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Lehrstuhl fur HochfrequenztechnikTechnische Universitat Munchen
Sparse Overcomplete Representation applied to FMCW Reflectometryfor Non-uniform Transmission Lines
Fengqing Bao
Vollstandiger Abdruck der von der Fakultat fur Elektrotechnik und Informationstechnik derTechnischen Universitat Munchen zur Erlangung des akademischen Grades eines
-Doktor-Ingenieurs-
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr.-Ing. Eckehard Steinbach
Prufer der Dissertation: 1. Univ.-Prof. Dr.-Ing. Dr.-Ing. habil. Jurgen Detlefsen (i.R.)
2. Univ.-Prof. Dr.-Ing. Norbert Hanik
Die Dissertation wurde am 22.04.2015 bei der Technischen Universitat Munchen eingereicht unddurch die Fakultat fur Elektrotechnik und Informationstechnik am 06.10.2015 angenommen.
i
Preface
This thesis was written during my scientific work at the chair of High Frequency Technology atthe Technical University of Munich. First of all, I would like to express my sincere thanks to mysupervisor Prof. Dr.-Ing. J. Detlefsen for fruitful discussions and continuous support during myfive years at HFS. Thanks for giving me the opportunity to work on the interesting topic of radarsensors. Special thanks are due to all the members at HFS as well as at the workshop for a verymotivating and pleasant working environment.
Furthermore, I would like to thank Prof. Dr.-Ing. N. Hanik for accepting to be the secondexaminer of my thesis and Prof. Dr.-Ing. E. Steinbach for heading the committee of examiners.
My thanks also goes to Prof. Dr.-Ing. E. Schrufer for his continuous support for CDHK program.His great work let us feel life in Munich is the same warm as that in our homeland.
Moreover, I would like to thanks all my Master students for their research work in the topic of in-verse scattering theory and in the field of compressive sensing. Without their scientific contributions,this research project would not have been possible.
In particular, I would like to thank my beloved family for continuous support and encouragement.
Last but not least, I appreciate Dr. E. Blumcke at Audi AG for creating the project and AutolivAG in Dachau for providing the laboratory.
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Abstract
In this thesis, microwave FMCW reflectometry is applied to inhomogeneous transmission linestructures for monitoring purposes. Compared to the conventional TDR (Time Domain Reflec-tometry) and VNA (Vector Network Analyzer) methods, the application of FMCW principleleads to a simplified hardware architecture. It provides a cost efficient solution for the detectionof rapidly time-varying impedance variations. The classical approach for FMCW signal pro-cessing typically uses Fourier analysis, since the time domain response, which gives informationabout the spatial structure of the inhomogeneities, and the frequency domain response obtainedby a VNA or by FMCW reflectometry are directly linked by a Fourier transform. Due to finitemeasurement bandwidth, small spectral responses can be masked by the occurring side-lobes.The inhomogeneities caused by distributed impedance variations are difficult to be localized andidentified in presence of spectrum overlap or dispersion effects. In principle, the impulse responseof a lossy transmission line, which is obtained by a FMCW reflectometer, can be represented bya deterministic convolution of the inhomogeneities with the point spread function. Assumingthat the distribution of those inhomogeneities can be represented by a superposition of basisfunctions with a sparse coefficient vector, the underdetermined inverse problem is possible tobe solved by using the sparse recovery algorithms. The methods are successfully applied to anonline monitoring application, where the prototype FMCW reflectometer successfully capturesreflections of a transmission line which is firmly attached to an airbag. The reflection patterns,which are rapidly changing during airbag deployment can be acquired with sufficient temporalresolution. It is further shown that the proposed sparse processing technique is additionallyable to provide spatial super resolution.
Zusammenfassung
In dieser Arbeit wird die Mikrowellen FMCW Reflektometrie auf inhomogene Ubertragungs-leitungen fur Uberwachungszwecke angewendet. Im Vergleich zu dem herkommlichen TDR(Time Domain Reflectometry) und VNA (Vector Network Analyzer) Methoden stellt das FMCW-Prinzip eine kostengunstige Losung fur die Erkennung von schnell zeitveranderlichen Impedanz-variationen auf der Leitung dar. Die Fourier-Analyse wird haufig als klassische Methode fur dieVerarbeitung von FMCW Signalen angewendet, da die Impulsantwort, die Aufschluss uber denortlichen Verlauf der Leitungsinhomogenitaten gibt, und die von ein VNA oder FMCW Reflek-tometrie erfasste Antwort im Frequenzbereich direkt uber eine Fourier-Transformation verbun-den sind. Aufgrund der endlichen Messbandbreite konnen kleine spektrale Anteile durch Neben-maxima verdeckt werden. Außerdem sind dadurch Reflexionen, die durch verteilte Impedanz-variationen verursacht werden, schwer zu lokalisieren und zu identifizieren. Auf einer verlust-behafteten Leitung kann die Impulsantwort durch Faltung der Verteilung der Inhomogenitatenmit der Punktbildfunktion dargestellt werden. Unter der Annahme, dass die Verteilung derInhomogenitaten als Uberlagerung von Basisfunktionen mit einem sparlich besetzten Koeffizien-tenvektor dargestellt werden kann, lassen sich Losungen fur das unterbestimmte inverse Problemfinden. Die Ansatze werden fur die Online-Uberwachung der Entfaltung eines Airbags einge-setzt. Dabei konnen die Anderungen der Reflexionen einer Leitung, die mit dem Airbag festverbunden ist, mit ausreichend hoher Datenrate erfasst werden. Es zeigt sich, dass die Meth-oden fur diese Aufgabe gut brauchbar sind und sich die angestrebte hohe raumliche Auflosungerreichen lasst.
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List of symbols
A amplitude variable
A sensing matrix
B signal bandwidth
~B magnetic flux density
C(x), S(x) Fresnel integrals
C ′, R′, L′, G′ (Per unit length) capacitance, resistance, inductance and conductance
D dictionary matrix
~D electric flux density
e(t) instantaneous voltage across the diode mixer
~E electric field
fL start frequency
fc cutoff frequency
fs physical sampling rate
F, F−1 operator of Fourier transform
h(t), h(n) impulse response (analog, discrete)
hH(), HH() operator of Hilbert transform in time domain and frequency domain
~H magnetic field
i electric current
i0 saturation current of a junction diode
i, j, n, k index variables
j imaginary unit, j2 = 1
I() Fisher information matrix
~l line element
k complex propagation constant of electromagnetic waves
N number of samples per sweep (FMCW)
N1 factor of gridding refinement
p() likelihood function
qloss attenuation term in Zakharov-Shabat system
q±() potential function in Zakharov-Shabat system
rd linear resistance of semiconductor junction
r nonlinear resistance of the diode mixer
iv
r0 residual
R range variable
Rss surface resistance of the conductors
Rmax maximum unambiguous detection range
Rs linear resistance equivalent of input impedance
RL linear resistance equivalent of load impedance
R the set of real numbers
∆R radar resolution by Fourier analysis
st, sr transmitted signal and reflected signal
Si diagonal selection matrix
S fixed open surface with boundary ∂S
S sparsity, the number of nonzero elements in a vector
~S line element
t time variable
Tm time domain spike sequences
uB(t), uB(n) FMCW beat signal (analog, discrete)
UB spectrum of uB
v instantanteous value of voltage
vs wave propagation velocity
wl, wr normalized wave (leftward propagating and rightward propagating)
~x, ~y, ~z unit vector of Cartesian coordinate system
xx traveling time
y measurement vector
Z characteristic impedance
Zin input impedance
α attenuation constant
αN nolinear coefficient
αt length of truncation
α bend angle
β phase constant
Γ,T reflection coefficient, transmission coefficient
∆Γ,∆T local reflection coefficient, local transmission coefficient
tan δ = ε′′
ε′ is the loss tangent of the material.
v
Γl,Γr left side reflection coefficient, right side reflection coefficient
ε noise, measurement fluctuations
ε permittivity
ε′, ε′′ real and imaginary part of permittivity
η0 intrinsic impedance of the dielectric material
Θ unbiased estimator vector
φ instantaneous phase
µ permeability
τ delay variable
µ mutual coherence of a marix
Ψ orthogonal projection matrix
ω angular frequency
[C] convolution matrix[FN,J
]partial Fourier matrix
[Pφ] phase shift matrix
[Ws] attenuation matrix
‖‖p p-norm
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Contents
1 Introduction 1
1.1 Microwave reflectometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objective and contents of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Forward and inverse problem of transmission line structures 5
2.1 Brief review of transmission line theory . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Perspective of field analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Time domain telegrapher equations . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3 Wave propagation and characteristic impedance . . . . . . . . . . . . . . . . . 9
2.2 Forward modeling of a non-uniform transmission line . . . . . . . . . . . . . . . . . . 11
2.3 Inverse scattering on non-uniform transmission line . . . . . . . . . . . . . . . . . . . 12
2.3.1 Frequency approach via solving Zakharov-Shabat equations . . . . . . . . . . 14
2.3.2 Discrete time domain model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Influence of finite observation bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Discretization error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Numerical simulation of an inverse scattering problem for a lossless non-uniform trans-mission line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6.1 Transmission line Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6.2 Recovery with sufficiently wide spectrum . . . . . . . . . . . . . . . . . . . . 23
2.6.3 Recovery with insufficient spectrum . . . . . . . . . . . . . . . . . . . . . . . 25
3 Microwave FMCW reflectometry 27
3.1 Fundamental of FMCW principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1 Linear sawtooth modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.2 Homodyne receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.3 Range resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Spurious intermodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Mixer intermodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.2 Analysis of intermodulation via diode current-voltage characteristic . . . . . 33
3.2.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 FMCW principle for lossless non-uniform transmission line . . . . . . . . . . . . . . 38
3.3.1 Discrete forward modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.2 Determination of discrete impulse response via DFT . . . . . . . . . . . . . . 39
3.4 Influence of measurement duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
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3.4.1 Cramer Rao Lower Bound for single frequency tone . . . . . . . . . . . . . . 40
3.4.2 Cramer Rao Lower Bound for two frequency tones . . . . . . . . . . . . . . . 44
4 Localization and identification via sparse overcomplete representation 46
4.1 Sparse representation techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1.1 Sparse overcomplete representation . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.2 Sparse recovery algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.3 Uniqueness conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.4 Running time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Sparse representation for FMCW responses . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.1 Representation in frequency domain . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.2 Convolution in time domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Regularization parameter and off-grid mismatch . . . . . . . . . . . . . . . . . . . . . 57
4.4 Analysis of the separation condition by numerical simulation . . . . . . . . . . . . . 61
4.4.1 A conflict of gridding refinement and uniqueness condition . . . . . . . . . . . 61
4.4.2 Minimum separation condition . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.3 Numerical study on approximate reconstruction . . . . . . . . . . . . . . . . . 63
4.5 Post processing based on time domain convolution model . . . . . . . . . . . . . . . 68
4.5.1 Efficient sparse representation via segmentation . . . . . . . . . . . . . . . . . 68
4.5.2 Deterministic deconvolution with range dependent spread function for lossytransmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.6 Sparse overcomplete representation for localization and classification of distributedimpedance variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6.1 Representation by equivalent spread function of the distributed impedancevariations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6.2 Classification of inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.7 Denoising by averaging compressive sensing . . . . . . . . . . . . . . . . . . . . . . . 75
4.8 Removing the amplitude bias by least square fitting . . . . . . . . . . . . . . . . . . 76
4.9 Simulative comparison of high resolution techniques . . . . . . . . . . . . . . . . . . 77
5 Construction of FMCW reflectometer 79
5.1 Frontend design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Reducing the VCO nonlinearity by a predistorted look-up table . . . . . . . . . . . . 80
5.3 Reconstruction of the complex beat signal via Hilbert transform . . . . . . . . . . . 82
5.3.1 Hilbert transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
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5.3.2 Reduction of the reconstructed error of the imaginary part by using a windowfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4 Calibration techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4.1 SOL (Short Open Load) calibration . . . . . . . . . . . . . . . . . . . . . . . 85
5.4.2 Phase compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.5 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.5.1 Inverse scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.5.2 Classification by sparse representation . . . . . . . . . . . . . . . . . . . . . . 93
6 Online acquisition of airbag deployment 95
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.1.1 Underlying physics and the sensing line . . . . . . . . . . . . . . . . . . . . . 96
6.1.2 Challenges and task description . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2 FMCW responses for bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2.1 Treatment of unwanted echoes . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2.2 Parameterized description of the bend . . . . . . . . . . . . . . . . . . . . . . 97
6.2.3 The equivalent spectral response for a bend . . . . . . . . . . . . . . . . . . . 98
6.3 Resolving multiple bends via adaptive sparse deconvolution . . . . . . . . . . . . . . 100
6.3.1 Influence of the regularization parameter . . . . . . . . . . . . . . . . . . . . 102
6.4 Equivalent measurement fluctuations of the FMCW reflectometer . . . . . . . . . . . 104
6.5 Adaptive sparse deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.6 Online measurements of airbag folding and unfolding patterns . . . . . . . . . . . . . 109
6.6.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.6.2 Rollfolding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.6.3 Crunch and compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.6.4 Deployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7 Conclusion 115
8 Appendix 117
8.1 Simulation result of sparse approximate reconstruction for point scatter . . . . . . . 117
8.1.1 By convex optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.1.2 By bayesian learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.1.3 By OMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.2 Autoregressive methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.3 MUSIC algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
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1 Introduction
Inhomogeneity identification on transmission lines has been extensively used for diagnostics and mon-itoring applications. In microwave network analysis, large inhomogeneity indicates poor impedancematching and efficiency loss in energy transmission. In electrical systems, the wire faults such asfray, chafing, short, open or connector problems might lead to failures related to electrical connec-tion. These faults will change the characteristic impedance profile of the electric wires and result ininhomogeneities, so that inhomogeneity detection becomes a standard and reliable method for wirehealth inspection [1,2]. Besides the changes of internal structure, many physical effects such as am-bient conditions, mechanical variations and changes of electrical parameter of the transmission linescould also generate impedance inhomogeneities. Based on this fact, versatile transmission line basedsensors have been developed. Liquid level sensing in oil tank [4, 5], modular and deformable touch-ing surface [6], landside monitoring [7, 8], soil humidity measuring [9] and dielectric characteristicinspection of material [10] are some typical applications.
To inspect the inhomogeneity on the transmission line, a number of methods have been proposed,such as visual inspection, X-ray, infrared imaging, ultrasonic guided wave and microwave reflectom-etry. In particular, microwave reflectometry based method has proven to be the best candidate andpromising technique [3] [11] for the capability of satisfying some contrasting requirements such asthe high versatility, low cost, reliability and real-time response.
1.1 Microwave reflectometry
The approaches of microwave reflectometry includes two main categories: time domain approachand frequency domain approach.
The representative time domain approach is Time Domain Reflectometry (TDR), which period-ically injects a step function or a short pulse into the line under test and measures the reflection si-multaneously. The round trip delay between the transmitted signal and the reflected signal indicatesthe location of those inhomogeneities. Other variation methods like Sequence TDR (STDR) [12,13]and Spread Spectrum TDR (STDR) [14, 15] inject a numerical pseudo noise sequence instead andimplement correlation operation at the receiver to determine the position of the inhomogeneities.
The most accurate frequency approach is by Vector Network Analyzer (VNA), which measuresthe S-parameter of the line under test. The impulse response can then be approximately recoveredby taking the inverse Fourier transform of the S11
1. Other cost effective methods like Standing WaveReflectometry (SWR) [16] and Phase Detection Frequency Domain Reflectometry (PDFDR) [17,18]have also been proposed to measure the length or determine the load of electrical cables.
The frequency domain analysis and the time domain analysis are theoretically equivalent, theirrelationship is given by the Fourier transform. Practically, the differences in sensitivity and inaccuracy of the electronic components, the different system architecture lead to different systemperformances. The inhomogeneity caused by large impedance variations is easy to be detected.To accurately identify and locate the inhomogeneities with small and smoothly varying impedancevariations however requires not only high dynamic range of the reflectometer but also wide spectrumcoverage.
1For TDR, S11 can be determined as the ratio between the Fourier transform of the reflected wave and injectedsignal.
1
On the one hand, TDR requires a narrow pulse to achieve a high range resolution. The powerspectrum of transmitted signal is inversely proportional to the frequency. Namely, a low SNR existsin high frequency components. In addition, direct sampling scheme leads to a very low spatialresolution due to the limitation of current ADC technology [20]. The equivalent time sampling[21,22]/interleaved time sampling [23] are alternatively used in modern TDR Oscilloscope, but againthese architectures are far from simple. Jitter behavior might significantly impact the accuracy ofTDR system if high precision positioning is of concern [24].
On the other hand, the measurement time of a VNA per frequency point is equivalent to 1/IFBW(IF bandwidth) [25], which depends on the settling time of the system internal components (IF filter,PLL, etc.). Because of the system complexity, a VNA is usually more expensive than a TDR andthe measurement of VNA normally takes much more time than that of TDR.
In this work, linear frequency modulated continuous wave (LFMCW) principle is studied for thenon-uniform transmission line, aiming to monitor a rapidly time varying characteristic impedanceprofile. Its operation philosophy exhibits several advantages:
1. Compared to TDR and VNA, a FMCW reflectometer has a simple system architecture, whichprovides a cost effective approach.
2. A FMCW system quickly transmits a swept frequency signal and measures the frequency shiftbetween the transmitted signal and reflections. As the range information is coded in thebeat frequency, the homodyne receiver permits a low physical sampling rate. Consequentlythe direct sampling can be implemented without negative effects. For a measurement of 300points, the modern commercial TDR needs at least 1.2ms(Agilent-54754A) [26] and the fastVNA needs 22ms(NI PXIe-5632) [27] while the proposed FMCW reflectometer takes only0.165ms with a physical sampling rate of only 2MS/s. Therefore, the FMCW reflectometry ismore suitable for online monitoring rapidly time varying inhomogeneities of the transmissionline.
3. FMCW radar uses principally a continuous transmission, a signal of higher energy can beevaluated compared with TDR.
One problem of FMCW reflectometry can be that the nonlinearity of VCO will seriously dis-turb the accuracy of localization. For long range application, a closed loop control of VCO [28]is recommended to counteract the VCO nonlinearity. Such negative feedback control is capable toreduce even the VCO nonlinearity due to the temperature drift over time. The lock time of PLLand transient response of the phase detector will however slow down tuning speed. In short rangeapplications, a predistortion lookup table [29], as applied in this work, is sufficient to correct for theinherent nonlinearity of VCO transfer function.
1.2 Objective and contents of the thesis
Early investigations on microwave FMCW reflectometry for detection of inhomogeneities on trans-mission lines date back to 1970s [30]. In 1990s, Geck [32] and Kamdar [31] use standard laboratoryequipments to measure the S-parameter of microwave network by FMCW principle. So far thereare still many theoretical and practical questions of microwave FMCW reflectometry need to be an-swered for transmission line based monitoring applications. For example, what is the inverse model
2
of FMCW principle for the transmission line structures? What factors influence the measurementaccuracy? Could FMCW reflectometry provide comparable results to that obtained by a VNA?
In this thesis, the theoretical analysis of FMCW principle for inhomogeneous transmission lines isworked out with respect to different aspects, including the inverse model, the measurement accuracyas well as the relevant signal processing. With the knowledge of impulse response, the characteristicimpedance profile of a lossless non-uniform transmission line can theoretically be reconstructed byusing inverse scattering methods. Due to the finite observation bandwidth, the measured impulseresponse is however in practice a low pass filtered version of the true impulse response. As theclassical Fourier analysis suffers from side lobe leakage, a number of methods are proposed overthe last decades for resolution enhancement. But many of them are based on point scatter model,which is invalid for monitoring distributed impedance variations. The problem might become morecomplex and a challenge on lossy transmission lines where dispersion effects appear. In [34], anadaptive filter based approach to solve for the inverse transfer function has been proposed to recoverthe local impedance of the transmission line in presence of dispersion effects. In [33], a responsemodel based on ABCD matrix has been established for inspect an impedance discontinuity on a lossycoaxial cables. But the performance of these algorithms is still a question for specified monitoringapplications, where serious spectrum overlap exists or the number of inhomogeneities is unknown.
Since the impedance response obtained by a FMCW reflectometer can be described by a convo-lution mechanism, naturally a deconvolution strategy could be applied to remove the convolutionkernel. The challenges lie in two points:
1. The dispersion effect results in a range dependent point spread function. The inverse problemcan not be easily solved by the conventional deconvolution techniques.
2. Due to the finite observation bandwidth, the inverse problem is intrinsically ill-posed. Theproper result selected from all possible solutions must be not only robust to measurementnoise, but also close to the reality.
In principle, the dispersion effect affects the convolution mechanism in a deterministic way, sothat the forward signal representation of FMCW reflectometry is possible to be correctly establishedwith the knowledge of the physical parameters of the transmission line or by taking some referencemeasurements.
Generally, there is however no universal solution to the second point. Preconditions are requiredto derive a proper solution. In this thesis, the precondition of ’sparsity’ is assumed for the dis-tribution of the inhomogeneities on the transmission lines. In other words, the distribution of theinhomogeneities can be represented by a set of basis functions, which contain the intrinsic informa-tion of those inhomogeneities, with a sparse coefficient vector indicating their reference locations.With a proper forward model, the sparse representation techniques can be used for solving theunderdetermined inverse problem.
The remainder of this thesis is structured as follows. In Chapter 2, the general forward andinverse problems on the transmission line are analyzed. A simulation study of the inverse scat-tering method for transmission line structures is presented in the end. Chapter 3 starts with theFMCW principle. The analysis on the localization accuracy is performed by examining the CRLB(Cramer Rao Lower Bound) of range estimation. Chapter 4 focuses on sparse representation ap-plied to FMCW signal processing. The forward signal representation of FMCW reflectometry forinhomogeneous transmission lines is established. The performance of sparse recovery algorithms
3
for deconvolution is analyzed in detail. The frontend design of microwave FMCW reflectometer,followed by the pertinent calibration techniques and phase compensation method, is presented inChapter 5. An industrial application for the monitoring the airbag deployment, where acquisitiontime is of concern, is demonstrated in Chapter 6. An adaptive strategy is proposed in order to derivethe appropriate sparse regularization parameter. Finally, the last chapter gives the conclusion.
4
2 Forward and inverse problem of transmission line struc-tures
It is well known that the distribution of electric and magnetic fields in time and space can bedetermined by solving the Maxwell’s equations, given the boundary conditions and the materialproperties. However, such field analysis usually offers much more information than one really wantor need. It is often that the circuit designers are only interested in voltage-current specification atsome terminals in microwave circuit analysis and take little care of the fields and charges distributionfor low frequency application. If every time a full field analysis by solving Maxwell’s equationsmust be performed, the analysis procedure becomes too complex and pays extra for unnecessarycalculations.
In 1874, Oliver Heaviside showed that when the fields propagate in the transverse electric andmagnetic mode (TEM) along two or more conductors, the voltage-current specification can be fullydescribed by the telegrapher equations. This is the transmission line theory, which bridges the gapbetween field analysis and basic circuit theory.
From the perspective of field analysis, the exclusive TEM mode is only valid on an ideal trans-mission line with uniform cross section and infinite length. Any conductive losses and geometricalchanges along the transmission line lead to longitudinal components of the electric field or magneticfield and excite additional modes consequently. As the field distributions become much more in-volved, oversimplified model might lead to erroneous results. In this case one has to apply the fullwave field analysis [35].
Nevertheless, at low frequency the higher modes can be neglected due to their rapid decaycompared to the dominant TEM mode on a transmission line in most applications. Such structureis called quasi-TEM mode, where the transmission line theory also works well and thus greatlyfacilitates the microwave circuit analysis.
2.1 Brief review of transmission line theory
2.1.1 Perspective of field analysis
The pure TEM mode can be supported by multiple infinitely long lossless conductors with identicalcross section in a uniform medium. Simple examples are the coaxial cable or parallel wires.
Fig. 1 depicts the electromagnetic fields around parallel wires in a homogenous medium. Withoutloss of generality, it is assumed that the field propagates along the z axis in a Cartesian coordinatesystem. In the source free region, the fields should satisfy Maxwells equations
5× ~E = −∂~B
∂t(1)
5× ~H =∂ ~D
∂t(2)
5 · ~D = 0 (3)
5 · ~B = 0 (4)
5
Figure 1: Parallel wires in a homogenous medium.
Under the assumption of TEM mode, the electric field and magnetic field can be expressed in ageneral form
~E = [~xEx(x, y) + ~yEy(x, y)]fe(z)~H = [~xHx(x, y) + ~yHy(x, y)]fh(z)
(5)
where fe(z) and fh(z) are scalar z dependent functions. Integrating the electric field along the Eline from point b to point a yields the voltage difference between two conductors
v = v(a)− v(b) =
∫ a
b
~Ed~l (6)
Similarly, the current can be obtained by integrating the H line along the boundary of an opensurface S enclosing a conductor
i =
∮∂S
~Hd~S (7)
6
Cross multiplying (1) with ~z and integrating it over the path ab yields∫ a
b
(5× ~E × ~z + jωµ ~H × ~z)d~l =∂v
∂z+
∫ a
b
(jωµ ~H × ~z)d~l
=∂v
∂z+
∫ a
b
−∂~E
∂zd~l
=∂v
∂z+
∫ a
b
−∂fe(z)∂z
(~xEx + ~yEy)d~l = 0
(8)
Since ∂ ~B∂t = jωµ ~H, it can be obtained from (1)
~H = −5×~E
jωµ=
1
jωµ
∂fe(z)
∂z(−~xEy + ~yEx) (9)
Substituting (9) into (7)∂fe(z)
∂z=
jωµi∮∂S
(−~xEy + ~yEx)d~S(10)
which can be inserted into (8) and gives
∂v
∂z+ jωL′i = 0 (11)
where L′ represents the per unit length series inductance, independent of z
L′ = µ−∫ ab
(~xEx + ~yEy)d~l∮∂S
(−~xEy + ~yEx)d~S(12)
Next, cross multiplying (2) also with ~z and integrating it over the boundary of the surface S yields∮∂S
(5× ~H × ~z − jωε ~E × ~z)d~S =∂i
∂z− jωεfe(z)
∮∂S
(~xEy − ~yEx)d~S = 0 (13)
where the z dependent function can be derived from (6)
fe(z) =v∫ a
b(~xEx + ~yEy)d~l
(14)
which is inserted into (13) and shows
∂i
∂z+ jωε
∮∂S
(−~xEy + ~yEx)d~S∫ ab
(~xEx + ~yEy)d~lv = 0 (15)
The imaginary part of the complex permittivity ε = ε′ − jε′′ = ε′(1 − j tan δ) contributes to thedielectric losses2, so (15) can be written as
∂i
∂z+ (jωC ′ +G′)v = 0 (16)
2See remark in list of symbols.
