signal processing for multicarrier modulation in
TRANSCRIPT
SIGNAL PROCESSING FOR MULTICARRIER MODULATION IN
UNDERWATER ACOUSTIC COMMUNICATION AND PASSIVE
RADAR
Christian R. Berger, Ph.D.
University of Connecticut, 2009
This dissertation focuses on advanced signal processing techniques for mul-
ticarrier modulation in two application scenarios: underwater acoustic (UWA)
communication and passive radar.
In UWA communication, multicarrier transmission promises a substantial in-
crease in data rate, following the path of recent success of broadband wireless
radio communications. However, UWA channels are much more challenging than
their radio counterparts, due to strong multipath and significant Doppler effects.
Advanced signal processing dedicated to the UWA environment is indispensable
to realize successful multicarrier modulation in underwater environments. In
this talk, I will present a receiver design where the channel estimator exploits the
sparsity nature of the UWA channel and the demodulator can effectively suppress
the inter-carrier interference (ICI). The channel estimators include subspace al-
gorithms from the array precessing literature, namely root-MUSIC and ESPRIT,
and recent compressed sensing algorithms in form of Orthogonal Matching Pur-
suit (OMP) and Basis Pursuit (BP). Results from a recent experiment organized
Christian R. Berger––University of Connecticut, 2009
by the Office of Naval Research (ONR) will be presented for performance demon-
stration.
In passive radar, multicarrier waveforms in the form of Digital Audio Broad-
cast (DAB) are used as illuminators of opportunity to detect and locate airborne
targets. As signal reflections off the targets compose additional time-varying
multipath components, target detection and localization are feasible through ad-
vanced channel estimation algorithms that can detect path variation. In this
scenario, super-resolution subspace methods like MUSIC, or BP from the field of
compressed sensing are proposed. These advanced methods can improve clutter
suppression and target resolution in the passive radar application.
SIGNAL PROCESSING FOR MULTICARRIER MODULATION IN
UNDERWATER ACOUSTIC COMMUNICATION AND PASSIVE
RADAR
Christian R. Berger
Dipl.-Ing., Universitat Karlsruhe (TH)
A Dissertation
Submitted in Partial Fulfillment of the
Requirements for the Degree of
Doctor of Philosophy
at the
University of Connecticut
2009
Copyright by
Christian R. Berger
2009
APPROVAL PAGE
Doctor of Philosophy Dissertation
SIGNAL PROCESSING FOR MULTICARRIER MODULATION IN
UNDERWATER ACOUSTIC COMMUNICATION AND PASSIVE
RADAR
Presented by
Christian R. Berger, Dipl.-Ing.
Major Advisor
Shengli Zhou
Associate AdvisorPeter K. Willett
Associate AdvisorYaakov Bar-Shalom
University of Connecticut
2009
ii
To my parents
iii
ACKNOWLEDGEMENTS
I would like to thank my advisors, Shengli Zhou and Peter Willett.
iv
TABLE OF CONTENTS
Chapter 1: Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 2: Sparse Channel Estimation for Multicarrier Under-
water Acoustic Communication:
From Subspace Methods to Compressed Sensing 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 ZP-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.3 Receiver Processing . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Subspace Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Root-MUSIC . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 ESPRIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Compressed Sensing . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.1 Non-Linear Estimation via Compressed Sensing . . . . . . 19
2.4.2 BP and OMP Algorithms . . . . . . . . . . . . . . . . . . 21
2.5 Effect of Time Resolution on Sparse Channel Estimation . . . . . 22
2.5.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . 22
v
2.5.2 Baseband sampling . . . . . . . . . . . . . . . . . . . . . . 24
2.5.3 High Time Resolution Dictionaries λ > 1 . . . . . . . . . . 25
2.5.4 Time Resolution vs. Composite Effect . . . . . . . . . . . 26
2.6 ICI Effects in Doppler Spread Channels . . . . . . . . . . . . . . . 27
2.6.1 ICI-Ignorant Receiver . . . . . . . . . . . . . . . . . . . . . 28
2.6.1.1 Equalizer Trade-Off for Mild Doppler Spread . . . . . . . 29
2.6.1.2 Effect of Mild Doppler Spread on Channel Estimation . 30
2.6.2 ICI-Aware Receiver . . . . . . . . . . . . . . . . . . . . . . 30
2.6.2.1 Equalizer Trade-Off for Severe Doppler Spread . . . . . . 32
2.6.2.2 Channel Estimation for Severe Doppler Spread Channels 33
2.7 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7.1 ICI-Ignorant Receivers for GLINT’08 Experiment . . . . . 34
2.7.2 ICI-Ignorant Receivers for SPACE’08 Experiment . . . . . 36
2.7.2.1 S1 Data (60 m) . . . . . . . . . . . . . . . . . . . . . . . 38
2.7.2.2 S3 Data (200 m) . . . . . . . . . . . . . . . . . . . . . . 42
2.7.2.3 S5 Data (1,000 m) . . . . . . . . . . . . . . . . . . . . . 42
2.7.3 ICI-Aware Receivers for SPACE’08 Experiment . . . . . . 44
2.7.3.1 S1 Data (60 m) . . . . . . . . . . . . . . . . . . . . . . . 44
2.7.3.2 S3 Data (200 m) . . . . . . . . . . . . . . . . . . . . . . 46
2.7.3.3 S5 Data (1,000 m) . . . . . . . . . . . . . . . . . . . . . 46
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
vi
Chapter 3: Signal Processing for Passive Radar Using OFDM
Waveforms 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 Passive Radar: Motivation & Challenges . . . . . . . . . . 51
3.1.2 Current State-of-the-Art . . . . . . . . . . . . . . . . . . . 53
3.1.3 Our Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.1 Transmitted Signal . . . . . . . . . . . . . . . . . . . . . . 57
3.2.2 Target/Channel Model . . . . . . . . . . . . . . . . . . . . 59
3.2.3 Matched Filter Receiver . . . . . . . . . . . . . . . . . . . 59
3.3 Efficient Matched Filter Based on Signal Approximation . . . . . 62
3.3.1 Small Doppler Approximation . . . . . . . . . . . . . . . . 62
3.3.2 Link to Uniform Rectangular Array . . . . . . . . . . . . . 63
3.3.3 Cancellation of Dominant Signal Leakage . . . . . . . . . . 64
3.4 2D-FFT MUSIC . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4.1 Subspace Construction via Spatial Smoothing . . . . . . . 65
3.4.2 Efficient Implementation as FFT . . . . . . . . . . . . . . 68
3.4.3 Pseudo-Code of the MUSIC Implementation . . . . . . . . 70
3.5 Compressed Sensing . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5.1 Non-linear Estimation via Sparse Estimation . . . . . . . . 71
3.5.2 Orthogonal Matching Pursuit . . . . . . . . . . . . . . . . 73
3.5.3 Basis Pursuit . . . . . . . . . . . . . . . . . . . . . . . . . 74
vii
3.6 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.6.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . 75
3.6.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . 78
3.7 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.7.1 Experimental Equipment . . . . . . . . . . . . . . . . . . . 80
3.7.2 Algorithm Performance . . . . . . . . . . . . . . . . . . . . 85
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Bibliography 91
viii
LIST OF TABLES
1 Parameters of ZP-OFDM in numerical simulation and SPACE’08
experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Parameters of ZP-OFDM in GLINT’08 experiment. . . . . . . . . 36
3 List of files decoded from SPACE’08 experiment. . . . . . . . . . . 38
4 Examples of channel responses from the SPACE’08 experiment,
taken from the LS estimate. . . . . . . . . . . . . . . . . . . . . . 39
5 OFDM signal specifications of DAB according to ETSI 300 401. . 76
6 Measurement setup of ELITE 2006 experiment . . . . . . . . . . . 82
ix
LIST OF FIGURES
1 Two example channels from the GLINT’08 experiment. . . . . . . 7
2 Simulation results comparing sparse channel estimators, assuming
baseband sampling rate delay resolution. . . . . . . . . . . . . . . 25
3 Simulation results comparing sparse channel estimators; BP and
OMP use increased delay resolution. . . . . . . . . . . . . . . . . 26
4 Simulation results comparing sparse channel estimators; the simu-
lated channel model is less sparse with three times as many paths
in the same delay spread. . . . . . . . . . . . . . . . . . . . . . . 27
5 Perfect channel knowledge, but only D off-diagonals from each side
are kept in the channel matrix for data demodulation. The channel
has a mild Doppler spread, i.e, the Doppler rates of the simulated
path-based model are generated using a uniform distribution with
σv = 0.1 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 Performance comparisons for ICI-ignorant receivers with different
channel estimation methods. . . . . . . . . . . . . . . . . . . . . . 29
7 Perfect channel knowledge, but only D off-diagonals from each
side are kept in the channel matrix for data demodulation. The
simulated channel has a severe Doppler spread with σv = 0.25 m/s. 31
x
8 Performance comparisons for ICI aware receivers, where the chan-
nel mixing matrix is assumed to have D off diagonals from each
side; full CSI case uses D = 5. . . . . . . . . . . . . . . . . . . . . 32
9 Performance results from the GLINT experiment using ICI-ignorant
receivers for two data rates, recorded over three days. . . . . . . . 35
10 Setup of the considered receivers for the SPACE’08 experiment. . 37
11 Environmental data for the SPACE’08 experiment. . . . . . . . . 39
12 Performance results using ICI-ignorant receivers at receiver S1
(60 m) on two Julian dates 295-300. . . . . . . . . . . . . . . . . 40
13 Performance results using ICI-ignorant receivers at receiver S3
(200 m) on two Julian dates 295-300. . . . . . . . . . . . . . . . . 41
14 Performance results using ICI-ignorant receivers at receiver S5
(1,000 m) on two Julian dates 295-300. . . . . . . . . . . . . . . . 43
15 Performance results using ICI-aware receivers at receiver S1 (60 m)
on two Julian dates 295-300. . . . . . . . . . . . . . . . . . . . . . 45
16 Performance results using ICI-aware receivers at receiver S3 (200 m)
on two Julian dates 295-300. . . . . . . . . . . . . . . . . . . . . . 47
17 Performance results using ICI-aware receivers at receiver S5 (1,000 m)
on two Julian dates 295-300. . . . . . . . . . . . . . . . . . . . . . 48
18 Summary across all considered days and side-by-side comparison
between ICI-ignorant and ICI-aware performance. . . . . . . . . . 49
19 Setup of a passive radar, including example bi-static range ellipses. 52
xi
20 The plot shows (a) target echoes within dense clutter; (b) time-
domain channel estimates for subsequent OFDM packets; the tar-
gets can only be detected due to their non-zero range-rate, leading
to phase changes over time that add constructively or destructively
with stationary clutter. . . . . . . . . . . . . . . . . . . . . . . . . 54
21 The phase rotation due to the Doppler shift is approximated as
constant over a block duration T ′. . . . . . . . . . . . . . . . . . . 62
22 Simulation setup of one receiver and three DAB stations illumi-
nating two closing targets; the markers are at the target starting
positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
23 Simulation results using conventional FFT processing. . . . . . . . 79
24 Simulation results for MUSIC and compressed sensing. . . . . . . 81
25 Photo of the antenna used to record the experimental data. . . . . 82
26 Overview of DAB stations and receiver in ELITE 2006 experiment. 83
27 Experimental data for conventional FFT based processing (a) with-
out clutter removal; (b) with adaptive clutter removal. . . . . . . 87
28 Experimental results for high resolution methods; (a) MUSIC, (b)
Basis Pursuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
29 Enlarged view to highlight the sidelobe suppresion of the high res-
olution methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
xii
Chapter 1
Introduction
1.1 Motivation
Multicarrier waveforms or multicarrier modulation has had an unprecedented
success in communications. Using this technology, broadband communication is
available for many applications. The history of multicarrier modulation spans
from Digital Subscriber Line (DSL) replacing dial-up modems for Internet access
at home, over wireless local area networks (WLAN) in the form of IEEE 802.11
a/g, all the way to new technologies such as Digital Audio/Video Broadcast
(DAB/DVB) replacing analog radio and TV broadcasters or Worldwide Interop-
erability for Microwave Access (WiMAX) making broadband access available in
rural areas.
The success of multicarrier modulation was largely fueled by the fact that
with the larger bandwidths necessary to provide broadband communications,
1
2
any communication channel will be significantly frequency selective, distorting
the transmitted signal. This requires communication systems to employ sophis-
ticated channel estimation and equalization techniques, to undo this effect. The
computational complexity of equalization became infeasible at high data rates;
this is a bottleneck for traditional singlecarrier modulation schemes. On the con-
trary, multicarrier modulation avoids this problem by using block transmission
in conjunction with frequency domain equalization, implemented efficiently using
Fast Fourier Transforms (FFT).
1.2 Overview
In the first part of the thesis, we will focus on the application of multicarrier
modulation to underwater acoustic (UWA) communication. Our group started
employing this modulation scheme in UWA communication a few years ago; the
first success was showing the feasibility of multicarrier modulation in a shallow
water environment [1]; next we addressed synchronization [2] and introduced
state-of-the-art forward error correction coding [3]. In this thesis, we will focus
specifically on channel estimation for UWA channels, which can be characterized
as doubly (time- and frequency-) spread channels. We will exploit the fact that
UWA channels are sparse in a joint time/frequency characterization and employ
compressive sensing [4] to formulate channel estimation algorithms.
