Test 105 Metrics Reassigned Spectrogram, 512, -3 ...................................283 Table 104: Test 106 Metrics Reassigned Scalogram, 512, 10 ......................................284 Table 105: Test 107 Metrics Reassigned Scalogram, 512, 0 ........................................285 Table 106: Test 108 Metrics Reassigned Scalogram, 512, -4.......................................286 Table 107: Test 109 Metrics RSPWVD, 512, 10..........................................................287 Table 108: Test 110 Metrics RSPWVD, 512, 0............................................................288 Table 109: Test 111 Metrics RSPWVD, 512, -3 ..........................................................289 Table 110: Test 112 Metrics HT of WVD, 512, 10 ......................................................291 Table 111: Test 113 Metrics HT of CWD, 512, -3 .......................................................293 Table 112: Test 114 Metrics HT of RSPWVD, 512, -4................................................295

TASK 4

xxx

Table 113: Test 115 Metrics WVD, 512, 10 .................................................................296 Table 114: Test 118 Metrics CWD, 512, 10 .................................................................298 Table 115: Test 119 Metrics CWD, 512, 0 ...................................................................299 Table 116: Test 120 Metrics CWD, 512, -1..................................................................300 Table 117: Test 121 Metrics Spectrogram, 512, 10 ......................................................301

Table 118: Test 122 Metrics Spectrogram, 512, 0 ........................................................302 Table 119: Test 123 Metrics Spectrogram, 512, -1 ......................................................303 Table 120: Test 124 Metrics Scalogram, 512, 10 .........................................................304 Table 121: Test 125 Metrics Scalogram, 512, 0 ...........................................................305 Table 122: Test 126 Metrics Scalogram, 512, -3 ..........................................................306 Table 123: Test 127 Metrics Reassigned Spectrogram, 512, 10...................................307 Table 124: Test 128 Metrics Reassigned Spectrogram, 512, 0.....................................308 Table 125: Test 129 Metrics Reassigned Spectrogram, 512, -2 ...................................309 Table 126: Test 130 Metrics Reassigned Scalogram, 512, 10 ......................................310 Table 127: Test 131 Metrics Reassigned Scalogram, 512, 0 ........................................311 Table 128: Test 132 Metrics Reassigned Scalogram, 512, -2.......................................312 Table 129: Test 133 Metrics RSPWVD, 512, 10..........................................................313 Table 130: Test 134 Metrics RSPWVD, 512, 0............................................................314 Table 131: Test 135 Metrics RSPWVD, 512, -3 ..........................................................315 Table 132: Test 136 Metrics HT of WVD, 512, 10 ......................................................317

Table 133: Test 137 Metrics HT of CWD, 512, -3 .......................................................319

Table 134: Test 138 Metrics HT of RSPWVD, 512, -4................................................321

1

1. Introduction

Today, radar systems face serious threats from anti-radiation systems [FAU02] and from

electronic attack [GAU02]. In order to be able to perform their functions properly, they

must be able to see without being seen [PAC09], [WIL06]. This necessitates that they

be low probability of intercept (LPI) radars. These radars typically have very low peak

power, wide bandwidth, high duty cycle, and power management, making them difficult

to be detected and characterized by intercept receivers. In the radar arena, there is a

current trend towards radars becoming LPI radars.

On the other side of the fence, the intercept receiver that intercepts these LPI radar

signals faces threats as well. It must be able to detect and characterize the signals from

these LPI radars in order to provide timely information about threatening systems, in

addition to providing information about defensive systems - which is important in

maintaining a credible deterrent force to penetrate those defenses. In this case, the

knowledge provided by the intercept receiver provides insight for determining the details

of the threat so that an effective response can be prepared [WIL06].

LPI radars attempt to detect specific targets at a longer range than the intercept receiver

can detect the LPI radar. This means that the success of the LPI radar is measured by

how difficult it is for them to be detected by intercept receivers. Consequently, the

2

properties of LPI radar, such as high duty cycle, low power, wide bandwidth, power

management, make the LPI radar signals difficult to be detected and characterized by

intercept receivers. Examples of intercept receivers are Electronic Support receivers,

Electronic Intelligence receivers, and Radar Warning Receivers.

