Fundamentals of Signal Enhancement andArray Signal Processing

Fundamentals of Signal Enhancement andArray Signal Processing

Jacob BenestyINRS, University of QuebecMontreal, Canada

Israel CohenTechnion, Israel Institute of TechnologyHaifa, Israel

Jingdong ChenNorthwestern Polytechnical UniversityXi’an, China

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v

Contents

Preface xiAbout the Companion Website xiii

1 Introduction . Signal Enhancement .. Speech Enhancement and Noise Reduction .. Underwater Acoustic Signal Enhancement .. Signal Enhancement in Radar Systems .. Signal Enhancement in Ultrasound Systems . Approaches to Signal Enhancement . Array Signal Processing . Organization of the Book . How to Use the Book

References

Part I Signal Enhancement

2 Single-channel Signal Enhancement in the Time Domain . Signal Model and Problem Formulation . Wiener Method .. Linear Filtering .. Performance Measures .. Optimal Filters . Spectral Method .. Joint Diagonalization and Reformulation of the Problem .. Noise Reduction with Gains .. Performance Measures .. Determination of the Gains from the Fullmode Output SNR

Problems References

3 Single-Channel Signal Enhancement in the Frequency Domain . Signal Model and Problem Formulation . Noise Reduction with Gains

vi Contents

. Performance Measures . Optimal Gains . Constraint Wiener Gains . Implementation with the Short-time Fourier Transform

Problems References

4 Multichannel Signal Enhancement in the Time Domain . Signal Model and Problem Formulation . Conventional Method .. Joint Diagonalization .. Linear Filtering .. Performance Measures .. Optimal Filtering Matrices . Spectral Method .. Temporal Joint Diagonalization and Reformulation of the Problem .. Spatial Joint Diagonalization .. Spatial Linear Filtering .. Performance Measures .. Optimal Filters . Case of a Rank Deficient Noise Correlation Matrix .. Eigenvalue Decompositions .. Maximization of the Output SNR .. Minimization of the Output SNR

Problems References

5 Multichannel Signal Enhancement in the Frequency Domain . Signal Model and Problem Formulation . Linear Filtering . Performance Measures .. Input SNR .. Output SNR .. Noise Rejection and Desired Signal Cancellation .. Desired Signal Distortion Index .. MSE Criterion . Optimal Filters .. Maximum SNR .. Wiener .. MVDR .. Tradeoff .. LCMV . Generalized Sidelobe Canceller Structure . A Signal Subspace Perspective .. Joint Diagonalization .. Estimation of the Desired Signal

Contents vii

. Implementation with the STFT Problems References

6 An Exhaustive Class of Linear Filters . Signal Model and Problem Formulation . Linear Filtering for Signal Enhancement . Performance Measures . Optimal Filters .. Wiener .. MVDR .. Tradeoff .. LCMV .. Maximum SINR .. Maximum SIR . Filling the Gap Between the Maximum SINR and Wiener Filters

Problems References

Part II Array Signal Processing

7 Fixed Beamforming . Signal Model and Problem Formulation . Linear Array Model . Performance Measures . Spatial Aliasing . Fixed Beamformers .. Delay and Sum .. Maximum DF .. Superdirective .. Robust Superdirective .. Null Steering . A Signal Subspace Perspective .. Joint Diagonalization .. Compromising Between WNG and DF

Problems References

8 Adaptive Beamforming . Signal Model, Problem Formulation, and Array Model . Performance Measures . Adaptive Beamformers .. Wiener .. MVDR .. Tradeoff .. Maximum Array Gain .. LCMV

viii Contents

. SNR Estimation . DOA Estimation . A Spectral Coherence Perspective .. Definitions .. Derivation of Optimal Beamformers

Problems References

9 Differential Beamforming . Signal Model, Problem Formulation, and Array Model . Beampatterns . Front-to-back Ratios . Array Gains . Examples of Theoretical Differential Beamformers . First-order Design .. Principle .. Design Examples . Second-order Design .. Principle .. Design Examples . Third-order Design .. Principle .. Design Examples . Minimum-norm Beamformers .. Principle .. Design Examples

Problems References

10 Beampattern Design . Beampatterns Revisited . Nonrobust Approach . Robust Approach . Frequency-invariant Beampattern Design . Least-squares Method . Joint Optimization

Problems References

11 Beamforming in the Time Domain . Signal Model and Problem Formulation . Broadband Beamforming . Performance Measures . Fixed Beamformers .. Delay and Sum .. Maximum DF .. Distortionless Maximum DF

Contents ix

.. Superdirective .. Null Steering . Adaptive Beamformers .. Wiener .. MVDR .. Tradeoff .. Maximum SNR .. LCMV . Differential Beamformers .. First Order .. Second Order .. General Order .. Hypercardioid .. Supercardioid

Problems References

Index

xi

Preface

Signal enhancement and array signal processing concern the problems of signalestimation, restoration, parameter estimation, and decision-making. These topics lie atthe heart of many fundamental applications, such as hands-free voice communications,sonar, radar, ultrasound, seismology, autonomous cars, robotics, and so on. This bookis designed as a textbook and its principal goal is to provide a unified introductionto the theory and methods of signal enhancement and array signal processing. Thetargeted readers are advanced undergraduate and graduate students who are taking – orinstructors who are teaching – courses in signal enhancement, array signal processing,and their applications. Of course, practitioners and engineers can also use this book asa reference in designing signal-enhancement and/or array systems.