7
where
C ′ = ε′∮∂S
(−~xEy + ~yEx)d~S∫ ab
(~xEx + ~yEy)d~l, G′ = tan δω
∮∂S
(−~xEy + ~yEx)d~S∫ ab
(~xEx + ~yEy)d~l(17)
which are the z independent capacitance and conductance per unit length, respectively.
Equation (11) and Equation (16) are the frequency domain form of telegrapher equations. Inpresence of small conductive loss, a new quantity R′ denoting series per unit length resistance willappear in the (12) so that
∂v
∂z+ (jωL′ +R′)i = 0 (18)
2.1.2 Time domain telegrapher equations
The time domain form of telegrapher equation can be obtained by applying the inverse Fouriertransform of (18) and (16). It could be also derived from the perspective of the basic circuit theory.Figure. 2 depicts a part of transmission line, on which the wave propagates along the longitudinaldirection z axis. The transmission line can be viewed as a cascade of infinitely many microscopicsegments with physical length ∆z. Suppose that the length of the segment is much smaller than thesignal wavelength, each segment can be modeled by a lumped element circuit.
Figure 2: Representation of the transmission line by lumped element circuit models.
8
Within a single segment, the circuit voltage and current should hold the Kirchhoffs voltage lawand Kirchhoffs current law, so that
v(z, t) = R′i(z, t)4z + L′∂i(z, t)
∂t4z + v(z +4z, t) (19)
and
i(z, t) = G′v(z +4z, t)4z + C ′∂v(z +4z, t)
∂t4z + i(z +4z, t) (20)
Dividing (19) and (20) by 4z and taking the limit as 4z → 0 gives the time domain form oftelegrapher equations
∂v(z, t)
∂z+ L′
∂i(z, t)
∂t+R′i(z, t) = 0 (21)
∂i(z, t)
∂z+ C ′
∂v(z, t)
∂t+G′v(z, t) = 0 (22)
2.1.3 Wave propagation and characteristic impedance
Substituting (16) and (18) into each other yields the wave propagation equations with respect tov(z, w) and i(z, w)
d2v(z, ω)
dz2− k2v(z, ω) = 0 (23)
d2i(z, ω)
dz2− k2i(z, ω) = 0 (24)
where the complex propagation constant is determined by
k = α+ jβ =√
(jωL′ +R′)(jωC ′ +G′) (25)
In addition, the characteristic impedance of the transmission line is defined as
Z(z) =
√jωL′ +R′
jωC ′ +G′(26)
In general, both of k and Z(z) are a function of frequency. However, the microwave transmissionline which is originally designed for carrying the electric power from one side to another with lowloss, can be properly modeled as a lossless line or low loss line.
For a low lossy line both conductor and dielectric loss will be small(R′ wL′, G′ wC ′). Thefirst order approximation of the complex propagation constant can be found in [35] as
k = α+ jβ ' 1
2(
R′√L′/C ′
+G′√L′/C ′) + jω
√L′C ′ (27)
By the same order of approximation, the characteristic impedance can be approximated equal tothat of the line in the absence of losses
Z(z) =
√jωL′ +R′
jωC ′ +G′'√L′/C ′ (28)
9
which shows that the characteristic impedance of a practical transmission line is approximatelyindependent of the frequency.
In a homogeneous transmission line which has a uniform characteristic impedance, the incidentwave will always propagate along the incident direction without any reflection. If there is a stepchange of the characteristic impedance, which is called impedance discontinuity, a fraction of theincident wave will reflect back off the discontinuity while the rest part will go through it. Whenthere are two impedance discontinuities on the transmission line, the incident wave will bounce backand forth between the discontinuities and generate multiple reflections.
Figure 3: Transmission and reflection of an impedance discontinuity.
Fig. 3 shows the transient behavior of a voltage wave vinc incident from z < 0 to an impedancediscontinuity at position zi on the transmission line. The incident voltage wave is splitted into twoparts. A fraction of the voltage wave will be reflected back traveling in the opposite direction whilethe rest voltage wave proceeds propagating in the incident direction. The local reflection coefficientgenerated by the impedance discontinuity at position zi is determined by
4Γ(zi) =vrefl(zi +4z)
vinc(zi)=Z(zi +4z)− Z(zi)
Z(zi +4z) + Z(zi)(29)
The transmission coefficient 4T(z) at the same position is related to the reflection coefficient by
1 = 4Γ(z) +4T(z) (30)
10
where 4Γ(z) is called local reflection coefficient or partial reflection coefficient, which represents theincremental reflection coefficient of the impedance step at position z.
Definition 1 (Inhomogeneity) An inhomogeneity is defined as a continuous region [z1, z2] wherelocal reflection coefficient ∆Γ(z), z ∈ [z1, z2] is nonzero.
Impedance discontinuity is a special case of inhomogeneity where z1 = z2. When an inhomo-geneity distributes only in a small region whose length is much smaller than the radar resolution∆R3, it can be approximated as an impedance discontinuity in practical applications. The leftsidereflection coefficient Γ represents the sum of all reflection effects at the measurement plane z = 0
Γ =Zin(0)− Z(0)
Zin(0) + Z(0)(31)
where Zin(0) is the input impedance seen looking towards the load and Z(0) represents the scoureimpedance.
If the characteristic impedance profile Z(z) is a smooth function and does not have a singularityin its domain, (29) changes for ∆z → 0 to
dΓ(z) =Z(z +4z)− Z(z)
Z(z +4z) + Z(z)/dz =
dZ(z)
2Z(z)=
1
2
d(lnZ(z))
dzdz (32)
By inverting (32), the characteristic impedance profile Z(z) along the transmission line can becalculated with the knowledge of the local reflection coefficient profile as well as the source impedance
Z(z) = Z(0)e
∫ z0
2dΓ(z)dz(33)
Equations (32) and (33) show that the characteristic impedance profile and local reflection coefficientprofile are acutally equivalent. One could be derived given the information of another.
2.2 Forward modeling of a non-uniform transmission line
The forward modeling methods address the topic how to determine the transient response on thetermination of a transmission line with known characteristic impedance profile. A good forwardmodel should not only be capable to describe the transmission line structure accurately, but also behelpful for understanding the pertinent inversion problem.
For a transmission line shown in Fig. 3, the input impedance seen looking toward the load atdistance z is defined as
Zin(z, ω) =v(z, ω)
i(z, ω)(34)
Substituting (34) into (16) and (18) yields the differential Riccati equation
dZin(z, ω)
dz= jωZ(z)− jωZin(z, ω)2
Z(z)(35)
With the knowledge of characteristic impedance profile, the input impedance at the source termi-nation can be calculated in a recursive way. The leftside reflection coefficient can be determined by(31), given the internal source impedance Z(0).
3z2 − z1 << ∆R
11
Table 1: Forward modeling techniques [36]
Generation Methods Applied techniques
FirstBounce diagram Graphical
Bergeron diagram Graphical
Second
Finite Difference Time Domain Time domain
Generalized bounce diagram Time domain, discrete
(Extended) Signal flow graph Analytical, graphical
ThirdS-parameter method Frequency domain, discrete
ABCD parameters Frequency domain, discrete
Wu has in his dissertation [36] made a survey of the various forward methods and classified theminto three generations as shown in Table 1.
The comparison work by Wu shows that the third generation methods offer excellent fidelity andcomputational efficiency. Therefore the S parameter method is utilized in this thesis.
The concept of S parameter forward model is to decompose a non-uniform transmission lineinto multiple cascaded uniform transmission line networks, whose physical length and characteristicimpedance can be arbitrary specified. An simple example is shown in Fig. 4, where the line is equallydivided into n segments with and identical physical length ∆l. The reflection coefficient Γr(1) atthe right side of segment.1 can be calculated by
Γr(1) =ZL − Z(1)
ZL + Z(1)(36)
so that the reflection coefficient in the left side of segment.1 becomes
Γl(1) = e−2jk∆lΓr(1) (37)
which indicates that the input impedance seen at the left side of segment.1 is equivalent to
Zin(1) = Z11 + Γl(1)
1− Γl(1)(38)
furthermore the reflection coefficient at the right side of segment.2 gives gives
Γr(2) =Zin(1)− Z(2)
Zin(1) + Z(2)(39)
Equations (36) to (39) can be applied in a recursive way until the input reflection coefficient Zin(n)at the measurement plane is obtained.
2.3 Inverse scattering on non-uniform transmission line
The inverse scattering problem deals with the reconstruction of the properties of the scatterer fromscattering data measured outside. Due to the finite measurement bandwidth, the reconstructed
12
Figure 4: A non-uniform transmission line with length L0 is decomposed into n small segments withuniform characteristic impedance.
scatters are a low pass filtered version of the true scatters. Namely, the inverse scattering problemis inherently ill-posed [37]. Therefore, a solution has to be chosen from many possible candidates insolving the inverse scattering problems.
In one dimensional case, the pioneering works were performed by Gel’fand and Levitan [38],Marchenko [39], in which one seeks to determine the coupling coefficients associated with time inde-pendent Schrodinger equation from the reflection and transmission coefficients. Based on these pre-vious work, Jaulent [40] first proved that the general inverse scattering problem for the Schrodingerequation, which is in form of a second-order differential equation, and the Zakharov-Shabat system,which are in form of two first-order differential equations, are equivalent. Later on, he established thetheoretical basis for the inverse scattering problem of non-uniform transmission line by transformingthe Telegrapher’s equations in terms of voltage-current relation to the Zakharov-Shabat system interms of normalized waves. Recently several numerical methods [41,42] have been proposed for solv-ing the inverse Zakharov-Shabat type system by solving the Gel’fand-Levitan-Marchenko (GLM)equations instead (Zakharov-Shabat system is related to GLM equations by a Fourier transform),since the analytical solutions are generally not easy to get.
For a lossy transmission line structure, the frequency form of Telegrapher’s equation can betransformed into Zakharov-Shabat equations involving two different coupling coefficients, which are
13
functions of the distributed L′, C ′, R′, G′ parameters. Because the system is underdetermined, it isnecessary to measure the reflection coefficients from both sides of the transmission line as well as thetransmission coefficients simultaneously. For the case of a lossless transmission line, two couplingcoefficients reduce to a single one, it is theoretically possible to obtain a unique solution from takingonly the reflection measurement but with an infinitive bandwidth.
2.3.1 Frequency approach via solving Zakharov-Shabat equations
Figure 5: A lossless transmission line with length L0 is connected to a reflectometer with internalimpedance of Z(0) at the measurement plane z = 0. The inverse scattering for transmission linestudies how to reconstruct the characteristic impedance profile Z(z) where z ∈ [0, L0] from themeasured input reflection coefficient Γ(0, ω).
Fig. 5 illustrates a part lossless transmission line with unknown characteristic impedance profileZ(z) distributed in region [0, L0] along z axis. At the termination z = 0, the line under test is con-nected to a microwave reflectometer with internal impedance Z(0). It is assumed that a normalizedprobe rightward propagating wave wr(0, ω) is injected onto the transmission line under test and themicrowave reflectometer monitors the leftward propagating reflection wl(0, ω) meanwhile.
To connect the Telegrapher’s equations to general inverse scattering problem, the position vari-
14
able z must be first transformed into traveling time by Liouville transform
xx(z) =
∫ z
0
√L′(z′)C ′(z′)dz′ (40)
From (12) and (17) it can be shown that√L′(z)C ′(z) =
√ε′(z)µ(z) is the wave propagation
velocity at position z, so that xx(z) indicates the time required for a wave traveling from originto the position z. In new coordinate system, the normalized rightward propagating wave and theleftward propagating wave are defined as
wr(xx, ω) =1
2[v(xx, ω)√Z(xx)
+√Z(xx)i(xx, ω)] (41)
wl(xx, ω) =1
2[v(xx, ω)√Z(xx)
−√Z(xx)i(xx, ω)] (42)
where Z(xx) =√L′(xx)/C ′(xx) is the characteristic impedance of the transmission line at the
position relevant to traveling time xx.
The leftside reflection coefficient is calculated from the measured ratio between the incident waveand reflected wave at the measurement plane4
Γ(0, ω) =Zin(0, ω)− Z(0)
Zin(0, ω) + Z(0)=
v(0,ω)i(0,ω) − Z(0)
v(0,ω)i(0,ω) + Z(0)
=wl(0, ω)
wr(0, ω)(43)
Inserting (41) and (42) into (16) and (18), the telegrapher equations are transformed intoZakharov-Shabat system:
dwl(xx, ω)/dxx+ iωwl(xx, ω) = qloss(xx)wl(xx, ω) + q+(xx)wr(xx, ω) (44)
dwr(xx, ω)/dxx− iωwr(xx, ω) = q−(xx)wl(xx, ω)− qloss(xx)wr(xx, ω) (45)
where
q±(xx) = −1
4
d
dxx(ln
L′
C ′)∓ 1
2(R′
L′− G′
C ′) (46)
qloss(xx) =1
2(R′
L′+G′
C ′) (47)
In case of lossless transmission line where R′ = G′ = 0, the potential function q±(xx) =
− 12d(lnZ(xx))
dxx differs from the local reflection coefficient dΓ(xx) with an amplitude scaling of fac-tor − 1
2 . The attenuation term qloss(xx) reduces to 0. The Zakharov-Shabat system [43] for losslesstransmission line is then reduced to
4In general one dimension inverse scattering problem, the theoretic left side reflection coefficient is defined as
limxx→−∞ Γ(xx, ω) =wl(xx,ω)wr(xx,ω)
e2jωxx, which is identical to the measured leftside reflection coefficient in (43) with
phase shift e2jωxx by extending the reflectometer to the infinity with a part of homogeneous transmission line.
15
dwl(xx, ω)/dxx+ iωwl(xx, ω) = −dΓ(xx)wr(xx, ω) (48)
dwr(xx, ω)/dxx− iωwr(xx, ω) = −dΓ(xx)wl(xx, ω) (49)
where the local reflection coefficient dΓ(xx) can be determined by solving the Gelfand-Levitan-Marchenko(GLM) integral equations for unknown kernels K1(xx, y) and K2(xx, y) (See [41,42])
K1(xx, y) +
∫ xx
−yK2(xx, l)h(y + l)dl = 0 (50)
−K2(xx, y) + h(xx+ y) +
∫ xx
−yK1(xx, l)h(y + l)dl = 0 (51)
in region xx ≥ |y| with boundary conditions
K1(xx,−xx) = 0K2(xx,−xx) = h(0)
(52)
where the function h() represents the impulse response , which is calculated by taking inverse Fouriertransform of the leftside reflection coefficient Γ(0, ω) over the whole spectrum
h(x) =1
2π
∫ ∞−∞
Γ(0, ω)e−iωxdω (53)
The unknown kernel K1(xx, y) and K2(xx, y) could be solved by numerical approach in [42]. First,the integral region in xx− y plane (xx ≥ |y|) can be transformed to the first quadrant of the ξ − ηplane through linear transformation:
ξ =xx+ y
2, η =
xx− y2
(54)
In this way, the equations become
B1(ξ, η) + 2
∫ ξ
0
B2(s+ η, ξ − s)h(2s)ds = 0 (55)
−B2(ξ, η) + h(2ξ) + 2
∫ ξ
0
B1(s+ η, ξ − s)h(2s)ds = 0 (56)
B1(0, η) = 0, B2(0, η) = h(0) (57)
∂2B1
∂ξ∂η= −2B2(s, 0) |s=ξ+η
∂B2
∂ξ− 2B2(ξ, η)
dB2(s, 0)
ds|s=ξ+η (58)
∂2B2
∂ξ∂η= 2B2(s, 0) |s=ξ+η
∂B1
∂ξ+ 2B1(ξ, η)
dB2(s, 0)
ds|s=ξ+η (59)
∆Γ(ξ) = −2B2(ξ, 0) (60)
By discretizing the ξ− η plane in rectangular grids, as shown in Fig. 6, (54)-(60) could be rewrittenas
B1(m,n) = −2d
m−1∑s=2
B2(s+ n− 1,m− s+ 1)h(2s− 1)
− dB2(m+ n− 1, 1)h(2m− 1)
(61)
16
Figure 6: Grid structure for the algorithm.
B2(m,n) = 2d
m−1∑s=2
B1(s+ n− 1,m− s+ 1)h(2s− 1)
+ dB1(m+ n− 1, 1)h(2m− 1) + h(2m− 1)
(62)
B1(1, n) = 0, B2(1, n) = h(0) (63)
B1(m+ 1, n+ 1) = B1(m,n+ 1) +B1(m+ 1, n)−B1(m,n)
− 2B2(m+ n, 1)[B2(m,n+ 1)−B2(m,n)]
− 2B2(m,n)[B2(m+ n+ 1, 1)−B2(m+ n, 1)]
(64)
B2(m+ 1, n+ 1) = B2(m,n+ 1) +B2(m+ 1, n)−B2(m,n)
+ 2B2(m+ n, 1)[B2(m,n+ 1)−B2(m,n)]
+ 2B2(m,n)[B2(m+ n+ 1, 1)−B2(m+ n, 1)]
(65)
∆Γ(m) = −2B2(m, 1) (66)
Equation (66) indicates that ∆Γ(x) at diagonal Dg = m+n are the same and (61)-(62) relate all thekernel values at the same diagonal, while the kernel values in the next diagonal could be calculatedby (64) and (65). Once the local reflection coefficient profile is reconstructed, the characteristicimpedance profile along the line can be recovered by using (33).
17
As the whole computational cost of above described algorithm is roughly of O(N4) (N is thedata size of h(n)), an efficient algorithm with computational cost of O(N2) for smoothly variedcharacteristic impedance profile using Born approximation (Neglecting the multiple reflections) couldbe refer to [41, 43]. This efficient algorithm will present large error when large values appear in theimpulse response.
2.3.2 Discrete time domain model
Figure 7: Discrete time model for a non-uniform transmission line. It is assumed that the signalsource is located in segment.0 with an uniform characteristic impedance profile and there is norightward propagating wave inside segment.0 after t = 0+. The sampling interval ∆τ = 2∆l
vs, where
vs is the wave propagation velocity on the transmission line.
The wave propagation on one dimensional transmission line can also be described in terms of the
18
discrete time domain model. By introducing the causal condition of a practical physical system, suchrepresentation provides a clear physical understanding. A well known inverse procedure is known asSchur algorithm [44] or ’layer peeling’ algorithm.
Fig. 7 illustrates the discrete time model for a non-uniform lossless transmission line, where theline is assumed to consist of N equally divided small homogenous segments with physical length ∆l.The reflections are assumed to take place only at the interface between two adjacent segments. Theimpulse response h(n∆τ) sampled at discrete time n∆τ consists of two parts: one is the wavefrontreflection directly from interface between nth segment and (n+1)th segment, the other is the multiplereflections generated by the interfaces before nth segment.
Suppose that an ideal time domain dirac pulse is rightward injected onto the left port of anon-uniform lossless transmission line at time instant 0−, the transmissions and reflections of thetraveling wave can be described by a causal lattice diagram. The wave flows out from lattice (n,m)contains a leftward propagating wave wLn,m as well as a rightward propagating wave wRn,m. The
leftward propagating wave wLn,m will become the left incident wave for the lattice (n,m− 1), while
the rightward propagating wave wRn,m will be the right incident wave for the lattice (n+ 1,m+ 1).This relationship can be generalized as
wLn,m = wRn−1,m−1∆Γ(m) + wLn,m+1(1 + ∆Γ(m)) (67)
wRn,m = wRn−1,m−1(1−∆Γ(m)) + wLn,m+1(−∆Γ(m)) (68)
where ∆Γ(m) is the local reflection coefficient of the interface m. According to the principle ofcausality, the boundary condition at the left measurement plane gives
wLn,1 = h((n− 1)∆τ), n = 1, 2, ..., NwR0,0 = 1,
wR0,m = 0, m = 1, 2, ..., N(69)
from which the rightward propagating wave at lattices in the first column and the leftward propa-gating wave at lattices in the second column can be calculated by
∆Γ(1) = wL1,1 = h(∆τ)wR1,1 = 1−∆Γ(1)
wLn,2 = wLn,1/(1 + ∆Γ(1)) = h(n∆τ)/(1 + ∆Γ(1)), n = 2, 3, ..., NwRn,1 = −wLn,2∆Γ(1), n = 2, 3, ..., N
(70)
The rightward propagating wave throughout the first interface becomes the source of the secondinterface, so that the information of lattices in the second column can be calculated from thatof lattices in the first column. In general, the information of lattices in the mth column can becalculated from that of lattices in the previous column by an recursive method
∆Γ(m) = wLm,m/wRm−1,m−1
wRm,m = wRm−1,m−1(1−∆Γ(m))wLn,m+1 = (wLn,m − wRn−1,m−1∆Γ(m))/(1 + ∆Γ(m)), n = m+ 1, ..., NwRn,m = wRn−1,m−1(1−∆Γ(m))− wLn,m+1∆Γ(m), n = m+ 1, ..., N
(71)
Once the local reflection coefficients ∆Γ(m) have been derived, the characteristic impedance profilecan be recovered by
Z(m) = Z(m− 1)1 + ∆Γ(m)
1−∆Γ(m)(72)
19
In case of small reflections where |∆Γ(z)| 1, the multiple reflection effects become so much smallerthan the wavefront reflections that they could be neglected. An approximation can be used via
∆Γ(z) ≈ h(z) = IDFT Γ(0, w) ,Z(z) = Z(0)e
∑2∆Γ(z) (73)
2.4 Influence of finite observation bandwidth
Figure 8: Convolution mechanism due to the finite observation bandwidth.
Theoretically, the impulse response can be uniquely determined by taking inverse Fourier trans-form of infinitive frequency samples. Any measurement equipment is in practice possible to cover afinite frequency bandwidth. The finite observation spectrum leads to a blurred version f ′1(t) of thetrue impulse response. Such mechanism is illustrated in Fig. 8, where the true impulse response intime domain f1(t) is assumed to contain 3 spikes. Once the frequency probing is implemented overa finite bandwidth B starting from a low frequency fL, it is identical to introduce an rectangularfrequency window F2(f) to truncate the spectrum F1(f). Such multiplication in frequency domainis equivalent to apply a convolution operation on the impulse response f1(t) with the point spreadfunction f2(t) in time domain. This spread function due to the rectangular frequency window can
20
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Normalized time t/B
Am
plitu
de
Magnitude of f
2(t)
Real part of f2(t), f
L=0;
Real part of f2(t), f
L=1GHz
Resolution
Figure 9: Spread function f2(t) in time domain. The start frequency fL will never change thecomplex magnitude envelope of f2(t) but affect its phase.
be determined byf2(t) = F−1(F2(f))
=
∫ fL+B
fL
1ej2πftdf
= ej2πfLtBsinc(πBt)[cos(πBt) + j sin(πBt)]
(74)
which has a magnitude envelope independent of the start frequency fL as illustrated in Fig. 9. Itsfirst zero-crossing point is located at t = ± 1
B5, indicating that the wider the observed spectrum,
the more narrow the main lobe of the convolution kernel and the closer f ′1(t) is to the originalsignal f1(t). Once the time distance between two distinct spikes of f1(t) is smaller than t = ± 1
B , theconvolution process will mask this feature in f ′1(t). This distance corresponds to the range resolutionin radar imaging.
The real part of the convolution kernel <f2(t) depends however on the start frequency fL.When fL = 0, the real part of the convolution kernel has a main lobe width which is a half to thatof the magnitude envelope.
5It could be rendered into the range domain as R = vs2B
for round trip propagation t = 2Rvs
.
21
2.5 Discretization error
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time (t/B)
Nor
mal
ized
am
plitu
de
Scattering on the grids of DFTScattering between the grids of DFT
Grid of DFT
Grid interval of DFT= Range resolution
|1−sinc(πB∆t)cos(πB∆t)|
Figure 10: Illustration of discretization error.
In frequency inverse scattering approach, the time domain impulse response is approximated bythe discrete form of f ′1(n), which is obtained by taking the IDFT (Inverse Discrete Fourier Transform)of finite frequency samples. This discrete representation is carried on a set of fixed time domain gridst = n
B , n ∈ Z. If the true impulse response f1(t) contains a spike exactly located on the discretizedgrid points, the approximation will be perfect as plotted by in Fig. 10. Otherwise, the amplitudedeviation will be introduced by the closest neighbour approximation
|1− sinc(πB∆t) cos(πB∆t)| , 0 < ∆t <1
2B(75)
which depends on the density of interpolation grids in time domain. In addition, the side lobes ofconvolution kernel, which decays along the range axis, will cast a leakage on the other grids. Anextreme example is shown as the gray curve in Fig. 10, where f1(t) contains a spike locating in themiddle of two grid points. Although the reflection coefficient owns a real value of 1, 0 appears inreal part of the reconstructed discrete time domain response.
In practical application, a gridding refinement by zero padding on the tail of the frequencysamples is recommended to make the time domain discretization more dense. One should note thatit will never change the magnitude envelope of the discretized approximation. Therefore, the sidelobe effect of convolution kernel still remains. One might use suitable window function to reducethe amplitude of side lobe at the cost of a wider main lobe width.
22
2.6 Numerical simulation of an inverse scattering problem for a losslessnon-uniform transmission line
In this section, the performance of inverse scattering methods as well as the influence of finiteobservation bandwidth will be investigated by the numerical simulation.
2.6.1 Transmission line Simulator
0 0.5 1 1.5 220
30
40
50
60
70
80True characteristic impedance profile
z(m)
Z(z
) (Ω
)
(a)
0 0.5 1 1.5 2−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4Measured impulse response
z(m)
Mo
du
lus
MagnitudeReal part
(b)
Figure 11: (a) Predefined characteristic impedance profile of a 1.7m long transmission line undertest. (b) Measured impulse response f ′1(n), recovered by frequency samples from 0 to 6GHz.
A discrete lossless transmission line with εeff = 1 is specified with a real-valued characteristicimpedance profile Z(z), z ∈ [0, 1.7]m, as shown in Fig. 11a. At the measurement plane z = 0m, thetransmission line is connected to a reflectometer with source impedance of 50Ω. At the terminationz = 1.7m, the transmission line is assumed to be matched with a 50Ω load.
