High resolution methods for small target detection and estimation in high frequency radar

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High Resolution Methods for Small Target Detection and

Estimation in High Frequency Radar

Jian Wang

B.S., Ocean University of Qingdao, 1997 MA.Sc., University of Victoria, 200 1

A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

in the Department of Electrical and Computer Engineering

O Jian Wang, 2004 University of Victoria

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Supervisors: Dr. R. L. Kirlin, Dr. A. Zielinski

ABSTRACT

The detection and tracking of small slow moving targets by High Frequency Surface Wave radar are limited by the presence of a dominate sea clutter spectrum. The ocean surface behaves as a distributed source in contrast to targets that are point sources. It is shown that by mapping data to eigenspaces, the sea clutter level decreases due to its non- deterministic behaviour while point targets' levels remain unchanged. The high resolution (subspace-based or eigenspace) methods and frequency tracking method for slowly time varying frequencies are evaluated to suppress this sea clutter to enhance detection of weak signals. Experimental results verify the advantage of subspace-based methods over the traditional processing techniques.

Conventional subspace methods can be utilized to enhance the detection, but they deteriorate dramatically in the presence of correlated sea clutter. In our thesis some adaptive sea clutter pre-filtering schemes are introduced which improve the threshold and accuracy of subsequent subspace methods. Both simulated and real ship targets are used to verify the effectiveness of our proposed method.

Furthermore, we propose another novel subspace algorithm to estimate the directions of arrival of superimposed cisoidal radar echoes from far-field targets in the radar pulse domain. The improvement provided by this algorithm is based on the use of a state space model that more accurately represents the received Doppler radar array signal prior to spatial processing such as MUSIC. A 2-d (spatial and temporal) pre-filtering matrix is structured and applied to the received array signal, which is finally combined with the high-resolution (MUSIC) method for DOA estimation. Lower resolution threshold and

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estimation variance are achieved by this algorithm compared to conventional beam-space MUSIC and sensor-space MUSIC. A simplified theoretical resolution threshold is derived, and both the theory and simulations verify the effectiveness of our proposed algorithm. Results from an experiment using a simulated target superimposed on real HF

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Abstract

ii

List of Tables

viii

List of Figures

ix

List of Symbols

xiv

Acknowledgments

xviii

Dedication

xix

1

Introduction and Thesis Outline

1 ...

1.1 Statement of Problems . 3

...

1.2 Main Contributions .6

...

1.3 Outline and Contents .7

2 HF Radar Target Detection in Sea Clutter

-

State of the Art

3 Theoretical Background and Foundation

31 ...

3.1 Introduction 31

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3.2 Array Signal Model 32

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5.3.3 GSC-MUSIC

...

93 5.3.4 Experimental Results

...

94

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5.4 Conclusions and Discussions -98

6 2-D Prefiltering Based MUSIC (2DP-MUSIC)

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7.2.1 Formulation of Sample Null Spectra for MUSIC, BS-MUSIC and 2DP-MUSIC.. . .

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8 Conclusions and Future Research

128

8.1 Summary

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128 8.2 Suggestions for Future Work

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Bibliography

Appendix A Theory of Multitaper Spectral Analysis

Appendix B Derivation of Eq. (6.24)

Appendix C Derivation of Eq. (6.25)

Appendix D Resolution of Spectral MUSIC

Appendix E Radar Parameters

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List of Tables

Table 4.1 Parameters for injected targets 5 1

Table 4.2 True parameters for experiments 3 and 4 62

Table 5.1 Targets at Doppler -0.08 Hz, range 160 km, 30 trials across range 98 Table 5.2 Targets at Doppler 0.3 Hz, range 80 km, 30 trials across range 99

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List of Figures

Figure 2.1 Figure 3.1 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 Figure 4.8 Figure 4.9 Figure 4.10 Figure 4.1 1 Figure 4.12 Figure 4.13 Figure 4.14

Measured HFSWR sea echo at 3.1 MHz transmitted carrier frequency. The zero Doppler frequency position corresponds to the carrier frequency. Bragg lines show at frequencies f 0.18 Hz

The incident relationship among different sensors The HF radar site sketch map

Block diagram of a HF radar receiving subsystem

Range-Doppler power spectrum image by conventional beamforming

Cells for estimating the array signal covariance matrix (Dark cell contains targets)

2-D spatial spectrum of prewhitened MUSIC

1-D spatial spectrum of prewhitened MUSIC at range 100 km 2-D spatial spectrum of standard MUSIC

1 -D spatial spectrum of standard MUSIC at range 100

km

2-D spatial spectrum of Root-MUSIC

1-D spatial spectrum of Root-MUSIC at range 100

km

2-D spatial spectrum of MVDR

1-D spatial spectrum of MVDR at range 100

km

2-D spectrum from conventional beam-forming

1-D spectrum from conventional beam-forming at range 100

km

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Figure 4.16 Figure 4.17 Figure 4.18 Figure 4.1 9 Figure 4.20 Figure 4.2 1 Figure 4.22 Figure 4.23 Figure 4.24 Figure 4.25 Figure 4.26 Figure 4.27 Figure 4.28 Figure 4.29 Figure 4.30 Figure 4.3 1 Figure 4.32 Figure 4.33 Figure 4.34 Figure 4.35 Figure 4.36 Figure 4.37 Figure 4.38

Spatial spectrum of standard MUSIC vs. DOA Spatial spectrum of MVDR vs. DOA

Spatial spectrum of conventional beam-forming vs. DOA 2-D spatial spectrum of prewhitened MUSIC

1-D spatial spectrum of prewhitened MUSIC at range 100 km

2-D spatial spectrum of standard MUSIC

1-D spatial spectrum of standard MUSIC at range 100 km 2-D spatial spectrum of Root-MUSIC

1-D spatial spectrum of Root-MUSIC at range 100 km 2-D spatial spectrum of MVDR

1 -D spatial spectrum of MVDR at range 100

km

2-D spectrum from conventional beam-forming

1 -D spectrum from conventional beam-forming at range 100 krn Spatial spectrum of prewhitened MUSIC vs. DOA

Spatial spectrum of standard MUSIC vs. DOA Spatial spectrum of MVDR vs. DOA

Spatial spectrum of conventional beam-forming vs. DOA 2-D spatial spectrum of prewhitened MUSIC

1-D spatial spectrum of prewhitened MUSIC at range 84.5 km

2-D spatial spectrum of standard MUSIC

1-D spatial spectrum of standard MUSIC at range 84.5 km 2-D spatial spectrum of Root-MUSIC

