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Waveform Optimization for Compressive-Sensing Radar Systems

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Waveform Optimization for Compressive-Sensing Radar Systems
Circuits and Systems
Mekelweg 4,
2628 CD Delft
The Netherlands
http://ens.ewi.tudelft.nl/
CAS-2013-03
M.Sc. Thesis
Waveform Optimization for
Compressive-Sensing Radar Systems
Lyubomir Zegov
Abstract
Compressive sensing (CS) provides a new paradigm in data acquisition and signal processing in radar, based on the assumptions of
sparsity of an unknown radar scene, and the incoherence of the transmitted signal. The resolution in the conventional pulse-compression
radar is foreseen to be improved by the implementation of CS. An
unknown sparse radar scene can then be recovered through CS with a
high probability, even in the case of an underdetermined linear system.
However, the theoretical framework of CS radar has to be verified in
an actual radar system, accounting for practical system aspects, such
as the signal bandwidth, ease of generation and acquisition, system
complexity, etc. In this thesis, we investigate linear frequency modulated (LFM), Alltop and Björck waveforms, which show theoretically
favorable properties in a CS-radar system, in the basic radar problem
of range-only estimation. The aforementioned waveforms were investigated through a model of a digital radar system - from signal generation in the transmitter, to sparse signal recovery in the receiver. The
capabilities of the CS-radar versus the conventional pulse compression radar were demonstrated, and the Alltop and Björck sequences
are proven to outperform the commonly used linear LFM waveform
in typical CS-radar scenarios.
c THALES NEDERLAND B.V. and/or its suppliers. Subject to restrictive legend on title page.
Goedkeuring afstudeerverslag van:
Lyubomir Zegov
Titel vers lag:
Opleidingsinstelling:
Waveform Optimization for Compressive-Sensing Radar
Systems
Group Circuits and Systems, Faculteit EWI van TU Delft
Afstudeerperiode:
13 augustus 2012 tIm 7 juni 2013
Afdeling:
Sensors Advanced Developments Deift
Stagebegeleider Thales:
Radmila Pribi~
Dit verslag is door de begeleider van Thales Nederland B.V. gelezen en becommentarieerd.
Hierbij heeft de begeleider, naast inhoudelijke zaken, gelet op gegevens die een vertrouwelijk
karakter hebben, zoals plattegronden, confidentiële informatie en organisatieschema’s waarin
namen staan vermeld.
Dit verslag valt in de categorie’:
0
Q
1. verslagen die vanwege veiligheidsredenen/commerciele aspecten intern (Thales)
moeten blijven. Het verslag blijft te allen tijde in het archief van Thales. Indien nodig kan
dit verslag door een afgevaardigde van de opleidingsinstelling bij Thales worden ingezien,
mits deze afgevaardigde een geheimhoudingsverklaring heeft ondertekend.
2. verslagen die beperkt openbaar zijn (binnen eigen hogeschool/universiteit of
studierichting).
3. verslagen die publiek en dus volledig openbaar zijn en dus ook op internet
gepubliceerd mogen worden.
In voorkomende gevallen moet een aangepast verslag voor de opleidingsinstelling worden gemaakt.
Akkoord:
Akkoord:
Radmila Pribi~
Willem Hol
Stagebegeleider Thales
P Authority!
Stagecoordinator/
Stagebureau Thales
Akkoord:
(alleen bij eerste categorie)
Opleidingsinstelling
Delft, 7 juni 2013
(Plaats I datum)
1
De stagebegeleider van Thales geeft aan in welke categorie het verslag wordt ingedeeld.
Het verslag wordt door de stagebegeleider en een medewerker van het HR stagebureau,
danwel de stagecoardinator afgetekend.
r i—i l\~ I.... E S
Waveform Optimization for Compressive-Sensing
Radar Systems
My Subtitle
Thesis
submitted in partial fulfillment of the
requirements for the degree of
Master of Science
in
Electrical Engineering
by
Lyubomir Zegov
born in Sofia, Bulgaria
c
Copyright 2013
THALES NEDERLAND B.V.
Alle rechten voorbehouden. Niets uit deze uitgave mag worden verveelvoudigd, opgeslagen in een geautomatiseerd gegevensbestand, of openbaar gemaakt, in enige vorm of
op enige wijze, zonder voorafgaande schriftelijke toestemming van bovengenoemden.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior written
permission of the above-mentioned.
c
THALES
NEDERLAND B.V. and/or its suppliers
Subject to restrictive legend on title page
Delft University of Technology
Department of
Telecommunications
The undersigned hereby certify that they have read and recommend to the Faculty
of Electrical Engineering, Mathematics and Computer Science for acceptance a thesis
entitled “Waveform Optimization for Compressive-Sensing Radar Systems”
by Lyubomir Zegov in partial fulfillment of the requirements for the degree of Master
of Science.
Dated: 31/07/2013
Chairman:
prof.dr.ir. Geert Leus, CAS, Delft University of Technology
Advisors:
prof.dr.ir. Geert Leus, CAS, Delft University of Technology
dr. ir. Radmila Pribić, Sensors Advanced Developments, Thales Nederland
Committee Members:
dr. ir. Radmila Pribić, Sensors Advanced Developments, Thales Nederland
prof. dr. ing. François. Le Chevalier, MSSS, Delft University of Technology
c THALES NEDERLAND B.V. and/or its suppliers. Subject to restrictive legend on title page.
iv
Abstract
Compressive sensing (CS) provides a new paradigm in data acquisition and signal processing in radar, based on the assumptions of sparsity of an unknown radar scene,
and the incoherence of the transmitted signal. The resolution in the conventional
pulse-compression radar is foreseen to be improved by the implementation of CS. An
unknown sparse radar scene can then be recovered through CS with a high probability,
even in the case of an underdetermined linear system. However, the theoretical framework of CS radar has to be verified in an actual radar system, accounting for practical
system aspects, such as the signal bandwidth, ease of generation and acquisition, system complexity, etc. In this thesis, we investigate linear frequency modulated (LFM),
Alltop and Björck waveforms, which show theoretically favorable properties in a CSradar system, in the basic radar problem of range-only estimation. The aforementioned
waveforms were investigated through a model of a digital radar system - from signal
generation in the transmitter, to sparse signal recovery in the receiver. The capabilities
of the CS-radar versus the conventional pulse compression radar were demonstrated,
and the Alltop and Björck sequences are proven to outperform the commonly used
linear LFM waveform in typical CS-radar scenarios.
v
c THALES NEDERLAND B.V. and/or its suppliers. Subject to restrictive legend on title page.
vi
c
THALES
NEDERLAND B.V. and/or its suppliers. Subject to restrictive legend on title page.
Acknowledgments
This thesis would not be a fact without the help of many people. First of all I would like
to express my greatest gratitude to my thesis supervisors Radmila Pribić and Geert
Leus for organizing this challenging and very interesting project. I must thank you
for introducing me to the topics of Compressed Sensing and your valuable criticism.
Radmila, thank you for granting me the opportunity to work with you in Thales, where I
learned a lot of new things about radar. My greatest thanks, Geert, for your dedication,
support, comments, patience and most of all valuable advice on my work. I would also
like to thank François Le Chevalier for being part of the MSc thesis committee. During
the past year working in Thales I met a lot of really nice colleagues and fellow students,
to which I thank for the pleasant working environment.
Of course, being away from home could be though, but thanks to the all my friends
Bubata, Giovanni, Stephan, Edo, Vasko, I felt like at home. Thank you guys for the
memories! Special thanks to Edo and Valerio for sheltering me for the last weeks.
Finally, but most importantly I express my many thanks to my family, my father
Todor, my mother Svetla, and Ani, for your daily support and encouragement! I
dedicate this thesis to you!
Lyubomir Todorov Zegov
Delft, The Netherlands
31/07/2013
vii
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viii
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Contents
Abstract
v
Acknowledgments
1 Introduction
1.1 Motivation . . .
1.2 Research Goals
1.3 Contributions .
1.4 Outline . . . . .
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2 Radar Basics
2.1 Range estimation .
2.2 Velocity estimation
2.3 The Matched Filter
2.4 Ambiguity function
2.5 Detection . . . . .
2.6 Conclusions . . . .
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3 Compressive Sensing Radar
3.1 Compressive sensing . . . . . . . . . . . . . . . . . . .
3.2 Measuring the Coherence . . . . . . . . . . . . . . . . .
3.2.1 Restricted Isometry Property . . . . . . . . . .
3.2.2 Mutual Coherence . . . . . . . . . . . . . . . .
3.3 Sparse Signal Recovery . . . . . . . . . . . . . . . . . .
3.4 Compressive Sensing Radar . . . . . . . . . . . . . . .
3.5 Relation between the coherence and the autocorrelation
3.5.1 Underdetermined system . . . . . . . . . . . . .
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
4 RF system, waveforms and waveform optimization
4.1 RF system . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Optimal Waveforms in CS Radar . . . . . . . . . . .
4.2.1 The LFM waveform . . . . . . . . . . . . . . .
4.2.2 The Cubic Alltop sequence . . . . . . . . . . .
4.2.3 The Björck sequence . . . . . . . . . . . . . .
4.3 Further optimization of the waveforms . . . . . . . .
4.4 Bandwidth considerations . . . . . . . . . . . . . . .
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . .
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5 Results
5.1 Simulation setup . . . . . . . . . . . . . . . . . . . . . . .
5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Targets with Different Strengths . . . . . . . . . . .
5.2.2 Higher resolution by oversampling of the estimation
5.2.3 Compression . . . . . . . . . . . . . . . . . . . . . .
5.2.4 Effect of the RF system on the recovery . . . . . .
5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Conclusions and future work
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A Comparison of the coherence
A.1 Up-sampling of the estimation grid . . . . . . . . . . . . . . . . . . . .
A.2 Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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B Minimum filter bandwidth
B.1 Oversampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Zero - order hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 Linear interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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C Cyclic Algorithm Pruned (CAP)
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List of Figures
2.1
2.2
Principle of operation of a pulse radar . . . . . . . . . . . . . . . . . .
Pulse compression for a linear frequency modulated input pulse. . . . .
3.1
Graphical interpretation of the structure of the signal model matrix S
in a range-only estimation problem. . . . . . . . . . . . . . . . . . . . .
A graphical interpretation of the signal model matrix S for a rangeDoppler estimation problem. . . . . . . . . . . . . . . . . . . . . . . . .
3.2
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
A block scheme of a general RF SDR system. . . . . . . . . . . . . . .
Simplified block scheme of a generalized digital transmitter and receiver.
Example of a LFM pulse: top) Time ; bottom) Frequency . . . . . . . .
Instantaneous frequency plot of a LFM waveform with Bs = 0.8. . . . .
Autocorrelation function of a LFM pulse in dB. . . . . . . . . . . . . .
Ambiguity function of a LFM pulse in dB. . . . . . . . . . . . . . . . .
Instantaneous frequency plot of a continuous-time Alltop. . . . . . . .
Example of an Alltop pulse s[n]: top) Time ; bottom) Frequency . . . .
Instantaneous frequency plot of an Alltop waveform. . . . . . . . . . .
Unwrapped Instantaneous frequency plot of an Alltop waveform . . . .
Autocorrelation function of an Alltop pulse in dB. . . . . . . . . . . .
Ambiguity function of an Alltop pulse in dB. . . . . . . . . . . . . . .
Example of an Björck pulse: top) Time ; bottom) Frequency . . . . . .
Autocorrelation function of a Björck pulse in dB. . . . . . . . . . . . .
Ambiguity function of an Björck pulse in dB. . . . . . . . . . . . . . .
Optimized waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ACF of the Alltop sequence as a function of the double sided BPF bandwidth Bf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.18 ACF of the Björck sequence as a function of the double sided BPF bandwidth Bf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
5.3
5.4
5.5
5.6
5.7
MF and SSR with two, differently strong targets (0 db and -12 dB),
separated by 10 reference cells, SNR = 15 dB for the strongest target. .
MF and SSR with two, differently strong targets (0 db and -12 dB),
separated by 10 reference cells, SNR = 10 dB for the strongest target. .
MF and SSR with two, differently strong targets (0 db and -12 dB),
separated by 10 reference cells, SNR = 5 dB for the strongest target. .
MF and SSR with two, differently strong targets (0 db and -12 dB),
separated by 10 reference cells, SNR = 0 dB for the strongest target. .
MF and SSR with two targets (0 dB), separated by one reference cell,
SNR = 15 dB and up-sampling factor Q = 2 . . . . . . . . . . . . . .
MF and SSR with two targets (0 dB), separated by one reference cell,
SNR = 10 dB and up-sampling factor Q = 2 . . . . . . . . . . . . . .
MF and SSR with two targets (0 dB), separated by one reference cell,
SNR = 5 dB and up-sampling factor Q = 2 . . . . . . . . . . . . . . .
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5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
5.25
5.26
5.27
5.28
5.29
xii
MF and SSR with 2 targets (0 dB), separated by one reference cell, SNR
= 0 dB and up-sampling factor Q = 2 . . . . . . . . . . . . . . . . . .
MF and SSR with two targets (0 dB), separated by half a reference cell,
SNR = 15 dB and up-sampling factor Q = 4. . . . . . . . . . . . . . . .
MF and SSR with two targets (0 dB), separated by half a reference cell,
SNR = 10 dB and up-sampling factor Q = 4. . . . . . . . . . . . . . . .
MF and SSR with two targets (0 dB), separated by half a reference cell,
SNR = 5 dB and up-sampling factor Q = 4. . . . . . . . . . . . . . . .
MF and SSR with two targets (0 dB), separated by half a reference cell,
SNR = 0 dB and up-sampling factor Q = 4. . . . . . . . . . . . . . . .
MF and SSR with two targets (0 dB) separated by two reference cells,
SNR = 15 dB per target , CF = 2. . . . . . . . . . . . . . . . . . . . .
MF and SSR with two targets (0 dB) separated by two reference cells,
SNR = 10 dB per target , CF = 2. . . . . . . . . . . . . . . . . . . . .
MF and SSR with two targets (0 dB) separated by two reference cells,
SNR = 5 dB per target , CF = 2. . . . . . . . . . . . . . . . . . . . . .
MF and SSR with two targets (0 dB) separated by two reference cells,
SNR = 0 dB per target, CF = 2. . . . . . . . . . . . . . . . . . . . . .
MF and SSR with two targets (0 dB) separated by two reference cells,
SNR = 15 dB per target, CF = 4. . . . . . . . . . . . . . . . . . . . . .
MF and SSR with two targets (0 dB) separated by two reference cells,
SNR = 10 dB per target, CF = 4. . . . . . . . . . . . . . . . . . . . . .
MF and SSR with two targets (0 dB) separated by two reference cells,
SNR = 5 dB per target, CF = 4. . . . . . . . . . . . . . . . . . . . . .
MF and SSR with two targets (0 dB) separated by two reference cells,
SNR = 0 dB per target, CF = 4. . . . . . . . . . . . . . . . . . . . . .
MF and SSR with two, differently strong targets (0 dB and -12dB) and
processed waveforms, Bf = 1, SNR = 10 dB for the strongest target. .
MF and SSR with two, differently strong targets (0 dB and -12dB) and
processed waveforms, Bf = 2, SNR = 10 dB for the strongest target. .
MF and SSR with two, differently strong targets (0 dB and -12dB) and
processed waveforms, Bf = 1.5, SNR = 10 dB for the strongest target.
MF and SSR with two, equally strong targets (0 dB) and processed
waveforms, separated by one reference cell, Bf = 1, Q = 2, SNR = 10 dB.
MF and SSR with two, equally strong targets (0 dB) and processed
waveforms, separated by one reference cell, Bf = 2, Q = 2, SNR = 10 dB.
MF and SSR with two, equally strong targets (0 dB) and processed
waveforms, separated by one reference cell, Bf = 1.5, Q = 2, SNR = 10
dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MF and SSR with two, differently strong targets (0 dB and -12 dB) and
processed waveforms, Bf = 1, SNR = 10 dB for the strongest target. .
MF and SSR with two, differently strong targets (0 dB and -12 dB) and
processed waveforms, Bf = 2, SNR = 10 dB for the strongest target. .
MF and SSR with two, differently strong targets (0 dB and -12 dB) and
processed waveforms, Bf = 1.5, SNR = 10 dB for the strongest target.
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5.30 MF and SSR with two targets (0 dB) separated by two reference cells,
SNR = 10 dB per target, CF = 2. . . . . . . . . . . . . . . . . . . . . .
5.31 MF and SSR with two targets (0 dB) separated by two reference cells,
SNR = 5 dB per target, CF = 2. . . . . . . . . . . . . . . . . . . . . .
5.32 MF and SSR with two targets (0 dB) separated by two reference cells,
SNR = 0 dB per target, CF = 2. . . . . . . . . . . . . . . . . . . . . .
5.33 MF and SSR with two targets (0 dB) separated by two reference cells
SNR = 10 dB per target, CF = 4. . . . . . . . . . . . . . . . . . . . . .
5.34 MF and SSR with two targets (0 dB) separated by two reference cells,
SNR = 5 dB per target, CF = 4. . . . . . . . . . . . . . . . . . . . . .
5.35 MF and SSR with two targets (0 dB) separated by two reference cells,
SNR = 0 dB per target, CF = 4. . . . . . . . . . . . . . . . . . . . . .
B.1 Power spectrum of the oversampled signal s[m], generated by a LFM
waveform. Up) Full- length spectrum. Down) Zoom in of the main lobe.
B.2 Power spectrum of the oversampled signal ŝ[m], generated by an Alltop
sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 Power spectrum of the zero-order hold interpolated signal ŝ[m], generated by a LFM waveform. Top) Full- length spectrum. Bottom) Zoom
