SIGNAL PROCESSING FOR MULTICARRIER MODULATION IN ...

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.



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



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


(200 m) on two Julian dates 295-300. . . . . . . . . . . . . . . . . 41


(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)


17 Performance results using ICI-aware receivers at receiver S5 (1,000 m)


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


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


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


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


(b) July 25, 64-QAM

1 2 3 410

−2

10−1

100

phones

BLE

R


(c) July 26, 16-QAM

1 2 3 410

−2

10−1

100

phones

BLE

R


(d) July 26, 64-QAM

1 2 3 410

−2

10−1

100

phones

BLE

R


(e) July 27, 16-QAM

1 2 3 410

−2

10−1

100

phones

BLE

R


(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

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(a) Julian date 295, S1

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(b) Julian date 296, S1

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(c) Julian date 297, S1

2 4 6 8 10 1210

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(d) Julian date 298, S1

2 4 6 8 10 1210

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BLE

R


(e) Julian date 299, S1

2 4 6 8 10 1210

−3

10−2

10−1

100

phones

BLE

R


(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

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phones

BLE

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2 4 6 8 10 1210

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2 4 6 8 10 1210

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2 4 6 8 10 1210

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BLE

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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

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10−2

10−1

100

phones

BLE

R



2 4 6 8 10 1210

−3

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phones

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2 4 6 8 10 1210

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2 4 6 8 10 1210

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2 4 6 8 10 1210

−3

10−2

10−1

100

phones

BLE

R



2 4 6 8 10 1210

−3

10−2

10−1

100

phones

BLE

R



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

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10−2

10−1

100

phones

BLE

R

LSOMPBP


2 4 6 8 10 1210

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10−2

10−1

100

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BLE

R

LSOMPBP


2 4 6 8 10 1210

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10−2

10−1

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BLE

R

LSOMPBP


2 4 6 8 10 1210

−3

10−2

10−1

100

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BLE

R

LSOMPBP


2 4 6 8 10 1210

−3

10−2

10−1

100

phones

BLE

R

LSOMPBP


2 4 6 8 10 1210

−3

10−2

10−1

100

phones

BLE

R

LSOMPBP


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

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10−2

10−1

100

phones

BLE

R

LSOMPBP


2 4 6 8 10 1210

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BLE

R

LSOMPBP


2 4 6 8 10 1210

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10−2

10−1

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phones

BLE

R

LSOMPBP


2 4 6 8 10 1210

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10−1

100

phones

BLE

R

LSOMPBP


2 4 6 8 10 1210

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10−2

10−1

100

phones

BLE

R

LSOMPBP


2 4 6 8 10 1210

−3

10−2

10−1

100

phones

BLE

R

LSOMPBP


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

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10−2

10−1

100

phones

BLE

R

LSOMPBP


2 4 6 8 10 1210

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phones

BLE

R

LSOMPBP


2 4 6 8 10 1210

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LSOMPBP


2 4 6 8 10 1210

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LSOMPBP


2 4 6 8 10 1210

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100

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BLE

R

LSOMPBP


2 4 6 8 10 1210

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10−2

10−1

100

phones

BLE

R

LSOMPBP


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

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10−2

10−1

100

phones

BLE

R


(a) ICI-ignorant, S1

2 4 6 8 10 1210

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10−1

100

phones

BLE

R

LSOMPBP

(b) ICI-aware, S1

2 4 6 8 10 1210

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100

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BLE

R


(c) ICI-ignorant, S3

2 4 6 8 10 1210

−3

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LSOMPBP

(d) ICI-aware, S3

2 4 6 8 10 1210

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100

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BLE

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(e) ICI-ignorant, S5

2 4 6 8 10 1210

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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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