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Adaptive Waveform Design and CFAR Processing for High FrequencySurface Wave Radar

by

Alison Cheeseman

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

Graduate Department of The Edward S. Rogers Sr. Department of Electrical &Computer EngineeringUniversity of Toronto

c© Copyright 2017 by Alison Cheeseman

Abstract

Adaptive Waveform Design and CFAR Processing for High Frequency Surface Wave Radar

Alison Cheeseman

Master of Applied Science

Graduate Department of The Edward S. Rogers Sr. Department of Electrical & Computer Engineering

University of Toronto

2017

High frequency surface wave radar (HFSWR), used for coastal surveillance, operates in a challenging

environment as the clutter signals returned from the ocean surface can be several orders of magnitude

larger than returns from targets. Reliably detecting small boats in severe sea states, is therefore quite

difficult.

In this thesis we take a two-fold approach to improving the detection performance of Raytheon

Canada’s third generation HFSWR system. First, we consider the design of transmit waveforms with

improved range resolution, thus reducing the area of the clutter cell. We develop an algorithm to design

practical spectrally-compliant waveforms which achieve high bandwidths while simultaneously avoiding

interference with concurrent communications users. We then propose a new detection algorithm based

on the best known statistical model for sea clutter, the K-distribution. We show that both the proposed

transmit waveforms and detection algorithm can lead to improved detection performance in a sea clutter

environment.

ii

Acknowledgements

First, I would like to thank my supervisor, Professor Raviraj Adve, for his guidance and support

throughout this project.

I would also like to acknowledge that this research has been made possible thanks to funding from

Raytheon Canada Limited, Defense Research and Development Canada (DRDC) and the Natural Sci-

ences and Engineering Research Council of Canada (NSERC). In addition, the spectrum monitoring

data used to produce Figure 2.1 was provided by DRDC and the measured data used to test the detec-

tors in Section 3.4 was provided by Raytheon Canada. In particular, I would like to thank Mr. Rick

McKerracher, of Raytheon Canada, for his valuable input and assistance throughout the completion of

this project.

iii

Contents

Acknowledgements iii

List of Tables vii

List of Figures viii

1 Introduction 1

1.1 Motivation and Problem Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The Design of Practical Spectrally-Compliant Pulses 4

2.1 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 The Waveform Design Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.1 Successive Convex Approximation and Projection Algorithm . . . . . . . . . . . . 12

2.4.2 Additional Filtering to Suppress the Out-of-band Emissions . . . . . . . . . . . . . 13

2.5 Extension of SCA-P to Multiple Near-Orthogonal Pulses . . . . . . . . . . . . . . . . . . . 15

2.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

iv

2.6.1 Comparison of Algorithms for the Single Waveform Case . . . . . . . . . . . . . . 16

2.6.2 Matched Filter Response in Sea Clutter Conditions . . . . . . . . . . . . . . . . . . 26

2.6.3 Multiple Near-Orthogonal Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7.1 Feasibility and Convergence of the SCA-P algorithm . . . . . . . . . . . . . . . . . 36

2.7.2 Alternative Exit Conditions for the SCA-P algorithm . . . . . . . . . . . . . . . . 38

2.7.3 Reduction of the ACF Peak Sidelobe Level . . . . . . . . . . . . . . . . . . . . . . 38

2.7.4 Computing the Average Spectrum of Multiple Pulses . . . . . . . . . . . . . . . . . 40

2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 CFAR Algorithms to Improve Ocean Clutter Rejection 42

3.1 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 CFAR Detectors for Heterogeneous Environments . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.1 Variability Index (VI) CFAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.2 Fuzzy VI-CFAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.3 Comparison on Simulated Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 Modifications to the Fuzzy VI-CFAR detector based on the K-distribution . . . . . . . . . 50

3.3.1 Simplified Membership Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3.2 Estimation of the K-distribution Parameters . . . . . . . . . . . . . . . . . . . . . 50

3.4 Detector Performance on Measured Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.1 Measurement Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.3 Characterizing the Clutter Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4.4 Masking Strong Clutter returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.5.1 Optimal Detector Footprint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.5.2 Optimal Masking Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

v

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 Conclusions 63

A Computational Complexity Calculations 66

A.1 SCA-P Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A.2 AP1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

A.3 AP2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

B Simulation of K-distributed Sea Clutter with Spatial Correlation 69

B.1 Uncorrelated K-distributed Clutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

B.2 Adding A Model for Spatial Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

References 71

vi

List of Tables

2.1 Execution times and resulting waveform properties for each solution method (spectral

environment 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Execution times and resulting waveform properties for each solution method (spectral

environment 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Execution times and resulting PAPR of each pulse generated using the SCA-P algorithm 30

3.1 Adaptive Threshold Generation Logic: VI-CFAR . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Lowest PFA which achieves 100% detection (Nrange = 18) . . . . . . . . . . . . . . . . . . 54

3.3 Lowest PFA which achieves 100% detection (Nrange = 6, NDopp = 3) . . . . . . . . . . . . 54

3.4 Lowest PFA which achieves 100% detection (Nrange = 6, NDopp = 11) . . . . . . . . . . . 55

3.5 Lowest PFA which achieves 100% detection (Nrange = 18, NDopp = 11) . . . . . . . . . . . 55

3.6 PFA of both detectors after clutter masking (Nrange = 6, NDopp = 3) . . . . . . . . . . . . 59

3.7 PFA of both detectors after clutter masking (Nrange = 6, NDopp = 11) . . . . . . . . . . . 59

vii

List of Figures

2.1 Typical spectral occupancy data taken on Sept. 8, 2008 (data provided courtesy of DRDC). 8

2.2 Example of the multiple pulse joint processing scheme for the case of M = 4 pulses . . . . 10

2.3 Time (left) and frequency (right) domain responses of a Kaiser window with parameter

β = 8.23 and L = 34. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Amplitude (left) and ACF of SCA-P waveform which illustrates the effect of filtering the

waveform (spectral environment 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 ESD of waveforms generated using AP1, AP2 and SCA-P algorithms after filtering with

a Kaiser window (spectral environment 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 19



2.7 ACF of waveforms generated using AP1, AP2 and SCA-P algorithms after filtering with


2.8 ACF mainlobe of waveforms generated using AP1, AP2 and SCA-P algorithms after

filtering with a Kaiser window (spectral environment 1) . . . . . . . . . . . . . . . . . . . 20

2.9 3D plot of the AF magnitude of the SCA-P generated waveform (spectral environment 1) 21

2.10 Contour plot of the AF power (in dB) of the SCA-P generated waveform (spectral envi-

ronment 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.11 ESD of waveforms generated using AP1, AP2 and SCA-P algorithms before filtering with




2.13 ACF of waveforms generated using AP1, AP2 and SCA-P algorithms after filtering with


viii

2.14 ACF mainlobe of waveforms generated using AP1, AP2 and SCA-P algorithms after

filtering with a Kaiser window (spectral environment 2) . . . . . . . . . . . . . . . . . . . 24

2.15 3D plot of the ambiguity function of the SCA-P generated waveform (spectral environment

2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.16 Contour plot of the ambiguity function of the SCA-P generated waveform (spectral envi-

ronment 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.17 ACF of SCA-P generated 170 kHz waveform compared to that of a 40 kHz LFM waveform

showing the FWHM of each peak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.18 Matched filter response of a 40 kHz LFM waveform in the presence of four targets with

K-distributed spatially correlated sea clutter in the background . . . . . . . . . . . . . . . 27

2.19 Matched filter response of SCA-P generated 170 kHz waveform (spectral environment 1)

in the presence of four targets with K-distributed spatially correlated sea clutter in the

background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.20 Matched filter response of AP1 generated 170 kHz waveform (spectral environment 1)


background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.21 Matched filter response of AP2 generated 170 kHz waveform (spectral environment 1)


background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.22 ESD of four spectrally disjoint LFM pulses used to initialize the SCA-P algorithm . . . . 29

2.23 ESD of multiple pulses generated using SCA-P algorithm after filtering with a Kaiser

window (spectral environment 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.24 Average ESD of multiple pulses generated using SCA-P algorithm after filtering with a

Kaiser window (spectral environment 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.25 Summed ACF of pulses generated using SCA-P algorithm and the mask used to constrain

the ACF (spectral environment 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.26 Summed ACF mainlobe of pulses generated using SCA-P algorithm and the mask used

to constrain the ACF (spectral environment 1) . . . . . . . . . . . . . . . . . . . . . . . . 32

2.27 Summed ACF and pulse-to-pulse cross-correlations (spectral environment 1) . . . . . . . . 33

2.28 Matched filter response after joint processing, with four targets in the first range interval

(spectral environment 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.29 Matched filter response after joint processing, with one target in each ambiguous range

interval (spectral environment 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

ix

2.30 Optimal value of the objective function from problem P3,i vs. iteration number for the

SCA-P algorithm (spectral environment 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.31 Error in the ACF vs. iteration number for the SCA-P algorithm (spectral environment 1) 37

2.32 PAPR vs. iteration number for the SCA-P algorithm (spectral environment 1) . . . . . . 37

2.33 Frequency response and ACF of a stepped frequency NLFM waveform . . . . . . . . . . . 39

2.34 ACF of the SCA-P output waveform when initialized with an NLFM waveform (compar-

ison of before and after Kaiser filtering) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.35 Frequency response of the output waveforms (filtered) when the SCA-P algorithm is ini-

tialized with 1) an LFM waveform, and 2) an NLFM waveform . . . . . . . . . . . . . . . 40

2.36 Average ESD of multiple pulses computed two different ways (spectral environment 1) . . 40

3.1 Leading and lagging windows for a general 1-dimensional CFAR detector . . . . . . . . . 43

3.2 Fuzzy membership function of the VI parameter. . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Fuzzy membership function of the mean ratio. . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 PD vs. SNR in a homogeneous environment, with PFA = 10−3. . . . . . . . . . . . . . . . 48

3.5 PD vs. SNR in the presence of a clutter edge (CNR = 5dB) in the first 10 cells, with

PFA = 10−3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6 PD vs. SNR in the presence of an interfering target (INR = SNR), with PFA = 10−3. . . 49

3.7 MSE vs. sample size (N) for ν = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.8 MSE vs. sample size (N) for ν = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.9 ROC with with detector footprint: Nrange = 6, NDopp = 11 . . . . . . . . . . . . . . . . . 55

3.10 ROC with with detector footprint: Nrange = 18, NDopp = 11 . . . . . . . . . . . . . . . . 56

3.11 Detections(red) plotted on top of the range-Doppler data for CPI 68 at a look angle of

−4 degrees with detector footprint: Nrange = 6, NDopp = 3 . . . . . . . . . . . . . . . . . 57

3.12 Detections (red) plotted on top of the range-Doppler data for CPI 68 at a look angle of

−4 degrees with detector footprint: Nrange = 6, NDopp = 11 . . . . . . . . . . . . . . . . . 58

3.13 Detections plotted on top of the range-Doppler data for CPI 68 at a look angle of −4

degrees with detector footprint: Nrange = 6, NDopp = 3 . . . . . . . . . . . . . . . . . . . 60

3.14 Detections plotted on top of the range-Doppler data for CPI 68 at a look angle of −4

degrees with detector footprint: Nrange = 6, NDopp = 11 . . . . . . . . . . . . . . . . . . . 60

x

Chapter 1

Introduction

According to the United Nations Convention on the Law of the Sea (UNCLOS), coastal nations have

sovereign rights over the 200 nautical miles (nm) of sea extending from their coast, known as the

Exclusive Economic Zone (EEZ). In return, these nations are responsible for monitoring and protecting

their EEZ, which of course leads to the question of how to effectively monitor such a vast region of

the ocean. Many existing radar technologies can be used to provide partial monitoring of the EEZ, for

instance, traditional land-based radars can be used, but are limited to ranges within their line-of-sight,

and therefore the maximum range at which targets can be detected using land-based radar is only around

50-60 km. Alternatively, a number of airborne radars can cover the full range of the EEZ, but they are

expensive to operate and cannot provide persistent monitoring [1].

Raytheon Canada, in collaboration with the Canadian Department of National Defence, has de-

veloped an optimal solution to this problem, an integrated maritime surveillance (IMS) system which

combines data from a number of complementary sensors. The primary sensor is a network of high fre-

quency surface-wave radars (HFSWR). HFSWR refers to a class of radar which operates in the HF band

(3 MHz to 30 MHz) and uses surface-wave propagation to effectively “see” beyond the horizon. The

HFSWR system has the capability to track both surface and airborne targets at ranges which surpass

the 200 nm boundary of the EEZ [1].

The HFSWR system operates in pulse-Doppler mode, meaning that the radar transmits a coherent

train of pulses and uses Doppler properties, i.e. the resulting shift in frequency due to target motion,

to further classify detections based on their velocity. Targets which are illuminated by the radar signal

reflect these pulses back to the radar, where the echoes are received by an array of antennas. The

received signals are processed and the resulting response is then compared against a detection threshold

to determine whether or not a target is present. Each detection is forwarded to a tracking algorithm

which combines consecutive detections into tracks of a single target.

1

Chapter 1. Introduction 2

1.1 Motivation and Problem Overview

The received signals at the array include target returns in the presence of interference from a number

of sources. The radar signal processor is responsible for determining the presence of targets amidst the

interference. In a coastal environment, possible sources of interference include external noise, returns

from the ocean surface (ocean clutter), ionospheric reflection (ionospheric clutter), and interference from

other users of the HF band. Of particular interest here, are the clutter returns, referring to returns

from objects not of interest to the radar, in this case the ocean surface, and ionospheric reflection. The

clutter response is dependent on the design of the transmit waveform, unlike other external sources of

interference, such as electronic noise.

The coastal environment that the HFSWR system operates in is particularly challenging, as the

clutter signals returned from the ocean surface can be several orders of magnitude stronger than typical

target returns. Additionally, the behaviour of ocean clutter is unique in that it is non-Gaussian, and

typically follows a heavy-tailed, or “spiky” distribution caused by the variation of the local ocean surface

shape [2]. These strong clutter returns lead to both target masking, where a real target is hidden by a

stronger clutter return, and false alarms, where the signal processor mistakes the clutter for a target.

The parameters of the clutter distribution are highly dependent on a number of environmental factors,

and thus are changing in time as the weather changes. This dynamic nature of the ocean clutter means

we do not necessarily have prior knowledge of the interference distribution and must make an educated

guess from the changing data. This, combined with the strong clutter power, makes it difficult to reject

the clutter returns while still detecting weak targets, leading to high rates of false alarm, particularly

when trying to detect small vessels in severe sea states. Large numbers of false alarms take up resources

which can lead to the tracker system being overloaded, and valid target detections being missed.

The types of vessels that may be detected by the HFSWR system can be classified into two main

categories; smaller vessels (typically less than 1000 tons) whose maximum detection range is limited

by the presence of ocean clutter, and larger vessels (typically greater than 1000 tons) whose detection

range is limited by the presence of external noise [3]. In the latter case, the detection range of large

vessels can be improved simply by increasing the radiated power, however in the case of detecting smaller

vessels, the signal-to-interference-plus-noise ratio (SINR) can only be improved by improving the range

resolution to effectively reduce the area of the radar clutter cell and thus reduce the clutter power. The

current HFSWR system is able to transmit at high enough power levels that the presence of external

noise is not a huge concern, and the detection performance is primarily limited by strong clutter returns

masking smaller vessels. Therefore, the focus of this project is to improve the ability of the HFSWR

system to detect small vessels, particularly in severe sea states, when the clutter power is strongest.

Two approaches to doing so are taken in this thesis; first we consider designing the transmit waveform

in a way that will increase the effective bandwidth and therefore reduce the size of the clutter cell, while

taking into account some strict spectral constraints which have previously limited the bandwidths which

can be transmitted by the system at hand. Second, we consider a receive processing algorithm which

improves on methods in the literature which are typically based on a Gaussian model for the background

interference. We propose a modified detector using a more accurate model for ocean clutter to achieve

Chapter 1. Introduction 3

better clutter rejection.

This research has been undertaken in the context of Raytheon Canada’s third generation HFSWR

system for Persistent Active Surveillance of the EEZ (PASE) located near Halifax, Nova Scotia [4].

Specifically, the PASE system operates as a secondary user in the HF band and must meet certain

limitations on the amount of interference it may cause to primary users in the same band. These

spectral constraints must be taken into consideration when designing transmit waveforms as in order to

achieve higher effective bandwidths it will be necessary to design waveforms which span across frequency

bands that are in use. Additionally, measured data taken with this system is used to test the performance

of various detection algorithms in realistic sea clutter conditions.

1.2 Organization of Thesis

This thesis is organized as follows. In Chapter 2 we discuss the design of spectrally-compliant waveforms

and develop an algorithm to design practical waveforms which achieve high bandwidths while simulta-

neously avoiding interference with communications users. We start with the simple case of designing

a single waveform and then extend the algorithm to the design of multiple pulses which can be jointly

processed to increase the unambiguous range of the system. In Chapter 3, we change our focus to receive

processing, introducing a new detection algorithm which estimates the parameters of the background

distribution using the best known statistical model for sea clutter amplitudes from the recent literature.

Chapter 4 concludes the thesis and offers suggestions for future work on this project.

1.3 Notation

Throughout this thesis we adopt the notation of using lower case italics for scalars (e.g. x), lower case

boldface for column vectors (e.g. x), and upper case boldface for matrices (e.g. X). The complex

conjugate, and conjugate transpose are denoted by (·)∗ and (·)H , respectively. Time-reversal of a signal

is denoted by (·). The p-norm of x will be written as ||x||p and the convolution of two vectors, x and

y, is denoted by x ∗ y. The letter j will be used to represent the imaginary unit, e.g. j =√−1, and for

any complex number x, |x| and ∠x represent the modulus and argument of x, respectively.

Chapter 2

The Design of Practical

Spectrally-Compliant Pulses

In this chapter we deal with the design of the radar transmit waveform for the HFSWR system at hand.