in of the main lobe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.4 Power spectrum of the oversampled signal ŝ[m], generated by an Alltop
sequence. Top) Full- length spectrum. Bottom) Zoom in of the main lobe.
B.5 Power spectrum of the oversampled signal ŝ[m], generated by a Björck
sequence. Top) Full- length spectrum. Bottom) Zoom in of the main lobe.
B.6 Power spectrum of linearly interpolated signal LFM waveform, generating ŝ[m]. Top) Full- length spectrum. Down) Zoom in of the main
lobe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.7 Power spectrum of linearly interpolated signal Alltop sequence, generating ŝ[m]. Top) Full- length spectrum. Bottom) Zoom in of the main
lobe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.8 Power spectrum of linearly interpolated signal Björck sequence, generating ŝ[m] Top) Full- length spectrum. Bottom) Zoom in of the main
lobe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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List of Tables
4.1
5.1
5.2
Comparison table for the parameters of the investigated waveforms, sampled at the reference sampling rate fs = 1. . . . . . . . . . . . . . . . .
32
Coherence of S in case of an up-sampled estimation grid. . . . . . . . .
Coherence µ(Θ) of Θ = ΦS, where Φ is a partial Fourier matrix, in dB
and absolute units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A.1 Coherence of S with up-sampled estimation grid as a function of Q. . .
A.2 Mutual coherence µ(S) with uniformly decimated rows, as a function of
CF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3 Mutual coherence µ(ΦS), where Φ selects random rows of S, calculated
over 10 independent realizations of Φ. . . . . . . . . . . . . . . . . . . .
A.4 Mutual coherence µ(ΦS), where Φ is a partial Fourier matrix, calculated
over 10 independent realizations of Φ. . . . . . . . . . . . . . . . . . . .
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Notation
(·)T
(·)H
(·)∗
|·|
|| · ||`
|| · ||F
b·c
Re(·)
Im(·)
I
Transpose of a vector or a matrix
Hermitian transpose of a vector or a matrix
Complex Conjugation
Absolute value
Vector `-norm
Frobenius norm
Floor function
Real part
Imaginary part
Identity matrix
Glossary
Symbol
c0
fd
s[n]
ŝ[m]
sa (t)
r[n]
r̂[m]
r̂IF [m]
Bf
Bs
fs
fIF
fs,IF
fN
fd
CF
N0
Q
R
σ2
A[k]
χ[k, fd ]
S
Θ
Φ
R
sk
φm
θk
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Speed of light
Doppler frequency
Discrete- time signal at reference sampling rate fs = 1 in transmitter
Discrete- time signal sampling rate fs IF = M fs in transmitter
Continuous time signal, analogous to s[n]
Discrete-time signal, sampling rate 1, in receiver
Discrete-time signal, sampling rate M , in receiver.
Discrete-time signal at IF, sampling rate M , in receiver.
Double sided filter bandwidth
Double sided signal bandwidth
Sampling frequency of s[n] and r[n]
Intermediate frequency
Sampling frequency of ŝ[m], sIF [m], r[m] and rIF [m]
Nyquist frequency
Doppler frequency
Compression factor
Noise power spectral density
Up-sampling factor of the estimation grid
Target range
Noise variance
Autocorrelation sequence
Ambiguity function
Model matrix
Sensing (measurement) matrix
Compression matrix
Autocorrelation matrix
Columns of S
Rows of Φ
Columns of Θ
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e
r
x
y
Noise vector
Measurements vector
Unknown signal of interest, range profile
Signal vector
Abbreviations
ADC
ACF
AF
AM
AWGN
BPF
CF
CS
CPI
CW
DAC
DDC
DUC
FFT
FIR
i.i.d
LASSO
LPF
LFM
MSK
MF
PD
PFM
PRF
PSD
PSK
RCS
RF
SD
SDR
SSR
SNR
WF
Analog to Digital Converter
Autocorrelation Function
Ambiguity Function
Amplitude Modulation
Additive White Gaussian Noise
Band Pass Filter
Compression Factor
Compressive Sensing
Coherent Processing Interval
Continuous Wave
Digital- to- Analog Converter
Digital Down Converter
Digital Up Converter
Fast Fourier Transform
Finite Impulse Response
Independent and Identically Distributed
Least Absolute Shrinkage and Selection Operator
Low Pass Filter
Linear Frequency Modulation
Minimum Shift Keying
Matched Filter
Pulse - Doppler
Partial Fourier Matrix
Pulse Repetition Frequency
Power Spectral Density
Phase Shift Keying
Radar Cross-Section
Radio Frequency
Software Defined
Software Defined Radio
Sparse Signal Recovery
Signal to Noise Ratio
Waveform
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1
Introduction
In this thesis, we explore optimal waveforms in a compressive sensing (CS) radar system
and investigate their performance over the whole radar system. This introductory
chapter provides the reader with the motivation for optimal waveforms specific to CS
radar, as well as with the outline and the main contributions of this thesis.
1.1
Motivation
The classical approach to range estimation in radar is based on the cross correlation
between the received echo signal r(t) = s(t − τ ) + e(t) and a time delayed replica of the
transmitted pulse s(t − τ ), where e(t) is white noise. This approach is known as pulse
compression or matched filtering (MF), and is widely used in conventional radar. The
target range R is related to the delay τ of the received signal as R = cτ /2, where c is the
speed of light. The range resolution ∆R is connected to the double- sided bandwidth
Bs of the transmitted pulse and is related to the time resolution ∆τ ∼ 1/Bs .
Modern radar systems process the acquired data in the discrete time domain. The
common way to represent a continuous complex signal s(t) with a discrete numeric
sequence consists of taking samples in the continuous domain of interest (time, space,
etc.) with a rate fs , to obtain the discrete signal s[n] = sa (n/fs ), where n is an integer
and −∞ < n < ∞. Conventional signal acquisition requires fs to be at least equal
to the bandwidth Bs of the complex signal of interest, i.e., fs ≥ Bs , according to the
Shannon - Nyquist theorem [1].
Thus, in order to achieve a higher time resolution ∆τ , radar needs to transmit and
recover a signal with a wider bandwidth, enforcing higher demands on the receiver
ADC sampling rates, and accordingly on the amount of data, which for today’s high
density radar arrays is highly undesirable.
The emerging theory of compressive sensing (CS) suggests that sparse signals can be
recovered from a reduced number of measurements, i.e. fs ≤ Bs , under the condition
that those measurements are incoherent with the signal. An area surveyed by radar
usually is coarsely populated with targets, thus it is sparse, making radar a good
application for the implementation of the CS theory.
In the noiseless case, the discrete-time signal y ∈ CP ×1 is modeled as
y = Sx,
(1.1)
where in radar x ∈ CN ×1 is a discretized version of the radar range profile, also referred
to as the radar scene, and S ∈ CP ×N the signal model matrix.
CS suggests that the heavy acquisition process (e.g. Nyquist rate sampling) can
be replaced by recording bP/CF c linear projections of the signal y, where CF ≥ 1
denotes the compression factor, and b·c the floor function. Compression by means of
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applying a compression matrix Φ ∈ CbP/CF c×P to the signal in matrix notation yields:
r = Φy = ΦSx = Θx,
(1.2)
where Θ ∈ CbP/CF c×N is the sensing (or measurement) matrix and r ∈ CbP/CF c×1 is
the measurements vector.
Research in CS suggests in many cases random signal acquisition [2]. Accordingly,
the compression matrix Φ can be constructed random, e.g. the entries are i.i.d. Gaussian or Bernoulli distributed random numbers, and for such random matrices a lower
bound on the number of measurements can be derived [3]. Since such random matrices
are very incoherent to any signal model matrix S, they are good candidates for CS
applications [4, 3].
The structure and content of the model matrix S is specific to the physics of the
modeled process, while the choice of Φ is user defined. A clear structure in S is present
in radar,e.g, due to the specific physical implications, the received signal is a time
- delayed version of the transmitted waveform, which enforces the specific structure
in S [3]. In the basic case of range only estimation, S has structure, related to the
time delayed nature of the measured signal. Because of this specific structure, the
incoherence of the model matrix S is directly related to the transmitted waveform.
A major disadvantage of random, unstructured compression matrices Φ is that no
fast matrix multiplication algorithms (e.g., by Fast Fourier Transform (FFT)) exist,
which is a major bottleneck for the speed of sparse signal recovery (SSR) algorithms,
especially in large scale problems. Moreover, a new realization of Φ has to be generated
for the performance guarantees of CS, a process which requires excessive computation
time, and large storage capacity. Thus, random data acquisition (e.g., random sampling, random demodulation, etc.) increases the complexity of the receiver hardware.
As a result of those considerations, a structured incoherent sensing matrix Θ = ΦS
in radar is required. The compression matrix Φ should be deterministic rather than
random. An example of deterministic Φ are matrices, for example, decimation matrices
or a partial Fourier matrices. The coherence of Θ depends on the model matrix S,
leading to the problem of waveform optimization, where waveforms which result in an
incoherent S, and accordingly Θ, need to be found.
Although in theory there exist waveforms, which yield an incoherent matrix [5],
those waveforms might be not suitable for implementation in an actual system, under
the constraints of the required transmission bandwidth or other specific requirements
such as the modulation type. For example, any amplitude modulation (AM) is unwanted because the power amplifier in the radar front end is operated in saturation
(class C), and AM can cause signal clipping and unwanted distortion.
1.2
Research Goals
The purpose of this project is to investigate and propose optimal waveforms for a CS
- based radar system, so that an optimal incoherent model matrix S is obtained. A
model of a generalized digital, software defined radio (SDR)-like system will be used,
in order to simulate the transmission-reception process. We define optimality of the
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waveform through the required transmission bandwidth and the incoherence of the
sensing matrix. Most early works related to CS [4] were concentrated on the use of
random signal acquisition (a random sensing matrix), rather than deterministic, as was
explained in Section 1.1. The intention is to utilize deterministic signal acquisition by
means of a deterministic sensing matrix Θ. Furthermore, the theoretically favorable
waveforms could be influenced by the RF transceiver system components, such as filters
and amplifiers, and those effects should be investigated. This work also intends to
further optimize the incoherence of the model matrix S. By minimizing the Frobenius
norm min||SH S − I||2F we try to match the autocorrelation matrix SH S to the ideal
autocorrelation matrix I. Since in the range only problem, S contains shifted copies of
the transmitted waveform, the ACF of the waveform when minimal solution for S is
found.
1.3
Contributions
This work provides a complete framework and a model of a CS based radar system.
Starting from signal generation at baseband, we model a simplified SDR-like system
without taking into consideration the effects of the RF amplifiers and antenna. Therefore, we analyze the signals only in the digital part of the radar transceiver. Consequently, all definitions are derived for discrete-time signals.
CS radar offers a way to reduce the sampling rates, and thus, the amount of data
to be processed at the radar receiver. Accordingly, the range resolution ∆R can be
increased with the implementation of large bandwidth waveforms, sampled at subNyquist rate. We have considered three theoretically favorable waveforms in our analysis: linear frequency modulated (LFM), cubic Alltop and Björck. We present optimized
versions of those waveforms by means of minimizing ||SH S − I||2F . We show that the
cubic Alltop, sequence, could be seen as naturally compressed due to specific folding.
Furthermore, we show the resolution capabilities of the waveforms through the radar
transmission - reception, and compare the performance through both the conventional
MF and SSR. Also, the incoherence of the sensing matrix in an underdetermined system
by up-sampling, and by compression, was analyzed.
1.4
Outline
This master thesis is organized as follows:
• Chapter 2 introduces the unfamiliar reader to the basic theory of radar. Starting
from the basics of range and radial velocity estimation, we give some further
insight in the principle of the MF and how it is applied in conventional pulse
compression radar. More fundamental aspects such as detection and the ambiguity
function are also presented.
• In Chapter 3 we review the CS framework and introduce the main criteria related
to the quality of the sensing matrix. Furthermore, we present the CS radar and
the major concepts for its implementation. At the end of the chapter we relate
the CS incoherence measures to the conventional radar ambiguity function.
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• Chapter 4 describes the model of the RF system involved in our analysis. It introduces the investigated waveforms and motivates their choice. Here the first results
are shown, concerning the bandwidth and autocorrelation of the waveforms. In
this chapter, we also describe a method adapted from [6] to further optimize the
incoherence, by means of minimizing the Frobenius norm ||SH S − I||22 , of the
investigated waveforms and present the optimization results.
• Chapter 5 lays out our main results and findings on the resolution performance
of the investigated waveforms through the MF and SSR estimation.
• Chapter 6 summarizes our findings and points out our conclusions, recommendations and suggestions for future work.
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2
Radar Basics
Radar (Radio Detection And Ranging) is an RF system for detection and ranging in
its basic version, together with velocity and angular location estimation, and tracking
of targets. Radar operates by detecting the echo, which results from the interaction of
a transmitted electromagnetic (EM) wave with a target. The operation frequency of
radar is very wide, ranging from a couple of MHz to a few hundred GHz, depending
on the application. Basically, radar systems can be divided in two major categories continuous wave (CW) and pulse-Doppler (PD) radar. In this introductory overview
we will focus only on PD radar but an interested reader can find further details on
CW radar in [7]. We base our formulations and analysis on discrete - time signals,
since in general, all processing is done digitally in the receiver after discretization of
the received analog signal.
Suppose a continuous-time signal sa (t) is sampled with a period Ts , corresponding
to a sampling rate fs = 1/Ts , normalized to fs = 1. The resulting discrete-time signal
after sampling s[n] = sa (n/fs ), n = 0, 1, . . . , L − 1, is a numerical sequence of (complex)
values. Stacking the values of s[n] into a vector we obtain the transmitted pulse vector
s = [s[0], s[1], . . . , s[L − 1]]T .
Figure 2.1: Principle of operation of a pulse radar
In PD radar, the transmitted signal sCP I [n], n = 0, 1, . . . , NCP I (N + L) − 1, is a
train of NCP I identical short pulses s[n], n = 0, 1, . . . , L − 1 , each of length L generated
at the transmitter signal generator and radiated through the transmit antenna with a
rate P RF = 1/TP RT as shown in Fig. 2.1:
sCP I [n] =
N
CP I
X
s[n − pTP RT ]
p=0
s[n] = 0,
(2.1)
L ≤ n < (L + N ),
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where L is the pulse length, N is the length of the off-period and TP RT = (N + L)/fs =
P/fs is the pulse repetition time (PRT).
The power received at the radar from a single target is given by the radar equation:
Pr =
Pt G2 λ2 ς
,
(4π)3 R4
(2.2)
where Pt , [W ] is the transmitted power, λ, [m] is the carrier wavelength, G, [abs units] is
the antenna gain (assuming a monostatic setup, where the transmit and receive antenna
have equal gain), ς, [m2 ] - the radar cross section (RCS) and R, [m] is the target range.
Radar is a very power-sensitive system since the received power decreases as R4 due
to the round trip power dissipation in free space, as we can see also from (2.2). The
transmitted power along with the receiver sensitivity play a major role in the maximum
range at which a target can be detected.
The received signal is corrupted by receiver noise, interference and clutter. Clutter
is a specific term in radar engineering, describing all unwanted echoes which arrive at
the radar. Clutter results from reflections from ground, sea, trees, animals and so on,
and can be orders of magnitude larger than the target echo [7].
2.1
Range estimation
Pulse radar transmits a train of short pulses, as shown in Fig. 2.1. The target range is
related to the round trip delay τ between the transmitted signal and the received echo:
c0 τ
,
(2.3)
2
where c0 ≈ 3 × 108 m/s is the speed of light in free space.
Pulses are transmitted at a rate P RF = 1/TP RT . Since the range is estimated
through the round trip delay and without additional measures, the radar system can
unambiguously estimate ranges from delays L < τ < TP RT , if no other techniques
(e.g. staggered PRF) are applied. In such a case, the maximum unambiguous range is:
Rua = c0 /(2P RF ).
The range resolution ∆R describes the minimal distance between two targets at
which they can be resolved as distinct, and not appearing as a single large one, and is
related to the time-resolution as ∆R = c0 /∆τ . The range resolution is limited by the
pulse bandwidth Bs , which expressed in distance equals c0 /(2Bs ), [m].
For discrete time signals, a delayed version of the transmitted waveform y[n] =
s[n − k] is described by a shift k. Each shift k, k = 0, 1, . . . , N − 1 corresponds to
