Analysis & Design of Multiple-Input Multiple-Output ... · PDF fileAnalysis & Design of Multiple-Input Multiple-Output Synthetic Aperture Sonar ... Input Multiple-Output Synthetic

Analysis & Design of Multiple-Input Multiple-Output SyntheticAperture Sonar Systems

by

Qiu Hua Tian

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science and Engineering

Graduate Department of Electrical and Computer EngineeringUniversity of Toronto

c© Copyright 2015 by Qiu Hua Tian

Abstract

Analysis & Design of Multiple-Input Multiple-Output Synthetic Aperture Sonar Systems

Qiu Hua Tian

Master of Applied Science and Engineering

Graduate Department of Electrical and Computer Engineering

University of Toronto

2015

A synthetic aperture sonar (SAS) system is a sophisticated invention which forms high resolution images

of the seafloor. The main issue of a SAS system is its low platform speed. This is due to the need to

accurately estimate the position of the sonar transducers in real-time. Using a single transmitter, multiple

receiver (SIMO) system reduces this problem. In an attempt to further augment the traveling speed,

we propose a SAS system with multiple transmitting elements placed along the cross-range direction,

and colocated with some of the receiving elements. The transmitters emit a set of orthogonal (or

nearly orthogonal) waveforms. We analyze the pros and the cons of this multiple input multiple output

(MIMO) design with a simulator that is built during this project, and look into ways to resolve some

of the complications. Specifically, we show that the MIMO configuration does allow for faster platform

speed. Further, the MIMO configuration can improve the near field cross-range resolution. Finally, we

show that these gains are possible if any cross-talk between the transmitted waveforms is suppressed.

ii

Dedication

To my family

iii

Acknowledgements

I would never be able to finish my dissertation without the guidance of my supervisor, help from

friends, and support from my family.

I would like to express my deepest gratitude to my supervisor, Prof. Raviraj S. Adve, for his excellent

guidance, trust, and providing me with a great atmosphere for doing research. He has always been patient

in his instructions and generous with sharing his expertise when I wander around seeking my way to

become a researcher. I couldn’t have asked for better supervisor.

I would like to thank the Defense Research and Development Canada (DRDC), especially Mr. Vincent

Myers, for technical and funding support, invaluable advices and guidance.

Many thanks to Arin Minasian, Zhe Cui, Sanam Sadr, Max Yuan and other colleagues in the lab for

all the stimulating discussions and fun times in the past two years.

I would also like to thank all my friends for their loving accompany and support in every aspect of

my graduate life. Thanks to Justin Wong for the many inspiring discussions. Thanks to Anna Yu for

being a great friend for the last six years. I always gain new perspectives by talking to you. It is truly

amazing to have you walk along with me on the exciting and adventurous journey of becoming who we

want to be, and to be each other’s support and inspiration.

Most importantly, a huge thanks to my parents and my grandparents for their support and encour-

agement, and for their never ending love through different stages of my life.

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Contents

1 Introduction 1

1.1 Overview of Synthetic Aperture Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Definition of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 Challenges Faced by the SAS System . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 SAS Simulator 8

2.1 System Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Linear FM Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Azimuth Beampattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Received Reflected Signal at Receiving Element . . . . . . . . . . . . . . . . . . . . 10

2.2 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Array Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 2D Matched Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Functionalities of the Simulation Package . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Sample Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Data Model with Non-Idealities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Multiple-Input Multiple-Output SAS system 25

3.1 Virtual Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Faster Platform Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Improved Azimuthal Resolution at Near Range . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.2 Derivation of Point Spread Function . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Cross-Talk Reduction and Waveform Selection 40

4.1 MIMO-SAR and its Focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Orthogonality Condition and Waveform Design . . . . . . . . . . . . . . . . . . . . . . . . 41

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4.3 Waveforms Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.1 Up- and Down-Chirps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.2 Short-Term Shift-Orthogonal Waveforms . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3.3 Frank Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4 Cross-Talk Reduction via Pulse Compression . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.1 Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.2 Adaptive MMSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Conclusion and Future Work 56

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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List of Tables

2.1 Default Values Based on the DRDC System . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 System Parameter for MIMO-SAS System Modeling . . . . . . . . . . . . . . . . . . . . . 28

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List of Figures

1.1 Generation of Phased Array through Time-Multiplexing, Recreation of Fig. 2 from [1] . . 2

1.2 Geometry of Simulation Ground, recreation of Fig. 2b from [2] . . . . . . . . . . . . . . . 3

1.3 Data Acquisition Geometry, recreation of Fig. 4.1 from [3] . . . . . . . . . . . . . . . . . . 4

2.1 Illustration of Coherent Transmitted Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Beampattern for the DRDC System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Azimuth Beampattern and its Impact on Received Signal Strength as the Platform Moves

Passes a Target Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Received signal at a single element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Sum of received data from all receiver elements . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 Range compressed signal for a point target . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.7 Target Recovering Block Diagram, recreation of Fig. 9 from [4] . . . . . . . . . . . . . . . 16

2.8 Post-processed image for a point target . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.9 A Complex Target Scene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.10 Platform Motion Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.11 Change of travel path at the presence of yaw and sway errors . . . . . . . . . . . . . . . . 19

2.12 Realizations of Yaw Error Along Track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.13 Intermediate Figures in Realizing One Target Point with Yaw Error . . . . . . . . . . . . 21

2.14 Intermediate Figures in Realizing One Target Point with Sway Error . . . . . . . . . . . . 22

2.15 Processed image with yaw errors with cutoff frequency = 0.05, variance = 9o. . . . . . . . 23

2.16 Illustration of speckle generation across region of interest . . . . . . . . . . . . . . . . . . 23

2.17 Example of generated background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.18 Example of Scene with background speckle and foreground objects . . . . . . . . . . . . . 24

3.1 Geometry of Phase Center Approximation (PCA) . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Virtual Array of a SIMO SAS system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Configuration 1: MIMO-SAS system with transmitters placed along range direction . . . 27

3.4 Configuration 2: MIMO-SAS system with transmitters placed along cross-range direction 27

3.5 System response of SIMO system at different speed of platform Distance to center of

target area = 50m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.6 Comparison of SIMO and MIMO system at different speed of platform Distance to center

of target area = 50m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7 Seafloor image - ship with debris simulated with SIMO and MIMO systems. MIMO

system annihilate ghosts images at high speed . . . . . . . . . . . . . . . . . . . . . . . . . 29

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3.8 Exposure time and aperture synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.9 At range = 50m, the MIMO system produces image with superior azimuth resolution

than SIMO system, both system contains 8 receiving elements . . . . . . . . . . . . . . . . 32

3.10 Far Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.11 Effective aperture length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.12 Point spread functions at near and far ranges . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.13 Simulated scene at 50m range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.14 Simulated scene at 500m range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.15 Simulated scene of ship debris at 50m range . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1 An illustration of instantaneous frequency of chirp signals . . . . . . . . . . . . . . . . . . 43

4.2 An illustration of chirp signals as obtained via frequency taken from segment A in Fig. 4.1 44

4.3 An illustration of chirp signals as obtained via frequency taken from segment B in Fig. 4.1 44

4.4 Different convolution results of different realizations of up and down chirps . . . . . . . . 45

4.5 Different configurations that MIMO Synthetic Systems Take . . . . . . . . . . . . . . . . . 45

4.6 Scene generated with up and down chirp as the transmitting signals Number of transmit-

ters = 2; number of receivers = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.7 MIMO-SAR geometry (Fig. 9 in [5]) targets can first be separated into range A and range

B by spatial filtering before using matched filtering to distinguish point scatterers (in this

case point i and point j) in each range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.8 MIMO scene. Transmitting waveforms: Frank codes . . . . . . . . . . . . . . . . . . . . . 48

4.9 Least Square Processing Magnitude of target points in all figures, from left to right: 0

dB, 0 dB, 0 dB, -10 dB, -20 dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.10 Demonstration of signal transmission and reception . . . . . . . . . . . . . . . . . . . . . . 53

4.11 MIMO scene processed with adaptive MMSE pulse length: 0.006second . . . . . . . . . . 54

4.12 MIMO scene processed with adaptive MMSE and implemented with short length pulse

pulse length: 0.002 seconds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1 Different processing on target scene injected with white Gaussian noise Amplitude of

target points: 0 dB, 0 dB, 0 dB, -10 dB, -20 dB noise level: -20 dB . . . . . . . . . . . . . 57

5.2 MIMO scene. Transmitting waveforms produced by Genetic algorithm . . . . . . . . . . . 58

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List of Symbols

Term Unit Description

PRF s−1 Pulse Repetition Frequency

t s fast time (in range direction)

η s slow time (in cross-range direction)

vp m/s platform velocity

K Hz linear range FM rate

Ka Hz linear azimuth FM rate

f0 Hz center frequency

λ m wavelength

k m−1 wavenumber

La m length of each antenna/hydrophone

Ne – number of antennas within array

D m distance between two consecutive pings

ρa m azimuth resolution

Xc m distance to center of target area

X0 m half target range

Tp m chirp pulse duration

B0 Hz baseband bandwidth

x

Chapter 1

Introduction

Many remote sensing applications, such as environmental monitoring, earth-resource mapping and mil-

itary surveillance require broad-area imaging at high resolution, at any time of the day or night, and

during adverse weather conditions. Radar transmits radiation in the microwave region of the electro-

magnetic (EM) spectrum. With appropriate choice of operating frequency, the EM wave can penetrate

clouds, rain, fog and precipitation with little attenuation and distortion. Being an active system, a radar

system provides its own illumination and permits operations regardless of day or night. Prior to the

development of imaging radar, most high resolution sensors were camera systems highly susceptible to

the ambiant operating conditions [6]. Radar systems were first used for detecting ships and airplanes.

First developed during World War II, imaging radar gradually evolved into Side Looking Airborne

Radar (SLAR), which corrected for severe distortions in the display, then into Synthetic Aperture Radar

(SAR) systems, which use signal processing to improve the azimuth resolution beyond the limitation of

the physical antenna aperture used [7].

High resolution imaging of the seafloor is also much needed in the underwater world for applications

such as offshore exploration, seabed mapping and environmental surveillance [8]. For a long time, side-

looking sonar was the most popular seabed imaging method. The main issue with this technique is

that its resolution along cross-range gets poorer as range increases [9]. A synthetic aperture system,

on the other hand, does not suffer from this problem. A SAR system is not directly applicable for

underwater condition. Its information carrier, the EM waves, suffers from severe propagation loss in

water caused by the high conductivity of water [10]. Acoustic waves, on the other hand, suffers much

less propagation loss, and can traverse much longer distances under water. A Synthetic Aperture Sonar

(SAS), with acoustic waves being its information carrier, is hence used to form seafloor images. A SAS

system bears close resemblance with a SAR system. Yet the two systems are drastically different due

to the physical constraints imposed by their respective operating media. After giving a preview on the

theory of operation behind synthetic aperture systems, we will elaborate on the differences between the

two systems, and some of the challenges faced by SAS system, which is the focus of this work.

1.1 Overview of Synthetic Aperture Systems

SAR and SAS are coherent imaging systems. The image of the static ground (in SAR) or seafloor (in

SAS) is formed by the coherent combination of a series of reflections collected using a moving platform.

1

Chapter 1. Introduction 2

Figure 1.1: Generation of Phased Array through Time-Multiplexing, Recreation of Fig. 2 from [1]

The overall length over which the data is collected essentially acts a single extremely large antenna or

aperture. This allows the generation of high resolution remote sensing imagery without complicated

post-processing.

Consider a side-looking SAS transceiver that insonates a region. If a small transmit aperture is used,

it will produce a beam that is much wider once it makes its way to the seafloor, as shown in Figure 1.1,

thus making it difficult to discern and accurately locate a target or isolate targets from the reflections.

In order to improve the resolution without complicated post-processing, a large aperture is needed to

generate a narrow beampattern and hence resolve the seafloor. Since building a very large array is

impossible, a platform carrying one single transducer of suitable size is mounted on a moving platform.

For now the motion is assumed to be linear and of constant velocity. Many beams, uniformly spaced in

time, are transmitted and received at the antenna. Processing these returns coherently is equivalent to

using a single large aperture. The SAS processor, therefore, stores and processes the complex baseband

signals corresponding to all returns. In general, these returns form a 3D complex matrix corresponding

to the number of transmissions (pings), the number of samples per ping (range bins) and the number of

elements in the receive transducer [1] [11] .

For convenience, we will assume, unless otherwise stated, that the the platform uses only a single

transducer. Extensions of this model to the case of multiple receivers is conceptually straightforward

and will be discussed as required.

1.1.1 Geometry

A synthetic aperture system is an active imaging system. Figure 1.2 illustrates the geometry of the

system under consideration. In this figure, the platform is moving vertically along the y-axis. The

shaded area in Figure 1.2 represents the area designated for imaging. Note that we assume a broadside

target area. The platform which carries the transducers moves along the cross-range (y-axis) with a

velocity vp; this dimension is also referred to as along-track. The figure also illustrates the range (along

the x -axis) and cross-range (along the y-axis) dimensions. The z -axis aligns with the vertical dimension

as shown in the 3D version in Figure 1.3. The platform velocity, together with the sampling period and

number of pings, decides the total distance traveled by the platform while coherently collecting data.