In a pulsed radar system, the system alternates between transmitting signals for short time duration,

known as the pulse width(T ), and receiving the returned signals while the transmitter is turned off. The

number of these cycles the radar completes each second is called the pulse repetition frequency (PRF).

The properties of the transmitted pulses play a key role in determining the detection performance of the

radar system, specifically the accuracy, resolution, and ambiguity of determining the range and radial

velocity of the target. It is important to note here that we typically discuss radar signals in terms of

their complex envelope; in the following sections when we discuss the radar waveform we will be referring

to its complex envelope.

The range of a target is determined by the delay of the returned signal. The relationship between

the range R and the signal delay τ is given by R = cτ/2, where c is the speed of light [5]. Typically,

radar systems use a matched filter (MF) to measure the delay of the returned signal. This means that

the returned signal is filtered with a time-delayed and complex conjugated version of itself, the result

being that the entire energy of the signal is concentrated into a single output peak at a predetermined

additional delay. The MF is optimal in terms of producing the highest output SNR, which is why it

is typically used. The output of the MF is also known as the autocorrelation function (ACF) of the

waveform, and is a very useful tool in characterizing the response of the waveform. The peak level of the

ACF depends only on the signal energy, however the shape of the response, i.e. peak width and sidelobe

properties, are highly dependent on the waveform itself. In general, the peak width of the ACF, which is

a measure of the range resolution, is inversely proportional to the waveform bandwidth [6]. This means

that by increasing the waveform bandwidth, the range resolution of the waveform is improved.

The radial velocity of a target is determined from the Doppler shift of the returned signal. The

Doppler shift is defined as the difference between the carrier frequency of the received signal, fR, and

the transmit signal, f0. We have the following relationship between the Doppler shift, fd, and the target

4

Chapter 2. The Design of Practical Spectrally-Compliant Pulses 5

radial velocity, fd = fR − f0 ≈ −2v/λ, where v is the radial velocity, and λ is the wavelength of the

transmit waveform [5]. The Doppler resolution is a function of the total signal duration. Typically, a

single pulse has poor Doppler resolution, however a train of multiple pulses processed coherently will

exhibit good Doppler resolution. To properly characterize the response of a waveform in terms of delay

and Doppler we look at the MF response as a function of both τ and v, which is known as the ambiguity

function (AF). In the following sections we will use both the ACF and AF as measures of waveform

performance.

In the case of a pulsed radar, the AF also illustrates the ambiguity of the range and Doppler mea-

surements. For instance, if the delay at which the pulse is received is greater than the interpulse period

(i.e. the time between pulses) the signal will not be received before the next pulse is transmitted. This

results in a range ambiguity, as the received pulse could be the result of the pulse which was just trans-

mitted, or a reflection from a previously transmitted pulse. The unambiguous range of a radar system

is given by Rua = c/(2PRF ), which is the maximum range at which the radar can measure target range

with no ambiguity. Unfortunately, there is a trade-off between the unambiguous range and the maxi-

mum Doppler shift which can be measured unambiguously. The maximum unambiguous Doppler shift

is proportional to the PRF, given by fua = ±PRF/2 [6], therefore in many practical applications no

single PRF can achieve the desired unambiguous region in both range and Doppler. There are, however,

techniques, such as the introduction of diversity between pulses, which can be leveraged to increase the

unambiguous range while maintaining the PRF. This will be discussed further in the following section

when we consider the design of multiple near-orthogonal pulses.

The particular radar system in question operates as a secondary user in the heavily congested HF

band. The transmitted signals are required to adhere to strict regulations on the interference they can

cause to concurrent communication links, meaning that the frequency spectrum must be designed so as

to avoid certain frequency bands. To ensure non-interference, the current HFSWR system includes a Pro-

active Remote Intelligent Spectrum Management system (PRISMs) which monitors channel occupancy

over the entire HF band and prevents transmission in a number of barred frequency bands. The PRISMs

has the ability to dynamically switch the radar carrier frequency, and adjust the carrier bandwidth, to

fit within the widest contiguous open frequency band available for use [4]. Spectral occupancy data

suggests that when the spectrum is particularly congested, which typically occurs overnight, the highest

available bandwidth is only in the 15-40 kHz range. In order to improve performance we would like

to design practical transmit waveforms with increased bandwidths in order to achieve improved range

resolution and therefore reduced clutter power due to the smaller size of the clutter cell. To achieve

this in a congested spectral environment, the resulting waveforms must span multiple disjoint frequency

bands, while meeting strict power constraints in multiple occupied frequency bands.

The remainder of this chapter is organized as follows. Section 2.1 contains a survey of the relevant

literature and 2.1 introduces the relevant system model. In Section 2.3 the problem of generating a

single feasible waveform is formulated mathematically as a non-convex optimization problem, and in

Section 2.4 an iterative solution to the problem is presented. In Section 2.5 the proposed algorithm is

extended to the design of multiple spectrally-compliant pulses with low pulse-to-pulse cross correlation,

and Section 2.6 presents some illustrative numerical results for both the case of a single waveform and

multiple pulses. Section 2.7 includes a discussion on the results in this chapter, and Section 2.8 provides


a summary of the chapter.

2.1 Literature Survey

Over the past few years, as demand for available bandwidth increases, spectral coexistence between

radar and communications systems has been gaining more attention among the research community. As

such, many approaches have been proposed in the recent literature for the design transmit waveforms

with disjoint spectral support. In [7], the problem is formulated as an optimization problem to maximize

the SINR with spectral constraints and a similarity constraint to a known good waveform, hoping to

retain some of the good properties in the resulting waveform. A similar approach is taken in [8] with

the addition of a constant modulus constraint. Both of these approaches are shown to be effective at

generating waveforms which meet given spectral constraints, however they do not allow any control over

the waveform ACF, instead relying on the convex similarity constraint to enforce some properties of

the reference waveform onto the solution. For our application we require greater control over the ACF,

therefore this approach is not suitable for our purposes.

Another prevalent approach in the literature is to use the method of alternating projections (AP).

Starting with a good template waveform, the AP approach alternately projects the waveform onto a set

meeting the required constraints in the frequency and time domains. In [9,10] the dynamic range of the

waveform is constrained in the time domain, and the spectrum in the frequency domain. In [11] a similar

method is proposed, with the addition of a constraint on the waveform periodic ACF. Again, in [12,13] a

method is proposed to constrain the amplitude and frequency spectrum through a series of AP, however

the authors propose a method to null specified frequency bands in the time domain, eliminating the need

to compute the discrete time Fourier transform (DTFT) within each iteration. This method tends to

converge much faster than the fast Fourier transform (FFT)-based methods, however completely nulling

the frequency bands does result in the unnecessary loss of phase information on each iteration. In [14]

and [15], the AP algorithm is used to enforce frequency gaps while shaping the spectrum with a Gaussian

window to maintain low ACF sidelobes.

The main drawback to the alternating projections algorithms is that there is no guarantee of con-

vergence to the desired waveform. In fact, we tested these algorithms with our required constraint sets

and found that when the constraints were made particularly strict we observed extremely long run times

before convergence to a solution. Often, allowing the waveform to have some amplitude modulation,

instead of forcing a constant amplitude solution, would lead to faster convergence, however the amount

of amplitude modulation necessary cannot be known a priori. This leads us to the approach which is

developed in this thesis. We formulate the design problem as an optimization problem and propose a

hybrid approach which combines iterative convex approximation to minimize the amplitude modulation

along with alternating projections onto the constraint set of the non-convex ACF constraint. Through-

out, we will use two AP-based methods as a baseline against which to compare our proposed algorithm;

the first being the method described in [11], referred to throughout as AP1, and the second, AP2, being

the frequency nulling method in [13] with the addition of an FFT-based projection to constrain the ACF.


2.2 System Model

We consider a pulsed radar system which transmits M distinct pulses in a given coherent processing inter-

val (CPI), and we let xm(t) be the baseband equivalent of the mth transmitted radar pulse. The number

of pulses, M , is chosen based on the system PRF in order to achieve the desired unambiguous range.

For convenience, we will work with the length-N discrete time radar code, xm = [x(1), ..., x(N)]T ∈ CN ,

assumed to be sampled at a sufficiently high sampling rate; that is, greater than the Nyquist rate. Each

pulse has length T , in seconds, given by T = N/Fs, where Fs is the rate at which the pulse is sampled.

We will design xm to meet a set of constraints in both the frequency and time domain, and then

recover the continuous time pulse, xm(t), from xm by interpolating the radar code with an appropriate

band-limited pulse. The corresponding transmitted pulse is modulated by the desired carrier frequency

before transmission.

2.3 The Waveform Design Problem

In this section the specific constraints imposed on the waveform are described in more detail, and

the waveform design problem is formulated as an optimization problem. This problem is found to

be non-convex and thus cannot be solved efficiently using existing methods. It is necessary to find a

good convex approximation and solve the resulting problem iteratively. This is done by combining two

known techniques from the relevant literature, successive convex approximation (SCA) to minimize the

amplitude modulation, and projections onto the ACF constraint set, as in the AP algorithm, to enforce

a constraint on the shape of the ACF.

2.3.1 Design Criteria

The HFSWR system at hand imposes some specific spectral constraints that have not been considered

in the previous literature. As we will see, the spectral environment is extremely challenging, the allowed

power level in the occupied frequency bands is very low, and occupied channels can be extremely narrow

in frequency and closely spaced. Additional regulations imposed by the Canadian Government also

require the waveform to meet strict constraints on the out-of-band emissions.

In addition to the aforementioned strict spectral constraints, we must also ensure that the designed

waveforms meet the following practical constraints:

1. minimal amplitude modulation

2. low ACF peak sidelobe level (PSL)

3. low pulse-to-pulse cross correlation


in order to be useful for the HFSWR system. The following sections describe both the spectral and

waveform constraints in more detail.

Finally, since the spectral environment is constantly changing, the proposed algorithm must be able

to generate a feasible waveform in real time, which for this application means within a few minutes, the

duration of one CPI . In this regard, we propose an iterative method which alternates between solving

successive convex approximations to an optimization problem, and projections onto the ACF constraint

set.

Spectral Constraints

Figure 2.1: Typical spectral occupancy data taken on Sept. 8, 2008 (data provided courtesy of DRDC).

The radar in question is a secondary user in the HF band, and as such, it operates in a very restrictive

spectral environment. It must avoid transmitting in occupied bands currently in use by primary radiators,

and also avoid a number of permanently prohibited frequency bands, ranging in width from a few kHz to

hundreds of kHz. Figure 2.1 presents one measurement of the spectral occupancy (using measurements

taken overnight in 2008), indicating the typical distribution of in-use channels when the spectrum is

particularly congested. We observe that occupied channels are typically 1 to 6 kHz wide, and may be

spaced as close as within a few kHz of each other. The relative transmissions in an occupied band must

be below -60 dB.

Not shown in Figure 1 are the additional permanently prohibited frequency bands which must also

be considered. The location of these prohibited bands is fixed and known ahead of time, typical gaps

between them being on the order of hundreds of kHz. Therefore, it is possible to achieve sufficiently

high bandwidths by transmitting a waveform which fits in between two prohibited bands and has low

out-of-band emissions. Specifically, the requirement is that the out-of-band emissions must also be at

least -60 dB below the maximum power, measured at twice the one-sided bandwidth away from DC.

Such low out-of-band emissions can be achieved by filtering the waveform, therefore this constraint is left

out of the waveform design problem for now, and simply enforced afterwards by filtering the designed

waveform.


Waveform Constraints

In addition to the spectral constraints which are required according to governmental regulations, the

following practical constraints on the radar waveform in the time domain should also be considered in

order for the designed waveform to be useful.

Amplitude Modulation: Most radar systems use non-linear power amplifiers operating at saturation

to achieve high power levels. Variable amplitude can only be achieved by backing off the peak power

level. For this reason, constant amplitude signals are highly desirable as radar waveforms. However, it

is not always possible to design a waveform which meets strict constraints on the frequency spectrum

or matched filter response without some amplitude modulation. For instance, there is often a trade-off

between amplitude modulation and ACF sidelobe levels, as well as the rate of spectral sidelobe decay [5].

For the HFSWR system at hand, it is reasonable to operate at slightly lower power levels and allow

for some amplitude modulation in order to improve the spectral and pulse compression properties of

the waveform. For this reason, we seek to minimize the amplitude modulation, as opposed to forcing a

constant amplitude waveform, as is done in most of the literature on this subject.

The amplitude modulation can be minimized by minimizing the waveform peak-to-average-power

ratio (PAPR), given by

PAPR(x) =||x||2∞1N ||x||

22

(2.1)

where x∞ denotes the l∞-norm of x, i.e. its peak value.

We will see later that the additional spectral shaping to meet the requirement on the out-of-band

emissions, which we achieve through filtering of the waveform, also necessitates some additional ampli-

tude modulation. In an environment with such strict spectral constraints this is unavoidable, however

we will target a final PAPR of approximately 3 dB or less for the final waveform, as this has shown to

be a reasonable level of amplitude modulation on the current system.

ACF Sidelobes: The HFSWR system must deal with large and dynamic clutter returns from the sea.

As such, the waveform ACF should have a low PSL to avoid target masking from high energy clutter

returns. We can write the nth element of the ACF as

g(n) = xHJnx, ∀ n ∈ −(N − 1), ..., N − 1 (2.2)

where the nth shift matrix Jn is given by [16]

Jn(l,m) =

1 if m− l = n

0 if m− l 6= n(2.3)

and then the corresponding PSL is given by

maxn∈1,...,N−1

|xHJHn x|2. (2.4)


Orthogonal Pulses: Radar transmissions in the HF band can propagate over very long distances, and

with substantial energy returns from these distances. Typically it is not possible to choose a PRF which

achieves a large enough unambiguous range for this application, while simultaneously resolving Doppler

ambiguities [17]. It is, however, possible to achieve a large unambiguous range by transmitting multiple

near-orthogonal pulses. By transmitting M near-orthogonal pulses and using a joint processing scheme

to differentiate between returns from ambiguous range intervals, the unambiguous range is effectively

increased by a factor of M . The processing scheme we use is detailed in [18] and requires all M pulses

to have low pulse-to-pulse correlation.

t

x0

x1

x2

x3

x2

x3

Processing Interval

Channel: 0

1

2

3

x0

x3

x2

x1

x1

x0

x3

x2

x2

x1

x0

x3

x3

x2

x1

x0

Matched filter to:

Figure 2.2: Example of the multiple pulse joint processing scheme for the case of M = 4 pulses

Figure 2.2 illustrates the joint processing scheme for a scenario with M = 4 pulses. We see that the

pulses are transmitted sequentially and the return signals are processed during the indicated processing

interval with filters matched to each of the M codes. After all M pulses have been transmitted, the

responses on each channel are time aligned and summed to get the overall channel response. The result

of using this scheme is that channel 0 is matched to the first range interval, and rejects the second, third,

and fourth time around returns due to the low cross-correlation between pulses. Similarly, channels 1,

2, and 3 are matched to the second, third, and fourth time around returns, respectively.

In practice, codes already exist with orthogonal pulse-to-pulse correlations, such as the Frank and

P4 codes described in [18]. These types of codes are relatively easy to generate, have orthogonal cross-

correlations and low ACF sidelobes, however there is, unfortunately, no simple way to modify these

pre-existing codes to meet the specific spectral constraints we require while still maintaining their other

desirable properties. Therefore, we must find a new way to design pulses which simultaneously have low

pulse-to-pulse correlation and span multiple disjoint frequency bands.

2.3.2 Problem Formulation

We first formulate the problem for the case of a single waveform, with discrete radar code x, spanning the

entire available bandwidth. The extension to designing multiple near-orthogonal pulses will be discussed

in a later section. We consider a spectral environment with K occupied frequency bands, denoted by

Ωk =[fk1 fk2

]∀ k = 1, ...,K. To constrain the power in each of the occupied bands, Ωk, we can enforce


an upper bound on the energy transmitted in the kth occupied frequency band, which is given by

∫ fk2

fk1

Sx(f)df = xHRksx (2.5)

where Sx(f) is the energy spectral density (ESD) of x and Rks is the N ×N matrix given by [7]

Rks (l,m) =

fk2 − fk1 if l = m

ej2πfk2 (l−m)−ej2πf

k1 (l−m)

j2π(l−m) if l 6= m(2.6)

Combining all of the aforementioned constraints leads to the following optimization problem in which

the PAPR of x is minimized subject to constraints on the spectral occupancy and PSL

P1 :

minx∈CN

||x||2∞1N ||x||

22

xHRksx ≤ Ek, k = 1, ...,K

|xHJHn x|2 ≤ γu(n), n = 1, ..., N − 1

(2.7)

where Ek is an upper bound on the energy transmitted in the kth occupied frequency band, and γu is a

vector defining upper bounds on each element of the ACF.

Problem P1 can be simplified by realizing that we can minimize the PAPR of x by equivalently

minimizing the maximum value, or l∞-norm, of x subject to a lower bound on the total energy. The

lower bound on the total energy is included to avoid the trivial solution of a waveform with zero energy.

The following is the modified problem

P2 :

minx∈CN ||x||2∞s.t. xHx ≥ E0

xHRksx ≤ Ek, k = 1, ...,K

|xHJHn x|2 ≤ γu(n), n = 1, ..., N − 1

(2.8)

where the second constraint is the additional constraint, with E0 the energy lower bound. The second

and third constraints impose the spectral and PSL constraints respectively.

Unfortunately, problem P2 is non-convex as a result of both the non-convex quadratic energy con-

straint, and the PSL constraint, which is, in general, non-convex, as shown in [19]. In order to solve

P2 efficiently, we must look for an approximation to the problem that can be solved with reasonable

computation time. In many cases, non-convex problems can be approximated by a convex optimization

problem, which can then be solved for a globally optimal solution in polynomial time using interior

point methods. Some common approaches in the literature for dealing with non-convex problems in-

clude convex relaxations, randomization algorithms, and iterative algorithms [20–22]. Here we will take

an iterative approach to solving the problem.


