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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.
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Dedication
To my family
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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
Chapter 1. Introduction 3
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.
Chapter 1. Introduction 4
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.
Chapter 1. Introduction 5
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
Chapter 1. Introduction 6
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
Chapter 1. Introduction 7
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.
Chapter 2. SAS Simulator 10
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
Chapter 2. SAS Simulator 11
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
Chapter 2. SAS Simulator 12
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.
Chapter 2. SAS Simulator 13
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.
Chapter 2. SAS Simulator 14
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 )),
Chapter 2. SAS Simulator 15
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
Chapter 2. SAS Simulator 16
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
Chapter 2. SAS Simulator 17
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
Chapter 2. SAS Simulator 18
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
Chapter 2. SAS Simulator 19
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.
Chapter 2. SAS Simulator 20
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
Chapter 2. SAS Simulator 21
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
(a) image of raw data from one receiver element
raw data, sum of all elements
range
cros
s−ra
nge
−30 −20 −10 0 10 20 30
−80
−60
−40
−20
0
20
40
60
80
(b) image of raw data from the sum of all receiver elements
range compressed data
range
cros
s−ra
nge
−1 −0.5 0 0.5 1 1.5 2−8
−6
−4
−2
0
2
4
6
8
(c) image of range compressed data
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
(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
Chapter 2. SAS Simulator 22
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
(a) image of raw data from one receiver element
Raw data, sum of all elements
range
cros
s−ra
nge
−30 −20 −10 0 10 20 30
−80
−60
−40
−20
0
20
40
60
80
(b) image of raw data from the sum of all receiver ele-ments
range compressed data
range
cros
s−ra
nge
−1 −0.5 0 0.5 1 1.5 2−8
−6
−4
−2
0
2
4
6
8
(c) image of range compressed data
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
(d) final processed image
Figure 2.14: Intermediate Figures in Realizing One Target Point with Sway Error
micronavigation work by Nathan V. Woudenberg [21].
Chapter 2. SAS Simulator 23
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 2.15: Processed image with yaw errors with cutoff frequency = 0.05, variance = 9o.
Figure 2.16: Illustration of speckle generation across region of interest
Chapter 2. SAS Simulator 24
range
cros
s−ra
nge
background
−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 2.17: Example of generated background
range
cros
s−ra
nge
Complete Scene
−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 2.18: 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.
Chapter 3. Multiple-Input Multiple-Output SAS system 27
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
Chapter 3. Multiple-Input Multiple-Output SAS system 28
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
cros
s−ra
nge
−15 −10 −5 0 5 10 15
−30
−20
−10
0
10
20
30
(a) Velocity = 1m/s
range
cros
s−ra
nge
−15 −10 −5 0 5 10 15−40
−30
−20
−10
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).
Chapter 3. Multiple-Input Multiple-Output SAS system 29
range
cros
s−ra
nge
−15 −10 −5 0 5 10 15−40
−30
−20
−10
0
10
20
30
40
(a) SIMO system, platform velocity = 2m/sAt high speed, ghost targets appear for SIMO system
range
cros
s−ra
nge
−15 −10 −5 0 5 10 15−20
−15
−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
Chapter 3. Multiple-Input Multiple-Output SAS system 30
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
Chapter 3. Multiple-Input Multiple-Output SAS system 31
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
Chapter 3. Multiple-Input Multiple-Output SAS system 32
range
cros
s−ra
nge
−15 −10 −5 0 5 10 15
−30
−20
−10
0
10
20
30
(a) case SIMO, Velocity = 1.6m/s
range
cros
s−ra
nge
−15 −10 −5 0 5 10 15
−30
−20
−10
0
10
20
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)
Chapter 3. Multiple-Input Multiple-Output SAS system 33
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)
Chapter 3. Multiple-Input Multiple-Output SAS system 34
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.
Chapter 3. Multiple-Input Multiple-Output SAS system 35
−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
Chapter 3. Multiple-Input Multiple-Output SAS system 36
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
Chapter 3. Multiple-Input Multiple-Output SAS system 37
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
Chapter 3. Multiple-Input Multiple-Output SAS system 38
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
Chapter 3. Multiple-Input Multiple-Output SAS system 39
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].
Chapter 4. Cross-Talk Reduction and Waveform Selection 42
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.
Chapter 4. Cross-Talk Reduction and Waveform Selection 43
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
Chapter 4. Cross-Talk Reduction and Waveform Selection 44
−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
Chapter 4. Cross-Talk Reduction and Waveform Selection 45
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
Chapter 4. Cross-Talk Reduction and Waveform Selection 46
−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
Chapter 4. Cross-Talk Reduction and Waveform Selection 47
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
Chapter 4. Cross-Talk Reduction and Waveform Selection 48
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.
Chapter 4. Cross-Talk Reduction and Waveform Selection 49
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
Chapter 4. Cross-Talk Reduction and Waveform Selection 50
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)
Chapter 4. Cross-Talk Reduction and Waveform Selection 51
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0.3
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0.6
0.7
0.8
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1
(a) Cross-section of scene generated with received signalfrom SIMO system processed with matched filter
−10 −5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b) Cross-section of scene generated with received signalfrom SIMO system processed with least squares method
−10 −5 0 5 10
−0.5
0
0.5
1
1.5
2
2.5
(c) Scene generated by processing received signal fromMIMO system with least squares methodTransmitting signals are generated using 100 bits Frankcode
−10 −5 0 5 100
0.1
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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
Chapter 4. Cross-Talk Reduction and Waveform Selection 52
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.
Chapter 4. Cross-Talk Reduction and Waveform Selection 53
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
Chapter 4. Cross-Talk Reduction and Waveform Selection 54
−20 −15 −10 −5 0 5 10 15 20
−15
−10
−5
0
5
10
15
(a) Initialization step
−20 −15 −10 −5 0 5 10 15 20
−15
−10
−5
0
5
10
15
(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.
Chapter 4. Cross-Talk Reduction and Waveform Selection 55
−20 −15 −10 −5 0 5 10 15 20
−15
−10
−5
0
5
10
15
(a) Initialization step
−20 −15 −10 −5 0 5 10 15 20
−15
−10
−5
0
5
10
15
(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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range
(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
−6
−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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