Without loss of generality, the point y = 0 is defined at the center of the swath. The target area is

centered at the point (Xc, 0, 0) and covers a rectangle of dimensions 2X0 in range (the swath) and 2Y0

in cross-range. Here we assume that the synthetic aperture, the distance over which data is coherently


moving path of platform

(Xc+Xo,Yo)(Xc-Xo,Yo)

(Xc-Xo,-Yo) (Xc+Xo,-Yo)

x(range)

y(cross-range)

Figure 1.2: Geometry of Simulation Ground, recreation of Fig. 2b from [2]

collected, is equal to the cross-range width, 2Y0.

Figure 1.3 presents a 3D model of the system geometry [3]. The altitude, denoted by h, aligns with

z-axis. For a target point located at (xt, yt, h), R0 =√x2t + h2 is the slant range and is perpendicular to

the cross-range; here xt is the range position of the given target point. For a receive antenna element at

point (0, yr, 0), R =√R2

0 + (yt − yr)2 is the distance from the antenna element to the target point; here

yt is the cross-range position of the target point. In the figure, θ is the angle measured from boresight

in the slant range plane, i.e., R0 = R cos(θ) and (yt − yr) = R sin(θ).

1.1.2 Definition of Terms

The terms used to describe a synthetic aperture system and its geometry are defined as follows.

Along-track: refers to the direction along the path of moving platform, used interchangeably with the

term azimuth and cross-range.

Azimuth: The term azimuth is used interchangeably with the term cross-range and along-track, and

refers to the direction along the path of moving platform on the ground plane.

Baseline: a line on the ground that extends to infinity, directly below the path of moving platform.

The baseline is marked by the y-axis in Fig. 1.3.

Cross-range: cross-range is along the path of moving platform, and is measured from the x -axis as

defined in Fig. 1.2.

Cross-track: refers to the direction perpendicular to the path of moving platform.

Hydrophone: a hydrophone is a device that can emits and receive acoustic signals. Hydrophone can

be used as a transmitter or a receiver in a SAS system.

Ping: denotes each sampling point along track, in slow time.


path of moving platform

xrange

y

cross-rangeh

swath

target

point

(azimuth)R 0

R

sensor location

Figure 1.3: Data Acquisition Geometry, recreation of Fig. 4.1 from [3]

Range: the term range can refer to either slant range or ground range. Slant range is measured from

the sensor location to a target point in the illuminated area, along the sensor’s line of sight. Slant

range is denoted R0 in Fig. 1.3. Ground range is measured from a certain point on the baseline to

the point in the target area along the ground, and perpendicular to the path of moving platform.

Range is the projection of slant range onto ground plane. Unless specified otherwise, the term

range refers to ground range in this thesis.

Resolution: the minimum separation distances between two equally strong target points can be before

they become inseparable to the imaging system.

Swath: the width of target area and is perpendicular to the path of moving platform, as illustrated in

Fig. 1.3. A target area has a swath size of 2X0.

Synthetic aperture: the distance illuminated by the antenna array, measured on the ground and along

the path of moving platform.

Target area: an area illuminated by transmitting element(s). This is a pre-determined area to be

imaged by a synthetic aperture system.

Transducer: a sensor element which combines a transmitting and a receiving element.

1.1.3 Challenges Faced by the SAS System

While the SAR system has been employed extensively in everyday life, the SAS system has seen slower

development for several reasons, and the slow speed of moving platform in a SAS is a limiting factor in

many real life applications. The main difference between a SAR and a SAS system is the use of different

information carrier, which in turn has an impact on the speed of moving platform.


The number of pings (transmissions) per second is the pulse repetition frequency (PRF), denoted as

f0. To avoid aliasing in both domains, the PRF is constrained by

2vpLa

< PRF <c

4X0(1.1)

where La is the length of the sensor in the cross-range/along-track dimension and c is the speed of

acoustic waves (electromagnetic waves in a SAR system). From (1.1), we can derive the maximum

unambiguous range that dictates conditions under which range and cross-range aliasing occurs [12]:

Rmax = 2X0,max =cLa4vp

(1.2)

Eqn. (1.2) shows that the desired swath sets an upper limit on the velocity of the platform. For a given

target area swath, a higher velocity is possible by using a receive array. For a SAS system employing a

physical antenna array wherein each element covers an aperture of La, Eqn. (1.2) becomes

Rmax =cD

4vp(1.3)

where D = NeLa/2 is the maximum displacement between two consecutive pings with Ne elements [13].

Now, v is bounded by

vp <cNeLa4Rmax

. (1.4)

For example, a SAS system was used in an attempt to discover the remains of the missing Malaysia

Airlines Flight MH370. The system takes five to seven days to comb a search area with a radius of 6.2

miles [14], a speed which only allows localized searches.

Sensor elements are expensive. It is undesirable, and sometimes not possible, to have a long sensor

array. The limitation of platform velocity motivates us to seek, from signal processing point of view,

means to provide higher system speed, with minimum modification to the original system. Specifically,

we will consider the use of multiple transmitters and receivers in a SAS system. This leads to a multiple

input multiple output SAS (MIMO-SAS) configuration.

1.2 Contribution

Multiple-Input Multiple-Output SAR system has been of recent research interest for the SAR community.

While the platform velocity is not a concern for the SAR system, SAR systems are used to image large

target scenes. It is of interest for the SAR community to increase the swath of imaging scene to avoid

multiple traversing. Inspired by the common interest to resolve the contradicting requirement of high

speed and large swath, this work represents an effort to introduce MIMO configuration to SAS system,

with the goal of increasing platform speed without compromising other system characteristics. In a

MIMO system, the multiple transmissions are orthogonal and are assumed to be so at the receiver.

Simple matched filtering is then used to separate individual transmissions.

To start, we build a simulator for a SAS system. We aim to make this a tool that allows users

to access simulation data through every stage of SAS imaging, from modeling the received signals at

receiver end by illuminating the target scene, to different stages of data processing to obtain the final


image. We want to provide maximum flexibility to users to modify the system and to visualize the

results of the modifications.

Then we move onto developing MIMO-SAS system. We implemented a MIMO-SAS system with the

goal of achieving higher platform velocity. Upon a closer examination of simulation results for systems at

near range, which is often time for imaging at shallow water, we discovered that the azimuth resolution

experiences an improvement with longer array as the target scene gets closer to the baseline. This was

unexpected since the generally accepted notion is that cross-range resolution is a function only of the

synthetic aperture. We will provide a mathematical derivation to support our claim.

Next we try to optimize the image quality produced by MIMO-SAS system. To our knowledge, only

up and down chirps are orthogonal. Signals that are not orthogonal for all time shifts will introduce

cross-correlation in the post processing stage and lower the SNR of the final image. We will use non-

adaptive and adaptive method to reduce the cross-talk, and will discuss and evaluate the challenges in

cross-talk reduction.

1.3 Literature Review

This section reviews background research on work related to MIMO-SAS systems. As each main section

in the body of this work has a different focus, more background related to each section will be detailed

in the respective section.

Few studies have been done on the topic of MIMO-SAS. Yan Pailhas and Yvan Petillot have done

extensive work on developing MIMO-sonar systems [15,16]. Their focus has mainly been on using MIMO

geometry, with array of receiving elements perpendicular to the array of transmitting elements, to ensure

statistically independent observations and to achieve super-resolution. However, this arrangement is

inappropriate for a MIMO-SAS application.

In Malphurs’ work, simulated images for MIMO-SAS system were examined using different sets of

waveforms [17]. He showed that MIMO-SAS system yields degraded image quality due to self-clutter

and claims that phase-coded waveforms can alleviate the problem. We will show later that more work

is needed to resolve the issue.

Teng et al. [18] published a work on MIMO-SAS and argued that MIMO-SAS system provides im-

proved range resolution. Their receiving array is placed along the cross-track direction, with transmitting

elements colocated with the receiving elements at each end of the array. In the physical system that we

analyze, the receiving array is along the along-track (azimuth) direction to maximize platform velocity

as explained in eq (1.4).

1.4 Organization

The thesis is organized as follows. chapter 2 introduces the required background theory about synthetic

aperture systems and post processing. This chapter will describe a SAS simulator that takes in user

inputs and outputs images. Simulation results will be presented to show different functions of the

simulator. This simulator will also be the testing ground for many of our ideas.

In an attempt to further increase the maximum speed of a moving platform, a SAS system with

multiple transmitting elements will be examined in Chapter 3. The concept of virtual array is behind

this design and will be introduced. Mathematical derivation and simulation results for a MIMO-SAS


system will be presented to show that compared to a SIMO-SAS system, MIMO system can achieve faster

platform speed and improved azimuth resolution at near range. The advantages and disadvantages of

the design will also be investigated.

Chapter 4 will take a closer look at different sets of transmitting waveforms. This chapter will

focus on the cross-talk effect resulting from non-orthogonal transmitting waveforms. A precise definition

of orthogonality will be given. Different waveforms will be analysed in terms of orthogonality and

applicability in the system. As most waveforms under analysis are non-orthogonal, the cross-correlation

between transmitting waveforms has a negative impact on the quality of the processed image. Non-

adaptive and adaptive methods, in an effort to reduce the effect of cross-talk, will be explored.

Chapter 2

SAS Simulator

Techniques proposed to improve SAS systems need to be validated experimentally. However, sea tri-

als are time-consuming, expensive and can only be performed during certain times of the year; many

modifications to the system requires sophisticated fine tuning of the hardwired physical system. To

avoid unnecessary trials and errors during testing, we have designed a simulator as per DRDC’s request.

The simulator can be configured according to the desired modification, either in system configuration

or system parameters, and it outputs raw data as each receiver would have collected over the course of

imaging. This simulator also provides a testing ground for the ideas that we will propose.

The first two sections of this chapter are devoted to the theory behind the construction of the SAS

simulator. It is worth emphasizing at this point that the data model developed is based on several

assumptions. Some of the key assumptions are:

• The model is 2D in the sense that the imaging region of interest is ”flat” and the point targets

within the region do not have any height.

• The transmitter elements are of size La along array with an inter-element spacing of La as well.

This assumption is used in forming the transmit and receive beampatterns.

• We ignore propagation loss.

• The propagation speed of sound within the water is a constant (1500m/s).

• In the ideal case (unless specified by user otherwise), the transducer platform moves along a straight

line and at a constant velocity and height above the seabed.

• All transmissions are broadside and the receive array is, ideally, parallel to the direction of motion.

• While the transducer platform is moving linearly, we use a stop-and-hop data model wherein the

platform transmits and receives at a fixed point and then hops to the next transmission point.

2.1 System Fundamentals

The simulator was developed under the request of Defense Research and Development Canada (DRDC).

The system parameters used by DRDC are listed in Table 2.1.

8

Chapter 2. SAS Simulator 9

Term Unit Description Default valuePRF s−1 Pulse Repetition Frequency 4HzVp m/s platform velocity 1.5m/sK Hz Linear range FM rate 4.3kHz/ms2

f0 Hz Center frequency 300kHzλ m wavelength 5mmLa m antenna length 33cmNe number of antennas within array 36D m distance between two consecutive pings 1.2mXc m distance to center of target area 38mX0 m half target range 35mY0 m half target area azimuth 85.25mTp m chirp pulse duration 14msB0 Hz baseband bandwidth 60kHz

Table 2.1: Default Values Based on the DRDC System

transmit period listening period

Tp

pulse repetition interval=1/PRF

platform traveling time

Figure 2.1: Illustration of Coherent Transmitted Pulses

2.1.1 Linear FM Signals

Linear FM pulses, also called chirp signals, are the most popular choice of pulses in synthetic aperture

systems. The instantaneous frequency of a chirp signal is a linear function of time. The main advantage

of linear FM pulses is in its constant envelope. The chirp signal essentially uniformly fills the available

bandwidth and provides excellent ambiguity function properties on matched filtering.

In the time domain, an ideal LFM signal is characterized with a duration Tp, a constant amplitude,

and a quadratic phase component; the complex low-pass equivalent signal is given by

s(t) = ejπKt2

, 0 < t < Tp

= rect(tTp

)ejπKt

2 (2.1)

where, rect(·) denotes the standard rect/rectangular function, t is the time variable and K the linear

FM rate since the instantaneous frequency is given by Kt, 0 < t < Tp. Note that K may be positive or

negative called up- or down- chirps respectively. Fig 2.1 presents a sample transmission of a sequence of

coherent up-chirps. Ignoring the effect of the finite time duration, the bandwidth of the pulse is given by

BW = |K|Tp. In our current implementation, the transmission is assumed to be a sequence of coherent

up-chirps.


0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Beampattern

ping number

mag

nitu

de

Figure 2.2: Beampattern for the DRDC System

2.1.2 Azimuth Beampattern

The system under discussion operates at a center frequency of f0 corresponding to a wavelength of

λ = c/f0 and a wavenumber of k = 2π/λ = 2πf0/c. Due to the size of the transmitting element, the

transmitted signal does not radiate isotropically, but follows a beampattern. Here we use the simplest

beampattern that assumes no tapering is used in the transmission. The transmit beampattern is given

by the so-called element pattern

Pe(θ) = sinc(kLa sin θ) (2.2)

where, as before k denotes the wavenumber and sinc(x) = sin(πx)/(πx). One could, similarly, write an

expression for the beampattern for the entire array, the array pattern Pa(θ). Based on our assumptions

stated earlier, this is equivalent to a single element of size NeLa in Eqn. (2.2). Clearly, the array pattern

would be much narrower (approximately by a factor of Ne narrower) than the element pattern. Fig. 2.2

illustrates a sample beampattern using the parameters of the current DRDC system. The parameters

are listed in Table 2.1.