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Figure 4.39 2-D spatial spectrum of MVDR 6 5

Figure 4.40 1-D spatial spectrum of MVDR at range 84.5 km 66

Figure 4.41 2-D spectrum from conventional beam-forming 66

Figure 4.42 1-D spectrum from conventional beam-forming at range 84.5 km 67

Figure 4.43 Spatial spectrum of prewhitened MUSIC vs. DOA 67

Figure 4.44 Spatial spectrum of standard MUSIC vs. DOA 6 8

Figure 4.45 Spatial spectrum of MVDR vs. DOA 6 8

Figure 4.46 Spatial spectrum of conventional beam-forming vs. DOA 69

Figure 4.47 2-D spatial spectrum of prewhitened MUSIC 69

Figure 4.48 1-D spatial spectrum of prewhitened MUSIC at range 125.8 km 70

Figure 4.49 2-D spatial spectrum of standard MUSIC 70

Figure 4.50 1-D spatial spectrum of standard MUSIC at range 125.8 km 7 1

Figure 4.5 1 2-D spatial spectrum of Root-MUSIC 7 1

Figure 4.52 1-D spatial spectrum of Root-MUSIC at range 125.8 km 72

Figure 4.53 2-D spatial spectrum of MVDR 72

Figure 4.54 1-D spatial spectrum of MVDR at range 125.8 km 73

Figure 4.55 2-D spectrum from conventional beam-forming 73

Figure 4.56 1-D spectrum from conventional beam-forming at range 125.8 km 74

Figure 4.57 Spatial spectrum of prewhitened MUSIC vs. DOA 74

Figure 4.58 Spatial spectrum of standard MUSIC vs. DOA 75

Figure 4.59 Spatial spectrum of MVDR vs. DOA 75

Figure 4.60 Spatial spectrum of conventional beam-forming vs. DOA 76 Figure 5.1 Doppler frequencies vs. time; means are [-0.1787 -0.0037 0.0795 8 1

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xii Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6 Figure 5.7 Figure 5.8 Figure 5.9 Figure 5.10 Figure 5.1 1 Figure 5.12 Figure 5.13 Figure 5.14 Figure 5.1 5

Doppler spectrum after suppressing the first order Bragg lines Doppler spectrum after beamforming but before suppression

Beamformed Doppler spectrum before (Dash line) and after (solid line) SCS. Target at -0.08 Hz. Note suppression of Bragg lines at

-

f 0.18 Hz and noise elsewhere

Pseudo spectra from MUSIC (Dash line) and SCS-MUSIC (Solid line). True DOA -40 deg., SCR 8 dB. Doppler: -0.08 Hz. Range: 160 km

Doppler spectrum from beamforming at -40 degrees. SCR 8 dB Pseudo spectra from MUSIC (Dash line) and SCS-MUSIC (Solid line). True DOA - 4 0 deg., SCR 16 dB. Doppler: -0.08 Hz. Range:

160 km

Doppler spectrum from beamforming at -40 degree. SCR 16 dB Pseudo spectra from MUSIC (Dash line) and SCS-MUSIC (Solid line). True DOA -50 degree. Doppler frequency is -0.0832 Hz. Range is 126 km

(a) The structure of generalized sidelobe cancellor. (b) Extended GSC

Beamformed Doppler spectrums from original array data (dashed trace) and GSC whitened data (solid trace) at DOA -50deg., SCR 6.6 dB

Pseudo spectrum vs. Doppler at -50 degrees between GSC-MUSIC (solid trace) and conventional MUSIC (dashed trace) at range 160 km, SCR 6.6 dB

Beamformed Doppler spectra from original array data (dashed trace) and GSC whitened data (solid trace) at DOA -50deg., SCR 9.4 dB Pseudo spectrum vs. Doppler at -50 degrees between GSC-MUSIC (solid trace) and conventional MUSIC (dashed trace) at range 160 km, SCR 9.4 dB

Pseudo spectrum vs. Doppler at -32 degrees between GSC-MUSIC (solid trace) and conventional MUSIC (dashed trace) at range 102.5 km

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...

X l l l

Figure 5.16 Pseudo spectra vs. DOA from MUSIC (Dashed line) and GSC- 97 MUSIC (Solid line). True DOA -30 deg., Doppler: -0.0896 Hz. Range: 102.5 krn.

Figure 6.1 Pseudo spatial spectrum from our 2dP-MUSIC method, MUSIC and 116 BS-MUSIC at SNR -5 dB. True DOA 1 degrees

Figure 6.2 Pseudo spatial spectrum from our 2dP-MUSIC method, MUSIC and 117 BS-MUSIC at SNR 5 dB. True DOA 1 degrees

Figure 6.3 Mean squared error. Solid line: our 2dP-MUSIC method. Dash-dot 117 line: BS-MUSIC. Dotted line: MUSIC. 100 trials

Figure 6.4 Doppler spectrum of sea clutter with target at range 100 Km and 118 sensor 1

Figure 6.5 Pseudo spatial spectrum from our 2dP-MUSIC method, MUSIC and 118 BS-MUSIC at SCNR 1 dB. True DOA 0 degrees

Figure 7.1 Probability of resolution versus SNR. Solid line: our 2dP-MUSIC 126 method. Dash-dot line: BS-MUSIC. Dotted line: MUSIC. 100 independent trials

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xiv

List of Symbols

Transpose Conjugate transpose Kronecker product Expectation operator Direction of signal arrival Radar carrier wavelength Eigenvalues of matrix Ryy Eigenvalues of matrix R, Eigenvalues of matrix R Variance of noise Time delay

Signal direction vector Signal directions matrix Beamformer matrix

Speed of light

Distance between two sensors Null spectrum

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g : Acceleration of gravity hi : Eigenvectors of matrix Ryy

h, : Eigenvectors of matrix Rxx h, : Eigenvectors of matrix R

Imxm : Identity matrix with dimension m

k : Number of signals

n ( t ) E Rm : Additive spatially white noise vector

m : Number of array sensors

m ' : Dimension of Beamspace

PMa : Spatial spectrum of MUSIC Rxx E Rm'xm' : Covariance matrix of vector x ( t )

Ityy E RmXm : Covariance matrix of sensor output y ( t )