2.4 Solution Method

In this section an approximate solution method to problem P2 is proposed. The proposed method is

based on successive convex approximations and alternate projections onto the constraint set, and has

been shown in simulation to converge to feasible waveforms with nearly constant amplitude in realistic

time frames. We then detail the filtering method used to achieve sufficient suppression of the out-of-band

emissions of the output waveform.

2.4.1 Successive Convex Approximation and Projection Algorithm

In order to design feasible waveforms in real time, we must be able to solve problem P2 efficiently;

we therefore must look for a convex approximation to the problem. First, we deal with the non-

convex quadratic energy constraint, as these type of constraints occur quite extensively in the literature.

From [21, 22] we see that one way to deal with such constraints is to use successive convex approxima-

tion (SCA). The idea behind the SCA method is to find a convex approximation for the non-convex

constraints in the problem, and solve the problem iteratively until convergence. As in [21] we can use

the following relationship

xHx ≥ 2RezHx

− zHz (2.9)

which holds for any reference waveform, z, to construct a convex lower bound approximation (CLBA)

for the non-convex quadratic energy constraint. If we ignore the PSL constraint for the time being, and

take the reference waveform to be the solution to the previous iteration of the optimization, then on the

ith iteration we have the following convex optimization problem:

P3,i :

minxi∈CN ||x||2∞s.t. 2Re

xHi xi−1

− xHi−1xi−1 ≥ E0

xHi Rksxi ≤ Ek, k = 1, ...,K

(2.10)

which can be reformulated as a second order cone program (SOCP) and solved in polynomial time via

interior point methods using, for example, the MATLAB toolbox CVX [23]. As a result of Equation (2.9)

any solution of P3,i will also satisfy the original non-convex energy constraint, that is xHi xi ≥ E0, and

therefore it can be shown that when problem P3,i is solved iteratively the optimal value of the objective

function will be non-increasing.

The PSL constraint, unfortunately, cannot be dealt with using the same approach, as even if we were

to find a convex approximation for this constraint, the dimension of this constraint becomes prohibitively

large for practical choices of N , resulting in long execution times for each algorithm iteration. We,

therefore, instead choose to incorporate this constraint by alternately solving P3,1 and projecting the


solution onto the ACF constraint set using the following projection,

(PGgx)n =

γu(n)∠gx(n) if |gx(n)| > γu(n)

gx(n) otherwise(2.11)

where gx denotes the ACF of x and γu is a vector which defines an upper bound on the value of the

ACF at each delay. The ACF projection can be executed efficiently, even for large values of N , by using

the FFT and inverse FFT (IFFT) operators, as in the AP1 method described in [11].

Combining the SCA method and the alternate ACF projections, we end up with what we will call

the SCA-P algorithm, Algorithm 1, which alternates between successive convex approximations and

projections onto the ACF constraint set. Algorithm 1 shows explicitly how the ACF projection is

computed efficiently and the resulting waveform with desired ACF is recovered. The computational

complexity of Algorithm 1, the SCA-P algorithm, is dominated by the worst-case complexity of solving

the resultant SOCP, P3,i, on each iteration, which turns out to be O(N3.5). For a detailed explanation

of this calculation, see Appendix A.

Algorithm 1 Alternate Successive Convex Approximation and Projection (SCA-P) Al-gorithm1: input xinit, γu, ε2: initialize x0 ← xinit, i← 0, maxdev ← (ε+ 1)3: while maxdev > ε do4: i← i+ 15: Compute DTFT: Xi−1 ← FFT (xi−1)6: Compute ACF: gxi−1 ← IFFT

(|Xi−1|2

)7: ACF Projection: gxi−1

← PGgxi−1

8: Set Xi−1 ← FFT(gxi−1

) 12 · ∠Xi−1

9: Set xi−1 ← IFFT (Xi−1)10: Solve P3,i, denote optimal solution by xi11: Compute ACF: gxi12: Compute max. deviation: maxdev ← maxn (gxi(n)− γu(n))13: end while

There are two key innovations in the SCA-P algorithm, as compared to other algorithms in the

literature. The first is the minimization of the PAPR, via the l∞-norm of the waveform, within each

iteration using a convex approximation; the second is enforcing the PSL constraint as a projection on

each iteration. As a result of this unique approach the output waveform on each iteration meets the

energy constraint in the occupied frequency bands, has minimal amplitude modulation, and should have

low ACF sidelobes as a result of the projection. It is also expected that the waveform ACF will continue

to improve with each additional iteration, similar to alternating projections based methods. This will

be confirmed by simulation results shown in a later section.

2.4.2 Additional Filtering to Suppress the Out-of-band Emissions

The final step to ensuring that the resulting waveform is suitable for the HFSWR application is to filter

the waveform with a Kaiser window having the appropriate parameters to suppress the out-of-band


emissions to −60 dB. The Kaiser window is defined by the following expression [24],

wK(n) =

I0(β(1−((n−α)/α)2)1/2)

I0(β) 0 ≤ n ≤ L− 1

0 otherwise(2.12)

where L is the length of the window, α = (L−1)/2 and I0(·) is the zeroth-order modified Bessel function

of the first kind. As can be seen in Equation (2.12), the Kaiser window is defined by two parameters,

β and L, which control the trade-off between main-lobe width and relative sidelobe amplitude. These

parameters should be computed for each output waveform of Algorithm 1 based on the measured peak

value of the out-of-band emissions prior to windowing. They can be computed using [24]

β =

0 Asl ≤ 13.26

0.76609(Asl − 13.26)0.4 + 0.09834(Asl − 13.26) 13.26 < Asl ≤ 60

0.12438(Asl + 6.3) 60 < Asl ≤ 120

(2.13)

L ≈ 24π(Asl + 12)

155∆ml+ 1 (2.14)

where Asl is the desired level of sidelobe suppression (in dB) of the window, and ∆ml is the desired

mainlobe width, given in radians. For our application we choose ∆ml so that the first null occurs at

twice the one-sided bandwidth of the designed waveform. Using the values for β from Equation (2.13)

will produce windows with sidelobe levels accurate to within 0.36 dB from the value of Asl [24].

Figure 2.3 shows the time and frequency domain response of a Kaiser window with parameters given

by β = 8.23 and L = 34. The waveforms output by the SCA-P algorithm are filtered by convolving

the time domain response of the filter with the given time domain waveform. The result is that the

frequency domain response of the window is multiplied by the frequency response of the given waveform.

For a given output waveform, the window parameters are chosen by measuring the transmitted power

at twice the one-sided bandwidth and determining the necessary amount of additional suppression to

lower the out-of-band emissions down to −60 dB below the peak power level.

Now, in practice, the application of a Kaiser window to suppress the out-of-band emissions to such

a low level has the unfortunate side effect of increasing both the waveform PAPR and the ACF sidelobe

levels of the filtered waveform. When enforcing such strict spectral constraints, some amount of ampli-

tude modulation is inevitable, and as was previously mentioned, can be handled by the system at hand.

In simulations, the resulting waveforms were found to have PAPR levels within reason for this applica-

tion. Similarly, although high range sidelobes are not desirable, the ESD and ACF are Fourier-transform

pairs, meaning that if we enforce a specific shape on the spectrum, a limitation is also enforced on the

shape of the ACF. We will see later that this results in an increase in the PSL after filtering.


Figure 2.3: Time (left) and frequency (right) domain responses of a Kaiser window with parameterβ = 8.23 and L = 34.

2.5 Extension of SCA-P to Multiple Near-Orthogonal Pulses

The SCA-P algorithm can be used to generate a feasible waveform which spans multiple disjoint frequency

bands, however we still must apply this to the construction of multiple pulses with low pulse-to-pulse

cross-correlation. The pulses will be processed according to the scheme in [18] where the filtered responses

corresponding to each individual pulse are averaged to get the final response. This means, that in order

to achieve good range resolution, the total effective bandwidth of the combined pulses should be high.

To see how the design of multiple near-orthogonal pulses can be achieved it is helpful to consider the

simplest case, of two continuous time pulses, x0(t) and x1(t). Their pulse-to-pulse cross-correlation is

given by

gx0,x1= x0 ∗ x∗1, (2.15)

which has frequency domain representation

Gx0,x1(f) = X0(f)X∗1 (f) (2.16)

and therefore, by Parseval’s Theorem, the total energy in the cross-correlation is given by

EGx0,x1 =

∫ fs/2

−fs/2|Gx0,x1

(f)|2 =

∫ fs/2

−fs/2|X0(f)|2|X1(f)|2df (2.17)

where fs is the sampling frequency. According to Equation (2.17) we see that in the ideal case where

the pulses have zero cross-correlation, the pulses must have disjoint support in the frequency domain.

This result also holds for the more general case of M orthogonal pulses.


More realistically, we can design M near-orthogonal pulses, i.e. with low pulse-to-pulse cross-

correlation, by designing M feasible waveforms with low spectral overlap. Each pulse can be designed

individually using the SCA-P algorithm, initialized with a set of spectrally disjoint (or nearly disjoint)

pulses, that span the total available bandwidth. The pulses must be designed so that the average

spectrum and average ACF meet the spectral and PSL constraints, respectively.

2.6 Numerical Results

In this section the performance of the SCA-P algorithm is assessed for the case of a radar transmitting

between two prohibited frequency bands spaced 170 kHz apart. For the single waveform case, the

algorithm is initialized with a linear frequency modulated (LFM) waveform, chosen for its good pulse

compression properties, with a two-sided bandwidth of 170 kHz and pulse length T = 750 µs. When

we deal with the extension to multiple pulses, the available 170 kHz is split amongst four spectrally

disjoint LFM pulses. The sampling rate is set to 2 MHz, i.e., the discrete time radar code has N = 1500

samples. Throughout this report we will consider two sets of occupied frequency bands, based on the

spectral occupancy data shown in Figure 2.1, which will be referred to as spectral environments 1 and

2. The spectral environments are defined as follows:

1. Ω1 = [−45.01,−44.00], Ω2 = [−36.88,−33.82] and Ω3 = [9.92, 12.97]

2. Ω1 = [−37.25,−35.22] and Ω2 = [21.75, 24.80]

where the frequencies are given in kHz at baseband. In both environments the ACF constraint set is

defined by a mask, γu, with a PSL of −30 dB and mainlobe width equal to that of the initial LFM

waveform. The ACF constraint is required to be satisfied to within a tolerance of ε = 10−1.5 on the

magnitude of the ACF, which corresponds to a tolerance of about 6 dB on the ACF power. The emissions

in the occupied frequency bands are constrained to be -60 dB below the energy of the initial waveform,

E0. The SCA-P algorithm, and all other algorithms included for comparison, are implemented on a

desktop PC using MATLAB. The optimization problem on each iteration of the SCA-P algorithm is

solved using the CVX toolbox.

2.6.1 Comparison of Algorithms for the Single Waveform Case

Here we consider the design one single waveform which occupies the entire 170 kHz bandwith while

avoiding the occupied frequency bands. The resulting waveform generated using the proposed SCA-P

algorithm is compared to waveforms generated using two variations of the alternating projections algo-

rithm from the relevant literature, which we refer to as AP1 and AP2. Here, AP1 refers to the method

described in [11] which uses FFT-based projections, and AP2, the method in [13] where the spectral

constraint is imposed by a projection in the time domain that nulls certain frequency components of the

waveform (and their derivatives). We include a slight modification to method AP2 to also include the

ACF projection using the FFT operation. We choose here to compare our algorithm to these alternating


projections based algorithms as they offer similar control over the spectral occupancy, ACF shape, and

amplitude modulation, resulting in waveforms which meet the same constraint sets prior to filtering,

allowing for a fair comparison. Both the AP1 and AP2 algorithms have complexity O(N logN) per

iteration, however the AP2 algorithm has been observed in practice to converge in significantly fewer it-

erations. A more detailed description of both algorithms and a complete derivation of the computational

complexities may be found in Appendix A. All three algorithms are compared based on their respective

execution times and the properties of the resultant waveforms.

The output waveform from each algorithm is required to satisfy the spectral constraint exactly, and

meet the ACF constraint within the prescribed tolerance, ε. The AP1 and AP2 algorithms are such

that the resulting waveform will be exactly constant amplitude, whereas the SCA-P algorithm only

minimizes the PAPR and the resulting waveform may have some amplitude modulation. In practice, for

both spectral environments, the SCA-P algorithm was found to converge to a near constant amplitude

waveform; in fact the resulting waveform has a PAPR within 10−4 of the ideal constant amplitude case

in both environments. As will be seen in the following sections, applying a Kaiser window to all of the

resulting waveforms causes some additional amplitude modulation. The amount of resulting amplitude

modulation depends on the specific filter parameters needed for each individual waveform.

Spectral Environment 1

All three algorithms were used to design feasible waveforms for the first spectral environment, with

three occupied frequency bands in the available bandwidth. Table 2.1 summarizes the results, giving

the execution time of the algorithm, and some key performance metrics. It should be noted here that

all algorithms were implemented in MATLAB R2014a and run on a desktop PC; therefore the following

execution times are included as a comparison between algorithms, not an indication of expected run times

using optimized software/hardware. The PC used was equipped with an Intel Core i7-4790 processor

with 16 GB of RAM and running a 64-bit version of Windows 7. We see here that the AP2 algorithm

is by far the fastest, with the SCA-P algorithm next, followed by the AP1 algorithm, which converges

much slower. Both the SCA-P and AP2 algorithms converge within a few minutes, and therefore would

be suitable for use in real time, however we will see that the SCA-P waveform has much more desirable

properties. The AP1 algorithm is found to take nearly 20 minutes to converge to a feasible waveform,

significantly longer than the other two algorithms.

Algorithm Execution Time Kaiser Window Parameters PAPR ISLR

SCA-P 2.96 mins L = 28, β = 6.62 2.51 dB -5.43 dBAP1 19.84 mins L = 30, β = 7.02 3.53 dB -5.26 dBAP2 0.50 sec L = 31, β = 7.24 4.65 dB -1.53 dB

Table 2.1: Execution times and resulting waveform properties for each solution method (spectral envi-ronment 1)

Table 2.1 also lists the necessary Kaiser window parameters required to suppress the out-of-band

emissions to −60 dB for each waveform, the resulting PAPR and the integrated sidelobe ratio (ISLR)

of each waveform after filtering. Each waveform is filtered with a different Kaiser window depending


on the level of the out-of-band emissions originally. This is why we see in Table 2.1 that although each

algorithm initially converged to a constant, or nearly constant, amplitude waveform, the resulting PAPR

values after filtering to reduce the out-of-band emissions are different for each waveform. It can be seen

that the SCA-P waveform required the least amount of filtering, and therefore this waveform has the

lowest resulting PAPR, and also the lowest ISLR. Specifically, only the SCA-P waveform meets our

target PAPR of less than 3 dB. The AP2 algorithm, although it converges much faster than the other

algorithms, results in a much poorer waveform, with much higher PAPR and ISLR, as compared to both

of the other algorithms.

Figure 2.4: Amplitude (left) and ACF of SCA-P waveform which illustrates the effect of filtering thewaveform (spectral environment 1)

The ISLR is included as a measure of the ratio of the power in the ACF sidelobes to the power in

the mainlobe, and is given by [25]

ISLR(g) =

∑α |g(k)|2∑θ |g(k)|2

(2.18)

where g is the waveform ACF, the sets α and θ are defined as the lags corresponding to the ACF sidelobes

and ACF mainlobe, respectively,

α := −N + 1 ≤ k < −τ ∪ τ < k < N − 1

θ := −τ ≤ k ≤ τ

and τ is half the width of the mainlobe in samples. We find that the application of the Kaiser window to

the output waveform tends to increase the PSL, however the resulting ISLR is not changed significantly,

sometimes increasing slightly, other times decreasing. Although the ISLR has not been specifically

considered in the formulation of the problem, a low ISLR is desirable as this corresponds to the total

power in the ACF sidelobes, which as with the PSL, affects the clutter response of the waveform.

Figure 2.4 is an illustration of the effect of applying the Kaiser window to the SCA-P waveform on

both the waveform amplitude and ACF sidelobes. The plot on the left shows the amplitude of the filtered

waveform. We can see that the filtering has introduced a significant amount of amplitude modulation,


as the original waveform was essentially constant amplitude. On the right hand side, we have the ACF

before and after the filter is applied, showing that the filter does increase the PSL, but also reduces the

sidelobe levels further from the mainlobe, resulting in minimal change to the overall ISLR. Similar results

are observed for the AP1 and AP2 waveforms as well, however the exact effect of the filter will depend

on both the necessary filter parameters and the phase modulation of each waveform. In particular, we

observe that when the parameter β is larger, indicating more out-of-band suppression due the filter, the

PAPR is increased more significantly.

Figure 2.5: ESD of waveforms generated using AP1, AP2 and SCA-P algorithms after filtering with aKaiser window (spectral environment 1)

Figure 2.6: ESD of waveforms generated using AP1, AP2 and SCA-P algorithms after filtering with a

Kaiser window (spectral environment 1)

Figures 2.5 and 2.6 are plots showing the ESD of all three waveforms, comparing the spectral shape

before and after filtering with a Kaiser window. The red vertical lines indicate twice the one-sided

bandwidth, the point at which the out-of-band emissions are measured. From Figure 2.5 we can see that

the SCA-P waveform has the lowest out-of-band emissions initially, which is why the resulting waveform


is filtered less and therefore has lower PAPR. Figure 2.6 is a plot of the ESD of all three waveforms after

they have been filtered using a Kaiser window, and here we can see that all three waveforms meet the

spectral constraints on both transmission in occupied bands, as well as the out-of-band emissions. We

also see that the AP1 waveform has very uneven transmission levels in the passband, and in particular,

has very low transmission levels in between the two leftmost notches, which are spaced within less than

10 kHz of each other. The result is that this waveform has a lower effective bandwidth than the other

two waveforms, the effects of which are seen when we look at the ACF.