a time delay τk = k/fs = k∆τ and is related to the target range as Rk = c0 k/(2fs ).
Thus, in the discrete-time case, the time resolution is equal to the sampling rate, i.e. ,
∆τ = 1/fs .
R=
2.2
Velocity estimation
Radar determines the radial velocity of a target by exploiting the Doppler effect. If
the received signal can be approximated as narrow band (NB), then the Doppler effect
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equates to the relative change in frequency between the transmitted and the received
signal due to the target motion. Otherwise, the received signal is a time scaled and
delayed replica of the transmitted signal, what is known as the wide band (WB) model.
This NB assumption results in a Doppler frequency fd
fd =
1 dφ
2 dR
2vr
=
=
,
2π dt
λ dt
λ
(2.4)
where φ is the phase of the received signal relative to the phase of the transmitted signal,
λ is the wavelength of the transmitted waveform. Accordingly the radial velocity of the
target is vr = fd λ/2. PD radar is ambiguous in range, as we saw in Section 2.1, but
also in Doppler. The maximum unambiguous Doppler frequency is |fdmax | ≤ P RF/2.
To extract the Doppler information, an FFT processing is employed. The Doppler
resolution depends on the observation time (or coherent processing interval), due to
the Fourier type of processing and is ∆fd = 1/(NCP I TP RT ) [1].
Exploiting the Doppler effect is essential in radar, first of all because it allows
estimation of the target velocity, and secondly since it provides a solution of isolating
stationary clutter.
2.3
The Matched Filter
The matched filter (MF) is a type of linear finite impulse response (FIR) filter that
maximizes the output SNRn [8] for an input signal y[n] corrupted by Additive White
Gaussian Noise (AWGN) e[n]:
SNRn = E/N0 ,
(2.5)
PN +L−1
|y[n]|2 = ||y||22 , the signal energy, and N0 , [W/Hz], the receiver
where E = n=0
noise power spectral density (PSD), which describes the noise power per unit of bandwidth.
The MF is the optimal pre-detection criterion for a single target in AWGN and is
implemented in all PD radar systems. In practice, MF can also be derived for nonwhite
noise [9] but this falls out of the scope of this work. The concept of MF in radar is
mostly known as pulse compression, where the long input pulse is compressed, and the
output is a narrower pulse with a higher amplitude as shown in Fig. 2.2. The width
of the compressed pulse is related to the bandwidth Bs and the shape of the input
pulse. The amplitude of the compressed pulse depends on the pulse length L. Since
the compressed pulse is narrower in time, the time resolution is increased.
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Figure 2.2: Pulse compression for a linear frequency modulated input pulse.
The impulse response h[n], and accordingly the frequency domain transfer function
H(ω), of the MF are determined by the specific signal that is transmitted. The transfer
function H(ω) = F{h[n]} of the MF is then given by:
H(ω) = S ∗ (ω),
(2.6)
where S ∗ (ω) is the complex conjugated Fourier transform of the transmitted signal
S(ω) = F{s[n]}. An interested reader is referred to [9] for further details about the
derivation of the MF.
Accordingly, the impulse response h[n] of the MF is:
h[n] = s∗ [−n].
(2.7)
The output of the MF, M Fout [n] = r[n] ∗ h[n], is the convolution of the received
signal plus noise r[n] = y[n] + e[n] with a (known at the receiver) complex conjugated
replica of the transmitted signal h[n] = s∗ [−n]:
M Fout [n] =
=
∞
X
k=−∞
∞
X
r[k]h[n − k] =
(2.8)
r[k]s∗ [k − n].
k=−∞
The MF does not preserve the shape of the waveform (see Fig. 2.2) at its output but
maximizes the SNRn , e.g., if the input is a square pulse then the output is a triangular
pulse. It is clear that a MF can be derived for any kind of signal no matter the pulse
shape, length or bandwidth.
For the case of noiseless measurements e[n] = 0, the expression at the right-hand
side of (2.8) is recognized as the autocorrelation function (ACF) A[k]. Reversing the
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roles of n and k, where k is the delay over the duration of one pulse s[n], the ACF of
a signal s[n] is:
L−k−1
X
∗
s[n]s [n − k] ,
A[k] = n=0
k = −L, −L + 1, . . . , L − 1.
(2.9)
The ACF has its maximum at k = 0 [8]. The power spectral density (PSD) of s[n]
is the DTFT F{A[k]} of the ACF. Accordingly, a highly localized A[k] is generated by
a waveform with a wide PSD.
2.4
Ambiguity function
As explained in Section 2.3 the output of the MF is the convolution of the received
signal r[n] = s[n − k] + e[n] with a complex conjugated replica of the transmitted signal
s∗ [−n]. In the noiseless case e[n] = 0, n = 0, 1, . . . , L − 1, the received signal r[n] will
be just a delayed replica of the transmitted signal r[n] = s[n − k] and the MF output
is then the ACF of s[n] as in equation (2.9).
In (2.9) the received signal is only delayed but not Doppler shifted. If the target
is in motion, the output of the MF, for a noiseless received signal, will be the cross
correlation between the transmitted signal and its time delayed and Doppler shifted
replica. The resulting two dimensional function χ[k, fd ] is the radar ambiguity function
(AF), given by:
L−1−k
X
χ[k, fd ] = s[n]s∗ [n − k]ej2πfd n (2.10)
n=0
The main properties of the AF, which determine the resolution of the radar system,
are listed below:
• The maximum of the ambiguity function occurs at the its origin:
|χ[0, 0]|2 ≥ |χ[k, fd ]|2 .
• The total volume under the AF is independent of the waveform:
XX
|χ[k, fd ]|2 = 1,
(2.11)
where the signal energy E = ||s||22 of is normalized to unity, i.e., ||s||22 = 1.
• The AF is symmetric w.r.t. the origin: |χ[k, fd ]|2 = |χ[−k, −fd ]|2
The autocorrelation function (ACF) from equation (2.9) is a special case of the AF
in (2.10). If we assume, that the radar echo originates from a stationary target, the
received signal will be only time delayed. This is the zero - Doppler cut in the AF
|χ[k, 0]| ≡ A[k] .
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2.5
Detection
Having r = [r[0], r[1], . . . , r[P − 1]]T , P = N + L, observations of the received signal,
we would like to make a decision, whether a target is present or not. The detection
procedure is based on a decision, whether or not the received signal r is noise only r = e
(no target) or signal plus noise r = y + e (target). We refer to those two scenarios as
hypotheses H0 and H1 :
H0 : r = e
H1 : r = y + e
(2.12)
In the classical framework, the procedure, which maximizes the detection probability
PD = Pr(H1 |H1 ) for a given probability of false alarm PF A = Pr(H1 |H0 ) is known as
the likelihood ratio test (LRT) specified by the Neyman-Pearson theorem [10]. We
basically decide that H1 is true if:
Λ(r) =
p1 (r|H1 )
> β.
p0 (r|H0 )
(2.13)
Assuming a AWGN process e ∈ CN (0, σ 2 I), the PDF’s under each hypothesis H0
and H1 are:
P PP −1 |r[n]|2
1
n=0
σ2
p0 (r|H0 ) =
e−
2
πσ
(2.14)
P PP −1 |r[n]−y[n]|2
1
− n=0 2
σ
p1 (r|H1 ) =
,
e
πσ 2
The optimal test statistic Λ(r) for white noise corrupted measurements is the output
of a matched filter Λ(r) = M Fout = yH r [10]. The detection scheme is then based on
comparison of M Fout to a predefined threshold α. If the output of the MF is larger then
the threshold M Fout [n] > α2 , a target is detected. The selection of α is crucial, since
setting α too low results in false alarms, and a too high α causes missed detections of
(weak) targets.
Given PF A and the noise variance σ 2 , the threshold is given by [10]:
1
2
2
α = σ ln
.
(2.15)
PF A
2.6
Conclusions
This chapter expounded some concepts of in radar technology, which will be used
throughout the next chapters. We explained the fundamental techniques for target
range and velocity estimation. Then, we lay down the theoretical principles of pulse
compression, implemented as the matched filter. We also presented the AF and ACF,
which are important tools in the radar waveform design. In the last Section 2.5 we
talked about the basic theory of classical detection, and related the MF threshold to
the probability of false alarm PF A .
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Compressive Sensing Radar
3
In this chapter, we introduce some preliminaries of CS and also show how CS can
be adopted in radar. We present the theoretical measures for the incoherence of a
sensing matrix, and the sparse signal recovery (SSR) framework. Furthermore, we
show two scenarios, where an underdetermined problem occurs- firstly by increasing
the resolution above 1/fs , and secondly by compression.
3.1
Compressive sensing
Almost all electronic systems rely on the principles laid by the Shannon - Nyquist
sampling theorem, stating that a continuous signal is perfectly described by discrete
measurements with a rate fs at least equal to the double-sided bandwidth fs ≥ Bs of
the complex signal of interest [1]. In an RF system, such as radar, the data conversion
is performed by an ADC, which uniformly samples the continuous-time input signal to
produce a number of discrete values at its output.
A typical lossy compression method, such as MP3 and JPEG, involves acquisition
of the full signal (video, music, etc.) and then storing, or transmitting, only the largest
coefficients. The novel paradigm of CS, introduced by Candès, Romberg and Tao [4],
suggests that one can directly measure or store a few mixtures of the largest signal
coefficients, under some assumptions on the signal and the mixtures.
CS conveys that sparse or compressible signals can actually be recovered from fewer
measurements, based on assumptions related to the signal sparsity and incoherence.
The number of required measurements is comparable to the sparsity, or put differently
it is proportional to the information content in the signal [11]. The specific requirements
for CS are:
• The signal should be sparse or compressible (sparse after projection on an appropriate basis). In such a case, it can be represented with only a few coefficients
compared to its dimensionality. Sparsity helps in isolating the original vector [11].
• The condition on the measurements is that they are incoherent, which ensures
that information is not damaged by the sensing approach. On the other hand,
coherent acquisition would yield similar measurements and an unique solution
cannot be obtained. The reason is that two (or more) signals x will be mapped
to a very similar measurements r.
As mentioned above, the discrete unknown signal of interest is represented by a
vector x = [x1 , x2 , . . . , xN ] ∈ CN ×1 , and it must be sparse or compressible. A vector is
sparse, if it has only a small number (k << N ) of non-zero coefficients.
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The `p norm of a vector x is defined as
p
(||x||p ) =
N
X
|xn |p ,
(3.1)
n=1
Formally, sparsity can be defined through the `0 pseudo-norm, which counts the number
of non zeros in x. A vector x is k -sparse if it has at most k nonzero entries, e.g.,
||x||0 = k, where k << N .
The acquisition process is modeled by recording bP/CF c, inner products between
bP/CF c
the unknown signal y and the sensing functions {φTp }p=1 , and CF ≥ 1, denoting
the compression factor. In the resulting linear CS model
r = ΦSx = Θx,
(3.2)
r ∈ CbP/CF c×1 are the noiseless measurements, Θ ∈ CbP/CF c×N the sensing matrix,
Φ ∈ CbP/CF c×P the compression matrix, and S ∈ CP ×N is the signal model matrix.
3.2
Measuring the Coherence
In this section we present two ways to compute the coherence of a matrix, namely the
restricted isometry property (RIP) and the mutual coherence µ(Θ) .
3.2.1
Restricted Isometry Property
A matrix Θ, with normalized columns ||θ̄ i ||2 = 1, i = 1, 2, . . . , N, satisfies the
RIP(k, δk ) of order k, and isometry constant δk if:
(1 − δk )||x||22 ≤ ||Θx||22 ≤ (1 + δk )||x||22 ,
(3.3)
for all k-sparse x ∈ CN ×1 .
Equation 3.3 states that all subsets of k columns of Θ are almost orthogonal.
RIP(k, δk ) can further be interpreted as a property of Θ, which ensures that the Euclidean distance between any k-sparse x is preserved by Θ. The RIP guaranties stability
of the CS problem solution under noise [12].
3.2.2
Mutual Coherence
The most common choice, in sense of computational complexity, for evaluating the
incoherence of a matrix is the mutual coherence µ(Θ). The mutual coherence evaluates
the inner products between the normalized columns of Θ, e.g., ||θ̄ k ||2 = 1
µ(Θ) = max |hθ̄ i , θ̄ k i|.
i6=k
(3.4)
Basically, µ(Θ) defines the biggest correlation between the columns of Θ. If the
coherence is nearly zero, then the columns of Θ are almost mutually orthogonal. The
incoherence is essential for the reconstruction algorithms and serves also as a lower
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bound on the number of measurements that need to be taken for reconstruction [3].
Put differently, the bounds on µ(Θ), expressed through the dimensionality of Θ are [13]
s
N − bP/CF c
≤ µ(Θ) ≤ 1
(3.5)
bP/CF c(N − 1)
Of course though, µ(Θ) depends both on the coherence of the model matrix S and
the coherence of the compression matrix Φ.
The RIP is connected with the mutual coherence through application of the Gershgorin circle theorem [11] and Θ satisfies RIP(k, δk ), for any k < µ−1 + 1. This relation
provides a way of calculating the RIP constant through the easily computable coherence
metric µ(Θ).
The procedure of generating a sensing matrix Θ with sufficient RIP is still an open
issue. However, some authors [6, 14] suggest algorithms for minimizing the coherence
of the S matrix. In our analysis we adopt the algorithm of [6], to minimize ||SH S−I||2F ,
based on some assumption about the structure in S. The matrix optimization procedure
is presented in Section 4.3.
3.3
Sparse Signal Recovery
In Section 3.1, we laid down the theoretical requirements on the signal and the measurements such that a solution of a possibly underdetermined problem can be found.
However, we did not explain how this solution is obtained.
In basic linear algebra, the solution of (3.2) in the underdetermined case (R < N )
has an infinite number of solutions. Since we have a priori knowledge about the signal
x (x is sparse), an intuitive approach is to find a solution to (3.2), by searching for the
sparsest x. This comes down to the `0 minimization problem
x̂ = arg min ||x||0
x
subject to r = Θx.
(3.6)
In general, the problem in (3.6) is NP - hard [12]. In other words, there does not
exist a tractable algorithm that solves (3.6) for any Θ and r [3].
A practical alternative to solve (3.2) is the `1 norm minimization problem:
x̂ = arg min ||x||1
x
subject to r = Θx.
(3.7)
Various techniques are available for solving (3.7),e.g., different basis pursuit algorithms or greedy algorithms, such as matching pursuit [15].
In radar, the measurements r are usually noise corrupted:
r = Φ(Sx + e) = Θx + Φe.
(3.8)
The noise vector e ∼ CN (0, σ 2 I) results from a AWGN process with zero mean and
variance σe2 . Tikhonov regularization would typically be used in inverse noise corrupted
problems, but it adopts a least squares solution, which is not sparse [15].
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If we assume, ||e||2 < , the problem from (3.8) can be solved for x as
x̂ = arg min ||x||1
x
subject to||Θx − r||22 < ,
(3.9)
where > 0 bounds the amount of noise in the data.
The Lagrangian formulation of (3.9), adopting the LASSO (least absolute shrinkage
and selection operator) operator (a `1 (sparsity promoting) penalized linear problem)
is used for the SSR
x̂ = arg min{||Θx − r||22 + α||x||1 }.
(3.10)
x
The parameter α balances between the noise energy and the sparsity of x. Various
computational methods [15] are available for solving (3.10), such as different basis
pursuit (BP), linear programing, or matching pursuit (MP) algorithms, or techniques
based on a Bayesian framework. In our analysis, we adopt the Bayesian approach,
implemented as a complex fast Laplace (CFL) algorithm because it is robust to noise
and is executed in nearly real time (FL from [16] adapted for complex signals in [17]).
3.4
Compressive Sensing Radar
The theory of CS is well suited for application in radar, since as stated in the introductory Chapter 1, the radar scene is usually sparse, which is one of the feasibility
requirements for CS.
To illustrate the concept of CS radar we start with the basic problem of range-only
estimation and at the end of the section, we shortly present the more general model
for the range-Doppler estimation problem.
Assume we are interested in detection of targets in the unambiguous range interval
[ρ1, ρ2]. Discretization of [ρ1, ρ2] into QN bins (range gates), each with size ∆τ =
1/(Qfs ), defines the estimation grid τ1 , . . . , τN (see Fig. 3.1). A target can be located
at any grid point τk , with a corresponding magnitude xk , related to the target RCS.
Each grid point corresponds to a time delay τk , where τk = k/(Qfs ), k = 1, . . . , QN .
The refer to reference estimation grid for Q = 1.
Obtaining P = N + L discrete measurements of r(t), with rate fs = 1 over the
observation variable t, we get the discrete received signal r[n], and define the observation
grid (see Fig. 3.1).
The signal reflected from a target on the grid τ1 , . . . , τN , is a time delayed replica
of the transmitted signal (see Chapter 2). In the reference case, fs = 1, the received
signal r[n], can be expressed as the superposition of shifted copies of the transmitted
waveforms s[n] corrupted by noise:
r[n] =
N
X
xk s[n − τk ] + e[n],
(3.11)
k=1
where τ1 = ρ1 and τN = ρ2 .
Stacking P = N + L measurements r[n] in a vector r we obtain a linear model of
the received signal:
r = Sx + e.
(3.12)
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A graphical interpretation of the structure of the signal model matrix S is given in
Fig. 3.1.
Figure 3.1: Graphical interpretation of the structure of the signal model matrix S in a rangeonly estimation problem.
Since the echo is a time delayed replica of the transmitted signal the columns sk of
S contain shifted copies of the transmitted waveform s. Only the first L elements of s1
describe the pulse, and the other columns sk , k > 1 are shifted copies of s1 as shown
in (3.13).