*YNYN : Covariance matrix of vector y , ( t )

s ( t ) E Rk : Temporal signal vector reflected by the k non-coherent scattered sources

t : Time

I? : Total number of snapshots

Us

: Signal subspace

U, : Noise subspace

v(e) : Steering vector

w : Beamformer weight vector

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xvi

y ( t ) E Rm : Sensor array snapshot output at time instant t Y N ( ~ ) : Nm -vector formed from N array signal vectors y ( t ) 2DP-MUSIC: 2-D spatial-temporal Prefiltering based MUSIC ASNR: BS-MUSIC: CBF: CFAR: CRB: DOA: Doppler: DPSS: EEZ : EM: ESPRIT: GRT: GSC: HF: HFOSR: MEM: ML: MUSIC: MVDR: nrn:

Array Signal to Noise Ratio Beamspace MUSIC

Conventional Beamforming Constant False Alarm Rate Cramer-Rao Bound Direction of Arrival

Equivalent to frequency domain Discrete Prolate Spheroidal Sequence Exclusive Economic Zone

Electromagnetic

Estimation of Signal Parameters via Rotational Invariance Techniques Gross Registered Tonnage

Generalized Sidelobe Canceller High Frequency

High Frequency Ocean Surveillance Radar Maximum Entropy Method

Maximum Likelihood

Multiple Signal Classificaiton

Minimum Variance Distortionless Response Nautical Mile (1 nm = 1.852 km)

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xvii

OTH: Over the Horizon

PEF: Prediction Error Filter

PSD: Power Spectral Density

Pulse domain: Equivalent to temporal domain

RCS: Radar Cross Section

SCR: Signal to Clutter Ratio

SCS-MUSIC: Sea Clutter Suppression MUSIC Snapshot: Instant array output

SNCR: Signal to Noise and Clutter Ratio SNR: Signal to Noise Ratio

SVD: Singular Value Decomposition S WR: Surface (ground) Wave Radar

UHF: Ultra High Frequency

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xviii

Acknowledgments

I would like to express my deepest gratitude to my supervisor, Dr. R. Lynn Kirlin, although I know any word is far away from enough. It has been my great pleasure to have met and worked with him, and from him I always get much more help than I expect. His kindness, generosity, patience and invaluable ideas help me out of every difficulty from my first day here. I respect him not only because of his specialty knowledge but also because of his personality. He definitely is a perfect supervisor.

I would like to thank Dr. A. Zielinski and Dr. P. Driessen for their important advice about my thesis work. Also I should thank Dr. S. Dosso and Dr. J. Ritcey for their kind help about the thesis revising and improvements. I wish to express my special thanks to Dr. W. S. Lu for his kind discussion and encouragement. I also appreciate the kind discussions and data providing from Dr. A. M. Ponsford and Dr. R. Dizaji in Raytheon Canada Ltd.

Many thanks to Mrs. Vicky Smith since her help and advice keep me on the right track of graduation, also thanks to Mrs. Moneca for her help. Thank Mr. Steve Campbell for his in time technical help.

I would like to express my gratitude from deep my heart to my dear parents who are a big part of my life. They are always encouraging me towards my goals. I am greatly indebted to my wife, Xiaoli, for her continuous support and encouragement. Also thanks her to feed me well.

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xix

To my dear parents, and my dear wife

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Chapter

1 Introduction and Thesis Outline

Radar techniques have been developed dramatically and applied widely in both military and civilian fields since World War 11. Radars at lower frequencies suffer a crowded, limited spectrum and wide beam width. In practice most radars operate at ultra high fi-equency (UHF) and microwave frequency due to their requirement for a relatively large portion of electromagnetic (EM) spectrum, but there are notable exceptions. In some fields such as surveillance radar over the sea, the range of the microwave radar is only 20 km, although that can be extended to 60 km by elevating the radar antenna. Under such circumstances high frequency (HF) radar (3-30MHz, decametric waves radar) has been given special attention due to the unique property that its EM radiation can propagate beyond the horizon [I] [2]. This is achieved by either surface wave diffraction around the curvature of the earth: surface (ground) wave radar (SWR), or sky wave refi-acted by the ionosphere: skywave radar. By this means the HF radar can sense far beyond the line of sight, and typically the range of HFSWR radar can be extended to the order of 400 krn, and HF sky wave radar to 4000 km or more. Therefore HF radar is also called over-the- horizon (OTH) radar.

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1. Introduction and Thesis Outline 2

Recently HF radar has been widely applied to ocean surface surveillance which monitors ships and aircraft within the 200 nautical mile (nm) Exclusive Economic Zone (EEZ). In addition more and more oceanographers have begun to apply HF radar for remote sensing of the sea states and mapping ocean surface parameters (currents, winds speed etc.), largely because this technique is land based and can be used in all weather conditions.

The main factor limiting the detection performance of HF radar is not the internal sensor thermal noise but external noise and clutter. When HF radar is applied to the ocean surroundings as mentioned above, sea clutter becomes the focus of study. Sea clutter or sea echo can be defined as the backscattered returns from a patch of sea surface illuminated by a transmitted radar signal. Sea clutter is either considered interference when the object is to detect and track surface targets (including low-flying aircraft, ships, navigation buoys, icebergs or growlers floating in the ocean) or considered signal when the object is to gather information about the sea surface. Thus the study of sea clutter is not only of theoretical but also of practical importance since it either severely deteriorates the detection performance of radar returns from targets on or near the sea surface or it contains information about the sea surface.

The experimental data processed in this thesis are from the East Coast HFSWR demonstration programme, which is the accumulation of more than 15 years of research and development into the use of HFSWR for the real-time, continuous, all weather surveillance of ships and aircraft within the 200 nm EEZ. This programme has been a collaborative, cost shared, project between the Canadian Department of National Defence

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I . Introduction and Thesis Outline 3

and Raytheon Canada Ltd. to develop and demonstrate the performance of HFSWR for monitoring activity within the 200 nm EEZ.

The system was initially designed to reliably detect targets greater that 3000 gross registered tonnage (GRT) out to 250 nm. Interest is now being expressed in tracking targets that are significantly smaller. The detection of small targets ( 4 0 0 0 GRT) is hampered by the presence of a continuum in the ocean clutter spectrum. This continuum is the result of second-order or high-order sea clutter. In a clutter limited situation the radar range equation becomes independent of radiated power and simplifies to the ratio of the target radar cross section to the radar patch area multiplied by the ocean clutter cross section at the target's Doppler. Detection can only be improved by either reducing the radar patch or suppressing the clutter. Alternatively, the radar can be operated at a higher frequency where the target may enter the resonance region and its cross section significantly enhanced. However, increasing the radar frequency may be constrained by the restricted frequency spectrum and also results in a limited detection range.

Typically a target must be 10 dB above the contending noiselclutter level in order to be successfully detected with a reasonable degree of accuracy and an acceptable false alarm rate. It is the intention of this thesis to evaluate and develop potential methods for improving the detection of small targets that fall below this detection threshold among clutter.