Figure 2.7: ACF of waveforms generated using AP1, AP2 and SCA-P algorithms after filtering with aKaiser window (spectral environment 1)

Figure 2.8: ACF mainlobe of waveforms generated using AP1, AP2 and SCA-P algorithms after filtering

with a Kaiser window (spectral environment 1)

Figures 2.7 and 2.8 show the ACF of each waveform for comparison. Figure 2.7 confirms what was

seen in Table 2.1; that the SCA-P waveform has the lowest ISLR, followed by the AP1 waveform, and

the AP2 waveform with the highest ISLR. We can also see that although the initial waveforms were

constrained to have a PSL below −30 dB, the use of the Kaiser window has raised the sidelobes by


around 10 dB. In Figure 2.8, a closer look at the mainlobe allows us to see that the PSL of the AP2

waveform is significantly higher than that of the other two waveforms. The SCA-P and AP1 waveforms

have a similar PSL of slightly below −20 dB. We can also observe that the AP2 waveform has a noticeably

wider mainlobe than the other two waveforms. This can be attributed to the lower effective bandwidth

of this waveform.

Figures 2.9 and 2.10 show the AF of the SCA-P waveform. The AF is shown here as a function of

the signal delay τ and Doppler shift ν. We use the following definition of the AF,

χ(τ, ν) =

∫ ∞−∞

x(t)x∗(t+ τ) exp(j2πνt) dt (2.19)

where x(t) is the complex envelope of the signal. For the HFSWR application, typical Doppler shifts

of moving targets are on the order of at most 10 Hz. Looking at both Figures 2.9 and 2.10 we can see

that the MF response of the waveform is nearly identical for all Doppler shifts shown, up to shifts of 50

Hz. This means that the response from targets moving at reasonable velocities will have minimal SNR

degradation and low range sidelobes. In fact, over a range of 50 Hz, the mainlobe amplitude decreases

by less than 0.1 dB and the range sidelobe level is nearly equivalent to that of a stationary target. As

was previously mentioned, in HFSWR, sufficient Doppler resolution is achieved by processing multiple

pulses coherently. As such, the AF properties, shown in Figure 2.9 for a single pulse are adequate for our

purposes. It is also noted here that the AF has similar properties to those of the initialization waveform,

in this case a 170 khz LFM waveform.

Figure 2.9: 3D plot of the AF magnitude of the SCA-P generated waveform (spectral environment 1)


Figure 2.10: Contour plot of the AF power (in dB) of the SCA-P generated waveform (spectral environ-ment 1)

Spectral Environment 2

Again, the AP1, AP2, and SCA-P algorithms were used to design feasible waveforms, this time for the

second spectral environment. Table 2.2 summarizes the results. Here we see very similar execution times

and waveform properties as in the first spectral environment. In general, execution times are shorter

and the waveform properties are slightly better, which is expected as this spectral environment has only

two occupied frequency bands, as compared to three in the first spectral environment. Typically, having

fewer occupied bands present in the environment has been found to result in shorter execution times

for all three algorithms. In addition, when fewer iterations are required for convergence, the resulting

waveform bears greater resemblance to the initial waveform, and tends to have better pulse compression

properties.

Algorithm Execution Time Kaiser Window Parameters PAPR ISLR

SCA-P 1.69 mins L = 29, β = 6.64 2.57 dB -6.64 dBAP1 8.33 mins L = 30, β = 7.12 3.35 dB -6.68 dBAP2 0.75 sec L = 29, β = 6.89 4.00 dB -3.33 dB

Table 2.2: Execution times and resulting waveform properties for each solution method (spectral envi-ronment 2)

Once again, we see that only the SCA-P and AP2 algorithms are able to generate a feasible waveform

in a reasonable time frame, and the resulting SCA-P waveform has much more desirable waveform

properties overall. We also see that the SCA-P waveform is again the only waveform to have a PAPR

below our 3 dB target, and the AP2 waveform has the highest PAPR, meaning in order to transmit this

waveform, the amplifier would have to operate significanly below its maximum efficiency.

Figures 2.11, 2.12, 2.13 and 2.14 plot the ESD and ACF of the resulting waveforms, confirming what

is shown in Table 2.2. As before, we can see that all waveforms meet the required spectral constraints,

and the resultant PSL of the ACF is approximately −20 dB after the Kaiser filter has been applied.


Figures 2.15 and 2.16 show the ambiguity function of the SCA-P waveform in this environment, and

similar to the previous case, we see that the SCA-P waveform has excellent Doppler tolerance over

typical frequency shifts observed by the HFSWR system.

It should be noted here that, as in the previous spectral environment, the AP2 waveform is found to

once again have the highest PAPR and ISLR, however the difference in performance is not as significant

in this case. This is likely a result of the fact that in this spectral environment there are fewer spectral

notches, and the notches are spaced further apart in frequency. If we compare the ESD in each environ-

ment, Figures 2.6 and 2.12, we can see that all waveforms use the available bandwidth more effectively

in the second environment, which results in a more consistent performance across all algorithms. From

these results, and additional simulations of other spectral environments, we observe that the SCA-P has

the most significant performance gain over the AP-based algorithms in severe spectral environments, as

the resulting waveform is better able to use the available bandwidth

Figure 2.11: ESD of waveforms generated using AP1, AP2 and SCA-P algorithms before filtering witha Kaiser window (spectral environment 2)

Figure 2.12: ESD of waveforms generated using AP1, AP2 and SCA-P algorithms after filtering with aKaiser window (spectral environment 2)


Figure 2.13: ACF of waveforms generated using AP1, AP2 and SCA-P algorithms after filtering with aKaiser window (spectral environment 2)

Figure 2.14: ACF mainlobe of waveforms generated using AP1, AP2 and SCA-P algorithms after filteringwith a Kaiser window (spectral environment 2)


Figure 2.15: 3D plot of the ambiguity function of the SCA-P generated waveform (spectral environment2)

Figure 2.16: Contour plot of the ambiguity function of the SCA-P generated waveform (spectral envi-ronment 2)


2.6.2 Matched Filter Response in Sea Clutter Conditions

As was previously mentioned, it is expected that the improved range resolution of the proposed higher

bandwidth waveforms will also improve the signal-to-clutter ratio (SCR) by decreasing the size of the

clutter cell. In this section this is confirmed through simulations comparing the matched filter response

of all three 170 kHz waveforms generated using the SCA-P, AP1 and AP2 algorithms with that of a

lower bandwidth 40 kHz LFM waveform. As we typically observe gaps of around 15− 40 kHz between

occupied bands, this 40 kHz waveform is representative of the largest bandwidth pulse that could be

transmitted using the current system in a congested spectral environment. For a fair comparison, all

waveforms have been filtered with a Kaiser window so as to meet the necessary requirements on the

out-of-band emissions.

Figure 2.17: ACF of SCA-P generated 170 kHz waveform compared to that of a 40 kHz LFM waveformshowing the FWHM of each peak

We begin by illustrating the improved range resolution of the 170 kHz waveform. Figure 2.17 is a

plot showing the ACF of both the 40 kHz LFM waveform and the 170 kHz SCA-P waveform, where it

can clearly be seen that the higher bandwidth waveform has a much narrower mainlobe, as expected. In

fact, the full width at half maximum (FWHM) of the 170 kHz waveform is found to be 3.92 times that

of the 40 kHz LFM waveform. This is consistent with what we expect since the effective bandwidth of

the SCA-P waveform should be approximately four times larger than that of the 40 kHz LFM waveform

once we account for some gaps due to the frequency notches.

We now illustrate the reduction in clutter power due to the reduced extent of the range cell. Since

the effective clutter is the integrated returns in a single cell, we expect a factor of 4 reduction in clutter

power. In the literature, it is widely accepted that radar sea-clutter can generally be modelled as a

compound non-Gaussian random process. One of the most popular models for sea-clutter amplitude is

the K-distribution [2]. The K-distribution is a compound distribution which consists of a fast-varying

complex Gaussian “speckle” component and a more slowly-varying Gamma distributed component which

represents the local sea clutter power. The pdf of the clutter amplitude, E, under the K-distribution is


given by

P (E) =4b(ν+1)/2Eν

Γ(ν)Kν−1(2E

√b) (2.20)

where ν is the shape parameter, which defines the “spikiness” of the clutter, and b, the scale parameter,

is inversely proportional to the mean clutter power [2, 26].

Figure 2.18: Matched filter response of a 40 kHz LFM waveform in the presence of four targets withK-distributed spatially correlated sea clutter in the background

Figure 2.19: Matched filter response of SCA-P generated 170 kHz waveform (spectral environment 1) inthe presence of four targets with K-distributed spatially correlated sea clutter in the background

Figures 2.18, 2.19, 2.20 and 2.21 plot the MF response of all four waveforms, respectively, in a

simulated sea clutter environment with four targets at the following ranges: 75 km, 153.75 km, 165

km, and 168.75 km. The background sea clutter is K-distributed with parameters given by ν = 3 and

b = 300. The Gamma-distributed local power is spatially correlated with correlation coefficient given

by ρk = e−k/10, k = 0, 1, 2, ..., N − 1. The method described in [26] was used to generate the correlated

Gamma-distributed power, from which the overall K-distributed clutter amplitudes were generated.


More details on how to simulate spatially correlated clutter amplitudes which follow a K-distribution

can be found in Appendix B.

First, we compare the response of the 40 kHz LFM waveform, Figure 2.18 to the higher bandwidth

waveform generated by the SCA-P algorithm in 2.19 in order to observe the effect of the increased

bandwidth. Two key observations can be made from these plots. First, we see that the improved range

resolution of the 170 kHz waveform means that the two targets at 165 km and 168.75 km can be resolved

separately, whereas the 40 kHz waveform merges these two targets. The second observation is that in

Figure 2.19 we see that all four targets can be easily distinguished above the clutter, however in Figure

2.18 the SCR is significantly lower and only the two strongest targets are discernible above the clutter

response. It turns out that when the average clutter power is computed over 10000 samples containing

only sea clutter returns the difference in clutter power between the response of the two waveforms is

found to be approximately 7 dB, confirming this expected benefit of using a higher bandwidth waveform.

Figure 2.20: Matched filter response of AP1 generated 170 kHz waveform (spectral environment 1) inthe presence of four targets with K-distributed spatially correlated sea clutter in the background

Figure 2.21: Matched filter response of AP2 generated 170 kHz waveform (spectral environment 1) inthe presence of four targets with K-distributed spatially correlated sea clutter in the background


In Figures 2.20 and 2.21 we plot the response of the waveforms resulting from the AP1 and AP2

algorithms, respectively. We see that the AP1 waveform has a very similar response to the SCA-P wave-

form, as expected since these waveforms had similar ACF properties. In the case of the AP2 waveform,

we observe that the wider mainlobe and higher sidelobe level does lead to a noticeable degradation in

the matched filter response in the presence of clutter. In Figure 2.21 we see that the SCR is worse than

the two previous waveforms, and the weaker targets are difficult to distinguish from the strong clutter

returns.

2.6.3 Multiple Near-Orthogonal Pulses

In this section the SCA-P algorithm is used to design multiple near-orthogonal pulses which meet the

same spectral and PSL constraints as the waveforms in the previous section. The number of pulses is

taken to be M = 4 which gives sufficient unambiguous range for this application [17]. We include results

from both spectral environments, however plots are only included for the first environment. The SCA-P

algorithm is initialized with four LFM pulses, each with a bandwidth of 42.5 kHz, and shifted so that

they span the total available bandwidth of 170 kHz, as shown in Figure 2.22.

Figure 2.22: ESD of four spectrally disjoint LFM pulses used to initialize the SCA-P algorithm

Since we will be considering the averaged ACF and ESD for the case of multiple pulses, we need to

consider how the PSL and spectral constraints might need to change to reflect this. The ACF is simple,

as at a delay of zero the ACF is always real, and equal to the waveform energy. Since all M pulses

have the same energy, the maximum value of the averaged ACF will be equal to the maximum value of

each individual pulse’s ACF, and thus the average PSL will be less than or equal to the maximum PSL

across all of the individual pulses. Unfortunately, taking the average ESD is a bit more complicated

since the pulses are spectrally disjoint, and do not necessarily add in phase. If we consider the worst

case scenario, that the waveforms are perfectly disjoint, i.e. the average passband power is 1M times that

of an individual waveform, and the stopband power levels are exactly −60 dB, then if we take E to be

the average power level in the passband of a single pulse, the average power (in dB) in each stopband


will be

Es = 10 log10

(10−6E2

E2/M2

)= 20 log10M − 60 (2.21)

which corresponds to an increase of approximately 12 dB for the case of M = 4.

Now, this is the worst-case scenario; in practice, the pulses are not perfectly disjoint and the stopband

power levels of the resulting waveforms are at most −60 dB, but typically are at least a few dB lower.

For the case of 4 pulses it has been observed that constraining the stopband power levels to −60 dB is

sufficient. It is expected that if the number of pulses were increased this would no longer be the case

and a stricter constraint would need to be enforced on each pulse in order for the average spectrum to

be feasible.

Table 2.3 gives the time required to generate each pulse using the SCA-P algorithm and the resulting

PAPR of each pulse after being filtered with a Kaiser window for both spectral environments. The

execution time for each pulse is found to depend on the location of the occupied frequency band within

that pulse. Typically, pulses with more occupied bands within their passband, such as pulse 1 in the

first spectral environment, take longer to generate, however all pulses are still generated within a few

minutes.

Spectral Env. Pulse Num. Execution Time PAPR

1

0 2.34 mins 3.00 dB1 4.10 mins 3.51 dB2 1.98 mins 3.23 dB3 2.52 mins 3.62 dB

2

0 1.91 mins 3.05 dB1 1.77 mins 3.13 dB2 1.34 mins 2.78 dB3 1.88 mins 2.43 dB

Table 2.3: Execution times and resulting PAPR of each pulse generated using the SCA-P algorithm

Once all M pulses have been designed to meet the spectral and ACF constraints, it is again necessary

to filter the waveforms so that the average spectrum meets the constraint on the out-of-band emissions.

The filter can, once again, be designed using Equations (2.13) and (2.14), however since the pulses should

be the same length we must design one single filter for all M pulses. The way this is achieved is to shift

all M pulses to baseband, and design a filter with the first null at fnull = B2 + B

2M , which is the distance

(in frequency) between the center of the outermost-pulse and the total bandwidth, B. We then design

a filter such that the maximum sidelobe level of any pulse is at or below -60 dB. As seen in Figure

2.23, this filter serves a dual purpose. Not only does this ensure that the average out-of-band emissions

are below -60 dB, but also reduces the spectral overlap between pulses which reduces the pulse-to-pulse

correlation. Figure 2.24 shows the average spectrum of all 4 pulses where it can be seen that the spectral

constraints are indeed satisfied on average.


Figure 2.23: ESD of multiple pulses generated using SCA-P algorithm after filtering with a Kaiserwindow (spectral environment 1)

Figure 2.24: Average ESD of multiple pulses generated using SCA-P algorithm after filtering with a

Kaiser window (spectral environment 1)

Figures 2.25 and 2.26 show the summed ACF of all four pulses and the mask which was used to

constrain each pulse. These plots were produced after applying a Kaiser window, which is why the

sidelobe level is seen to surpass the −30 dB constraint level. Prior to filtering, the constraint was met to

within the specified tolerance, ε ≈ 6 dB. In Figure 2.26 we see that the mainlobe width of the summed

pulses is much narrower than the mainlobe of ACF mask, due to the increased bandwidth obtained by

summing all M pulses, and that this leads to fairly high range sidelobes near the mainlobe, as they are

not constrained by the mask. This is one unfortunate side effect of constraining each pulse individually,

as the mask must be designed according to the lower bandwidth of each individual pulse. The summed

pulses have a combined bandwidth of approximately B, and thus when they are jointly processed they

should achieve a range resolution comparable to that of a single waveform with bandwidth B.


Figure 2.25: Summed ACF of pulses generated using SCA-P algorithm and the mask used to constrainthe ACF (spectral environment 1)

Figure 2.26: Summed ACF mainlobe of pulses generated using SCA-P algorithm and the mask used to

constrain the ACF (spectral environment 1)

In practice, the pulses will be processed using the joint processing scheme described previously [18].

For the case shown here of M = 4 this means that returns will be processed on 4 channels, corresponding

to 4 previously ambiguous range intervals. Figure 2.27 shows the summed ACF along with the various

summed pulse-to-pulse correlations for each possible pulse shift. These plots correspond to the filter

responses on each channel for a target in the first range interval under the joint processing scheme.

We see that the pulse-to-pulse correlations are low for all pulse shifts, as desired, in fact they are

generally equal to, or lower than the ACF sidelobes. It should be noted that the returns on Channel

2, corresponding to third time around returns, have significantly lower power than the other channels.

This is because this channel collects the filtered responses from the pulses which are furthest apart in

frequency, and therefore have the least spectral overlap.


Figure 2.27: Summed ACF and pulse-to-pulse cross-correlations (spectral environment 1)

The following figures show the filtered, time-aligned and summed response on each of the four channels

when four targets are present in the environment and again, the background clutter is modelled by a

spatially correlated K-distribution. In Figure 2.28 all four targets are in the first range interval, at the

same locations as in the previous simulation for a single waveform. Figure 2.29 again shows the same

four channel responses, however in this scenario the targets have been relocated so that there is one

target present in each ambiguous range interval. Here, we can see that each target is resolved on the

corresponding channel and not on any of the other channels, as desired. This allows for range ambiguities

to be resolved, effectively multiplying the nominal unambiguous range by the number of pulses, M .