The physical length of each discrete segment is assumed of 2.5cm, which means that a measure-ment bandwidth of (6q)GHz, where q is natural, can perfectly capture such discrete features. Withthe knowledge of the characteristic impedance profile, physical length of segement and the effectivepermittivity, the input reflection coefficient Γ(0, ω) can be calculated by S-parameter method.
2.6.2 Recovery with sufficiently wide spectrum
In the first simulation, supposing that our measurement of the input reflection coefficient Γ(0, w)covers a finite bandwidth from 0 − 6GHz, the objective of inverse scattering is to recover thecharacteristic impedance profile of the transmission line under test from such bandlimited reflectioncoefficient.
23
The reconstruction procedure can be described as
1. Measure the reflection coefficient ∆Γ(z), ω ∈ [0, 6]GHz.
2. Calculate the measured impulse response by taking discrete inverse Fourier transform
f ′1(n) =1
N
J−1∑m=0
ei2πmnJ Γ(0, ωm), ωm ∈ [0, 6]GHz, n = 1, 2, ...N − 1 (76)
as shown in Fig. 11b, where the measured impulse response presents three large value in therear region z ∈ [1, 1.3]m. Such features come from the three distinct discontinuities of thecharacteristic impedance profile as shown in Fig. 11a.
3. The profile of local reflection coefficient ∆Γ(z) can be calculated by discrete inverse scatteringapproach. A comparison of the recovered local reflection coefficient profile and the measuredimpulse response in time domain is plotted in Fig. 12a. In the front region, where impedancesmoothly varies, ∆Γ(z) and the measured impulse response f ′1(n) are almost identical. Namely,the multiple reflections can be almost neglected for small impedance variation (Born approxi-mation). While in the rear region, where distinct impedance discontinuities exist, the multiplereflections lead to a discrepancy between the profile of local reflection coefficient and the mea-sured impulse response.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
z(m)
Mo
du
lus
Recovered local reflection coefficient vs impluse response
Real part of impulse responseReal part of recoveredlocal reflection coefficient
(a)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.620
30
40
50
60
70
80
z(m)
Z(z
) (Ω
)
Reconstructed characteristic impedance profile
Inverse scatteringIntegration
(b)
Figure 12: (a) Comparison of the local reflection coefficients ∆Γ(z) by using inverse scatteringapproach with the measured impulse response. (b) A comparison of reconstructed characteristicimpedance profile by inverse scattering approach and by directly integral of the measured impulseresponse f ′1(n).
4. The characteristic impedance profile Z(z) can be calculated by (33) from ∆Γ(z), as shown inFig. 12b, where the gray curve is the reconstructed characteristic impedance profile by usingthe inverse scattering approach. It is identical to the true characteristic impedance profile asshown in Fig. 11a. The black dashed line is reconstructed by taking a simple integration ofthe measured impulse responses according to (73).
24
2.6.3 Recovery with insufficient spectrum
In this simulation, the measurement bandwidth of the input reflection coefficient is reduced to0− 2GHz. Namely, the discrete DFT grid is now larger than the features of specified characteristicimpedance profile.
As shown in Fig. 13a, the measured impulse response with standard IDFT now is not sufficientto cover the whole features of specified characteristic impedance profile. This leads to an enormouserror in the recovered characteristic impedance profile by using the inverse scattering approach asshown in Fig. 13b.
0 0.5 1 1.5 2−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
z(m)
Mod
ulus
f1’ (n)
Zero paddingNo zero padding
(a)
0 0.5 1 1.5 240
50
60
70
80
90
100
110
z(m)
Z(z
)(Ω
)
Recovered characteristicimpedance profile byinverse scattering(No zero padding)
(b)
0 0.5 1 1.5 220
30
40
50
60
70
80
Z(m)
Z(z
) (Ω
)
Recovered characteristic impedanceprofile by inverse scattering(Zero padding)True characteristic impedance profile
(c)
Figure 13: Reconstructed transmission line characteristics. (a) Signal response using no zero paddingin comparison to zero padding by a factor of 6. (b) Reconstructed characteristic impedance profileusing no zero padding. (c) True characteristic impedance profile in comparison to the reconstructedcharacteristic impedance profile by using discrete inverse scattering method.
A better reconstruction might be by gridding refinement. As shown in Fig. 13a, the measured
25
impulse response with zero padding, denoted by the gray curve, is still possible to contain themajority of original features of the characteristic impedance variations. The relevant reconstructedcharacteristic impedance profile by the inverse scattering approach is plotted in Fig. 13c. Comparingthis result with that of Fig. 13b, the discrepancy from the true characteristic impedance profileand the reconstructed characteristic impedance profile by using inverse scattering approach hasbeen greatly reduced. The remaining error comes from low pass filtering effect (sidelobe ripples ofconvolution kernel), which is hard to be further reduced in general. However, once the characteristicimpedance profile presents some special features such as sparseness, further improvement might bepossible using sparse overcomplete representation, which will be discussed in Chapter 4.
26
3 Microwave FMCW reflectometry
3.1 Fundamental of FMCW principle
0 1 2 3 4 5 60
1
2
3
4
5
6
Time(S)
Inst
ant
freq
uen
cy(H
z)
Operating frequency over time
(a)
0 1 2 3 4 5 6−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time(S)
Am
plit
ud
e o
f o
utp
ut
vo
lta
ge
(V)
Transmitted signal
(b)
Figure 14: Time and frequency representation of transmitted LFMCW signal.
FMCW radar (Frequency Modulated Continuous Wave Radar) is a special type of CW radar(Continuous Wave Radar), which continuously transmits an unmodulated oscillation of frequency.Simple CW radar has poor ability to measure the distance due to the narrow bandwidth of the trans-mitted waveform. Usually the accurate localization needs wide spectrum coverage of the transmittedsignal. This relation is regulated by the properties of the Fourier transform.
The spectrum of a CW transmission can be broadened by several methods such as AM (AmplitudeModulation), FM (Frequency Modulation) and PM (Phase Modulation) [45]. For instance, pulsemodulation is an example of AM wherein the carrier frequency is gated at a pulsed rate. Thenarrower a pulse in time domain, the broader spectrum in frequency domain so that the moreaccurate position result can be expected. In contrast to AM, FM sweeps the operating frequencyover a wide bandwidth during the measurement time T . The echo signal from a stationary pointscatter is a delayed version of the transmitted signal. The transmitted signal as well as the reflectedsignal present a difference in instantaneous frequency as a function of the delay f(τ) related tothe modulation type. A homodyne receiver is often used to measure such instantaneous frequencydifference so that the delay τ can be calculated by f−1(). With the knowledge of the propagationvelocity of electromagnetic wave in the medium as well as the estimated delay, the difference of thewave propagation distance can be determined.
The frequency modulation may take many forms, e.g, linear modulation6 (LFMCW), steppedfrequency modulation, sinusoidal modulation depending on the applications [45]. Fig. 14 lists the
6Linear modulation is versatile and includes several patterns: sawtooth modulation, triangular modulation etc.
27
time and frequency representation of linear modulation. In practice, LFMCW is the most extensivelyused due to the possibility of a wide variety of hardware realization and efficient digital signalprocessing based on FFT (Fast Fourier Transform) [46]. For that reason almost all recent FMCWradar applications use LFMCW, on which this thesis concentrates.
3.1.1 Linear sawtooth modulation
Figure 15: Sawtooth modulation.
Fig. 15 depicts a saw-tooth form linear frequency modulation, where the instantaneous frequencyof nth sweep is determined by
f(t) =
fL + B
T (t− nTt), t ∈ [nTt, nTt + T ]fL, t ∈ [nTt + T, (n+ 1)Tt]
(77)
where fL denotes the start frequency, T is the modulation duration, Tt represents the sweep durationand B is the bandwidth. In case of successive measurements, the time interval [nTt + T, (n+ 1)Tt]between two adjacent repetitions depends on the settling time of frequency source. It should bechosen large enough that ensures a stable frequency hopping from fL+B to fL. The samples duringthis period are not used and will be discarded in data processing. The phase of transmitted signalφ of one ramp can be determined via integration of (77)
φ(t) = 2π
∫ t
0
f(t′)dt′ = 2πfLt+ πB
Tt2 + φ0 (78)
where the integral constant φ0 represents an arbitrary initial phase term.
The transmitted LFMCW signal with normalized amplitude can be expressed as
st(t) = cos(φ(t)) = Reejφ(t)
(79)
28
−10 −5 0 5 10−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
x
Am
plitu
de
C(x)S(x)
Figure 16: Normalized Fresnel integrals: S(x) and C(x).
whose spectrum can be calculated by Fourier transform
St(f) =
∫ T
0
st(t)e−j2πftdt =
1
2(
∫ T
0
ejφ(t)e−j2πftdt+
∫ T
0
e−jφ(t)e−j2πftdt) (80)
where the first integration term represents the positive spectrum while the second term denotes thenegative spectrum. With substitution α = πB
T , β1 = 2π(fL − f) and β2 = 2π(fL + f), the positivespectrum and the negative spectrum can be calculated by
St(f+) =1
2
∫ T
0
ejφ(t)e−j2πftdt =1
2
√π
2αe−j(
β21
4α−φ0)[C(x) + jS(x)] |√
2πα (αT+
β12 )
β12
√2πα
(81)
St(f−) =1
2
∫ T
0
e−jφ(t)e−j2πftdt =1
2
√π
2αej(
β22
4α+φ0)[C(x)− jS(x)] |√
2πα (αT+
β22 )
β22
√2πα
(82)
where C(x) and S(x) are known as Fresnel Integrals [47] and defined as
C(x) =
∫ x
0
cos(π
2t2)dt (83)
S(x) =
∫ x
0
sin(π
2t2)dt (84)
29
−6 −4 −2 0 2 4 6−50
−40
−30
−20
−10
0
10
Frequency (GHz)
Nor
mal
ized
am
plitu
de (d
B)
T=1µ sT=1msT=1s
1 1.2−1
0
1
Figure 17: Power spectrum of transmitted LFMCW signals under various modulation duration T .In this simulation, the parameters of FMCW signal are fL = 1GHz, B = 4GHz.
As shown in Fig. 16, both Fresnel Integrals are odd functions. When x → ∞, they converge toconstant 1
2
√π2 . The squared magnitude of the spectrum is calculated by
|St(f)|2 = |St(f+)|2 + |St(f−)|2
=T
8B
[C(x1)− C(x2)]2 + [S(x1)− S(x2)]2
+
T
8B
[C(x3)− C(x4)]2 + [S(x3)− S(x4)]2
(85)
where x1 =√
2πα (αT + β1
2 ), x2 = β1
2
√2πα , x3 =
√2πα (αT + β3
2 ) and x4 = β2
2
√2πα . Fig. 17 shows the
normalized power spectrum of LFMCW signals. Although the transmitted frequency span rangesfrom 1GHz to 5GHz, large outband leakages as well as inband fluctuations can be observed inthe spectrum of transmitted signal with shortest modulation duration T = 1µs. However, as themodulation duration increases, the power spectrum of LFMCW signals becomes flat and comes closeto a rectangular frequency window.
30
3.1.2 Homodyne receiver
The echoes reflected from the stationary point scatters can be seen as round trip delayed versionsof the transmitted signal. They arrive at the receiver and make up the receiving signal
sr(t) = ΣAist(t− τi) = ΣAi cos(2πfL(t− τi) + πB
T(t− τi)2 + φ0) (86)
where Ai represents the reflection coefficients of the ith scatter, the delay τi = 2Ri/vs reveals therelationship between the relative distance from the ith scatter to the measurement plane and thewave propagation velocity vs in the medium.
Figure 18: Simplified block diagram of a homodyne receiver, where the transmitted signal and thereceiving signal can be described by (78) and (85) respectively.
Fig. 18 shows the principle of homodyne demodulation, where the echoes are directly mixed witha fraction of transmitted signal. An ideal mixing product takes the form
u(t) =1
2ΣAi[cos(2π
B
Ttτi + 2πfLτi − π
B
Tτ2i )
+ cos2πBT
(t2 − tτi) + πB
Tτ2i + 2πfL(2t− τi) + 2φ0]
(87)
The second cosine term is a FM chirp, whose carrier frequency is usually far beyond the cut offfrequency of the baseband filter. Hence, this term is not of interest. The first cosine term describesa beat signal, whose instantaneous frequency components fb,i are linearly proportional to the round
31
trip delay of the scatters. This delay is caused by path length difference between reference signaland reflected signal at the mixer. By evaluating the instantaneous frequency of the beat signal, therelative position of ith scatter to the calibration plane can be determined by
Ri =fb,iTvs
2B(88)
3.1.3 Range resolution
A common definition of range resolution is the smallest distance between two point scatters withidentical reflection coefficient that a radar system can resolve. As the finite measurement bandwidthcontributes to a sinc function with a main lobe width of ∆f = 1/T in the time domain related bythe Fourier transform, the time response of two point scatters spacing equal to this boundary willshow only one big overlapped peak in magnitude. This time span corresponds to a spatial distanceof
∆R = ∆fbTvs2B
=vs2B
(89)
which is the range resolution evaluated by classical Fourier analysis for a FMCW system. It dependson only one system parameter, the bandwidth of transmitted signal B. In some cases, the pointscatters spacing even smaller than ∆R are still possible to be resolved by using appropriate signalprocessing techniques, which are called high resolution methods.
3.2 Spurious intermodulation
3.2.1 Mixer intermodulation
(a) (b)
Figure 19: (a) Spectrum of the mixer input. (b) Spectrum of the mixer output.
Mixers are electronic devices which translates the electromagnetic signal from one frequencyrange to another. In a mixing process, LO signal and RF signal are combined together through anonlinear device to realize the frequency transform. The nonlinear responses contain however not
32
Table 2: Composition of intermodulations.
IM product Composition Orderf1 |fRF1 − fRF2| 2f2 |2fRF1 − 2fRF2| 4f3 |fLO + fRF1 − 2fRF2| 4f4 |fLO + fRF2 − 2fRF1| 4f5 |2fLO − 2fRF2| 4f6 |2fLO − fRF1 − fRF2| 4f7 |2fLO − 2fRF1| 4
only the desired IF signal but also harmonics of input signals as well as their intermodulations.These unwanted signals appearing at the output stage are called spurious responses. For example,the input of a diode mixer is assumed containing two RF tones fRF1, fRF2 and one LO signal fLO,whose spectrum are shown in Fig. 19a. In the spectrum of the mixer output, fIF1 = fLO − fRF1
and fIF2 = fLO − fRF2 are the desired IF signals coming from the second order mixing product asillustrated in Fig. 19b. Meanwhile, plenty of additional frequency components might be observed inthe mixer output spectrum. The number of those spurious frequency components goes in principletowards infinity, but the power of each spurious frequency decreases dramatically when it is generatedby a high order intermodulation.
Table 2 lists some intermodulation products in region of baseband and their orders. In frame ofhomodyne FMCW, the spurious intermodulations appearing in the baseband are normally generatedfrom the even order while those from odd order locate in the RF range and can be therefore filteredby the baseband filter.
3.2.2 Analysis of intermodulation via diode current-voltage characteristic
The earlier circuit modeling on the diode mixer [49–52] provides an efficient way to analyze themechanism of the spurious intermodulation. The current-voltage characteristic of a junction diodecan be expressed as
i = i0(eαNv(t) − 1) (90)
where i0 is the saturation current typically ranging between 10−6A and 10−15A. αN is a coefficient,which depends on the temperature and the material property of the junction. Equation (90) indicatesthat the diode can be modeled as an equivalent lumped element nonlinear resistor. A representationof the microwave circuit of homodyne receiver in terms of equivalent lumped elements is shown inFig. 20, where
33
v(t) Instantaneous input voltage.i(t) Instantaneous current induced in the load.e(t) Instantaneous voltage across the diode mixer.Rs Linear resistance equivalent of input impedance.rL Linear resistance equivalent of load impedance.rd Linear resistance of semiconductor junction.r Nonlinear resistance of the diode mixer.Cd Shunt capacitance.
The voltage across the diode mixer with nonlinear resistance can be expressed as
e(t) = v(t)− i(t)Rw (91)
whereRw = Rs + rd +RL (92)
Figure 20: Equivalent circuit of nonlinear resistance mixer diode.
Substituting (91) into (90), it yields
i(t) = i0(eαNv(t)−i(t)Rw − 1) (93)
34
which indicates that v(t) can be described in an explicit form
v(t) =1
αNln(
i
i0+ 1) + iRw (94)
At the quiescent operating point, the output current and the input voltage can be representedby Taylor series expansion as
∆i = a1 |v=vb ∆v + a2 |v=vb ∆v2 + a3 |v=vb ∆v3 + a4 |v=vb ∆v4 + · · · (95)
where ∆i = i− ib and ∆v = v−vb, ib and vb are the DC bias current and the DC bias voltage acrossthe diode mixer. The kth order coefficients of the Taylor series can be calculated by reversion ofthe power series of voltage-current representation in (94). The first six coefficients in Taylor seriesexpansion as derived in [50] turn to be
a1 =[
1x+1
](i+ i0)αN
a2 =[
1(x+1)3
](i+i0)
2! α2N
a3 =[−2x+1(x+1)5
](i+i0)
3 α3N
a4 =[
6x2−8x+1(x+1)7
](i+i0)
4! α4N
a5 =[−24x3+58x2−22x+1
(x+1)9
](i+i0)
5! α5N
a6 =[
120x4−444x3+328x2−52x+1(x+1)11
](i+i0)
6! α6N
(96)
where x = αN (i+ i0)Rw. The most significant contribution to a kth order intermodulation productof small input signals comes from the kth order Taylor series expansion coefficient ak. Therefore,the behavior of these coefficients relates directly to the mixer intermodulation product levels. Thealgebraic expression of Taylor series coefficients in (96) shows that ak varies with the bias current ibor the bias voltage vb across the diode. Therefore, the power level of kth order mixer intermodulationproducts are expected to fluctuate as the diode bias varies.
As aforementioned that, the desired IF mixing output at homodyne FMCW receiver comes fromthe second order mixing product while the undesired spurious intermodulation products are gener-ated by the other order mixing products, a good intermodulation suppression ratio to a particularorder undesired response |a2(ib)/ak(ib)| can be achieved by appropriately choice the operating con-ditions.
Fig. 21 computes the intermodulation suppression ratio term 20 log(∣∣∣ a2
ak
∣∣∣), k = 3, 4, 5, 6 versus
the bias current ib across the diode for αN = 23V −1, i0 = 10−6A,R = 115Ω. In small input signalcase, most significant spurious intermodulation terms come from the 4th order. The choice of biascurrent in the vicinity of 80µA can greatly reduce the receiver sensitivity to the 4th order undesiredspurious intermodulations.
Some studies have reported that the effect of LO level play an important role on the spuriousintermodualtion product output levels. This is because the LO voltage incident to the diode willalso generate an effective DC bias across the diode due to the rectification. It in turn means thatthe intermodulation suppression ratio to a particular order undesired response can also be reducedby properly controlling the LO levels.
35
0 0.1 0.2 0.3 0.4−60
−50
−40
−30
−20
−10
0
10
20
30
ib(mA)
20lo
g|a 2/a
n| (dB
)
(a2,a
4)
(a2,a
6)
(a2,a
3)
(a2,a
5)
Figure 21: Influence of the bias current on the intermodulation suppression ratio [50].
3.2.3 Measurement
Fig. 22 illustrates the measurements of intermodulation responses on a low barrier Schottky diodeAgilent 8473C, which is used as the detector of proposed microwave FMCW reflectometer describedin Chapter 5. During the measurement, the LO and RF signal flow through a power combiner tothe diode, which is connected to a spectrum analyzer (RS, ZVL5). The power of RF signal is fixedto −12dBm while the LO power is varied from −20dBm to +4dBm. The frequency of RF signaland that of LO signal are configured as
fRF = 1GHz, fLO = 1.1GHz (97)
The desired IF signal should appear at 100MHz while the undesired fourth order intermodulationproduct will emerge at 200MHz.
The output power of the second order intermodulation product will almost linearly increase untilthe LO power reaches PLO = −3dBm, after that the diode response curve enters a square law region.The 1dB compression point appears at PLO = −1dBm, which could be used to separate the dioderesponse into linear region and non linear region. The fourth order intermodulation product firstappears at PLO = −13dBm, it then almost linearly increases until PLO = −6dBm and presentsa local minimum at PLO = −4dBm. After that, it will increase shortly again and then decreasesmonotonically as PLO further increases.
36
Figure 22: Measured second order and fourth order responses of Agilent 8473C detector. PRF =−12dBm. The sixth order response can be first detected and exceed the power of fourth orderresponse until PLO > 7dBm, whose effect can be neglected because the position of which is farbeyond the 1dB compression point.
The selection of PLO and the power of transmitted signal PRF should follow:
1. The total input power should lie in the linear region of diode response curve. Namely, PLO +PRF < −1dBm for any measurement.
2. On one side, PLO should be set greater than PRF . The larger ratio PLOPRF
, the better suppressionof undesired second order intermodulation product in case of multiple reflections as discussedin Section 3.3.1. Besides, an appropriate large PLO should be chosen for an optimal suppressionof the fourth order intermodulation. This is in practice of more importance, because the fourthorder intermodulation affects the accuracy of the calibration technique in Chapter 5.
3. One the other side, PRF needs to be large for a high SNR.
In practice, a trade off must be unavoidable made. An optimal operating configuration for proposedFMCW reflectometer is chosen at PLO = −4dBm and PRF = −12dBm, this configuration ensuresthe best suppression of the fourth order intermodulation.
37
3.3 FMCW principle for lossless non-uniform transmission line
3.3.1 Discrete forward modeling
It is well known that the impulse response can be determined by taking the Fourier transform ofthe input reflection coefficient
h(t) =1
2π
∫ ∞−∞
Γ(0, ω)e−jωtdω (98)
where Γ(0, ω) is identical to S11 in frequency domain.
If a LFMCW signal is injected into a transmission line rather than an ideal dirac impulse, thereceiving signal follows the convolution operation in time domain
sr(t) =
∫h(τ)st(t− τ)dτ (99)
Using the discrete time domain model to represent the non-uniform transmission line, the receivingsignal can be described as a superposition of several distinct delayed version of transmitted signal
sr(t) =∑
h(i∆τ)st(t− i∆τ) (100)
which contains both of the wavefront reflections and multiple reflections.
Here we use the homodyne receiver structure shown in Fig. 44 as an example, the instantaneousvoltage at the input stage of diode mixer takes the form
v(t) = VLO cos(φLO(t)) +∑
Vsh(i∆τ) cos(φLO(t− i∆τ)) (101)
where VLO and Vs represent the voltage level of the LO and of the transmitted signal, respectively.According to the analysis of diode behavior in Section 3.2.2, the low frequency voltage at the loadimpedance can be expressed as
ub(t) = a0vbRL + ∆iRL = a0vbRL
+ a2VLOVsRL∑
h(i∆τ) cos(φLO(t)− φLO(t− i∆τ))
+ a2V2s RL
∑∑h(i∆τ)h(p∆τ) cos(φLO(t− i∆τ)− φLO(t− p∆τ))
+ a4a4,1V4s RL
∑∑h(i∆τ)2h(p∆τ)2 cos(2φLO(t− i∆τ)− 2φLO(t− p∆τ)) + ...)
+ a4a4,2VLOV3s RL
∑∑h(i∆τ)h(p∆τ)2 cos(φLO(t) + φLO(t− i∆τ)− 2φLO(t− p∆τ))
+ a4a4,3V2LOV
2s RL
∑h(i∆τ)2 cos(2φLO(t)− 2φLO(t− p∆τ))
+ a4a4,4V2LOV
2s RL
∑∑h(i∆τ)h(p∆τ) cos(2φLO(t)− φLO(t− i∆τ))− φLO(t− p∆τ)))
+ a6...(102)
where ak is kth order Taylor series coefficient of diode mixer voltage-current specification. It isevident that the second term in (102) is the desired beat signal, which contains the pure informationof impulse response. The other terms are the spurious intermodulation products and will disturb
38
the measurement results. Among them there are two main terms: 1) a dc bias a0vbRL, which mustbe compensated for by applying the calibration technique. 2) the second-order intermodulationa2v
2sRL, whose influence will be reduced in case of small reflections or by increasingthe ratio VLO
Vs
In case of small reflection coefficients |∆Γ(i)| 1, the magnitude of the spurious intermodu-lation products becomes much smaller than the desired first term, so that beat frequency can beapproximated as
ub(t) ' a2VLOVsRL∑
h(i∆τ) cos(φLO(t)− φLO(t− i∆τ))
= a2VLOVsRL∑
h(i∆τ) cos(2πB
Ti∆τt+ 2πfLi∆τ − πB
T(i∆τ)2)
(103)
whose complex form can be obtained in a simplistic way by taking Hilbert transform
uB(t) =∑
h(τi)ej(2πBT τit+2πfLτi−πBT τ
2i ), for ∆Γ(i) 1 (104)
where the amplitude is scaled by a factor of 1a2VLOVsRL
, this procedure can be automatically ac-complished by the calibration technique as discussed in Chapter 5. One problem of using Hilberttransform is that it will lead to a reconstructed error of the imaginary part, which can be howeverreduced by using a suitable windows function (See Section 5.3.2). For convenience, the notation i∆τis changed to τi in the later part of this thesis.
Equation (104) shows that the impulse response h(τi) now is coded into the amplitude of therelevant low frequency components with range/delay dependent phase shift.
3.3.2 Determination of discrete impulse response via DFT
Assumed that the beat signal is sampled at N equally time space ∆t = T/N within one modulationduration, the discrete complex form of beat signal gives
uB(n) =∑
h(τi)ej(2π nBN τi+2πfLτi−πBT (τi)
2)), n = 1, ..., N − 1 (105)
For short range application, the third phase term of (105) can be usually neglected since τ T .The DFT spectrum with zero-padding factor N1 is determined by
UB(k) =
NN1−1∑n=0
uB(n)e−j2πNN1
nk
=∑
h(τi)sin(π(τiB − k
N1))
sin( πN (τiB − kN1
))ejπ
N−1N (τiB− k
N1)ej(2πfLτi−π
BT τ
2i ), k = 0, 1, ..., NN1 − 1
(106)
With the knowledge of the operational parameters such as fL, B and T , the phase shift can becompensated for by multiplying a conjugate term e−j(2πfLτi−π
BT τ
2i ) at the point of each spectral
grid, so that the discrete impulse response can be evaluated by the amplitude of the relevant spectrallines7
h(τi) =UB(k)e−j(2πfLτi+π
BT τ
2i )
N|k=
τiB
N1
(107)
where discrete impulse response h(τi) can be complex valued when the transmission line under testcontains capacitive or inductive elements.