In the second part of the thesis, a quite interesting link is made to a research
area in radar signal processing. In passive radar illuminators of opportunity
3
are used to detect and track airborne targets, see e.g. [5]. This was based on
analog radio and TV transmissions in the past, but with the arrival of digital
broadcasters such as DAB/DVB, this radar signal processing problem can be
formulated as channel estimation for multicarrier waveforms as well. Both parts
share similar signal processing algorithms, as we are dealing with mutlicarrier
modulation across a channel described as a time-varying linear system.
1.3 List of Publications
During the course of the Ph.D. program the following articles have been pub-
lished or submitted for publication. The body of this thesis corresponds largely
to the work in articles J11 and J12.
Journal Papers (Appeared/To Appear)
J1. C. R. Berger, M. Guerriero, S. Zhou, and P. Willett, “PAC vs. MAC for De-
centralized Detection using Noncoherent Modulation,” IEEE Trans. Signal
Process., accepted for publication, Mar. 2009.
J2. S. F. Mason C. R. Berger, S. Zhou, and P. Willett, “Detection, Synchroniza-
tion, and Doppler Scale Estimation with Multicarrier Waveforms in Under-
water Acoustic Communications,” IEEE J. Select. Areas Commun., Vol. 26,
No. 9, pp. 1638-1649, Dec. 2008.
4
J3. D. F. Crouse, C. R. Berger, S. Zhou, and P. Willett, “Optimal Memoryless
Relays with Noncoherent Modulation,” IIEEE Trans. Signal Process., Vol.
56, No. 12, pp. 5962-5975, Dec. 2008.
J4. C. R. Berger, S. Zhou, Y. Wen, P. Willett, and K. Pattipati, “Optimizing
Joint Erasure and Error-Correction Coding for Wireless Packet Transmis-
sion,” IEEE Trans. Wireless Commun., Vol. 7, No. 11, pp. 4586-4595, Nov.
2008.
J5. C. R. Berger, S. Zhou, Z. Tian, and P. Willett, “Performance Analysis on
an MAP Fine Timing Algorithm in UWB Multiband OFDM,” IEEE Trans.
Commun., Vol. 56, No. 10, pp. 1606-1611, Oct. 2008.
J6. C. R. Berger, S. Zhou, P. Willett, and Lanbo Liu, “Stratification Effect
Compensation for Improved Underwater Acoustic Ranging,” IEEE Trans.
Signal Process., Vol. 56, No. 8, pp. 3779-3783, Aug. 2008.
J7. C. R. Berger, M. Daun, and W. Koch, “Low Complexity Track Initialization
from a Small Set of Non-Invertible Measurements,” EURASIP Journal on
Advances in Signal Processing, vol. 2008, Article ID 756414, 15 pages, 2008.
J8. C. R. Berger, M. Eisenacher, S. Zhou, and F. Jondral, “Improving the UWB
Pulseshaper Design Using Non-Constant Upper Bounds in Semidefinite Pro-
gramming,” IEEE J. Select. Topics Signal Process., Vol. 1, No. 3, pp.
396-404, Oct. 2007.
5
J9. C. R. Berger, P. Willett, S. Zhou, and P. Swaszek, “Deflection-Optimal Data
Forwarding Over a Gaussian Multiaccess Channel,” IEEE Commun. Lett.,
Vol. 11, No. 1, pp. 1-3, Jan. 2007.
Journal Papers (Submitted/In Review)
J10. C. R. Berger, S. Choi, S. Zhou, and P. Willett, “Channel Energy Based Es-
timation of Target Trajectories Using Distributed Sensors with Low Com-
munication Rate,” IEEE Trans. Signal Process., submitted for publication,
Dec. 2008.
J11. C. R. Berger, B. Demissie, J. Heckenbach, S. Zhou, and P. Willett, “Signal
Processing for Passive Radar using OFDM Waveforms,” IEEE J. Select.
Topics Signal Process., submitted to the special issue on MIMO radar, Feb.
2009.
J12. C. R. Berger, S. Zhou, J. Preisig, and P. Willett, “Sparse Channel Esti-
mation for Underwater Acoustic Communication using Multicarrier Wave-
forms: From Subspace Methods to Compressed Sensing,” IEEE Trans. Sig-
nal Process., submitted for publication, May 2009.
J13. S. Mason, C. R. Berger, S. Zhou, K. R. Ball, L. Freitag, and Peter Willett,
“Receiver Comparisons on an OFDM Design for Doppler Spread Channels,”
IEEE J. Ocean Eng., submitted for publication, Jun. 2009.
Chapter 2
Sparse Channel Estimation for Multicarrier
Underwater Acoustic Communication:
From Subspace Methods to Compressed Sensing
2.1 Introduction
Underwater acoustic (UWA) communication and networking has been under
extensive investigation in recent years [6–8]. At the physical layer, UWA channels
pose grand challenges for effective communications, featuring long delay spreads
and significant Doppler effects due to internal waves, platform and sea-surface
motion [9]. The long channel delay spread leads to significant inter-symbol-
interference (ISI) in single-carrier transmissions [10]. The receiver complexity for
channel equalization becomes one major burden when the symbol rate increases.
Multicarrier approaches like orthogonal frequency division multiplexing (OFDM)
6
7
0 5 10 15 20 250
10
20
30
40
delay [ms]
ampl
itude
0 5 10 15 20 250
5
10
15
20
25
delay [ms]
ampl
itude
Figure 1: Two example channels from the GLINT’08 experiment.
can equalize the channel at low complexity, but the aforementioned Doppler
effects destroy the orthogonality of the sub-carriers and lead to inter-carrier-
interference (ICI).
The combination of large delay spread and significant Doppler effects qualify
UWA channels as doubly (time- and frequency-) spread channels. One known
approach to this class of channels is the use of a basis expansion model (BEM)
to reflect the time-varying nature of the UWA channel, see e.g., [11–13]. Another
approach is to exploit the fact that UWA channels are naturally sparse, meaning
that most channel energy is concentrated on a few delay and/or Doppler values
[14,15]. A plot of two estimated channel impulse responses from data recorded at
the GLINT’08 experiment is shown in Fig. 1; clearly the energy is concentrated
around a few significant paths.
Sparse channel estimation has been extensively studied for frequency selective
radio channels based on, e.g., subspace fitting [16], model order fitting using a
generalized Akaike information criterion [17], zero-tap detection [18], or Monte
Carlo Markov Chain methods [19]. More recently, advances in the new field
of compressive sensing [4, 20–22] have led to numerous applications on sparse
8
channel estimation, e.g., [23–29]. Specifically on UWA channels, the matching
pursuit (MP) algorithm and its variants have been used both in [14, 30] for a
single carrier system and in [31] for a multicarrier system.
We in this work deal with sparse channel estimation for multicarrier systems.
We focus on our previously used OFDM design [1–3] using a block-by-block re-
ceiver, where each OFDM symbol is separately, coherently demodulated based
on pilot subcarriers inserted between the data. The contributions of this work
are the following:
• We suggest a path-based channel model, amenable to sparse estimation,
where the UWA channel is parameterized by a number of distinct paths,
each characterized by a triplet of delay, Doppler rate, and path attenuation.
We derive the exact ICI formulation at the output of the block-by-block
OFDM receiver after proper time-domain Doppler compensation.
• We link well known algorithms from the array processing literature to
the sparse channel estimation problem, namely Root-MUSIC and ESPRIT
[32]. These algorithms can be applied when the channel has small Doppler
spread, where the residual ICI can be ignored after proper Doppler com-
pensation.
• We use compressed sensing techniques, specifically Orthogonal Matching
Pursuit (OMP) and Basis Pursuit (BP) algorithms, to deal with channels
with larger Doppler spread. Relative to existing work based on baseband
9
delay sampling, we work on dictionaries based on finer delay and Doppler
resolutions.
• We use extensive numerical simulation and experimental data to investigate
the performance of the proposed sparse channel estimators.
The experimental data was recorded as part of the GLINT 2008 experiment
in the Mediterranean, south of the island Elba, Italy, in July 2008, and as part
of the SPACE 2008 experiment off the coast of Martha’s Vineyard, MA, from
Oct. 14 to Nov. 1, 2008. We have the following observations.
• Root-MUSIC and ESPRIT channel estimators outperform the conventional
least-squares (LS) scheme on sparse channels, but perform worse when most
energy arrives as “diffuse” multipath.
• Both OMP and BP can well handle sparse and diffuse multipath, performing
uniformly the best, with BP having a slight edge over OMP.
• On channels with mild Doppler spread, receivers that operate ICI-ignorant
can achieve sufficient performance and still take advantage of a sparse chan-
nel delay profile.
• Using compressed sensing in conjunction with an ICI-aware receiver leads to
drastic performance improvement in channels with severe Doppler spread.
The rest of this chapter is as follows. In Section 2.2 we introduce the signal
model. In Sections 2.3 and 2.4 we present the subspace and compressed sensing
10
algorithms, respectively. In Sections 2.5 and 2.6 we use numerical simulation
to investigate effects of time resolution and Doppler spread on channel estima-
tion performance. Section 2.7 contains experimental results, and we conclude in
Section 2.8.
Notation: We will use the following notations throughout this chapter: Col-
umn vectors and matrices will be denoted by lower case, x, and upper case, A,
bold face symbols respectively. AT , AH denote the transpose and the Hermitian,
the complex conjugate transpose. The Moore-Penrose pseudo inverse is denoted
as A†.
2.2 System Model
2.2.1 ZP-OFDM
We consider zero-padded (ZP) orthogonal frequency division multiplexing
(OFDM) as in [1, 33]. Let T denote the OFDM symbol duration and Tg the
guard interval for the ZP. The total OFDM block duration is T ′ = T + Tg and
the subcarrier spacing is 1/T . The kth subcarrier is at frequency
fk = fc + k/T, k = −K/2, . . . , K/2 − 1, (1)
where fc is the carrier frequency and K subcarriers are used so that the bandwidth
is B = K/T . Let s[k] denote the information symbol to be transmitted on the
kth subcarrier. The non-overlapping sets of data subcarriers SD, pilot subcarriers
11
SP, and null subcarriers SN satisfy SD∪SP∪SN = {−K/2, . . . , K/2−1}; the null
subcarriers are used to facilitate Doppler compensation at the receiver (see [1]).
The transmitted signal is given by
x(t) = 2Re
{[ ∑k∈SD∪SP
s[k]ej2π kT
tq(t)
]ej2πfct
}, t ∈ [0, T + Tg], (2)
where q(t) describes the zero-padding operation, i.e.,
q(t) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
1 t ∈ [0, T ],
0 otherwise.
(3)
2.2.2 Channel Model
The underwater acoustic (UWA) time-varying channel impulse-response is
often defined as
c(τ, t) =∑
p
Ap(t)δ (τ − τp(t)) . (4)
The time varying delays are caused by motion of the transmitter/receiver as well
as scattering off of the moving sea surface or refraction due to sound speed vari-
ations. The path amplitudes change with the delays as the attenuation is related
to the distance traveled as well as the physics of the scattering and propagation
processes.
For the duration of an OFDM symbol, the time variation of the path delays
can be reasonably approximated by a Doppler rate as,
τp(t) = τp − apt, (5)
12
and the path amplitudes are assumed constant Ap(t) ≈ Ap. Furthermore we
assume that the UWA channel can be well approximated by Np dominant discrete
paths, what we denote in the following as a “path-based” channel model. With
this, the channel model can be simplified to
c(τ, t) =
Np∑p=1
Apδ (τ − [τp − apt]) , (6)
where we specifically keep the path dependent Doppler rates ap. The received
passband signal is then
y(t) =
Np∑p=1
Apx([1 + ap] t − τp) + w(t), (7)
where w(t) is additive noise.
2.2.3 Receiver Processing
A two-step approach to mitigating the channel Doppler effect was proposed
in [1].
1. The first step is to resample y(t) in passband with a resampling factor a
that corresponds to a rough Doppler estimate, leading to z(t), c.f. (9).
2. The second step is to perform fine Doppler shift compensation on z(t) to
obtain z(t)e−j2πεt, where ε is the estimated residual mean Doppler shift.
13
The resampling can be written as the following:
z(t) =
Np∑p=1
Apx
((1 + ap
1 + a
)t − τp
)+ w (t/(1 + a)) , (8)
=
Np∑p=1
Apx((1 + bp)
(t − τ ′
p
))+ w (t/(1 + a)) . (9)
To simplify notation, we have defined the new residual Doppler rates and scaled
delays
1 + bp = 1 +
(ap − a
1 + a
)=
1 + ap
1 + a, (10)
τ ′p =
τp
1 + bp. (11)
Comparing (7) with (9), we see that the received waveform after resampling
is equivalent to one that passed through a channel with Doppler rates bp. In
channels with a single dominant Doppler, e.g., from platform motion, this can
reduce the channel to an ICI free system. In practice this operation will let us
assume that the Doppler spread is centered around zero, as a non-zero mean of
the ap is removed by the resampling. The use of scaled delays only exchanges the
order of scaling and delaying in the definition of the channel impulse-response in
(6).