s[0]
···
0
..
...


.




s[0] 
S = s[L − 1]
.
(3.13)


.
.
..
..


0
· · · s[L − 1] P ×N
S as in (3.13), is the model matrix used in our analysis to model the (received) signal
in the problem a range estimation problem. The resulting linear model r = Sx + e
is equivalent to the general CS model from (3.2), with noise corrupted observations
r = y + e and no compression (Φ = I of size P × P ).
Now the problem of range estimation boils down to finding the solution for (3.12),
where xk is related to the target RCS and the corresponding grid cell τk describes the
target range.
A conventional radar detection scheme starts with the MF x̂M F = SH r. The output
of the matched filter is the optimal test statistic for a likelihood based detection. However, in SSR the MF approach is not optimal, due to the assumed sparsity structure in
the range profile x. Thus, CS radar would utilize an SSR estimator, as in (3.10), which
can then be seen as a refinement of the MF. It should be noted that the MF is kept,
because of its optimality to maximize the SNR. In (3.10), the parameter α balances
between the sparsity constraint and the noise regularization. In the CFL algorithm [17],
α can be seen as the threshold for a given PF A and according to [18], α is calculated
as in (2.15).
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Figure 3.2: A graphical interpretation of the signal model matrix S for a range-Doppler
estimation problem.
One can extend the model to allow the simultaneous estimation of two or more
parameters (i.e. range - Doppler, range- angle, etc. ). We are going to briefly talk
about the most common case of range-Doppler estimation [13]. In that case the model
matrix S ∈ CNCP I P ×NCP I N is formed by stacking together, NCP I Doppler-shifted copies
of S from (3.13), generating a Gabor frame [19] as illustrated in Fig. 3.2. The resulting
matrix can be underdetermined. Then the SSR prefers simultaneous estimation of
both range and Doppler. Furthermore, in conventional PD radar the estimation of the
separate target parameters is done in a sequential manner for optimal gain (e.g., first
range, then Doppler, etc.), while CS offers a framework, where all parameters are to be
simultaneously estimated. A graphical interpretation of the signal model matrix S for
a joint range-Doppler estimation is presented in Fig. 3.2 .
3.5
Relation between the coherence and the autocorrelation
In the case of range only estimation and reference estimation grid, there is a clear
connection between the mutual coherence µ(S) and the ACF A[k]. Taking the product
SH S, we obtain the autocorrelation matrix R ∈ CN ×N .
The elements of R are the inner products between the columns of S i.e. Ri,k =
hs̄i , s̄k i, which contain shifted copies of the transmitted waveform s[n] (see Fig. 3.1).
The largest off diagonal value of R is therefore equal to the mutual coherence µ(S)
defined by 3.4. Furthermore, due to the Toeplitz structure in S, µ(S) is related to the
ACF. Each column of R is the ACF of s[n] at a certain delay τ . In the context of
radar, the mutual coherence µ(S) is the highest sidelobe in the ACF. In other words,
µ(S) can be seen as the highest off-peak correlation coefficient in A[k].
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3.5.1
Underdetermined system
In the case of range-only estimation, an underdetermined CS problem can be formulated
in two ways. Keep the same number of measurements P and up-sample Q > 1 times
the estimation grid ∆τ = 1/(Qfs ), allowing for a higher resolution than the reference
∆τ = 1/fs . Or keep the reference resolution ∆τ = 1/fs and reduce the number of
measurements r ∈ CbP/CF c×1 , with CF > 1.
• One can increase the maximum attainable resolution to ∆τ = 1/(Qfs ). Accordingly, the estimation grid τ1 , . . . , τN Q is finer, with cells size ∆τ = 1/(Qfs ). The
unknown signal vector becomes of dimensionality x ∈ CN Q × 1. In this case, the
(n, k)th element of S is given by
Sn,k = sa (n/fs − k/(Qfs )).
(3.14)
Therefore S has a block Toeplitz structure and the system in (3.2) is underdetermined, if P ≤ QN .
• To reduce the number of measurements by a compression factor CF > 1, while
keeping the same resolution, a compression matrix Φ ∈ C(bP/CF c×P ) is applied to
the measurements:
r = ΦSx + Φe.
(3.15)
Φ can be constructed in various ways. A common choice is a partial Fourier
matrix [3], or an operator that selects random elements of r [5, 20]. A partial
Fourier matrix is constructed by randomly choosing bP/CF c rows from a square
P × P discrete Fourier matrix. The benefit of such matrices, is that an FFT
algorithm can be used to compute matrix-vector multiplication operations.
3.6
Conclusions
In this chapter, we presented the main concepts of CS, specifying the conditions on the
unknown signal x and the measurements r. To sum up, a stable solution is provided,
given a sparse or compressible signal and incoherent measurements. The reconstruction
process is specific for signals in noise and can be formulated as the LASSO problem.
Furthermore, the theoretical framework of CS radar was developed. Particularly, we
observe a specific (Toeplitz or block Toeplitz) structure in the model matrix. The target
scene is recovered through SSR such as in (3.10), which can be seen as a refinement of
the MF in the conventional PD radar.
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RF system, waveforms and
waveform optimization
4
This chapter starts with an introduction of the RF system model used in our analysis of
the performance of the waveforms through radar transmission and reception. Following
is the analysis of several waveforms, chosen because of their theoretically favorable
properties defined in Section 4.2 . We investigate the instantaneous frequency of the
analyzed waveforms along with their ACF and the mutual coherence of the resulting
model matrix S. A comparison of the waveform properties is presented in a summarizing
Table 4.1. In Section 4.3 an algorithm [6] is adopted to further minimize the matrix
coherence, and accordingly the off-peak ACF correlations. The concluding Section 4.5
exposes an overview of our findings.
4.1
RF system
Each modern RF system can be roughly divided in two parts - digital and analog. The
intention in modern RF systems is to avoid as much as possible the analog processing
and move all processing (filtering, up- and down- conversion, signal transforms) towards
the digital domain. It is desirable to steer to a more or less software defined radio (SDR)
type of radar systems due to the flexibility of the architecture of such systems [21].
Figure 4.1: A block scheme of a general RF SDR system.
A block scheme of a general SDR system is given in Fig. 4.1. A typical realization
of a SDR system will require a fast digital- to-analog converter (DAC), analog-todigital converter (ADC), digital up-converter (DUC) and a DDC digital down converter
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(DDC). SDR systems ease the generation of different and various waveforms, without
the need of new hardware, even at RF [22]. Additionally, SDR type of radar system
supports very stable and precise digital oscillators and digital filtering. All front-end
components such as the power amplifiers and data conversion modules (DACs, ADCs)
are available as external modules.
In our work we omit the modeling of the RF power amplifiers and the antenna
from the analog part of the system in Figure 4.1. We concentrate on the digital part
of the system and model the digital band-pass filters (BPF), the Hilbert transform,
interpolation and digital up-conversion. In our system model, we adopt a digital radar
system as in Fig. 4.2 and model all the blocks.
Figure 4.2: Simplified block scheme of a generalized digital transmitter and receiver.
The transmitter takes care of everything from signal generation in the baseband to
signal transmission at RF. The initial waveform s[n] is in the discrete-time domain,
the reference at rate fs = 1. The next step is to up-sample s[n] with a factor M >
1, generating ŝ[m], m = 0, 1, . . . , M L − 1 with a corresponding signal vector ŝ =
[ŝ[0], ŝ[1], . . . , ŝ[M L − 1]]H , in order to prepare it for up conversion to IF. The upsampling can be done in three ways: directly generate ŝ[m] from the analog signal sa (t)
sampled at high rate M > 1, or by sample and hold (zero order hold) interpolation, or
by linear (first order hold) type of interpolation:
• Oversampling - introduce extra samples by sampling the generating function sa (t)
at a higher rate fs,IF = M fs :
ŝ[m] = sa (m/(M fs ))
(4.1)
• Sample and hold - introduce M − 1 samples with the same value in between two
of the samples of s[n]:
ŝ[m] = s[bm/M c],
(4.2)
where b·c denotes the floor function.
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• Linear interpolation - introduce M − 1 samples on the straight line connecting
two adjacent samples of s[n]:
ŝ[m] = (s[bm/M c + 1] − s[bm/M c])(m modM )/M + s[bm/M c].
(4.3)
The up-converted signal at IF sIF [m] is then generated by mixing the up-sampled
baseband signal ŝ[m] with the carrier exp(j2πfIF m/fs,IF ) at IF, obtaining sIF [m] =
ŝ[m]exp(j2πfIF m/fs,IF ) . Conversion to IF is implement in order to investigate the
effect (if any) of the Hilbert transform in reception.
The double-sided bandwidth Bf of all the linear phase filters is tunable and is
relative to the reference sampling frequency fs = 1.
The receiver generally does the reverse job of the transmitter, starting with downsampling of the received waveform r(t). The sampling rate of the analog to digital
converter (ADC) is chosen Kfs , K > M , which is the same as the one used for
simulation of the DAC process. At the output of the ADC, we obtain the received
signal r̂IF [m] at IF. The IF signal r̂IF [m] is then passed through a BPF with the
same bandpass Bf as the one in the transmitter, to restrict its bandwidth and filter
noise. Since the received signal is real, a Hilbert transformer reconstructs the complex
envelope of the received signal. The next step in the receiver down-convert r̂IF [m] to
baseband, obtaining an up-sampled version of the received signal r̂[m] at rate M . The
final sequence r[n] is obtained by decimation of r̂[m] with a factor M .
4.2
Optimal Waveforms in CS Radar
In Chapter 2, we presented the received signal parameters (delay τ and Doppler shift
fd ) which translate to the target’s range and velocity. This section presents a selection
of waveforms, which show theoretically favorable properties for implementation in a CS
radar system. The performance criteria on which the waveforms are compared are:
• Localization of the mainlobe in the ACF.
• Sidelobe level in the ACF. In Section 3.5 we established a relation between the
highest sidelobe level and the matrix coherence for reference sampling rate fs = 1,
and we are going to use µ(S) and A[k] interchangeably.
• Required double - sided transmission bandwidth.
• Ease of generation, transmission and reception.
Since the power amplifier in the radar transmitter is operated in saturation (Class C),
all amplitude modulation is unwanted, and if allowed can cause signal clipping and
an unwanted signal distortion. This feature puts also a constraint on the modulation
scheme. The preference for a modulation scheme is for one, that does not modulate
the amplitude of the transmitted waveform, e.g., PSK or MSK.
The linear frequency modulated waveform (LFM, linear chirp) is known since the
middle of the XXth century and still is one of the most popular pulse-compression
waveforms in modern radar systems [8]. This is a good reason to choose it as a reference,
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to which we compare the performance of the other waveforms. A literature study [23,
8] suggests an enormous amount of discrete sequences, as well as modifications of
existing waveforms. Due to this large number and also the infeasibility (i.e., OFDM
waveforms introduce amplitude modulation) with our design criteria, we have restricted
our analysis to three waveforms. From the theoretically optimal favorable (e.g. [13]
and [24]) we choose two (discrete) phase sequences with constant unit amplitude and
good ACF properties, namely the cubic Alltop sequence and the Björck sequence.
The cubic Alltop sequence originates from a continuous phase signal, while the
Björck sequence is a truly discrete binary sequence. The implementation of the cubic
Alltop sequence in a stylized CS radar provides good SSR results (better resolution
than with a MF) [13]. However, in [13], the transmission-reception effects are neglected,
although in a practical application, the RF system might have significant effects. We
are going to show that the cubic Alltop sequence is not severely damaged by the RF
transmission - reception process, only under the condition that an appropriate, wider,
e.g., Bf > 1 bandpass filter (BPF) is involved.
Constant Amplitude Zero Auto Correlation (CAZAC) sequences are also proposed
for radar due to their good properties - highly localized autocorrelation peak and low
off peak correlations (zero in the periodic case [24]), along with constant amplitude [24].
The Björck sequence falls in the class of CAZAC and is a good candidate for CS radar.
Another example of CAZAC sequences, widely used in radar are the Barker codes [8],
but they fall out of the scope of our work due to their limited length availability.
4.2.1
The LFM waveform
During a pulse interval T , the frequency is linearly swiped through the frequency band
Bs (e.g., from f0 = −Bs /2 to f1 = Bs /2), where the chirp rate β = Bs /T defines the
slope of the frequency sweep.
Figure 4.3: Example of a LFM pulse: top) Time ; bottom) Frequency
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An example of an LFM chirp of length L = 100 and Bs = 0.8, in the time domain
and its corresponding power spectrum, calculated through FFT, is shown in Fig. 4.3.
The complex envelope of the normalized LFM waveform is given by:
1
1
sa (t) = √ ejφLF M (t) = √ ejπ(2f0 +tβ)t ,
T
T
0 ≤ t < T.
(4.4)
The instantaneous frequency of the LFM pulse is calculated by differentiation of the
phase of the complex envelope:
f (t) =
1 dφLF M
= f0 + βt,
2π dt
0 ≤ t < T.
(4.5)
which is clearly a linear function of time as also shown in Fig. 4.4.
Figure 4.4: Instantaneous frequency plot of a LFM waveform with Bs = 0.8.
In the discrete - time case, the expression in (4.4) is evaluated at instances s[n] =
sa (n/fs ), where fs = 1. Thus the normalized discrete time linear chirp of length
L = T /fs is given by:
1
1
s[n] = √ ejφLF M [n] = √ ejπ(2f0 +nβ)n ,
L
L
n = 0, 1, . . . , L − 1.
(4.6)
Accordingly, the discrete-time instantaneous frequency of the LFM waveform is computed by taking the finite difference of the phase of s[n]:
f [n] =
1
(φLF M [n + 1] − φLF M [n]) = f0 + β(n + 1),
2π
n = 0, 1, . . . L − 2
(4.7)
In the following Fig. 4.11 and Fig. 4.12 respectively we present the ACF and the AF
of the LFM.
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Figure 4.5: Autocorrelation function of a LFM pulse in dB.
Figure 4.6: Ambiguity function of a LFM pulse in dB.
The range resolution properties of the LFM can be derived, from the first null in
the ACF, which occur at 1/Bs [8]. The ACF of the LFM has the sinc structure, where
the first sidelobe level is −13.4 dB [8], causing weak targets to be masked.
4.2.2
The Cubic Alltop sequence
The cubic Alltop sequence is a member of the family of periodic sequences with low
correlation magnitude, derived in [25]. From this family of sequences [25] we choose
the cubic phase sequence. In continuous-time domain, the equivalent of the Alltop
sequence is given by:
1
1
3
sa (t) = √ ejφAlltop (t) = √ ej2πt /T ,
T
T
0 ≤ t < T.
(4.8)
The instantaneous frequency can then calculated as:
f (t) =
1 dφAlltop
= 3t2 /T.
2π dt
(4.9)
In (4.9) we observe a quadratic behavior in f (t). Thus, in continuous- time, the cubic
Alltop sequence is equivalent to a quadratic frequency modulated waveform (quadratic
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chirp), as shown in Fig. 4.7. However, (4.8) is only an assumption of what the discrete
Alltop sequence [25], would be in the continuous-time domain. This assumption is
hammered due to the fact that in [25] the author only derived discrete sequences of odd
length.
Figure 4.7: Instantaneous frequency plot of a continuous-time Alltop.
Evaluating (4.8) at time instances t = n/fs , and fs = 1, we come to an expression
for the Alltop sequence of odd length L as in [25]:
1 j2πn3
s[n] = √ e L , 0 ≤ n ≤ L − 1,
L
(4.10)
By taking the finite difference of the phase in (4.10) we obtain:
1
(φAlltop [n + 1] − φAlltop [n])(mod2π)
2π
3n2 + 3n + 1
=
(mod2π), n = 0, 1, . . . L − 2
L
f [n] =
(4.11)
Figure 4.8: Example of an Alltop pulse s[n]: top) Time ; bottom) Frequency
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In Fig. 4.8, we present the Alltop sequence s[n] from (4.10) of length L = 101 in the
time and frequency domain. The time domain sequence is noise - like and no specific
structure is present. The same is true also in the frequency (Fourier) domain.
As known from the Fourier theory [1], noise like signals have wide frequency content.
The frequency domain plot in Fig. 4.8 does not provide sufficient information on the
frequency content of the Alltop sequence, since we can only observe the signal in the
range −fs /2 to fs /2. Due to the almost flat spectrum of the Alltop sequence, we might
suspect that the actual frequency content Bs (the bandwidth of the signal) is larger
than fs = 1 in this case. Thus, we look at the instantaneous frequency (4.11) of the
Alltop sequence in Fig. 4.9.
Figure 4.9: Instantaneous frequency plot of an Alltop waveform.
In Fig. 4.10 we show the unwrapped instantaneous frequency of the Alltop, which
is computed by calculating the finite difference of the unwrapped phase φAlltop [n]. The
unwrapping process, equates to correcting for the phase jumps of the phase φAlltop larger
than π, by adding multiples of ±2π.
Figure 4.10: Unwrapped Instantaneous frequency plot of an Alltop waveform
The Alltop sequence from (4.10) can be seen as an under-sampled version of a
quadratic chirp waveform. It is rather interesting that due to the specific folding of the
Alltop signal, recovery of the wide bandwidth waveform with sub-Nyquist sampling
is possible. This can be seen as a natural compression. In other words, the Alltop
sequence contains an information of a higher bandwidth signal in a low dimensional
signal.
The Alltop sequence also yields a highly localized AF, because of its large bandwidth
Bs . Also no particular structure is present in the ACF and in the AF. The off peak
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behavior is flat, where the largest off-peak correlation coefficient is −16.5 dB
Figure 4.11: Autocorrelation function of an Alltop pulse in dB.
Figure 4.12: Ambiguity function of an Alltop pulse in dB.
4.2.3
The Björck sequence
The Björck sequence falls in the family of constant amplitude zero autocorrelation
(CAZAC) sequences. The Björck sequence is known to have a zero periodic ACF and
nearly zero aperiodic ACF with uniformly low sidelobes [24]. It is a discrete-valued,
almost binary sequence of prime length L, given by:
1 j2π n arccos 1+1√L
s[n] = √ e [( L )]
, 0 ≤ n ≤ L − 1,
(4.12)
L
where L ≡ 1(mod 4) prime and Ln is the Legendre symbol, taking values ±1 and
defined as:

, if n(L−1)/2 = 1(mod L)
h n i  1
=
(4.13)
−1 , if n(L−1)/2 = −1(mod L)