7It is assumed that B → ∞ and N1 → ∞ in ideal case such that there are no sidelobe response from nearbyspectral lines.
39
3.4 Influence of measurement duration
To fulfill Nyquist-Shannon sampling theorem, the physical A/D sampling rate fs must be at leasttwice as the highest frequency appearing in the baseband signal to prevent spectrum aliasing8.Therefore, the shortest measurement time Tmin of a FMCW reflectometer depends on several op-erating parameters, such as the signal bandwidth B, the maximum unambiguous detection rangeRmax as well as the physical sampling rate fs. This relationship can be described by
fs2≥ B
T
2Rmaxvs
(108)
Rearranging the (108), the shortest measurement time turns to be
Tmin ≥ B4Rmaxvsfs
(109)
In a concrete application, where the maximum unambiguous detection range, signal bandwidthand the physical sampling rate are given, the measurement time can be chosen as an arbitraryvalue T ∈ [Tmin,∞]. When T goes towards the lower boundary, the system range estimationuncertainty will however increase. Due to the measurement noise, the range estimation result differsfrom measurement to measurement and its deviation to the true value can be characterized as astatistical variable. A stable system and a good range estimator should show low variance of therange estimation deviation. The relation of range estimation uncertainty and related parameterscan be checked by the Cramer Rao Lower Bound (CRLB) [53,54] of the applied range estimator.
3.4.1 Cramer Rao Lower Bound for single frequency tone
Assumed that Θ = [θ1, θ2, ..., θp]T is a vector containing a set of unknown deterministic parameters
which are to be estimated from the measurements y, CRLB allows to place a lower bound on thevariance of each element θi. As derived in [53], the CRLB is found as the (i, i)th element of theinverse of a matrix or V ar(Θi) = [I−1(Θ)]i,i, where I(Θ) is the p× p Fisher information matrix:
[I−1(θ)]i,j = −E[∂2 log p(y; Θ)
∂θi∂θj], for i = 1, 2, ..., p; j = 1, 2, ..., p. (110)
where log p(y; Θ) is the natural logarithm of the likelihood function given the measurement vectory and the to be estimated parameters.
In general the analog beat signal of coherent FMCW response for a single scatter takes thecomplex form
uB(t) = Aej(2πBT τt+2πfLτ−πBT τ
2), for 0 < t < T (111)
In one modulation period, assume that the receiver takes the complex samples y(n) = uB(n)+ε(n)from a complex AWGN (Additive White Gaussian Noise) channel with zero mean and variance σ2.The likelihood function of the measurement samples y with the unknown local reflection coefficientA and the time delay τ can be described by
p(y,A, τ) = (1
2πσ2)Ne−
12σ2
∑N−1
n=0[<(y(n))−<(uB(n))]2
· e−1
2σ2
∑N−1
n=0[=(y(n))−=(uB(n))]2
(112)
8For orthognal I/Q sampling, fs must be at least equal to the highest frequency of the signal.
40
Taking logarithms and differencing both sides with respect to unknown variables A and τ yields
∂ ln p(y,A,τ)∂A = 1
σ2
∑[<(y)−<(uB)]∂<(uB)
∂A + [=(y)−=(uB)]∂=(uB)∂A
∂ ln p(y,A,τ)
∂τ = 1σ2
∑[<(y)−<(uB)]∂<(uB)
∂τ + [=(y)−=(uB)]∂=(uB)∂τ
(113)
Differentiating again with respect to unknown variables produces
∂ ln p(y,A, τ)
∂A∂τ=
1
σ2
∑[<(y)−<(uB)]
∂2<(uB)
∂A∂τ− ∂<(uB)
∂A
∂<(uB)
∂τ
+
1
σ2
∑[=(Y )−=(uB)]
∂2=(uB)
∂A∂τ− ∂=(uB)
∂A
∂=(uB)
∂τ
(114)
∂ ln p(y,A, τ)
∂A2=
1
σ2
∑[<(y)−<(uB)]
∂2<(uB)
∂A2− (
∂<(uB)
∂A)2
+
1
σ2
∑[=(y)−=(uB)]
∂2=(uB)
∂A2− (
∂=(uB)
∂A)2
(115)
∂ ln p(y,A, τ)
∂τ2=
1
σ2
∑[<(y)−<(uB)]
∂2<(uB)
∂τ2− (
∂<(uB)
∂τ)2
+
1
σ2
∑[=(y)−=(uB)]
∂2=(uB)
∂τ2− (
∂=(uB)
∂τ)2
(116)
Since E[<(y)− <(uB)] = E[<(ε)] = 0 = E[=(y)− =(uB)] = E[=(ε)], substituting (111) into (114)-(116) yields
∂ ln p(y,A,τ)∂A∂τ = 0
∂ ln p(y,A,τ)∂A2 = − N
σ2
∂ ln p(y,A,τ)∂τ2 = −4π2A2
σ2
∑N−1n=0 (BT n+ fL − B
T τ)2
(117)
In case that the beat frequency is much smaller than the RF frequency BT τ fL, the Fisher
Information matrix can be derived as
I(Θ) = −
[E[∂
2 log p(y,A,τ)∂A2 ] E[∂
2 log p(y,A,τ)∂A∂τ ]
E[∂2 log p(y,A,τ)
∂τ ] E[∂2 log p(y,A,τ)
∂τ2 ]
]
=
[Nσ2 0
0 4A2
σ2 π2(Nf2
L + fLBN + N(2N−1)6(N−1) B
2)
] (118)
The diagonal elements of inversion of 2× 2 matrix can be calculated by
I−1[θ]1,1 =I[θ]2,2detI
I−1[θ]2,2 =I[θ]1,1detI
(119)
Since R = τvs2 , the maximum likelihood FMCW range estimator for single scatter has a lower bound
V ar(R) =v2s
4V ar(τ) ≥ v2
s
4[I−1(Θ)]22 =
v2s
4
1
4A2
σ2 π2(Nf2L + fLBN + N(2N−1)
6(N−1) B2)
(120)
41
which shows that the range estimation variance using maximum likelihood method is related tothe SNR, the signal bandwidth, the start frequency as well as the measurement duration/number ofsamples per sweep. For broadband FMCW reflectometry (fL B), the range estimation variance isalmost inversely proportional to the SNR and the measurement bandwidth B, inversely proportionalto the square root of the measurement duration T .
The amplitude estimation variance is σ2
N , which depends only on the measurement duration aswell as the noise variance (Not SNR).
DFT based range estimator
In FMCW radar signal processing, usually the range R is estimated from the DFT spectrum of thecomplex sample vector
Y (k) =1
N
N−1∑n=0
y(n)e−j2πnkN , k = 0, 1, .., N − 1.
= UB(k) + ε(k) = A0(k)ejφ + ε(k)
(121)
where A0(k) = Asin(π( 2RB
vs− kN1
))
sin( πN ( 2RBvs− kN1
))and φ = 2πfL
2Rvs− πN−1
N ( 2RBvs− k
N1). This relation suggests that
approximate estimation of the beat signal frequency f and A can be made directly from the DFTof y as was done by Palmer [55]. Since R = vsT
2B f , the standard range estimation deviation σR ofDFT frequency based estimator
σR =vsT
2Bσf (122)
where σf is CRLB of frequency estimation for single tone and is given in [56] as
σ2f =
3(2πfs)2
2π2 A2
2σ2N(N2 − 1)(123)
so that the variance of DFT based range estimator is
σ2R = (
vsT
2B)2 3(2πfs)
2
2π2 A2
2σ2N(N2 − 1)(124)
which is also inversely proportional to the SNR and the measurement bandwidth B, inversely pro-portional to the square root of the measurement duration T .
Numerical results
Fig. 23 plot the CRLB of maximum likelihood range estimators for FMCW principle using single
scatter model under varying SNR = ∆Γ2
σ2 and different number of samples per ramp N . The rangeestimation deviation of DFT based estimator is plotted as a comparison, which exhibits a largerdeviation than the range estimator using maximum likelihood method. This is because that thephase term φ = 2πfL
2Rvs− πN−1
N ( 2RBvs− k
N1) has not been used in DFT range estimation. With a
fixed SNR, the more samples per measurement acquires (The same as extending the measurementtime since N = T
fs.), the smaller localization uncertainty can be achieved. But this benefit decreases
42
as the T goes large. This is important for monitoring rapidly time varying impedance variationsbecause taking average of several measurements (smoothly processing in time domain, often used inmeasurement by TDR) to improve the stability of results is not possible. In an application wherehigh accurate position is of concern, one could estimate the minimal measurement time accordingto (119) or (123).
0200
400600 0
20
400
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
SNR (dB)
CRLB of range estimators for FMCW principle
N
CR
LB
of
rang
e es
timat
or (
Nor
mal
ized
to 1
/ ∆R
)
0.02
0.04
0.06
0.08
0.1
0.12
0.14
DFT range estimator
Maximum likelihood range estimator
Figure 23: CRLB of maximum likelihood range estimator and DFT range estimator for FMCWprinciple using single scatter model. The simulation parameters: B = 4.4GHz, fL = 100MHz,εr = 2.1, fs = 2Ms/S. The resolution is ∆R = 2.54cm on a coxial cable.
43
3.4.2 Cramer Rao Lower Bound for two frequency tones
In presence of multiple frequency tones, more variables will be involved such as the location ofscatters and the spacing between them. Using two point scatters model as example, the FMCWbeat frequency can be expressed as
uB(t) = A1ej(2πBT τ1t+2πfLτ1−πBT τ
21 ) +A2e
j(2πBT τ2t+2πfLτ2−πBT τ22 ), for 0 < t < T (125)
where the unknown parameters include the amplitudes and the delays Θ = [A1, A2, τ1, τ2]T . TheFisher Information matrix can be derived as
I(Θ) =
Nσ2 f1(τ1, τ2) 0 0
f1(τ1, τ2) Nσ2 0 0
0 0 f2(τ1, τ2) f3(τ1, τ2)0 0 f3(τ1, τ2) f2(τ1, τ2)
(126)
where
f1(τ1, τ2) =1
σ2
∑cos(2π
B
T(τ1 − τ2)t+ 2πfL(τ1 − τ2)− πB
T(τ2
1 − τ22 )) (127)
f2(τ1, τ2) = 4A2
σ2π2(Nf2
L + fLBN +N(2N − 1)
6(N − 1)B2 (128)
f3(τ1, τ2) =4π2A1A2
σ2
∑[cos(2π
B
T(τ1 − τ2)t+ 2πfL(τ1 − τ2)− πB
T(τ2
1 − τ22 ))
·(BTt+ fL −
B
Tτ1) · (B
Tt+ fL −
B
Tτ2)]
(129)
The CRLB of range estimator using maximal likelihood method could be numerically calculated by
V ar(R) =v2
4V ar(τ) ≥ [I−1(Θ)]jj, j = 3, 4 (130)
Fig. 24 plots the CRLB of maximum likelihood range estimator for two scatters. As one scatterlocates close to another, the range estimation uncertainty arises rapidly. In case that two scattersare spaced larger than 0.2∆R, the CRLB of range estimator using maximal likelihood method withat least 50 samples is smaller than 0.02∆R. This result seems much better than the radar rangeresolution, because it benefits from the assumptions that the number of scatters are known as aprior information in this simulation but which is not easy to know in many practical measurements.
Especially when two scatters are too close to each other (< 0.02∆R), the range estimationdeviation of one scatter will becomes infinitive large. It means that the range estimation uncertaintyfor distributed scatters using point scatter model will also be extremely large.
Furthermore, as the location of one scatter R1 reaches the margin of detection range 0 or Rmax,the range estimation deviation will increase slightly compared to that in the middle of detectionrange.
44
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
One scatter located at R1=0 or R
1=R
max
Spacing between two scatters (Normalized to resolution ∆R)
CRLB
of r
ange
est
imat
ion
(Nor
mal
ized
to re
solu
tion
∆R)
N=50N=100N=500
(a)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
One scatter located at R1=0.5R
max
Spacing between two scatters (Normalized to resolution ∆R)
CRLB
of r
ange
est
imat
ion
(Nor
mal
ized
to re
solu
tion
∆R)
N=50N=100N=500
(b)
Figure 24: CRLB of range estimation deviation for the scatter R2 under varying spacing to thelocation of another scatter R1 and varying N . SNR = 10dB.
45
4 Localization and identification via sparse overcomplete rep-resentation
0 100 200 300 400 500 600 700−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Range bin
No
rma
lize
d r
efle
ctio
n
FFT−MagnitudeFFT−RealTrue locationsCS reconstruction
Figure 25: A simulation scene of 6 point scatters spacing equidistant.
Theoretically, increasing the signal bandwidth can linearly improve the resolution, it is in mostcase not preferred because the hardware cost will raise exponentially. Since the signal processing ofFMCW principle is intrinsically a spectral analysis problem, an alternative solution is to use highresolution methods.
Fig. 25 shows an example of the Fourier spectrum response of a FMCW ranging scene consistingof 6 point scatters, where each point scatter is specified equidistant to its adjacent scatters and thespacing is equal to the radar resolution ∆R. In the Fourier spectrum, the position and the amplitudeof 6 largest peak’s still deviate from the ground truth due to the spectral overlap. Besides, manyspurious peaks due to the side lobe extends over the whole range profile. As the dynamic rangeof the reflections increases, the side lobes of the response of strong scatters will easily mask theresponse of weak scatters nearby. Therefore, to correctly identify the number of scatters as well astheir amplitude of reflections is often a challenge issue.
Table 3 gives an overview of advanced high resolution techniques, which are classified into threegroups: parametric methods, eigenanalysis methods as well as the sparse overcomplete representationmethods.
46
Table 3: Overview of high resolution methods for spectral analysis
Categories Methods
ParametricAutoregressiveMoving averageAutoregressive moving average
EigenanalysisMUSICESPRIT
Sparse representationBasis pursuitGreedy algorithmsBayesian learning
The parametric methods assume that the structure of a signal can be fully or approximatelydescribed by mathematical models. With a proper model, the unknown data values outside theobservation interval can be extrapolated. As a consequence, it provides better frequency resolutionthan the nonparametric methods9. On the other side, accurate localization requires an appropriateestimation of the model order in advance. Akaike Information Criterion (AIC), Final PredictionError criterion (FPE), Criterion Autoregressive Transfer (CAT) are some standard methods formodel order estimation (See Appendix 8.2).
The subspace based methods use the eigen decomposition to reveal the mechanism between thesignal subspace and the noise subspace. It has been reported in numerous literatures [57–59] thatsuch methods are able to provide super frequency resolution. To make this possible, the number ofscatters must be however accurately estimated in advance.
Recently, the development in Compressive Sensing Theory offers a new glance at the radar signalprocessing. Once the imaging scene of interest presents compressible or even sparse structure10, superresolution can be achieved by sparse representation of the radar responses in term of overcompletebasis [60–62].
This chapter mainly focuses on the sparse representation technique for FMCW signal processing.In the end of this chapter, the performance of different high resolution methods will be compared bya simulation study. The theory of parametric methods and eigenanalysis methods could be referredto Appendix 8.2 and Appendix 8.3.
4.1 Sparse representation techniques
In contrast to redundant representations, sparse representations attempt to represent most or allinformation of a signal in a compact way, namely with a linear combination of only a small numberof basis functions selected from an overcomplete dictionary. Such compact and sparse represen-tation has many uses and attracts a great of research effort in diverse areas such as denoising orsmoothing [63, 64], image/video compression [65–67], data reconstruction [68, 69] as well as pat-tern classification [70,71]. Recently, the Compressive Sensing theory (CS) provides a new samplingparadigm in data acquisition [72–76]. CS asserts that a sparse signal or a sparse image might be
9The nonparametric methods are principally based on the discrete Fourier analysis, they present robust to thenoise effect and usually do not require prior information. These methods are still used in many FMCW applicationsdue to the efficient computational effort provided by FFT.
10If a signal vector y is sparse, it has only a small number of non zero elements.
47
perfectly reconstructed from sampling rate far below the Nyquist rate. This has driven rapid devel-opment of sparse overcomplete representations applied to radar [60]. The power of CS lies in that,it could not only correctly reconstruct the data from incomplete samples but also can exhibit superresolution properties [77, 79] once the assumptions of sparsity and incoherence of the dictionarymatrix have been met.
4.1.1 Sparse overcomplete representation
Supposing that we obtain a measurement vector of a imaging scene y = [y1, y2, ..., yn]T ∈ Rn, witha sensing matrix A ∈ Rn×m the measurement vector y can be projected into a signal of interestg ∈ Rm by
y = Ag (131)
where g is assumed to be a sparse or compressible signal, which itself may contain only few zerocoefficients but can be transformed into a sparse coefficient vector x ∈ Rm with S non-zero orsignificant elements by an orthonormal projection x = Ψg. In this way, the measurement vector ycan be compactly represented by
y = Ag = AΨHx = Dx (132)
where the dictionary matrix D = AΨH has a dimension of n×m, the column vector di of dictionarymatrix D is called atom or basis function. The objective of sparse representation seeks to find a xwith few nonzero elements to fully or approximately represent y.
If m > n, the representation of (132) is underdetermined, so that the solution of x given themeasurement y and the dictionary D will be non-unique and ill-posed in general. Namely, thereare many representations for the measurement vector y =
∑xidi as a linear combination of atoms
in this dictionary. In this case, D is called as an overcomplete dictionary. However, the theory ofsparse overcomplete representations [80, 81] shows that11, when the underlying representation x issufficient sparse and the dictionary D presents low coherence or incoherence, the ill-posed inverseproblem can disappear. In other words, a sparse coefficient vector x can be accurately and efficientlyapproximated given y and D.
The mathematical inverse problem to find the sparsest representation possible in an overcompletedictionary is equivalent to an optimization problem
min ‖x‖0 , subject to y = Dx (133)
where the l0 norm ‖x‖0 simply counts the number of nonzero elements in x. To solve (133) directly,one must sift through all possible realizations of the nonzero elements in x. Such algorithm costs atleast O(2m) flops to carry out, which is intractable for combinatorial approaches in high dimensionalspace. Alternative algorithms are therefore applied to solve the l0 norm based optimization problem.
11The CS theory shows that once the number of measurements follows n ≥ O(Sµ2(A,Ψ) logm), the sparse recon-struction via (134), (135), (136) is exact with overwhelming probability. The coherence between A and Ψ is defined
by µ(A,Ψ) =√m max
1≤k,j≤m
∣∣⟨ak, ϕj
⟩∣∣, where ak is the kth row of the sensing matrix A and ϕj is the jth row of the
orthogonal basis ϕ. S is the sparsity of x.
48
4.1.2 Sparse recovery algorithms
In the literature of sparse representation, there is a variety of heuristic approaches aiming to solvethe optimization problem of (133) under the assumption x is sparse. They roughly fall into twomain categories:
1. Convex relaxation: These methods replace the l0 norm by the closest convex norm, l1 norm(‖x‖1 =
∑mi=1 |xi|), so that the problem of sparse representation is converted into the frame-
work of an convex optimization problem. The global optimal solution can be obtained effi-ciently by interior-point methods [82].
2. Greedy methods: These techniques iteratively refines the current estimate for the coefficientvector x by making locally optimal choices at each step. Examples include MP (MatchingPursuit) [83], OMP (Orthogonal Matching Pursuit) [84] and CoSaMP (Compressive SamplingMatching Pursuit) [85].
Other method like FOCUSS (FOCal Underdetermined System Solver) [86] and bayesian sparse learn-ing [87] have also been proposed to solve the optimization problem. The FOCUSS method uses lpnorm (0 < p < 1) in order to better approximate the l0 norm but losing convexity. Sparse bayesianlearning assumes a prior distribution of the unknown vector x that favors sparsity and develop amaximum a posteriori probability (MAP) estimator to solve the optimization problem.
Convex optimization methods
The convex relaxation methods use l1 norm regularization as a heuristic for finding the sparsestrepresentation in presence of noise, which can be formulated as a constrained optimization problem
min ‖x‖1 , subject to ‖y −Dx‖2 ≤ β (134)
The main advantage of such approach comes from convexity. Namely, the global optimal solutionto (134) is tractable even in large scale problems due to the modern interior-point methods [82].
Two variant forms of the optimization problem (134) are often to see in the literature. One isknown as LASSO (Least Absolute Shrinkage and Selection Operator)
min ‖y −Dx‖2 , subject to ‖x‖1 ≤ γ (135)
and the other is described as an unconstrained formulation
min ‖y −Dx‖2 + λ ‖x‖1 (136)
where λ > 0 is a regularization parameter used to trade off the quality of the data fitting (l2 norm)and the sparsity of the coefficient vector x (l1 norm).
In principle, the (134)-(136) are equivalent. Given one regularization parameter, for example, βof (134), it always exist a λ and a γ such that all three optimization problems yield the identicalsolution. Therefore, we focus in this thesis only on solution to (134) because the regularizationparameter β has a intuitive physical meaning relevant to the noise whose energy has a bounded l2norm.
49
l1 norm minimization of complex valued problems
Although the optimization problem of (134) considers only real valued data, it is often to deal withcomplex valued data in microwave engineering. Once the complex values are involved, (134) canno longer be directly applied. However, it is still possible to transform a complex valued l1 normminimization problem to the second order cone programming in the following way [78].
By introducing a new variable t = [t1, t2, ..., tm]T ∈ Rm, the optimization problem (134) forcomplex valued data is equivalent to
min 1′t
subject to ‖y −Dx‖2 ≤ β and
∥∥∥∥<(xi)=(xi)
∥∥∥∥2
≤ ti, i = 1, ...,m(137)
Furthermore, the complex valued data can be decoupled into their real and imaginary parts asfollows:
y =
[<(y)=(y)
]∈ R2n, x =
[<(x)=(x)
]∈ R2m (138)
and
D =
[<(D) −=(D)=(D) <(D)
]∈ R2n×2m (139)
so that the optimization problem of (137) can be further expressed as
min 1′t
subject to
∥∥∥∥<(xi)=(xi)
∥∥∥∥2
≤ ti, i = 1, ...,m∥∥∥y − Dx∥∥∥2≤ β
(140)
where the inequality constraints can be interpreted containing m second order cones of dimensionthree (ti,<(xi),=(xi)) as well as one second order cone (β,<(y − Dx),=(y − Dx)) of dimension2m+ 1. Consequently, the computational effort has been greatly increased.
Since the optimization problem of (140) belongs to the second order cone programming (SOCP),it can be efficiently solved by some non-commercial numerical solver like SeDuMi [88], SDPT3 [89]or CVX [90].
Greedy methods
Greedy methods use iteration approaches that seek to build up a best possible approximation at eachstep by making a locally optimal choice at each iteration. The rule of directional update determinesthe type of greedy methods. For instance, the concept of OMP can be summarized as follows:
Input: D,y,β.Initialize: residual r0 ← y, support index Ω←, x0 ← 0 and maximal iteration kmaxIteration: k ← 01) k ← k + 1
50
2) i = arg maxj∈Ωc|〈dj, rk−1〉|12
3) Ω = Ω ∪ i4) xs = arg min
z‖Dz − y‖2, s.t. supp(z) ∈ Ω.
5) rk ← y −Dxs.6) Break, if ‖rk‖2 < β or k == kmax and output x? ← xs, otherwise return to 1).
By using the similar matrix decomposition of the real and imaginary parts as (138) and (139), theGreedy methods and other algorithms like bayesian learning could also solve the complex valuedproblem (min ‖x‖1 , subject to y = Dx). Compared to the solution of SOCP, such simple decom-position has however a slight disadvantage as real part and imaginary part of the original coefficientsare decoupled.
4.1.3 Uniqueness conditions
To ensure the unique recovery of sparse representation problem of (133), Donoho [91] show that thesparsity of the solution x? should satisfy
‖x?‖0 <Spark(D)
2(141)
where the Spark of a matrix D is the smallest number n0 such that there exists a set of n0 columnsin D which are linearly dependent. Formally,
Spark(D) = minη 6=0‖η‖0 , subject to Dη = 0 (142)
Since the l0 norm is NP-hard, a nature question arises: Do the sparse recovery algorithms providenecessarily the identical sparsest solution as that of the optimization with l0 norm? The sufficientcondition for this to hold true is based on the principle: incoherence of the dictionary matrix. TheSpark(D) [92] exhibits the relationship as
Spark(D) >= 1 +1
µ(D)(143)
where µ(D) is mutual coherence of the dictionary matrix and defined by
µ(D) = maxi 6=j
|〈di, dj〉|‖di‖2 ‖dj‖2
(144)
where di and dj are the ith and jth column of D, respectively. Rearranging the (141)-(144), we obtainthe mutual coherence based sparse condition
‖x?‖0 <1 + µ−1
2(145)
which guarantees the unique sparsest reconstruction by all sparse recovery algorithms. Such rela-tionship is plotted in Fig. 26.
12For non-negative OMP, the condition changes to i = arg maxj∈Ωc
(⟨dj, rk−1
⟩> 0)
51
In practice, the measurement signal is however contaminated by the noise as y = Dx + ε. Awell known condition ensuring the robust recovery of sparse representations in presence of noise isrestricted isometry property (RIP) [75] introduced by
(1− δξ) ‖ξ‖22 ≤ ‖Dξ‖22 ≤ (1 + δξ) ‖ξ‖22 (146)
where S-sparse vector ξ has the identical support of S-sparse signal x. The smaller δξ, of the higherpossibility the sparse signal x can be reconstructed from noisy measurement. The interpretation ofthis property should be clear: The S-column submatrices of D should be of low coherence to preservea unitary transformation and guarantees the energy always distributing on the identical support ofx.
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
16
18
µ(D)
Spar
sity
Figure 26: Relationship of sparsity and mutual coherence under uniqueness condition.
4.1.4 Running time
Table 4: Time consumption of sparse recovery algorithms. S: sparsity of signal, n,m: dimension ofthe sensing matrix.
Convex optimization Greedy methods Bayesian learningRunning time O(n2m1.5) O(Smn) O(Smn)
52
400 600 800 1000 1200 14000
0.5
1
1.5
2
2.5
3
n
Run
ning
tim
e(s)
Convex
OMP
Bayesian
Figure 27: Running time of sparse representation algorithms in MATLAB platform. In this simula-tion, the signal sparsity S is 5, the dimension of the sensing matrix is n× n.