1.1 Statement of Problems

The problems addressed and studied in this thesis are listed and described in this section, which relate mostly to the signal processing aspects rather than EM theory or antenna

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1 . Introduction and Thesis Outline 4

design. Our emphases in this thesis focus on array signal processing techniques that can enhance the detection of a slow moving target (such as ships) using HF Ocean Surveillance Radar (HFOSR).

Detrimental Detection Factors

The main factor limiting the detection performance of HFOSR in the final 2-D range- Doppler image is the external noise and clutter. Those factors mainly include impulsive noise, ionospheric clutter and sea clutter. The impulsive noise will give rise to the total noise floor of the 2-D image and blank most of the targets, but it is comparatively easier to deal with since it appears as a strong short duration pulse in the temporal domain. The ionospheric clutter has a complicated nature and is highly non-stationary. It is usually negligible during daytime due to the existence of an absorbing ionospheric layer: D layer, but it severely hampers the effective detection range during nighttime. In our thesis we will focus on signal processing techniques to suppress sea clutter in daytime data, whose characteristic is well observed in the Doppler domain. The first order Bragg lines [3] dominate the Doppler power spectrum of the sea clutter; there is also a continuum spectrum due to second or higher order scattering. In general the Bragg lines are much stronger than all targets and result in two blind zones. However, because Bragg lines' frequency width is very narrow, the detection and direction of arrival (DOA) estimation of ship targets with small radar cross-section (RCS) are mainly hampered by the presence of the high order sea clutter spectrum.

Array Processing Techniques

In HFOSR the original array processing technique to form a narrow beam is conventional digital beamforming (Bartlett or FFT beamformer). The Bartlett

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I . Introduction and Thesis Outline 5

beamformer is a natural extension of classical Fourier-based spectral analysis to sensor array data. From an optimization point of view, this algorithm maximizes the power of the beamforming output for a given input signal. In spite of the available data quality, the Bartlett beamformer has the same resolution limitation as Fourier analysis, which is physically determined by the number of receiving array sensors.

Due to the supper-resolution (equivalent to a smaller radar patch area) characteristic of subspace methods, we would expect an advantage of applying them in improving the weak target detection. The ocean surface behaves as a distributed source in contrast to targets that are point sources. It will be shown that by mapping data to eigenspaces, the sea clutter level decreases due to its non-deterministic behaviour while point targets' levels remain unchanged.

Preprocessing

Sea clutter is known to be correlated in both spatial and temporal domains. In the presence of correlated clutter, conventional subspace methods give a high threshold and degrade considerably compared to white background noise. We will propose and study sea clutter pre-suppression based subspace methods which combat the correlated sea clutter and achieve better estimation accuracy and lower threshold.

We list the specific objectives studied in this thesis as follows:

Objective 1 To develop a novel application of subspace methods to the detection of weak ship targets in sea clutter. By taking advantage of the super-resolution of subspace methods, we hope to enhance the detection of weak targets which are usually buried in the sea clutter in the convention digital beamforming processing.

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Objective 2 To propose and analyze various preprocesses in order to suppress or pre- whiten the sea clutter before applying subspace methods. Preprocessing will help subspace methods to detect the targets at a much lower threshold and with better accuracy.

Objective 3 To propose a 2-D (spatial and temporal) pre-filtering based Multiple Signal Classification (MUSIC) algorithm for DOA estimation in the temporal (pulse) domain instead of the Doppler domain in order to overcome the problem of little data (lack of snapshots).

Objective 1 is addressed in chapters 3 and 4. Objective 2 is addressed in chapters 5, and objective 3 is addressed in chapters 6 and 7.

1.2 Main Contributions

The main contributions of this thesis are:

A novel use of the 2-D range-Doppler MUSIC pseudo spectrum map. This map is shown to be more appropriate for target detection than the conventional 2-D power spectrum map from digital beamforming.

Combinations of subspace methods (MUSIC and Root-MUSIC) with singular value decomposition (SVD) based suppression techniques for the temporally and spatially correlated sea clutter and noise.

Presentation of a generalized sidelobe canceller (GSC) like adaptive space-time whitening process which combines with the subspace method to provide a lower threshold detector for a weak ship target signal embedded in sea clutter.

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1. Introduction and Thesis Outline 7

o A new 2-D spatio-temporal pre-filtering matrix that, when combined with the

subsequent high-resolution DOA estimation algorithm, provides higher resolution, lower detection threshold, and lower estimation bias and variance than those of conventional high-resolution methods that use only spatial pre-filtering. A performance analysis for the 2-D spatio-temporal pre-filtering based MUSIC, where a simplified derivation is made possible by considering the specific physical configuration. A close fit is observed between the theoretical values and simulations.

1.3 Outline and Contents

A literature survey on state of the art of HF radar target detection in sea clutter is presented in chapter 2. Different statistical models have been reviewed for sea clutter; these include Rayleigh, Weibull and K distributions. Discussion on chaos and fractal theories have also been presented as possible sea clutter models. In chapter 2, we also review signal processing techniques for target detection in sea clutter, which include high resolution spatio-spectral methods, time-frequency analysis and AR modeling etc. We place the emphasis on the signal processing techniques.

In chapter 3, we build the theoretical background and foundation for the thesis. Firstly we will give a brief description of a widely adopted linear array snapshot model, based on which we will discuss the conventional processing procedures in HF radar based on digital beamforming techniques. We then present the background and principles of subspace methods with the focus on sensor domain traditional MUSIC, which is believed to be the most promising representative among its class.

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Next in chapter 4 we address a novel use of the 2-D range-Doppler MUSIC pseudo spectrum map instead of the conventional 2-D digital beamforming power spectrum map for the target detection. Both simulation and experimental results from an investigation into the use of above technique to enhance target detection are shown and verify our procedures.

In chapter 5 we present a novel scheme for using HFSWR radar to detect slow weak targets embedded in sea clutter having a continuous spectrum and temporal correlation. We propose adaptive space-time algorithms which first pre-whiten or suppress the sea clutter in the sensor-time domain and then follow with a subspace method (MUSIC) in the Doppler domain to detect the targets. Simulation results demonstrate the effectiveness.