Figure 2.28: Matched filter response after joint processing, with four targets in the first range interval(spectral environment 1)


Figure 2.29: Matched filter response after joint processing, with one target in each ambiguous rangeinterval (spectral environment 1)


2.7 Discussion

In this chapter we have proposed a new algorithm for the generation of spectrally disjoint waveforms

for HFSWR, and have shown through simulations that the proposed SCA-P algorithm is capable of

generating feasible waveforms for this application. In this section some aspects of the SCA-P algorithm

which have been previously glossed over are discussed in greater detail.

2.7.1 Feasibility and Convergence of the SCA-P algorithm

In the previous sections we did not discuss the feasibility of the SOCP, however we observe that the

probability of any iteration resulting in an infeasible problem will be negligible. For instance, given any

starting waveform, x0, so long as x0 has a non-zero transmission level in at least one of the unoccu-

pied frequency bands a feasible solution to problem P3,i can be constructed from x0 by projecting the

transmission level in all occupied bands down to −60 dB and increasing the power in any one of the un-

occupied bands until the linear energy constraint is met. In order to have a high probability of feasibility

on each iteration we initialize the algorithm with an LFM waveform which transmits nearly equal power

in each frequency band within the useable bandwidth. As a result of the choice of starting waveform,

and the energy constraint, which acts somewhat as a similarity constraint to the reference waveform, it

is highly unlikely that any iteration will result in an optimization problem that is not feasible.

Figure 2.30: Optimal value of the objective function from problem P3,i vs. iteration number for theSCA-P algorithm (spectral environment 1)

Now, given that the resultant SOCP on each iteration should be feasible for reasonable choices of

initial reference waveform, we should also consider the convergence of the overall algorithm. As was

previously mentioned, the SCA algorithm to approximate a non-convex constraint has the property that

the optimal value is non-increasing on each iteration. This non-increasing property of the optimal value

results from the fact that the solution of the previous iteration will always be feasible for the current

iteration. Unfortunately, the addition of the ACF projection on each iteration means that this no longer

necessarily holds and there is no longer a guarantee that the optimal value will be non-increasing. In


practice, however, the optimal value has been observed to be non-increasing in simulations using practical

constraint sets for our HFSWR system.

Figure 2.31: Error in the ACF vs. iteration number for the SCA-P algorithm (spectral environment 1)

Figure 2.30 plots the value of the objective function output from CVX on each iteration for the

first spectral environment, i.e., ||xi||∞, which shows that although there is no theoretical guarantee,

the optimal value does in fact decrease with each additional iteration. This same trend was observed

for all other constraint sets we implemented. In fact, we were unable to find a counter-example in

which the optimal value increased on any given iteration, although that is not to say that one does not

exist. We also observe, in Figure 2.31, that the l2 distance between the current ACF and the previous

iteration’s ACF, d(gi − gi−1), decreases on each iteration. This is consistent with what we expect from

the alternating projections algorithm [11–13], however in this case there is no guarantee that we will still

see this behaviour. As it turns out, this same behaviour was also observed in all simulations using our

constraint sets.

Figure 2.32: PAPR vs. iteration number for the SCA-P algorithm (spectral environment 1)

Figure 2.32 shows the resultant PAPR on each iteration, again for the first spectral environment.


Here we see that the PAPR does increase on some iterations, however the general behaviour is that it

tends to decrease with additional iterations. It is possible for the PAPR to increase, even as the value of

the objective function is decreasing, because the approximation of the energy constraint means that the

total energy may also decrease between iterations, leading to a higher PAPR, despite a lower maximum

value. As the algorithm converges, the approximation becomes closer to the desired energy constraint,

and we expect to see the PAPR level out, which is what we do see here. Similar results were observed

for other spectral environments which were implemented. Based on these observations, it is expected

that so long as the initialization waveform used is reasonable, i.e. spans the appropriate bandwidth, the

algorithm should converge to a waveform with low amplitude modulation.

2.7.2 Alternative Exit Conditions for the SCA-P algorithm

In the previous section, all algorithms were terminated when the maximum deviation between the wave-

form ACF and the ACF mask was within a set tolerance level, ε, in order to achieve a fair comparison

between algorithms. In practice, the SCA-P algorithm could be implemented with a number of different

exit conditions, depending on what is most important for the given application.

For instance, if having an exactly constant amplitude waveform is the most important property for a

given application, the algorithm could be set to terminate based on the PAPR value instead, or a hybrid

condition based on both the ACF and PAPR. In both of these cases, of course, there is no guarantee

that the algorithm will converge to within the specified tolerance, so a second condition should be added

to exit after a given number of iterations, or run time.

For the HFSWR application, where the spectral environment is dynamic, and feasible waveforms

must be generated in real time, it may make the most sense to use a time-based condition to exit the

algorithm. Figures 2.30, 2.31 and 2.32, above, show that in general the output waveform of the SCA-

P improves with respect to PAPR and ACF PSL with additional iterations of the SCA-P algorithm,

therefore when a change is detected in the spectral environment, the best choice might be just to run the

SCA-P algorithm until the end of the current CPI, at which point the new waveform will be transmitted.

In this way, all of the available time can be used to improve the waveform properties.

2.7.3 Reduction of the ACF Peak Sidelobe Level

In the simulations shown in this report we are able to achieve peak ACF sidelobe levels as low as −20

dB, an improvement over the standard LFM waveform. For the HFSWR environment, which is clutter-

limited, it is desirable to have the range sidelobes as low as possible to avoid interference from clutter

sources. Unfortunately, there is a trade-off between spectral shape, and ACF shape, as the two are

Fourier-transform pairs. We see this effect when we attempt to constrain the ACF even further to have

a lower PSL. This can be achieved prior to filtering, however the application of the Kaiser window to

meet the out-of-band emission constraints increases the sidelobes, bringing the PSL back up to around

−20 dB or so.

Typical techniques to reduce range sidelobes, such as windowing or the addition of non-linear fre-


quency modulation (NLFM) are all based on spectral shaping, and so are not compatible with the strict

spectral constraints which our waveform must satisfy. The results in [14,15] are a good example of how

spectral shaping can be used to significantly reduce range sidelobes, however we see that the result is

out-of-band emissions which are much too high for this application. As an example, we consider initializ-

ing the SCA-P algorithm with a stepped frequency NLFM waveform. This waveform has approximately

the same time-bandwidth product as the 170 kHz LFM waveform, and much lower ACF sidelobes, as

shown in Figure 2.33, however the spectral shape is such that the out-of-band emissions are quite high.

As a result, even if the PSL is constrained to −40 dB, the application of the Kaiser window increase the

sidelobe levels up to around −20 dB, as seen in Figure 2.34.

Figure 2.33: Frequency response and ACF of a stepped frequency NLFM waveform

Figure 2.34: ACF of the SCA-P output waveform when initialized with an NLFM waveform (comparisonof before and after Kaiser filtering)

Based on these results, it is reasonable to assume that the spectral shape of the Kaiser window

enforces a limitation on the minimum PSL which can be achieved. In Figure 2.35 we can see that the

spectral output of the SCA-P waveform initialized with an NLFM waveform after filtering is nearly

identical to that of the LFM initialization, which explains the similarity in the resulting MF responses.


Further work in this area could include looking into other window shapes which might allow for lower

ACF sidelobe levels while still meeting the out-of-band emission requirements, however it is unlikely that

this will yield significant improvements since the strict constraint on the out-of-band emissions does not

leave a lot of flexibility in potential spectral shapes.

Figure 2.35: Frequency response of the output waveforms (filtered) when the SCA-P algorithm is ini-tialized with 1) an LFM waveform, and 2) an NLFM waveform

2.7.4 Computing the Average Spectrum of Multiple Pulses

In the previous section, the average spectrum was computed by summing the average voltage output

of each pulse and then squaring the result to get the average power, however the regulations from the

Canadian Government are not clear as to how the spectral occupancy is actually measured in practice,

therefore we should account for other possible scenarios. We could realistically consider two different

Figure 2.36: Average ESD of multiple pulses computed two different ways (spectral environment 1)

methods of measuring the average spectrum, the first being to average the voltages, and then square the


result to get power, as was used in the previous section. The second method is to compute the power of

each pulse and then take the average of the power values. Specifically, we can write the average power

using each method as

Pavg,1(f) =

(1

M

M∑i=1

|Xi(f)|

)2

(2.22)

and

Pavg,2(f) =

(1

M

M∑i=1

|Xi(f)|2)

(2.23)

respectively.

Figure 2.36 shows the average spectrum computed using both methods. We observe that the two

average spectra are very similar, in fact they almost perfectly overlap. Additionally, both spectra meet

the constraints, so it is reasonable to assume that the proposed method for designing multiple pulses

will be sufficient regardless of which method is actually used to measure the average spectrum.

2.8 Summary

In this chapter we developed an iterative algorithm for the generation of practical spectrally-compliant

waveforms for HFSWR. Using the proposed SCA-P algorithm, we were able to generate feasible wave-

forms with effective bandwidths of 170 kHz, nearly four times the current system capability. The SCA-P

algorithm was compared to two AP-based algorithms from the recent literature, and was found to give

better performance in terms of convergence speed, PAPR and matched filter response. We also addressed

the use of additional filtering techniques to meet constraints enforced on the out-of-band emission level

and discussed some of the trade-offs involved with meeting such strict constraints.

We confirmed through simulation of a sea clutter environment that the new waveforms do, in fact,

have improved range resolution, leading to a significant reduction in clutter power, as compared to

a lower bandwidth waveform. Finally, we extended the SCA-P algorithm to the case of multiple near-

orthogonal waveforms and showed that by transmitting four such pulses we are able to effectively increase

the unambiguous range by a factor of four while maintaining the necessary PRF.

Chapter 3

CFAR Algorithms to Improve Ocean

Clutter Rejection

The environment in which the HFSWR system operates is particularly challenging due to the dynamic

nature of the interfering signals. The ocean clutter spectrum, which results from the interaction of

the radar waveform with the ocean waves, can vary significantly depending on the sea state (weather

conditions) and is therefore hard to model ahead of time [1,3]. Trying to detect smaller vessels among all

of this interference results in a large number of false detections due to the strong clutter returns. False

detections are costly to a radar system as they trigger the system to take action and use up resources,

in this case tracking capability. When the sea state is particularly calm, the tracking algorithm may

be sufficient to reject clutter returns, however in severe sea states the amount of false detections can

overwhelm the tracker, leading to missed detections. For this reason we seek a detection scheme which

can achieve a low false alarm rate even in the presence of strong sea clutter returns.

In a radar system, targets are detected by comparing the radar measurements to a threshold, and

declaring a target present if the measurements exceed the threshold. We typically characterize a detector

in terms of the likelihood of detecting a target, or the probability of detection (PD), and the probability

of false alarm (PFA). Given a threshold, PD and PFA can be computed from the target-plus-interference

and interference-only probability density functions (PDFs), respectively [6].

Ideally, we want to choose the detection threshold so as to maximize PD for a given PFA, using what

is called the Neyman-Pearson (NP) detector. The NP detector is derived under the assumption that

the interference is independent and identically distributed (IID) and the parameters of the interference

PDF are known. In many practical cases, however, the parameters of the background distribution are

unknown and may be changing in time and/or space, and therefore the NP detector is no longer useful,

as it requires explicit knowledge of the interference PDF [6].

When the interference statistics are changing, this often leads to changes in the false alarm rate.

False alarms cause the radar to take action, using up finite resources. A large number of false alarms

can overload the system, and may result in dropper detections. When the interference statistics are

42

Chapter 3. CFAR Algorithms to Improve Ocean Clutter Rejection 43

CUT G GG G

Leading

Window

Lagging

Window

CFAR Window

Figure 3.1: Leading and lagging windows for a general 1-dimensional CFAR detector

unknown, or may be changing, it is desirable for a detector to maintain a constant false alarm rate,

or fixed PFA. Constant false alarm rate (CFAR) detectors, are detectors which do exactly this, by

estimating the interference statistics from the measured data and dynamically computing the necessary

detection threshold to achieve a desired PFA [6]. Figure 3.1 illustrates the basic architecture of a simple

one-dimensional CFAR detector. The CFAR window consists of the cell under test (CUT), a number of

guard cells (G) surrounding the CUT, and the leading and lagging reference windows from which the

interference statistics are estimated. It is possible to extend this framework to data which spans multiple

dimensions, which will be discussed in a later section, however for now we will consider the simple case

of a detector operating on one-dimensional data.

The simplest CFAR detector is a cell-averaging (CA) CFAR detector. The CA-CFAR detector uses

the data in two reference windows, the “leading” and “lagging” windows, surrounding the CUT to obtain

an estimate of the local interference power. Often guard cells, denoted as G in Figure 3.1, are included

between the CUT and the reference windows as target responses may span multiple cells. The threshold

is then computed based on this estimate so as to maintain a desired PFA. In homogeneous environments

the CA-CFAR detector can achieve near-optimal detection performance. We consider a homonogenous

environment to be one which meets the following assumptions:

1. The interference in the reference windows and in the CUT is IID.

2. When a target is present in the CUT, there are no targets present in either reference window to

bias the threshold.

In practice, it is difficult to meet these conditions, and thus a number of robust CFAR detectors exist to

deal with specific heterogeneous conditions. For instance, a greatest-of (GO) CFAR detector addresses

clutter edge false alarms when the interference power undergoes a sudden change by estimating the

interference power only from the reference window with the largest mean. Similarly, smallest-of (SO)

CFAR takes the estimate from the smaller window and is effective in cases where another target is

present in one of the reference windows [6].

The remainder of this chapter is organized as follows. Section 3.1 provides a literature survey on

state-of-the-art CFAR methods and Section 3.2 provides an in-depth look at two such methods, which

will be used throughout this chapter. In Section 3.3 we propose a modified detector based on a statistical

model for sea clutter, which is then compared to the current state-of-the-art in Section 3.4. In Section 3.4

we also consider the effects of clutter from other sources, such as ionospheric clutter and Bragg peaks,

and devise a simple method to separate these regions from the rest of the data, leading to improved


detector performance. Finally, the chapter concludes with a discussion on the results in Section 3.5 and

a summary in Section 3.6.

3.1 Literature Survey

The current state-of-the-art for CFAR detection in heterogeneous background environments is the vari-

ability index CFAR (VI-CFAR) detector, proposed by Varshney and Smith in 2000 [27]. The VI-CFAR

detector involves a composite approach based on the CA-CFAR, GO-CFAR, and SO-CFAR detectors.

This detector characterizes the background data before choosing which of the aforementioned detectors

to use and has been found to perform robustly in non-homogeneous environments.

In recent years, applications of fuzzy logic to CFAR detection have been making an appearance

in the literature [28–30]. In particular, we are interested in a fuzzy version of the VI-CFAR detector

which was proposed in [31] in 2011. This detector is very similar to the traditional VI-CFAR detector,

however the data is characterized in a non-binary way, and the fuzzy contribution from each of the

robust CFAR detectors is accounted for. The motivation for applying fuzzy logic to detection is that

by replacing binary decisions with fuzzy membership functions more information about the background

signal is retained up until the final detection decision. In [31] the fuzzy VI-CFAR detector was shown

to outperform the VI-CFAR detector in a number of simulated environments.

Both the VI-CFAR and fuzzy VI-CFAR detectors make the assumption that the interference statistics

are exponentially distributed, an assumption which is reasonable in many radar environments. Sea

clutter, however, has been found to follow a more heavy-tailed distribution, resulting in “spikier” clutter.

In this thesis we propose a modified fuzzy VI-CFAR distribution based on a more accurate statistical

model of the sea clutter returns in order to achieve better clutter rejection in a coastal environment.

3.2 CFAR Detectors for Heterogeneous Environments

Both the VI-CFAR and fuzzy VI-CFAR detection algorithms have been formulated based on the as-

sumption that the background data is random noise which has been square law detected, meaning that

the resulting background data follows an exponential distribution, with PDF given by f(x;λ) = λeλx

for x ≥ 0 and cumulative distribution function (CDF), F (x;λ) = 1 − e−λx, where λ−1 is the mean

of x. In this section both detectors are introduced and their performance is compared on simulated

exponential data to verify the algorithm implementation, and that the fuzzy VI-CFAR detector does

indeed outperform the VI-CFAR detector in heterogeneous environments.

3.2.1 Variability Index (VI) CFAR

The VI-CFAR detector, as described in [27] combines the desirable properties of the three aforementioned

CFAR detectors, by dynamically choosing which detector to use based on the characteristics of the


background data in each reference window. Two statistics are computed in order to characterize the

background, the variability index (VI) and the mean ratio (MR) of the two reference windows. To

simplify notation, the leading window is referred to as window A and the lagging window is referred to

as window B. The sample in the CUT is denoted as QCUT and there are N reference samples in each

window, QiNi=1. The VI statistic is a second order statistic, defined as

VI =1

N − 1·N∑i=1

(Qi − Q)2/(Q)2 (3.1)

and is a measure of how variable the data is in a particular reference window. The second statistic is

the ratio of the means of the two reference windows, given by

MR = QA/QB . (3.2)

which indicates how homogeneous the background environment is. In the case of a homogeneous envi-

ronment, the mean ratio will be close to one and the VI statistic close to zero. The following hypothesis

tests based on two fixed thresholds, KV I and KMR are used to characterize the background data in each

reference window:

VI ≤ KV I → Nonvariable

VI > KV I → Variable

K−1MR ≤MR ≤ KMR → Same Means (A & B)

MR < K−1MR or MR > KMR → Different Means (A & B)

The values for these constants are chosen as in [27] such that the probability of hypothesis test error in

a homogeneous environment is low. They are KV I = 4.76 and KMR = 1.806, and these are the values

used throughout this report.

Based on these two statistics, the VI-CFAR detector chooses which of CA-CFAR, GO-CFAR or

SO-CFAR, to use for each CUT. The adaptive threshold generation logic for the VI-CFAR detector is

summarized in Table 3.1. The quantity in brackets following the name of each detector indicates which

reference window, or windows, the detector uses to compute the threshold.

Leading WindowVariable?

Lagging WindowVariable?