L
1
, if n = 0(mod L)
Since the Björck sequence is defined as a discrete sequence, we cannot derive its
continuous-time equivalent. Being a discrete phase code, in radar the Björck sequence
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can be used as a spreading code or transmitted as a QPSK, or MSK modulated signal.
CAZAC sequences are CAZAC also in the frequency domain [24], which results in a
noise-like spectrum, shown in Fig. 4.13.
Figure 4.13: Example of an Björck pulse: top) Time ; bottom) Frequency
The AF and ACF of the Björck sequence are also highly localized as shown in
Fig. 4.14 and Fig. 4.15.
Figure 4.14: Autocorrelation function of a Björck pulse in dB.
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Figure 4.15: Ambiguity function of an Björck pulse in dB.
4.3
Further optimization of the waveforms
Let us consider the problem of range-only estimation, where S is as in (3.13) having
a Toeplitz structure. As noted earlier in Section 3.4, this assumption is only valid in
case the estimation grid ∆τ = 1/fs , e.g, reference estimation grid. In such case, the
autocorrelation matrix R = SH S. In the ideal case, the ACF is unity for zero delay and
all off-peak correlations are zero. Thus in the ideal case, the autocorrelation matrix
will be equal to an identity matrix R = I. Then the problem of optimization the ACF
of a sequence boils down to minimizing the Frobenius matrix norm:
min ||SH S − I||2F ,
(4.14)
Since in the range only problem, S contains shifted copies of the transmitted waveform,
finding the minimum in (4.14), would result in optimization of the ACF of the waveform. The problem of (4.14) is a quadratic form, which is difficult to handle. Several
algorithms [6] relax the criterion in (4.14) aiming to minimize the average coherence of
S. The average coherence is defined through the integrated sidelobe level (ISL) metric.
The ISL is the “average” power in the sidelobes of the ACF:
ISL =
L−1
X
|A[k]|2 ,
(4.15)
k=1
where A[k] is defined by equation (2.9). We adopt the constant amplitude pruned
(CAP) algorithm from [6], which relaxes the criterion (4.14). The CAP algorithm is
initialized with the initial sequence s[n], used to generate S. A detailed description
of the algorithm is provided in Appendix C, while below we show the ACF of the
optimized waveforms. In Fig. 4.16 we show the ACF of the optimized waveforms along
with the ACF of the initial input waveforms for the algorithm, used to generate the
sensing matrix S.
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Figure 4.16: Optimized waveforms
Optimization of the initial waveforms by means of minimizing the ISL is shown in
Fig. 4.16. Clearly a decrease in the sidelobes is notable in Fig. 4.16. The sinc function
structure of the ACF of the LFM is lost, and additionally there is a decrease in the
width of the main lobe. For the optimized cubic Alltop and Björck sequences, the sharp
response at zero delay is contained and additionally, a decrease of a couple of dBs in
the off-peak correlations is attained.
4.4
Bandwidth considerations
The effects of the RF blocks on the ACF of the optimal waveforms defined in Section 4.1
are presented in this subsection. By varying the BPF bandwidth Bf we are trying to
graphically match the initial ACF to the one of the processed waveform. We start with
Bf = 1. The cubic Alltop higher rate equivalent ŝ[m] is linearly interpolated with a
factor M = 100 as in equation (4.3). In Fig. 4.17 one can observe an increase of the
sidelobes of about 4 dB. Additionally, the main lobe is wider due to the filtered high
frequency components of the waveform as shown in Fig. 4.17. Increasing Bf to Bf = 2,
the ACF of the received cubic Alltop matches the initial one well.
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Figure 4.17: ACF of the Alltop sequence as a function of the double sided BPF bandwidth
Bf .
For the Björck sequence, the higher rate equivalent ŝ[m] is generated as in (4.2).
This type of transmission is equivalent to a BPSK type of modulation, where the phase
of the carrier frequency is changed by the phase code, which in this case is the Björck
sequence. The required Bf for good reception, without increase of the sidelobe level, is
Bf = 1, as shown in Figure 4.18 since the transmitted binary sequence can be recovered
at the receiver.
Figure 4.18: ACF of the Björck sequence as a function of the double sided BPF bandwidth
Bf .
4.5
Conclusions
In this chapter we provided an RF system model and presented three optimal waveforms
for CS radar application - LFM, Alltop and Björck. The LFM waveform is well known
and easy to generate radar signal, with a rather high peak sidelobe amplitude of the
ACF in range. Furthermore, it experiences the sinc structure in its ACF, and a knife
edge structure in the AF.
The Alltop and the Björck sequences have a highly localized ACF and AF, with no
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specific structure of the sidelobes in range and Doppler profiles. Moreover, the Alltop
sequence is a type of an under sampled quadratic chirp, and due to the specific folding its
good correlation properties are preserved through the transmission - reception process.
We summarize our findings of the coherence (ACF) and the minimum filter width
of the investigated waveforms in Table 4.1, where we show the largest correlation coefficients, corresponding to the mutual coherence µ(S) in dB’s and absolute units (in
the brackets).
Waveform
Min. BPF width Bf
µ(S)
LFM
1
-3.9 dB (0.64)
Alltop
2
-16.5 dB (0.15)
Björck
1
-18 dB (0.13)
Table 4.1: Comparison table for the parameters of the investigated waveforms, sampled at
the reference sampling rate fs = 1.
As shown in Fig. 4.5, the mainlobe of the LFM is not an impulse like, but rather
wide. Thus, the ACF A[1], is the largest off- peak correlation, equal to −3.9 dB.
The Alltop and the Björck have a impulse - like (thumbtack) ACF (see Fig. 4.11 and
Fig. 4.14) and are significantly more incoherent than the LFM chirp.
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5
Results
This chapter contains the simulation results through which we compare the performance
of the waveforms from Section 4.2. We examine the behavior of those waveforms in the
basic problem of range-only estimation, through SSR and MF-type of detection, over
several simulation scenarios, e.g., different SNR, up-sampling of the estimation grid and
compression. Furthermore, we investigate the effect of the RF system from Section 4.1
on the performance of the different waveforms, again in the conventional MF-type of
detection and SSR.
Higher than the reference resolution is allowed by up-sampling of the estimation
grid ∆τ = 1/(Qfs ), with a factor Q > 1, while keeping the same number of observations P = N + L. Compression (reduction of the number of measurements) is achieved
by applying a compression matrix Φ to the received signal y plus noise e, keeping the
reference estimation grid ∆τ = 1/fs with N range cells but processing less measurements bP/CF c, CF > 1. We direct the reader back to Section 3.5.1, where we explain
in more detail how those aspects are addressed in the range estimation problem. We
compare the resolution capabilities of the examined waveforms through SSR and the
conventional MF-type of detection.
5.1
Simulation setup
In the RF system model of Fig. 4.2, the signal ŝ[m] is generated by interpolation with a
factor M = 100. The intermediate frequency is chosen fIF = M/4. The analog signal
s(t) at the output of the ADC, which is a linearly interpolated version of ŝIF [m] as in
Eq. (4.3) with an interpolation factor K = 100. The transmitted LFM pulse (4.6) is
of length L = 100 samples. The Alltop (4.10) and Björck (4.12) pulses are of length
L = 101, because of the odd length requirement on the Alltop and the prime length
requirement on the Björck. We choose N = 400 range cells in the reference grid case,
which increase to QN when the grid is up-sampled.
The reconstruction results are averaged over 100 independent noise realizations, and
different SNR values. The total AWGN captured by the receiver in a bandwidth Bf
is Bf N0 [26], where Bf is the double-sided BPF bandwidth of the receiver. In our
simulations the SNR is defined per sample as
SNR =
E{|s[n]|2 }
1/L
= 2 ,
2
E{|e[n]| }
σ
(5.1)
where σ 2 = Bf N0 is the noise variance per sample, e.g., e[n] ∈ CN (0, σ 2 ). We fix
the noise PSD N0 , e.g., SN R = 1/L
(similar to (2.5), but defined per sample), and
N0
2
accordingly set σ = Bf N0 . In this way, we can incorporate the effect of capturing
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more noise by increasing Bf . Furthermore, the power of the strongest target in x is
normalized to 1 (0dB).
We incorporate two models of the measurements r, whether if the RF system effects are taken into account or not. In our models, grid mismatch [27] effects are not
considered, and the measurements r are modeled as follows:
• In the case where the RF system effect is not included:
r = Sx + e,
(5.2)
where the columns of S are formed by shifted copies of s[n].
• In the case where the RF system effect is included:
r = Ax + e
(5.3)
where the columns of A are formed by shifted copies of the received (processed)
s[n].
In the high-resolution tests we define an up-sampling factor Q, as in (3.14) from
Section 3.5.1, such that the estimation grid cell size is ∆τ = 1/(Qfs ). Simulations
are performed with Q = 1 (reference estimation grid), Q = 2 and Q = 4. In the RF
system model we examine the performance over two BPF widths Bf = 2, Bf = 1 and
Bf = 1.5.
In the following figures on the x-axis is the estimation grid, where the thicks are
marking the reference grid points. In case of an up-sampled grid (Q > 1), targets can
also occupy positions in between the reference estimation grid points. On the y-axis is
the power of the ground truth and the estimates in dB. The dashed black line in the
figures represents the MF threshold level, for PF A = 10−6 , calculated as in (2.15). The
MF output (dashed red line) exceeding the threshold is a hit, e.g., a target. Hits that
do not coincide with the true target locations are called false alarms.
5.2
Simulation Results
First, we show the performance of the waveforms in the reference estimation grid ∆τ =
1/fs , Q = 1, and inspect the capabilities of the WFs through the SSR and the MF, in
the case two targets, with different powers, separated by ten reference cells ∆τ occupy
the range profile. We show the performance for SNR values for the strongest target
SNR = {0, 5, 10, 15} dB. Next, we show the performance of the waveforms in a high
resolution test, where ∆τ = 1/(Qfs ), Q = 2, and Q = 4. Then, we apply compression
through a PFM Φ, with CF = 2 and CF = 4, and show the features of the proposed
waveforms. Finally, we investigate the effect of the double-sided BPF bandwidth on
the recovery for SNR= 10dB.
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5.2.1
Targets with Different Strengths
The dynamic range of the targets in a typical radar scenario can be quite large. Thus,
we examine a scenario with two differently strong targets (0 dB and −12 dB), separated
by 10 reference cells ∆τ = 1/fs . The weaker target is chosen, such that its power is
−12 dB lower than the power of the strongest target. The −12 dB level is chosen
because it is close to the sidelobe level of the LFM waveform, which is −13.7 dB, and
sidelobes of the strong target in MF-type of detection can mask weak targets. By this
test we intend to show the capability of the SSR to recover weak targets in masked by
the high sidelobe of a stronger target or buried in noise. This capability of SSR is due
to its feature of taking the contribution of the whole ACF into account, instead of only
single correlation peaks above a predefined threshold.
In Fig. 5.1, Fig. 5.2, Fig. 5.3 and Fig. 5.4 we show the results, for the case when
no RF system effects are taken into account, e.g., the measurements r are generated as
in (5.2).
As we can see from Fig. 5.1, a large portion of the MF response is above the predefined threshold. As a consequence large amount of false alarms would appear through
MF. This trend cannot be overcome with any of the waveforms, even by averaging the
MF output over several observations, due to the deterministic nature of the “noise”,
produced by the high sidelobes. However, though SSR the targets are resolved better with all three waveforms. Similar results are obtained in case of lower SNR, e.g.
SNR= 10 dB and SNR= 5 dB from Fig. 5.2, and Fig. 5.3.
We can also see that the bias of the estimates increases with SNR. The reason is
that when α increases with the noise variance [18] the weak estimates are considered
as noise.
Figure 5.1: MF and SSR with two, differently strong targets (0 db and -12 dB), separated by
10 reference cells, SNR = 15 dB for the strongest target.
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Figure 5.2: MF and SSR with two, differently strong targets (0 db and -12 dB), separated by
10 reference cells, SNR = 10 dB for the strongest target.
Figure 5.3: MF and SSR with two, differently strong targets (0 db and -12 dB), separated by
10 reference cells, SNR = 5 dB for the strongest target.
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Figure 5.4: MF and SSR with two, differently strong targets (0 db and -12 dB), separated by
10 reference cells, SNR = 0 dB for the strongest target.
For the lowest SNR= 0 dB in Fig. 5.4, both the SSR and MF fail in detection of
the weak target due to the low SNR, regardless of the waveform. However, the strong
target is still recovered with all waveforms.
5.2.2
Higher resolution by oversampling of the estimation grid
Here we show the results for an up-sampled estimation grid ∆τ = 1/(Qfs ) with Q > 1
allowing higher resolution. The procedure of generating the model matrix S, and
accordingly modeling the measurements r, in case of an up-sampled grid was presented
in detail in Section 3.5.1.
The estimation grid is up-sampled with a factor Q = 2 and Q = 4, resulting in
N Q resolution cells of size ∆τ = 1/(Qfs ). Loss of incoherence would yield false alarms
between the true target locations in the SSR. To evaluate the resolution performance
of the waveforms we choose to position the targets such that they are separated by one
grid cell ∆τ = 1/(Qfs ), i.e., the targets are separated by one reference cell, e.g., 1/(fs )
for Q = 2, and by half a reference cell, e.g., 1/(2fs ) for Q = 4. In this way the targets
do not occupy adjacent grid cells and if false alarms occur, they can be observed on a
grid point between the true target locations.
First, let us consider a two-times up-sampled estimation grid with Q = 2, e.g.,
∆τ = 1/(2fs ), and show the reconstruction results in Fig. 5.5 to Fig. 5.8.