In chapter 6 we propose a new 2-D (spatial and temporal) pre-filtering based MUSIC (2DP-MUSIC) algorithm to estimate the DOAs of superimposed cisoidal radar echoes from far-field targets. The improvement provided by this algorithm is based on: the use of a state space model that more accurately represents the received Doppler radar array signal prior to spatial processing, and the 2-D pre-filtering which reduces the interference from groups of signals, noise and clutter having spatial or temporal spectra outside the space-frequency region of interest. Lower resolution threshold and estimation variance are achieved by this algorithm compared to conventional beam-space MUSIC and sensor- space MUSIC. Both the theory and simulations verify the effectiveness of our proposed algorithm. Results from an experiment using a simulated target superimposed on real HF radar sea clutter also confirm the algorithm.

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I . Introduction and Thesis Outline 9

We analyze the asymptotic performance (number of snapshots goes to infinity) of our proposed 2DP-MUSIC algorithm in chapter 7. We first give a simplified theoretical derivation of the resolution capability which is characterized by the lowest possible threshold above which the two closely spaced emitters can be resolved. We then compare these asymptotic expressions to simulations in order to assess their accuracy for smaller number of snapshots.

The thesis results are summarized in Chapter 8 where we also provide suggestions for future research work.

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2. HF Radar Target Detection in Sea Clutter - State of the Art 10

Chapter 2 HF Radar Target Detection in Sea

Clutter

-

State of the Art

2.1 Background

Sea clutter was first observed by HF radar during World War 11, and the early processing techniques were simply to null out the slow time-varying clutter by filter banks and thereafter to concentrate on aircraft detection. After that Fourier analysis was utilized to study the temporal structure of the back-scattered sea clutter. In 1955, Crombie [3] first found that there were two well-defined spikes symmetrically placed around the radar carrier frequency in the Doppler power spectrum of the sea clutter. In addition Crombie found that the displacement of these spikes appeared to vary with the square root of the carrier frequency and their amplitudes were dependent on wind and sea state conditions. These are called the first order Bragg lines since they are similar to the X-ray scattering mechanism in crystals identified by W. L. Bragg. Since all gravity sea waves of given wavelength are known to have a given velocity,

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2. HF Radar Target Detection in Sea Clutter - State of the Art 1 1

v = & Z G , (2.1)

where g is the acceleration of gravity and L is the wavelength of the gravity waves, we can calculate the Doppler shift (

Af

) by

Af

= 2 v r / i l (2.2)

where il is the radar wavelength. Combining equations (2.1) and (2.2) and assuming v = vr (for ground wave), Crombie found that although all sea waves interacted with the radar wave, only the gravity waves with exactly one half of the radar wavelength and which are moving toward (positive Doppler shift) and away from (negative Doppler shift) behaved as a diffraction grating and reinforced echoes. In fact this finding forms the basis for remote sensing of sea states.

Figure 2.1 is an example of the measured surface-wave sea clutter Doppler spectrum at carrier frequency 3.1 MHz. The back-scattered clutter were taken at range 100 km from the receiver antenna, and the Fourier transform was taken over 164 s, providing a Doppler frequency resolution 0.006 Hz. The first order Bragg lines exist at f,=

i,/G

= iO.18 H z , where c is the EM wave velocity in vacuum and

f,

is the radar

carrier frequency. It is obvious that there is a continuum spectrum around the first order Bragg lines, and this continuum is of second or higher order scattering.

2.1.1. Sea Clutter and Oceanography

Although Crombie (1955) suggested using HF radar for measuring sea states, few efforts appeared in the following decade. After 1966 a couple of theoretical discoveries

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2. HF Radar Target Detection in Sea Clutter - State of the Art 12

([4] [5] [6]) indicated the relationship between the first and higher order scattering and sea states parameters. An excellent review over 12 previous years of early HF radio oceanography experiment and theoretical discovery was published by Barrick [7] in 1978. He summarized how to extract the information of wave height spectrum, surface winds and currents from the sea scattered echoes spectrum. The wave height spectrum is related to second order Bragg spectrum. The first order Bragg lines are in general not of equal amplitude and the difference can be used to measure wind direction.

Figure. 2.1. Measured HFSWR sea echo at 3.1 MHz transmitted carrier frequency. The zero Doppler frequency position corresponds to the carrier frequency. Bragg lines show

at frequencies 5-0.18 Hz.

The Doppler shift calculated above is under the assumption that there are no surface currents changing the motion of the gravity waves. If however there is surface current with nonzero velocity, the frequency will be shifted further depending on the magnitude and direction of the current's velocity. So the current can be found by measuring the further frequency shift from the original Bragg lines caused by only gravity wave motion,

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2. HF Radar Target Detection in Sea Clutter - State of the Art 13

but the velocity calculated is only radial velocity (velocity projection onto the ray pointing away from the receiver). In order to get full current velocity vector information, it is common to employ two radars with overlapping beams.

In 1983 Shearman [8] provided another illuminating review on radio science and oceanography, where he presented a good description of the physical underlying mechanisms of various clutters, other sea-associated phenomena and Doppler measurements that can be used to estimate those parameters. He also summarized three reasonable physical models for second order scattering:

The sea waves are not sinusoidal but trochoidal (sharp crest and broad trough) caused by the circular motion of water particles, which can be decomposed as a fundamental sinusoid with its harmonics travelling at the same velocity. Therefore, the second order scattering will take place at

f,

, where n=2, 3,

. .

. and

f,

is the Doppler frequency of the first Bragg lines.

Radar waves are scattered from two sea waves travelling at a right angle difference in direction; this phenomenon will generally form a spectral peak at 2314 .

fb

.

Sea waves interact with each other and result in a wave with the exactly half wavelength of the radar wave, which contributes to the continuum second order spectrum. Shearman also pointed out that the first mechanism could be viewed as a special case of this mechanism when the wave is interacting with itself.

These mechanisms are helpful to understand how to invert the radar Doppler spectrum to yield the wave height directional spectrum mathematically.

Kingsley [9] provided background on the relationship of the sea-wave to the Bragg lines in the Doppler spectrum 2nd target features for ice, ships, currents and wind. He

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also discussed the possible reasons for the Bragg line's broadening and the appropriate dwell time for the Bragg process to remain coherent for a given wavelength and resolution cell size, but only tentative conclusions were presented. A modem study in the coherence and broadening of sea echoes is given by Parkinson in his paper [lo]. Based on the analysis of a large number of experiments he summarized all kinds of broadening mechanisms and suggested that coherence broke down in shallow water and with higher carrier frequency.

2.1.2. Physical Scattering Model and Radar Cross Section

Wright [ l l ] derived the radar cross section (RCS) for sea clutter by applying first order scattering, and he verified his theory by experimenting with P and X-band radars. However, these results are not applicable to the HF band.