Performing ZP-OFDM demodulation, the output zm on the mth subchannel
is
zm =1
T
∫ T+Tg
0
z(t)e−j2πεte−j2π mT
tdt, (12)
14
where z(t) is the baseband version of z(t). Plugging in z(t) and carrying out the
integration, we simplify zm to
zm =
Np∑p=1
Ap
1 + bpe−j2π(fm+ε)τ ′
p
∑k∈SD∪SP
�(p)m,ks[k] + vm, (13)
where vm is the additive noise and
�(p)m,k =
sin(πβ
(p)m,kT
)πβ
(p)m,kT
e−jπβ(p)m,kT , (14)
β(p)m,k = (m − k)
1
T+
ε − bpfm
1 + bp. (15)
Defining a stacked received vector z, data vector s, and noise vector v across
all subcarriers, we can write the following input-output relationship:
z = Hs + v. (16)
where the channel mixing-matrix H has entries
[H]m,k =
Np∑p=1
Ap
1 + bp
e−j2π(fm+ε)τ ′p�
(p)m,k. (17)
The channel estimation methods in this paper use a baseband formulation
where each path has a complex path gain. Specifically, the mixing matrix H is
now expressed as
H =
Np∑p=1
ξpΛpΓp, (18)
where the complex path gain for the pth path is
ξp =Ap
1 + bp
e−j2π(fc+ε)τ ′p, (19)
15
the matrix Γp has an (m, k)th entry as
[Γp]m,k = �(p)m,k, (20)
and the matrix Λp is a diagonal matrix with
[Λp]m,m = e−j2π mT
τ ′p . (21)
The formulation in (18) clearly specifies the contribution from each discrete path
with delay τ ′p and Doppler scale bp toward the channel mixing matrix that defines
the ICI pattern.
2.3 Subspace Methods
When all the paths have similar Doppler scales, proper choices of a and ε can
render H close to diagonal, which is the rationale for the receiver design in [1].
Specifically, the residual ICI is ignored, and Γp in (18) is approximated by an
identity matrix.
Let us now relate this simplified setup to the direction finding problem from
the array processing literature. Dividing the measurements, zm, by the trans-
mitted symbol on each subcarrier, s[m], (in practice, only pilot subcarriers are
considered, as will be clear later on), the estimated frequency responses can be
collected into a vector, where we ignore the noise at this moment. Collecting the
diagonal entries of H into a vector h, we obtain
h =
Np∑p=1
ξpw(τ ′p
), (22)
16
where w(τ ′p) has the mth entry
[w(τ ′
p)]m
= e−j2π mT
τ ′p. (23)
The formulation in (22) is thus equivalent to a direction finding problem in the
array processing literature; each arrival from a certain direction has a steering
vector in a similar form to w(τ ′p). Hence, subspace methods from array process-
ing can be applied to identify the distinct path arrivals. Specifically, from the
collected measurements, one needs to estimate the covariance matrix
Rh = E[hhH
]=
Np∑p=1
E[|ξp|2
]w(τ ′p
)w(τ ′p
)H. (24)
The delays {τ ′p}, in our channel estimation problem correspond to the direc-
tions of arrival in array processing, which can be identified based on eigenvalue-
decomposition of the covariance matrix Rh.
Usually, a number of OFDM symbols (let’s say I) need to be observed to
approximate the covariance matrix,
Rh ≈ 1
I
I∑i=1
hihHi . (25)
In our work, we assume a block-by-block receiver as in [1]. Hence, we need
to estimate the covariance matrix based on one OFDM symbol only. This is
possible via spatial smoothing (see e.g. [34] or [32]). In a nutshell, as long as the
elements of the steering vectors w(τ ′p
)exhibit a shift invariance property, we can
exchange the observation of a large array for multiple “independent” observations
of a smaller array, but generated by the same τ ′p.
17
Specifically, let us assume that the pilots are spaced uniformly within each
OFDM symbol, i.e., m = Δ, 2Δ, . . . and introduce partial vectors hba, wb
a, which
includes pilots a through b of the original vector:
wba
(τ ′p
)=
[e−j2π aΔ
Tτ ′p e−j2π (a+1)Δ
Tτ ′p · · · e−j2π bΔ
Tτ ′p
]T
. (26)
Therefore, we have
hb+δa+δ =
Np∑p=1
ξpwb+δa+δ
(τ ′p
)(27)
=
Np∑p=1
(ξpe
−j2πδ ΔT
τ ′p
)wb
a
(τ ′p
)(28)
which can be interpreted as a second observation of hba with new amplitudes
ξpe−j2πδ Δ
Tτ ′p . We can approximate the covariance matrix of size NC = b − a as,
RNC
h≈ 1
I
I∑i=1
hi+NCi
(hi+NC
i
)H
(29)
where I = K/Δ − NC + 1 depends on the number of available observations
(pilots). Clearly there is a trade off: a larger NC leads to better resolution of
the τ ′p, while a larger I approximates the covariance matrix better. In any case
both dimensions have to be larger than the assumed maximum number of paths,
as the rank of the covariance matrix limits the maximum number of identifiable
components.
2.3.1 Root-MUSIC
We choose the unitary implementation of Root-MUSIC, to reduce computa-
tional complexity (for details see [32]). The order selection problem is solved in
18
the following way: after matrix decomposition of the covariance matrix, we choose
all eigenvectors corresponding to eigenvalues less than twice the noise variance
to compose the noise space.
Once the {τ ′p} are estimated, the channel response on the data subcarriers
is estimated by using the LS solution to (22) based on the channel frequency
responses on the pilot subcarriers.
2.3.2 ESPRIT
As for Root-MUSIC, we choose the unitary implementation for ESPRIT, fol-
lowing the details in [35] or [32]. The signal space is determined complementary to
the noise subspace in MUSIC; we choose all eigenvectors corresponding to eigen-
values larger or equal to twice the noise variance. To improve robustness against
model mismatch (especially caused by Doppler), we solve for the unknown delay
parameters τ ′p using a total-least-squares (TLS) formulation. Then the channel
response on the data subcarriers is determined as in Sec. 2.3.1.
2.4 Compressed Sensing
Although H in (18) has K2 entries, it is defined by Np triples of (ξp, bp, τ′p).
Since UWA channels are sparse, the value of Np is small, hence, it is possible that
those Np paths can be identified by compressed sensing methods based on only
a limited number of measurements.
19
To facilitate implementation, we rewrite z as
z =
[Λ1Γ1s · · · ΛNpΓNps
]⎡⎢⎢⎢⎢⎢⎢⎣
ξ1
...
ξNp
⎤⎥⎥⎥⎥⎥⎥⎦
+ v. (30)
If the parameters(bp, τ
′p
)were available, we could construct the (K ×Np)-matrix
in (30) and solve for the ξp using the least-squares solution.
2.4.1 Non-Linear Estimation via Compressed Sensing
A brute force approach to solve (30) would be to try all possible combinations
of{(
bp, τ′p
)}Np
p=1and choose the solution with the best fit. Of course the fit always
improves as a function of Np, which is also unknown. Similar estimation problems
have been solved using compressed sensing (see tutorial in [4] and references
therein). An observed signal is defined as a linear combination of an unknown
number of structured signals, each defined by an unknown parameter(s). This
problem is solved by constructing a so-called dictionary, made of the signals
parameterized by a representative selection of possible parameters (or parameter
sets). In this model, parameter sets not part of the solution will be assigned a
zero weight coefficient. Since a large number of such sets is necessary to construct
an accurate dictionary, most weights will be zero and the problem is sparse.
20
We follow this approach and choose representative sets of (b, τ ′) as,
τ ′ ∈{
0,T
λK,
2T
λK, · · · , Tg
}, (31)
b ∈ {−bmax,−bmax + Δb, · · · , bmax} . (32)
The discretization in τ ′ is based on the assumption that after synchronization all
arriving paths fall into the guard interval, where we choose the time resolution
as a multiple, λ, of the baseband sampling time T/K, leading to Nτ = λKTg/T
tentative delays. For the residual Doppler rates, we assume that they are spread
around zero after compensation by a, and bmax can be chosen based on the as-
sumed Doppler spread, with resolution 2bmax/(Δb) + 1 = Nb. Hence, a total of
NτNb candidate paths will be searched, and we expect Np � NτNb significant
paths due to the channel sparsity.
With this, we form vectors x(i)A = [ξ
(i)1 , . . . , ξ
(i)Nτ
]T , corresponding to all delays
associated with Doppler scale bi, and form a vector x = [(x(1)A )T , . . . , (x
(Nb)A )T ]T .
The linear formulation of the problem is that
z =
[Λ1Γ1s · · · ΛNτ Nb
ΓNτ Nbs
]x + v
:= Ax + v
(33)
where A is a fat matrix with NτNb columns, and most of entries of x are assumed
to be zeros since the channel is sparse. Without the assumption that most entries
are zero, the problem would be ill defined, i.e., estimation of the parameters would
be impossible.
21
2.4.2 BP and OMP Algorithms
To solve the sparse estimation problem with the measurement model in (33),
we focus on two popular algorithms:
1. Basis Pursuit, for an efficient implementation, see e.g. [36].
2. Orthogonal Matching Pursuit, see e.g. [14, 37].
Due to lack of space, we skip the implementation details. For implementation of
these algorithms, it is important to consider that multiplying by the matrix A
can be done efficiently using FFTs.
To reduce the complexity of computing the dictionary set with a large size, we
choose to retain only D off diagonals on the templates Γp, (therefore also on H).
This means that only ICI from D directly neighboring subcarriers on each side
are considered. The symbol vector s contains known pilot symbols, and zeros,
but also unknown data symbols. The unknown data symbols are set to zero to
compute the matrix A.
Once the channel mixing matrix is constructed, a zero-forcing receiver (see
e.g. [38]) is applied for data demodulation
s = H†z ≈ s + H†v, (34)
followed by channel decoding for data recovery. Again, the banded matrix struc-
ture of H leads to reduced complexity by allowing efficient matrix inversion. The
special case of D = 0 corresponds to an ICI-ignorant receiver, where bmax in (32)
will be set to zero correspondingly.
22
2.5 Effect of Time Resolution on Sparse Channel Estimation
To investigate channels that are sparse in the time domain, we will first fo-
cus on linear time invariant channels, and will consider channels with Doppler
spread in Section 2.6. This is motivated by the fact that previous work on sparse
channel estimation has focused only on channels that are sparse in the equivalent
discrete baseband representation. Although this representation can capture the
full channel effect, corresponding to a complete basis, considering an increased
time resolution will render a more sparse channel representation, which in turn
improves channel estimation accuracy.
2.5.1 Simulation Setup
For purpose of numerical simulation, we approximate the continuous time
operations in (12) with a sampling rate being twice the bandwidth. We start
with a sparse channel with Np = 15 discrete paths, where the inter-arrival times
are distributed exponentially with mean E [τp+1 − τp] = 1 ms. Hence, the average
channel delay spread is about 15 ms. The amplitudes are Rayleigh distributed
with the average power decreasing exponentially with delay, where the difference
between the beginning and the end of the guard time of 24.6 ms is 20 dB.
The ZP-OFDM specifications in numerical simulation are deliberately chosen
to match the settings used in the SPACE’08 experiment. The carrier frequency,
bandwidth, number of subcarriers, inter carrier spacing, and symbol interval are
summarized in Table 1.
23
Table 1: Parameters of ZP-OFDM in numerical simulation and SPACE’08 ex-periment.
carrier frequency fc = 13 kHzbandwidth B = 9.77 kHznumber of subcarriers K = 1024symbol length T = 104.86 mssubcarrier spacing 1/T = 9.54 Hzguard interval Tg = 24.6 ms
The data rate, R, depends also on the modulation scheme and the number
of subcarriers used for channel estimation. We adopt the subcarrier allocation
from [3]. Out of the K = 1024 subcarriers, there are |SP| = 256 subcarriers
carrying pilot symbols, distributed on every fourth subcarrier, and |SN| = 96
zeros, half at the band edges and half inserted randomly between the data. The
remaining 672 data subcarriers are encoded using a rate 1/2 nonbinary LDPC
code (see [3] for details). With a 16-QAM constellation, the spectral efficiency α
and the data rate R are
α =T
T + Tg· 672
1024· 1
2· log2 16 = 1.1 bits/s/Hz, (35)
R = αB = 10.4 kb/s. (36)
We use block-error-rate (BLER) as our performance measure, which is the
average number of error-free OFDM blocks after LDPC decoding. We see this as
a reasonable performance criterion, since on unreliable channels such as UWA,
it can be expected that there is a mechanism in place to recover lost blocks,
e.g., automatic repeat-request (ARQ) or a higher layer block erasure code. In
this context it has been recently shown that BLER’s around 10−1 to 10−2 achieve
24
optimal overall spectral efficiency [39], when combined with a higher layer erasure
code.
2.5.2 Baseband sampling
The compressed sensing algorithms use a dictionary only in the delay dimen-
sion (i.e., bmax = 0); furthermore the delay grid is at first spaced at baseband
sampling rate:
τ ′ ∈{
0,T
K,2T
K, · · · , Tg
},
which corresponds to λ = 1. These are typical assumptions that have been
made in previous work on sparse channel estimation, see [14,23–29,31]; where a)
Doppler spread is ignored, and b) the channel is assumed sparse in the equivalent
discrete baseband representation. We designate this implementation as OMP(1)
and BP(1) to reflect the value of λ.