DifferentMeans?

EquivalentCFAR Method

No No No CA-CFAR(A,B)No Yes – CA-CFAR(A)Yes No – CA-CFAR(B)No No Yes GO-CFAR(A,B)Yes Yes – SO-CFAR(A,B)

Table 3.1: Adaptive Threshold Generation Logic: VI-CFAR

The first scenario is a homogeneous environment, and so the standard CA-CFAR detector can be

used, and the threshold is computed using all 2N reference cells. The second two scenarios have high


variability in just one of the reference windows, likely due to an interfering target in that window, so in

this case it is better to use CA-CFAR, but only use the N reference cells in the non-variable window.

When windows A and B have different means, we assume that there is a clutter edge present and use a

GO-CFAR detector to avoid clutter edge false alarms. Finally, when both windows have high variability,

there may be an interfering target present in both reference windows. In this case it is best to use an

SO-CFAR detector. The threshold which achieves a desired probability of false alarm, PFA,allowed, for

a reference window containing N cells is

TN,i = CN · Σi (3.3)

where CN = (PFA,allowed)−1/N − 1, and Σi =

∑Ni=1Qi is the sum of the N reference samples in window

i [6]. The choice of reference window depends on which detector is being used, based on the logic in

Table 3.1. If both reference windows are used, then N in Equation (3.3) should be replaced with 2N .

3.2.2 Fuzzy VI-CFAR

Fuzzy sets were first defined by Lofti A. Zadeh in 1965, as sets whose elements may have degrees of

membership to the set [32]. Typically, a membership function, µ(x), is defined where µ(x) = 0 indicates

that x is not in the set, µ(x) = 1 indicates that x is completely in the set, and values in between indicate

to what degree x is partially in the set.

Figure 3.2: Fuzzy membership function of the VI parameter.

The fuzzy VI-CFAR detector, proposed by Cheikh and Soltani in 2011, is a modification of the afore-

mentioned VI-CFAR detector, with the distinction that it treats the characterizations of the reference

windows as “variable/not variable” and “same means/different means” as fuzzy sets. Instead of making

a binary decision as to whether or not the data in the reference windows is variable, and the means

are the same, the fuzzy VI-CFAR detector computes fuzzy membership functions for the VI and MR

statistics based on the data in the reference windows.

In addition, the different CFAR detectors themselves are also “fuzzified”. Figures 3.2 and 3.3 show


Figure 3.3: Fuzzy membership function of the mean ratio.

the trapezoidal membership functions which are used for the VI and MR statistics. In the figures, VI var

denotes the degree to which the reference window is variable and VI nvar the degree to which it is non-

variable. Similarly, Mmoy represents how similar the means of the two windows are, and Dmoy, how

much they differ.

The membership functions of the fuzzy detectors are defined based on the distribution of Z := QCUT /∑Ni=1Qi,

as the complement of the CDF,

µ(Q) = Pr (Z > z|H0) (3.4)

where each of the Qi and QCUT are considered to be IID unit mean exponential random variables. The

particular reference window, or windows, the Qi belong to is determined by the choice of detector. A

detailed derivation of these membership functions can be found in [30], the final results are as follows:

CA-CFAR detector:

µ(z) =1

(1 + z)N(3.5)

GO-CFAR detector:

µ(z) = 2(1 + z)−M − 2

M−1∑k=0

(M + k − 1

k

)(2 + z)−(M+k) (3.6)

SO-CFAR detector:

µ(z) = 2(2 + z)−MM−1∑k=0

(M + k − 1

k

)(2 + z)−k (3.7)

where M = N/2. These membership functions are then combined to produce a final “defuzzified” global

membership function, which acts as our traditional detection statistic.


The contributions from each detector are computed using the same logic found in Table 3.1, however

the AND logic is replaced by the arithmetic product to provide a number between 0 and 1 representing

the respective contribution of each detector. The final global membership function is computed from

the detector contribution and membership function using the centre of gravity method, as follows:

µ(z) =

(contCA(A,B) · µCA(A,B)(z) + contGO · µGO(z) + contCA(A) · µCA(A)(z)

+ contCA(B) · µCA(B)(z) + contSO · µSO(z)

)(contCA(A,B) + contGO + contCA(A) + contCA(B) + contSO)

(3.8)

A target is declared present if µ < T , where the threshold, T is determined by simulation so as to

achieve the desired PFA. Unfortunately, for this more complex fuzzy detector there does not exist a

known convenient analytical expression to compute T based on the desired PFA.

3.2.3 Comparison on Simulated Data

In this section the performance of the VI-CFAR and fuzzy VI-CFAR detectors is compared using sim-

ulated data, where the interference is assumed to follow an exponential distribution. All figures in this

section are reproductions of the figures from [31] which have been replicated here in order to verify the

performance of both detectors. The total number of reference cells used for both detectors is 36, with

18 in each reference window, and the PFA is set to 10−3. To compare performance we plot PD versus

the SNR, for a fixed PFA.

Figure 3.4: PD vs. SNR in a homogeneous environment, with PFA = 10−3.

Figure 3.4 shows the detection performance of both the VI-CFAR and fuzzy VI-CFAR detectors

in a homogeneous environment with exponentially distributed interference data. We can see that both

detectors have very similar performance here, however the fuzzy detector does perform marginally better.

In Figures 3.5 and 3.6 we now consider two common heterogeneous environments. In Figure 3.5 a

clutter edge with a clutter-to-noise (CNR) ratio of 5 dB in the first 10 cells is introduced. In Figure 3.6

there is an interfering target present in one of the reference windows. Again, in both figures we see that

the fuzzy detector achieves slightly better performance than the conventional VI-CFAR detector, with


Figure 3.5: PD vs. SNR in the presence of a clutter edge (CNR = 5dB) in the first 10 cells, withPFA = 10−3.

Figure 3.6: PD vs. SNR in the presence of an interfering target (INR = SNR), with PFA = 10−3.

the most improvement occurring in the clutter edge environment.


3.3 Modifications to the Fuzzy VI-CFAR detector based on the

K-distribution

Both the VI-CFAR and fuzzy VI-CFAR detectors were derived based on the assumption that the under-

lying background distribution is exponential, yet as we saw in the previous chapter, sea clutter environ-

ments are better modelled by the K-distribution [2]. The PDF of the intensity under the K-distribution

is given by,

fQ(q) =2b(ν+1)/2q(ν−1)/2

Γ(ν)Kν−1(2

√bq) (3.9)

which has two parameters, the shape parameter b and the order parameter ν. The parameters of

the distribution are related by E[Q] = ν/b, and therefore the mean clutter intensity depends on both

parameters.

This PDF is significantly more complex than the exponential PDF, and therefore the computation

of the PDF of Z = QCUT /∑Ni=1Qi when the Qi are K-distributed would be quite complex, if a closed

form even exists. For this reason in order to design a detector based on the K-distribution we propose,

first, to simplify the fuzzy membership function.

3.3.1 Simplified Membership Function

As the number of samples in the reference window, N , gets large, we have that∑Ni=1Qi → N · E[Q].

In order to simplify the computation of the fuzzy membership functions given in Equation (3.4) we

can approximate this sum in the denominator as a constant. Under the assumption of the exponential

distribution, this results in the following membership function for all of the detectors,

µ(z) = Pr(Z > z) ≈ Pr(QCUT > (N · E[Q])z)

= exp(−Nz) (3.10)

where z = qCUT /∑Ni=1 qi and the value of N is determined by the choice of reference window. This

simplified detector was tested on simulated exponential data and it was found that in a homogeneous

environment the performance was nearly identical to the original fuzzy VI-CFAR detector. In the

presence of a clutter edge or interfering target some loss was observed, however performance was still

reasonable considering that this involves a significant simplification of the membership functions.

3.3.2 Estimation of the K-distribution Parameters

The membership function derived above, in Equation (3.10), for the exponential distribution turns out

to be equivalent to

µ(z) = 1− FQ(q; λ) (3.11)


where FQ(q) is the CDF of the exponential distribution and λ is the maximum likelihood (ML) estimator

of λ, in this case given by 1λ

= 1N

∑Ni=1 qi. Since this approximate membership function performed

reasonably well on exponentially distributed data, we derive a new membership function based on the

K-distributon CDF,

µ(z) =2bν

Γ(ν)(q)

ν2Kν(2

√bq) (3.12)

where b and ν are estimates of the parameters b and ν. Ideally, the ML estimates would be used, however

it has been shown that obtaining the ML estimates of the K-distribution parameters is analytically

intractable [33]. For this reason sub-optimal alternatives must be used. A number of methods have been

proposed in the literature, three of which are examined here. It should be noted that these methods

estimate the order parameter, ν, and the shape parameter can then be found as

b =ν

[q](3.13)

where the notation [h(q)] is used to denote the sample mean of the function h(q).

A common method of estimating the K-distribution parameters is to use the relationship between

two moments of the distribution, called method of moments (MoM) estimation. Often the first and

second moments of the intensity are used, which results in the following expression [33] that can be

inverted to solve for ν,

V =

⟨q2⟩

〈q〉= 2

(1 +

1

ν

)(3.14)

where 〈h(q)〉 is used to denote the expectation of h(q). Estimates for the moments can be found from

the data, using sample means to come up with an estimator V from which the estimate ν can be found.

The main drawback of MoM estimators is that sometimes they result in an equation with no solution,

for instance in (3.14) if[q2]

= 2 [q] then there is no finite solution for ν. In [34] it is shown that the

probability of this occurring is not insignificant and increases when the sample size is small or ν is large.

In [33] a method based on moments of the form 〈qr log(q)〉 is considered as previous results have

shown improved accuracy from using the log of the data. The proposed estimator is

X =〈qlog(q)〉〈q〉

− 〈log(q)〉 = 1 +1

ν(3.15)

and as with the V estimator, X can be estimated from the data to solve for ν.

In [34], a composite MoM-Bayesian approach is suggested where a uniform prior is introduced on

the moment ratio and is set to zero past the maximal theoretical value to prevent situations with no

valid solution. In this approach the data is assumed to be envelope detected, so x =√q is used, and the

moment ratio is D = X2/σ2. We note that here X refers to the random variable X, not the estimator

described above. Under the uniform prior an estimate of D can be found as the mean of the posterior

pdf using an analytical approximation for the likelihood function as described in [34]. This estimator is

thus referred to as the MoM-Bayes-AA estimator. Using this estimate for D, an approximation for ν is


given by

ν ≈ 1

4log[(1 + 1/D)π/4]. (3.16)

Figure 3.7: MSE vs. sample size (N) for ν = 0.1

Figure 3.8: MSE vs. sample size (N) for ν = 2

Figures 3.7 and 3.8 show the mean squared error (MSE) of the three estimators as a function of

the sample size, N . These figures were produced using Monte Carlo simulations with 10000 trials on

simulated K-distributed interference data. In both figures we see that the estimation accuracy improves

as N is increased, and that for all three estimators the accuracy is better for ν = 0.1 than it is when

ν = 2. In general, the accuracy decreases with increasing values of ν, in particular the performance of

X and V estimators degrades much more rapidly with ν, as can be seen upon comparison of Figures 3.7

and 3.8. We note that for values of ν greater than one, the X and V estimators were both observed to

occasionally return negative values for ν leading to an invalid PDF. Based on these results, the MoM-

Bayes-AA estimator was found to be the most reliable, particularly at lower sample sizes, and therefore

was implemented to estimate the parameters for this modification to the fuzzy VI-CFAR detector.


In implementing this modified detector, referred to as the fuzzy VI-CFAR II detector in the following

section, it was observed that due to the large order of magnitude of the data there were some numerical

issues in which the magnitude of the membership function was too small to be distinguished from zero.

To avoid this, the input square-law detected data has been scaled by a factor of 10−6 before applying

this detector. Since the K-distribution is a compound distribution resulting from the exponential and

Gamma distributions, and both these distributions have the scaling property, the K-distribution does as

well, meaning that, assuming the input data is K-distributed, the scaled data will still be K-distributed.

It should be noted here, that although we are referring to this as the fuzzy VI-CFAR II detector, because

of its similarity to the fuzzy VI-CFAR detector, the approximations made to the membership function

mean that this is no longer a true CFAR detector.

3.4 Detector Performance on Measured Data

3.4.1 Measurement Set-up

In order to test the detectors in real sea clutter conditions, measured data from the HFSWR system was

provided by Raytheon Canada, along with a tracker file containing information on all known targets in

the data set. The data was taken early in the morning on March 31, 2010. The array which took this

data has A = 16 receive antenna elements, and operates at a centre frequency of 3.25 MHz. Each CPI

comprises M = 256 pulses, with an effective PRF of 0.97656250 Hz, and contains returns from R = 270

range cells, each covering 1.5 km in distance. The first range cell occurs at 34.84 km, therefore the

furthest measured range cell is at a distance of 438.3 km [35].

The data is organized into data cubes, which have dimension A×M ×R, however the known targets

are characterized by the look angle (in degrees), range (in km) and the measured Doppler shift. In order

to compare the known targets in the tracker file with the detections resulting from the proposed detectors

the data cubes must be transformed into the angle-Doppler domain, so that for each angle-Doppler bin

there is a corresponding vector of range data to which the detectors can be applied. This is done by

implementing a 2-dimensional DTFT with the desired resolution in each direction.

3.4.2 Numerical Results

To compare their performance, all three detectors, the VI-CFAR detector, fuzzy VI-CFAR and fuzzy

VI-CFAR II, were implemented in MATLAB and applied to the measured data in the first CPI in the

data set, hereafter referred to as “CPI 68”. For the first trial, all parameters were kept the same as in

the simulations from Section 3.2, aside from the addition of two guard cells surrounding the CUT, since

target returns in the measured data have been observed to span two or three range bins.


Performance Comparison

Table 3.2 compares the performance of the detectors on the data in CPI 68. In order to obtain a fair

comparison of detector performance, the minimum PFA at which each detector detects all known targets

in CPI 68 was found using a bisection method to search over possible threshold values. In Table 3.2 it

can be seen that when applied to the measured data the VI-CFAR and fuzzy VI-CFAR detectors have

identical false alarm rates, while the fuzzy VI-CFAR II detector has a noticeably higher false alarm rate,

despite being based on a better model for the sea clutter amplitudes.

Detector Type Probability of False Alarm

VI-CFAR 2.7 · 10−3

Fuzzy VI-CFAR 2.7 · 10−3

Fuzzy VI-CFAR II 3.3 · 10−3

Table 3.2: Lowest PFA which achieves 100% detection (Nrange = 18)

It turns out that although using Nrange = 18 made sense on the simulated data, in the measured

data a single range cell spans a total distance of 1.5 km, and thus the reference window contains returns

spanning a distance of 27 km. Now, CFAR detection relies on the assumption that the cells in each

reference window are homogeneous, which may no longer be valid when the reference window spans such

a large distance. We can decrease the number of cells in the reference window in order to maintain

homogeneity, but of course, decreasing the number of samples in the reference window also decreases

the accuracy of estimating the state of the background interference. In order to compensate for this we

can reduce the number of range cells in the reference window and compensate by including additional

samples at neighbouring Doppler frequencies. Typically, samples are highly uncorrelated across Doppler

frequencies, therefore these samples should maintain the homogeneity assumption.


VI-CFAR 1.4 · 10−3



Table 3.3: Lowest PFA which achieves 100% detection (Nrange = 6, NDopp = 3)

Table 3.3 shows the performance of all three detectors when the reference windows have Nrange = 6

and NDopp = 3, for a total of 18 cells in each window. Again we observe that the VI-CFAR and fuzzy

VI-CFAR detectors have nearly identical false alarm rates, which have been reduced by almost a factor

of 2 from the previous result. The fuzzy VI-CFAR II detector has the same false alarm rate with this

new detector footprint.

Tables 3.4 and 3.5 display the performance of these two detectors with larger reference window sizes.

In Table 3.4 the total number of cells in the reference window is 66 and in Table 3.5, 198 cells are used. It

can be observed from these results that as more range cells are added to increase the size of the reference

window, the performance of the VI-CFAR detector begins to degrade, and it is actually outperformed by

the fuzzy VI-CFAR II detector. Considering the strong dependence between the accuracy of estimating



VI-CFAR 1.3 · 10−3





VI-CFAR 1.7 · 10−3




the parameters of the K-distribution on the sample size used that was observed in Section 3.3, this result

makes sense.

Receiver Operating Characteristics

We can also compare detector performance on the measured data by computing PD and PFA for a number

of detector thresholds. The resulting plot is commonly known as the receiver operating characteristic

(ROC). In this section we show the ROC for two different detector footprints, Nrange = 6, NDopp = 11

and Nrange = 18, NDopp = 11. To produce each plot, the detector thresholds were varied so that PD

goes from 0− 100%, and the resulting PFA is computed.

Figure 3.9: ROC with with detector footprint: Nrange = 6, NDopp = 11

Figure 3.9 shows the ROC for a detector footprint containing 6 range bins and 11 Doppler bins, which

corresponds to the results shown in Table 3.4. We observe, as in Table 3.4, that the fuzzy VI-CFAR II

detector has a higher PFA at PD = 100%, and similarly has a higher PFA than the other two detectors

for high detection rates. At lower detection rates, we actually observe a switch in behaviour, with the

fuzzy VI-CFAR II detector having lower false alarm rates for the same value of PD.


Figure 3.10: ROC with with detector footprint: Nrange = 18, NDopp = 11

In Figure 3.10 we have the same plot for the larger detector footprint of 18 range bins and 11 Doppler

bins, corresponding to the results in Table 3.5. Here we can see how the fuzzy VI-CFAR II achieves a

higher PD than the other two detectors for most values of PFA. We observe in both figures the nearly

identical performance of the VI-CFAR and fuzzy VI-CFAR detectors.