Fig. 2.3 plots bampatterns at three points as the transducer moves along-track. The figure also shows

the effective portion of the beampattern ’seen’ by a single target point. The azimuth 3dB beamwidth is

well approximated by

βbw =0.886λ

La(2.3)

2.1.3 Received Reflected Signal at Receiving Element

In our data model, the reflected signal is assumed to be due to the superposition of the reflections from

several point targets. At each element, the signal received is two-dimensional. The range dimension

corresponds to the ”fast time” domain, t, which in turn corresponds to the time of pulse travel. The


A B C

cross-rangeslant

range

R0

target

Figure 2.3: Azimuth Beampattern and its Impact on Received Signal Strength as the Platform MovesPasses a Target Point

cross-range corresponds to the ”slow time” domain, η, which corresponds to the time that platform has

traveled (pings transmitted).

Consider the i -th point target located at (xi, yi, h) defined in the coordinate system in Fig. 1.2 and

a single receiver at location (0, yr, 0). As before, the distance between the target and the receiver is

Ri =√x2i + (yr − yi)2 + h2, (2.4)

corresponding to a round-trop time delay of τi = 2Ri/c. If the complex amplitude of the target reflector

is Ai, the reflectivity is scaled by the transmit and the receive beampattern of the single receive element

(the element pattern Pe(θ), given by Eqn. (2.2)). The complex baseband signal received from the point

target is given by

sr(t) = AiP2e (θi)s(t− τi)ej2πf0τi , (2.5)

corresponding to the real signal

sr(t) = |Ai|P 2e (θi) cos

(2πf0(t− τi) + πK(t− τi)2 + φi

)(2.6)

where Ai = |Ai| ejφi . In these equations, the element pattern Pe(θ) is squared because the element

contributes a two-way amplitude weighting. Since we use the array for receive, but use an individual

element for transmit, the array pattern only contributes a one-way amplitude weighting.

The discussion so far has considered a single receive element located at (xi, yi, h). This element is,

in general, one of an array of Ne elements in the hydrophone receiver. The distance R as defined in

Eq. (2.4), is different from one receiver element to the next. These differences in distance are, most

importantly, reflected in the phase term in Eqn. (2.5).

Taking the center point of the array as the reference point and for the ideal case of uniform linear

motion, at the p-th ping, the center element is located at (0, pvp/f0, 0). In this case, assuming that the

number of elements in the array is odd, the n-th receiver is located at (0, pvp/f0 + (n− (Ne+ 1)/2)La, 0)

where n = 1 corresponds to the tail of the array and n = 1, 2, . . . , Ne. In this case, the distance to the

n-th element is again given by Eqn. 2.4 with ynp = pvp/f0 + (n−Ne + 1)/2)La replacing yr, i.e., on the


Raw data, 1 element

range

cros

s−ra

nge

−30 −20 −10 0 10 20 30

−80

−60

−40

−20

0

20

40

60

80

Figure 2.4: Received signal at a single element

p-th ping the distance between i-th target and the n-th receive element is given by

Rnpi =√x2i + (ynp − yi)2 + h2, (2.7)

It is worth noting that, in the case of SAR, it is common to assume that the inter-element spacings are

small enough, as compared to the range to the target area (xi) such that the square-root expression in

Eqn. 2.4 can be linearized. This simplifies the data model considerably. However, given the lower speeds

and far shorter distances involved, this approximation may not be valid for SAS systems.

Given the discussion above, the continuous time signal received at the n-th element on the p-th ping

due to Nt point reflectors (targets) is given by

snpr (t) =

Nt∑i=1

AiP2e (θnpi ) s (t− τnpi ) ej2πf0τ

npi , (2.8)

where τnpi = 2Rnpi /c and Rnpi given by equation (2.7). Furthermore, θnpi is the azimuth angle to the i-th

target as before, but in relation to the array element position. Due to the attenuation associated with

the beampatterns, for a scene with many targets distributed over the region of interest, most targets

effectively do not contribute to the signal.

The raw receive signal given by Eqn. (2.8) is sampled every ∆Ts to obtain a data cube corresponding

to Ne elements (the index n), P pings (the index p) and R ranges (the index r indicates for received

signal - with a slight abuse of notation):

snpRr = sr (R∆Ts)

=∑Nt

i=1AiP2e (θnpI ) s (r∆Ts − τnpi ) ej2πf0τ

npi

(2.9)

Usually, ∆Ts is proportional to 1/B0, where B0 is the bandwidth of the transmitted signal. The

range resolution is given by 1/2B0; however, one can choose to oversample by choosing a higher sampling

rate.

Fig. 2.4 plots the received signal for a single element (n = 1) as a function of ping p (cross-range)

and sample number r (range) for a single point target at the center of the scene. The spread along the

range domain is due to the length of the linear FM pulse whereas the slight broadening in the cross-range

domain is due to the non-zero beamwidth of the array pattern.


Raw data, sum of all elements

rangecr

oss−

rang

e

−30 −20 −10 0 10 20 30

−80

−60

−40

−20

0

20

40

60

80

Figure 2.5: Sum of received data from all receiver elements

2.2 Data Processing

The last section has presented the equations describing the signal received at an individual element

of the transducer array. In this section, we describe the standard approach to image formation - this

approach is used to validate the data model and to test the associated MATLAB programs.

2.2.1 Array Processing

The description in this chapter is based on [3], modified for synthetic aperture sonar. The data processing

comprises a series of matched filters - matched to the elements, matched to the range profile and matched

to the cross range profile.

In the previous section, we saw that the signals were received at an array of transducer elements. The

first step in processing the signals is to combine the signals at the Ne elements into a single signal. Using

the fact that array is transmitting in the broadside direction and is ”looking” broadside, the optimal

array processor, in white noise, is to add all elements together - this is equivalent to matched filtering

to the spatial profile of broadside signals. The signal input to the next step of the data processing block

is, therefore,

ssum(t) =

Ne∑n=1

Nt∑i=1

AiP2e (θnpi )s(t− τnpi )ej2πf0τ

npi . (2.10)

Note that the summation can also be applied to the sampled data in Eqn. (2.9).

This matching to the spatial profile effectively converts of the element pattern terms into an array

beampattern n the received signal, i.e., the signal in Eqn. (2.10) can also be written as

ssum(t) =

Nt∑i=1

AiPa(θpi )Pe(θpi )s(t− τpi )ej2πf0τ

pi . (2.11)

where τpi = 2R0i/c is the delay to the center element (with similar definitions for θpi ).

Figure 2.5 plots the summed signal over all elements for the same scenario as in Fig. 2.4. The effect

of the much narrow array pattern is clear - the element pattern is approximately a factor of Ne times

narrower than the element pattern.


2.2.2 2D Matched Filtering

Processing a synthetic aperture system involves processing in both the range (fast time) and cross-range

(slow time) directions.

Pulse Compression

We assumed that the transmitted signals were linear FM pulses - such pulses spread their energy over

time allowing for a relatively large signal energy with limited power. The use of pulse compression

allows for all the energy to be gathered at a single time sample providing range resolution that is

inversely proportional to bandwidth, i.e., two targets that are spaced 1/2B0 apart can be distinguished

in the resulting range profile. The range resolution is dependent on the bandwidth of the signal by

ρ =cTp2

=c

2B, (2.12)

where c is the speed of transmission wave, B is the bandwidth.

Pulse compression is essentially matched filtering in the range domain - the pulse compression filter

is matched to the transmitted linear FM waveform s(t).

Using the standard theory of matched filtering, the impulse response of the pulse compression filter,

h(t), is given by h(t) = s∗(−t). For the linear FM signal, this implies the matched filter is

h(t) = rect

(t

Tp

)exp

{jπK(−t)2

}= rect

(t

Tp

)exp

{jπK(t)2

}. (2.13)

Using the fact that when the input is a time-shifted linear FM pulse, s(t− t0), the output of the matched

filter is given by

pt0(t) ≈ Tp sinc(KTp(t− t0)), (2.14)

using Eqn. (2.8), the output at each array element after range pulse compression is given by

src(t) = Tp

Nt∑i=1

AiPa(θnpi )Pe(θpi )sinc(KTp(t− τpi ))ej2πf0τ

pi . (2.15)

with range resolution given by1

2

1

|K|Tp.

For the point target at the center of the scene of interest, as in Figure 2.4 and 2.5, the range

compressed signal is plotted in Figure 2.6. Note that to provide a better understanding, the figure

zooms into a much shorter extent in cross-range. On comparing this figure to Fig. 2.5 the effect of range

compression is clear. The spread in the target point in range is due to the non-zero range resolution.

Using the definitions of τpi , the locations of the platform at the p-th ping, the spatially processed

receive signal can also be written as

src(t, η) =

Nt∑i=1

AiPa(θpi )Pe(θpi )×

rect

(t− 2

R0i

c

)exp

{−j 4πf0R0i

c

}exp

{−jπ

2v2pλR0i

η2

}sinc(KTp(t− τpi )),


range compressed data

rangecr

oss−

rang

e

−1 −0.5 0 0.5 1 1.5 2−8

−6

−4

−2

0

2

4

6

8

Figure 2.6: Range compressed signal for a point target

where, as indicated earlier, η = p/f0 represents ’slow-time’ in the cross-range domain.

Cross-range/Azimuthal Matched Filtering

Equation (2.16) shows that as the transducer platform moves along-track, each reflector induces a

quadratic azimuthal phase modulation. This is equivalent to the quadratic phase modulation due to

the linear FM transmission in the range domain. To form a final image, synthetic aperture processing

compresses the cross-range data (azimuthal data) using an azimuth-compression filter.

The reference signal to which the azimuthal-compression filter is matched is given by the quadratic

term

hp = exp

{−jπ

2v2pλR0

η2

}, (2.16)

where, as before, η = p/f0 denotes slow-time/cross-range.

While this approach is valid, in implementation this implies summing over all targets in the scene -

while only a narrow range of the scene contributes to the signal. To reduce the computation load, we

use an exposure time Tar given by

Tar =2× (Xc +X0)

vptan

(0.886λ

La

). (2.17)

Based on this exposure time, the reference signal used is given by

hp = rect

(η

Tar

)exp

{−jπ

2v2pλR0

η2

}. (2.18)

Azimuthal processing is performed as a discrete-time convolution of the signal in Eqn. (2.16) and hp:

simage(t, η) = src(t, η) ? hp. (2.19)

The output of the convolution is a 2D discrete time signal that is also the image.

The procedure for recovering an image from the raw data is summarized in Fig. 2.7. To minimize

the computation load, the pulse compression and the azimuthal compression are implemented using a


Fast Fourier Transform (FFT). The transmit pulse is represented as the ’Range Reference Signal’ and is

used for range pulse compression. The azimuth reference signal is then used for azimuthal compression

- this is the final step and results in a single final image.

Range Reference Signal

Received raw data

FFT

Range IFFTRange FFT

Range Azimuthal

Signal

Range Compressed

SignalAzimuth

FFT

Azimuth Reference

Signal in Frequency

Domain

Image

Figure 2.7: Target Recovering Block Diagram, recreation of Fig. 9 from [4]

For the same example as in the previous figures, the range compressed signal is plotted in Figure 2.8.

As with Fig. 2.6, to avoid the appearance of a blank figure, this plot focuses on a short extent in

cross-range.

range

cros

s−ra

nge

Processed image

−1 −0.5 0 0.5 1 1.5 2−8

−6

−4

−2

0

2

4

6

8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2.8: Post-processed image for a point target

2.3 Functionalities of the Simulation Package

The previous two sections developed the data model (Section 2.1 and presented the SAS processing

approach (Section 2.2 undertaken in this project. The data model and the data processing were imple-

mented as separate MATLAB programs. This choice is to allow for a user to modify the data processing

scheme used as they see fit.

This section will not detail the use of the MATLAB programs. The details are found in [19] and the

comments in the MATLAB programs. In this section, we will give a brief descriptions of functionalities


foreground scene

range

cros

s−ra

nge

−30 −20 −10 0 10 20 30

−80

−60

−40

−20

0

20

40

60

80

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2.9: A Complex Target Scene

of the package, and show some results.

2.3.1 Sample Results

This package forms object-like shapes on the seafloor by placing densely spaced target points into the

scene. Running time is linearly proportional to the number of target points and size of the raw data

matrix (in all three dimensions - fast time, slow time and number of receiving elements). Each pixel is

of size cross-range bin (inter-ping) by range bin. The cross-range bin size is determined by the PRF and

the platform speed of the system. The range bin size is determined by bandwidth. In the figures in this

report, target points used to form a shape are spaced 4 bins in range (corresponding to 0.05m for the

parameters in Table 2.1 and one cross-range bin distance (corresponding to 0.3857m for the parameters

in Table 2.1). These numbers are chosen as a compromise between the visual effect of the final image

and running time. However, a user is free to change these values.

The building blocks of target scene consists of triangles and rectangles (details in [19]). After all the

building blocks are specified, the simulator will produce a raw data file as well as an image produced

after post-processing the raw data according to algorithms described in Section 2.2. As an example, Fig.

2.9 shows the a complex scene which simulates a hull of a boat with other random items of different

shape.

2.3.2 Data Model with Non-Idealities

We have so far been focused on presenting constructing and processing data for the ideal case of a

uniform linear motion. this package also allows the generation of a more realistic data model which


allows for translational (sway, surge and heave) and rotational (yaw, pitch and roll) motion errors and

speckle effect.