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Figure 5.5: MF and SSR with two targets (0 dB), separated by one reference cell, SNR = 15
dB and up-sampling factor Q = 2
Figure 5.6: MF and SSR with two targets (0 dB), separated by one reference cell, SNR = 10
dB and up-sampling factor Q = 2
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Figure 5.7: MF and SSR with two targets (0 dB), separated by one reference cell, SNR = 5
dB and up-sampling factor Q = 2
Figure 5.8: MF and SSR with 2 targets (0 dB), separated by one reference cell, SNR = 0 dB
and up-sampling factor Q = 2
Regardless of the SNR, the LFM waveform causes false alarms in the SSR, although
the targets are resolved, since those false alarms are at least 20 dB lower. A reason
for those false alarms is the poor incoherence of the LFM (see Table 5.1), and the
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mainlobe width its ACF. The Alltop and Björck sequences on the other hand result
in a more incoherent measurement matrix, thus both of the closely spaced targets are
resolved through SSR. Here again in all cases the targets are blurred because the wide
MF response.
For Q = 4 the grid size is ∆τ = 1/(4fs ). We position the targets, such that they are
separated by only half a reference cell. The performance of the Alltop and the Björck
sequences only partially overwhelms the LFM for Q = 4, since the matrix coherence
becomes large as shown in Table 5.1. For SNR= 15 dB from Fig. 5.9 and SNR= 10 dB
from Fig. 5.10, false alarm of about −20 dB is notable between the true targets for the
Alltop and Björck. The same false alarm also appears for the LFM but with power of
about −10 dB.
Figure 5.9: MF and SSR with two targets (0 dB), separated by half a reference cell, SNR
= 15 dB and up-sampling factor Q = 4.
For the case of even lower SN R ≤ 10 dB as in Fig.5.11 and Fig. 5.12, the LFM is
incapable to resolve the targets, as we can see false alarms, with power comparable to
the correct estimates. The Alltop and the Björck sequences for SNR≤ 5 dB yield false
alarms with increased power (about −10 dB) but the targets still could be resolved.
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Figure 5.10: MF and SSR with two targets (0 dB), separated by half a reference cell, SNR
= 10 dB and up-sampling factor Q = 4.
Figure 5.11: MF and SSR with two targets (0 dB), separated by half a reference cell, SNR
= 5 dB and up-sampling factor Q = 4.
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Figure 5.12: MF and SSR with two targets (0 dB), separated by half a reference cell, SNR
= 0 dB and up-sampling factor Q = 4.
For Q = 4 and all SNR values we can observe false alarms between the true target
locations. This is no surprise because with Q = 4, the columns of S become very
correlated as shown in Table 5.1.
Up-sampling increases the correlation between the samples, and therefore the mutual
coherence of S deteriorates. Table 5.1 summarizes our findings on the coherence of the
model matrix µ(S) and clearly shows the trend of decreasing µ(S) due to up-sampling.
We can also observe for Q = 1 and Q = 2 a better incoherence of S, containing the
Alltop and Björck sequences, then the LFM. However, further up-sampling ends up in
comparable coherence of all three waveforms, which explains the false alarms appearing
in the simulations from Fig. 5.9 to 5.12.
Waveform
LFM
Alltop
Björck
Q=1
-3.9 dB (0.64)
-16.6dB (0.15)
-18 dB (0.13)
Q=2
-0.91 dB (0.9)
-3.18 dB (0.7)
-3.25 dB (0.68)
Q=4
-0.22 dB (0.97)
-0.55 dB (0.94)
-0.6 dB (0.93)
Q=8
-0.05 dB (0.99)
-0.13 dB (0.98)
-0.14 dB (0.98)
Table 5.1: Coherence of S in case of an up-sampled estimation grid.
5.2.3
Compression
The received signal is compressed by application of a partial Fourier (PFM) compression
matrix Φ ∈ C(bP/CF c×P ) on the received signal plus noise r = ΦSx + Φe. The reason
for choosing Φ a PFM, is because it results in an incoherent sensing matrix Θ. To
achieve an optimal performance, a new realization of Φ must be generated for each
noise realization. However, because of the structure in the Fourier matrix, from which
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the PFM was derived, a fast generation of PFMs is possible.
Fig. 5.13 to Fig. 5.16 expose the reconstruction results with CF = 2. We present
the mutual coherence µ(Θ) of the resulting measurement matrix in Table 5.2.
It should be noted that by compression, e.g., r = ΦSx + Φe, the variance σc2 of the
compressed noise ec = Φe, e.g., ec = CN (0, σc2 I) is boosted:
||E{Φe}||22 = Φσ 2 ΦH ≈ CF σ 2 ,
(5.4)
where e = CN (0, σ 2 I) . Accordingly, the SNR decreases CF times, e.g., SNRc =
SNR/CF . However to keep the simulations consistent, we do not scale the SNR, and
use the definition given by (5.1).
For high SNR≥ 10 dB from Fig.5.13 and Fig.5.14, the targets are resolved with no
false alarms appearing with all waveforms. However, SSR with an LFM and SNR=5
dB, produces a false alarm (−10 dB) between the targets.
Figure 5.13: MF and SSR with two targets (0 dB) separated by two reference cells, SNR = 15
dB per target , CF = 2.
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Figure 5.14: MF and SSR with two targets (0 dB) separated by two reference cells, SNR = 10
dB per target , CF = 2.
Figure 5.15: MF and SSR with two targets (0 dB) separated by two reference cells, SNR = 5
dB per target , CF = 2.
In the lowest SNR= 0 dB case and LFM waveform, the true targets are not resolvable, as shown from 5.16.
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Figure 5.16: MF and SSR with two targets (0 dB) separated by two reference cells, SNR = 0
dB per target, CF = 2.
The advantage of the Alltop and Björck sequences is notable for SNR ≤ 5 dB from
Fig. 5.15 and Fig. 5.16, where the LFM is unable to resolve the closely spaced targets.
It is also interesting to notice that the targets are not blurred by pulse compression
with the Alltop and Björck sequences.
Figure 5.17: MF and SSR with two targets (0 dB) separated by two reference cells, SNR = 15
dB per target, CF = 4.
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Increasing, CF = 4, from Fig. 5.17 to Fig. 5.20, the number of measurements is
reduced to P/4. Good recovery is attained with the Alltop and Björck, while the LFM
fails to resolve the targets for SNR= 0dB (see Fig. 5.20). For CF = 4 we can also
see that sharp matched filter response with the Björck and Alltop pules, which also
resolves the targets.
Figure 5.18: MF and SSR with two targets (0 dB) separated by two reference cells, SNR = 10
dB per target, CF = 4.
Figure 5.19: MF and SSR with two targets (0 dB) separated by two reference cells, SNR = 5
dB per target, CF = 4.
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Figure 5.20: MF and SSR with two targets (0 dB) separated by two reference cells, SNR = 0
dB per target, CF = 4.
By compressing with Φ a PFM, the incoherence of the sensing matrix µ(Θ) is not
so severely deteriorated as in case of up-sampling the estimation grid. Table 5.2 gives
a clear indication of the better incoherence µ(Θ) achieved with the Alltop and the
Björck waveforms, even for CF = 4 and CF = 8. Thats why even by formulating an
underdetermined system with approximately CF = 4 times less rows and columns a
sparse the true target locations are found.
Waveform
LFM
Alltop
Björck
CF = 1
-3.9 dB (0.64)
-16.6dB (0.15)
-18dB(0.13)
CF = 2
-3.82 dB (0.64)
-12.53 dB (0.23)
-14.3 dB (0.19)
CF = 4
-3.68 dB (0.64)
-9.61 dB (0.33)
-12.17 dB (0.25)
CF = 8
-3.63 dB (0.67)
-6.83 dB (0.45)
-8.25 dB (0.38)
Table 5.2: Coherence µ(Θ) of Θ = ΦS, where Φ is a partial Fourier matrix, in dB and
absolute units.
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5.2.4
Effect of the RF system on the recovery
In this section, we repeat the experiments from Sections 5.2.1, 5.2.2, 5.2.3 for the cases
SNR= 10 dB, and examine the effects of the RF system on the SSR, by varying the
double-sided BPF bandwidth Bf .
The effects of the double-sided BPF bandwidth Bf on the ACF of the waveforms
were analyzed in Section 4.4, where we showed that the ACF of the Alltop sequence
can be recovered with Bf ≥ 2, and Bf ≥ 1 for Björck. However, the conclusions drawn
in Section 4.4 about the required Bf were based only on visual comparison between the
ACF’s of the processed and non-processed waveforms. Thus it is interesting to evaluate
the performance of the MF and SSR for the cases Bf = {1, 1.5, 2}.
The noiseless received signal y is modeled as r = Ax, where the columns āk of A
contain shifted copies of the processed s[n] as in (5.3). Furthermore, as indicated in
Section 5.1, the noise contribution is dependent on the double-sided BPF bandwidth as
σ 2 = N0 Bf , which increases with increasing Bf . Accordingly, increasing Bf increases
the noise, captured by the receiver (see (5.1)). Then the matched filter threshold (2.15)
(dashed black line) is accordingly adjusted, which is the reason that in the cases of
Bf = 2 it is 3 dB higher than in the cases of Bf = 1.
Figure 5.21: MF and SSR with two, differently strong targets (0 dB and -12dB) and processed
waveforms, Bf = 1, SNR = 10 dB for the strongest target.
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Figure 5.22: MF and SSR with two, differently strong targets (0 dB and -12dB) and processed
waveforms, Bf = 2, SNR = 10 dB for the strongest target.
Figure 5.23: MF and SSR with two, differently strong targets (0 dB and -12dB) and processed
waveforms, Bf = 1.5, SNR = 10 dB for the strongest target.
For the reference estimation grid ∆τ = 1/fs and Bf = 1 in Fig. 5.21, the LFM
resolves both targets through SSR without causing false alarms. The RF system does
not have any significant effect on the performance of the LFM waveform for Bf = 1,
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because its bandwidth is very well controlled during generation in the reference grid
case. However, due to Bf = 1, the Alltop and Björck sequences result in false alarms
near the strong target, which are with power close to the one of the weak target.
Thus, those waveforms require a somewhat larger Bf . Increasing Bf to Bf = 2, lets
through the high frequency components in both the Alltop and Björck, increasing their
incoherence and resulting in a good reconstruction without any false alarms, as shown
from Fig. 5.22. In the intermediate case, Bf = 1.5, from Fig. 5.23, we also notice the
good reconstruction capabilities of the SSR, for all three waveforms.
Next, we research the effect of Bf in the high-resolution tests, where the estimation
grid is up-sampled ∆τ = 1/(Qfs ). The experiment is defined as in (3.14), again for
SNR= 10 dB and Q = 2, 4. The results are shown in Fig. 5.24 to Fig. 5.29.
Figure 5.24: MF and SSR with two, equally strong targets (0 dB) and processed waveforms,
separated by one reference cell, Bf = 1, Q = 2, SNR = 10 dB.
False alarms appear when Alltop and Björck are transmitted, as shown in Fig. 5.24,
due to the insufficient filter bandwidth. The same observation is valid also for the LFM
in this case, but the false alarms are caused as a result from the poor incoherence of
the up-sampled LFM (see Table 5.1).
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Figure 5.25: MF and SSR with two, equally strong targets (0 dB) and processed waveforms,
separated by one reference cell, Bf = 2, Q = 2, SNR = 10 dB.
Figure 5.26: MF and SSR with two, equally strong targets (0 dB) and processed waveforms,
separated by one reference cell, Bf = 1.5, Q = 2, SNR = 10 dB.
For Bf = 1 from Fig.5.30 the filter bandwidth also plays a role in the reconstructions
performance depending on the waveform, as false alarms are observed for Bf = 1 in
the Alltop and Björck waveforms. Increasing Bf to Bf = 2 (see Fig. 5.25), and even
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to the lower Bf = 1.5 (see Fig. 5.26), yields optimal performance for the Alltop and
Björck, similar to the case when no processing through the RF system is involved, as
in Section 5.2.2.
Also with Q = 4 the filter bandwidth Bf = 1 from Fig. 5.27 is not sufficiently large to
allow the high-frequency components of the Alltop and Björck sequences. Furthermore,
the LFM does not manages to reconstruct the true targets. We see many false alarms
and the wrongly estimated target locations.
Figure 5.27: MF and SSR with two, differently strong targets (0 dB and -12 dB) and processed
waveforms, Bf = 1, SNR = 10 dB for the strongest target.
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Increasing Bf to Bf = 2 in Fig. 5.28 improves the SSR performance with the Alltop
and Björck, but still the LFM fails in both the sparse recovery and pulse compression.
We also observe low power false alarms of about −20 dB, caused by the poor incoherence
due to oversampling (see Table 5.1).
Figure 5.28: MF and SSR with two, differently strong targets (0 dB and -12 dB) and processed
waveforms, Bf = 2, SNR = 10 dB for the strongest target.
Figure 5.29: MF and SSR with two, differently strong targets (0 dB and -12 dB) and processed
waveforms, Bf = 1.5, SNR = 10 dB for the strongest target.
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Simulations results for compressed, with a factor CF = 2, measurements and processed waveforms waveforms are shown in Fig. 5.30, Fig. 5.31 and Fig. 5.32. Pulse
compression with an LFM pulse does not resolve the two targets, separated by one
range cell. Next we show that the same requirement B ≥ 1 for good target recovery,
without false alarms, with the processed Alltop and Björck sequences holds in case of
compression with CF = 2, as shown from the following Fig. 5.31 to Fig. 5.32.
Figure 5.30: MF and SSR with two targets (0 dB) separated by two reference cells, SNR = 10
dB per target, CF = 2.
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Figure 5.31: MF and SSR with two targets (0 dB) separated by two reference cells, SNR = 5
dB per target, CF = 2.
Figure 5.32: MF and SSR with two targets (0 dB) separated by two reference cells, SNR = 0
dB per target, CF = 2.
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Finally, we show in Fig. 5.33 to Fig. 5.35 that by compressing with CF = 4, SSR
is still capable to resolve the targets. In this case also Bf = 1 is insufficient for good
reception of the Alltop and Björck from Fig. 5.33.
Figure 5.33: MF and SSR with two targets (0 dB) separated by two reference cells SNR = 10
dB per target, CF = 4.
Figure 5.34: MF and SSR with two targets (0 dB) separated by two reference cells, SNR = 5
dB per target, CF = 4.
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Figure 5.35: MF and SSR with two targets (0 dB) separated by two reference cells, SNR = 0
dB per target, CF = 4.
5.3
Conclusions
The resolution limit ∆τ = 1/fs can be achieved in classical MF type of detection, if