In [85], Needell has analyzed the worst-case cost of sparse recovery algorithms to recover areal-valued signal, given a normal sampling matrix. As listed in Table 4, the main computationalcomplexity of convex optimization is equal to solve a linear program with m variables and n con-straints, which is O(n2m1.5) for advanced interior-point method. The Greedy methods13 and theBayesian sparse learning takes the same order cost O(Smn), which relates to the signal sparsity leveland the dimensional of the sensing matrix.
Fig. 27 gives a comparison of the time consumption of the convex algorithm, OMP method aswell as bayesian learning, where the simulation is implemented in the MATLAB of a desk PC with3.37G RAM and Pentium(R) dual core CPU(@3.2GHz). The results show that the Greedy methodand the bayesian approach (local optimal) are much more efficient than the convex optimizationmethod (global optimal).
13Theoretically, the computational complexity of CoSaMP is related only to O(mn), but the signal sparsity level Sshould be estimated in advance as prior information.
53
4.2 Sparse representation for FMCW responses
As discussed in Chapter 3, the transmission line under test can be discretized into J small seg-ments with piecewise homogeneous characteristic impedance profile along the propagation direction.Namely, the reflections take place at those interfaces between the segments R = [R0, R1, ...RJ−1]T .
The location of each reflections corresponds to a propagation time τj =2Rj
vs.
We begin with a simple propagation model on the lossless transmission line. The measurementvector is the discrete complex FMCW beat signal uB = [uB(0), uB(1), ..., uB(N −1)]T . Each sampleis related to the discrete time domain impulse response h(τi) by
uB(n) =
J−1∑i=0
h(τi)ej(2π nBN τi+2πfLτi−πBT (τi)
2)
=
J∑i=1
h(τi)ejΨ(τi,n), n = 0, ..., N − 1
(147)
Defining a vector H = [h(τ0), h(τ1), ..., h(τJ−1)]T , which indicates the complex impulse responseat each discrete range grids, the response of FMCW beat signal for the lossless transmission line canbe expressed as
uB = EΨH (148)
where EΨ can be considered as a N × J dictionary matrix
EΨ =
ejΨ(τ0,0) ejΨ(τ1,0) · · · ejΨ(τJ−1,0)
ejΨ(τ0,1) ejΨ(τ1,1) · · · ejΨ(τJ−1,1)
......
. . ....
ejΨ(τ0,N−1) ejΨ(τ1,N−1) · · · ejΨ(τJ−1,N−1)
(149)
Equation (148) shows that
1. When J < N , the system is overdetermined. In general there won’t be a solution. Instead tosolving the solution to (148), an approximation H that minimizes the residual ‖uB − EΨH‖2by least squares is often used.
2. When J = N , H can be uniquely determined. This is used in Fourier analysis without griddingrefinement.
3. On the one side, J > N means that a fine gridding is used in the detection range of in-terest. Only in such case, super resolution is possible. One the other side, (148) becomesan underdetermined problem. An approximate solution must be found from many possiblesolutions.
The sparse overcomplete representation states that, once H possesses a compressible or sparse struc-ture, the reconstruction problem of (134) is able to provide a solution close to the true vector H. Inthe following our analysis focuses only on the sparse overcomplete representation.
54
Non-uniform gridding
If the distribution of inhomogeneity is known as a prior information, non-uniform gridding strategyshown as in Fig. 28 could be applied to enhance the computational efficiency. The idea is to usea loose gridding for the homogeneous region and a dense gridding for the inhomogeneous region ofinterest.
Figure 28: Graphic illustration of non-uniform gridding.
4.2.1 Representation in frequency domain
Note that, if the second and third phase terms of (147) can be compensated, the relation becomes
u(n) =
J−1∑i=0
h(τi)ej(2π nBN τi), n = 0, ..., N − 1 (150)
the determination of vector H becomes to a spectral estimation problem, which has a linear range/delayinvariant spread function in the domain associated by the Fourier transform. In fact, the overcom-plete dictionary EΨ in (148) can be interpreted as phase shifted discrete Fourier tranform by
EΨ =[F∗N,J
][Pφ] (151)
55
where[F∗N,J
]is N × J partial inverse Fourier matrix
[F∗N,J
]=
F0t F0
t · · · F0t
F0t F−1×1
t · · · F−1×(J−1)t
.... . .
. . ....
F0t F
−(N−1)×1t · · · F
−(N−1)×(J−1)t
,Ft = e−j2πJ (152)
and the J × J phase shift matrix is a diagonal matrix
[Pφ] =
ej0 0 · · · 0
0 ej[1×2πfLTfsBJ −π
BT (1×TfsBJ )2] · · · 0
.... . . 0
0 · · · 0 ej[(J−1)×2πfLTfsBJ −π
BT ((J−1)×TfsBJ )2]
(153)
where in short range application the RVP (Reside Video Phase) term can be neglected while for farrange application it must be kept. A precise phase compensation for practical measurement will bediscussed in Chapter 5.
Rewriting (148) yields the representation of FMCW responses in frequency domain
uB =[F∗N,J
][Pφ] H (154)
4.2.2 Convolution in time domain
Taking the Fourier transform in both sides of (154) yields[FN,J
] [F∗N,J
][Pφ] H =
[FN,J
]uB (155)
where[FN,J
]is partial Fourier matrix with dimension of J × N . Rearranging Equation ( 155)
produces the time domain convolution representations
[C] H = [Pφ]∗ [
FN,J]
uB = U (156)
where the inverse phase shift matrix is also the diagonal matrix, whose diagonal elements are theconjugate of those in the phase shift matrix. The right side of (156) can be easily obtained byfirst taking the DFT of the measurement data and then by applying a phase compensation. Afterphase compensation, the response of a FMCW reflectometer for impedance discontinuity on losslesstransmission lines is linear range/time invariant. Namely, it follows a circular convolution mechanismin the time domain associated by the Fourier transform.
Fig. 29 illustrates such shift invariant circular convolution mechanism for two point scatters in
time domain. [C] =[FN,J
] [F∗N,J
]is the convolution matrix, whose basis functions are shifted
sinc function with a decaying side lobe extending over all grids. For symmetrical spread function, itsmiddle point is usually used as the reference location, which indicates the position of Dirac sequencesand stays exactly in the diagonal of [C]. If the reference location is chosen not in the middle point ofthe spread function, a location bias exists in the support detection of the Dirac sequences. It shouldbe noted that the convolution mechanism of Fig. 29 can also be interpreted as a row shifted of thereversing point spread function.
56
Figure 29: Graphic illustration of circular convolution of two point scatters. The basis functions ofthe convolution matrix [C] are generated in a deterministic way by a shift-invariant point spreadfunction.
4.3 Regularization parameter and off-grid mismatch
In fact, the physical responses of interest are usually analog functions. They are independent of thediscrete representation one applied in the signal processing. With an infinitive fine discretizationJ →∞, the representation of (156) is identical to the true physical model, where the basis functionsof convolution dictionary matrix [C] are analog functions. In practice, the gridding error due to thediscrete representation of an analog signal always exists.
In presence of measurement noise, the optimization problem of (134) guarantees a robust recon-struction results when RIP has been satisfied. Theoretically the regularization parameter should beat least large enough to cover all measurement fluctuations
β ≥√Pn + Pg (157)
where Pn represents the sum of noise power, Pg corresponds to an equivalent noise power comingfrom the off-grid mismatch of the basis function and true analog signal.
57
Figure 30: Illustration of nearest neighbour approximation for an analog delay, where the analog de-lay signal ej2π
nBN τti is approximated by its closest discrete form ej2π
nBN τi . Such approximation results
in a gridding error of |τti − τi|, which corresponds to a off-grid mismatch of∣∣∣ej2π nBN τti − ej2π nBN τi
∣∣∣in sampling domain.
It has been reported [87] that the regularization parameter is critical for deriving appropriatesolutions since it controls the sparseness of the solutions. If the regularization parameter β is chosenmuch smaller than the noise power, smooth recovery instead of a sparse recovery will be generated.A reconstruction of zero vector will appear if β exceeds the input signal power. In fact, it neithermeans that only the optimal βop is able to provide the appropriate support detection. An examplebased on the practical measurement, showing the influence of the regularization parameter, is givenin Fig. 62 in Chapter 6, where the β is allowed varying in a dynamic region for the exact supportreconstruction. A good sparsity recovery algorithm should be able to provide a relative large dynamicregion of β for correct support detection.
Different from the analysis in [87], the gridding error should also be taken into considerationfor determination of the regularization parameter. Using the signal model of (150) as example, ananalog delay signal is always approximated on a set of regular discrete grids. If the analog delaylies in between two grid points, it is naturally to be approximated by the nearest grid point asshown in Fig.30. Such approximation has gridding error, which leads to an off-grid mismatch ofthe basis function and the true analog function. The small gridding error requires that the objectsare a priori close to the grid points. But this precondition can usually not be assumed in practicalmeasurements. Therefore, only a fine gridding could reduce this error during the inverse procedure.
Fig. 31a and Fig. 31b illustrate the real part of N samples’ off-grid mismatch between an analogdelay signal u(n) = ej(2π
nBN τti ) and its nearest neighbour approximation u′(n) = ej(2π
nBN τi) under
58
0 20 40 60 80 100−1.5
−1
−0.5
0
0.5
1
1.5
Time samples
No
rma
lize
d a
mp
litu
de
Nearest neighbor approximationOriginal
(a)
0 20 40 60 80 100−1.5
−1
−0.5
0
0.5
1
1.5
Time samples
No
rma
lize
d a
mp
litu
de
Nearest neighbor approximationOriginal
(b)
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
Gridding refinement factor N1
e g
(c)
Figure 31: (a)(b) Off-grid mismatch of the basis function (nearest neighbour approximation) and thetrue analog delay signal by using Fourier transform and by using Fourier transform with griddingrefinement of factor 10. (c)The maximal relative off-grid mismatch vs gridding refinement factor.
IDFT (The maximal off-grid mismatch presents when τti − τi = ∆τ2 ) and IDFT with gridding
refinement of factor 10 (The maximal off-grid mismatch reduces to τti − τi = ∆τ20 ).
Here, the relative off-grid mismatch is defined as
eg =‖u− u′‖2‖u‖2
(158)
Chi has analyzed such off-grid mismatch from the perspective of basis mismatch in [93] and showsthat it depends also on the signal sparsity14.
14The maximal mismatch exists when a single analog delay lies exactly in the middle of two grid points. Themismatch will slightly go down as the sparsity increases, so that the maximal relative off-grid mismatch by the singleanalog delay provides an upper bound of the off-grid mismatch.
59
The relative off-grid mismatch of examples in Fig. 31a and Fig .31b are 0.9755 and 0.1082 respec-tively. The maximal relative off-grid mismatch vs gridding refinement is illustrated in Fig. 31c, wherethe maximal relative off-grid mismatch is roughly inverse proportional to the gridding refinementfactor15.
0 50 100 150 200 250−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Range bin
No
rma
lize
d a
mp
litu
de
Sparse reconstruction with β excluding gridding error
OMP reconstructionOriginal signal
(a)
0 50 100 150 200 250−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Range bin
No
rmal
ized
am
pli
tud
e
Sparse reconstruction with β including gridding error
OMP reconstructionOriginal signal
(b)
Figure 32: (a) Spare reconstruction of an analog signal with 6 random specified spikes by using OMPalgorithm, where β excludes the off-grid mismatch. (b) The same reconstruction with β includingthe off-grid mismatch. In the simulation, N1 = 5, SNR = 40dB.
Fig. 32 gives a comparison of sparse reconstruction with and without considering off-grid mis-match. The analog signal is assumed to contain 6 randomly specified spikes
u(n) =
6∑i=1
Atiej2π nBN τti , n = 0, 1, ..., N − 1 (159)
The received signal is polluted by a Gaussian noise with SNR = 40dB. When the regularizationparameter β only includes the noise power, plenty of spurious spikes arise in the sparse reconstructionresult as shown in Fig. 32a. But once β is large enough to cover both the noise power and the off-gridmismatch, a well approximation reconstruction results has been realized as shown in Fig. 32b.
15The gridding refinement is done by zero padding in the tail of the measurement vector uB for FMCW reflectometry.
60
4.4 Analysis of the separation condition by numerical simulation
4.4.1 A conflict of gridding refinement and uniqueness condition
0 20 40 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Correlation pattern of columns in FN,J
Column spacing
Corr
elatio
n
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Mutual coherence of FN,J
Gridding refinement facotor N1
µ (F)
ComplexReal
2∆ R∆ R
Figure 33: (a) Mutual coherence of partial Fourier matrix FN,J vs N1. (b) Correlation of columnsin FN,J, where N1 = 20. N1:gridding refinement factor.
It has been reported in plenty of works [77, 79, 94, 95] that compressive sampling could providesuper resolution properties for radar imaging. Some works attempt to realize the sparse samplingby simply taking a fraction of measurement samples. Usually this is not a good strategy becausethe accumulated SNR reduces. Without throwing away any samples, the sparse reconstructioncould also show super resolution when the spacing of imaging grids is chosen smaller than theradar resolution [77,79]. In such cases the mutual coherence of the sensing matrix will however risedramatically.
Fig. 33a shows the mutual coherence of partial Fourier matrix vs the gridding refinement factorN1. Even with a factor of 4, the mutual coherence of partial Fourier matrix exceeds 0.9 for complex-valued signal and 0.6 for real-valued signal16. In such case, at most one spikes is hopeful to beexactly reconstructed according to the uniqueness condition of (145).
Fig. 33b illustrates the columns coherence pattern of partial Fourier matrix with N1 = 20. Infact, the IDFT matrix without gridding refinement only chooses a fraction of columns with eachspacing 20 columns, so that the incoherence of the submatrix has been achieved. When the griddingpoints goes infinity fine, the nearby columns in the partial Fourier matrix are almost parallel, sothat the adjacent elements in H become unresolvable. Fortunately, the very recent works [96,97] on
16Once the impulse response H is real-valued data, the real part of the measurement vector <uB and the real
part of partial Fourier matrix <[FN,J
] is sufficient for sparse reconstruction.
61
sparse recovery in highly coherent frames provide some answers to this conflict.
4.4.2 Minimum separation condition
The precondition of sparsity alone is in principle not enough. For instance, if two time spikes aretoo close to each other, the separation is almost hopelessly ill-posed. This can be also examined byCRLB analysis of the range estimation uncertainty for two scatters. To avoid such ill-posedness,a minimum spacing between the elements of the support of spikes need to be satisfied [96–98].In [96,97], Candes has proved that once analog spikes h =
∑j ajδtj are separated sufficiently (tj are
locations in normalized delay domain [0, 1]), satisfying the minimum separation distance, the sparsereconstruction by convex optimization is able to achieve infinitive accurate support detection fromlow frequency Fourier measurements
h(f) =
∫ 1
0
he−j2πftdt =∑
j
aje−j2πftj , |f | < fc (160)
where fc is the cutoff frequency of low pass filtering, the amplitude of Dirac spikes aj could becomplex-valued.
Definition 2 (Minimum separation) The minimum separation of the support of a spike sequencesTm is defined as
∆Tm = inf(t,t′)∈Tm,t6=t′
|t− t′| (161)
where |t− t′| is the wrap-around l∞ distance17.
It is shown in [96,97] that ∆Tm should be at least 2fc
for complex valued spikes and 1.87fc
for real
valued spikes.18 For FMCW reflectometry, it means that the minimal separation distance should beat least 4∆R for a complex valued impedance discontinuity and 3.74∆R for a real valued impedancediscontinuity.
Once the analog spikes h obey the minimal separation conditions, the sparse reconstruction hunder a noise ε, which has a bounded l1 norm (‖ε‖1 ≤ ε1), obeys∥∥∥h− h∥∥∥
1≤ C0N
21 ε1 (162)
where C0 is some positive constant.
Robust recovery in presence of bounded l2 norm noise
Suppose that the support Tm of time spikes h satisfies the minimum separation condition and thelow frequency measurement is perturbed by a noise ε, which has a bounded l2 norm ‖ε‖2 ≤ ε2, thesparse recovery by the convex optimization takes the form
h =∑tj∈Tm
ajδtj (163)
17l∞ norm: ‖x‖∞ = max|x|1 , |x|2 , ..., |x|m.18In another very recent work [99], V.Duval shows that the minimal separation for real valued measurements should
be 12fc
and he conjecture that the minimal separation for complex valued measurement is 1fc
based on numerical
simulations.
62
where δtj is a Dirac spike at time moment tj. The reconstructed support Tm of nonzero elements
has been proved to obey the following properties in [100]
i)∣∣∣aj −
∑tl∈Tm,|tl−tj|≤C1
fc al
∣∣∣ ≤ C2ε2,∀tj ∈ Tm
ii)∑tl∈Tm,tj∈Tm,|tl−tj|≤C1
fc |al| (tl − tj)
2 ≤ C3ε2f2c
iii)∑tl∈Tm,tj∈Tm,|tl−tj|>C1
fc |al| ≤ C4ε2
where C2, C3 and C4 are positive numerical constants and C1 = 0.1649.
Properties i) and ii) indicate that the estimate tightly clusters around the support of the originalsignal due to the high correlation of the nearby columns in the partial Fourier matrix as shownin Fig. 33b. Property iii) shows that the spurious spikes have small amplitude far away from thesupport of original signal. Their amplitude is proportional to the noise power and is l1 norm bounded.Intuitively speaking, once the support of spikes are spacing large enough, the support detection errorunder bounded l2 norm noise for a single spike is independent of the value of the signal at otherlocations.
4.4.3 Numerical study on approximate reconstruction
Although perfect reconstruction is attractive, a relaxed approximation might be of more practicaluse for radar imaging and sensing applications. The minimal separation distance of 4∆R are notsatisfied in [77, 79], where the reconstruction by compressive sensing based on point scatter modelstill provides precise super resolution. Therefore, it arise another question: how is the sparse recoveryunder the noise having bounded l2 norm when the minimal separation condition fails? The problemis studied by some numerical simulation here.
The simulation model is based on the phased compensated FMCW spectrum response for pointscatter of (150), where measurement vector y has a dimension of N = 100, the grid refinement is doneby zero padding with factor of N1 = 28, other simulation parameters are defined by B = 4.4GHz,fs = 2GS/s, fL = 0Hz. The reconstruction by convex optimization is carried under varying dynamicrange19, noise level and spacing distance. Total 13 real-valued spectral spikes with equally spacingserves as the worst case situation, namely
y(n) =
13∑j=1
afjej2π
nBN τfj + ε, n = 0, 1, ...N − 1 and afj
∈ R; (164)
The reconstruction of a single spectral spike is counted as a success if a recovered spectral spike δfj
of the top 13 largest magnitude is within a relaxation range ρ∆R around the original support δfj,
namely fj ∈ [fj − ρ∆R, fj + ρ∆R], and satisfies a threshold detection (Corresponding to Propertiesi) in Section 4.4.2.) ∣∣∣∣∣∣
fj+ρ∆R∑fj−ρ∆R
aj
∣∣∣∣∣∣ ≥ Threshold (165)
19In this simulation, the dynamic range refers to max(∣∣aj
∣∣)/min(|ak|), j, k = 1, 2, ..., 13 and j 6= k
63
where ρ ∈ [0, 0.5) regulates the relaxation range with domain, the threshold is chosen as 0.5 min(∣∣afj
∣∣),j = 1, 2, ...13 in the simulation. A successful reconstruction of all spikes is equivalent to the criterionthat the Bottleneck distance [98, 101] between the recovered support and original support is lessthan ρ∆R.
Definition 3 (Bottleneck distance) In one dimension the Bottleneck distance is defined as
dB(A,B) = maxj|aj − bj| (166)
where A = [a1, a2, ...an] and B = [b1,b2, ...bn] are two subsets listed in the ascending order.
The detection algorithm is performed as
Input: the sparse recovered signal hIteration: for k = 1, 2, ..., S0
f = Ø
1) Find the maximum element max |aj| and its support fj
2) If∣∣∣∑fj+ρ∆R
fj−ρ∆Raj
∣∣∣ ≥ Threshold
Sk = Sk−1 ∪ fjai ← 0, for fi ∈ [fj − ρ∆R, fj + ρ∆R]k ← k + 1, go to 1).
elsebreak.
Output the support Sf .
Simulation 1: rate of success
Fig. 34 illustrates the rate of successful support detection by using convex optimization under varyingSNR and varying spacing between the original spikes. The sparse overcomplete representation isbased on the time domain convolution model of (155), where only the real part of the representationhas been used since the aj is assumed to be real valued data. The gridding refinement is configuredas a factor of 28, which is a trade-off between the simulation time as well as the gridding error. Inthis case, the maximum relative off-grid mismatch can be considered as an equivalent noise power of26dB SNR, as shown in Fig. 31c. The regularization parameter β could be then estimated by (158)under casted noise of varying SNR.
Even in the case of dynamic range equal to 0dB, exact support detection is hopeless to achieveunder the condition that the elements spacing of the spikes ranges from 0.5∆R to 2∆R, as shownin Fig. 34a. If a relaxation range of 6/28∆R is taken into account, the rate of success reaches 100%when the spacing of original spikes is larger than 0.8∆R (Worst case), as shown in Fig. 34b. In otherwords, all 13 specrtal spikes can be successfully detected and approximately located. The maximumlocalization error is bounded up to 6/28∆R.
The same support detection is also performed for the case of dynamic range equal to 14dB, asshown in Fig. 34c and Fig. 34d. Now the original spikes are assumed to have different amplitude.When the orignal spikes are spaced larger than 0.8∆R to each other, approximate reconstructionwith relaxation range of 6/28∆R shows 100% successful support detection in high SNR (≥ 25dB).In low SNR case, around 60% successful support detection has been achieved. Actually it comes
64
from those spikes with large amplitude. This is consistent with the fact that the strong frequencycomponents are more likely to be detected.
0
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Figure 34: Rate of successful sparse reconstruction by convex optimization under varying spacingand SNR. (a) Dynamic range 0dB, relaxation range 0. (b) Dynamic range 0dB, relaxation range628∆R. (c) Dynamic range 14dB, relaxation range 0. (d) Dynamic range 14dB, relaxation range628∆R.
Simulation 2: comparison of various sparse recovery algorithms
The identical support detection is performed by other sparse recovery algorithms. The rate of successas well as the rate of false alarm are listed in Appendix 8.1.
The results show that the convex optimization has the most accurate performance and stability,which should be the best choice for the monitoring applications where the accuracy is of the firstpriority. Bayesian learning is able to provide comparable successful support detection, but with toomuch false alarms.
65
Simulation 3: Bottleneck distance
As Bottleneck distance calculates the maximum deviation of support detection, it clearly gives aupper bound of localization error. In this simulation, the measured signal model is similar to (163),but aj is assumed to be complex valued. The sparse representation is based on the frequency domainform of (149), where the complex partial Fourier matrix as well as the complex measurement vectorare used.
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(b)
Figure 35: (a) The average Bottleneck distance of sparse reconstruction by using convex optimiza-tion. (b) The average Bottleneck distance of FFT based peak detection method.
The largest 13 elements in the reconstructed support are used to calculate the Bottleneck distanceunder varying dynamic range, varying spacing of original spikes and varying SNR. The simulationis repeated for 50 random trials and the averaged Bottleneck distance is plotted in Fig. 35a.
66
In low SNR cases (10dB), the Bottleneck distance increases as the dynamic range of the spikesincrease. This is independent of the spacing of the original spikes. In fact, the large supportdetection deviation comes from those spikes with small amplitude. When all the spikes have identicalamplitude and are spacing at least 0.9∆R, the largest deviation of the reconstructed support to thetrue support is around only 0.2∆R. It is noted that in Simulation 1, this minimal spacing distanceis around 0.8∆R, because both the real valued sensing matrix and real valued measurement vectorare used in that simulation.
When the SNR arises, the spikes of small amplitude becomes easily to be correctly localized.For instance, under dynamic range of 14dB, the support detection presents 0.2∆R and 0.5∆R whenthe spikes are spaced from each other at least 0.9∆R and ∆R.
Fig. 35b shows the support detection deviation in averaged Bottleneck distance of FFT basedpeak detection method. Compared this result to that in Fig. 35a, the side lobe of the spikes withlarge amplitude causes spurious support detection.
Discussion
The simulations show that in the optimal case where all spikes have an uniform reflection coefficientthe average maximum range detection error will be smaller than 0.2∆R as long as the minimal dis-tance between adjacent spikes is larger than 0.9∆R. As the dynamic range of reflection coefficientsrises, the range detection error will arise. Usually this happens on those spikes with low reflection co-efficient. Generally speaking, as long as the reflection of those spikes is larger than the measurementfluctuation and they are spacing larger than the system resolution, the maximum support detectionerror can be controlled smaller than 0.2∆R.
67
4.5 Post processing based on time domain convolution model
Theoretically the frequency domain representation of (150) and the time domain representation(156) linked by convolution are identical. In practice, the targets’ locations are indicated in thetime domain for FMCW principle and the precondition assumes that the distribution of targetsshow somehow sparseness. It implies that some post processing techniques are much easier to beintegrated into the time domain convolution model.
4.5.1 Efficient sparse representation via segmentation
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Figure 36: Illustration of range filtering by segmentation. (a) Measured real and imaginary part ofFMCW beat signal. (b) Phase compensated Fourier spectrum U, see (156).
Fig. 36 shows an example of FMCW imaging. In the frequency domain, as shown in Fig. 36a,little information can be directly read from the measurement vector uB(n) . Of course one coulduse the frequency domain sparse representation of (150) to perform the support detection. But asthe number of samples N becomes large, the time consumption of such sparse reconstruction arisesrapidly. For example, the Greedy methods needs a computational effort of O(SN1N
2) for this sparsereconstruction problem.
Fig. 36b illustrates the time domain FMCW responses with a gridding refinement factor of 16,where the targets cluster in four groups [G1, G2, G3, G4] with large spacing. Directly performingtime domain convolution will be even more time consuming with computational effort of O(SN2
1N2)
by Greedy methods. A simple strategy could be however done by first doing threshold detectionand then applying segmentation. Only in the area of interest sparse reconstruction need to beimplemented. Assume that the maximum data volume of the time domain segmentations is Ns(Ns N), the computational effort of segmentation based sparse recovery in time domain convolutionform by Greedy methods is maxO(SN2
s ), O(Nlog(N1N))20. Such segmentation strategy could beactually applied to all sparse recovery algorithms if segmentation is possible. The range filtering
20The second term is the time consumption of FFT.