Platform Error

x (ra

nge)

y (cross-range)

z (altitude)

x (ra

nge)

y (cross-range)

z (altitude)

x (ra

nge)

y (cross-range)

z (altitude)

x (ra

nge)

y (cross-range)

z (altitude)

x (ra

nge) y (cross-range)

z (altitude)

x (ra

nge)

y (cross-range)

z (altitude)

sway surge heave

pitch roll yaw

Translational Error

Rotational Error

Figure 2.10: Platform Motion Errors

Figure 2.10 illustrates the 3 translational and 3 rotational platform motion errors. However, note

that since the elements are assumed to be zero-thickness, platform roll has no effect on the data model.

The motion errors are generated to follow a Gaussian distribution with mean zero and a chosen standard

deviation (the variance is chosen by the user and can be set to 0 to eliminate the effect of the related

error). To ensure that a realistic error profile is generated, the program generates as many independent,

identically distributed (i.i.d.) Gaussian random (with chosen variance) variables as there are pings.

These random variable are then low-pass filtered to generate errors that are correlated (across slow-

time) from ping to ping. Figure 2.11 illustrates the motion of the array over three pings in the presence

of yaw and sway errors.

The translational errors do not affect the relative position between antenna elements. So Eqn. (2.7)

is still valid, except xi, yi and zi needs to be modified from ping to ping according to the errors specified.

For example, with a sway of xs, the position of the individual element moves from (0, yi, h) to (xs, ynp, h)

and the distance changes to Rnpi =√

(xi − xs)2 + (ynp − yi)2 + h2. Translational errors are therefore


Figure 2.11: Change of travel path at the presence of yaw and sway errors

easy to model and to include in the data model.

Rotational errors are a little more complicated. As mentioned before, roll does not effect the data

model as currently developed and so is ignored. An important issue is that in calculating the location of a

specific element at ping p, we must account for the location errors from previous pings - Figure 2.11 shows

this effect in that the range and cross-range positions of any element array in the array is dependent on

the yaw error from all previous pings. This is also true for pitch errors. Therefore, with a yaw error of θyp

and a pitch error of θpp at the p-th ping, the n-th array element has

xnp =vpf0

P−1∑i=1

cos θpi sin θyi + dn sin θyp cos θpp

ynp =vpf0

P−1∑i=1

cos θpi cos θyi + dn cos θyp cos θpp + yi − vp × dur/2 (2.20)

znp =vpf0

P−1∑i=1

sin θpi + dn sin θpp

Rpn =√

(xnp − xi)2 + (ynp − yi)2 + (znp + h)2 (2.21)

where

dn = La

(1

2+ n− Ne

2− 1

)(2.22)

represents the distance of each element from the center of the array, θpi represents the azimuth angle of

target i in relation to the baseline (the ideal straight line from which the errors are measured).

Based on this expression, we can re-write the demodulated received signal for at a single target at

the array of hydrophones as

sr(t, p, n) = A0rect

(t− 2Rpn/c

Tp

)P 2e (θpn)exp {−j4πf0Rpn/c} exp

{jπK (t− 2Rpn/c)

2}

(2.23)

where, as before, η = p/f0, Rpn is defined as in Eqn. (2.21), and θpn is the azimuth angle obtained with

the coordinates given in Equation 2.20 as defined in Fig. 1.3.


As mentioned earlier, the errors are generated by passing a realization of i.i.d. Gaussian random

variables through a low-pass filter (LPF). Figure 2.12a plots a single realization of yaw errors; this plot

uses a 4-th order Butterworth filter with a fractional cutoff of 0.05 and a standard deviation of 3o.

Figure 2.12b provides another realization of the yaw error, in a similar configuration, but with a cutoff

frequency of 0.3. This example is meant to recreate a case where the yaw errors are fluctuating rapidly.

0 50 100 150 200 250 300 350 400 450−1

−0.5

0

0.5

1

1.5

ping #

yaw

err

or (

deg)

One Realization of Yaw Error along track with std=3 deg

(a) A single realization of yaw errors with cutoff fre-quency = 0.05, variance = 9o.

0 50 100 150 200 250 300 350 400 450−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

ping #

yaw

err

or (

deg)

One Realization of Yaw Error along track with std=3 deg

(b) A single realization of yaw errors with cutoff fre-quency = 0.3, variance = 9o.

Figure 2.12: Realizations of Yaw Error Along Track

Figure 2.13 plots the intermediate and final images in the generation and post-processing of the same

examples as in Fig. 2.8. The single target is located at at the center in the case with yaw error of standard

deviation of 3 degrees and a fractional cutoff frequency of 0.3. As is clear, presence of errors defocuses

the target and the processed image is corrupted. It is worth noting that such a result is consistent with

similar results in the literature for SAR [20].

To provide another example, Fig. 2.14 plots the intermediate and final images of the same example,

in the case with sway errors with a standard deviation of 1 meter. Since the sway is in range dimension,

the defocusing is largely in range.

The previous plots used a cutoff frequency of 0.3 (as in Fig. 2.12b). The final example in this section

is for a milder yaw error with a normalized cutoff frequency of 0.05. Figure 2.15 plots the processed

image. As is clear by comparing this figure to Fig. 2.13, the corruption of the figure is relatively benign

as well.

Speckle Generation

The discussion so far has focused exclusively on ”targets”, i.e., objects of interest. However, in a real

SAS system, the returned signal comprises the reflection from the target objects and that from the

background, a random component called the speckle. Clearly high fidelity is required for the former; on

the other hand, speckle is largely random. So,creating a speckle-like signal by using a superposition of

‘small’ targets would be overkill and is extremely time consuming. In this regard, we avoid unnecessary

computation time by generating speckle using random combinations of speckle created off-line and stored

in a data file.

Our approach to generating speckle signals is illustrated in Figure 2.16. First, raw signals for a dense

group of random, weak target points situated at the center of the target scene are generated. This


raw data, 1 element

range

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−30 −20 −10 0 10 20 30

−80

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−20

0

20

40

60

80

(a) image of raw data from one receiver element

raw data, sum of all elements

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−20

0

20

40

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80

(b) image of raw data from the sum of all receiver elements


range

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−6

−4

−2

0

2

4

6

8

(c) image of range compressed data

range

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Processed image

−1 −0.5 0 0.5 1 1.5 2−8

−6

−4

−2

0

2

4

6

8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(d) final processed image

Figure 2.13: Intermediate Figures in Realizing One Target Point with Yaw Error

process is repeated until we have several blocks at hand. Then random linear combinations of these

blocks are formed moved across the entire scene. Then the scene is windowed in range and cross-range

for each element to produce correlation between adjacent blocks.

As an example, Figure 2.17 plots a background scene is generated using five 4m × 4m blocks. A

complete scene (e.g., in Fig. 2.18) can be formed by adding the background scene to the foreground

objects (e.g, in Figure 2.9). The background signal is very weak compared to that of the foreground.

The speckle signals can be scaled to enhance or weaken them in comparison to the desired objects.

2.4 Summary

This chapter includes a brief overview of the background theory behind the simulator and an explanation

on the functionalities of the simulation package. The package takes in user input and produces images

that can incorporate non-idealities according to user’s need. The two non-idealities that we are concerned

with are platform error, a motion error as the platform travels, and speckle, a result of reflective noise

in the image background. The simulator is used to test ideas for the rest of this thesis, and is used for


Raw data, 1 element

range

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0

20

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(a) image of raw data from one receiver element

Raw data, sum of all elements

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0

20

40

60

80

(b) image of raw data from the sum of all receiver ele-ments


range

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−4

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0

2

4

6

8

(c) image of range compressed data

range

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Processed image

−1 −0.5 0 0.5 1 1.5 2−8

−6

−4

−2

0

2

4

6

8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(d) final processed image

Figure 2.14: Intermediate Figures in Realizing One Target Point with Sway Error

micronavigation work by Nathan V. Woudenberg [21].


range

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Processed image

−1 −0.5 0 0.5 1 1.5 2−8

−6

−4

−2

0

2

4

6

8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2.15: Processed image with yaw errors with cutoff frequency = 0.05, variance = 9o.

Figure 2.16: Illustration of speckle generation across region of interest


range

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background

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20

40

60

80

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2.17: Example of generated background

range

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Complete Scene

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0

20

40

60

80

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2.18: Example of Scene with background speckle and foreground objects

Chapter 3

Multiple-Input Multiple-Output

SAS system

Synthetic aperture system offers high resolution in range and cross-range directions, and they are inde-

pendent of each other [13]. This property allows the system to move as fast and image a scene as wide as

possible without compromising the quality of image in either direction. However, as we have described

in Section 1.1.3, wide swath and platform velocity pose contradicting requirements on the system design.

We need a method that allows the speed of the system to increase without reducing the swath width and

with minimum modification to the system (i.e., with minimum addition to the number of transducers).

3.1 Virtual Array

The relationship between the speed of platform and target area is related according to Eqn. (1.4),

reproduced here for convenience.

vp <cNeLa4Rmax

(3.1)

It is worth noting that the maximum velocity cannot always be reached before aliasing becomes a

problem.

In Eqn. (3.1), vp and La are both system parameters. We want to maximize vp without reducing

Rmax and we want to maintain a reasonable Ne, and yet, this seems to be the only parameter that

we can change. To explain why MIMO system can help with the situation and resolve the apparent

contradiction, we need to first introduce the idea of a virtual array.

A transmitter receiver pair physically located together are known as a colocated transmitter/receiver

pair. In a SAS system, only one receiver is colocated with the transmitter and the rest are known as

displaced transmitter/receiver pairs, resulting in a bistatic configuration. Using phase center approxima-

tion [22], we can replace the bistatic situation with a monostatic one. As illustrated in Fig. 3.1, take the

middle point between the transmitter and the receiver to be the ”phase center”, the transmission and

reception distances can be approximated by the distance from phase center to the target point [22]. This

approximation is called the phase center approximation (PCA). Recall that the maximum displacement

between two consecutive pings is D = NeLa/2. This is exactly the length of the virtual array of the

system. We can effectively substitute Ne in Eqn. (3.1) with the number of phase centers in a system.

25

Chapter 3. Multiple-Input Multiple-Output SAS system 26

Rx

Tx

phase center

Tx/Rx

Figure 3.1: Geometry of Phase Center Approximation (PCA)

Tx

Rx

virtual

array

Figure 3.2: Virtual Array of a SIMO SAS system

The PCA holds when ∆2/4r � λ, where ∆ is the distance from the transmitter to the receiver, and r is

the distance from the receiver to the scatterer. This condition almost always holds, and is true for the

parameters chosen in this paper.

For the system that we have considered so far, a (SIMO) SAS system has the same number of phase

centers as the number of receiving elements but half in length. The array formed by these phase centers

is a virtual array (Fig. 3.2). The number of phase centers dictates the value of Ne, and the goal of

our MIMO system is to add additional transmitters to increase the number of phase centers (in the

cross-range direction) and therefore the length of the virtual array.

There are two possible configurations to a MIMO-SAS system. In configuration 1, as illustrated

in Fig. 3.3, two transmitting elements are placed in line with the first receiving element along range

direction. The virtual array of this configuration has the same length as is the SIMO case.

In configuration 2, two transmitting elements are placed at each end of the receiving array. In our

simulation, the two transmitting elements are colocated with the two receiving elements at each end of

the array. These two elements are transducers. This configuration and its corresponding virtual array

are illustrated in Fig. 3.4. The length of virtual array has extended by (Ne − 1)La/2 compare to the

SIMO scenario and configuration 1.

Since the SIMO system and configuration 1 MIMO system share the same virtual array, it is expected

that these two systems also share the same maximum platform speed. Configuration 2 yields a virtual

array almost twice as long, we expect the corresponding MIMO system to have a higher limit in the

speed of platform.


Tx1

Rx

virtual

array

Figure 3.3: Configuration 1: MIMO-SAS system with transmitters placed along range direction

Tx1

Rx

virtual

array

Figure 3.4: Configuration 2: MIMO-SAS system with transmitters placed along cross-range direction


Parameter Value UnitNumber of hydrophones 8Hydrophone size 0.05 mPRF 8 HzPlatform speed variesCenter frequency(f0) 100 kHzBandwidth 30 kHzchirp length 50 msTime shift 22 msDistance to center of target area 50/500 m

Table 3.1: System Parameter for MIMO-SAS System Modeling

3.2 Faster Platform Speed

Using the simulator described in Chapter 2, a simulation was performed to prove this conjecture. The

parameters used in the simulation are listed in Table 3.1. In the MIMO case, the transmitters transmit

up and down chirps.

range

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10

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30

(a) Velocity = 1m/s

range

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0

10

20

30

40

(b) Velocity = 2m/s

Figure 3.5: System response of SIMO system at different speed of platformDistance to center of target area = 50m

Fig. 3.5 presents the results for the SIMO system. Ghost targets are produced at high speed but

not at low speed. When a second transmitter is added to the other end of the array, the ghost targets

disappear (Fig. 3.6), confirming our assumption that longer virtual array allows superior platform

velocity.

To show some more realistic scenes, the simulator that we have constructed is used to make a scene

of a ship with debris. Since we have illustrated cases with target area at 50m away from the baseline,

we choose Xc = 500m for the ship scene to demonstrate that the speed improvement introduced by a

MIMO system is generally applicable. Fig. 3.7 illustrates the results.