Simulation results are plotted in Fig. 2. Clearly all sparse channel estimation
schemes outperform the simple least-squares (LS) channel estimator (see [1] for
details), gaining about 1.5 dB. We also include a plot based on full channel state
information (CSI) as a lower bound. All sparse channel estimation methods per-
form similar well, where ESPRIT is slightly preferable for low SNR, but lagging
as SNR increases.
25
7 8 9 10 11 12 1310
−3
10−2
10−1
100
SNR per Symbol [dB]
BLE
R
LSESPRITMUSICOMP(1)BP(1)full CSI
Figure 2: Simulation results comparing sparse channel estimators, assuming base-band sampling rate delay resolution.
2.5.3 High Time Resolution Dictionaries λ > 1
We next increase the dictionary size of the compressed sensing methods, to
reflect the discrete nature of the channel in continuous time, corresponding to
our path-based channel model. We find that a λ > 1 increases performance
significantly, but the improvement saturates quickly. We plot the same simulation
with λ = 4 for OMP and λ = 2 for BP (see Fig. 3). Although the delays at
baseband sampling (λ = 1) form a complete basis to explain the channel effect,
the use of over-complete dictionaries improves performance by almost 1 dB. This
is intuitive, as the path delays are generated from a continuous time distribution,
the dictionaries with higher time resolution can explain the observations with
fewer non-zero elements.
26
7 8 9 10 11 12 1310
−3
10−2
10−1
100
SNR per Symbol [dB]
BLE
R
LSESPRITMUSICOMP(4)BP(2)full CSI
Figure 3: Simulation results comparing sparse channel estimators; BP and OMPuse increased delay resolution.
2.5.4 Time Resolution vs. Composite Effect
Based on our reasoning on time resolution, the subspace methods Root-
MUSIC and ESPRIT should outperform the compressed sensing methods, as
they inherently operate on a continuous estimation space, while the compressed
sensing methods can only approximate the continuous time operation. We spec-
ulate that the super-resolution properties of subspace methods do not work well
when several paths fall too close to be resolved, leading to a known bias in sub-
space estimators [32]. In these cases the compressed sensing methods model the
composite effect, which is ultimately the rationale behind the equivalent baseband
model. In UWA, this is often termed “diffuse” multipath.
To verify this hypothesis, we run the same simulation with a denser channel
model. We increase the number of paths to Np = 45, while keeping the total
delay spread constant, leading to closer spaced arrivals. The simulation results
27
in Fig. 4 support our hypothesis, as while all sparse estimators gain less over the
LS approach, the subspace methods suffer considerably more.
7 8 9 10 11 12 1310
−3
10−2
10−1
100
SNR per Symbol [dB]
BLE
R
LSESPRITMUSICOMP(4)BP(2)full CSI
Figure 4: Simulation results comparing sparse channel estimators; the simulatedchannel model is less sparse with three times as many paths in the same delayspread.
2.6 ICI Effects in Doppler Spread Channels
We now consider the effect of Doppler spread on the system performance.
First, we will generate data corresponding to a low degree of Doppler spread and
continue using the receiver previously used on the linear time invariant channels,
see Section 2.5 (also used in [1–3]). This reflects well the conditions in UWA
communication on days of calm sea, as there will always be a certain degree of
Doppler spreading present, even when assumed negligible. As Doppler effects
are not addressed, any ICI is treated as additional additive noise, therefore the
28
receiver operates ICI-ignorant. We will afterward proceed to more severe Doppler
spread channels, which can only be handled by directly addressing the ICI.
2.6.1 ICI-Ignorant Receiver
To simulate Doppler spread using the path-based channel model, each path
is assigned a Doppler rate drawn from a zero mean uniform distribution (we use
again Np = 15). With the velocity standard deviation σv, the maximum possible
Doppler is√
3σvfc/c (the sound speed is set to c = 1500 ms). We choose a zero-
mean Doppler distribution, because a non-zero mean could be removed through
the resampling operation.
8 9 10 11 1210
−3
10−2
10−1
100
SNR [dB]
BLE
R
D = 0D = 1D = 3D = 5D = 10
Figure 5: Perfect channel knowledge, but only D off-diagonals from each sideare kept in the channel matrix for data demodulation. The channel has a mildDoppler spread, i.e, the Doppler rates of the simulated path-based model aregenerated using a uniform distribution with σv = 0.1 m/s.
29
8 10 12 14 1610
−3
10−2
10−1
100
SNR [dB]
BLE
R
LSESPRITMUSICOMP(4)BP(2)full CSI
Figure 6: Performance comparisons for ICI-ignorant receivers with different chan-nel estimation methods.
2.6.1.1 Equalizer Trade-Off for Mild Doppler Spread
To assess the need for equalization to suppress ICI, we first assume that
the receiver has perfect knowledge of all path amplitudes, delays, and Doppler
rates. However, the channel mixing matrix H in (16) will be approximated with
a banded structure keeping D off-diagonals to each side (i.e., a total of 2D + 1
diagonals are retained). We then suppress ICI by using a zero-forcing equalizer,
see (34). This is a trade-off in the sense that by choosing a larger D we can
remove more ICI, but will have to accept higher computational complexity in the
associated matrix inversion.
Fig. 5 shows the performance for different D, where the channel has mild
Doppler spread with σv = 0.1 m/s. We observe that what corresponds to the
ICI-ignorant receiver (D = 0) works well, being about 1.5 dB away from the full
30
matrix case. Most of the ICI can be captured by a banded matrix approximation
with D = 3; for D = 10 the ICI is practically removed and the performance
matches closely the full CSI curve for Doppler free channels, see e.g. Fig. 4.
2.6.1.2 Effect of Mild Doppler Spread on Channel Estimation
In Fig. 6, we compare the ICI-ignorant receivers (D = 0). That means the
channels are estimated the same as on the Doppler free channels in the previous
sections, and no ICI is equalized. We find that all receivers can still achieve a
low BLER, but at different levels of SNR. This reflects that the level of ICI is
below the necessary SNR for the LDPC code to decode successfully. The loss in
performance is about 1.5 dB compared to the ICI-free case in Fig. 3. We posit that
the performance loss is due to the unaddressed ICI, but that channel estimation
is not significantly affected by the model mismatch of the linear time invariant
channel assumption. Between the sparse channel estimators, the compressed
sensing based algorithms still outperform the subspace algorithms, but less so
than on the Doppler free channel.
2.6.2 ICI-Aware Receiver
We now consider channels with more severe Doppler spreads. To improve the
channel estimation performance in the presence of severe ICI, we convert 96 data
subcarriers into additional pilots by assuming that 96 data symbols are known a
priori. The additional pilots are grouped in clusters between zero subcarriers and
31
6 7 8 9 10 11 12 1310
−3
10−2
10−1
100
SNR [dB]
BLE
R
D = 0D = 1D = 3D = 5D = 10
Figure 7: Perfect channel knowledge, but only D off-diagonals from each side arekept in the channel matrix for data demodulation. The simulated channel has asevere Doppler spread with σv = 0.25 m/s.
existing pilots, creating groups of five consecutive known subcarriers. Adjacent
observations are needed as to effectively estimate the Doppler rate bp of each path
by observing the ICI.
Since 96 coded symbols are assumed known while the same LDPC code
structure is used (code truncation), this leads to an equivalent coding rate of
(336− 96)/(672− 96) ≈ 0.4. With 16-QAM constellation, the spectral efficiency
and the data rate are
α =T
T + Tg· 336 − 96
1024· log2 16 = 0.76 bits/s/Hz, (37)
R = αB = 7.4 kb/s. (38)
32
2.6.2.1 Equalizer Trade-Off for Severe Doppler Spread
We first assume that the channel is known to assess the need for equalization.
The numerical simulation results are depicted in Fig. 7, where σv = 0.25 m/s.
Clearly ICI-ignorant receivers (D = 0) will have very poor performance, which
indicates the need for ICI-aware receivers. This means in turn that the ICI
needs to be estimated as part of channel estimation, so that equalization can be
performed. We also notice that in the full CSI case, once we remove sufficient
levels of ICI the performance is about 1 dB better than in Fig. 5, due to the
change in coding rate.
8 9 10 11 12 13 14 1510
−3
10−2
10−1
100
SNR [dB]
BLE
R
LSOMPBPfull CSI
D = 3
D = 5
D = 0
Figure 8: Performance comparisons for ICI aware receivers, where the channelmixing matrix is assumed to have D off diagonals from each side; full CSI caseuses D = 5.
33
2.6.2.2 Channel Estimation for Severe Doppler Spread Channels
The channels with significant Doppler spread can only be handled by the com-
pressed sensing based estimators. In addition to delay, we introduce dictionaries
that also consider fifteen different Doppler rates uniformly distributed within
[−bmax, bmax], where bmax = vmax/c = 5 · 10−4. As comparison we include the LS
and the OMP/BP algorithms that assume no Doppler as previously (D = 0), but
benefit from the increased number of pilots 1 . Simulation results are in Fig. 8.
We observe that performance significantly improves by considering ICI explicitly
through the increase of D. For channels with large Doppler spread, we notice
that the improvement of BP over OMP increases.
2.7 Experimental Results
As numerical simulation can only capture some of the effects of real UWA
communication, we next use data experimentally recorded in two different en-
vironments: i) during the GLINT’08 experiment; and ii) during the SPACE’08
experiment. We will start with the GLINT’08 experiment as it corresponds more
so to the mild Doppler spread scenario, then proceed to the SPACE’08 experi-
ment, as it included stormy days with strong wind and wave activity leading to
what we call severe Doppler spread.
1The same is not possible for the subspace algorithms, as the pilot pattern no longer has itsshift invariance property.
34
2.7.1 ICI-Ignorant Receivers for GLINT’08 Experiment
The first data we consider was recorded during the GLINT’08 experiment, in
the area around Pianosa, just south of Elba, off the coast of Italy, in July 2008.
At this point of the Mediterranean, the water depth is about 90 m, and the data
was recorded by a hydrophone array with four elements. We will focus on data
recorded on three days of the experiment, July 25 to July 27 of 2008.
Although the general OFDM structure is the same as in Section 2.5, i.e., total
number of subcarriers, split into data, pilots, and zeros, the carrier frequency,
bandwidth, and symbol duration are different, as specified in Table 2. With this
the spectral efficiency for 16-QAM is the same, but the data rate is slightly less,
due to the smaller bandwidth:
α =T
T + Tg
· 672
1024· 1
2· log2 16 = 1.1 bits/s/Hz, (39)
R = αB = 8.6 kb/s. (40)
We will additionally consider 64-QAM for increased data rate:
α =T
T + Tg· 672
1024· 1
2· log2 64 = 1.65 bits/s/Hz, (41)
R = αB = 12.96 kb/s. (42)
Two recorded channel impulse responses are plotted in Fig. 1; we notice that
the channels are extremely sparse, with about four noticeable clusters, and feature
a total delay spread of about 20 ms. The data from the three days was recorded
under the following conditions,
35
1 2 3 410
−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(4)BP(2)
(a) July 25, 16-QAM
1 2 3 410
−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(4)BP(2)
(b) July 25, 64-QAM
1 2 3 410
−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(4)BP(2)
(c) July 26, 16-QAM
1 2 3 410
−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(4)BP(2)
(d) July 26, 64-QAM
1 2 3 410
−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(4)BP(2)
(e) July 27, 16-QAM
1 2 3 410
−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(4)BP(2)
(f) July 27, 64-QAM
Figure 9: Performance results from the GLINT experiment using ICI-ignorantreceivers for two data rates, recorded over three days.
36
Table 2: Parameters of ZP-OFDM in GLINT’08 experiment.
carrier frequency fc = 25 kHzbandwidth B = 7.8125 kHznumber of subcarriers K = 1024symbol length T = 131.072 mssubcarrier spacing 1/T = 7.63 Hzguard interval Tg = 25 ms
• July 25: Recorded at a distance of 905 m, drift negligible.
• July 26: Recorded at a distance of 1,720 m, drifting at 0.7 knots (0.36 m/s).
• July 27: Recorded at a distance of 1,500 m, drifting at 0.6 knots (0.31 m/s).
For each day, we use five recorded files, for each file 15 OFDM blocks are trans-
mitted, leading to a total of 75 transmitted blocks to assess the BLER.
Inspecting the performance results in Fig. 9, we notice that almost all blocks
can be decoded correctly, for both 16-QAM and 64-QAM. Generally BP is the
best, followed by OMP; the subspace methods can be better or worse than the LS
estimator at times. The overall good performance makes differentiation difficult.
The transmitter motion seems to be well compensated by the resampling and
fine Doppler shift compensation, as it does not degrade the performance. We
conclude that the calm water surface during the experiment does not lead to
noticeably Doppler spread channels.
2.7.2 ICI-Ignorant Receivers for SPACE’08 Experiment
The SPACE’08 experiment was held off the coast of Martha’s Vineyard, MA,
from Oct. 14 to Nov. 1, 2008. The water depth was about 15 meters. We consider
37
S1 S3
transmitter
60 m
200 m
S5
receivers
1000 m
Figure 10: Setup of the considered receivers for the SPACE’08 experiment.
three receivers, labeled as S1, S3, and S5, which were 60 m, 200 m, and 1,000 m
from the transmitter, respectively (cf. Fig. 10). Each receiver array has at least
twelve hydrophones. We plot the performance combining an increasing number
of phones to increase the effective SNR and show performance differences.