a waveform with a highly peaked ACF, e.g., Alltop or Björck, is transmitted. However, this is possible only in the case when only one target is present in the range
profile and the transmitted pulse has very narrow ACF. In the case of several differently strong targets clustered in proximity, the interference from the off-peak portions
of the ACF would mask the weak targets in the classical MF type of detection, as we
saw in Section 5.2.1. Moreover, this “noise” from the off-peak MF response is deterministic, and cannot be resolved by averaging over several pulses. We demonstrate this
in Section 5.2.1, assuming two, differently strong targets, where the weaker target has
−12 dB lower power, which is close to the sidelobe level of the transmitted waveform.
We showed that through SSR the weak target can be resolved with all investigated
waveforms.
CS radar can also allows for a higher range resolution than the classical pulsecompression radar. By defining an up-sampled estimation grid in our CS model, where
∆τ = 1/(Qfs ), Q = 2, 4 the size of the range cells can be decreased. However, upsampling increases the correlation between the columns s̄ of the model matrix S, as
indicated in Table 5.1. In an example experiment, with a twice denser grid Q = 2,
the Alltop and Björck resolve the closely spaced targets, even for low SNR. Since CS
relies on the incoherence of the model matrix, for an up-sampling factor Q = 4, the
correlations become too large as was shown in Table 5.1. Targets separated by half a
reference cell ∆τ = 1/(2fs ) are still resolved, but only with the very incoherent Alltop
and Björck sequences although false alarms start to appear between the true estimates.
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The LFM show similar behavior but only in the cases of high SNR, e.g., SN R ≥ 10
dB and fails in resolving the targets for low SNR.
Next in Section 5.2.3, we saw that due to their better incoherence (see Table 5.2), the
Alltop and Björck in case of low SNR≤ 5 dB, overwhelm the LFM, when compression
through partial Fourier compression matrix Φ is applied.
Finally we showed the effects of varying the BPF bandwidth Bf on the reconstruction. We could conclude that a BPF bandwidth Bf = 1 (see Fig. 5.21, Fig. 5.24,
Fig. 5.27, Fig. 5.30 and Fig. 5.33), appears to be filtering some of the high-frequency
components of the Alltop and Björck, deteriorating the incoherence, and accordingly
causing false alarms, adjacent to the true targets. Although in Section 4.4 we concluded
by graphical inspection that the Bf = 1 is sufficiently large for transmission-reception
without increasing the ACF coefficients of the Björck, this in not the case in SSR and
a slightly wider filter, e.g., Bf ≥ 1.5 is required. With a properly chosen double-sided
BPF bandwidth, e.g., Bf = 2, or even Bf = 1.5, the simulations from Section 5.2.4
yield similar results to the case of the non-processed signals of Sections 5.2.1 to 5.2.3.
Despite the larger noise variance σ 2 = N0 Bf , the SSR could reconstruct the range profile for SNR values larger than 0 dB, and the increased noise variance does not cause
any additional undesirable effects, e.g, resolution loss.
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Conclusions and future work
6
In this chapter we briefly summarize our findings, draw the final conclusions and suggest
some future steps for the development of CS radar system.
6.1
Conclusions
In this thesis, we studied several waveforms for implementation in a CS radar system.
Due to its specific structure in radar, the coherence of the model matrix S is directly
related to the transmitted waveform.
We investigated three theoretically favorable waveforms - the LFM, the cubic Alltop
sequence and the Björck sequence. The motivation to choose those waveforms is based
on the width of the main lobe in the ACF, low off-peak ACF coefficients, ease of
generation and reception, and constant amplitude. Furthermore, we examined the
influence of the radar transmission - reception process on the aforementioned waveforms,
and more precisely on their ACF. A simplified model of a general SD-radar system was
provided to simulate the major building blocks of an actual system.
In Chapter 3 we presented the concept of CS radar, and showed how it can be
extended to increase the range resolution by reducing the range cell size ∆τ = 1/(Qfs ),
formulating an underdetermined system. It was also shown, that keeping the same
resolution, one can reduce the amount of data to be processed by the SSR algorithm
by applying a compression matrix Φ to the received signal.
The Alltop and the Björck sequence were considered as an alternative of the classical
LFM waveform. Those two phase sequence posses the desirable properties for CS
implementation, such as highly peaked (thumbtack) ACF and flat, non structured
off-peak correlation structure. Moreover, adapting an optimization algorithm [6] we
demonstrated that the average coherence of the investigated waveforms, defined through
the ISL metric, can be improved. Furthermore, because the Alltop sequence originates
from an under-sampled cubic chirp signal, it can be viewed as a naturally compressed
waveform, in the sense that its bandwidth is higher than the discrete time sampling
rate. However, a thumbtack response at the receiver, without distortion in the ACF,
is only attainable if the double-sided bandwidth Bf of the system’s BPF is chosen
somewhat larger than the reference sampling frequency, e.g., Bf ≥ 1.5. Nevertheless,
the increase of the filter bandwidth comes to the price of decreasing the SNR, since
more noise is captured. However, in Section 5.2.4 that this increase does not affect the
reconstruction. Also, the evaluation of the effects of the BPF’s bandwidth Bf on the
ACFs only by graphical inspection (as in Section 4.4), can be misleading, as shown in
Section 5.2.4.
Our simulation results, presented in Chapter 5 showed, that the CS radar can outperform the conventional MF in resolving closely spaced targets in typical scenarios.
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CS radar is capable to reliably resolve weak targets, whose power is close to the sidelobe
level in the ACF, in case there is also a strong target in the radar scene. Although
increasing the resolution by up-sampling the estimation grid introduces higher correlation between the columns of S, it is still possible to reliably resolve closely spaced
targets. With a twice denser grid ∆τ = 1/(2fs ), we can provide successful SSR with
all waveforms. The situation changes with four times higher resolution ∆τ = 1/(4fs ),
where only the newly proposed Alltop and Björck manage to resolve adjacent targets,
and only for reasonably high input SNR≤ 10 dB.
6.2
Future work
• Extension to range - Doppler - angle estimation
Our research was only focused on the basic case of range-only estimation, considering stationary targets. Of course, this is not the case in general. A proper model
of the physical process, reflected in the model matrix S, as shown in Section 3.4,
would allow a joint range-Doppler estimation. The model can be extended even
further, to range-Doppler- angle reconstruction. In Chapter 4 we showed that the
Alltop and Björck show also very nice properties in the time-frequency plane, thus
they could also be good candidates for the extension to range-Doppler CS radar.
• Grid mismatch
In our analysis we looked at targets, which are positioned exactly in the center of
a cell in the discretized domain. This assumption is not valid in practice, because
a target can occupy every position in the continuous domain of interest. The
resulting mismatch causes energy leakage in the rest of the cells [27] and loss of
the sparsity. Thus, it is fruitful to investigate the effects of grid mismatch on the
SSR estimates.
• Refinement of the RF system
In Chapter 4 we provided a model of the digital part of the RF system, e.g., the
components which appear after the signal discretization. Although, this model
gives insight to the required double-sided BPF bandwidth, there are effects, which
were not modeled, e.g., quantization noise. It might be also interesting to include
a model of the analog part of the RF system,e.g., power amplifier and the antenna
effects, to further study the effects on the transmitted waveform, especially when
the Alltop sequence is implemented, due to the natural compression assumption.
• Further optimization of the waveform
Although we presented results for optimizing the coherence of the waveforms in
Section 4.3, their bandwidth and behavior in the RF system are still an open
issue. A consistent analysis of those optimized waveforms, similar to the one
in Chapter 4, should be provided, and their performance through MF-type of
detection and SSR evaluated.
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Comparison of the coherence
A
In this Appendix, we present and summarize the results on the coherence µ(S) of
the model matrix S, if the estimation grid is up-sampled, or a compression matrix
Φ is applied. We present the results in both dB and absolute units (in brackets).
Table 5.1 from Section 5.2.2 is equivalent to Table A.1, and also Table 5.2 is equivalent
to Table A.4.
A.1
Up-sampling of the estimation grid
Table A.1: Coherence of S with up-sampled estimation grid as
Waveform
Q=1
Q=2
Q=4
LFM
-3.9 dB (0.64) -0.91 dB (0.9) -0.22 dB (0.97)
Alltop
-16.6dB (0.15) -3.18 dB (0.7) -0.55 dB (0.94)
Björck
-18 dB (0.13) -3.25 dB (0.68) -0.6 dB (0.93)
a function of Q.
Q=8
-0.05 dB (0.99)
-0.13 dB (0.98)
-0.14 dB (0.98)
Up-sampling increases the correlation between the samples, and therefore the coherence
of S does not improve. Table A.1 summarizes our findings on the coherence of the model
matrix µ(S) and clearly shows the trend of decreasing µ(S) due to up-sampling. We
can also observe for Q = 1 and Q = 2 a better incoherence of S, containing the Alltop
and Björck sequences, then the LFM.
A.2
Compression
Table A.2: Mutual coherence µ(S) with uniformly decimated rows, as a function of CF .
Waveform
CF = 1
CF = 2
CF = 4
CF = 8
LFM
-3.9 dB (0.64)
-3.9 dB (0.64)
-3.9 dB (0.64) -2.28 dB (0.77)
Alltop
-16.6 dB (0.15) -9.77 dB (0.32) -4.95 dB (0.57) -2.1 dB (0.79)
Björck
-18 dB (0.13) -11.9 dB (0.25) -6.04 dB (0.5) -1.73 dB (0.82)
Table A.3: Mutual coherence µ(ΦS), where Φ selects random rows of S, calculated over 10
independent realizations of Φ.
Waveform
CF = 1
CF = 2
CF = 4
CF = 8
LFM
-3.9 dB (0.64) -2.9191 dB (0.71) -2.2269 dB (0.77) -1.26 dB (0.87)
Alltop
-16.6dB (0.15)
-8.17dB (0.39)
-3.22dB (0.69)
-1.82dB (0.81)
Björck
-18dB(0.13)
-8.37db (0.38)
-3.84dB (0.64)
-1.06db (0.89)
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Table A.4: Mutual coherence µ(ΦS), where Φ is a partial Fourier matrix, calculated over 10
independent realizations of Φ.
Waveform
CF = 1
CF = 2
CF = 4
CF = 8
LFM
-3.9 dB (0.64) -3.82 dB (0.64)
-3.82 dB (0.64) -4.06 dB (0.67)
Alltop
-16.6dB (0.15) -12.53 dB (0.23) -9.61 dB (0.33) -6.83 dB (0.45)
Björck
-18dB(0.13)
-14.3 dB (0.19) -12.17 dB (0.25) -8.25 dB (0.38)
By compressing with a PFM Φ, the coherence of the sensing matrix µ(Θ) is not so
severely deteriorated as in case of up-sampling the estimation grid. Table A.4 gives a
clear indication of the larger incoherence µ(Θ) achieved with the Alltop and the Björck
waveforms, even for CF = 4 and CF = 8.
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Minimum filter bandwidth
B
Here is an overview of the bandwidth and the minimum required BPF bandwidth Bf
for the up-sampled waveforms ŝ[m]. For the LFM and the Alltop sequences the spectra
is examined through all three up sampling procedures - oversampling (4.1), zero- order
hold (4.2) and linear interpolation (4.3). The Björck sequence is only examined through
up-sampling by zero order hold and linear interpolation, because of its discrete origin
(an almost binary sequence) and no continuous time equivalent available. For each
scenario we also provide a comparison plot of the minimum required Bf .
B.1
Oversampling
The oversampling process is described in Section 4.1 and it equates to sampling the
analog equivalent sa (t) of the given waveform with high rate, e.g. fs IF = M , where
M >> 1.
Figure B.1: Power spectrum of the oversampled signal s[m], generated by a LFM waveform.
Up) Full- length spectrum. Down) Zoom in of the main lobe.
The power spectrum of the oversampled LFM waveform is shown in Figure B.1.
Since the initial sequence s[n] is generated such that the Nyquist criteria is fulfilled
(fs > Bs , where Bs = 0.8), the spectrum of the oversampled version ŝ[m], remains the
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same.
For the Alltop sequence, however the sampling rate of s[n] is much lower that the
Nyquist rate for such a signal. As also shown in Figure B.1, oversampled introduces
high frequency components to the signal ŝ[m]. The discrete Alltop sequence s[n] is a
form of an under sampled quadratic chirp signal.
Figure B.2: Power spectrum of the oversampled signal ŝ[m], generated by an Alltop sequence.
B.2
Zero - order hold
Figure B.3: Power spectrum of the zero-order hold interpolated signal ŝ[m], generated by a
LFM waveform. Top) Full- length spectrum. Bottom) Zoom in of the main lobe.
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Figure B.4: Power spectrum of the oversampled signal ŝ[m], generated by an Alltop sequence.
Top) Full- length spectrum. Bottom) Zoom in of the main lobe.
Figure B.5: Power spectrum of the oversampled signal ŝ[m], generated by a Björck sequence.
Top) Full- length spectrum. Bottom) Zoom in of the main lobe.
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B.3
Linear interpolation
Figure B.6: Power spectrum of linearly interpolated signal LFM waveform, generating ŝ[m].
Top) Full- length spectrum. Down) Zoom in of the main lobe.
Figure B.7: Power spectrum of linearly interpolated signal Alltop sequence, generating ŝ[m].
Top) Full- length spectrum. Bottom) Zoom in of the main lobe.
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Figure B.8: Power spectrum of linearly interpolated signal Björck sequence, generating ŝ[m]
Top) Full- length spectrum. Bottom) Zoom in of the main lobe.
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C
Cyclic Algorithm Pruned
(CAP)
Here is a detailed description of the CAP algorithm from [6] for minimization of the
off-peak correlations of the ACF of the waveforms in Section 4.2.
Let s = [s[0], s[1], . . . , s[L − 1]]T be a discrete phase sequence,e.g., s[n] = ejφ[n] , n =
0, 1, . . . , L − 1, which ACF
L−k−1
X
s[n]s∗ [n − k] , k = −L, −L + 1, . . . , L − 1,
(C.1)
A[k] = n=0
we would like to optimize by means of minimizing the ISL metric, defined as
X
ISL =
= 1L − 1|A[k]|.
(C.2)
k
Furthermore, s is normalized, such that ||s||2 = 1 .
Stacking L shifted copies of s in a matrix S