As seen in Fig. 3.7a, at low platform velocity, the SIMO configuration allows for a reconstruction of

the target scene. However, again, ghost targets appear at high speed for a SIMO system (Fig. 3.7b),

whereas under the same platform speed, the ghost targets are annihilated when simulated under the

MIMO system setup (Fig. 3.7c).


range

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0

10

20

30

40

(a) SIMO system, platform velocity = 2m/sAt high speed, ghost targets appear for SIMO system

range

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−10

−5

0

5

10

15

20

(b) MIMO System, platform velocity = 2m/sTransmitting signals: short-term shift-orthogonal sig-nals with a time shift of Tp/2

Figure 3.6: Comparison of SIMO and MIMO system at different speed of platformDistance to center of target area = 50m

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

(a) SIMO imaging at low speed, platform velocity = 1m/s

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

(b) MIMO imaging at high speed, platform velocity = 2m/s

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

(c) MIMO imaging at high speed, platform velocity = 2m/sTransmitting signals: short-term shift-orthogonal signalswith a time shift of Tp/2

Figure 3.7: Seafloor image - ship with debris simulated with SIMO and MIMO systems. MIMO systemannihilate ghosts images at high speed


3.3 Improved Azimuthal Resolution at Near Range

It is well-accepted that the azimuth resolution of a synthetic aperture radar/sonar (SAR/SAS) system

is determined exclusively by the size of the synthetic aperture, not the element or array size, nor range

or operating frequency. However, when we extend the SIMO system to a MIMO system, we discovered

that a MIMO system could be used to enhance azimuth resolution, but only at near range.

This section attempts to resolve the apparent contradiction, re-deriving the original resolution and

using the exposure time to indicate the relative contributions of the element, array and aperture beam-

patterns. As the results show, for a SAS system, at short ranges, the MIMO array and effective synthetic

aperture sizes may be comparable and the increased effective array size (compared to a SISO/SIMO ar-

ray) can help improve resolution. At further ranges, the resolution is, as expected, determined by the

synthetic aperture size and is not affected by MIMO processing.

3.3.1 Background

By integrating over large apertures formed by platform motion, synthetic aperture radar/sonar(SAR/SAS)

systems use active coherent imaging to obtain fine along-track (or azimuth) spatial resolution. The

conventional definition of azimuth spatial resolution is that two equally strong point targets are distin-

guishable if are separated by a distance greater than the half-power beamwidth of the antenna element.

Another interpretation is that the resolution is the 3dB spread of the point spread function (PSF) of a

single point target. The half-power (or -3dB) width is given by [6]

θ3dB =λ

La(3.2)

where λ is the operating wavelength. The length of the footprint at range R (and therefore the resolution

of the system) is then

ρa = Rθ3dB = λR/La (3.3)

In contrast to the conventional antenna, it is generally accepted that the azimuth resolution achievable

by a stripmap SAR/SAS system is independent of range and frequency of operation [13]. The best

achievable azimuth resolution, ρa, for large aperture size, is determined by the size of antenna element

La [6, 23]:

ρa =La2

(3.4)

.

This result can be derived using the notion of an exposure time. As the platform moves along track,

a point target is effectively illuminated by one transmitter (radar or sonar) as long as it falls within the

3dB beamwidth of the transmitter. A point target, at range R, is therefore illuminated for a time period

(the exposure time) for which the platform traverses a distance

L = Rθ3dB = λR/La, (3.5)

which is equivalent to resolution of conventional radar in Eqn. 3.4. For a platform moving at speed vp,

the exposure time is Ts = L/vp.

Now if we observe a target point at range R, by coherently integrating all pings, with np being


θ3dB

L

La

target

L

a) b)

R

Figure 3.8: Exposure time and aperture synthesis

the number of samples in azimuth direction that can observe the target point, then L = npLa (Fig.

3.8b)) and the target can be seen for a time period of Ts. In other words, the successive transmit-

receive positions of the system constructs an array of length L. The azimuth resolution of the system

therefore corresponds to an array of elements of length L, and L is called a synthetic aperture. The

optimum resolution of a synthetic aperture is twice as good as that of a real aperture [23] because each

transmit-receive path exists at a different time. Therefore, substituting La in Eqn. (3.4) with 2L we

obtainρa = λR

2L

= La

2

, (3.6)

where the last step was obtained by substituting Eqn. (3.5).

A more detailed derivation of this result is given in [23]1. A different approach in deriving this limit

is detailed in [6]. The results described here are applicable to single transmitter/receiver (single-input,

single-output) systems..

3.3.2 Derivation of Point Spread Function

In Section 3.1, we have extended SIMO SAS systems to MIMO SAS. As was shown, this extends the

array, effectively doubling its size as compared to a SIMO system. Interestingly, we noticed that some

simulation results (for example Fig. 3.9) showed improved resolution. However, as was shown in Section

3.3.1, the azimuth resolution of a SAS system is determined by the synthetic aperture and therefore

should not improve.

This section attempts to resolve this apparent contradiction. As we will see, both the well-accepted

analysis and the simulations that we will present are correct - essentially, the improved azimuth resolution

only applies to near ranges. We will start with a derivation of received signals.

In this analysis, we use a system comprising one transmitter and Ne receivers (a SIMO system).

With a pulse repetition interval (PRI) of Tr, the transmitter transmits a series of waveforms of duration

Tp; the transmitter element has an element pattern of Pe(θ). Stop-and-go approximation is used. Wer

are interested in the point spread function, therefore the focus will be on the simplest case of a single

point target located in the center of the swath and at y = 0 along-track.

Denoting the baseband transmitted signal, of duration Tp, as STx(t), the baseband signal received

1For a platform velocity of v and a wide swath Wg , the azimuth resolution is also lower bounded by ρa > (2vWg) /c.Here we will assume that Eqn. (3.6) limits resolution


range

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30

(a) case SIMO, Velocity = 1.6m/s

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0

10

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30

(b) case MIMO, Velocity = 1.6m/s

Figure 3.9: At range = 50m, the MIMO system produces image with superior azimuth resolution thanSIMO system, both system contains 8 receiving elements

at receiver n at ping p can be expressed as

snpr (t) = APe (θpTx)Pe (θnpRx) sTx (t− τnp) e−j2πf0τnp

, n = 0, 1, . . . , Ne − 1, (3.7)

where, for ping p, pTr < t < (p + 1)Tr, Rnp is the distance from the n-th receiver to the target point,

θpTx denotes the azimuth angle relative to the transmitter and θnpRx denotes azimuth angle relative to the

n-th receiver. And,

τnp =2Rnp

c, (3.8)

where c is the speed of sound and, for an element of size La,

Pe(θ) =sin (kLa sin θ)

kLa sin θ, (3.9)

where k = 2π/λ is the wavenumber.

If the target, as shown in Fig:3.10 is sufficiently far away (an assumption we will make throughout this

document), we have that θnpRx ' θpTx = θp, i.e., the relative azimuth angles are approximately independent

of the receiver; further, we have Rnp ≈ R0p−nLa sin θ = Rp−nLa sin θ. For the term sTx (t− τnp), we

can further approximate τnp to be independent of receive element number and τnp ' τp = 2Rp/c.

Based on these approximations, the received signal at element n at ping p is given by

snpr = AP 2e (θp) sTx (t− τp) e−j2πf0τp

ej2πf0nLa sin θp/c, n = 0, 1, . . . , Ne − 1,

= AP 2e (θp) sTx (t− τp) e−j2πf0τp

ejnkLa sin θp ,(3.10)

Before SAS image processing, beamforming combines the received signals. Since we are imaging at

broadside, beamforming is equivalent to summing the Ne received signals to obtain

spr(t) =∑Ne−1n=0 snpr (t, τ)

= AP 2e (θp)sTx (t− τp) e−j2πf0τp ∑Ne−1

n=0 ejnkLa sin θp ,(3.11)


cross-range

La sin

La

target point

broadside

array at

ping p

(0,y ,h)0p

(0,y ,h)n-1p

(0,y ,h)1p

R0p

R(N

-1)p

R1p

Figure 3.10: Far Field Approximation

The beamforming therefore introduces an array factor Pa (θp) where

Pa(θ) =

Ne−1∑n=0

e2jnkLa sin θ =sin (NkLa sin θ)

sin (kLa sin θ)(3.12)

and the received signal can be expressed as

spr(t) = AP 2e (θp)Pa(θp)sTx (t− τp) e−j2πf0τ

p

(3.13)

As before, this signal corresponds to the returns at ping p, i.e., pTr < t < (p+ 1)Tr.

The matched filter for pulse compression is h(t) = s∗(−t) where * denotes the conjugate. The pulse

compressed signal is given by

sprc(t, η) = APe(θp)2Pa(θp)wrc (t− τp) e−j2πf0τ

p

. (3.14)

Here wrc(t) denotes the transmitted pulse after pulse compression.

Let η = pTr denotes slow time. Recall from the geometry section (Fig. 1.3)that R0 is the minimum

range to the target and Rp =√R2

0 + (vη)2, where, for symmetry, we consider η = 0 corresponding

to the platform location closest to the target (i.e. η = 0 corresponds to x -axis in Fig.1.2). Using the

approximation Rp ≈ R0 + (η2v2)/2R0, and λ = c/f0, we have

sprc(t, η) ≈ APe(θp)2Pa(θp)wrc(t− τp)e−j2πf0R0/cexp

(−jπ 2v2

λR0η2). (3.15)

The phase term e−j2πf0R0/c is a constant that can be absorbed into the amplitude A and, since we are

mainly interested in resolution, we will drop amplitude from here on.

Applying the azimuth matched filter h(η) = exp(jπ 2v2

λR0η2)

with all the terms that contain η yields

the point spread function of the target, as a function of slow-time η (equivalently, along-track)

PSF (η) =

∫ ∞−∞

P 2e (θu)Pa(θu)exp

(−jπ 2v2

λR0u2)exp

(jπ

2v2

λR0(η − u)2

)du. (3.16)


Impact of Array Factor in Azimuth Resolution

It is now that we see the potential impact of the array factor Pa(θ) on the azimuth resolution. In Section

3.3.1, we have set the effective aperture size L = R0θ3dB where the 3dB beamwidth, θ3dB , was due to

the element pattern, Pe(θ). The integration in Eqn. (3.16) is over the range (−L/2v, L/2v), i.e.,

PSF (η) =

∫ L/(2v)

−L/(2v)P 2e (θu)Pa(θu)exp

(−jπ 2v2

λR0u2)exp

(jπ

2v2

λR0(η − u)2

)du. (3.17)

Compared to the array factor, Pe(θ) is essentially constant and can be dropped from the integral. Note

that the original derivation of the azimuth resolution in Eqn. (3.6) assumes a SISO system (no array

factor) and that L is large.

The easiest way to interpret the PSF arising from Eqn. (3.17) is to assume the array factor is also

effectively constant. This assumption is valid for a large synthetic aperture L. Using the fact that∫ L/(2v)

−L/(2v)exp

(−jπ 2v2

λR0u2)exp

(jπ

2v2

λR0(η − u)2

)du =

L

vsinc

(L

v

v2

πλR0η

)=L

vsinc

(vη

πLa

), (3.18)

where we have used the fact that L = λR0/La. Dropping constant terms, we have

PSF (η) = P 2e (θp(η))Pa(θp(η))sinc

(vη

πLa

), (3.19)

where we have made the dependence of θp on the slow-time η explicit. Specifically, θp = tan−1(vη/R0) 'vη/R0.

Defining an aperture factor, Paper(θ) = sinc(vη/(πLa)), we have that the PSF is a product of three

contributions, from the element, array and aperture factors:

PSF (η) = P 2e (θp(η))Pa(η)Paper(θ

p(η)). (3.20)

While it is worth emphasizing that the expression in Eqn. (3.20) is an approximation, it illustrates the

essential characteristics of the overall PSF - and hence azimuth resolution. As is clear from its definition,

the aperture factor is not a function of range. For large L, this term is much narrower than the element

and array factors and hence determines overall resolution. Specifically, the resolution is as given by Eqn.

(3.6).

On the other hand, if L is relatively small, then Paper is not the single dominant term in Eqn. (3.20).

The overall PSF - and resolution - is a somewhat complex interplay between the array factor (Pa(θ)) and

the effective aperture factor. Specifically, this could lead to a range dependent resolution. Essentially, if

L, the effective aperture, and the size of the SIMO array are comparable, this effect may be noticeable.

Next we present a few simulation results to illustrate cases where the use of a SIMO array can improve

azimuth resolution (near ranges) and where it would not (far ranges). Furthermore, since this work was

motivated by an investigation into MIMO-SAS, we present results corresponding to the MIMO case as

well.


−80 −60 −40 −20 0 20 40 60 80−140

−120

−100

−80

−60

−40

−20

0

Cross−range position

Sig

nal s

tren

gth

(dB

)

Range = 500mRange = 50m

Figure 3.11: Effective aperture length

3.3.3 Simulation Results

Here we present results of simulations based on the SAS simulator we have previously developed and

explained in Chapter 2. Here we use the same parameters as provided in Table 3.1. We use up LFM

chirps in the SISO and SIMO cases, and an up-chirp coupled with a time-shifted up-chirp in the MIMO

case (we will explore the potential of different waveforms in depth in the next chapter).

We begin by identifying the length of the effective aperture as a function of range. Fig. 3.11 plots

the magnitude of the received signal due to a single point target at the receive array for the two ranges

of 50m and 500m. As the figure shows, as expected, the effective aperture is linearly proportional to the

range - here, the 3dB effective synthetic aperture is 1.3m at 50m range and 14m at 500m.

Consider the array being simulated, which is an 8-element array with inter-element spacing of 0.05m,

i.e., a physical array of size 0.4m. this size is comparable to the effective aperture size at when the

range is 50m, but not so when the range is 500m. When using a MIMO array, the effective array size

is doubled, i.e., we effectively have an array of 0.8m, again comparable to an aperture size of 1.4m at a

50m range.