We consider recorded data from six consecutive days, Julian dates 295 through
300, where most days have rather calm sea and day 300 has severe wind activ-
ity. Wind speed and wave height environmental data for the whole duration of
the SPACE’08 experiment is plotted in Fig. 11. For each day, there are twelve
recorded files consisting of twenty OFDM symbols each. Due to various equip-
ment failures or sporadic ship noise, some files turn out to be corrupted beyond
possible synchronization or decoding. On day 300, the files recorded during the
afternoon were severely distorted by the increasing weather conditions and we
focus on the files recorded during the morning. An overview of the included files
is given in Table 3.
38
Table 3: List of files decoded from SPACE’08 experiment.
Julian date 295 296 297 298 299 300S1 (60 m) 11 11 11 7 8 6S3 (200 m) 12 12 11 7 10 5S5 (1,000 m) 12 12 11 7 10 4
The OFDM parameters are identical to those in Sec. 2.6.1, given in Table 1;
hence, the achieved spectral efficiency and the data rate are in (35) and (36),
respectively.
In this subsection, we test ICI-ignorant receivers. The sample channel re-
sponses based on the LS estimators at different receiver locations are shown in
Table 4.
2.7.2.1 S1 Data (60 m)
At a short distance of only 60 m and considering the shallow water depth,
we expect rich multipath and significant Doppler variation due to the geometry.
This makes this receiver the most challenging in terms of its channel response,
but the easiest in terms of received signal strength or SNR. From Table 4, we
notice that there are three to four significant clusters of similar strength. The
total delay spread is around 10 ms.
In Fig. 12 we see the BLER performance for Julian dates 295 through 300.
As in the numerical simulation the order of compressed sensing, subspace, LS
stays the same, although MUSIC and ESPRIT switch places, and LS sometimes
outperforms the subspace methods when combining only few phones.
39
288 290 292 294 296 298 300 3020
0.05
0.1
0.15
0.2w
ind
spee
d [m
/s]
Julian date
288 290 292 294 296 298 300 3020
1
2
3
4
wav
e he
ight
[m]
Julian date
Figure 11: Environmental data for the SPACE’08 experiment.
Table 4: Examples of channel responses from the SPACE’08 experiment, takenfrom the LS estimate.
Julian date 297 Julian date 300
S1–
60 m0 5 10 15 20
0
0.02
0.04
0.06
delay [ms]
ampl
itude
0 5 10 15 200
0.02
0.04
0.06
delay [ms]
ampl
itude
S3–
200 m0 5 10 15 20
0
0.01
0.02
0.03
delay [ms]
ampl
itude
0 5 10 15 200
0.01
0.02
0.03
delay [ms]
ampl
itude
S5–
1000 m0 5 10 15 20
0
1
2
3x 10
−3
delay [ms]
ampl
itude
0 5 10 15 200
0.5
1x 10
−3
delay [ms]
ampl
itude
40
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(a) Julian date 295, S1
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(b) Julian date 296, S1
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(c) Julian date 297, S1
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(d) Julian date 298, S1
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(e) Julian date 299, S1
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(f) Julian date 300, S1
Figure 12: Performance results using ICI-ignorant receivers at receiver S1 (60 m)on two Julian dates 295-300.
41
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(a) Julian date 295, S3
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(b) Julian date 296, S3
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(c) Julian date 297, S3
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(d) Julian date 298, S3
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(e) Julian date 299, S3
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(f) Julian date 300, S3
Figure 13: Performance results using ICI-ignorant receivers at receiver S3 (200 m)on two Julian dates 295-300.
42
2.7.2.2 S3 Data (200 m)
The middle distance might be the best trade off between channel difficulty and
received SNR. The example channel responses in Table 4 seem to be more con-
tained, with a more dominating first cluster. The BLER performance in Fig. 13
is much better compared to the S1 receiver, where on calm days like Julian date
297 and 298 LS outperforms the subspace methods and is close to the compressed
sensing algorithms. Otherwise we again have the “crossing” phenomenon, where
LS performs better with few phones, but the subspace methods perform better
combining more phones. This could be a trade off between over fitting and under
fitting, corresponding to LS and subspace respectively.
2.7.2.3 S5 Data (1,000 m)
At the 1 km distance only one significant cluster can be spotted in the channel
estimates, and at the stormy day (Julian date 300) the received energy seems to
be vanishingly small, c.f. Table 4. Now the LS channel estimator is always clearly
ahead of the subspace methods and often close to the compressed sensing algo-
rithms. This concurs with our previous observations that the subspace methods
perform worse if the discrete paths cannot be well resolved. On the stormy day
the performance is generally bad, with the best being BP successfully recovering
about 80 % of the OFDM blocks. This is also reflected somewhat on Julian date
299, as the last files already experience stiffening weather conditions.
43
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(a) Julian date 295, S5
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(b) Julian date 296, S5
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(c) Julian date 297, S5
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(d) Julian date 298, S5
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(e) Julian date 299, S5
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(f) Julian date 300, S5
Figure 14: Performance results using ICI-ignorant receivers at receiver S5(1,000 m) on two Julian dates 295-300.
44
2.7.3 ICI-Aware Receivers for SPACE’08 Experiment
We saw in the previous section that the performance was the best at receiver
S3, even though it was significantly farther from the transmitter. This is a clear
indication that the ICI is the limiting factor compared to the environmental noise
and we expect significant gains by employing the ICI-aware receiver.
The OFDM parameters for the ICI-aware signal design are identical to those
in Section 2.6.2, given in Table 1; hence, the achieved spectral efficiency and the
data rate are in (37) and (38), respectively. When plotting the ICI-aware receiver
performance, we include the ICI-ignorant receivers as comparison: i) LS and ii)
OMP/BP with D = 0. The ICI-aware design will use D = 3 to keep complexity
low.
2.7.3.1 S1 Data (60 m)
From the previous results, we know that S1 is the receiver with the highest ICI
effect. Even though the SNR should be high, we observed significant error floors
on Julian dates 295, 299, and 300. In the ICI-aware results for D = 3, see Fig. 15 ,
we see that all OFDM blocks can be decoded. The ICI-ignorant receivers (D = 0
are the dashed lines) also gain due to the reduced coding rate and additional
pilots, but still experience error floors (same for LS). Comparing ICI-aware BP
and OMP, the gap becomes larger. This is related to the difficulty of estimating
a correct noise variance that is needed in OMP as a stopping criterion. Since the
observed noise includes ICI, it is unclear what level of noise should be used to stop
45
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(a) Julian date 295, S1
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(b) Julian date 296, S1
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(c) Julian date 297, S1
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(d) Julian date 298, S1
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(e) Julian date 299, S1
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(f) Julian date 300, S1
Figure 15: Performance results using ICI-aware receivers at receiver S1 (60 m)on two Julian dates 295-300.
46
the algorithm. In comparison, BP seems to be more robust to some mismatch in
this regard.
2.7.3.2 S3 Data (200 m)
At receiver S3 the gains are generally small, as already the ICI-ignorant design
provided good performance. The exception being Julian date 300, where a sizable
improvement can be observed.
2.7.3.3 S5 Data (1,000 m)
Receiver S5 experiences both days with zero BLER with just one phone (Ju-
lian date 298) and the most challenging environmental conditions (Julian dates
299 and 300). While on the first four days no significant improvement can be seen
(or is even necessary), the last two days showcase the advantage of addressing
ICI in an UWA OFDM receiver.
A summary across all days is plotted in Fig. 18.
2.8 Summary
In summary, this part of the thesis considered sparse channel estimation for
multicarrier underwater acoustic communication. Based on the novel path-based
channel model, we linked well-known subspace methods from the array-processing
literature to the channel estimation problem. Also we employed recent com-
pressed sensing methods, namely Orthogonal Matching Pursuit (OMP) and Basis
47
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(a) Julian date 295, S3
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(b) Julian date 296, S3
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(c) Julian date 297, S3
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(d) Julian date 298, S3
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(e) Julian date 299, S3
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(f) Julian date 300, S3
Figure 16: Performance results using ICI-aware receivers at receiver S3 (200 m)on two Julian dates 295-300.
48
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(a) Julian date 295, S5
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(b) Julian date 296, S5
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(c) Julian date 297, S5
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(d) Julian date 298, S5
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(e) Julian date 299, S5
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(f) Julian date 300, S5
Figure 17: Performance results using ICI-aware receivers at receiver S5 (1,000 m)on two Julian dates 295-300.
49
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(a) ICI-ignorant, S1
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(b) ICI-aware, S1
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(c) ICI-ignorant, S3
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(d) ICI-aware, S3
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSESPRITMUSICOMP(2)BP(2)
(e) ICI-ignorant, S5
2 4 6 8 10 1210
−3
10−2
10−1
100
phones
BLE
R
LSOMPBP
(f) ICI-aware, S5
Figure 18: Summary across all considered days and side-by-side comparison be-tween ICI-ignorant and ICI-aware performance.
50
Pursuit (BP). Based on the continuous time characterization of the path delays,
we suggested the use of finer delay resolution overcomplete dictionaries. We
also extended the compressed sensing receivers to handle channels with different
Doppler scales on different paths, supplying intercarrier interference (ICI) pat-
tern estimates that can be used to equalize the ICI. Using extensive numerical
simulation and experimental results, we find that in comparison to the LS re-
ceiver the subspace methods show significant performance increase on channels
that are sparse, but perform worse if most received energy comes from diffuse
multipath. The compressed sensing algorithms do not suffer this drawback, and
benefit significantly from the increased time resolution using overcomplete dic-
tionaries. When accounting for different Doppler scales on different paths, BP
and OMP can effectively handle channels with very large Doppler spread.
Chapter 3
Signal Processing for Passive Radar Using
OFDM Waveforms
3.1 Introduction
3.1.1 Passive Radar: Motivation & Challenges
In passive radar, illuminators of opportunity are used to detect and locate air-
borne targets. This is essentially the same as a bi-static radar setup, as sender and
receiver are not co-located, and time difference of arrival (TDoA) measurements
localize targets on ellipses around the sender-receiver axis, c.f. Fig. 19 and [5].
It is the differences, though, that make passive radar attractive; i) as the illu-
minators are not part of the radar system, its presence is virtually undetectable;
ii) illuminators of opportunity are often radio and TV stations, broadcasting in
the VHF/UHF frequency bands otherwise not available to radar applications.
The first point, in conjunction with the bi-static setup, makes it impossible for
51
52
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10
x−axis [km]
y−ax
is [k
m]
receiverilluminatorstargetbistatic range
Figure 19: Setup of a passive radar, including example bi-static range ellipses.
targets to know if they have been detected, while the operation in the radio/TV
VHF/UHF frequency bands needs no frequency allocation, gives frequency diver-
sity, and can help to detect low-flying targets beyond the horizon [40, 41].
Challenges connected to implementing a passive radar system are mostly due
to using broadcast signals, which are not under control, for illumination. There-
fore the transmitted signals are not known a priori, which means that a regular
matched filter based receiver cannot be implemented easily. Second, although
broadcast antennas are sectorized at times, since broadcast signals have to cover
a broad area the transmit antennas are approximately isotropic and there is no
significant transmitter gain. This can lead to constraints on the achievable range
of a passive radar, if the transmit signal does not belong to a high power regional
broadcast station. Last, since the illumination is continuous, there is no easy way
53
to separate the direct signal from reflections off targets in the time domain, as is
typically done in bi-static settings.
3.1.2 Current State-of-the-Art
First systems working with analog broadcast (TV/FM) used the direct sig-
nal as a noisy template to implement an approximate matched filter [42–46].
Newly available digital broadcast systems give passive radar receivers the unique
opportunity to perfectly reconstruct the transmitted signal after successful de-
modulation and forward error correction (FEC) coding [47–52]. A big challenge
is also to excise the direct signal. Also, the received signal has a dynamic range
of easily 100 dB between direct signal and targets, due to possibly small target
radar-cross-section (RCS) and large coverage area, which cannot be handled by
analog-to-digital converters. This makes additional analog pre-compensation of
the direct signal necessary, e.g., in the form of null-steering or directional anten-
nas, see [45].
A current state-of-the-art system has the following structure, see e.g. [48],
1. The digital broadcast signal is decoded and perfectly reconstructed based
on the direct signal.
2. Null steering attenuates the direct signal to the level of clutter, reducing
the required dynamic range to below 70 dB.
3. The signal is divided into segments.
54
delay
signalstrength
direct signals
target reflections
(a) ground truth
delay
delay
channelpacket 1
channelpacket K
detect targets usingFourier transformacross packets
(b) observations
Figure 20: The plot shows (a) target echoes within dense clutter; (b) time-domain channel estimates for subsequent OFDM packets; the targets can onlybe detected due to their non-zero range-rate, leading to phase changes over timethat add constructively or destructively with stationary clutter.
4. Matched filtering is performed efficiently in the frequency domain using the
fast Fourrier transform (FFT).
5. A second Fourier transform is executed across segments to separate low
SNR targets from the dominant direct signal and clutter based on Doppler
information.
The last three steps are illustrated in Fig. 20; the outputs of such a processing
chain are bi-static range and range-rate, locating targets on ellipses around the
55
transmitter/receiver axis; see Fig. 19. This implementation is especially applica-
ble in digital multi-carrier broadcast systems, such as digital audio/video broad-
cast (DAB/DVB), as the transmit signal is specifically designed for frequency
domain equalization.