s[0]
0
...
 ..
 .

s[0]
X =  sL−1

..
.
..

.
0
s[L − 1]







(C.3)
(2L−1)×L
the autocorrelation matrix R is given by

A[0]
A∗ [1] · · · A∗ [L − 1]
..
...

A[0]
.
 A[1]
H
S S=
.
.
.
..
..
..

A∗ [1]
A[L − 1] · · · A[1]
A[0]





(C.4)
L×L
The intention is to have an autocorrelation matrix R equivalent to an identity matrix
R = I, which boils down to minimizing the following criterion:
min||SH S − I||2F
(C.5)
Since the problem in (C.5) is a quadratic function of s, the minimization criterion
is substituted with the simpler
min||S − Q||2F ,
(C.6)
where Q ∈ C(2L−1)×L is a semi-unitary matrix, e.g., QH Q = I.
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The matrix Q is defined as Q = VUH , with U and V being the left and right
singular matrices of S, obtained through the singular value decomposition of S =
UΣVH .
Then the minimizer of s[n], n = 0, 1, . . . , L − 1, is given by:
min
L−1
X
|s − µk |,
(C.7)
k=0
where µk are the elements on the k th diagonal of Q and s is the corresponding k th
element of s[n] (also corresponding to the k th diagonal of S) [6]. The minimizer of
(C.7) is given by:
!
L−1
X
s = ejφ , φ = arg
µk
(C.8)
k=0
−1
The algorithm is initialized by a unimodal sequence {xn }N
n=0 , ||s||2 = 1 and the
S matrix, having a Toeplitz structure is generated. Next the matrix Q is computed
and the minimizer as in (C.7) is generated iteratively, where at each next iteration the
entries of S are replaced by the minimizer (C.8) and the Q matrix is re-computed based
on the new S. Since no residue is defined, the algorithm is run only over a predefined
number of iterations.
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