68
could be versatile and depend on the individual application. For example, according to the range ofinterest or applying threshold detection.
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Figure 37: (a) Convolution kernel of sinc function. (b) Energy ratio of truncated sinc kernel tothe full kernel. (c)(d): Bottleneck distance of sparse deconvolution by convex optimization usingtruncated sinc kernel with varying element spacing and dynamic range.
The convolution kernel due to the finite observation time extends over the whole range profile,as shown in Fig. 37a. The range profile’s dimension of N1N is definitively larger than that ofthe segmented data Ns. To make the sparse deconvolution possible on the segmented data, anapproximation of the convolution kernel by truncated transfer function muss be used.
The key point is how to determine the truncation length of the convolution kernel in order topreserve nearly identical support detection results to those when full kernel has been used. Here,single side truncated length αt in unit ∆R is used as a parameter to describe the truncation length.As shown in Fig. 37a, the sinc kernel has a rapidly decaying side lobe, which means that the mostenergy of the convolution kernel concentrates on the region near its main lobe. The energy ratio ofthe truncated kernel to the full kernel is calculated in Fig. 37b. As long as the truncation length αtis sufficient large, the discarded energy is very small so that a good approximation is likely to be
69
achieved. For example, when the αt reaches 5∆R, the discarded energy is less than 1.4%, which isequivalent to a power bounded noise with SNR of 37dB.
Fig. 37c and Fig. 37d calculate the reconstruction error in Bottleneck distance using truncationconvolution kernel under varying αt and elements spacing. The simulation configuration is identicalto Simulation 3 in Section 4.4.3. The results show that when the spikes are spacing larger than ∆Runder noise of SNR = 20dB, a truncation length of 7∆R is enough for identifying the spikes withdynamic range equal to 0dB and a truncation length of 9∆R is enough for identifying the spikeswith dynamic range equal to 14dB.
4.5.2 Deterministic deconvolution with range dependent spread function for lossytransmission lines
Figure 38: Graphic illustration of sparse deconvolution strategy for range dependent system response.
In long range monitoring applications, the dispersion effects will lead to range dependent systemresponses. For instance, the attenuation on a lossy coaxial cable can be [35] calculated by
An = e−αR, α =1
2[Rss
η0 ln ba
(1
a+
1
b) + wε
′′η0] (167)
70
where η0 =√µ/ε′ is the intrinsic impedance of the dielectric material of the coaxial cable, Rss is
the surface resistance of the conductors which might be also frequency dependent due to the skineffect. Parameters a and b denote the radius of the inner and outer conductor respectively.
Equation (167) shows that the loss is not only range dependent but also frequency dependent.As a result, the phase compensated FMCW beat signal is extended into a general case
uB(n) =
J−1∑i=0
h(τi)ej(2π nBN τi+2πfLτi−πBT (τi)
2)ws(τi, n(fn)), n = 0, ..., N − 1 (168)
where ws(τi, n(fn)) represents a range and modulation frequency dependent attenuation. It maycontains three terms:
1. Attenuation due to the conductive loss as well as dielectric loss.
2. The transfer function of the lowpass filter in the baseband.
3. The window function used in the data processing.
Rearranging (168) in matrix form yields
([F∗N,J
]· [Ws]) [Pφ] H = uB (169)
where · represents the element-wise product, attenuation matrix [Ws] has a dimension of N × J
[Ws] =
ws(τ0, f0) ws(τ1, f0) · · · ws(τJ−1, f0)ws(τ0, f1) ws(τ1, f1) · · · ws(τJ−1, f1)
......
. . ....
ws(τ0, fN−1) ws(τ1, fN−1) · · · ws(τJ−1, fN−1)
(170)
Taking the Fourier transform on the (169) yields
[C] H =[FN,J
]([F∗N,J
]· [Ws])H =
[P∗φ] [
FN,J]
uB (171)
which results in a time domain convolution process with range dependent point spread function asshown in Fig. 38. Obviously, if this range dependent spread function is known as a prior, the sparsedeconvolution could be again performed to accurately locate the point scatter spacing sufficient faraway. In practice there are two methods to derive such spread function.
1. If the electrical parameters of the transmission line under test, e.g. Rss, a, b, η0 in Equation. 167are available from the data sheet, the range and frequency dependent attenuation matrix [Ws]could be determined. Consequently, the new convolution matrix [C] with range dependentspread function can be estimated.
2. If the electrical parameters of the transmission lines are unknown, the attenuation matrix is inprinciple also unavailable. But the range dependent point spread functions could be determinedfrom taking the some baseline measurements of a reference load at various positions on thetransmission line under test. Since such range dependent point spread functions are the basisfunctions of the convolution matrix, a time domain representation could be established as(171).
71
Fig. 39 shows the FMCW responses for an open termination on a coaxial cable. As the positionof the open termination leaves far away from the calibration plane, the frequency responses (mea-surement vectors) present clear range and frequency dependent attenuation as shown in Fig. 39a.This results in range dependent point spread function as shown in Fig. 39b, where the amplitude ofthe spread function decreases due to the attenuation and the main lobe of the spread function alsoslightly widens as the range increases (seeing the black dashed line). Suppose that the point spreadfunction not rapidly changes along the range axis and the spread functions ci and cj relevant to twoclosely spacing locations are obtained from the reference measurements, an interpolation methodcould be used to approximate those point spread functions in the region in between, as shown inFig. 38.
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Figure 39: Range and frequency dependent responses of an open termination on coaxial line (RG59). (a) Calibrated measurement vector |uB(n)|. (b) Spectrum responses with a 6th order Kaiserwindow.
4.6 Sparse overcomplete representation for localization and classificationof distributed impedance variations
4.6.1 Representation by equivalent spread function of the distributed impedance vari-ations
The minimal separation condition of impedance discontinuity definitively fails for distributed impedancevariations. Namely, The sparse reconstruction is not able to guarantee a proper result close to thereality. In monitoring applications, distributed impedance variations might be caused by variousphysical effects of interest such as the mechanical deformation on the transmission line, the fray orchafe on the electrical wires. Usually, the same kind of physical effects produces a similar profileof impedance variations, which continuously distributes in a region R ∈ [RLB , RUB ] and can be
72
described by an local characteristic impedance profile function ∆Γdis(R)21. The local reflection co-efficient profile of multiple same kind of physical effects at various positions on a lossless transmissionline can also seen as a convolution of a dirac impulse sequences indicating the reference positions ofeach physical effect Ri with the local characteristic impedance profile function
u = ∆Γdis(R) ∗∑
δs(R−Ri) (172)
Although the inhomogeneities due to the distributed impedance variation are themself not sparse, butthey present block sparse. Namely, the Dirac impulse sequences indicating their reference positionsmight be sparse. On a lossy transmission lines, the spectrum responses related to the distributedimpedance variation can be described as
([F∗N,J
]· [Ws]) [Pφ] [Γdis] H = uB (173)
where [Γdis] is a J×J matrix, whose basis functions are range shifted local characteristic impedanceprofile function of the distributed impedance variation. It should note that the to be determinedsparse vector H now indicates the reference locations of distributed impedance variations. Takingthe Fourier transform yields the time domain convolution form
[C] H =[FN,J
]([F∗N,J
]· [Ws]) [Γdis] H =
[P∗φ] [
FN,J]
uB (174)
where the basis functions of convolution matrix [C] is called equivalent spread function for dis-tributed impedance variation, whose basis function comes from the convolution of local characteristicimpedance profile function with the range dependent point spread function on a lossy transmissionline.
4.6.2 Classification of inhomogeneities
It is often that various inhomogeneities simultaneously exist on the transmission line in monitoringapplications. Some of them might be appropriately modeled as point scatter as their physical lengthis much smaller than the radar resolution. Others might be characterized by distributed impedancevariations. When the inhomogeneities locate far away from each other, little spectral overlap appearsin the time responses so that they are easy to be identified and located from the spectrum responses.But the task will becomes very difficult in presence strong spectral overlap.
Fig. 40 shows a time domain response of multiple inhomogeneities, which consists of threeimpedance discontinuities and two distributed impedance variations. Two distributed impedancevariations are assumed to be generated from the same kind of physical effect so that they presentssimilar equivalent spread function differing only in amplitude. Because of the serious spectrumoverlap, from the overall spectrum responses it is hard to directly interpret the locations of eachinhomogeneity as well as to identify them.
The relation between overall FMCW spectrum response and two-class impedance variations canbe described as
[C1] H1 + [C2] H2 + ε = U (175)
21∆Γdis(R) = 0, R /∈ [RLB , RUB ].
73
where [C1] =[FN,J
]([F∗N,J
]· [Ws]) represents the convolution matrix for point spread function,
H1 represents the range grid of the impedance discontinuities, [C2] =[FN,J
]([F∗N,J
]·[Ws]) [Γdis]
is the convolution matrix, which is composed of equivalent spread function for distributed impedancevariation.
Figure 40: Responses for two-class inhomogeneities.
Combing two matrices yields
[C1 | C2]
[H1
H2
]+ ε = U (176)
Once the combined coefficient vector
[H1
H2
]is sparse and its element spacing is sufficient enough, the
positions of each class impedance variations can be automatically identified by applying the sparserecovery algorithm with the knowledge of spread functions. The sparse reconstruction result basedon the two-class model (176) is shown in Fig. 41b, where not only the position of each impedance
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variation but also the amplitude of each impedance variation have been correctly identified, evenin presence of serious spectrum overlap. In contrast, the directly using the point spread model asstated in (156) produces uninterpretable identification results (See Fig. 41a) due to the violation ofthe minimal separation condition.
The two-class model can be further extended in the same way to M -class model for monitoringpurpose
[C1 | C2... | CM ]
H1
H2
...HM
+ κUBL = U (177)
where UBL is the baseline spectrum indicating the intrinsic impedance variation of the transmis-sion line under test, variable κ represents the power variation of transmitted signal over time.When more class of inhomogeneities are involved, the column of the combined convolution matrix[C1 | C2... | CM ] becomes more and more correlated and the minimum separation condition must bereestimated based on the mutual coherence of [C1 | C2... | CM ]
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Figure 41: (a) Sparse reconstruction by using point spread model.(b) Sparse reconstruction usingtwo-class model.
4.7 Denoising by averaging compressive sensing
In general, the received signal is contaminated by the noise. The performance of the standard robustreconstruction of (134) might degenerate under a low SNR level. In case of additive white noise, abenefit could be obtained by using correlation of signal components and noncorrelation of the noise.The ideal is similar to that of Welch method. Namely, the measurement vector U now is dividedin to several segments. In each segments, only a fraction of the data are randomly selected and is
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independently preformed by sparse reconstruction
min ‖Hi‖1 , subject to ‖Ui −DiHi‖2 ≤ βi,Ui = SiU,Di = SiD (178)
where Si is a diagonal selection matrix, whose element is 1 for selected data and 0 for unselecteddata. A combination of every reconstruction result H = 1
M
∑Mj=1 Hi lead to positive construction of
the signal components and cancelling of the noise component.
Fig. 42 shows a denoising example of 1D FMCW imaging, where the measurement vector uB(n)is contaminated by the noise of 10dB and has 200 samples. In each time, only 100 samples fromuB(n) are randomly selected to implement CS reconstruction. The signal recovery of averagingof 30 trails has been denoted by the blue circles. Compared with standard BP denoising, 5dBimprovement of spurious suppression has been achieved.
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Figure 42: Denoising by combination of Welch method and sparse overcomplete representation byBasis Pursuit. The gridding refinement factor is 4.
4.8 Removing the amplitude bias by least square fitting
As the l1 minimization tends to underestimate the value of true coefficients, the solution of theoptimization problem (133) is always biased in amplitude. One can perform an l2 norm fitting(which is not biased) upon the support of solution to remove such bias.
76
Assuming that the support of sparse vector is solved by a sparse recovery algorithm using athreshold detection
Q = i|hi 6= Threshold (179)
a general least square fitting could be applied to remove the amplitude bias by
h = (DTQDQ)−1DT
QuB (180)
where DQ is a selected dictionary matrix composed of those columns relevant to the support in Q.
4.9 Simulative comparison of high resolution techniques
Table 5: Simulation parameters of FMCW reflectometer
Start frequency fL = 100MHz Propagation velocity c = 3× 108m/s
Bandwidth B = 4.4GHz Physical sampling 2MS/s
Number of samples N = 300 Resolution ∆R = 3.3cm
Table 5 lists the simulation parameters of FMCW reflectometry. For simplification, it is as-sumed that the measurement is carried on an air-filled coaxial cable with εeff = 1, where the wavepropagation velocity is identical to the speed of light. The resultant theoretical range resolution ofthe system is 3.3cm.
In addition, the characteristic impedance profile is assumed consisting of 3 discontinuities locatedin R1 = 100cm,R2 = 102cm,R3 = 200cm with real valued local reflection coefficient ∆Γ(τ1) =0.2,∆Γ(τ2) = 0.2,∆Γ(τ3) = 0.1.
Substituting these parameters into (147) and introducing white Gaussian noise yields the noisymeasurement of FMCW beat signal uB(n). Applying various high resolution techniques to uB(n)produces the detection results as shown in Fig. 43, where the amplitude of all results are normalizedto the highest value and displayed in dB unit.
Since the spacing between the first two discontinuities is only 0.6∆R, the standard Fourierspectrum are not able to resolve them whatever how high is SNR. This could be checked in Fig. 43a,where serious side lobe effect and casted white noise clearly degenerates the detection result at thelow SNR.
The detection result by AR-Burg algorithm and by MUSIC algorithm are shown in Fig. 43band in Fig. 43c respectively. In both cases, although the optimal model order has been used, thefront two discontinuities could neither be separated. Compared to Fourier spectrum, better signal tonoise suppression could be observed in high SNR cases. However, some range estimation deviationpresents in low SNR cases, especially in the localization result by MUSIC algorithm.
The detection result by sparse overcomplete representation is shown in Fig. 43d, where not onlyall discontinuities are well identified, but also the side lobe effect and the noise interference havebeen significantly reduced. Moreover, the support detection show a very robust results, which almostindependent of SNR when the regularization parameter is appropriately derived. This is very usefulfor applications where high precisely localization is required.
77
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Figure 43: (a) Fourier spectrum. (b) Results of AR-Burg algorithm with estimated optimal modelorder p = 13. (c) Results of MUSIC algorithm with order p = 3. (d) Results of sparse reconstruction,where convex optimization is used.
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5 Construction of FMCW reflectometer
5.1 Frontend design
Figure 44: Block diagram of FMCW reflectometer.
Fig. 44a illustrates the block diagram of broadband homodyne reflectometer, where the openloop control of the FMCW source ensures the shortest modulation duration. A Ku−band varactor-tuned VCO (Voltage Controlled Oscillator) produces a frequency modulation from 12.6GHz to17GHz. The tuning voltage is generated by a Digital/Analog Converter (NI USB-6366) with thehelp of a predistorted look up table to reduce the non-linearity of the VCO. Since the VCO requiresa tuning voltage ranging from 0V to 15V while the maximal voltage output of DAC is up to 10V ,a Voltage/Voltage amplification following a lowpass filter are inserted in between. The frequencymodulated Ku−band signal is then down mixed with a reference tone of 12.5GHz produced by alow phase noise dielectric resonator oscillator. The mixing output of the double balanced mixerwill go through a low pass filter by using Defected Ground Structure (DGS) [102, 103] to form thedesirable LFMCW signal from 100MHz to 4.5GHz22. After amplification, the RF signal is dividedby a power splitter. The majority of signal power is coupled through a 10dB directional coupler
22A broadband VCO with direct frequency output from 100MHz to 4.5GHz is technically difficult to realize sincethe wavelength has changed 45 times.
79
to a low barrier Schottky diode (Agilent 8473C) as LO signal (−4dBm), while the remainder flowsthrough the right directional coupler onto the transmission line under test as the transmitted signal(−12dBm). An adjustable attenuator could be placed between two couplers to suppress interferencesdue to the low directivity of directional coupler near the diode.
The echoes reflected by those inhomogeneities on the transmission line under test are then downconverted with the LO signal to generate the beat signal, which will be first filtered, then amplifiedand finally sampled by a Analog/Digital Converter (ADC) for further data processing. Because theDA/AD Converter operates at physical rate up to 2MS/s, the cutoff frequency of the low pass filterat receiver is planed as 650kHz to prevent aliasing. The frontend related to the prototype FMCWreflectometer is in Fig. 45.
Figure 45: Frontend of the prototype FMCW reflectometer.
5.2 Reducing the VCO nonlinearity by a predistorted look-up table
The VCO nonlinearity will seriously limit the performance of a FMCW system. Both hardwareand software solutions have been proposed to solve this problem. Hardware approaches focus onthe signal generating stage to eliminate the VCO nonlinearity. Examples include using predistorted
80
4 6 8 10 12 1412
13
14
15
16
17
18
19
Tuning Voltage(V)
Ou
tpu
t fr
equ
ency
(GH
z)
VCO characteristic
9.5 9.6 9.715.58
15.615.6215.6415.6615.68
∆f
(a)
0 100 200 300 400 500 6003
4
5
6
7
8
9
10
11
12
13
Output sequence
Vol
tage
(V)
(b)
Figure 46: (a) Three measured VCO characteristic curves: tuning voltage vs output frequency ofVCO. (b) The predisdorted voltage look up table used to generate a linear frequency tuning from12.5GHz to 17GHz.
compensation circuits [104], predistorted lookup table [106] and PLL based closed loop control ofVCO [105]. Software approaches attempt to analyze the transmitted nonlinearity by estimationalgorithms. For instance, Meta [108] and Jian [109] apply a predefined delay line structure toextract the model of transmitted nonlinearity and then use the estimated nonlinearity function tocompensate the transfer function in a post processing stage. Assume that the noise-free transmittedsignal, receiving signal and beat signal of FMCW system are
st lin = ej(2πfLt+πBT t
2+φ0)
sr lin = ej(2πfL(t−τ)+πBT (t−τ)2+φ0)
ub lin = ej(2πBT τt+2πfLτ−πBT τ
2)
(181)
In presence of VCO nonlinearity, the transmitted signal and receiving signal can be modeled with anonlinear phase error term ε(t), i.e.,
st = ej(2πfLt+πBT t
2+φ0+ε(t))
sr = ej(2πfL(t−τ)+πBT (t−τ)2+φ0+ε(t−τ))(182)
After demodulation, the beat signal can be expressed as
ub = ej(2πBT τt+2πfLτ−πBT τ
2+ε(t)−ε(t−τ)) (183)
where the phase error can be approximated as
∆Φe = ej2π(ε(t)−ε(t−τ)) ≈ ej2π(τε′(t)) (184)
81
which indicates that in short range the phase error due to the VCO nonlinearity is approximatelyproportional to the scatters’ range. For long range application, a PLL-based closed loop controlof VCO is recommended for reduce this error. Such negative feedback control is capable to reduceeven the VCO nonlinearity over time due to the temperature drift. On the other side, the lock timeof PLL, the transient response of the phase detector will slow the tuning speed. In short rangeapplication, a predistorted technique is usually sufficient to counteract the inherent nonlinearity ofVCO transfer function.
Fig. 46a depicts three VCO characteristic curves measured at different time. They are almostparallel to each other. By using a predistorted look up table as shown in Fig. 46b, the linear frequencyoutput over time can be realized. In short time, the small phase error due to the temperature drift isonly determined by ε′(t), which could be considered as a constant frequency offset ∆f . In practice,this frequency offset is possible to be estimated by taking some reference measurements discussedlater in this chapter.
5.3 Reconstruction of the complex beat signal via Hilbert transform
In coherent radar signal processing, the complex raw data ua(t) can be derived from the real valuedsamples. A better solution, which requires more hardware effort, is by I/Q demodulation. In onechannel (I channel), the RF signal is mixed with in phase carrier frequency so that the real value ofthe baseband signal uRe(t) can be determined. In a second channel (Q channel), the same RF signalis mixed with a π
2 phase shifted carrier frequency to generate the imaginary part of the basebandsignal uIm(t):
ua(t) = uRe(t) + j · uIm(t) (185)
The problem for this approach is that for broadband application a π2 phase shifter with sufficient
accuracy cannot be easily realized. In addition, the calibration process and the removal of DC offsetson the I and Q channels require intense computational efforts. An alternative solution is to representthe complex data ua(t) by the analytic signal ua(t), which can be reconstructed from the samples ofthe I channel via Hilbert transform. Such strategy not only simplifies the system architecture, butalso makes it faster and cheaper. On the other side, there is a reconstructed error of the signal’simaginary part, which can be reduced to a tolerate level by several methods.
5.3.1 Hilbert transform
In any causal system, an analytic signal ua(t) can be explicitly determined either in terms of uRe(t)or in terms of uIm(t), namely, uRe(t) and uIm(t) can be expressed as functions of each other. Thisrelationship is referred as Hilbert transformation:
uIm(t) = hH(uRe(t)) = uRe(t) ∗1
πt=
1
π
∫ ∞−∞
uRe(t)
t− τdτ (186)
uRe(t) = −hH(uIm(t)) = −uIm(t) ∗ 1
πt= − 1
π
∫ ∞−∞
uIm(t)
t− τdτ (187)
where hH() denotes the Hilbert transform operator. Given a real valued measurement signal uRe(t),its entire analytic signal ua(t) could be uniquely recovered by
ua(t) = uRe(t) + j · hH(uRe(t)) (188)
82
Taking the Fourier transform on the both side of (188), we obtain
Ua(f) = URe(f) + j ·HH(f) · URe(f) (189)
where HH(f) = −j · sfn(f) and sgn is the signum function
sgn(f) =
−1, f < 00, f = 01, f > 0
(190)
The analytic signal can be for an infinite large frequency domain efficiently calculated by
ua(t) = F−1 F (URe(t)) · (1 + j ·HH(f)) (191)
5.3.2 Reduction of the reconstructed error of the imaginary part by using a windowfunction
The finite measurement time is equivalent to employ a rectangular time window on the measurementsamples, which leads to a spread function in the spectral domain related to the Fourier transform.Due to this convolution kernel, a spectral spike presents not only a high magnitude in its trueposition, but also side lobes extending over all circular spectrum. In other words, the negativespectrum of a nature complex signal may not be zero. However, the negative spectrum of theanalytic signal, which is reconstructed by the real part signal by Hilbert transform, is automaticallyforced to zero. The deviation of the imaginary part of the analytic signal via Hilbert transform tothat of the true complex signal is defined as the reconstructed error
∆uIm(t) = uIm(t)− uIm(t) = uIm(t)− hH(uRe(t)) (192)
Obviously, the less imaginary side lobes reside in the negative spectrum, the lower the reconstructederror of the imagionary part between the exact complex signal and its analytic representation byHilbert transform. Fig. 47a-c plots the range detection results of a point reflection under variousdistance, where the green curve is the spectrum of the true complex FMCW beat signal by I/Qdemodulation and the blue dashed curve is the spectrum of the reconstructed analytic signal usingHilbert transform. As the point scatter is located close to the detection range boundary 0m, morediscrepancy between the exact spectrum and the spectrum of reconstructed analytic signal by Hilberttransform could be observed.
Two methods could be used to minimize such deviation:
1. Using a matched delay line in the measurement path, which shifts the appearing frequencycomponent far away from the DC and thus reduce the leakage in the negative spectrum.
2. By applying a window function win(t) prior to the Hilbert transform and afterwards makingan inverse filtering
ua(t) =F−1 F (uRe(t) · win(t)) · (1 + j ·HH(f))
win(t)(193)
A flexible window function can be chosen for example like Kaiser-Bessel window, Hammingwindow, Turkey window and so on.
83
Figure 47: (a)-(c): Exact spectrum responses (green dashed lines) and reconstructed spectrumresponses (blue dashed lines) of a point reflection at different distances. (d)-(f): Imaginary partdeviation |=ua(t) − =ua(t)| of the exact beat signal and the reconstructed complex beat signalin time domain. Simulation parameters of FMCW principle: fL = 0Hz, B = 4.5GHz, fs = 2MS/s,N = 300.
Fig. 47e-f compares the imaginary part deviation |=ua(t) − =ua(t)| of true complex signaland reconstructed analytic signal by Hilbert transform (red curve) as well as that of by Hilberttransform using a Kaiser window with order 4 (blue curve). Clearly, the window function caneffectively reduce such reconstructed imaginary deviation. It is also possible to improve the accuracysignificantly by combining two methods.
5.4 Calibration techniques
It is well known that ambient variations have impact on the performance of analog devices. Thetemperature drift, humidity and mechanical vibrations may cause deviations on the measurementresults. In most cases, these ambient variations are not so easy to be predicted that the resultantdeviations could be categorized into dynamic error.
Unlike the ambient variations, the systematic error usually influences the measurement results ina deterministic way. It can be classified into non-linear error and linear error [25]. For instance, themixer HF-specification accounts for the non-linear error. Fig. 22 shows the square law response of a
84
diode, any input power PLO+PRF > −1dBm leads to an output saturation. The mismatch betweenthe connections and the leakages due to the low directivity of the directional couplers however resultin linear error.
In practice, the non-linear error can be avoided by properly operating the system in the linearregion while the linear error could be reduced by calibration techniques.
5.4.1 SOL (Short Open Load) calibration
Figure 48: Error propagation model on proposed homodyne FMCW structure.
In the past, several calibration techniques for FMCW principle have been proposed for differ-ent applications. For instance, the free space calibration for a FMCW Synthetic Aperture Radarcould be found in the dissertation of A. Dallinger [110]. Hauschild [111] uses a network analyzercalibration techniques to calibrate a short range FMCW radar. In FMCW MIMO radar, the calibra-tion processes aim to compensate the systematic error due to the different physical channels [112].Principally, the individual calibration process is developed on the error propagation model of anindividual system structure. Therefore, the analysis on the error propagation model of the proposedFMCW system is of the first importance.
Fig. 48 illustrates the homodyne structure of proposed FMCW system, which mainly consistsof two cascaded directional couplers. Assume that a DUT with zero length is connected directly toFMCW system on the calibration plane. In ideal case, the normalized output signal of a zero-lengthDUT on the calibration plane should take the form uB(t) = |∆Γ(0)| ejφ(0). However, the measuredsignal might be distorted by several normalized inferences:
85
1. Different path lengths and attenuations between the measurement path and the reference pathwill lead to an error
up(t) = |rp| ej2πfpt+jφp (194)
2. Low directivity of the two directional couplers (index DL for left directional coupler, DR forright directional coupler) might lead to some leakages
uDL(t) = |rDL| ej2πfDLt+jφDLuDR(t) = |rDR| ej2πfDRt+jφDR
(195)
3. The port mismatch between the FMCW system and the DUT results in a fraction of backscat-tered signal being reflected back to the DUT again
uR(t) = |rR| ejφR (196)
Figure 49: Error term flowgraph for one port measurement used in VNA. a and b denote the incidentwaves and reflected waves, respectively. ΓM is measured reflection coefficient and Γ denotes the truereflection to be determined.