Further challenges include target localization and tracking; as in the present
system angle-of-arrival (AoA) information is often unreliable, localization has to
be accomplished by finding the intersection of the ellipses from different transmit-
ters. This highlights another unique feature of DAB/DVB, the operation in what
is termed a “single frequency network” (SFN). This means that the same signal
is transmitted by a network of broadcast stations in the same frequency band.
For the purpose of target localization and tracking this delivers multiple “free”
measurements per target within the same operating bandwidth. This offers the
opportunity to gain diversity with respect to RCS fluctuations, but also poses an
additional association problem, as it can not be determined which target echo
originated from which transmitter. Suggested approaches include target track-
ing based on the probability hypotheses density (PHD) filter [53] and multiple
hypotheses tracker (MHT) [54].
3.1.3 Our Work
We are interested in investigating passive radar using digital multicarrier mod-
ulated signals, as in the DAB scenario considered in [48,49,51,52]. The signal is
56
modulated using orthogonal frequency division multiplex (OFDM), which is es-
pecially amenable to the FFT based approach outlined above. Our contributions
are the following:
1. We derive the exact matched filter formulation for OFDM waveforms, which
was not available before. We reveal that the practical approach, outlined
in Section 3.1.2, is equivalent to the matched filter, based on a piecewise
constant approximation of the Doppler induced phase rotation in the time
domain.
2. We investigate two signal processing schemes for passive radar: we show
a link to two-dimensional direction finding and apply MUSIC; and we for-
mulate the receiver as a sparse estimation problem to leverage the new
compressed sensing framework to detect targets.
3. In addition to simulations, we test both the MUSIC and the compressed
sensing based receivers on experimental data and point out practical im-
plementation issues.
In a detailed simulation study we find that the piecewise constant approxi-
mation decreases the receiver performance by less than 3 dB for high Doppler
targets. We also compare both receiver architectures against the current state-
of-the-art approach, where we find that while more costly in complexity, the
new algorithms offer advantages in target resolution and clutter suppression by
removing sidelobes.
57
In the experimental data, we find that the biggest challenge is in handling
the dominant clutter and direct signal, which can be easily 50 dB above the
target signal strength. Both the conventional FFT processing based approach and
the approach based on MUSIC need an additional step to remove the dominant
clutter and direct signal, before these algorithms can work successfully. While
in compressed sensing this is not necessary, we find that the lower complexity
algorithm Orthogonal Matching Pursuit (OMP) [37, 55] cannot handle direct
arrivals in the experimental data, but had to be replaced by the computationally
more expensive Basis Pursuit (BP) [20–22].
The rest of this chapter is organized as follows, in Section 3.2 we explain the
signal model and derive the matched filter receiver, in Section 3.3 we show which
approximations change the matched filter receiver into the FFT based receiver
outlined above. Then in Section 3.4 we show how to apply subspace algorithms,
in Section 3.5 we leverage compressed sensing for improved performance, in Sec-
tion 3.6 we use numerical simulation, while in Section 3.7 we take a look at
experimental data, and conclude in Section 3.8.
3.2 Signal Model
3.2.1 Transmitted Signal
The Digital Audio Broadcast (DAB) standard [56], uses orthogonal frequency
division multiplex (OFDM), which is a multicarrier modulation scheme, using N
frequencies that are orthogonal given a rectangular window of length T at the
58
receiver,
xi(t) =
N/2−1∑n=−N/2
si[n]ej2πnΔftq(t). (43)
Accordingly each block carries N data symbols si[n]; the frequencies are or-
thogonal because the frequency spacing is Δf = 1/T , whereby the transmitted
waveform is extended periodically by Tcp to maintain a cyclic convolution with
the channel, i.e.,
q(t) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
1 t ∈ [−Tcp, T ],
0 otherwise.
(44)
We define a symbol duration as T ′ = T + Tcp. The broadcast signal
x(t) =
∞∑i=−∞
xi(t − iT ′) (45)
is continuous. The data symbols si[n] vary with each block, but we assume they
can always be decoded without error for our purposes1 . Some of the data
symbols si[n] might be deactivated for various reasons (protection of bandwidth
edges, Doppler estimation, etc.) and, also, a complete Null symbol is inserted
periodically for synchronization (all si[n] are zero). The baseband signal is up-
converted to the carrier frequency,
x(t) = Re{ej2πfctx(t)
}. (46)
1This is reasonable due to the application of error correcting codes in digital broadcastsignals.
59
3.2.2 Target/Channel Model
When a waveform is emitted by a transmitter, we expect to receive a direct
arrival as well as reflections off targets that are characterized by a delay τ and
a Doppler shift fd. We adopt a narrow-band model here where a signal x(t)
of center frequency fc, will only experience a phase rotation or Doppler shift
fd = afc; time compression or dilation is assumed negligible and a is the ratio
of range-rate to speed of light. Indexing the return of the pth arrival and its
associated Doppler shift and delay, the received signal is
y(t) =∑
p
Apej2πapfctx(t − τp) + w(t), (47)
where w(t) is additive noise and Ap is the attenuation including path loss, re-
flection, and any processing gains. The delays τp and Doppler shifts apfc are
assumed to be constant during the integration time. In (47) we assume that down-
conversion has been performed at the receiver, such that y(t) = Re{ej2πfcty(t)
}.
We only refer to the baseband signals in the following.
3.2.3 Matched Filter Receiver
The standard approach is to “search” for targets using a bank of correlators
tuned to the waveform given a certain Doppler shift and delay, i.e., a matched
filter. As an example, the kth correlator will produce for every τ and a fixed
Doppler shift akfc,
zk(τ ) =
∫ Ti
0
e−j2πakfctx∗(t − τ )y(t) dt. (48)
60
Due to limitations in signal processing complexity, the delay dimension τ is usu-
ally only evaluated at discrete points, as well. As waveforms with varying pa-
rameterizations are not orthogonal, for a given target multiple non-zero correlator
outputs are generated, which can be described by the ambiguity function [57],
U(τ , a) =
∫ ∞
−∞e−j2πafctx∗(t − τ)x(t) dt. (49)
The integration time in (48) can be chosen within bounds, limited from below
by the necessary integration gain to detect targets and from above by the target
coherence time (time variability of Ap) and target motion (τp and ap).
As the transmission x(t) is divided into blocks of length T ′, see (45), each
consisting of a signal of length T and a cyclic extension of length Tcp, assuming
that the largest possible delay is smaller than the cyclic extension τmax < Tcp,
the correlator in (48) can be implemented as2 ,
zk(τ) =
Ti/T ′∑i=0
∫ iT ′+T
iT ′e−j2πakfctx∗(t − τ)y(t) dt (50)
=
Ti/T ′∑i=0
e−j2πakfciT ′z
(i)k (τ ). (51)
The integration time Ti is chosen as an integer multiple of T ′, which means
we coherently combine a certain number of OFDM blocks, and we define the
correlator output of the ith block as,
z(i)k (τ) =
∫ T
0
e−j2πakfctx∗(t + iT ′ − τ )y(t + iT ′) dt. (52)
2We point out that by not using the signal information in the cyclic extension, the SNRis reduced by T/T ′, but the processing is greatly simplified by making the output a cyclicconvolution with the channel impulse response; this is the standard approach in OFDM receiverprocessing.
61
For OFDM signals the block correlation operation in (52) can be efficiently
implemented using the fast Fourier transform (FFT). This is further simplified
since due to the cyclic prefix, the correlation operation is actually cyclic in an
interval of length T . We write this as,
z(i)k (τ ) =
∫ T
0
e−j2πakfctx∗i (t − τ)y(t + iT ′) dt (53)
=
∫ T
0
e−j2πakfct
⎛⎝ N/2−1∑
n=−N/2
s∗i [n]e−j2πnΔf(t−τ )
⎞⎠ y(t + iT ′) dt (54)
=
N/2−1∑n=−N/2
(ej2πnΔfτs∗i [n]
∫ T
0
e−j2πnΔft[e−j2πakfcty(t + iT ′)
]dt
)(55)
In words, there are four steps, corresponding to the parentheses, from inside out:
1. compensation for the phase rotation in the time domain caused by the
Doppler shift;
2. integration over t - in practice an FFT operation of the sampled signal -
giving N outputs for each subcarrier;
3. compensation of the (assumed known) data symbols s∗i [n]; and
4. inverse FFT operation across various delays.
The output will be correlation values for given delay τ and Doppler akfc for the
ith OFDM block, the outputs for all blocks have to be combined as given in (51).
62
t
phase
T 2T 3T 4T
ap fc T
linear phase
approximation
Figure 21: The phase rotation due to the Doppler shift is approximated as con-stant over a block duration T ′.
3.3 Efficient Matched Filter Based on Signal Approximation
3.3.1 Small Doppler Approximation
Often, the integration time is almost on the order of a second, this means
that a very large number of OFDM blocks are included T ′ � Ti. When the
product between T ′ and the Doppler shifts is small compared to unity, we can
approximate the phase rotation within one OFDM block as constant,
e−j2πakfct ≈ e−j2πakfc(T/2)∀t ∈ [0, T ]. (56)
Then the Doppler shift has to be estimated based on the increasing accu-
mulated phase shift between consecutive blocks (see Fig. 21), and only a single
correlator is needed, as (55) can be simplified to
z(i)k (τ ) =
N/2−1∑n=−N/2
(ej2πnΔfτs∗i [n]
∫ T
0
e−j2πnΔft[e−jπakfcT y(t + iT ′)
]dt
)(57)
= e−jπakfcT
N/2−1∑n=−N/2
ej2πnΔfτH(i)n , (58)
63
where H(i)n corresponds to the channel estimate of the nth frequency in the ith
block ignoring inter-carrier-interference (ICI), and the phase rotation out front is
constant and can usually be ignored. With this, the final matched filter output
can be written as,
|zk(τ )| =
∣∣∣∣∣∣Ti/T ′∑i=0
N/2−1∑n=−N/2
e−j2π(iakfcT ′−nΔfτ)H(i)n
∣∣∣∣∣∣ , (59)
which is a two-dimensional discrete Fourier transform (DFT) of the OFDM chan-
nel estimates that can be efficiently implemented as an FFT.
3.3.2 Link to Uniform Rectangular Array
As the operation in (59) is identical to direction finding with a uniform rect-
angular array (URA), we take a closer look at the channel estimates. Assuming
no noise and only a single target present with amplitude A0, delay τ0 and Doppler
a0fc, we calculate
H(i)n = s∗[n]
∫ T
0
e−j2πnΔfty(t + iT ′) dt (60)
= s∗[n]
∫ T
0
e−j2πnΔft(A0xi(t − τ0)e
j2πa0fc(t+iT ′))
dt (61)
= A0
N/2−1∑m=−N/2
s[m]s∗[n]e−j2πmΔfτ0
∫ T
0
e−j2π(n−m)Δftej2πa0fc(t+iT ′) dt (62)
Using the approximation in (56), all frequencies are orthogonal, hence
H(i)n ≈ A0T |s[n]|2 ej2π(ia0fcT ′−nΔfτ0), (63)
64
and the channel estimates have the same form as the receiver elements of a URA,
with equivalent element spacing of T ′ in Doppler and Δf in delay, the total array
aperture size is Ti and B = NΔf respectively.
3.3.3 Cancellation of Dominant Signal Leakage
Due to the fairly long integration time, corresponding to a large URA, the
ambiguity function will be relatively sharp. So interference from other targets
will not be an issue except for very close targets. Of concern is the clutter – since
the ambiguity function has a “sinc” like shape – the attenuation is relatively slow,
leading to significant leakage into the non-zero Doppler bins [45]. As the clutter
stems from direct and almost direct arrivals that are easily 50 dB stronger than
the targets, the leakage will affect even targets of significantly non-zero Doppler.
One approach is to evaluate the matched filter only at what corresponds to
the zeros of the sinc shape, avoiding leakage, but greatly reducing resolution.
Another approach is to remove the direct signals using adaptive signal processing
on the digital data. This can be done simply by least-squares fitting the received
data to a template assuming no time variation (nulling only zero Doppler) or
a very limited degree of change (fitting can be easily achieved using a Fourier
basis). After the signal components corresponding to these Doppler values have
been approximated, we simply subtract them out of the digitally available signal.
This leads to a blind spot of variable size (depending on the least-squares model),
but significantly limits the leakage of the dominant signal components.
65
We will see later that the combination of efficiently implemented matched
filter with adaptive signal processing works reasonably well in practice, at low
complexity. This will therefore serve as our baseline comparison in regard to
other algorithms.
3.4 2D-FFT MUSIC
3.4.1 Subspace Construction via Spatial Smoothing
As outlined in Section 3.3.2, we have a signal model that is completely equiva-
lent to the one of Np wavefronts impinging on a grid of sensors, where the steering
vectors have amplitudes Ap.
H(i)n =
Np∑p=1
Apej2π(iapfcT ′−n�fτp) (64)
The azimuth and elevation direction angles are just displaced by delay and
Doppler. In order to use subspace methods like “multiple signal classification”
(MUSIC), see e.g. [32], several snapshots of the wavefronts are required. We have
i = 1, . . . , L (L = Ti/T′) OFDM symbols, each consisting of n = 1, . . . , N channel
estimates, corresponding to our virtual URA. Since i corresponds to time, the
time variations of the multi-path amplitudes Ap could be exploited, to generate
independent snapshots at a cost of a smaller equivalent aperture. Typically the
alteration in time is not significant enough on the time scale we are considering,
therefore we will apply spatial smoothing instead (see e.g. [34] or [32]).