−0.2 0 0.2 0.4 0.6 0.8 1 1.2−6

−5

−4

−3

−2

−1

0

cross−range (m)

ampl

itude

(dB

)

50m SISO50m SIMO (8 Rx)50m SIMO (16 Rx)50m MIMO

(a) Resolution at 50m

−0.2 0 0.2 0.4 0.6 0.8 1 1.2−6

−5

−4

−3

−2

−1

0

cross−range (m)

ampl

itude

(dB

)

50m SISO50m SIMO (8 Rx)50m SIMO (16 Rx)50m MIMO

(b) Resolution at 500m

Figure 3.12: Point spread functions at near and far ranges

The two figures in Fig. 3.12 illustrates this effect, plotting the PSF of a point target at a near range of


50m (Fig. 3.12a) and at the far range of 500m (Fig. 3.12b). Each figure consists of four cases: the SISO

case, the SIMO case with 8 receive elements, a SIMO case with 16 elements and the MIMO case with 8

physical receive elements (which effectively equivalent to the SIMO case with 15 receive elements). Fig.

3.12a illustrates two key points: first, at the near ranges, using an array does improve resolution - this is

seen by the thinner corresponding PSFs; second, the 8-element MIMO system does indeed behave like

a 15-element SIMO system, since they have the same virtual array.

Fig. 3.12b confirms that this improved resolution is available only at near ranges - at the far ranges

all four PSFs are essentially the same. This is because the SISO/SIMO/MIMO receivers all have effective

sizes far smaller than the effective aperture size - the overall PSF is dominated by the aperture factor.

The sharper PSFs available at the near ranges lead to sharper images as well. To illustrate this, we

begin with a simple scene comprising 4 targets closely spaced in cross-range. As before, we consider two

cases where the targets are at ranges of 50m or 500m.

We simulate a target scene with 4 target points, spaced 0.8m apart in cross-range at a range of 50m

away. Fig. 3.13d plots the SAS images resulting from the same systems considered in Fig. 3.12. As

it is clear from the figures, moving from a SISO system (simulated at 1/8th speed of multiple receiver

elements system) to an 8-element SIMO system improves resolution. As suggested by Fig. 3.12a, further

increasing effective array size by using either a 16-element SIMO system or an 8-element MIMO system

further improves resolution. With similar array sizes, the results of 16-element SIMO system (Fig. 3.13c)

and 8-element MIMO system (Fig. 3.13d) are similar.

The setup is then used to simulate a target scene at a range of 500m (Fig. 3.14). The SAS images

are formed from the same four systems as Fig. 3.13. It is as expected that the PSFs are similar at this

range, so are the images. Again, we see that the improved resolution is only available at the near ranges.

Last but not least, we show the simulation results of a more realistic scene with ship debris at range

of 50m (Fig. 3.15). The horizontal slit in the ship is fairly small. So we can see a very clear contrast

between the scene imaged by SIMO and MIMO system.

In brief, while it is generally accepted that the resolution of a synthetic aperture radar/sonar system is

independent of array size and range, recent work investigating MIMO SAS suggested improved resolution

was possible. This chapter has attempted to resolve this seeming contradiction. Specifically, we derive the

point spread function of a single point target - the derivation yields three beamwidth related components

in the point spread function - the element beampattern, the array factor and an effective aperture factor.

The effect of the array factor is only apparent if the effective aperture is of the same order as the size

of the effective array. As seen in the results, at near ranges, using a large SIMO (or effectively large

MIMO system) does improve resolution (as compared to a SISO system). However, at far ranges, where

the effective aperture size is large, there is no discernible improvement.

3.4 Summary

With two transmitters on each end of the hydrophone array, a virtual array of size NeLa is created. This

virtual array is longer than the virtual array of a SIMO SAS system by (Ne−1)La. The elongation of the

virtual array contributes to two advantages of a MIMO-SAS system, a higher platform traveling speed

and an improved azimuth resolution at near ranges. Higher platform speed is possible because longer

virtual array provides more sampling point along cross-range at each ping. Further, it is well-accepted

that the azimuth resolution is determined by the size of antenna element. We have shown that the point


range

cros

s−ra

nge

−15 −10 −5 0 5 10 15−20

−15

−10

−5

0

5

10

15

20

(a) 8 elements SISO system

range

cros

s−ra

nge

PRF=8, v=1, Xc=50, 8 Rx 1 Tx

−15 −10 −5 0 5 10 15−20

−15

−10

−5

0

5

10

15

20

(b) 8 elements SIMO system

range

cros

s−ra

nge

−15 −10 −5 0 5 10 15−20

−15

−10

−5

0

5

10

15

20

(c) 16 elements SIMO system

range

cros

s−ra

nge

PRF=8, v=1, Xc=50, 8 Rx 2 Tx

−15 −10 −5 0 5 10 15−20

−15

−10

−5

0

5

10

15

20

(d) 8 elements MIMO system

Figure 3.13: Simulated scene at 50m range


range

cros

s−ra

nge

−15 −10 −5 0 5 10 15−20

−15

−10

−5

0

5

10

15

20

(a) 8 elements SISO system

range

cros

s−ra

nge

PRF=8, v=1, Xc=500, 8 Rx 1 Tx

−15 −10 −5 0 5 10 15−20

−15

−10

−5

0

5

10

15

20

(b) 8 elements SIMO system

range

cros

s−ra

nge

PRF=8, v=1, Xc=500, 16 Rx 1 Tx

−15 −10 −5 0 5 10 15−20

−15

−10

−5

0

5

10

15

20

(c) 16 elements SIMO system

range

cros

s−ra

nge

PRF=8, v=1, Xc=500, 8 Rx 2 Tx

−15 −10 −5 0 5 10 15−20

−15

−10

−5

0

5

10

15

20

(d) 8 elements MIMO system

Figure 3.14: Simulated scene at 500m range

−15 −10 −5 0 5 10 15

−10

−8

−6

−4

−2

0

2

4

6

8

10

(a) SIMO scene

−15 −10 −5 0 5 10 15

−10

−8

−6

−4

−2

0

2

4

6

8

10

(b) MIMO scene with the same parameters

Figure 3.15: Simulated scene of ship debris at 50m range


spread function is a product of array factor and aperture factor. The point spread function is dominated

by the aperture factor at far range. At near ranges however, the size of the antenna array is comparable

to the length of synthetic aperture. The array factor and aperture factor both contribute to the azimuth

resolution. Longer virtual array leads to sharper array factor, and therefore finer resolution.

Chapter 4

Cross-Talk Reduction and Waveform

Selection

In Chapter 3, we established the geometry of MIMO-SAS system and have justified the feasibility of the

system. A part of the problem that we have yet to explore is the different sets of waveforms that can

be used as transmitting waveforms for the MIMO-SAS system. Conventional radar systems strive to

resolve targets in time and in frequency, which cannot be achieved perfectly for a given time-bandwidth

product [24]. Researchers introduced a function called the ambiguity function to capture the inherent

resolution properties of SIMO radar systems in range and Doppler and to evaluate their performance:

|χ(τ, υ)| =∣∣∣∣∫ ∞−∞

u(t)u∗(t+ τ)ej2πυtdt

∣∣∣∣ (4.1)

One of the fundamental assumptions of SAS systems is that all the targets to be imaged are still,

and therefore, Doppler resolution does not concern us. In our discussion, we will therefore set u = 0.

Ideally, ambiguity function would approach a thumbtack which peaks at (0,0). The same concept

applies to a MIMO system on a more perplexed level by taking into account the effect of multiple

waveforms on spatial resolution [24].

On the other hand, while it is common practice to many MIMO radar systems to select waveforms

that occupy different frequency band to ensure the orthogonality of signals, this is not applicable to

synthetic aperture systems. The signals of synthetic aperture systems are to be combined coherently

in post processing, which dictates that all transmitting waveforms ought to occupy the same frequency

band. In addition, using different frequency bands implies loss of bandwidth and hence range resolution.

In this chapter, we are going to define what we mean by orthogonal waveforms, what are some of

the waveforms that we have analysed and taken into consideration, what are some issues that was

encountered using multiple transmitter systems and what has been attempted to resolve the problem.

4.1 MIMO-SAR and its Focus

Extensive work on MIMO-SAR system has been done. Their focus and geometry adopted is different,

but yet closely related to this project. Before diving into waveform analysis further, we would like to

give a brief introduction on MIMO-SAR systems for further comparison later on.

40

Chapter 4. Cross-Talk Reduction and Waveform Selection 41

Wide unambiguous swath and high azimuth resolution pose contradicting requirement for the design

of synthetic systems. Several configuration/systems were designed to improve the situation, includ-

ing multi-beam, displaced phase center antenna (DPCA), quad-array, and high-resolution wide-swath

(HRWS) SAR imaging systems (for a more detailed description of these systems, please refer to [25,26]).

The most successful method is the HRWS configuration, which uses multiple transmitters and receivers

in the range direction (with more receiver subapertures in height or z-dimension), with the array slightly

elevated with respect to the ground plan, to increase maximum swath. The main difference between the

HRWS system and the MIMO-SAS system is the geometry. The HRWS system creates a virtual array

in the range direction and the MIMO-SAS system in the cross-range direction. The importance of this

difference will become clear later.

4.2 Orthogonality Condition and Waveform Design

While conventional MIMO radar uses the ambiguity function to evaluate the performance of each set

of transmitting waveforms, orthogonality (in time) is what matters for synthetic aperture systems as

Doppler resolution is not a concern. Orthogonal waveforms should allow the system to produce unam-

biguous images (i.e. images free of ghost targets) with minimum sidelobe level. We will see the precise

definition of orthogonality below.

The orthogonality conditions are crucial in evaluating the performances of signals during the post

processing of the signals. A description of pulse compression was given in Section 2.2.2 for a SIMO sys-

tem. For a MIMO system, since all transmitting elements transmit simultaneously, assuming orthogonal

signals, it suffices to substitute the compression filter in Eqn. (2.13) with

h(t) =

NTx∑k=1

s∗k(−t), (4.2)

where NTx is the total number of transmitting elements in the MIMO system and sk(t) is the kth

transmitted pulse. For now, we will work with a system with two transmitting signals. Assume that the

system looks broadside and that the two transmitting signals arrive simultaneously at the receiver end

after a delay of t0 seconds (this is an over-simplification of the system model, we will later deal with a

more realistic case). When the received signal∑2k sk(t− t0) is convolved with the reference signal h(t)

we get2∑k

sk(t− t0) ∗ h(t) = R11(t0) +R22(t0) + 2R12(t0), (4.3)

which consists of the sum of two autocorrelation and two cross-correlation terms. The autocorrela-

tion terms are what specify the location and amplitude of targets, and the cross-correlation terms will

deteriorate the quality of the image. If the transmitting signals are orthogonal to each other, the cross-

correlation terms will vanish.

Three orthogonality conditions have been proposed by the MIMO SAR community. The description

of this section is mainly drawn from [5].


Orthogonality Without Shift

The first proposed orthogonality condition is orthogonality without shift, which requires that∫s∗i (t) · sj(t) · dt = 0 if i 6= j (4.4)

where si(t) and sj(t) are any pair of transmitted signals.

This condition ensures perfect separation of target points assuming that the width of the target

scene is less than the pulse length. For spatially extended scattering scenarios, it is said that when the

orthogonality is not ensured for arbitrary shifts between the different transmit signals, the energy from

a distributed scene would appear smeared [5]. Up- and down-chirps form such a pair of signals that

satisfy Eqn. (4.4) and that are not orthogonal for arbitrary time shifts. We will analyze its applicability

further in Section 4.3.1.

Orthogonality for Arbitrary Time Shift

The other ”extreme” of the orthogonality condition is orthogonality for arbitrary shifts, which states

that ∫s∗i (t) · sj(t+ τ) · dt = 0 ∀ τ ∈ <, i 6= j (4.5)

While this condition allows perfect signal separation, it also requires that si(t) and sj(t) occupy nonover-

lapping frequency bands. To see this, note that the magnitude of the Fourier transform of the left hand

side of Eqn. 4.5 is [S∗i (f) · Sj(f)]. Hence (4.5) requires that the product of Si(f) and Sj(f) vanishes for

all f .

This condition however violates the fundamentals of array processing in a synthetic aperture system,

which requires that the received signals from all pings are to be combined coherently, i.e., the received

signals need to have spectral overlapping. Orthogonality for all time shifts is hence not a viable solution.

Short-Term Shift-Orthogonality

Krieger [5] found a middle ground to the two conditions above and came to propose using short-term

shift-orthogonal signals which satisfy∫h(τ) · s∗i (t) · sj(t+ τ) · dt = 0 ∀τ ∈ TD, i 6= j (4.6)

where TD is a range of interest. This condition ensures that signals si(t) and sj(t) are orthogonal for a

time period τ . Targets cτ apart can therefore be separated perfectly, where c is the speed of transmitted

waveform. For a target scene that has width larger than cτ , Krieger suggests to use digital beamforming

on receive. This method takes advantage of the elevation of transmitting array to spatially separate

the scene into slices of width less than cτ . Krieger further suggests to use an up-chirp and an up-chirp

shifted by τ as the transmitting waveforms to ensure orthogonality within the required time period. This

set of waveforms are explored in Section 4.3.2.

While this design comes to be very handy for the MIMO-SAR system, it is not realizable for the

configuration that we have proposed. We will see why in Section 4.3.2.


A

B

f

t

(a) Instantaneous frequency of up chirp

A

B

f

t

(b) Instantaneous frequency of down chirp

Figure 4.1: An illustration of instantaneous frequency of chirp signals

4.3 Waveforms Choices

In this section, we will introduce and analyse a few waveforms most pertinent to our work.