Different from the older pulsed radar systems, which were relatively easy for an intercept

receiver to detect due to their high peak power, the LPI radar transmitter makes use of

sophisticated phase and frequency modulation to spread the signal bandwidth, making the

signal difficult to intercept. The LPI radar receiver is able to make use of the appropriate

matched filter so that the radars performance is similar to that of traditional pulsed radar

radiating the same amount of average power [GAU02].

To make matters even more challenging for intercept receivers, most of the intercept

receivers currently in the fleet are analog intercept receivers that were designed to

intercept older (non-LPI) radar signals, and which perform poorly when faced with LPI

radar signals. Digital receivers, which are becoming a growing trend, are seen as the

eventual solution to the intercept receivers LPI challenge [PAC09].

In this dissertation, the use of the reassignment method and the Hough transform are

explored as methods for improving the readability of classical time-frequency analysis

representations of LPI radar signals collected by intercept receivers. This improved

readability may lead to more accurate signal detection and parameter extraction metrics

of these signals.

3

The following sections give a brief overview of the current intercept receiver signal

analysis techniques, the classical time-frequency analysis techniques, the reassignment

method, and the Hough transform. Each of these will be explained in more detail in the

background section of this dissertation.

1.1. Intercept Receiver Signal Analysis Techniques

Currently, most of the intercept receivers in the fleet are analog, and since analog

intercept receivers perform poorly when faced with LPI radar signals, digital intercept

receivers represent the growing trend for LPI signal analysis.

To identify the emitter parameters, Fourier analysis techniques using the FFT have been

used as the basic tool of the digital intercept receiver, and make up a majority of the

digital intercept receiver techniques that are currently in the fleet [PAC09]. From a

Fourier Transform representation, the evolution of the frequency content in time cannot

be determined. This is due to the fact that the Fourier Transform is a decomposition of

complex exponentials, which are of infinite duration and completely un-localized in time.

Time information is in fact encoded in the phase of the Fourier Transform (which is

simply ignored by the energy spectrum), but its interpretation is not straightforward and

its direct extraction is faced with a number of difficulties such as phase unwrapping

[ISI96]. Consequently, when a practical non-stationary signal (such as an LPI radar

signal) is processed, the Fourier transform cannot efficiently analyze and process the

time-varying characteristics of the signals frequency spectrum [XIE08], [STE96]. In

4

order to have a more informative description of such signals, it would be better to directly

represent their frequency content while still keeping the time description parameter

which is what time-frequency analysis accomplishes [ISI96], [SOL00].

1.2. Classical Time-Frequency Analysis Techniques

From the basic tool of the FFT, more complex signal processing techniques have

evolved, such as time-frequency analysis techniques. In many operational radar

environments, the non-stationary nature of the received radar signal mandates the use of

some form of time-frequency analysis technique, which can clearly bring out the non-

stationary behavior of the signal. The important virtue of time-frequency analysis is that

it provides an identification of the specific times during which certain spectral

components of the signal are observed. Time-frequency techniques can also be used to

detect and extract the parameters from LPI modulations [MIL02]. For digital intercept

receivers, classical time-frequency analysis techniques are currently at the lab phase,

being analyzed by industry in an attempt to port the time-frequency analysis algorithms

that have been developed to hardware (FPGAs, DSPs) for the purpose of running them in

real-time on realistic IF bandwidths [PAC09].

Four of the classical time-frequency analysis techniques (Wigner-Ville Distribution

(WVD), Choi-Williams Distribution (CWD), Spectrogram, Scalogram) will be described

briefly in this section, and then in more detail in the background section of this

dissertation.

5

WVD: One of the most prominent members of the time-frequency analysis techniques

family is the WVD. The WVD satisfies a large number of desirable mathematical

properties. In particular, it is always real-valued, preserves time and frequency shifts,

and satisfies marginal properties [ISI96], [QIA02]. The WVD, which is a transformation

of a continuous time signal into the time-frequency domain, is computed by correlating

the signal with a time and frequency translated version of itself, making it bilinear. The

WVD exhibits the highest signal energy concentration in the time-frequency plane

[WIL06]. By using the WVD, an intercept receiver can come close to having a

processing gain near the LPI radars matched filter processing gain [PAC09]. The WVD

also contains cross term interference between every pair of signal components, which

may limit its applications [GUL07], [STE96], and which can make the WVD time-

frequency representation very unreadable, especially if the components are numerous or

close to each other, and the more so in the presence of noise [BOA03]. This lack of

readability can in turn translate into poor signal detection and parameter extraction

metrics, potentially placing the intercept receiver signal analyst in harms way.