66
Spatial smoothing can be used to generate the necessary snapshots, where
the time variation of the amplitudes is replaced by exploiting shift invariances
between the steering vectors corresponding to certain subarrays of the full URA.
In a nutshell, when considering two subarrays of a certain shift, they will only
vary in a phase shift, but which is different for each signal component, allowing us
to construct a full-rank set of observation vectors, again at the cost of a smaller
equivalent aperture.
To define a steering vector, we denote the response of one array element to a
wave of (τ , a) as
bn,i(τ , a) = ej2π(iafcT ′−n�fτ) (65)
and define a subarray matrix, indexed by n′ and i′, of reduced dimension N ′×L′:
Bn′,i′(τ , a) =
⎡⎢⎢⎢⎢⎢⎢⎣
bn′,i′ . . . bn′,i′+L′−1
......
bn′+N ′−1,i′ . . . bn′+N ′−1,i′+L′−1
⎤⎥⎥⎥⎥⎥⎥⎦
. (66)
The total number of subarrays we generate this way, must be larger than the
number of multipath components, and is given by
Nsub = (N − N ′ + 1)(L − L′ + 1) > Np. (67)
Next, we use the vec{}-operation, which takes a matrix column-wise in order to
construct a vector from it, and define a steering vector
bn′,i′(τ , a) = vec {Bn′,i′(τ , a)} . (68)
67
We have the shift invariances
bn′+1,i′(τp, ap) = e−j2π�fτpbn′,i′(τp, ap) (69)
bn′,i′+1(τp, ap) = ej2πapfcT ′bn′,i′(τp, ap) , (70)
and, consequently, all vectors bn′,i′(τp, ap) are linearly dependent.
In the same way, we can group the channel estimates H(i)n in subarrays and
stack the columns of each matrix on top of each other. We can write the signal
model for these vectors as
hn′,i′ =
Np∑p=1
Apbn′,i′(τp, ap) . (71)
Due to the above shift-invariances we find
hn′+m,i′+l =
Np∑p=1
Apej2π(lapfcT ′−m�fτp)bn′,i′(τp, ap) , (72)
i.e. we have a new ’snapshot’, where the signals
Ap = Apej2π(lapfcT ′−m�fτp) (73)
passed the same subarray.
From these snapshots we can build an (N ′ · L′) × Nsub observation matrix,
A =
[h1,1 · · · hN−N ′+1,1 · · · hN−N ′+1,L−L′+1
]. (74)
The signal and noise subspace, U(s) and U(n), can be obtained via a SVD,
A = UDV, (75)
68
with U and V being size (N ′ · L′) and Nsub unitary matrices respectively, and
U = [U(s) U(n)] is split such that the vectors in U(s) correspond to the Np largest
singular values. The MUSIC cost-function can be given either in terms of the
noise subspace or signal subspace respectively,
fMUSIC(τ , a) =(∣∣bH(τ , a)U(n)
∣∣2)−1
(76)
=(N ′L′ − ∣∣bH(τ , a)U(s)
∣∣2)−1
, (77)
where we have dropped the indices of the subarray steering vector b, as due to
the shift invariance any one of the subarrays could be chosen.
3.4.2 Efficient Implementation as FFT
As a first step, there is an advantage of expressing the MUSIC cost function
in terms of the signal subspace, due to the fact that the dimension N ′L′ ≈
105, whereas the number of signal eigenvectors Np is only on the order of a few
hundred, i.e., under these conditions it is much “cheaper” to work with the signal
subspace which is obtained via a “short” SVD of the subarrays.
We further change the evaluation of the MUSIC cost-function to use the FFT.
Let the Np-dimensional signal subspace be partitioned into
U(s) =[u1 · · ·uNp
], (78)
then we have
fMUSIC(τ , a) =
(N ′L′ −
Np∑p=1
∣∣bH(τ , a)up
∣∣2)−1
. (79)
69
As in the case of the matched filter, we will not practically be able to evaluate
the cost-function for any combination of τ and a, but instead consider possible
discrete values, commonly arranged in a grid fashion. We next consider the
individual terms in the sum in (79) and show that when evaluating them for τ
and a values on a grid, the computation can be put into the format of a two-
dimensional Fast Fourier Transform that allows efficient evaluation.
In the following we make use of a formula involving the Kronecker product
and the vec-operation, which was derived by Magnus and Neudecker [58]:
vec{ABC} = (CT ⊗ A)vec{B} . (80)
Defining the matrix Up by up = vec{Up} and using the fact that the steering
vector of the rectangular array can be written as the Kronecker product of two
steering vectors with elements,
b(N ′)(τ) =
[e−j2π�fτ · · · e−j2πN ′�fτ
]T
(81)
b(L′)(a) =
[ej2πafcT ′ · · · ej2πL′afcT ′
]T
, (82)
we find that
bH(τ , a)up = (b(L′)(a) ⊗ b(N ′)(τ ))Hup (83)
= (b(L′)(a) ⊗ b(N ′)(τ ))Hvec{Up} (84)
= (b(L′)H
(a) ⊗ b(N ′)H
(τ))vec{Up} (85)
= vec{b(N ′)H
(τ)Upb(L′)∗(a)} (86)
= b(N ′)H
(τ )Upb(L′)∗(a) . (87)
70
When inspecting the definition of b(N ′)(τ ) and b(L′)(a), we see that they are
columns of a DFT matrix if we define τ = 0, T/N ′, . . . , (N ′ − 1)T/N ′ and a =
0, 1/(fcT′L′), . . . , (L′−1)/(fcT
′L′). Accordingly, the MUSIC cost-function can be
evaluated using Np two-dimensional FFTs, which are summed up in magnitude.
The FFTs can be performed with additional zero-padding to evaluate a denser
grid of tentative values of τ and a.
3.4.3 Pseudo-Code of the MUSIC Implementation
Define:
hm = vec(Hm)
1. Remove the direct-blast with Doppler high-pass filtering
2. Project on the space orthogonal to the stationary components: Perform an
eigen-decomposition of the matrix of all channel estimates
R = [h1, ...,hM ]
RHR = UΣUH
Ur = RU(:, 1 : Mr)Σ−1/2
hm =(I −UrU
Hr
)hm
3. 2D-FFT-MUSIC
Xm = reshape(hm, N, L
)
71
Select K shifted sub-arrays X(k)s and ’vectorize’ them
x(k) = vec(X(k)s )
Perform an eigen-decomposition of the matrix of all sub-array-vectors
Rs = [x(1), ...,x(K)]
RHs Rs = UsΣsU
Hs
Bs = Us(:, 1 : Nev) · Σ−1/2s
F =Nev∑n=1
FFT2(reshape(Rs · Bs(:, n), N ′, L′))2
3.5 Compressed Sensing
3.5.1 Non-linear Estimation via Sparse Estimation
Similar to the subspace approach, we use the OFDM channel estimates as
measurements. This has no loss in information, as the bandlimited signal can
be exactly represented by a sufficient number of frequency estimates. Using the
same simplifications from (63) and the definition of steering vectors in (66) and
(68), we write the measurement model as in (71),
h =
Np∑p=1
Apb(τp, ap) + w, (88)
but where we drop the indices connected to subarrays and include noise explic-
itly. We can reasonably assume that the noise is still circular-symmetric complex
72
Gaussian of Power N0 (the information symbols si[n] are unit amplitude). Defin-
ing the following notation,
BNp =
[b(τ1, a1) . . . b(τNp , aNp)
](89)
aNp =
[A1 . . . ANp
]T
(90)
we can rewrite the model as
h = BNpaNp + w. (91)
We see that if we knew the Np pairs of (τp, ap), we could construct BNp and solve
for aNp via a simple least-squares solution.
aNp = arg minaNp
∣∣h −BNpaNp
∣∣2 (92)
Of course if we had a set of (τp, ap) we already knew where the targets were
and wouldn’t need the amplitude values Ap, but estimating the amplitudes lets
us confirm targets, e.g., if we had a larger list of possible targets constituting
a larger matrix B. This is essentially what the matched filter does, we look at
the “energy” at potential target locations, whereby the least-squares solutions is
replaced by the correlation operation as in,
aMF = BHh, (93)
since we generally have more tentative target tuples (τp, ap) as measurements,
making the matrix B “fat”. One special case is when we choose just as many
tentative tuples (τp, ap) as there are observations on an evenly spaced grid. As
73
pointed out in the previous section on MUSIC, the columns of B are then the coef-
ficients of a two-dimensional Fourier transform and therefore orthogonal, making
the least-squares and correlation solutions equal.
A different approach to solve (92) without explicit knowledge of the (τp, ap)
is using sparse estimation. In a nutshell, we solve the least-squares problem by
additionally enforcing that the solution should be based on the assumption that
there are only few targets, i.e., the solution aCS should be sparse (few non-zero
entries).
3.5.2 Orthogonal Matching Pursuit
In our earlier work [51], we employed a low complexity algorithm, Orthogonal
Matching Pursuit (OMP) [37,55]. This greedy algorithm uses the matched filter
outputs to detect the strongest target and associated (τp, ap), solves (92) and
subtracts the influence of this target from all correlator outputs, similar to serial
interference cancellation. This is repeated until “enough” targets have been iden-
tified, usually determined based on all adjusted correlator outputs being lower
than a threshold.
Although good results on simulation data were achieved, [51], OMP proved
to have problems on experimental data. This seems to have been due to two
reasons: i) there are a large amount of clutter and direct signals, leading to
high complexity since the algorithm’s run time scales with the number of targets
(clutter count as stationary targets); ii) more importantly, there is always some
74
modeling inaccuracy, e.g., due to only considering a grid of possible (τp, ap). The
modeling inaccuracy is usually a minor concern, but since the direct arrivals
are more than 50 dB stronger than the targets, when the correlator outputs are
adjusted, these inaccuracies lead to residuals on the same order or larger than the
targets. Furthermore these residuals do not decrease in value quickly with each
iteration of the algorithm as they do not match the vectors in B well. Removing
the clutter as in the conventional FFT based processing did not lead to significant
improvement either, which lead us to employ Basis Pursuit instead.
3.5.3 Basis Pursuit
Instead of trying to contruct the matrix BNp by identifying targets iteratively,
Basis Pursuit (BP) uses the so-called l1-norm regularization term [20–22],
|x|1 =∑
i
|xi|. (94)
With this the problem is formulated as,
minimize |h− Ba|2 + λ|a|1, (95)
where λ determines the “sparsity” of the solution and a can have significantly
higher dimension than h without detrimental effect on the solution.
The problem formulation in (95) is a convex optimization problem. Various
efficient implementations have been suggested in the literature [36, 59]. Since
baseband data is generally complex valued, the definition in (94) becomes,
|x|1 =∑
i
√|Re{xi}|2 + |Im{xi}|2. (96)
75
We implemented the algorithm outlined in the appendix of [36] as an extension to
the real case. It is based on an interior point method using approximate Newton
search directions.
Both the OMP and BP can be implemented more efficiently by noticing that
the multiplication with B can be implemented using FFT operations as long as
the τ and a are chosen on an evenly spaced grid (see Sect. 3.4.2). This leads to
an almost linear complexity in the number of observations (N · L) and number
of tentative target parameters (τp, ap), whereby due to zero-padding in the FFT
operation the larger number dominates (number of tentative target parameters).
3.6 Numerical Simulation
3.6.1 Simulation Setup
The signal is simulated as,
y(t) =∑
p
Apx (t − τp(t)) + w(t) (97)
where τp(t) is the exact bi-static delay. The definition of bi-static delay for a
signal transmitted from xs, received at xr, and reflected off a target at x(t) is:
τ(t) =1
c(|x(t) − xr| + |xs − x(t)| − |xr − xs|) . (98)
For simulation purposes we generate receiver data at a sampling rate of 2.048
MHz, the bi-static delays are updated at the same rate. The target is simulated
as a point target, but the auto-correlation of Ap(t) over time is modeled based
76
Table 5: OFDM signal specifications of DAB according to ETSI 300 401.
carrier frequency fc 227.36 MHzsubcarrier spacing Δf 1 kHzno. subcarriers N 1537bandwidth B 1.537 MHzsymbols length T 1 mscyclic prefix Tcp 0.246 msblock length T ′ 1.246 msblocks per frame L 76Null symbol TNULL 1.297 msframe duration TF 96 ms
on a five-point extended target assumption, similar as in [60]. The target size
is assumed at a diameter of about 30 m (only for the auto-correlation of Ap(t)).
The RCS with respect to different transmitters is assumed to be independent, as
these are several kilometers apart.
The DAB signal is specified in [56], for convenience we reproduce most pa-
rameters in Tab. 5, notation matching ours in Section 3.2. We see that due
to the bandwidth of B = 1.537 MHz, the spatial resolution is approximately
c/B ≈ 195 m (the speed of light is c = 3 × 108 m/s). Therefore for assumed
targets of 30 m diameter, the point target model seems reasonable. This could
be quite different when using a DVB signal of larger bandwidth.