Since the error term uR(t) may generate multiple reflections between DUT and FMCW systemfor a general measurement, the relation of normalized measured signal uM (t) and all error termscan be presented as
uM (t) = uDR(t) + (rk + uDL(t))up(t)uB(t)·1 + uB(t)uR(t) + [uB(t)uR(t)]2 + ...
= uDR(t) +
(rk + uDL(t))up(t)uB(t)
1− uB(t)uR(t)
for |uB(t)uR(t)| < 1
(197)
where rk denotes the ratio between the transmitted signal and the LO signal in time domain.Equation (197) can be rearranged with u′p(t) = (rk + uDL(t))up(t) and gives
uB(t) =uM (t)− uDR(t)
u′p(t) + uR(t)(uM (t)− uDR(t))(198)
86
Comparing (198) with three term error model of a vector network analyzer as shown in Fig. 49, arelationship emerges uD(t) ∼ e11, u′p(t) ∼ e21e12 and uR(t) ∼ e22.
Therefore, these time dependent error terms can be determined by measuring three knownstandards, open, short and matched load (SOL standard) at an identical plane, which is calledcalibration plane:
1 =uopen(t)− uD(t)
u′p(t) + uR(t)(uopen(t)− uD(t))(199)
−1 =ushort(t)− uD(t)
u′p(t) + uR(t)(ushort(t)− uD(t))(200)
0 =uload(t)− uD(t)
u′p(t) + uR(t)(uload(t)− uD(t))(201)
Substituting (199)-(201) into (198), the calibrated FMCW beat signal for an arbitrary measurementis then determined by
uB(t) =(uM (t)− uload(t))(ushort(n)− uopen(t))
2P (t)−Q(t)(202)
whereP (t) = (uopen(t)− uload(t))(ushort(t)− uload(t))
Q(t) = (uopen(t) + ushort(t)− 2uload(t))(uM (t)− uload(t))(203)
and uopen(t), ushort(t) and uload(t) correspond to the measured complex FMCW response of a knownopen, short and 50Ω load at the calibration plan respectively.
0 50 100 150 200 250 300−0.4
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
Samples
Vo
lta
ge
(V)
OpenShort
50Ω load
(a)
0 10 20 30 40 50 0
5
100
0.2
0.4
0.6
0.8
1
x 10−3
Time(Second)Distance(cm)
No
rma
lize
d m
ag
nitu
de
(b)
Figure 50: (a) The measured fmcw beat signals of open, short and load terminations. Their complexforms can be calculated by Hilbert transform. (b) The fluctuations over time and range due to thethermal noise.
87
The measured real valued fmcw beat signals are plotted in Fig. 50a. Due to simple diode mixer,the dc bias exists on the measured signal as well as their complex forms. But such bias can be alsoremoved from the calibration process. uM (t) is measured uncalibrated complex FMCW beat signalfor the transmission line under test.
Fig. 50b illustrates the amplitude deviation of a 50Ω load at 1m from the calibration plane.Such deviation are due to the thermal noise of the system, which presents a highest value of 0.0007close to the dc region and corresponds to a maximal ±0.07Ω dynamic error.
5.4.2 Phase compensation
0 1 2 3 4 5
x 105
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
∆ φ (
Rad)
∆φ= 3.4×10−12f2 + 3.4×10−6f + 0.011
Measured phase profile Quadratic fitting
a2 a
1 a0
Figure 51: Measured phase response of a offset short at various positions.
The phase compensation aims to remove the range dependent phase shift so that the FMCWreflectometry has a linear range invariant spectrum response. Namely, the range dependent phaseterm in (105), Pr = ej(2πfLτ−π
BT τ
2), must be accurately compensated. By doing so, the impulseresponse is exactly coded in the frequency component of the beat signal.
Theoretically, known the start frequency fL, the range varying phase term Pr could be uniquelydetermined at each range grid. Practically, due to the open loop control of VCO, the start frequencyfL might present a small offset ∆f , which results in a range dependent spread function in DFTspectrum. In order to precisely compensate for this phase term, one could measure the phasedeviation ∆φ = [∆φ1,∆φ2, ...,∆φp]
T of a offset short in comparison to the true phase at various
88
positions R. Since the phase term has a theoretically quadratic response with respect to the range23,the frequency offset ∆f can be determined by applying the quadratic fitting to the measured phaseresponses. If this were not the case, either the VCO nonlinearity is not correctly compensated orthe offset transmission line for calibration presents dispersion.
Fig. 51 shows a measured phase responses of an offset short at various distances. Beforemeasurement, the range varied phase term Pr is first multiplied with the predefined start fre-quency e−j2πfLτ , so that the remaining phase shift comes from the frequency offset as well as
RVP term:P′r=e
j(2π∆fτ−πBTτ2)
. This ensures the phase shift not exceeding the first period [0, 2π) byperforming range sliding, because the frequency offset ∆f usually lies within only several tens ofMHz. Applying the quadratic curve fitting to the measured phase response yields three coefficientsa0,a1 and a2. The fitted second order coefficient a2 = 3.4×10−12 is almost identical to the theoreti-
cal value πBT ( TB )2 = π 0.15×10−3
4.4×109 = 3.409× 10−12, which inversely indicates a good VCO nonlinearitycorrection. The fitted first order coefficient relates to the frequency offset by
2π∆fτ = 2π∆fT
Bf = a1f (204)
from which the frequency offset can be determined by
∆f =a1B
2=
3.4 ∗ 4.4× 109
2π ∗ 0.15× 10−3= 15.8MHz (205)
The small dc phase offset a0 mainly comes from to the gridding error and flicker noise.
Once the range dependent phase shift has been compensated, the impulse response is correctlycoded into the relevant spectral component of the FMCW beat signal:
uB(n) =∑
h(τi)ej(2π nBN τi), n = 1, ..., N − 1 (206)
which leads to a range-invariant convolution kernel in the delay domain/range domain.
A comparison of the FMCW range response with phase compensation and without phase com-pensation is illustrated in Fig. 52. By using the phase compensation, the real valued characteristicimpedance profile can be calculated via the real part of FMCW spectrum response. This could notonly improve the resolution but also facilitate the deconvolution process.
5.5 Measurements
Table 6: Operational parameters
Start frequency fL = 100MHz Bandwidth B = 4.4GHz
Number of samples N = 300 Physical sampling 2MS/s
The operational parameters of FMCW reflectometer are listed in Table 6.
23Range, delay and beat frequency have the same meaning here, because they are linear dependant.
89
60 80 100 120 140 160 180
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
Distance (cm)
No
rma
lize
d a
mp
litu
de
Real partMagnitude
(a)
60 80 100 120 140 160 180
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
Distance (cm)
No
rma
lize
d a
mp
litu
de
Real partMagnitude
(b)
Figure 52: Spectrum responses of FMCW principle for three bends on a sensing coaxial cable. (a)Without phase compensation. (b) With phase compensation.
5.5.1 Inverse scattering
Fig. 53 shows a 30cm long coplanar waveguide under test, which is terminated with a 50Ω load atthe left side and connected to the FMCW reflectometer at the right side. The grounding as well asinner conductor are both made of 8mm wide copper tape. The copper tapes are glued on the topsurface of a PVC board with εr = 2.6. So the effective dielectric constant can be approximately as
εeff ≈εr + 1
2= 1.8 (207)
The characteristic impedance profile of the coplanar waveguide can be theoretically calculatedby the formulae in [113]
Z = 30π√εeff
K′(k)K(k)
K(k)K′(k) = π
ln( 1+√k′
1−√k′
)for k ∈ [0, 1/
√2]
K′(k)K(k) =
ln( 1+√k
1−√k
)
π for k ∈ [1/√
2, 1]
(208)
where k′ =√
1− k2, k = WW+2g , W and g are the width of inner conductor and the width of gap
respectively.
Since the mechanical parameters of the coplanar waveguide are known, the theoretical charac-teristic impedance profile could be calculated according to Equation. 208.
Fig. 54a shows the spectrum response of the FMCW reflectometry, where the calibration hasbeen applied. If the attenuations due to the conductive loss and electric loss are neglected in suchshort range, the characteristic impedance profile can be reconstructed by using the discrete inversescattering method as discussed in Chapter 2. A comparison of the reconstructed characteristicimpedance profile by inverse scattering method and the theoretical characteristic impedance profile
90
Figure 53: The coplanar waveguide under test.
calculated by the mechanical parameters is given in Fig. 54c. A maximal absolute impedance devi-ation of 7Ω happens in the region of the load while a good consistency can be found elsewhere. Thedeviation of gap width due to manual fabrication and the transform of SMA connector/coplanarwaveguide by soldering might account for this discrepancy. The ripples in the reconstructed char-acteristic impedance profile with gridding refinement come from the convolution kernel due to thefinite observation bandwidth. As previously discussed, for practical inverse scattering application,the interpolation of the spectrum response by gridding refinement is recommended. Without doingso, a loose gridding by the standard Fourier transform is easy to lose significant features althoughthe range resolution reaches in 2.54cm in this experiment. So that enormous reconstruction errorcould be observed in the inverse scattering results without gridding refinement.
The measured impulse response of the same waveguide by a VNA(RS, ZVL5) is plotted inFig. 54b for comparison. The reconstructed characteristic impedance profile by using the inversescattering method is shown in Fig. 54d. For such small distributed impedance variations, the pro-posed prototype FMCW reflectometer is able to provide comparable result to that by a VNA.
91
0 5 10 15 20 25 30 35 40−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Distance(cm)
No
rma
lize
d a
mp
litu
de
|U| with gridding refinement of factor 16
ℜ U with gridding refinement of factor 16
ℜ U without gridding refinement
Coaxial to coplanar
Coplanar to coaxial
(a)
40 50 60 70 80 90 100−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Distance(cm)
No
rma
lize
d a
mp
litu
de
Magnitude of impulse responsewith gridding refinement of factor 16Real part
Coplanar to coaxial
Coaxial to coplanar
(b)
0 5 10 15 20 25 30 35 4030
40
50
60
70
80
90
Distance(cm)
Cha
ract
eris
tic
impe
danc
e(Ω
)
Inverse scattering(gridding refinement)TheoreticalInverse scattering(without gridding refinement)
(c)
40 50 60 70 80 90 10045
50
55
60
65
70
75
80
85
Distance(cm)
Cha
ract
eris
tic
impe
danc
e(Ω
)
Inverse scatteringTheoretical
(d)
Figure 54: (a)(c): The calibrated spectrum response of FMCW reflectometry of the coplanar waveg-uide under test and the reconstructed characteristic impedance profile by using the inverse scatteringmethod. (b)(d): The measured impulse response of the coplanar waveguide under test by a VNAwith the same measurement bandwidth from 0 to 4.4GHz and the reconstructed characteristicimpedance profile by using the inverse scattering method.
92
5.5.2 Classification by sparse representation
Figure 55: Transmission line structure under test.
In this experiment, both impedance discontinuities and distributed impedance variations existon the transmission line structure under test, which consists of a microstrip line and a 50cm longcoaxial cable with dielectric constant εr = 1.8. The microstrip line contains a 4.5cm long homogenousregion of 75Ω so that there are two impedance discontinuities. Besides, the microstrip line is createdon a substrate Rogers R04350 with εr = 3.5. Its effective dielectric constant could be calculatedaccording to the formulae [114] as εeff = 2.61. The coaxial cable contains three bends at differentpositions and each bend results in distributed impedance variations. Such combined transmissionline structure contributes to a FMCW spectrum response as shown in Fig.56c. Suppose that thespectrum responses for the bend and for the point spread function are prior information (They couldbe obtained from measurement), a sparse deconvolution strategy as discussed in Section 4.6 couldbe used to provide identification and localization results, which are shown in Fig.56d. It could beobserved that not only the locations of each bend center (red squares) but also the locations of twopoint discontinuities (blue circles) are correctly identified at the true positions.
93
0 50 100 150 200 250
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Range bin
Am
plitu
de
(a)
0 50 100 150 200 250−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
0.025
Range bin
Am
plitu
de
(b)
0 20 40 60 80 100
−0.1
−0.05
0
0.05
0.1
0.15
Distance(cm)
Am
plitu
de
|U|ℜ U
(c)
0 20 40 60 80 100−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Distance(cm)
Am
plitu
de
Impedance discontinuitiesBend
17cm 9cm5.5cm−>4.58cm
(d)
Figure 56: (a) Point spread function at 0.5cm. (b) Spectrum response for a bend on the coaxialcable. (c) Measured FMCW spectrum response of the transmission line under test. (d) Sparsedeconvolution results using two class classification model. The identified spacing of two point dis-continuities should be divided by a factor of
√2.6/√
1.8 to 4.58cm due to the change of propagationvelocity.
94
6 Online acquisition of airbag deployment
6.1 Introduction
As an essential safety device, airbag has been extensively used for protection of occupants fromvehicle collisions. Since the introduction in early 1970s, the airbag system has been continuouslyimproved for true safety. Smart sensors and sophisticated algorithms [115–118] are employed inadvanced airbag systems to predict the occupants position when the airbag is completely deployed.An ignition decision is made once the deployment is more likely to benefit rather than to harm theoccupant.
Although the airbags have saved many lives, the predefined airbag deployment often leads toinadvertent injuries in case that the occupants are out of position or the occupants are improperlyrestrained children [119, 120], because the algorithms based prediction is easily distorted by thecomplicated noisy environment of a crash. It is therefore desired to develop a controllable deploymentproviding at all times an indication if the deployment should be inhibited even when the ignitionhas been triggered.
In order to realize an optimal predictable deployment and to reduce the inadvertent injury risks,a precisely understanding the airbag deployment mechanism is necessary [121–123]. The deploymentprocess can be sensed by optical devices like high speed camcorders or high speed stereo cameras.With the help of dynamic photogrammetry [124], accurate and detailed evaluation results can beobtained. However, the optical approach is restricted. An example is given in Fig. 57a, from whichit is really hard to determine the airbag surface state since the sight is blocked by the cover disc. Inorder to explore the missing information, a non-optical approach has to be applied.
Figure 57: (a) Video frame of airbag deployment. At that moment the airbag is still inside thehousing but the deployment already starts. (b) Sensing principle. (c) Visualization by computertomography. The position of sensing lines is fixed on the fabric of a driver airbag module.
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6.1.1 Underlying physics and the sensing line
Fig. 57b describes the basic sensing principle. A thin sensing transmission line is fixed onto theairbag fabric and is supposed to undergo the deformation similar to those of the airbag fabric.The geometry along the fixing path can be described by a few bends locating at different positionswith different opening angles. The generation and modification of those bends cause changes incharacteristic impedance and generate or modify the inhomogeneities on the sensing line. It isassumed that the change of the reflections, measured in real time, gives insight into the deploymentprocess.
On the one side, the transmission line theory works well on those line structures supporting TEMmode or quasi-TEM mode. Microstrip line, parallel wires and coaxial cable seem to be potentialcandidates as the sensing line. On the other side, the sensing environment of an airbag duringdeployment is very harsh. Strong collisions, vibrations, high pressure and other unwanted effectssimultaneously occur during a short period of time. In this process, the geometry of the sensing linehas been changed quite a lot. The coupling effect as well as the signal radiation will take place onunshielded line structures and result in additional propagation modes, for which the transmissionline theory is not able to provide a complete description. In order to suppress the unexpectedinterferences, shielded lines must be used. Therefore a thin coaxial cable RG178 is finally applied.
6.1.2 Challenges and task description
There are three challenges for this application:
1. A driver airbag has a radius of 30cm, where multiple folds might simultaneously generate. Inorder to identify and resolve the individual fold, a broadband reflectometry is required.
2. Although the coaxial cable provides a perfect shielding to the interferences from the ambient, itpresents less sensitivity to the bending. This requires a high dynamic range of the measurementsystem to inspect the small impedance variations.
3. The deployment of an airbag normally finishes in a few milliseconds, depending on the type ofthe airbag. To track the reflection coefficient changes on the sensing line in such short time,the applied reflectometry requires a very rapid acquisition.
The analysis work of the microwave reflectometries in Chapter 1 shows that the broadband FMCWreflectometry is very suitable for monitoring such fast varying impedance variations. As a frequencyapproach, it also provides a better SNR to TDR.
In the preliminary step of work, the sensing line will undergo a series roll folds during thefolding process and it ends with a clear structure featuring a predefined number of bends as shownin Fig. 57c. The major task is to identify where these bends are located and how they develop duringthe airbag deployment.
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6.2 FMCW responses for bending
6.2.1 Treatment of unwanted echoes
Due to the fabrication technique, intrinsic inhomogeneities exist on the characteristic impedanceprofile of the sensing line, which leads to unwanted echoes in the FMCW spectral response. Suchresponse, which can be considered as clutter, might be in some cases even larger than the impedancevariations due to the bending. In order to suppress this, the difference signal ud between the referenceline status and a real time sensing line status is used as observed data for further processing
ud(t) = uB(t, realtime)− uB(t, reference) (209)
(a) (b)
Figure 58: (a) Parameterized description of a bend. (b) The slot board for collection of the spectralresponse for bends with various parameters.
6.2.2 Parameterized description of the bend
As shown in Fig. 58a, each bend on the sensing line can be uniquely determined by any two ofthe parameters from the bend radius r, the bend opening angle α and bend arc length L sinceL = r(π − α). In order to measure the spectral response for the bends, the sensing line is placedinto a set of slots with predefined radius and opening angle as shown in Fig. 58b. If each bendwith parameter (R,Li, αi) leads to an unique waveform wi(R,Li, αi) in DFT spectrum, the set ofwaveforms W provides a complete description for the bend. Conversely, the parameters of a bend inthis case can also be uniquely estimated from a measured waveform wi(R,Li, αi) in DFT spectrum.
On one side, practical measurements show that such uniqueness is hard to be observed withrespect to bend arc length L. As shown in Fig. 59a, the main lobe width of the spectral responsesfor two bends differs less than 0.2cm from each other, despite the actual bend arc length is L = 1cmand L = 2cm. This indicates that the bend arc length L has little influence on the shape of themeasured waveform for the bend.
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85 90 95 100 105−1.5
−1
−0.5
0
0.5
1
Distance (cm)
Am
plit
ud
e
1cm2cm
92.2 92.4 92.6 92.8
−0.02
0
0.02
0.04
(a)
78 79 80 81 82 83 84−1.5
−1
−0.5
0
0.5
1
Distance(cm)
No
rma
lize
d a
mp
litu
de
180°150°120°90°75°60°45°30°15°0°α =30°
(b)
Figure 59: (a) The real part of spectral responses for two bends with different bend arc length. (b)Spectral response for bends under various opening angles.
On the other side, the bend opening angle α affects the amplitude of the spectral response asshown in Fig. 59b. When the opening angle of a bend α decreases, the amplitude of the spectralresponse arises. If the attenuation due to the sensing line and due to the multiple reflections can becorrectly compensated, the opening angle of a bend could be inferred from the amplitude of responsewaveform in DFT spectrum inversely.
6.2.3 The equivalent spectral response for a bend
Since the bend arc length L has little influence on the response while the bend angle α only deter-mines the amplitude of the spectral response, the bend dependent spectral response w(R,α) can bedescribed by the normalized spectal response w0(R) multiplied by a factor x(α) which only dependson the bend angle.
A sufficiently large number of measured spectral responses for single bends at different positionsand bend angles are plotted together in Fig. 60, where the radar resolution is 2.35cm and a griddingrefinement of factor 16 is used. The amplitude of the spectral responses are normalized to theirhighest positive value. As the shape of these shifted normalized spectral responses are nearly iden-tical, the mean curve is used as the equivalent normalized response w0(R), which is given as the redcurve in Fig. 60.
Neglecting line attenuation, the spectral response for a bend at location Pi can be described as
w(R− Pi, α) = x(α)w0(R) ∗ δ(R− Pi)
= x(α)Di(R) ∗∆Γbend(R− Pi)(210)
where Di(R) is the sinc kernel due to the limited observation bandwidth. The local reflectioncoefficient variations ∆Γbend(R−Pi) due to the bending are supposed to have nonzero value within
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−5.88 −4.41 −2.94 −1.47 0 1.47 2.94 4.41 5.88
−1.5
−1
−0.5
0
0.5
1
Range(cm)
No
rma
lize
d a
mp
litu
de
w0(R)
w0(0)w
0(−16) w
0(16)
Figure 60: Normalized spectral response for bends. The red curve represents the equivalent normal-ized spectral response for the bend, which has slowly decaying side lobes. Its main lobe extension isabout 6cm in space.
the bend arc length while the sinc kernel Di(R) extends over the complete range profile R. Thismakes the resultant spectral response w(R − Pi, α) due to the bend centered at distance Pi violatethe ’Causality’ property.
The calibrated spectral response for multiple bends at different locations becomes a superpositionof the individual bend responses
U(R) = Ud(R)e−j2πfl2Rvs
= w0(R) ∗ (
m∑i=1
x(αi) · δ(R− Pi)) + ε(R)(211)
where m is the number of bends, x(αi) determines the opening angle of the ith bend and Pi cor-responds the location of ith bend center. ε(R) represents all dynamic fluctuations of measurement,which includes
1. Thermal noise, jitter, phase noise due to the electronic components.
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2. Gridding error due to the discretizing representation of the continuous spectral response (off-grid mismatch).
3. Model error by assuming the normalized spectral response of bends to be independent of bendarc length L.
The dynamic error of the measurements can be treated as an equivalent measurement noise,whose statistical characteristica are possible to be estimated from the measured data, shown inFig. 60.
6.3 Resolving multiple bends via adaptive sparse deconvolution
In Fig. 60, the main lobe extension of equivalent normalized spectral response is larger than aresolution cell. It displays therefore distributed characteristic impedance variations. When morethan 5 bends simultaneously take place on a driver airbag of 30cm radius, their main lobe will overlapeach other. Our previous work [125] has already shown that the conventional high resolution methodslike MUSIC [126], AR-Burg [127], compressive sensing method based on Fourier basis, are not ableto identify the true location of multiple closely located bends, because the bending causes distributedimpedance variations and its spectral response can not be considered as a point scatter. However,the relationship of (211) indicates a potential solution by employing a deconvolution technique.Rewriting (211) into matrix form
w0(0) w0(1). . . w0(−1)
w0(−1) w0(0). . . w0(−2)
......
. . ....
... w0(−J). . .
...
w0(−J) 0. . .
...
0...
. . . w0(−J)... 0
. . . 0
0 w0(J). . .
. . .
w0(J)...
. . . 0...
.... . . w0(J)
......
. . ....
w0(1) w0(2). . . w0(0)
x1
x2
...
...
...
...
...
...
...
...xp−1
xp
+ ε =
U1
U2
...
...
...
...
...
...
...
...Up−1
Up
(212)
where p = NN1 is the number of spectral grid points. 2J + 1 determines the truncated length of theequivalent spectral response for the bend w0(R) and the elements outside the truncation windowwill be set to 0, which make up the column vector of the p × p Toeplitz matrix W0 and will beshifted one grid by each column to perform the circular convolution. In this work, J is chosen 65
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satisfying
∑J
i=−J(w0(i))2∑
(w0(i))2= 97%. If a bend occurs at ith range grid, xi is a coefficient which depends
on the opening angle of that bend. If no bend occurs at that position, xi equals to zero. Vector Uis the real part of the calibrated spectrum after phase compensation
U = <
P∗φFN,pud
(213)
where ud is the vector form of the discrete signal in (209), the phase compensation matrix is adiagonal matrix
Pφ =
ej0 0 · · · 0
0 ej(1×2πfLTfsBp ) · · · 0
.... . . 0
0 · · · 0 ej[(p−1)×2πfLTfsBp ]
(214)
and FN,p is the N × p partial Fourier matrix. Given U and w0(R), a l1 norm based nonnegativedeconvolution technique can be applied to recover the sparse vector X via
min ‖X‖1 , subject to ‖U−W0X‖2 ≤ β and xi > 0. (215)
where β is called the regularization parameter, which controls the sparseness of the solutions. Theoptimization problem of (215) can be either solved by basis pursuit using CVX toolbox [90] or bymatching pursuit. Basis pursuit usually need much computational effort to find the global optimumwhile Greedy method seeks the local optimum with a great reduction of the computational effort. Acomparison of both methods is given in a measurement example shown in Fig. 61e and f, where thebasis pursuit provides robust results closer to the reality. For high accurate localization, the basispursuit is therefore recommended.
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6.3.1 Influence of the regularization parameter
Figure 61: (a) The sensing transmission line is slowly twisted around a ruler. The bends generatesfrom point 1 about 80cm far away from the calibration plane. (b) Complex magnitude plot ofthe spectral response. The horizontal axis indicates the position to the calibration plane while thevertical axis indicates the slow time, which contains totally 50 ramps with interval of 0.2 seconds. (c)Real part of the spectral response. (d) Real part of the spectral response after phase compensation.(e) Non-negative deconvolution by OMP method with the same regularization parameter β as in(f). (f) Non-negative adaptive sparse deconvolution by using basis pursuit.
Fig. 61 illustrates an experiment, during which the sensing line is slowly twisted around a woodenruler sequentially from position 1 → 2 → ... → 8. The entire process is monitored by the FMCWreflectometer. Using the spectral response after phase compensation at time instant 5.2s, denotedby the gray dashed line in Fig. 61d, as an example, the deconvolution results solved by (215) presentdifferent support detection as β varies. It is possible to identify more bends for a small value β < 0.02while a smaller number of bends can be identified for a large value β > 0.06, as shown in Fig. 62a.Only in the region between two horizontal gray dashed line, the correct number of bends can be
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identified at the true position. The similar effect can be observed on the spectral response afterphase compensation at time instant 1.6s, which corresponds to the orange dashed line in Fig. 61d.At that moment only two bends existed on the sensing line and the relevant deconvolution resultsunder varying β are plotted in Fig. 62b. Therefore how to determine the regularization parameterβ becomes critical for deriving a proper support detection.
(a)
(b)
Figure 62: (a) Support detection with different regularization parameter β for the spectral responseat time instant 5.2s. At that time, 8 bends exist. (b) Support detection with different regularizationparameter β for the spectral response at time instant 1.6s. At that time, 2 bends occur.