3.4.3 Characterizing the Clutter Returns

In addition to simply comparing false alarm rates, it is also interesting to consider the characteristics of

the false alarms which are detected by each detector. For instance, in HFSWR, under certain conditions

we may see additional clutter returns other than the expected K-distributed sea clutter which may lead

to additional false detections. For instance, we will typically observe strong peaks at a single Doppler

frequency which span many kilometres in range. These are the result of Bragg resonant scatter from

ocean surface waves with a wavelength exactly one-half of the radar carrier wavelength. These Bragg

lines occur at Doppler shifts, fB , which are proportional to the phase velocity of the Bragg-matched

ocean waves, given by

fB = ±√gfcπc

(3.17)

where g is the acceleration due to gravity, fc is the radar carrier frequency, and c the speed of light. For

the parameters of this particular experiment we have that fB ≈ 0.1839 Hz [1, 3].

In addition to the Bragg lines, we may also see ionospheric clutter returns in the data set. Ionospheric

clutter occurs when some of the energy emitted from the radar is directed upwards and reflects from the

ionosphere. Under certain conditions, especially at night, these returns can be quite strong. Ionospheric

clutter typically appears as a strong peak over a narrow band of ranges corresponding to the height of

either the E- or F-layers of the ionosphere, and a wide spread of Doppler shifts. Depending on the time

of day, ionoshperic scatter may appear at ranges from approximately 100 − 300 km [36]. The returns

from ionospheric clutter may also be significant in magnitude, much larger than the returns from the


ocean.

Figure 3.11: Detections(red) plotted on top of the range-Doppler data for CPI 68 at a look angle of −4degrees with detector footprint: Nrange = 6, NDopp = 3

In Figure 3.11 the detections resulting from the VI-CFAR (left) and fuzzy VI-CFAR II (right) detec-

tors are plotted, as red circles, on top of the range-Doppler intensity data for a single look angle of −4

degrees. It should be noted here that the VI-CFAR and fuzzy VI-CFAR detectors were observed to have

nearly identical detections, and so here we compare only the VI-CFAR and fuzzy VI-CFAR II detectors

for the sake of brevity. The detector footprint used to generate these plots was 6 range bins by 3 Doppler

bins, a total of 18 reference cells. According to the tracker file, there is one target located at this angle,

at a range of 84.34 km and a Doppler shift of −0.1148 Hz. Therefore, all other detections shown are

false detections which serve only to distract the tracker from the real target. This single target can be

seen in Figure 3.11 in the bottom left of the image, as a few detections spread over multiple range cells,

and a single Doppler bin.

A couple of key observations should be made from Figure 3.11. We can see a single Bragg line with

a positive Doppler shift, as well as a band of ionospheric clutter at ranges of nearly 300 km. It can

be seen that both detectors pick up a large number of false detections due to the ionospheric clutter

and we observe that the VI-CFAR detector does does a much better job at rejecting the Bragg lines.

We also see that the fuzzy VI-CFAR II detector picks up far more false detections in the region near

zero Doppler where the ionospheric clutter spans a larger set of ranges. The number of false alarms in

these regions is seen to make up the majority of detections, with only a few additional false detections

scattered throughout the remainder of the data. Figure 3.12 shows the same range-Doppler data, this

time with detections generated using a detector footprint of 6 range bins by 11 Doppler bins. With

the increased amount of Doppler bins included in the reference window, we can see that performance is

noticeably better for both detectors, however there are still quite a few false detections occurring due to

the ionospheric clutter and Bragg lines. Based on these observations, we can infer that if these regions of


Figure 3.12: Detections (red) plotted on top of the range-Doppler data for CPI 68 at a look angle of −4degrees with detector footprint: Nrange = 6, NDopp = 11

particularly strong clutter could be identified and either excluded or treated differently, the performance

of both detectors would be improved. The question which remains, is if we exclude these regions of non

K-distributed clutter, which detector will perform better?

3.4.4 Masking Strong Clutter returns

As a proof of concept we implement a simple masking method, using tools from MATLAB’s Image

Processing Toolbox, to demonstrate the improved performance of both the VI-CFAR and fuzzy VI-

CFAR II detectors when detections resulting from regions of high ionospheric clutter and prominent

Bragg lines are excluded. The predictable appearance of these forms of clutter when we look at the

range-Doppler data allows for the use of straight forward image processing methods to segment out

these regions reliably. As was seen in Figures 3.11 and 3.12, ionospheric clutter appears as bright

rectangular regions extending horizontally across all Doppler values and Bragg lines occur at a single

known Doppler frequency and span multiple ranges. Due to the noisy background data, although these

regions are fairly easily identified by the human eye, some additional processing is necessary to ensure

that only clutter is masked, and strong targets are not also excluded from the data. This is done using

two separate 2D Gaussian smoothing filters, each designed to smooth out clutter extending in either the

range or Doppler direction, while leaving the other direction intact. The filters are anisotropic, i.e., not

circularly symmetric, and the radii are chosen based on the smallest region we want to detect in each

direction.

The result is two filtered images, one in which the horizontal ionospheric clutter is most prominent,

while bright vertical regions have been blurred out, and the second in which the Bragg lines and other

strong vertical regions of clutter are easy to pick out. The adaptive threshold function in MATLAB,


which computes a locally adaptive threshold using first-order statistics, is then used to produce a binary

image with strong clutter regions separated from the background clutter and targets. Since many false

detections tend to occur right on the edges of these regions, one additional isotropic Gaussian filter is

applied to the binary mask image which blurs the masked regions. This blurred image is converted once

again to a binary image, which as a result has slightly widened masked regions.

This filtering and thresholding process requires a number of parameters which must be supplied in

advance; the variance of each anisotropic Gaussian filter (σh and σv), the sensitivity of the adaptive

thresholds (sh and sv), and the variance of the isotropic Gaussian filter (σmask). In order to be con-

sistent with previous simulations, the parameters used here are chosen to maintain PD = 100%. The

following parameters were determined by searching over a subset of reasonable parameters determined

by observation of the clutter properties: sh = 0.5, sv = 0.42, σh = [8, 1], σv = [3, 22], and σmask = [2, 2].

Detector Type PFA (without masking) PFA (with masking)

VI-CFAR 1.4 · 10−3 7.7 · 10−4

Fuzzy VI-CFAR 1.3 · 10−3 7.7 · 10−4

Fuzzy VI-CFAR II 3.3 · 10−3 8.9 · 10−4

Table 3.6: PFA of both detectors after clutter masking (Nrange = 6, NDopp = 3)

Detector Type PFA (without masking) PFA (with masking)

VI-CFAR 1.3 · 10−3 8.2 · 10−4

Fuzzy VI-CFAR 1.3 · 10−3 8.3 · 10−4

Fuzzy VI-CFAR II 1.4 · 10−3 5.3 · 10−4

Table 3.7: PFA of both detectors after clutter masking (Nrange = 6, NDopp = 11)

Tables 3.6 and 3.7 show the resulting probabilities of false detection before and after the masked

regions are excluded for two different detector footprints. The same binary mask, which achieves 100%

detection, is used for both footprints. When the detector footprint includes 6 range bins and 3 Doppler

bins, we see that the mask significantly improves the performance of both detectors. The fuzzy VI-

CFAR II detector sees the most improvement which makes sense, since in the previous section we saw

that this detector was most negatively affected by strong regions of clutter. Similarly, when the footprint

is increased to include 11 Doppler bins, the fuzzy VI-CFAR II sees the most improvement, and actually

has a lower false alarm rate than the other detectors when the ionospheric clutter and Bragg lines are

masked.

Figures 3.13 and 3.14 show the range-Doppler data, again at a look angle of −4 degrees, with

detections plotted as red circles and masked regions in dark blue for both detector footprints discussed

above. Here we compare the VI-CFAR and fuzzy VI-CFAR II detectors once again. From these images

we can see that the mask effectively covers the majority of the strong clutter returns, while the target

at 84.34 km and −0.1148 Hz remains unmasked. In both figures we can see that a significant number of

false detections lie within the masked regions, accounting for the improved performance after masking

that was observed in Tables 3.6 and 3.7. In particular, the fuzzy VI-CFAR II detector appears to

pick up more detections within the masked regions, which accounts for it having the most significant

improvement in performance when clutter masking is implemented.


Figure 3.13: Detections plotted on top of the range-Doppler data for CPI 68 at a look angle of −4degrees with detector footprint: Nrange = 6, NDopp = 3

Figure 3.14: Detections plotted on top of the range-Doppler data for CPI 68 at a look angle of −4degrees with detector footprint: Nrange = 6, NDopp = 11


3.5 Discussion

In this chapter we have compared the performance of the state-of-the-art VI-CFAR and fuzzy VI-CFAR

detectors with the proposed fuzzy VI-CFAR II detector which is based on the assumption that the

background data is K-distributed. We have also discussed a simple method to mask strong regions

of clutter to prevent false tracks overloading the detector. In this section some further discussion is

included on choosing the best parameters for each CFAR detector along with a discussion of the choice

of parameters used in the masking algorithm.

3.5.1 Optimal Detector Footprint

After implementing the three proposed detectors on the measured data, we observe that detector perfor-

mance is highly dependent on the detector footprint used. For instance, we see that all three detectors

perform better when more Doppler bins are added to the window and that performance can actually

degrade when too many range bins are included. Ideally, the number of samples included in the reference

window should be as large as possible to improve the accuracy of estimating the background parameters,

however this must be balanced with the requirement that the samples in the reference window should be

identically distributed. Due to the large distance covered by a single range bin, this puts a limitation on

the number of range bins which should be included. We also want to avoid scenarios where target returns

are present in both reference windows, the probability of which is lowered by using smaller reference

windows.

This is actually illustrated in Tables 3.4 and 3.5 where we see that when NDopp is fixed at 11 and

Nrange is increased, the false alarm rate of each detector increases, despite the larger reference window

size. If necessary, further simulations could be done to find the “tipping point” for each detector, i.e.

how many additional bins can be included in the reference window before performance begins to degrade

as a result of non-homogeneous data in the reference windows.

3.5.2 Optimal Masking Parameters

In order to be consistent when comparing detectors, the clutter mask parameters were chosen so that

each detector still achieved 100% detection of known targets, i.e., no targets were included in the masked

regions. These parameters were found by searching over a subset of reasonable parameters for those which

gave PD = 100%, and choosing a set of these parameters with low PFA for all three detectors. In this

way the same mask could be applied with all three detectors to compare performance.

In reality, target locations are not known a priori, and optimal parameters cannot be found in this

way. The variance of each Gaussian filter can, however, be chosen based on observations of the shape

and size of clutter regions desired to be masked. As a rule of thumb, the smallest region resolved in

each direction will be approximately twice the variance of the filter. Since the HFSWR system has a

tracker system which requires target returns in multiple CITs before declaring a detection, in practice,

using perfectly optimal parameters which maintain a 100% detection rate is not necessary. As long as


the parameters are reasonable based on observations of the clutter properties, the tracker can deal with

the occasional missed detection in one CIT, and can reject any remaining clutter returns that are not

covered by the mask.

3.6 Summary

In this chapter we focused on the design of a new detection algorithm to better discriminate against

sea clutter returns. The fuzzy VI-CFAR II detector was developed based on similar detectors in the

literature, with modifications based on the assumption of K-distributed clutter amplitudes. The pro-

posed detector was compared to the current state-of-the-art in CFAR detection using measured data

from the HFSWR system. Detector performance was found to depend significantly on the dimensions of

the detector footprint used, and when a large enough footprint was used, the fuzzy VI-CFAR II detector

was found to have the lowest false alarm rate.

We also looked into some other sources of strong clutter returns, such as ionospheric clutter, and

Bragg peaks, both of which present as bright rectangular regions when we look at the range-Doppler data.

We developed a simple and efficient method to mask these regions in the data and saw improvements in

detector performance when these regions were excluded. The fuzzy VI-CFAR II detector was found to

improve the most when the masked regions were excluded from the data, as the majority of false alarms

lay within these regions.

Chapter 4

Conclusions

In this thesis we developed multiple improvements to the HFSWR system at hand, taking into consid-

eration a number of unique constraints which apply to this particular system. Specifically, the detection

performance in the presence of strong sea clutter returns has been addressed using a comprehensive ap-

proach in which we consider both transmit waveform design and, on the receive side, detection algorithms

with improved clutter rejection performance.

First, we proposed a new algorithm, SCA-P, for the design of high bandwidth spectrally compliant

radar waveforms with minimal amplitude modulation and low ACF sidelobes. The SCA-P algorithm, a

hybrid between SCA and AP-based algorithms, is found to consistently generate feasible waveforms with

good pulse compression properties within a few minutes when run in MATLAB on a desktop PC. In fact,

when we compare the proposed algorithm to the common AP algorithms in the recent literature, the

SCA-P algorithm is found to outperform these other methods with respect to both ISLR and resulting

PAPR after necessary filtering is applied. The resulting high bandwidth waveforms are feasible for

the strict spectral environments encountered by the HFSWR system and have better range resolution

than what can be transmitted with the current technology. We also show simulations of performance

in correlated K-distributed sea clutter in which it is clearly seen that the additional bandwidth of the

SCA-P waveforms leads to higher SCR by decreasing the size of the clutter cell. This, in turn will lead

to improved detection performance in strong sea clutter environments.

We also show how the SCA-P algorithm can be used to generate multiple pulses with low pulse-to-

pulse correlation. By designing each pulse individually we can ensure that the average spectrum and

ACF meet the desired constraints, while keeping the spectral overlap between pulses low to minimize

their cross-correlation. Again, we see that the resulting pulses can be generated within a few minutes and

have the desired properties. A joint processing scheme is implemented for the case of four pulses and it

is shown that using this scheme we are able to distinguish between ambiguous range intervals, effectively

increasing the unambiguous range of the system by a factor of four. Since the HFSWR system often

sees strong returns from relatively large distances, this multiple pulse processing scheme is necessary in

order to accurately detect and track targets while maintaining the required PRF.

63

Chapter 4. Conclusions 64

In addition to the proposed SCA-P algorithm, we also consider the use of filtering to suppress the

out-of-band emissions down to −60 dB. This is a unique constraint that is not covered in any of the

related literature, and although it is easily achievable with a Kaiser window, it is important to consider

the trade-off between achieving such low out-of band-emissions and other desirable waveform properties.

For instance, we see that applying this window not only increases the PAPR, but increases the ACF

sidelobe levels. This puts a limit on how low the ACF sidelobes can be, while still meeting the necessary

specrtal emission levels. In this section we also saw that the SCA-P algorithm has the benefit that the

waveform deviates the least from the LFM initialization waveform, and thus requires the least amount

of filtering to reduce the out-of-band emissions. This leads to a lower resulting PAPR in the end.

In addition to reducing the clutter power with higher bandwidth waveforms, we also look at detection

algorithms to improve clutter rejection and avoid tracker overload in particularly severe sea states. We

propose a modified version of the fuzzy VI-CFAR detection algorithm, which although technically no

longer a CFAR algorithm, we refer to as the fuzzy VI-CFAR II detector. In this detector, the membership

functions are simplified and modified so as to reflect the fact that the background is dominated by

K-distributed sea clutter. This algorithm relies on estimating the K-distribution parameters using a

composite MoM-Bayesian approach, which is highly dependent on the number of samples included in

the estimate. For this reason, we observe that the performance of the fuzzy VI-CFAR II detector improves

significantly as more samples are added to the reference windows. In fact, when enough reference samples

are used, we find that this detector outperforms both the VI-CFAR and fuzzy VI-CFAR algorithms in

terms of false alarm rate on the given data set.

It turns out that for each detector footprint, but particularly the smaller ones, all three detectors are

overwhelmed by false alarms due to strong Bragg peaks and ionospheric clutter. The clutter returns in

these regions can be so strong that they overwhelm the tracker, leading to missed detections in other

regions. As a proof of concept we implement a simple algorithm based on 2D Gaussian filtering to mask

these strong clutter regions and exclude detections in these regions from the data. The result is that all

three algorithms see significant improvements in the false alarm rate. In particular, the performance of

the fuzzy VI-CFAR II detector sees the most improvement as this detector previously had significantly

higher rates of false alarm caused by strong clutter regions. In fact, with the implementation of clutter

masking, this detector now outperforms the other two using smaller reference windows.

Although the fuzzy VI-CFAR II detector relies on an approximation of the membership function, it

can lead to significant improvements in performance in clutter-limited systems such as this one where

the background data is not exponentially distributed. More sophisticated algorithms do exist for image

segmentation which could lead to even further reduction in false alarm rate, however the simple method

implemented here is sufficient to show that when we are able to separate and exclude returns from

ionospheric clutter and Bragg peaks, the fuzzy VI-CFAR II detector performs quite well in the sea

clutter environment

It has been the goal throughout this project to develop techniques to improve the performance of

the HFSWR system in severe sea states which can be implemented as is on the current system. For this

reason it has been necessary to consider a number of practical constraints, such as algorithm run time

and out-of-band emissions which are not typically mentioned in the literature. So far we have been able

to address all known considerations to come up with two algorithms, the SCA-P algorithm for waveform

Chapter 4. Conclusions 65

design and the fuzzy VI-CFAR II algorithm for detection in sea clutter environments. Both of these

algorithms are feasible in their current state, however, there is of course room for improvements in the

future.

For instance, on the transmit side, we chose to initialize the SCA-P algorithm with an LFM waveform

known for its good pulse compression properties and simple implementation, however other good radar

codes do exist. It would be interesting to see how both the convergence time and performance changes

when the algorithm is initialized with other known radar codes. We saw that the PSL was not reduced

significantly when initializing with an NLFM waveform, however the convergence time with the same

PSL constraint was significantly faster, and the resulting ISLR was lower.

On the receive side of the problem, we observed that the fuzzy VI-CFAR II detector performance is

highly dependent on the detector footprint used. This detector should also see changes in performance

depending on the parameters of the background data, as the accuracy of estimating the K-distribution

parameters was shown to depend on their values. Unfortunately we only had access to a single data

set for this project, and so the background distribution was relatively consistent over all data. It would

be beneficial to obtain more data sets taken at different time intervals, and on different days, in order

to analyse detector performance in different sea states. Ideally, simulations could be done to determine

an ideal threshold for certain ranges of sea states, and then applied in real time based on available

information about current environmental conditions.