4.3.1 Up- and Down-Chirps

The up- and down- chirps are one of the few, if not the only set of waveforms that are nearly orthogonal

to each other with almost no sidelobes [27].

In Section 2.1, we have defined the up chirp, reproduced here for convenience:

sup(t) = ejπKt2

, 0 < t < Tp, (4.7)

and down-chirp is defined as

sdown(t) = e−jπKt2

, 0 < t < Tp. (4.8)

The phase of the chirp signal is defined as

φ(t) = ±πKt2, (4.9)

with the + sign for up chirp and − sign for down chirp, and with corresponding instantaneous frequency

of ±Kt.To get a better understanding of whether up and down chirp signals work for MIMO synthetic system,

we will first appeal to some visual explanation.

Fig. 4.1 illustrates the instantaneous frequency of an up and a down chirp. Each graph is divided

into three segments with equal length. Each segment is of length Tp (i.e. of a pulse length). Plotting

up and down chirp using frequency range A in Fig. 4.1a and 4.1b gives two identical and symmetrical

realizations, as illustrated in Fig. 4.2. When these two signals are used as transmitting waveforms, the

cross-correlation persists with a non-fading amplitude envelop for a duration of 2Tps (Fig. 4.4a). If

frequency range B is used, we get two different realizations for up and down chirp (Fig. 4.3). Their


−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 4.2: An illustration of chirp signals as obtained via frequency taken from segment A in Fig. 4.1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(a) One realization of up chirp

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(b) One realization of down chirp

Figure 4.3: An illustration of chirp signals as obtained via frequency taken from segment B in Fig. 4.1


time

mag

nitu

de

2Tp

(a) Convolving up and down signals in Fig. 4.2

time

mag

nitu

de

(b) Convolving up and down signals in Fig. 4.3

Figure 4.4: Different convolution results of different realizations of up and down chirps

range

crange

Tx1 Tx2

t1 t2Rx

(a) Cross-talk inducing configuration of MIMO-SARsystem

range

crange

t1

t2Rx/Tx1

Tx2

(b) Controllable, cross-talk free configuration of MIMO-SAS system

Figure 4.5: Different configurations that MIMO Synthetic Systems Take

cross-correlation function (Fig. 4.4b) is almost a thumbtack and has almost no sidelobes.

The difference of the two figures in Fig. 4.3 and Fig. 4.2 is just a time shift. During post processing,

the received signal∑2k=1 sk(t − tk) with the reference signal h(t), where tk, for k = 1, 2 denotes the

time it takes signal sk to travel from transmitter k to a certain receiver. For a certain time difference

δt = t1− t2, the cross-correlation can produce sidelobes with undamped amplitude. The question is then

whether we can control δt to avoid sidelobes.

The answer is yes, and no. The determinate factor is the geometry of the system and the beamwidth

of the antenna elements. Fig. 4.6a illustrates the MIMO-SAR configuration with two transmitting

elements and one receiving element. This illustration is applicable to all receiving elements along track.

The transmitting elements have very wide field of view to illustrate a large area, so the difference

between t1 and t2 (from the figure) can take on a very wide range. It will be very hard to control the

transmitting signals to avoid the cross-talk inducing time delay between the two transmitted signals.

In the MIMO-SAS configuration that we have proposed, the transmitting and receiving elements are

placed in cross-range direction. Since it suffices to image the few pings at broadside at each sampling

point along track, we can make the beamwidth of each antenna element very small. This in turn means

t1 − t2 is also very small. We can therefore pick the frequency range for the two transmitting signals to

make sure that they do not generate unwanted sidelobes.

Applying this concept, Fig. 4.6 shows a scene plotted by using the up- and down-chirp as transmitting


−20 −15 −10 −5 0 5 10 15 20

−15

−10

−5

0

5

10

15

(a) Scene generated with unwanted sidelobes between upand down chirps

range

cros

s−ra

nge

−20 −15 −10 −5 0 5 10 15 20

−15

−10

−5

0

5

10

15

(b) Scene generated with no unwanted sidelobes betweenup and down chirps

Figure 4.6: Scene generated with up and down chirp as the transmittingsignals

Number of transmitters = 2; number of receivers = 8

waveforms, which are chosen to be orthogonal. The image is clean and clear, free of any sidelobes, as

expected.

4.3.2 Short-Term Shift-Orthogonal Waveforms

In his paper [5], Krieger suggested use of an up-chirp and a shifted version of the up-chirp as the

transmitting waveforms for a MIMO system. This set of transmitting waveforms is orthogonal only for

the duration of the time shift. If the width of the target scene exceeds the length of the time shift,

matched filter will create a ghost target on each side of the real target, with half the amplitude, one

time shift away. Krieger suggests to use this set of transmitting waveforms in combination with digital

beamforming on receive. In the MIMO-SAR geometry, reflections at different ranges are associated with

different signal delays due to the elevated transmitting array in the range direction (Fig. 4.7). Targets

can first be separated into different ranges using spatial filtering before further processing.

For the MIMO-SAS system, the transmitting elements are placed along track. It is hence not appli-

cable to use spatial filtering to separate different range of the target scene. When the width of target

scene exceeds the length of time shift, short-term shift-orthogonal waveforms will produce sidelobes.

4.3.3 Frank Code

Apart from up- and down- LFM chirp signals, other sets of transmitting waveforms will have sidelobes

generated during post-processing. A detailed and extended analysis of different kind of waveforms for

MIMO radar is described in [28]. For our purpose, a good set of transmitting waveforms need to satisfy

two criteria

• Autocorrelation function with high peak and minimal side-lobe level

• Minimal cross correlation


Transmitting

array

A B

Figure 4.7: MIMO-SAR geometry (Fig. 9 in [5])targets can first be separated into range A and range B by spatial filteringbefore using matched filtering to distinguish point scatterers (in this case

point i and point j) in each range

We have first considered the phased code that gives ideal autocorrelation function, the Barker code.

For each code length, there is only one Barker code. We need at least two sets of code for MIMO

system, and since Barker code is only up to length 13, it is very restrictive to divide the code down

further. Therefore, we have focused on another type of popular code, the Frank code.

A Frank code is a type of phase-coded pulse that offers coherent integration of received signals. It

does not satisfy our two criteria the best, but is used as an example of non-orthogonal waveforms. We

will show how cross-correlation affects the quality of the image.

Frank code is constructed by first forming a Frank matrix of size L× L:

0 0 0 . . . 0

0 1 2 . . . L− 1

0 2 4 . . . 2(L− 1)...

......

. . ....

0 L− 1 2(L− 1) . . . (L− 1)2

A Frank code of length L×L is formed by concatenating the rows of the Frank matrix and multiplying

by 2π/L.

The Frank code is then concatenated into NTx segments of length M = L×L/NTx, and each sequence

is implemented into each transmitting signal

s(t) = cos(2πf0t+ πKt2 + πc(t)), 0 ≤ t ≤ Tp, (4.10)

where c(t) is a code sequence of length M .

Figure 4.8 displays a scene with 4 points in the cross-range direction with a length 100 Frank code.

The system has two transmitters, each transmits a length 50 Frank code. The shadows in the range

direction are a result of cross-correlation from post-processing. When we have multiple target points in

range direction, the superposition of these shadows will worsen the image quality. We will discuss ways


rangecr

oss−

rang

e

−15 −10 −5 0 5 10 15−6

−4

−2

0

2

4

6

Figure 4.8: MIMO scene. Transmitting waveforms: Frank codes

to reduce these shadows in the next section.

4.4 Cross-Talk Reduction via Pulse Compression

The main concern in SAS community on the MIMO system is the problem of cross-talk. Apart from

linear up and down chirp signals, which are nearly orthogonal to each other, other sets of waveforms

(such as coded and noise modulated waveforms) produce strong cross-talk that interfere with the target

echo [27]. There is an issue with using linear up and down chirp - there are no other orthogonal signals.

This limits the MIMO system to use only two transmitters. Cross-talk has to be dealt with to incorporate

more transmitters into the system. In this section, we are going to explore methods to reduce the cross-

talk generated by non-orthogonal waveforms. We borrow much of this approach from the work of Blunt

et al. [29].

4.4.1 Least Squares

We will start with the most simplest model. Say that we have a range profile x of length M (M is the

number of time samples in range, in other words, the number of range bins) and a reference signal h of

length N (where N is the number of time samples of transmitting signal). We obtain a received signal

y of length M +N − 1. At each time instant l we have

y(l) = xT (l)h, (4.11)

where x(l) = [ x(l) x(l+1) ··· x(l+N−1) ]T is a vector of N contiguous samples, and h = [ h0 h1 ··· hN−1 ].

Applying matched filter we get

xMF (l) = hHy(l), (4.12)

where xMF (l) is the matched filter estimate of the lth sample in the range profile, and y(l) = [ y(l) y(l+1) ··· y(l+N−1) ]T

is a vector of N contiguous samples of the received signal.


We can write the received vector y(l) of length N as

y(l) = [y(l) y(l + 1) · · · y(l +N − 1)]T

= [xT(l)h xT(l + 1)h · · · xT(l + N− 1)h]T

=

x(l) x(l − 1) · · · x(l −N + 1)

x(l + 1) x(l) · · · x(l −N + 2)...

.... . .

...

x(l +N − 1) x(l +N − 2) · · · x(l)

h

= AT(l)h

(4.13)

Substituting Eqn. 4.12, we can apply matched filter to Eqn. 4.13 to estimate x by using

xMF (l) = sHAT(l)h (4.14)

However, the matched filter is optimal if only the diagonal elements of A matrix are non-zero. In other

words, if two target points are present simultaneously within one processing window of length N , mutual

interference will occur as a result of range sidelobes and can affect the quality of the image. Least square

(LS) solutions can be used to alleviate the problem [29]. To formulate the LS problem, we can write the

received signal vector y as

y = Hx

=

h0 0 · · · · · · 0... h0

...

hN−1...

. . ....

0 hN−1. . .

.... . . 0

. . . h0...

...

0 · · · · · · 0 hN−1

x(4.15)

where H is a banded (M+N−1)×M matrix. Here we have a slight abuse of notation. When the letter H

is not in superscript, it denotes the matrix described in Eqn. 4.15; when it is in the superscript position,

it denotes the Hermitian operation. The LS formulation then yields an estimation of the reflections of

all the range bins from the ping under consideration

x = (HHH)-1HHy (4.16)

Efficiency wise, LS method needs to process received signal ping by ping. The processing speed is a

linear function of the number of pings.

The LS method works very well when there is one transmitter. Fig. 4.9a shows the cross-section

along range of a target scene generated with SIMO system and processed with matched filter. Fig. 4.9b


process signals from the same setup but using least squares during the processing stage. The performance

of least squares is slightly superior to that of matched filter as least squares offers a sharper resolution.

When LS is applied to a MIMO system using Frank codes (Fig. 4.9c), the strongest three target points

can be seen, but the weaker two are hard to distinguish. Taking a closer look at Fig. 4.9c, we can see

that LS helps reducing the cross-correlation and if we take a cross-section along broadside, we can see

all five targets well displayed, but LS does not perform so well away from broadside. Fig. 4.9d shows the

cross-section of the scene 5 pings away from broadside. The second weakest target barely made it above

the interferences, and the weakest target is not distinguishable. When the target point falls outside the

array’s broadside but is still within the 3dB beamwidth (i.e. can still be seen by the receiver elements),

the paths from each transmitter to the target points and back to a certain receiver are have different

lengths. The transmitting signals then arrive at the receiver with different time delays.

For a single target point, the reference signal for pings off the broadside should hence be the sum of

shifted version of two transmitting waveforms. For an extended scene, however, the received signals at

each ping comprises influences from several pings. We cannot distinguish return signals from broadside

and off-broadside. It is hence difficult to come up with a general reference signal that takes care of all

time shifts resulting from all scatterers within the beamwidth. It is of interest to find a better reference

signal that can accommodate the variation in the reference signals. In the next section, we will try to

address this issue and use adaptive method to improve the estimation of range profiles.

4.4.2 Adaptive MMSE

Adaptive pulse compression is designed to reach superresolution in radar systems [29, 30]. The idea is

to use estimation from previous iterations to refine the estimate at the current iteration. This concept

is transferable to a MIMO-SAS system. The adaptive minimum mean square error (MMSE) method

minimizes the difference between the estimation of range profile x and the received signal y by iteratively

updating the reference signal.

We will start with the single transmitter case [29]. The matched filter estimation of x is given by

xMF (l) = sHAT(l)h (4.17)

as we have seen in last section. We will now use w as the MMSE filter, to represent a modified and

constantly updated version of h. At each range bin l, an optimal reference function w can be found by

minimizing the MMSE cost function

J(l) = E[x(l)−wHy(l)

](4.18)

at each range bin l. Here, E[·] denotes expectation. We assume that neighbouring impulse response

terms are uncorrelated, i.e. E[x(l)x∗(l + j) = 0]. By minimizing the MMSE cost function, the MMSE

filter takes the form

w(l) = (E[y(l)yH(l)])−1E[y(l)x∗(l)], (4.19)

where * denotes complex conjugation.