CWD: The CWD is a member of the Cohens class of time-frequency distributions

which use smoothing kernels [GUL07] to help reduce cross-term interference so

prevalent in the WVD [BOA03], [PAC09], [UPP08]. The reduction in cross-term

interference can make the time-frequency representation more readable and can make

signal detection and parameter extraction more accurate. The down-side is that the

CWD, like all members of Cohens class, is faced with an inevitable trade-off between

cross-term reduction and time-frequency localization. So the signal detection and

6

parameter extraction benefits gained by the cross-term reduction may be offset by the

decrease in time-frequency localization (smearing or widening of the signal).

Spectrogram: The Short-Time Fourier Transform (STFT) will be examined first, and

then connected to the Spectrogram. The Spectrogram is defined as the magnitude

squared of the STFT [HIP00], [HLA92], [MIT01]. The STFT was the first tool devised

for analyzing a signal simultaneously in both time and frequency. The basic idea of the

STFT is to analyze a small part of the signal around the time of interest based on Fourier

methods in order to determine the frequencies at that time. Since the time interval is

short compared to the whole signal, the process is called taking the Short-Time Fourier

Transform [STE96]. The observation window allows localization of the spectrum in

time, but also smears the spectrum in frequency in accordance with the uncertainty

principle, leading to a trade-off between time resolution and frequency resolution. In

general, if the window is short, the time resolution is good, but the frequency resolution is

poor, and if the window is long, the frequency resolution is good, but the time resolution

is poor. For the Spectrogram, its robustness and ease of implementation have made the

spectrogram a popular tool for speech analysis [BOA03], [COH95]. Since the

Spectrogram is a quadratic (bilinear) representation, the Spectrogram of the sum of two

signals is not the sum of the two Spectrograms (quadratic superposition principle); there

is a cross-Spectrogram part and a real part. Therefore, as for every quadratic distribution,

the Spectrogram presents interference terms, however, the interference terms are

restricted to those regions of the time-frequency plane where the signals overlap. If the

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signal components are sufficiently far apart so that their spectrograms do not overlap

significantly, then the interference terms will be nearly identically zero [ISI96].

Scalogram: The wavelet transform will be examined first, and then connected to the

Scalogram. The Scalogram is defined as the magnitude squared of the wavelet transform,

and can be used as a time-frequency distribution [COH02], [GAL05], [BOA03]. The

wavelet transform is of interest for the analysis of non-stationary signals, because it

provides still another alternative to the STFT and to many of the quadratic time-

frequency distributions. The basic difference between the STFT and the wavelet

transform is that the STFT uses a fixed signal analysis window, whereas the wavelet

transform uses short windows at high frequencies and long windows at low frequencies.

This helps to diffuse the effect of the uncertainty principle by providing good time

resolution at high frequencies and good frequency resolution at low frequencies. This

approach makes sense especially when the signal at hand has high frequency components

for short durations and low frequency components for long durations. The signals

encountered in practical applications are often of this type. The wavelet transform allows

localization in both the time domain via translation of the mother wavelet, and in the

scale (frequency) domain via dilations. The wavelet is irregular in shape and compactly

supported, thus making it an ideal tool for analyzing signals of a transient nature; the

irregularity of the wavelet basis lends itself to analysis of signals with discontinuities or

sharp changes, while the compactly supported nature of wavelets enables time

localization of a signals features [BOA03]. Unlike many of the quadratic functions such

as the WVD and CWD, the wavelet transform is a linear transformation; therefore cross-

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term interference is not generated. There is another major difference between the STFT

and the wavelet transform; the STFT uses sines and cosines as an orthogonal basis set to

which the signal of interest is effectively correlated against, whereas the wavelet

transform uses special wavelets which usually comprise an orthogonal basis set. The

wavelet transform then computes coefficients, which represents a measure of the

similarities, or correlation, of the signal with respect to the set of wavelets. In other

words, the wavelet transform of a signal corresponds to its decomposition with respect to

a family of functions obtained by dilations (or contractions) and translations (moving

window) of an analyzing wavelet.

A filter bank concept is often used to describe the wavelet transform. The wavelet

transform can be interpreted as the result of filtering the signal with a set of bandpass

filters, each wi

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