Appendix A

Computational Complexity

Calculations

In this Appendix, the computational complexity of the alternating projections algorithm, AP1 and AP2,

and the proposed SCA-P algorithm are derived. As all methods are iterative, in each case we compute

the complexity per iteration.

A.1 SCA-P Algorithm

The SCA-P algorithm is shown in detail in Algorithm 1 where it can be seen that the main computations

are the multiple FFT and IFFT operations, the ACF projection, solving the SOCP, and the error

calculation. To determine the overall complexity of the algorithm we must work out the complexity of

each operation.

The FFT and IFFT operations compute the DTFT and inverse DTFT of a signal in O(N logN)

operations, while the ACF projection and error calculation both require just O(N) operations. The

complexity of solving the SOCP using CVX on each iteration is slightly more difficult and involves

reformulating the original problem in the standard form of an SOCP.

When solved using interior points methods, the number of iterations required to decrease the duality

gap to a constant fraction of itself is bounded above by O(√

N)

and the complexity of each iteration

is O(N2∑j nj

), where nj is the dimension of the jth constraint [37]. Therefore to determine the

complexity of our algorithm we must determine the dimension of all constraints when problem P3,i is

formulated as a standard SOCP. A standard SOCP constraint is written as

||Ajx+ bj || ≤ cTj x+ dj (A.1)

where x ∈ RN is the optimization variable and the problem parameters are Aj ∈ R(nj−1)×n, bj ∈ Rnj−1,

66

Appendix A. Computational Complexity Calculations 67

cj ∈ Rn and dj ∈ R.

To reformulate problem P3,i as an SOCP we first rewrite the problem in terms of real variables, and

then as an SOCP, as follows

minv∈R2N ,t t

s.t. ||Anvi|| ≤ t ∀ n = 1, ..., N

0 ≤ 2vTi−1vi − vTi−1vi−1 − E0

||Qks

12vi|| ≤ Ek

12 ∀ k = 1, ...,K

(A.2)

where the new optimization variable is vi = [<xTi=(xTi )]T ∈ R2N , An ∈ R2×2N is given by

An(l,m) =

1 if (l,m) = (1, n), (l,m) = (2, N + n)

0 otherwise

and Qks ∈ R2N×2N by

Qks =

[<Rks

−=

Rkx

=Rks

<Rks

] . (A.3)

The total constraint dimension is thus∑j nj = 3N + 1 + K(2N + 1), making the work per iteration

O(N3). Including the number of iterations required to converge on a solution, the overall complexity

of solving the SOCP is O(N3.5). Clearly, the complexity of the SCA-P algorithm is dominated by the

SOCP step on each iteration, and thus we say that its complexity per iteration is also O(N3.5).

A.2 AP1 Algorithm

The AP1 algorithm involves three projections to constrain the ACF, the frequency spectrum, and the

amplitude in the time domain. Each is computed using the FFT and IFFT operations which have

complexity O(N logN), and the respective projections are done on an element-by-element basis which

has worst case complexity O(N).

Algorithm 2 shows the AP1 algorithm. The inputs fu, γu and au and al are masks to impose the

spectral, ACF and amplitude constraints, respectively. The corresponding projections, PF , PG and PA,

are implemented based on these masks. In Algorithm 2 we see that each iteration of the AP1 algorithm

consists entirely of FFT and IFFT operations, projections, and two maximum error computations. Both

the projections and the maximum error calculations can each be completed with worst case complexity

of O(N) operations. Therefore, the amount of work per iteration is dominated by the FFT and IFFT

operations, making the overall complexity of the AP1 algorithm O(N logN) per iteration.

Appendix A. Computational Complexity Calculations 68

Algorithm 2 Alternating Projections 1 (AP1) Algorithm

1: input xinit, fu, γu, au, al, ε1, ε22: initialize x← xinit, maxdev1 ← (ε1 + 1), maxdev2 ← (ε2 + 1)3: while maxdev1 > ε1 OR maxdev2 > ε2 do4: Compute DTFT: X ← FFT (x)5: Compute ACF: gx ← IFFT

(|X|2

)6: ACF Projection: gx ← PGgx7: Set X ←

√FFT (gx) · ∠X

8: DTFT Projection: X ← PFX9: Set x← IFFT (X)

10: Amplitude projection: x← PAx11: Compute DTFT: X ← FFT (x)12: Compute ACF: gx ← IFFT

(|X|2

)13: Compute max. deviation in DTFT: maxdev1 ← maxn (gxi(n)− γu(n))14: Compute max. deviation in ACF: maxdev2 ← maxn (gxi(n)− γu(n))15: end while

A.3 AP2 Algorithm

The AP2 algorithm is nearly identical to the AP1 algorithm, with the only difference being that the

spectral constraint is implemented in the time domain by subtracting the relevant component of the

signal to null specific frequencies, instead of as a projection in the frequency domain. This is reflected

in Algorithm 3, below. Instead of a mask to constrain the frequency spectrum, the matrix Q is defined,

which contains information on which frequencies to null, and the multiplicity of each null.

Algorithm 3 Alternating Projections 2 (AP2) Algorithm

1: input xinit, Q, γu, au, al, ε1, ε22: initialize x← xinit, maxdev1 ← (ε1 + 1), maxdev2 ← (ε2 + 1)3: while maxdev1 > ε1 OR maxdev2 > ε2 do4: Compute DTFT: X ← FFT (x)5: Compute ACF: gx ← IFFT

(|X|2

)6: ACF Projection: gx ← PGgx7: Set X ←

√FFT (gx) · ∠X

8: Set x← IFFT (X)9: DTFT Projection: x← x−QQHx

10: Amplitude projection: x← PAx11: Compute DTFT: X ← FFT (x)12: Compute ACF: gx ← IFFT

(|X|2

)13: Compute max. deviation in DTFT: maxdev1 ← maxn (gxi(n)− γu(n))14: Compute max. deviation in ACF: maxdev2 ← maxn (gxi(n)− γu(n))15: end while

The DTFT projection consists of the multiplication of two matrices and the vector x and the sub-

traction of two vectors. The matrix Q is an orthogonalization of an N ×KM matrix, where K is the

number of frequency nulls, and M is the multiplicity of each null, therefore the dimension of Q is N ×Lfor some L ≤ KM . The computation of the product QQHx therefore requires 2LN operations, making

the complexity of this matrix multiplication O(N). Similarly, the subtraction can be carried out in O(N)

operations. As a result, the complexity of the AP2 algorithm is also dominated by the FFT and IFFT

operations, meaning that it is O (N logN)

Appendix B

Simulation of K-distributed Sea

Clutter with Spatial Correlation

In Sections 2 and 3 the performance of the radar is simulated in a sea clutter environment where the

clutter amplitudes follow the K-distribution. In Section 3 we consider the simple case where the clutter

amplitudes are assumed to be uncorrelated, however in Section 2 the simulations are done at a much

higher sampling rate and therefore it is necessary to account for some amount of spatial correlation

between samples. In this Appendix, the method used to simulate the clutter amplitudes is described,

starting first with the simplest case, uncorrelated clutter.

B.1 Uncorrelated K-distributed Clutter

The K-distribution is a compound distribution consisting of a fast varying “speckle” component with

amplitude, E, which is Gaussian distributed and a slow varying local power which follows a Gamma

distribution. The amplitude of the fast-varying component follows a Rayleigh distribution with PDF

f(E|z) =2E

zexp(−E2/z); 0 ≤ E ≤ ∞ (B.1)

which depends on the local power, z. As we know, z has a Gamma distribution, with PDF

f(z) =bν

Γ(ν)zν−1 exp(−bz); 0 ≤ z ≤ ∞ (B.2)

where b is the scale parameter of the distribution, and ν, the shape parameter. When we combine these

two distributions, the resulting distribution of E is the K-distribution [2]. The shape parameter is related

to the “spikiness” of the clutter, with lower values of ν corresponding to spikier clutter, and thus higher

sea states. The scale parameter, b can be expressed in terms of both ν and the average clutter power,

as b = ν/Pc, where Pc is the average clutter power.

69

Appendix B. Simulation of K-distributed Sea Clutter with Spatial Correlation 70

Now, instead of trying to generate samples which follow the K-distribution directly, we can make

use of the simpler compound form of the distribution. We generate each component separately, taking

advantage of some of MATLAB’s built-in functionality for generating random vectors, and combine them

to form a K-distributed sequence. First, a sequence, z[n], n = 1, ..., N − 1, of independent Gamma-

distributed values is generated with desired shape and scale parameters. The value of z[n] represents the

local clutter power at the nth sample. Using the z[n] as inputs, a vector of Rayleigh-distributed clutter

amplitudes, E, can then be generated with varying local power as desired.

B.2 Adding A Model for Spatial Correlation

For our purposes we consider the Rayleigh component of the clutter to be independent and the local

clutter power to be spatially correlated, meaning that the samples are correlated across range bins.

Therefore, if we are able to generate a sequence of Gamma-distributed values with a given spatial

correlation, then we can use the same method as described in the previous section to generate the

corresponding correlated K-distributed sequence.

Typically, the simulation of correlated non-Gaussian processes is fairly complicated and requires the

implementation of a memoryless non-linear transform (MNLT), however if we take the shape parameter,

ν, to be an integer, then there exists a simple transform from a correlated Gaussian process to a correlated

Gamma process [26]. For our purposes, it is sufficient to only consider integer values of ν, and so we

can make do with this simple method. If more precise values of ν were required, the hybrid method

which is also described in [26] could be used to generate correlated Gamma-distributed sequences with

an arbitrary shape parameter.

As in [26], we first consider a sequence, x[n], which follows the Complex Normal distribution,

X ∼ CN (0, 1) with correlation coefficient ρNk, k = 0, 1, .... Now, if we let z[n] = |x[n]|2, then it

will be true that Z ∼ Γ(1, 1) and Z will have correlation coefficient ρΓk = ρN2k. It should be noted here

that this method does impose some restrictions on the correlation coefficient of the Gamma distribution,

i.e. the following conditions must be satisfied:

1. ρk ≥ 0,

2. ρ0 = 1 and |ρk| ≤ 1∀k > 0, and

3. the correlation matrix Mz must be positive semi-definite

where Mz(n, k) = ρ|n−k|. For our purposes, these restrictions are not an issue. Now, for ν = m > 1,

where m is an integer, we can generate m sequences, xk, k = 1, ...,m and then z[n] =∑mk=1 |xk[n]|2 will

follow a Γ(m, 1) distribution with desired correlation ρΓk. The shape parameter can easily be re-scaled

since the Gamma distribution has the following property, if Z ∼ Γ(ν, 1) then θZ ∼ Γ(ν, θ).

Bibliography

[1] L. Sevgi, A. Ponsford, and H. C. Chan, “An integrated maritime surveillance system based on

high-frequency surface-wave radars. Part 1.” IEEE Antennas and Propagation Magazine, vol. 43,

pp. 28–43, August 2001.

[2] K. Ward, R. Tough, and S. Watts, Sea Clutter: Scattering, the K Distribution and Radar Perfor-

mance. The Institution of Engineering and Technology, 2006.

[3] A. Ponsford and J. Wang, “A review of high frequency surface wave radar for detection and tracking

of ships,” Special Issue on Sky and Ground-wave High Frequency (HF) Radars: Challenges in

Modelling, Simulation and Application, Turk J Elec. Eng. and Comp. Sci., vol. 18, pp. 409–428,

2010.

[4] P. Moo, T. Ponsford, D. DiFilippo, R. McKerracher, N. Kashyap, and Y. Allard, “Canada’s third

generation high frequency surface wave radar system,” Journal of Ocean Technology (JOT) - special

issue on Maritime Domain Awareness, vol. 10, pp. 22–28, 2015.

[5] N. Levanon and E. Mozeson, Radar Signals. John Wiley and Sons Inc., 2004.

[6] M. Richards, J. Scheer, and W. Holm, Principles of Modern Radar: Vol. 1 - Basic Principles.

SciTech Publishing Inc., 2010.

[7] A. Aubry, V. Carotenuto, and A. D. Maio, “Forcing multiple spectral compatibility constraints in

radar waveforms,” IEEE Signal Processing Letters, vol. 23, pp. 483–487, April 2016.

[8] O. Aldayel, V. Monga, and M. Rangaswamy, “Successive QCQP refinement for MIMO radar wave-

form design under practical constraints,” IEEE Trans. Signal Processing, vol. 64, pp. 3760–3774,

2016.

[9] W. Rowe, P. Stoica, and J. Li, “Spectrally constrained waveform design,” IEEE Signal Processing

Magazine, pp. 157–162, 2014.

[10] S. U. Pillai, K. Y. Li, and H. Beyer, “Reconstruction of constant envelope signals with given fourier

transform magnitude,” in Proc. 2009 IEEE Radar Conference, May 2009.

[11] R. Kassab, M. Lesturgie, and J. Fiorina, “Alternate projections technique for radar waveform

design,” in Proc. 2009 International Radar Conf., Oct. 2009.

71

Bibliography 72

[12] I. Selesnick, S. U. Pillai, and R. Zheng, “An iterative algorithm for the construction of notched

chirp signals,” in Proc. 2010 IEEE Radar Conf., May 2010.

[13] I. W. Selesnick and S. U. Pillai, “Chirp-like transmit waveforms with multiple frequency notches,”

in Proc. 2011 IEEE Radar Conf., May 2011.

[14] J. Jakabosky, S. Blunt, and A. Martone, “Incorporating hopped spectral gaps into nonrecurrent

nonlinear FMCW radar emission,” in Proc. IEEE Intl. Workshop on Computational Advances in

Multi-Sensor Adaptive Processing, Dec 2015.

[15] J. Jakabosky, B. Ravenscroft, S. Blunt, and A. Martone, “Gapped spectrum shaping for tandem-

hopped radar/communications & cognitive sensing,” in Proc. IEEE Radar Conference (RadarConf),

May 2016.

[16] A. D. Maio, Y. Huang, M. Piezzo, and S. Zhang, “Design of radar receive filters optimized according

to Lp-norm based criteria,” IEEE Trans. Signal Processing, vol. 59, pp. 4023–4029, August 2011.

[17] R. McKerracher, personal correspondence, 2016-2017.

[18] F. F. Kretschmer and K. R. Gerlach, “New radar pulse compression waveforms,” in Proc. 1988

IEEE Radar Conference, Apr. 1988.

[19] A. Aubry, A. D. Maio, B. Jiang, and S. Zhang, “Ambiguity function shaping for cognitive radar via

complex quartic optimization,” IEEE Trans. Signal Processing, vol. 61, pp. 5603–5619, July 2013.

[20] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.

[21] O. Mehanna, K. Huang, B. Gopalakrishnan, A. Konar, and N. D. Sidiropoulos, “Feasible point

pursuit and successive approximation of non-convex QCQPs,” IEEE Signal Processing Letters,

vol. 22, pp. 804–808, July 2015.

[22] T. Wang and L. Vandendorpe, “Successive convex approximation based methods for dynamic spec-

trum management,” in Proc. 2012 IEEE International Conference on Communications (ICC), Jun.

2012.

[23] M. Grant and S. Boyd, “CVX”: matlab software for disciplined convex programming,” online, 2008.

[24] A. Oppenheim and R. Schafer, Discrete-Time Signal Processing. Pearson Higher Education Inc.,

2010.

[25] S. Frost and B. Rigling, “Sidelobe predictions for spectrally-disjoint radar waveforms,” in Proc.

IEEE Radar Conference (RadarConf), May 2012.

[26] Y. Dong, L. Rosenberg, and G. Weinberg, “Generating correlated gamma sequences for sea-clutter

simulation,” Defence Science and Technology Organisation (DSTO), Australian Government De-

partment of Defence, Tech. Rep., March 2012.

[27] M. Smith and P. Varshney, “Intelligent CFAR processor based on data variability,” IEEE Trans.

Aerospace Elec. Syst., vol. 36, July 2000.

Bibliography 73

[28] S. Leung, “Signal detection by fuzzy logic,” in Proc. Joint Conference of the Fourth IEEE Interna-

tional Conference on Fuzzy Systems and the Second International Fuzzy Engineering Symposium,

vol. 1, 1995.

[29] S. Leung and J. Minett, “Signal detection using fuzzy membership functions,” in Proc. SYS’95 Int.

AMSE Conf., July 1995.

[30] Z. Hammoudi and F. Soltani, “Distributed CA-CFAR and OS-CFAR detection using fuzzy spaces

and fuzzy fusion rules,” in IEE Proc. Radar and Sonar Navigation, vol. 151, June 2004.

[31] K. Cheikh and F. Soltani, “Performance of the fuzzy VI-CFAR detector in non-homogeneous envi-

ronments,” in Proc. IEEE International Conference on Signal and Image Processing Applications,

2011.

[32] L. A. Zadeh, “Fuzzy Sets,” Information and Control, vol. 8, pp. 338–353, 1965.

[33] D. Blacknell and R. Tough, “Parameter estimation for the K-distribution based on [zlog(z)],” in

IEE Proc. Radar and Sonar Navigation, vol. 148, 2001.

[34] D. Abraham and A. Lyons, “Reliable methods for estimating the K-distribution shape parameter,”

IEEE Journal of Oceanic Engineering, vol. 35, April 2010.

[35] M. Ravan and R. Adve, “Effective space-time adaptive processing for weak target detection in sea

clutter using high frequency surface-wave radar,” University of Toronto, Tech. Rep., May 2011.

[36] B. Zolesi and L. Cander, Ionoshperic Prediction and Forecasting. Springer Geophysics, 2014.

[37] M. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, “Applications of second-order cone program-

ming,” Linear Algebra and its Applications, pp. 193–228, 1998.

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