The first literature which studied HF ground wave radar cross sections of the ocean surface is presented by Barrick [12] in 1972. Another modem paper [13] by Gill et al. presents the theoretical analysis of second-order RCS of the ocean surface for bistatic radar. These two papers are all based on the Walsh scattering theory but they lack experimental verification.

2.1.3. HF Radar Detection and Sea Clutter

Based on a mathematical model, the complete sea echo Doppler spectrum is a function of sea state and radar carrier frequency as presented in [5] [12]. Maresca et al.

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1141 studied the theoretical limitation on ship detection performance imposed by the sea clutter. They concluded that the detection performance is improved by increasing the working frequency, while lower frequencies are more sensitive to the sea state. Larger ship detection is only limited by strong first order echoes over a very limited portion of the Doppler spectrum which is non-sensitive to sea state. However for small ship detection the sea state is an important parameter. Detailed detectability and blind (unobservable) speeds are described as a function of several parameters: radial velocity of ships, sea state, operating frequency, coherent integration time and spatial resolution of the radar. However, all the theoretical analysis and experiments were based on skywave radar and high signal to noise ratio (SNR) with minimum 10 dB. They also did not consider the ionospheric multipath effects.

Although mature auto detection and tracking algorithms have long been developed for microwave radar, Ralph 1151 first introduced those algorithms for HF groundwave radar in 1988. He presented the required algorithms for thresholding, plot extraction, tracking and plot association, which have formed a good basis for further advanced techniques. Only simulated data were tested; there is yet a need for real data to test those algorithms.

As we have mentioned above, for remote sensing of sea states we need to study and analyze sea clutter to extract the desired information. For radar target detection in sea clutter it is necessary to remove or suppress sea clutter. In addition, the optimization of the detection performance heavily depends on a detailed knowledge and understanding of the back scattering echo's statistical properties, which will enable the development of

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2. HF Radar Target Detection in Sea Clutter - State of the Art 16

suitable signal processing techniques. These have formed the driving force in the development of models of sea clutter.

2.2 Statistical Models for Sea Clutter and Related

Techniques

When radar illuminates a large patch of the sea, the probability distribution of the envelope of the return signal can be well approximated by the Rayleigh distribution [16]. This can be well understood according to the central limit theorem since the signal can be considered as the sum of randomly phased components from a large number of independent scatterers. However, for high-resolution radar the Rayleigh model cannot fit the data well. In an effort to provide better fits to the amplitude distribution of sea clutter log-normal [17], Weibull [IS] and K- [19] distributions were proposed in the 1970s. Among them it is believed that K-distribution is the most promising one to date for high- resolution radar and it has consequently attracted more and more attention. Hence the conventional receivers based on the Rayleigh model are no longer optimal, and their detection performances degrade greatly. Furthermore, CFAR detection is not guaranteed.

The K-distribution was first introduced by Jakeman et al. [19] to model microwave sea echo in 1976. Ward [20], Watts [21] and others followed with a series of papers that provided empirical evidence supporting of the K-distribution for modeling sea clutter. The basics of the compound K- distributions [20] are as follows:

Two sea clutter components with different correlation times contribute to the amplitude distribution. The fast varying component can be identified with the changing interference

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between scatters and has a correlation time on the order of 10ms. It can be decorrelated from pulse to pulse by the use of frequency agility. The slowly varying component can be associated with a bunching of scatters and has a correlation time in the order of seconds; it is unaffected by frequency agility. Thus suppose the overall amplitude z of sea clutter is represented by the product of two independent random variables: z = xy , where x has a long correlation time and y is decorrelated by frequency agility. y has a Rayleigh distribution, and x is observed to have the chi distribution with the density function

It is shown that lzl = R has K-distribution. The K-distribution function is defined in terms

of K-Bessel function

Kv

( c ) by

where we define c = 2R&, b is the scale parameter and v is the shape parameter. Ward et al. provided a detailed discourse on the evidence and statistical analysis which leads to the K-distribution model in their papers [22] [23]. They also described the

performance prediction for a cell-averaging CFAR in real sea clutter. A more recent one can be found in Watts's paper [30].

Conte et al. [24] presented an extensive study on the parametric detector in K-

distributed clutter using a generalized Neyrnan-Pearson strategy. However such a detector could be too complex to implement in a real system, and an alternative procedure based on a rank test was presented by Zoubir in his paper [25] for HF radar.

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2. HF Radar Target Detection in Sea Clutter

-

State of the Art 18

Farina et al. [26] performed a detailed analysis of experimental data. They concluded that the sea clutter exhibits K behavior for vertical-vertical polarization, but log-normal for HH polarization. Their conclusions are based on microwave radar. Authors in [27] tried to build the mathematical model based on K-distribution to simulate the sea clutter.

The K-distribution model has been given a deeper theoretical foundation by Bisceglie et al. [28]. They showed the relationship between the K-distribution, the random walk model, and the physical processes which regulate the scattering inside the radar cell. At the same time some effort on how to translate to normal distribution for sea clutter has been done by Smith [29].

However, for very high-resolution radars the K-distribution model also fails; this may be due to the reason that the K-distribution model or its extensions have not accounted for the underlying dynamics responsible for the generation of sea clutter.

2.3 AR Based Methods and SVD Methods

An early paper that suggested adaptive techniques to improve target detection was presented by Gjessing [31], who suggested multiple frequencies chosen according to modeled or known target geometries for enhancing detection, implying a priori knowledge about both targets and backgrounds. The author concentrates on the multifrequency radar system that is similar to a spaced antenna system, but there was no experimental verification.

Bourdillon et al. [45] adopted maximum entropy method (MEM) to improve the ship detection by correcting the phase of a HF skywave sea echo signal contaminated by the

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2. HF Radar Target Detection in Sea Clutter

-

State of the Art 19

ionosphere. The MEM is equivalent to fitting an autoregressive model to a random process. After the correction and FFT processing a better quality average power spectrum results, and hence the ship target is more easily detected.

Wang et al. showed the advantage gained by using frequency diversity signaling and coherent wide-band high-resolution processing for tone frequency estimation in radar system with nonfluctuating targets [46]. Simulation results and analytical analyses were presented to demonstrate that this method could provide more accurate estimation of the tone frequency.

It is apparent that Moses and Carl [32] first applied autoregressive techniques to model radar data without considering the effects of clutter. The use of an AR model for radar data can be easily justified for microwave radars since for such wavelengths radar targets are often well approximated by a small number of scattering centers, and temporal or spatial AR models are effective at representing signals that are narrow band in frequency or narrow angle in azimuth, respectively. These authors extracted the features of radar targets via their estimated M coefficients, and used those as the basis of classification. Different sets of features for two sets of aircraft data were presented based on X-band radar data. More discussion about AR techniques in radar classification can be found in [3 31.