By assuming that neighboring impulse responses are uncorrelated and by substituting Eqn. 4.13, the

MMSE filter becomes

w(l) = ρ(l)(C(l) + R)−1h. (4.20)


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(a) Cross-section of scene generated with received signalfrom SIMO system processed with matched filter

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(b) Cross-section of scene generated with received signalfrom SIMO system processed with least squares method

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(c) Scene generated by processing received signal fromMIMO system with least squares methodTransmitting signals are generated using 100 bits Frankcode

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NOT a targettarget point

(d) Cross-section of scene generated with received signalfrom SIMO system processed with least square methodTransmitting signals are generated using 100 bits FrankcodeCross-section is taken at 5 pings away from broadside

Figure 4.9: Least Square ProcessingMagnitude of target points in all figures, from left to right: 0 dB, 0 dB, 0 dB, -10 dB, -20 dB


where ρ(l) = |x(l)|2 and C(l) =∑N−1nr=−N+1 ρ(l+nr)hnh

Hn . hn is obtained by zero padding and shifting

h by n samples. For instance, h2 = [ 0 0 h0 ··· hN−3 ]T . It is clear from this equation that the reference

signal is updated based on (2N − 1) point estimations around range bin l. Then this updated signal

w(l) is used to obtain the current estimate of the point x(l) by performing xcurr(l) = wHy(l).

As a side note, for most of the simulations discussed in this work, noise is not added to the simulated

data. It is possible to do so, and simulation results from adding complex random Gaussian noise show

that LS and MMSE perform poorly in terms of noise tolerance, compared to matched filter. Noise

tolerance is a topic that needs further research. Hence we will focus on a noise free environment for now.

To apply Eqn. (4.20), a prior estimate of the range profile is needed to account for the ρ(l) term.

Therefore we assume that the impulse responses are equal (i.e. having a magnitude of 1) for the initial-

ization stage. Eqn. (4.20) then reduces to

w ∼=

(N−1∑

n=−N+1

hnhHn

)h. (4.21)

Without any prior knowledge, the initialization stage MMSE filter in Eqn. 4.21 processes data in a

way very similar to the LS method. These two methods are equivalent if the length of the processing

window equals the length of reference signal.

There are two options for reference signals. Option 1 is the sum of all transmitting waveforms (Eqn.

4.2). Option 2 is to process received data with matched filter first. The range compressed signal then

consists of autocorrelation and cross-correlation components. We can use the sum of autocorrelation

and cross-correlation across all transmitting waveforms as the new reference signal. Simulation shows

that option 1 yields results in lower sidelobes. Hence we will use the sum of all transmitting waveforms

as the reference signal h.

The invertibility of the matrix C(l) + R has to be taken care of when using Eqn. 4.20. In the

absence of noise (or if noise level is very low), the rank of the matrix to be inverted is dominated by

the number of non-zero ρ(l) elements in constructing C matrix. C comprises (2N − 1) ρ(l) elements

and has a maximum rank of N . There can be a maximum of N − 1 data points with magnitude of zero

at each processing window. This observation has two consequences. First, adaptive MMSE performs

better on a scene with extensive scatterers than a scene with few scatterers; second, the estimation of

N − 1 points at both ends of each ping cannot be updated/refined through iterations. Therefore, we

expect large scene with short transmitting signals to have better performance.

In [30], the adaptive pulse compression method was modified for a multistatic case. We have men-

tioned in the previous section that the received signal at each receiver at each ping is the sum of reflections

from scatterers across several pings within the beamwidth of the array, and that transmitting signals

reflected from off-broadside scatterers reach receivers at different times, it is hard to find a reference

signal that accounts for the time shift. The idea of processing returns from multistatic radar is very

interesting. There is a difference between our problem and the issue that the paper is trying to deal with.

The paper focuses on the case where the target(s) position and angle have been accurately estimated,

and better resolution is needed to separate closed by targets. The known angle of the target(s) with

respect to the transmitters are then used to construct steering vectors to help refining estimation and

to achieve better target resolution. We do not know the exact positions of the target point(s) - we will

most likely have an extensive target scene with various incident angles. Fortunately, our system (and

any SAS systems in general) has a relatively small 3dB beamwidth with only a few pings within sight.


Tx1 Tx2

nth

receiver

RR

1nPr

2nPr

Figure 4.10: Demonstration of signal transmission and reception

The target points are therefore approximately at broadside. To start the estimation, we will use the

sum of transmitting waveforms as the reference signal. At the following iterations, we use estimations

from different pings and take their influence on the ping under estimation into account to refine h and

to make the current estimation.

To put this into mathematical terms, we first consider the case of the transmitter transmits a delta

function. Fig. 4.10 shows the transmission and reception path of the target array with respect to the

target point. R1nPl and R2nPl denotes the distance from transmitter 1 and 2 to receiver n, respectively,

where r denotes the range bin, n is the index of receiving element, P denotes the index for the current

ping. The signal received at the nth element from transmitter 1 is then

P+Pb∑p=P−Pb

xple−jkR1npl , (4.22)

where the influences of target points from Pb pings ahead and after the current ping will be taken into

account. Due to the small beamwidth, and for computational efficiency, Pb usually takes on the value

of 2 or 3.

Now if we extend this to transmitting a signal array s1 from transmitter 1 is then

P+Pb∑p=P−Pb

xTpls1p (4.23)

where s1p = s1 ⊗∑Ne−1n=0 e−jkR1np and xpl =

xplxp(l−1)

...xp(l−N+1)

.

Expanding the sum in Eqn. 4.22 we get

y1(l) = [ x(P−Pb)lx(P−Pb+1)l ··· x(P+Pb−1)l x(P+Pb)l ]

s1(P−Pb)

s1(P−Pb+1)

···s1(P+Pb−1)

s1(P+Pb)

= X(l)S1

. (4.24)

Here y1(l), the received signal, is considered as a lump sum of scatterers across pings illuminated by

transmitter 1. The received signal is y(l) = y1(l) + y2(l). The reference signal is then h = S1 + S2.

We now illustrate the concepts developed with some simulations. A Frank code of length 100 is

used to implement transmitting waveforms. With two transmitters, each transmitter transmits a 50 bit

segment of the Frank code. Fig 4.11 shows two realizations of a target scene before and after using

adaptive MMSE. The system parameters are the same as the ones from Table 3.1. At the initialization


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(a) Initialization step

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(b) Simulation after two iterations

Figure 4.11: MIMO scene processed with adaptive MMSEpulse length: 0.006second

stage, the cross-correlation is rather severe in Fig. 4.11a. Two iterations after, the target scene becomes

much clearer, and the sidelobe effects are much less pronounced, although not completely eliminated. It

is interesting to note there is less of a sidelobe effect where there are more scatterers due to destructive

interference. More iterations always help reducing sidelobes further. On the other hand, the target

scene is to be updated range bin by range bin, ping by ping. Additional iterations significantly increase

computational load. We usually use two iterations as a balance between performance and computational

efficiency.

Fig. 4.12 plots the same scene under the same setting, except with a shorter pulse that is of a third

of the pulse length used to construct Fig. 4.11. Short pulses lead to shorter duration of sidelobes to

start with, and gives cleaner image after processing.

The Frank code was divided into two segments to provide transmitting waveforms for two trans-

mitters. Although the Frank code has good autocorrelation properties, the code segments do not have

good cross-correlation properties and therefore produce severe sidelobes at the initialization step. As

the simulation results show, adaptive MMSE can greatly reduces the cross-talk effect, and produces a

relatively clean image for large scenes.

Despite successful attenuation of sidelobes via adaptive MMSE, sidelobes cannot be eliminated com-

pletely. Adaptive MMSE helps finding a good reference signal, but does not cancel the cross-correlation.

Waveform design that takes cross-correlation into account is therefore very important. Since Frank code

segments have severe sidelobes, we can expect good performance from any set of waveforms that take

low cross-correlation as one of the design criteria.

Extension to MIMO System with Multiple Transmitters

More simulations have been performed on adding additional transmitters in the array along the cross-

correlation direction. Generally speaking, more tranmitters induce more severe is the cross-talk effect.

Cross-correlation between every pair of transmitting waveforms are added up together during the pro-

cessing of the data and hence affect the final image.


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(a) Initialization step

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(b) Simulation after two iterations

Figure 4.12: MIMO scene processed with adaptive MMSE and implemented with short length pulsepulse length: 0.002 seconds

4.5 Summary

This chapter gives an evaluation of different waveforms and their applicability to a MIMO-SAS system

was analyzed based on orthogonality conditions. For a MIMO-SAS system, the transmitted waveforms

are required to occupy the same frequency band to ensure coherent integration of received signals.

With this criterion in mind, apart from the up and down chirps, there is no other waveforms that are

orthogonal and free of cross-correlation. Least squares and adaptive MMSE investigated to reduce the

effect of cross-correlation when post-processing the image. While both methods help reducing the cross-

talk at broadside, the adaptive MMSE, which iteratively update the reference signal, is more efficient at

dealing with cross-talk off-broadside.

Chapter 5

Conclusion and Future Work

In this work, we studied multiple-input multiple-output synthetic aperture sonar (MIMO-SAR) systems.

While we extended the single-input multiple-output (SIMO) existing SAS system, the biggest challenge

that we faced is the cross-talk effect between non-orthogonal waveforms.

5.1 Summary

This thesis started by elaborating the details over the construction of a SAS system simulator in Chapter

2. This simulator provides a testing ground for the construction of a MIMO-SAS system.

Compared to the SIMO-SAS system, a MIMO-SAS system offers two advantages: faster platform

speed and improved azimuth resolution at near ranges. The mathematics behind these concepts was

inversely elaborated upon in Chapter 3 Both improvements are a consequence of an elongated virtual

array. The number of phase centers in a virtual array is directly related to the maximum platform speed.

Moreover, the azimuth resolution of a system is dominated by what we call the aperture factor. At near

range however, the effective aperture size is comparable to the array size. The azimuth resolution is

hence inversely proportional to the length of the array. The azimuth resolution becomes range dependent

at near ranges. In some ways, a MIMO system is an economical alternative to a longer SIMO system.

MIMO-SAS system also comes with a side effect, the cross-talk between transmitting waveforms.

Chapter 3 introduced two methods, the least square method, which is non-adaptive, and the adaptive

MMSE method, to reduce the cross-talk. Both methods perform target estimation ping by ping, and

are less computationally efficient compared to the matched filter, but they both show better estimation

results. The least square method provides good resolution in a SIMO setting, but does not handle cross-

talk very well. The adaptive MMSE iteratively uses previous estimations to refine the matched filter to

create better estimation. It is shown to be very effective in reducing the artifacts created by cross-talk

in image scenes with extended scatterers.

5.2 Future Work

This work represents one of the first exploring MIMO concepts in a SAS system. In this regard, there

are several issues that are yet unexpected. Some examples are:

56

Chapter 5. Conclusion and Future Work 57

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(a) Matched filter processing

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(b) Least square processing

Figure 5.1: Different processing on target scene injected with white Gaussian noiseAmplitude of target points: 0 dB, 0 dB, 0 dB, -10 dB, -20 dB

noise level: -20 dB

1. Noise Tolerance: It came to our attention that matched filter is the most tolerant to white Gaussian

noise, compare to LS and MMSE. In Fig. 5.1, a five targets and white Gaussian noise scene was

generated for a SIMO system, matched filter and least square method were used to post process the

data. While all five targets are well identified after matched filter processing, the weakest target is

masked out by noise after processing with least square. It is also the case for MMSE processing.

Therefore, in order to fully explore the advantages of different processing methods, we must lower

the impact of noise.

2. 2D MIMO Systems: So far we have been focusing on increasing the platform velocity of SAS

systems, and therefore additional transmitters have been added in the cross-range direction. Po-

tentially, we can construct a 2D transmitting system with additional transmitters in the range

direction as it is the case for MIMO-SAR system. We expect 2D MIMO-SAS system to also have

widened maximum unambiguous range, on top of improved maximum platform velocity. Waveform

design will again be very important to 2D MIMO sytem. The system geometry may need careful

design to separate waveforms in the range direction.

3. Waveform Design: As we have discussed in Section 4.4, although it is possible to reduce the cross-

talk generated during the post-processing of images, cross-talk cannot be completely removed. So

it is important to use transmitting waveforms with low cross-correlation. Here are a few ideas

worth exploring.

Recently, a novel class of waveforms based on OFDM techniques has been proposed to replace the

conventional LFM chirp pulses [31]. This requires dividing the available bandwidth and assign

subsequent frequency components to either one or the other waveform. Coherence is ensured by

limiting the scene extension to less than half of the length of the transmitted OFDM waveforms.

The work of Wang [32] is particularly relevant here.

Noise modulated waveforms or waveforms obtained using heuristics have never been the focus of

study in conventional MIMO radar systems as they do not have a good ambiguity function. Now

that Doppler resolution is no more a concern, we may re-explore this option. As an example,

we have used a set of waveforms generated by Genetic algorithm [33] to produce Fig. 5.2. The

maximum of the sidelobe level is lower than the image generated by Frank code, but the shadow

Chapter 5. Conclusion and Future Work 58

rangecr

oss−

rang

e−15 −10 −5 0 5 10 15

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−4

−2

0

2

4

6

Figure 5.2: MIMO scene. Transmitting waveforms produced by Genetic algorithm

extends farther away and its amplitude do not fade fast enough. Although this set of waveforms

are not adopted, there is potential in exploring heuristics to generate desired waveforms given our

constraint.

Last but not least, Pailhas and Petillot [34] suggested a set of low cross-correlation functions using

micro-chirps.

In summary, this thesis has established the potential of MIMO concepts to enhance the efficiency

of SAS systems. In particular, the MIMO-SAS system allows for faster platform speed and improved

resolution at near ranges, and the use of adaptive MMSE can effectively mitigate problems with cross-

talk.

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