One of the main assumptions to test in the simulation is the small Doppler
approximation. Accordingly we simulate targets at a relatively high speed that
will lead to significant Doppler shifts. The scenario is shown in Fig. 22, where
three radio towers illuminate two targets, the receiver is placed at the origin. The
targets are moving at constant velocity of about 180 m/s, approaching each other
77
slowly with simulation time, c.f. Fig. 22. We simulate 200 DAB frames, for a
total time of 18.2 s, in which the targets cover about 3.5 km.
The SNR of the direct signal is about 20 dB, at which any regular DAB
receiver would operate virtually error free, this makes our assumption of perfect
signal reconstruction well justified. We fix each target at -30 dB (per sample),
leading to a difference of 50 dB between direct arrival and target signatures. We
do not specifically consider any transmitter or receiver gain, attenuation based
on distance traveled or signal frequency, as we directly generate digital samples
at the output. Knowing that the targets follow a Swerling I model (due to the
extended target model), we will need about 20 dB SNR to detect the targets
reliably. We therefore coherently combine one OFDM frame, which leads to an
integration gain of TF · B ≈ 50 dB, but doesn’t affect the ratio between targets
and direct blast (see [45] for a detailed discussion of integration gain calculation).
−15 −10 −5 0 5 10 15−15
−10
−5
0
5
10
15
20
x−axis [km]
y−ax
is [k
m]
receiverilluminatorstarget 1target 2
Figure 22: Simulation setup of one receiver and three DAB stations illuminatingtwo closing targets; the markers are at the target starting positions.
78
To see the performance of the super-resolution methods, we use two targets
which move on trajectories bringing them closer during simulation time. This
will let us evaluate the target resolution.
3.6.2 Simulation Results
We first take a look at the results using conventional FFT processing, see
Fig. 23, the figure shows the superposition of the algorithm outputs over all
frames. After subtracting the direct signal, the results look fairly “clean”. Some
speckle indicates the noise floor, making the subtracted region show up clearly.
We notice that even the target signatures appearing at high range-rate are easily
detected: a range-rate of r = 400 m/s corresponds to a phase rotation of about
r/c · fcT · 2π ≈ 0.6π. In further simulation studies we found that even for close
to a half phase rotation during one OFDM symbol, the target loses only about
3 dB.
In the zoomed view of Fig. 23(b), we also notice that the signal amplitude
fades due to our extended target model. Using conventional FFT processing
the targets are not resolved, due to the large sidelobes. On the contrary, the
super-resolution methods both fully remove the sidelobes, see Fig. 24. While
MUSIC needs to use the same direct signal subtraction as the conventional FFT
processing, our compressed sensing implementation via Basis Pursuit can handle
the direct blast within its framework.
79
range−rate [m/s]
rang
e [k
m]
−500 0 500
10
20
30
40
50
60
70
(a) Conventional FFT Processing
range−rate [m/s]
rang
e [k
m]
−350 −300 −250 −200 −150 −100 −50 015
20
25
30
35
(b) Zoom
Figure 23: Simulation results using conventional FFT processing.
80
The run times are as follows (all on a regular desktop PC using MATLAB),
the beamforming algorithm needs about 0.2 s per frame of data, the MUSIC
approach is in the tens of seconds, while Basis Pursuit is in the hundreds of
seconds. Another comment on the compressed sensing algorithms is that OMP
runs on the same order as MUSIC (higher tens of seconds) for this simulation
data, with identical results, but did not work at all on the experimental data.
3.7 Experimental Data
3.7.1 Experimental Equipment
The experimental data was acquired during a measurement campaign con-
ducted by the German Research Establishment for Applied Science (FGAN).
The Research Institute for High Frequency Physics and Radar Techniques (FHR)
build CORA (Covert Radar), a passive radar receiver, for the purpose of tech-
nology demonstration [48]. In CORA, a circular antenna array with elements for
the VHF- (150-350 MHz) and the UHF-range (400-700 MHz) is used to exploit
alternatively DAB or DVB signals for target illumination. A fiber optic link
connects the elevated antenna and RF-front-end with the processing back-end,
consisting of a cluster of high power 64-bit processors. Thus, CORA is also a
demonstration of the so called “software-defined-radar” principle. Fig. 25 shows
the antenna and front-end of the CORA system during installation at the military
electronic warfare exercise ELITE 2006.
81
range−rate [m/s]
rang
e [k
m]
−350 −300 −250 −200 −150 −100 −50 015
20
25
30
35
(a) MUSIC
range−rate [m/s]
rang
e [k
m]
−350 −300 −250 −200 −150 −100 −50 015
20
25
30
35
(b) Basis Pursuit
Figure 24: Simulation results for MUSIC and compressed sensing.
82
Figure 25: Photo of the antenna used to record the experimental data.
Table 6: Measurement setup of ELITE 2006 experiment
lat. long. alt. baselineRx 48.14◦ 9.06◦ 9,200 m N/A
Tx 0 47.67◦ 9.18◦ 471 m 52.9 kmTx 7 47.51◦ 9.24◦ 564 m 71.7 kmTx 12 47.37◦ 8.94◦ 1164 m 86.0 kmTx 39 49.54◦ 8.80◦ 737 m 157 km
A circular array antenna with 16 element panels has been realized to avoid
mechanically rotating parts. The reflector planes of the panels approximate a
cylinder. Each panel holds two element planes. In the current configuration,
the lower plane is equipped with crossed butterfly dipoles for horizontal and
vertical polarization, which cover the 150 to 350 MHz frequency range and are
thus suited for DAB reception. The 16 elements, feeding the 16 receiver channels
of the front-end, allow 360◦ beam forming.
83
8 8.5 9 9.5 1047
47.5
48
48.5
49
49.5
50
latit
ude
[deg
ree]
longitude [degree]
receiver
illuminatorsTx−ID: 39
Tx−ID: 0
Tx−ID: 12Tx−ID: 7
Figure 26: Overview of DAB stations and receiver in ELITE 2006 experiment.
The upper plane is equipped with 16 vertically polarized UHF-broad band
dipoles for DVB. Due to the higher frequencies, each panel holds two of these
dipoles, horizontally spaced, to allow for beam forming within a field of 0◦ to
180◦ . The back half of the upper plane is equipped with spare dipole elements.
Alternatively, both planes can be equipped with crossed butterfly dipoles, which
can be combined to sharpen the beam in elevation. The individual dipole elements
in front of the reflector plane each have a cardioid element diagram, providing for
approximately 3 dB gain. All elements which are not used in the measurement
configuration are terminated with 50 ohms resistors mounted inside the central
tower of the array.
The HF-front-end consists of 16 equal receiver channels. Each of the receiver
channels comprises of a low-noise amplifier (LNA), a tunable or fixed filter and
an adaptive gain control for optimum control of the Analog to Digital Converter
84
(ADC). The LNAs have a noise figure of 1.1 dB and a gain of 40 dB. In the
current configuration fixed DAB band-pass filters are being used with a pass
band of 220 to 234 MHz. A chirp signal with a bandwidth of 1.536 MHz, centered
around 227.36 MHz (226.592-228.128 MHz, channel 12C), generated by a separate
signal generator and transmitted to the front-end by coaxial cable, is used for
calibration. A bank of switches provides for calibration of each receiver channel
chain from the LNA to the ADC, excluding only the antenna element.
A/D conversion is realized with 4 FPGA boards, each processing 4 receiver
channels. The FPGAs currently provide only for serializing the 4 channels. Each
FPGA board is equipped with 4 ADC-modules with 14 bit 100 Msamp/sec max-
imum sample rate ADC chips. For the processing of DAB signals, a sample
rate of 18.432Msamp/sec is used. It is matched to sample all sub-bands of TV-
channel-12 and is also a multiple of 512 kHz, the basic clock rate of the DAB
signal.
Each ADC output is fed to an electro-optic converter and linked to the signal
and data processing unit via a fibre-optic cable. In the signal and data pro-
cessing unit, the optical signals are converted back to digital data streams of 4
serial channels, each. Four high performance Quad-Opteron computers handle
the four data streams. The opto-electric conversion and the feeding of the data
to PCI-X-bus are performed on 64-bit-boards, hosting 4 FPGAs. The FPGAs
can additionally be used for pre-processing the raw data. A data control pro-
cess controls the storing of the raw data on two 1 TByte hard disc drives per
85
Quad-Opteron and drives a FiFo RAM, which serves the ’pre-view’ real time vi-
sualization process. Hence, the signal and data processing unit provides 8 Tbyte
of hard disc recording space. For further data storage a raid-array is used with
15.3 Tbyte hard disc space, where the measured raw data can be saved after each
measurement sequence. In the visualization master processor, which is served
by the four data streams from the Quad-Opterons via 10 Gbit Infiniband links,
performs the processing control, beam forming and detection processing. The
’pre-view’ display is processed on a separate high performance computer in the
signal and data processing unit.
The experimental data available was recorded during a measurement cam-
paign in the southern part of Germany, the precise locations can be seen in Ta-
ble 6. There were four active DAB transmitters in the area, the geometry of the
setup is depicted in Fig. 26, where we see that one station is to the north (close
to Mannheim, Germnay), and three more towards the south (around the Swiss
border). About six hundred DAB frames, or roughly one minute of recorded data
is available. Currently no ground truth in form of radar or air traffic control data
is available at this point.
3.7.2 Algorithm Performance
The DAB specifications and algorithm settings are identical as in the simu-
lation study, as it was designed to mirror this scenario. The major difference is
that received SNR is lower, due to the quite far observation range. We try to
86
compensate the low SNR by increasing the integration time, instead of combining
one DAB frame, we will consider two or four frames (about 200-400 ms).
We first examine the results using the conventional FFT processing, see
Fig. 27, where we again plot the superposition over all processed frames. To
show the effect of leakage due to severe clutter, we also include a plot without
the adaptive clutter removal, see Fig. 27(a). After adaptive clutter removal, a
number of tracks can be observed. The axis in Fig. 27 and all following figures
are limited to 250 m/s, as all target detections occur at lower range-rates.
In the case of MUSIC, the direct signal and clutter removal can also be handled
in beamspace. Since the stationary signal components do not change across
frames, we simply take a large number of frames and choose the two-hundred
largest eigen-vectors. This also benefits from the fact that the targets change
across frames, diminishing their effect compared to the stationary signal parts.
Each frame is then projected onto this space to remove direct signal and other
clutter. In Fig. 28(a), we see that this different approach gives as a softer “gap”
around the zero range-rate region compared to Fig. 27. Unfortunately a different
type of artifact surfaces as vertical lines. Using compressed sensing in form of
Basis Pursuit, the experimental data was processed, see Fig. 28(b).
To point out the advantages of the super-resolution methods, we enlarge a cen-
tral area with several tracks, see Fig. 29. Comparing the results for conventional
FFT processing, Fig. 29(a), to MUSIC and Basis Pursuit, see Fig. 29(b) and (c),
we can clearly see that the super-resolution methods do not suffer from the same
87
range−rate [m/s]
rang
e [k
m]
−200 −100 0 100 200
10
20
30
40
50
60
70
(a) No Clutter Removal
range−rate [m/s]
rang
e [k
m]
−200 −100 0 100 200
10
20
30
40
50
60
70
(b) Adaptive Clutter Removal
Figure 27: Experimental data for conventional FFT based processing (a) withoutclutter removal; (b) with adaptive clutter removal.
88
range−rate [m/s]
rang
e [k
m]
−200 −100 0 100 200
10
20
30
40
50
60
70
(a) MUSIC
range−rate [m/s]
rang
e [k
m]
−200 −100 0 100 200
10
20
30
40
50
60
70
(b) Basis Pursuit
Figure 28: Experimental results for high resolution methods; (a) MUSIC, (b)Basis Pursuit.
89
sidelobes like the conventional FFT processing. In addition, Basis Pursuit can
detect targets with significantly smaller Doppler values, due to not utilizing any
clutter removal.
range−rate [m/s]
rang
e [k
m]
−60 −40 −20 0 20 40 6025
30
35
40
(a) Conventional FFT Processing
range−rate [m/s]ra
nge
[km
]
−60 −40 −20 0 20 40 6025
30
35
40
(b) MUSIC
range−rate [m/s]
rang
e [k
m]
−60 −40 −20 0 20 40 6025
30
35
40
(c) Basis Pursuit
Figure 29: Enlarged view to highlight the sidelobe suppresion of the high resolu-tion methods.
3.8 Summary
In this paper, we illustrated the passive radar concept and described the
current state-of-the-art. We derived the exact matched filter receiver, which was
90
not available before. We showed that a current efficient FFT based approach
is equivalent to matched filtering based on a piece-wise constant approximation
of the Doppler induced phase rotation on the received waveforms. Using the
same approximation we developed efficient implementations of receiver algorithms
using subspace concepts, namely MUSIC, and compressed sensing implemented
as Basis Pursuit. We discussed the implementation and various benefits of these
algorithms, and tested them using numerical simulation and experimental data.
We find that in complexity the subspace approach is one order of magnitude
higher than the current approach, followed by the Basis Pursuit formulation
that is again one order of magnitude more costly in complexity. Nevertheless,
high-complexity algorithms achieved higher target resolution by avoiding common
sidelobes found in conventional FFT based processing. Additionally Basis Pursuit
does not require adaptive removal of clutter and the direct signal, leading to better
detectability of small Doppler targets.
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