Nohara et al. presented an M-based detector for growler detection in sea clutter and compared the detection results with a non-coherent detector based on real X-band radar data in their paper [34]. In such working frequencies the sea clutter can be considered as a wide band signal with a fixed center frequency. The authors point out that the spectral width is an ideal statistic to discriminate a growler from sea clutter since in general a

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2. HF Radar Target Detection in Sea Clutter - State of the Art 20

growler has narrower spectral width than does sea clutter. The basic procedure is to estimate the AR coefficients from a short data record first and then compare the pole with the largest magnitude to a threshold, since the larger the magnitude of the pole the narrower the corresponding width. If the threshold is exceeded, a growler is declared. The authors here arbitrarily choose 6 as the order of the AR model.

Bouvier et al. [41] proposed a new technique for modeling and simulating the sea clutter by multiplying a modulating random variable and a correlated complex Gaussian process which is obtained by filtering a complex white Gaussian process through a low order AR model. The simulated sea clutter is compared with real S-band radar sea clutter data.

Martin et al. 1421 applied an AR technique to remote sea current sensing for HF radar. The approach is more accurate in determining the radial component of sea current than is the conventional FFT. The algorithm can be implemented in real time and can form the basis for further signal processing improvements in an HF radar system.

An important model of sea clutter for HF radar was introduced by Khan 1351 in 1991, where the dominant components of sea clutter were modeled by two narrow-band signals with slowly time-varying frequencies (angle modulated), centered on the first Bragg frequencies. This time-varying model has a significant impact on the processing of HF radar data in the background of ocean, and a series of papers [36] [37] [38] followed this model to improve target detection or sea state parameters measurement. The author concludes that within a short observation time (less than one minute) the sea clutter signals are strictly non-stationary. It is obvious that optimal processing of such HF radar data can be achieved by a low-order linear prediction filter with time-varying

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2. HF Radar Target Detection in Sea Clutter - State of the Art 2 1

coefficients. Since the prediction error filter (PEF) is an all-zero filter, it can be shown [39] that the transfer function of the PEF has M zeros on the unit circle at the locations corresponding to the angular frequencies of the M sinusoids in the signals. By this means the time-varying model for sea clutter is verified by Khan through an adaptive PEF. The experimental results coincide with the theory very well. Sea clutter suppression for target detection is also exhibited in [35] by use of an adaptive lower order prediction filters. The results are promising since the first order sea clutter signals are suppressed by about 40 dB, and high order sea clutter signals are smoothed and suppressed. All experiments are based on the CODAR (Coastal Ocean Dynamics Applications Radar) system, whose working frequency is 25.4 MHz. The author explored only the basic LMS adaptive algorithm. We expect better results from more advanced adaptive techniques [43] which thus deserve future study.

DiMonte and Arun [40] gave one of the first papers dealing with the temporal variation of sea clutter statistics. They show that the complex data covariance matrix from multiple sinusoids has a rank equal to the number of sinewaves present even if the frequencies are slowly changing, and therefore the instantaneous Doppler frequencies can be estimated in a time-varying fashion using eigenstructure methods.

In their paper [36] for suppressing sea clutter in HF radar, Poon and Khan combined the sea clutter model [35] and the new method [40] of tracking time-varying frequencies of superimposed harmonics by singular value decomposition (SVD) and rank reduction of a Hankel matrix of the time series data. The clutter suppression technique presented there is based on the theory that the SVD of the Hankel matrix of radar signals (complex, time domain) which contain a finite of narrow-band time-varying sinusoidals can be

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approximated by the same finite number of dominant singular values even if the sinusoidal frequencies are varying slowly with time. Nevertheless, the method is only suitable for small ocean range cells with dimensions around a kilometer; for larger cells there may be need for a larger number of singular values. The frequency components corresponding to the sea clutter can be removed from the original radar signals, and another reduced rank Hankel matrix is constructed, giving a new time series. By this means the sea clutter can be suppressed in order to improve the target detection performance. The simulation results show that both first and higher order sea clutter are suppressed substantially more than 20 dB while at the same time keeping target strength unaffected. From this point of view, this method is better than the method presented in [35] wherein only first order sea clutter is suppressed well. But there is no real radar data testing for this technique.

The same methodology was adopted in [37] to estimate the ocean surface currents with HF radar. The results show that this parametric estimation outperforms the conventional FFT-based method since it provides a significantly lower variance. Again, the reason is that the Hankel rank reduction method in association with the SVD is very efficient in tracking the Bragg components of sea clutter. The results are impressive but the method is computationally intensive. The authors also suggest combining this method with other high-resolution spectral analysis techniques to achieve better estimation results.

In 1997, Khan et al. [38] applied their method of [36] to real radar data to suppress the sea clutter. The experimental results demonstrate that this SVD method is effective

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2. HF Radar Target Detection in Sea Clutter - State of the Art 23

even when the Doppler frequencies of the target and clutter are very close to each other (the velocity difference between target and first order sea clutter is only 1.6 knots).

The above-discussed methods have a common advantage in that they are computationally efficient, and a real time implementation of these techniques for currently operational systems is practical. They are especially suitable for short time series and close tonal frequency components. However, it should be pointed out that AR modeling normally requires a high SNR, therefore the processes to be modeled must be first filtered from background interference and noise.

An alternative using an extended Kalman filter as demonstrated by Hedstrom and Kirlin [44] allows separation of at least 2 totally co-channel (co-Doppler) angle modulated signals if the state models of their modulations are distinct and carrier amplitudes are within 10dB.

2.4 High Resolution Methods

A pioneering paper that applies super resolution methods (signal subspace) to HF radar was given by Barnum [47], who discusses sea clutter limitations in HF Skywave radars for enhancing the Doppler spectral resolution. He presented an overview of the techniques needed for ship detection by HF skywave radar. He has also shown that shorter times may be used by considering spectral estimation methods other than the FFT thus permitting faster scanning for targets.

Yang et al. [48] applied a maximum entropy spectral analysis technique to the received HF radar data having correlated outer noise and sea clutter in order to achieve

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higher spatial angular resolution. This technique was supposed to resolve the targets within one beam width. All figures in their paper were based on real sea clutter plus simulated targets. The authors concluded that this method was not suitable for lower signal to noise ratio; it failed when SNR was lower than 12 dB. In addition sea clutter itself will try to decrease the spatial resolution due to its correlation.