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6.4 Equivalent measurement fluctuations of the FMCW reflectometer
The equivalent measurement fluctuations include several terms: the phase deviation due to thetemperature drift, the model errors due to the reproducibility of a bending (The response for eachbending is analog signal), gridding error (Off-grid mismatch), the jitter effect, the thermal noise andso on. Among them, only the thermal noise has a nearly independent Gaussian distribution whilethe other effects are more or less related to the power of return signal. For the proposed FMCWsystem, the thermal noise shows a standard deviation of 0.0005 in terms of the reflection coefficient,which is much smaller than the other fluctuations.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
1/SNR
Prob
abili
ty d
ensit
y
HistogramGamma distribution(k=1.86,θ=0.0396)
Figure 63: The histogram of 1/SNR|single from large number of measurements presents a near gammadistribution with expectation of 0.073. The Gamma distribution using data fitting is presented bydashed curve.
To quantify the behavior of equivalent measurement fluctuations, 1/SNR around a single bendis calculated via
1/SNR|single =
∑Ji=−J(UN (i)− w0(i))2∑J
i=−J(w0(i))2(216)
where UN represents the normalized measured spectral response for the bend. After calculatingthe 1/SNR|single from collected data as shown in Fig. 60, the histogram is shown in Fig. 63 and itexhibits a near gamma distribution.
The power of the ideal spectral response for a single bend centering at position Pi with opening
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angle αi is defined as
Pbend = (x(αi))2
J∑i=−J
(w0(i))2 (217)
The power of fluctuations around a single bend is assumed to linear proportional to the powerof the ideal bend response and follows the Gamma distribution
Pflu = Pbend · 1/SNR|single = Pbend ·Gamma(k, θ) (218)
Furthermore, the bends are assumed to be independent of each other. In presence of m bendson the sensing line, the expectation E of the 1/SNR|mul is
E[1/SNR|mul] = E(
∑Pf i∑Pbi
) = E[Gamma(k, θ)] (219)
while the variance V ar of 1/SNR|mul is given by
V ar(1/SNR|mul) = V ar(
∑Pf i∑Pbi
) =
∑V ar(Pf i)
(Pb1 + Pb2 + ...+ Pbm)2
=x(α1)4 + ...+ x(αm)4
(x(α1)2 + ...+ x(αm)2)2V ar(Gamma(k, θ))
(220)
From (219) and (220) it can be concluded that
1. The expectation of 1/SNR|mul remains the same as 1/SNR|single, whatever how many bendsoccur.
2. The variance of 1/SNR|mul is a scaled version of the variance of the Gamma function. Thefactor depends on the number of bends on the sensing line as well as the opening angle of theindividual bend.
6.5 Adaptive sparse deconvolution
Defining a lower boundary L Bound and a upper boundary U Bound which satisfy
L Bound∑i=0
p(1/SNR) = 2.5%
∞∑i=U Bound
p(1/SNR) = 2.5%
(221)
In presence of a single bend, 1/SNR will be located in the region [L Bound = 0.003, U Bound =0.26] with a probability of 95%. The value of two boundaries are numerically calculated from thehistogram in Fig. 63. In presence of multiple bends, two dynamic boundaries shift towards theexpectation. The confidence interval narrows and the probability density of 1/SNR clusters aroundthe expectation. Accurate calculating the value of L Bound and U Bound for multiple bends turns
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Figure 64: Probability distribution of 1/SNR converges to the expectation value as the number ofbends increase.
to be very complex, because the sum of multiple scaling gamma distributions with various parameterscan not be described as an explicit analytic form. An approximation can be made by
L Bound ≈ E[1/SNR]− q(E[1/SNR]− 0.003)
U Bound ≈ E[1/SNR] + q(0.26− E[1/SNR])(222)
where q =√
x(α1)4+...+x(αm)4
(x(α1)2+...+x(αm)2)2 . As shown in Fig. 62 these dynamic boundaries indicate a proper
region (1/SNR ∈ [L Bound, U Bound]), where the support of nonzero element in X? is consistentwith the ground truth. They can be used to validate the deconvolution results. For an appropriatedeconvolution result solved by (215), the reestimated 1/SNR with a high probability of 95% shouldlocate inside these boundaries.
Fig. 65 presents the flow diagram of the adaptive sparse processing which consists of the followingsteps:
1. Initializing the sparseness regularization parameter β, maximal iteration number imax and thestep value ∆β.
2. Solving the optimization problem in (215) and getting the X?.
3. Using a threshold to filter those small coefficients in sparse vector X? obtained from Step.2, thethreshold is set to be 3σ where σ = 0.0005 denotes the standard deviation of the system thermalnoise measured by a matched load in the calibration plane. Since the l1 minimization willalways bias the amplitude of elements in vector X?, each element x?i is point-wise compensatedby a the reweighting factor ρi which is defined in (223).
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Figure 65: Flowchart of the adaptive sparse deconvolution.
4. Reestimating 1/SNR, L Bound as well as U Bound based on the ’debiased’ sparse vectorX?ρT .
5. If the reestimated 1/SNR locates in the boundaries, outputing the support of the sparse vectorand applying the nonnegative least square fitting to recover the amplitude bias. Otherwise,updating the sparseness regularization parameter β and return to Step.2.
Fig. 66 illustrates the determination of the reweighting factor ρi. After deconvolution andthreshold filtering, the position of each nonzero element x?i indicates a bend center. At the samegrid on DFT spectrum the values U(i) and U′(i) are very close to zero due to the response for singlebend, where U′ = W0X?. Directly make division of each other will enlarge the error in presence ofstrong system noise. Hence within the 7 grids around the ith bend center, the maximal value andminimal value on both spectrum will be searched and the reweighting factor is determined via
ρi =|max Ai|+ |min Ai||max Bi|+ |min Bi|
(223)
107
100.5 101 101.5 102 102.5 103 103.5 104
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
Distance(cm)
Ref
lect
ion
coef
ficie
nt
UU’X*
MaxAi
MaxBi
MinAi
MinBi
Figure 66: Determination of the reweighting factor ρi.
Once the output condition has been satisfied, the support of nonzero elements in X? will berecorded. A new vector X(i) is established, whose length is determined by the number of nonzeroelements in X?. Meanwhile the columns of W0 relevant to nonzero elements X? form a new matrixW0(i). The non-negative least square fitting performs as
X′(i) = arg min
X(i)
∥∥∥U−W0(i)X(i)∥∥∥l2,
subject to X(i) > 0.
(224)
The optimal solution X′(i) to the (224) will then be mapped to the sparse vector X and forms the
final results.
Applying this technique to 50 online measurements, the identification results for the experimentare plotted in Fig. 61f, which allows for correctly identifying when and where the bends occur aswell as how they develop over time and range.
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6.6 Online measurements of airbag folding and unfolding patterns
6.6.1 Experimental setup
Figure 67: Three standard steps to fold a driver airbag. The sensing line is placed exactly alongthe rollfolding direction. Because the sensing line turns around near the airbag edge, every foldgenerates two bends on the sensing line, which leads to a symmetrical patterns as shown in Fig. 68.
Before deployment three steps are implemented to fold the airbag in a predefined way: rollfolding,crunch as well as compression, shown in Fig. 67. At first, the airbag is folded 4 times towards themiddle along ex direction manually. After that, it will be crunched along ey axis by the foldingmachine. In last step, the crunched airbag is compressed down to a cover disc along the negative ezaxis.
Table 7 summarizes the scanning configurations of FMCW reflectometer in various processes.During deployment the system repetition time is set to be 0.165ms24, while in other processes therepetition time is extended to 0.2s due to the loose temporal requirement.
24Equation (109) indicates a shortest measurement time of 0.128ms in case of Rmax = 3m.
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Table 7: Scanning configuration of FMCW system
Process Tt/T Total ramps Overall time
Rollfolding 0.2s/0.15ms 100 20s
Crunch 0.2s/0.15ms 100 20s
Compression 0.2s/0.15ms 100 20s
Deployment 0.165ms/0.15ms 300 49.5ms
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6.6.2 Rollfolding
Figure 68: Detected folding pattern for the rollfolding process.
In the beginning of rollfolding, the airbag is placed flat on a folding machine. It will be thenmanually folded from the edge to the airbag mouth in the middle by using a metallic bar with 5cmwidth as shown in Fig. 68. The deconvolution results show a clear folding pattern: At 2s afterstart, the first fold a occurs 78.7cm far away from the calibration plane. 1s later emerges the secondfold b in the front 5.5cm and then follow the fold c as well as the last fold d. It is noted that thefourth fold pattern d almost disappears from 9s, which is marked by gray dashed circle in Fig. 68.It is because the folded airbag is inserted into the middle slot of the folding machine in the end ofrollfolding. Once the worker releases her hands, the force originally maintaining the fold structurealso disappears so that the folded airbag will act oppositely. The outmost fold d enlarges its openingangle and becomes flat.
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6.6.3 Crunch and compression
After folded in both sides along ex direction, the airbag is crunched by the folding machine. Sincethe sensing line does not cover the crunch region along ey axis, no change patterns has been detected.This could be checked by the initial pattern of compression and the final pattern of the roll folding,they are identical to each other. The top left photo of Fig. 69 shows the compression process,where the crunched airbag is down compressed by the folding machine into a cover disc. Duringcompression, the first three folds stay almost at rest, while the previously disappeared fourth foldd emerges again. Besides, a new microfold dc develops, whose center varies during the compressionand finally locates 3cm away from the fold d. From the CT of a compressed airbag used in theprevious experiment with the parallel wire, it could be concluded that the effect e is due to theinteraction of the sensing line and the internal structure of the cover disc.
Figure 69: Measured folding pattern for the compression process by adaptive sparse deconvolution.The change patterns between two gray dashed line were due to the compression.
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6.6.4 Deployment
(a) (b)
Figure 70: (a) Magnitude of Fourier spectrum. (b) Phase compensated Fourier spectrum <U.
After compressed into the cover disc, the airbag under test is fixed onto a basis over a gasgenerator. During deployment, the FMCW reflectometer synchronizes with other laboratory devicesby the same ignition trigger. When the trigger comes, every subsystem starts to work and thereflectometer periodically injects a LFMCW signal onto the sensing line. The detected Fourierspectrums are plotted in Fig. 70, where the number of folds as well as their locations are difficultlyidentified because of the overlap. The adaptive sparse deconvolution results are shown in Fig. 71,whose initial folding patterns is identical to the final patterns of the compression process. Thefolding patterns of the deployment start to change at about 2.5ms after triggering. This indicatesthe exact moment the gas flowing into the airbag. All the folding patterns release until 12.2ms witha clear order d → b → c → a. At time instant 12.2ms, the last existing fold a disappears, whichcould be confirmed by the video frame of 11ms. The development of individual fold can be obtainedby revealing the patterns in the region of interest. For example, the region denoted by the blackdashed rectangular in Fig. 71 shows that the third fold c developing into 2 closely spaced microfoldswith an interval of 2cm at 6.3ms. Such information can neither be resolvable from the DFT spectralnor from the video record. The region marked by the green dashed rectangular in Fig. 71 indicates anew deformation event. The frame of 39.5ms confirms that a part of the sensing line in the vicinityof the fold a falls away from the airbag surface because of the loose fixation.
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Figure 71: Unfolding pattern for airbag deployment by adaptive sparse deconvolution. In videoframes, the sensing line is fixed exactly along the bottom edge of the deployed airbag.
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7 Conclusion
Over the years, microwave reflectometry has been successfully implemented in various applicationfields. This thesis focuses on the application of FMCW principle for transmission line based monitor-ing applications. Compared to TDR and VNA approaches, a FMCW reflectometer has a simplifiedhardware structure and reduced requirement of physical sampling rate. It offers a cost efficientsolution for detecting the rapidly time varying impedance variations.
Theoretically, the inhomogeneities can be located from the impulse response with the knowledgeof propagation velocity, and the characteristic impedance profile of a lossless transmission line canbe determined from the impulse response by the inverse scattering methods. Due to the finite obser-vation bandwidth, the measured impulse response is only a blurred version of true impulse response.Namely, the ’high-frequency’ features of true impulse response are masked by the convolution kernel.
The time domain response of distributed inhomogeneities on a lossy transmission line can berepresented by a general convolution of the delta sequences, which indicate the reference locationsof those inhomogeneities, with a set of basis functions, which corresponds to the spread functionsfor an inhomogeneity located at different positions. Such spread functions are possible either to becalculated with the knowledge of physical parameters of the transmission lines and the inhomogeneityor can be simply obtained by taking reference measurements.
On the one side, high resolution requires the imaging grids having a spacing smaller than thestandard radar resolution. Namely, the measured discrete FMCW beat signal has to be representedby an overcomplete representation in terms of the equivalent spread functions. On the other side,such gridding refinement makes the inverse problem underdetermined. In this thesis, sparse over-complete representation has been investigated for solving such ill-posed inverse problem. To achievea proper solution, two issues needs to be taken care of:
1. A minimum separation distance, which depends on the coherence of the convolution matrix,has to be satisfied by two adjacent inhomogeneities. Our numerical simulations show that, forreal valued and complex valued impedance discontinuities the minimum separation distanceare at least 0.8∆R and 0.9∆R to guarantee an approximate support detection for sufficientlarge SNR. For distributed impedance variations, the numerical study can be performed in thesimilar way to obtain the relevant minimum separation distance.
2. The regularization parameter plays an important role on the support detection. The opti-mal regularization parameter should be chosen just large enough to cover the power of allmeasurement fluctuations including off-grid mismatch as well as the model error. In practicalapplication where fluctuations are non-gaussian, an appropriate regularization parameter canbe determined by the proposed adaptive scheme in Chapter 6.
In addition, the overcomplete representation of discrete FMCW beat signal in frequency domainform is theoretically equivalent to that in time domain form, the later form could be however easilyintegrated into the post process and shows the following advantages:
1. Range filter can be applied, which improves the efficiency of sparse recovery algorithms and issuitable for the large-scale inverse scattering problem.
2. For practical application, the convolution kernel may contain several compositions such as theVCO nonlinearity and the transfer function of the baseband filter, the time domain represen-
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tation reduces the requirement of precisely error modeling of the individual composition. Forinstance, the influence of imperfect VCO nonlinearity correction as well as the influence of thetransfer function of baseband filter do not need to be separately determined.
3. The equivalent spread functions could be simply obtained by taking the reference measurementsof the relevant physical effects of interest. The closely spaced impedance variations caused bydifferent physical effects become possible to be separated and automatically classified with theknowledge of the equivalent spread functions.
4. In long range monitoring applications, a range dependent regularization parameter could beapplied to the segmentation based time domain representation for more accurate supportdetection, because the signal to noise/interference ratio of echoes is also range dependentin practice. This implementation is difficult to be implemented in the frequency domainrepresentation directly.
Moreover, the numerical simulation shows that the sparse reconstruction by using convex opti-mization method shows a high rate of success and lowest rate of false alarms at the cost of muchmore computational effort. It is therefore strongly recommended for the monitoring applicationswhere precise localization is of first importance.
Furthermore, a broadband microwave FMCW reflectometer has been shown demonstrated forinverse scattering on transmission line structures. In cases of small reflections, it is capable topresent accurate result comparable to that measured by a VNA. To make this possible, two pertinentcalibration techniques must be carried out. The SOL (Short Open Load) calibration is importantfor precisely determination of small impedance variations in presence of systematic error, while thephase compensation is essential to derive the discrete impulse response.
By combining the FMCW reflectometry with the sparse overcomplete representation, the re-quirements of high temporal resolution, high spatial resolution and high dynamic range can besimultaneously satisfied. In an online monitoring application for airbag deployment, where measure-ment time is of concern, the proposed methods successfully capture the rapidly changing reflectionsof a sensing line attached to the airbag. By applying the proposed sparse representation to themeasured reflections produces the deployment patterns of the airbag, which present both sufficienttemporal resolution and spatial super resolution.
116
8 Appendix
8.1 Simulation result of sparse approximate reconstruction for point scat-ter
8.1.1 By convex optimization
0
10
20
30
40
0
0.5
1
1.5
20
0.5
1
SNR(dB)
Dynamic range 0dB, relaxation range 6/28∆ R
Spacing (∆ R)
Rat
e of
suc
cess
0
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1
(a)
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40
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SNR (dB)
Convex optimization, dynamic range 0dB
Spacing(∆ R)
Rat
e of
fal
se a
larm
0
0.1
0.2
0.3
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0.6
0.7
(b)
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40
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0.6
0.8
1
SNR(dB)
Dynamic range 14dB, relaxation range 6/28∆ R
Spacing (∆ R)
Rat
e of
suc
cess
0.55
0.6
0.65
0.7
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0.8
0.85
0.9
0.95
1
(c)
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40
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0.2
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SNR (dB)
Convex optimization, dynamic range 14dB
Spacing(∆ R)
Rat
e of
fal
se a
larm
0
0.1
0.2
0.3
0.4
0.5
0.6
(d)
Figure 72: Success probability and false alarm probability of sparse reconstruction by convex op-timization under varying spacing and SNR. The relaxation range is chosen as 6/28∆R. (a)Successrate under dynamic range 0dB. (b)False alarm rate under dynamic range 0dB. (c)Success rateunder dynamic range 14dB. (d)False alarm rate under dynamic range 14dB.
117
8.1.2 By bayesian learning
0
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40
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1
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20.2
0.4
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SNR (dB)
Bayesian learning, dynamic range 0dB
Spacing(∆ R)
Rat
e of
suc
cess
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1
(a)
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2
4
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8
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Bayesian learning, dynamic range 0dB
Spacing(∆ R)
Rat
e of
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se a
larm
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7
(b)
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40
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Bayesian learning, dynamic range 14dB
Spacing(∆ R)
Rat
e of
suc
cess
0.7
0.75
0.8
0.85
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0.95
1
(c)
0
10
20
30
40
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1
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20
1
2
3
SNR (dB)
Bayesian learning, dynamic range 14dB
Spacing(∆ R)
Rat
e of
fal
se a
larm
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
(d)
Figure 73: Success probability and false alarm probability of sparse reconstruction by bayesianlearning under varying spacing and SNR. The relaxation range is chosen as 6/28∆R. (a)Successrate under dynamic range 0dB. (b)False alarm rate under dynamic range 0dB. (c)Success rateunder dynamic range 14dB. (d)False alarm rate under dynamic range 14dB.
118
8.1.3 By OMP
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1
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0.5
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OMP, dynamic range 0dB
Spacing(∆ R)
Rat
e of
suc
cess
0
0.1
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1
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SNR (dB)
OMP, dynamic range 0dB
Spacing(∆ R)
Rat
e of
fal
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larm
0
0.1
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(b)
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40
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1
SNR (dB)
OMP, dynamic range 14dB
Spacing(∆ R)
Rat
e of
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cess
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1
(c)
0
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0.5
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SNR (dB)
OMP, dynamic range 14dB
Spacing(∆ R)
Rat
e of
fal
se a
larm
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(d)
Figure 74: Success probability and false alarm probability of sparse reconstruction by OMP undervarying spacing and SNR. The relaxation range is chosen as 6/28∆R. (a)Success rate under dynamicrange 0dB. (b)False alarm rate under dynamic range 0dB. (c)Success rate under dynamic range14dB. (d)False alarm rate under dynamic range 14dB.
119
8.2 Autoregressive methods
It has been show [128] that a wide-sense stationary random process can be represented as the outputof a linear filter driven by white noise with variance σ2. Conversely, a stationary random processwith power spectral density P (f) can be transformed into a white noise process by passing throughan inverse filter with the transfer function 1
H(z) . The pair of relations is illustrated in Fig. 75
Meanwhile, the transfer function of a linear filter can be approximated by a parametric model.Once the model parameter of the filter has been determined from autocorrelation of the signalsequence, the power spectral density of the scattering process can be estimated by an inverse com-putation since the spectral power density of the input white noise is a constant.
In digital signal processing, the output sequence of a linear system x(n) can be described by adifference equation
x(n) = −p∑
i=1
aix(n− i) +
q∑i=0
biw(n− i) (225)
where w(n) is the input sequence to the system.
The transfer function of the linear filter can be obtained by z transform
H(z) =B(z)
A(z)=
∑qi=0 biz
−i
1 +∑p
i=1 aiz−i(226)
A random process generated by the pole zero model in (226) with nonzero parameter p, q is calledautoregressive moving average (ARMA) process of oder (p, q). In special case where q = 0, b0 = 1,the transfer function reduces to H(z) = 1
A(z) , which becomes an autoregressive (AR) process of
order p. Once p = 1, a1 = 0 so that H(z) = B(z), it becomes a moving average (MA) process oforder q.
Generally the AR model has a simple linear equations for the AR parameters and is suitable forrepresenting spectral with narrow peaks, so that the AR model is so far extensively used.
With the AR model, the corresponding system differential (225) is reduced to
x(n) = −p∑
i=1
aix(n− i) + w(n) (227)
where the model parameters ai can be derived from the signal autocorrelation by many algorithmsuch as the Yule-Walker method, the Burg method, the Least-Squares method.
Once the model parameter is determined, the spectral estimator of AR model is computed by
PAR(f) =σ2
|A(f)|2=
σ2
|1 +∑p
i=1 aiej2πfk |2 (228)
In the past, many research works [129–131] has been addressed on how to select a proper modelorder p, which plays an important role on the estimation accuracy. A much low order results in ahighly smoothed spectrum while a too high order introduces spurious peaks. Some better-knowncriteria for model order selection are Akaike Information Criterion (AIC)
AIC(p) = lnσ2p +
2p
N(229)
120
(a)
(b)
Figure 75: Innovation representation of a random process. (a) the random process x(n) from thewhite noise through a linear filter. (b) the inverse filter.
and minimizes the description length (MDL) principle
MDL(p) = N lnσ2p + p lnN (230)
and final prediction error (FPE) criterion
FPE(p) =N + P + 1
N − P − 1σ2p (231)
and criterion autoregressive transfer (CAT)
CAT (p) =1
N
p∑k=1
N − kNσ2
p
− N − pNσ2
p
(232)
where σ2p is the estimated noise variance under model order p. Jiang has in his dissertation [132]
made thorough simulation comparison of the influence of model order selection methods on the
121
FMCW ground penetrating radar (GPR). His work shows that the correct selection of model orderstill remains a challenging issue for real radar data samples in presence of non-Gaussian and nonwhitenoise. Nevertheless, for nearly Gaussian noise model, AIC and MDL provide the same comparableresults. Thus in simulation of this thesis, the model order is derived from AIC criterion.
122
8.3 MUSIC algorithm
The multiple signal classification (MUSIC) method is a well known high resolution frequency es-timator proposed by Schmidt [126]. It detects frequencies in a signal by performing a subspacedecomposition on the covariance matrix of the samples of the received signal. Assumed that thediscrete samples at the radar receiver consist of p complex sinusoids in noise
y(n) = x(n) + w(n) =
p∑i=1
Aiej(2πfin+φi) + w(n) (233)
where Ai is the amplitude of ith sinusoid and fi is the frequency of the ith complex sinusoid, the phaseφi is a random phase variables uniformly distributed on [0, 2π), w(n) is a sample of a zeros-meanwhite noise process with spectral density σ2
w. Then the autocorrelation function of the noise freestationary process x(n) gives
rxx(m) =
p∑i=1
A2i ej2πfin (234)
The autocorrelation function for the received samples y(n) satisfies
ryy(m) = E [x(n) + w(n)]∗[x(n+m) + w(n+m)]= E x∗(n)x(n+m) + x∗(n)w(n+m) + x(n+m)w∗(n) + w∗(n)w(n+m)= rxx(m) + σ2
wδ(m), m = 0, 1, ...,M − 1
(235)
where rxx is the autocorrelation function for noise free signal x(n).
Hence, the M ×M autocorrelation matrix for y(n) is a Toeplitz matrix
Ryy =
ryy(0) ryy∗(1) · · · ryy∗(M − 1)ryy(1) ryy(0) · · · ryy∗(M − 2))
......
. . ....
ryy(M − 1) ryy(M − 2) · · · ryy(0)
=
p∑i=1
|Ai|2 eieHi + σ2
wI
= Rxx + Rww
(236)
whereei = [1, ej2πfi , ej4πfi , ..., ej2π(M−1)fi ]T (237)
is a signal vector of dimension M and H denotes the Hermitian transpose (conjugate transpose) andI is the identity matrix. (237) indicates that the autocorrelation matrix Ryy is composed of the sumof signal autocorrelation matrix Rxx as well as noise autocorrelation matrix Rww.
Note that if p < M , the rank of signal autocorrelation matrix Rxx is not of full rank, becauseits rank is equal to the number of sinusoids p. Due to the full rank of Rww, Ryy is full rank.
If we perform the eigen-decomposition of the autocorrelation matrix Ryy, it should own p nonzeroeigenvalues while the other M − p eigenvalues are equal to zero in absence of noise [128].
123
Let the eigenvalues of Ryy be arranged in decreasing order with λ1 ≥ λ2 ≥ ... ≥ λM so thatthe vi is the eigenvector associated to the λi. In absence of the noise, all of the eigenvalues can beformulated as
λ1 ≥ λ2 ≥ ... ≥ λp > 0λp+1 = λp+2 = ... = λM = 0
(238)
so that the eigenvectors vi, i = 1, 2, ..., p span the signal subpace as do the signal vector ei, i =1, 2, ..., p.
Figure 76: The eigen decomposition of the autocorrelation matrix Ryy separates the space into twosets: the signal subspace and the noise subspace.
In presence of noise, the autocorrelation matrix Ryy can be decomposed into
Ryy =
p∑i=1
(λi + σ2)vivHi +
M∑i=p+1
(σ2)vivHi (239)
where the space spanned by the eigenvectors vi, i = 1, 2, ..., p is called signal subspace while the spacespanned by vi, i = p+ 1, p+ 2, ...,M belongs to the noise subspace.
124
Since the eigenvectors are orthonormal to each other, any complex sinusoidal vector e(fi) = ei
should orthogonal to the noise subspace. In other words, the spectral estimate P (f) at any psinusoidal frequency should be equal to zero
P (fi) =
M∑k=p+1
|eivk|2 = 0, i = 1, 2, ..., p (240)
which means the reciprocal of P (f) is peaked at the frequencies of the sinusoidal components.
The MUSIC pseudospectrum is proposed by Schmidt as
PMUSIC(f) =1
eH(f)(∑Mk=p+1 vkvH
k )e(f)(241)
125
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