In 1998 Xie, working with data from the Weihai HFSWR system 1491 in Shandong Province, China, and his colleagues from the Harbin Institute proposed and demonstrated a super-resolution technique [50] based on a Pre-whitened MUSIC algorithm to enhance the detection accuracy of targets embedded in sea clutter and atmospheric noise. By utilizing the neighborhood ranges and Doppler cells [15], the authors first estimated the

A

spatial noise covariance matrix R, of the background noise and sea clutter and the

A

spatial covariance matrix of the array signal R . In principle atmospheric noise is spatially white, non-coherent noise, while sea clutter is a coherent noise. Letting the

A

Cholesky decomposition of the estimated spatial covariance matrix be R, = c H c . Then

A

the pre-whitening process is embodied by letting = C - ~ R C - ' = C - ~ R ~ C - '

+

I,, where

R,

is the covariance matrix of the signal and I, is an M by M identity matrix. The

spatial spectrum is given as follows:

where M is the number of sensors, K is the number of signals estimated by AIC [51], a(@ = [a, (4) a, (4)

...

a, (@IT, a, (4) = exp[- j2x(i - l)d sin

4

/ A] ,

V i

is the ith eigenvector

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2. HF Radar Target Detection in Sea Clutter

-

State ofthe Art 25

superscripts H and T denote conjugate transpose and transpose respectively. The spatial spectrum is plotted vs. DOA for sea clutter echoes containing one real target and one simulated target in aircraft mode. The experimental results are promising for aircraft detection since the pre-whitened MUSIC can provide higher spatial resolution than the other methods such as conventional beamforming (CBF), minimum variance distortionless response (MVDR) and typical MUSIC. Further study for ship detection is still needed since over the frequency band used, the sea clutter has serious impact. A post processing following the pre-whitened MUSIC process is also presented in [50]; it divides the MUSIC spatial spectrum by the MVDR spatial spectrum of noise in order to eliminate the unnecessary sidelobes.

We point out that improving the angular resolution is equivalent to improving the signal to clutter ratio (SCR), since when we use high-resolution techniques, we indeed look at target within a smaller area of sea clutter. It means that with a fixed target level, we get higher SCR.

If we apply some other super-resolution methods such as ESPRIT [52] [53] for the unknown background noise we might get improved detection or higher spatial resolution since the ESPRIT algorithm has a lower threshold than that of MUSIC. Further study along this line is suggested but not pursued in this thesis.

A more recent paper by Agrawal et al. [54] presented a modified maximum likelihood (ML) estimator of DOA of the multiple source signals in the presence of unknown spatially correlated Gaussian noise. The proposed method does not impose any constraints on the structure of the signal and noise covariance matrices and provides a framework for obtaining an estimate of the unknown noise covariance matrix by

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projecting the observed covariance matrix onto a noise subspace. However this method is restricted to uniform linear arrays and is computationally intensive since it requires multivariate nonlinear optimization for reaching the maximum. The formulation of the likelihood function is somewhat heuristically defined and only simulation results are presented.

Another paper presented by Gini et al. in 2000 [55] studies the estimation of the Doppler frequency of targets from the radar signal with K-distributed clutter by deriving a new joint ML technique. It is verified by experiment with real sea clutter data. The authors show that the sub-optimal ML method is equivalent to the estimation results of ESPRIT and periodogram estimation methods but with much less computation. Unfortunately this method will deteriorate dramatically when the SCR is low.

2.5. The Chaos Model

Chaos is the irregular and complex behavior of a simple deterministic nonlinear dynamical system. This word also represents a system with relatively few degrees of freedom. Chaotic signals, as a special case of signals, lie between regular periodic signals and unpredictable, irregular signals, and the latter is in general referred to as random noise. A chaotic signal is in essence deterministic and short-term predictable, regardless of its pseudo random waveform. It has three basic ingredients: nonlinearity, determinism and positive Lyapunov exponent(s) [62]. The random-looking waveform of a chaotic process and its short- term predictability come from those three basic properties.

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In a study by Haykin and Leung [56] (1991), radar sea echo (sea clutter) was first modeled by using chaos theory. This research uses real radar data to demonstrate the applicability of chaos theory for modeling sea clutter. A finite-dimensional deterministic model was developed using a radial basis function (RBF) network as a predictor of the next value of the dynamic process; it was shown to fit the sea clutter waveform well. More importantly the authors make the case that sea clutter is not a stochastic process but the result of the nonlinear interaction of a deterministic dynamical system (chaotic system). In fact almost all natural processes that exist in the universe are nonlinear in essence, and sea clutter is argued by Haykin et al. to be one such natural process. This new point of view uses a deterministic approach for explaining the underlying sea clutter dynamics instead of the stochastic approach where sea clutter is considered as a random process with a large number of degrees of freedom.

Following paper [56] considerable efforts ([57]-[63]) are made along this line. It is clear that by using a nonlinear predictive model designed to capture the underlying dynamics of sea clutter, the SCR can be improved considerably, and detectability of a target in sea clutter is thereby significantly improved.

Some detection techniques for targets in chaos are discussed in the literatures [58]- [60] [63]. Among them [62] is a good summary of all the related techniques on chaos theory in sea clutter application.

However, methods utilizing chaos as a clutter model have not given promising results in actual application and are computationally intensive. Also there is a debate as to whether sea clutter is stochastic or chaotic. Unsworth et al. [64] argue that sea clutter should not be considered as chaos since the standards which Haykin et al. [62] had

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chosen were redundant and not sufficient. They verified their conclusion by generating a stochastic time series with a K-distribution and letting it pass the same test as Haykin et al. used in [62]. The test suggested the generated process to be chaos but it was obviously not.

2.6. Other Methods

Time-Frequency Method

More recent innovations are also worth noting. Much use is being made of time- frequency (T-F) signal analysis for distinguishing transient events in data, including [65]

- [67]. Although the ideas are somewhat similar to AR modelling, AR requires semi or real stationarity, whereas T-F analysis will find nearly instantaneous features relevant to detecting transients due to targets within background noise processes that are either transient or stationary and occupy the same bandwidth. This analysis finds intensive applications in target classification.

Fractal Method

Fractal theory first introduced by Mandelbrot [68] [69] provides a good mathematical description for many complex forms in nature which are hard to describe with Euclidean geometry. Nevertheless, these complex forms often possess a remarkable simplifying invariance under change of magnification. This statistical self-similarity is the basic property of fractals in essence. The natural figures can be quantified by the fractal dimension which is the number that agrees with our intuitive concept of dimension but does not need to be an integer.

High resolution methods for small target detection and estimation in high frequency radar