Since the primary users of this book may come from many different fields, withdifferent background knowledge, we choose to focus on the key principles, theory andmethods of signal enhancement and array signal processing from a signal processingperspective without discussing in detail the introductory material related to specificapplications. Students are encouraged to read background material from the specializedacademic books and research papers in their own field while studying this book.

In most, if not all, application systems, signals are acquired in the time domain.Therefore, comprehensive coverage of the formulation, methods, and algorithms ofsignal enhancement and array beamforming is provided for this domain. Likewise, thor-ough coverage of the material in the frequency domain is also presented as the formula-tion, derivation, analysis, and implementation of signal-enhancement and beamformingalgorithms are often carried out in this domain. Readers are assumed to be familiarwith Fourier transforms and the short-time Fourier transform (STFT) by which a time-domain signal is mapped to an equivalent sequence in the frequency domain. Readersare also assumed to have some prior knowledge on discrete-time linear systems, linearalgebra, statistical signal processing, and stochastic processes.

A solid theoretical understanding always goes hand-in-hand with practical imple-mentations. Therefore, this textbook includes a large number of examples to illustrateimportant concepts and show how the major algorithms work. MATLAB functions forall the examples can be found on the authors’ websites. Besides examples, exercises andproblems are provided at the end of every chapter to challenge readers and facilitatetheir comprehension of the material.

xii Preface

With the long experience of the authors (especially the first one) in both the industryand academia, we hope that this book has been written in such a way that it is easy andpleasant to read without compromising on the rigor of the mathematical developments.

Jacob BenestyIsrael Cohen

Jingdong Chen

Trim size: mm x mm Single Column_BW book.tex // : page xiii

xiii

About the Companion Website

Don’t forget to visit the companion website for this book:

www.wiley.com/go/benesty/arraysignalprocessing

There you will find valuable material designed to enhance your learning, including:

) Matlab codes used in the book) Slides for lectures

Scan this QR code to visit the companion website

1

1

Introduction

Signal enhancement is a process to either restore a signal of interest or boost the relevantinformation embedded in the signal of interest and suppress less relevant informationfrom the observation signals. Today, there is almost no field of technical endeavor thatis not impacted in some way by this process.

Array signal processing manipulates the signals picked up by the sensors that forman array in order to estimate some specific parameters, enhance a signal of interest, ormake a particular decision. The main purpose of this chapter is to:● define the scope of the field that we call signal enhancement● present a brief historic overview of this topic● give some examples of fields where signal enhancement is needed and used● discuss briefly the principal approaches to signal enhancement● explain how array signal processing works.

1.1 Signal Enhancement

We human beings rely on our senses to sense the environment around us. Based onthis information we build and expand intelligence in our brain to help make decisionsand take actions. Similarly, we strive to build systems to help us “see” or “hear” distantevents that cannot be reached by our senses. For example, nowadays, sonar systemscan hear ships across hundreds of miles of ocean, radar devices can see airplanes from athousand miles over the horizon, telecommunication systems can connect two or moreusers from different corners of the world, and high-definition cameras can see eventshappening on our planet from space. These systems use sensors to measure the physicalenvironment of interest. Signal processing is then applied to extract as much relevantinformation as possible from the sensors’ outputs. Generally, sensors’ outputs consist ofthe signal of interest, which carries very important information, and also a compositionof unwanted signals, which is generally termed “noise”. This does not contain usefulinformation but interferes with the desired signal. To extract the useful information inthe presence of noise, signal enhancement is needed, the objective of which is to:● enhance the signal-to-noise ratio (SNR)● restore the signal of interest

Fundamentals of Signal Enhancement and Array Signal Processing, First Edition.Jacob Benesty, Israel Cohen, and Jingdong Chen.© John Wiley & Sons Singapore Pte. Ltd. Published by John Wiley & Sons Singapore Pte. Ltd.Companion website: www.wiley.com/go/benesty/arraysignalprocessing

2 Fundamentals of Signal Enhancement and Array Signal Processing

● boost the relevant information while suppressing less relevant information● improve the performance of signal detection and parameter estimation.

Signal enhancement is a specialized branch of signal processing that has been aroundfor many decades and has profound impact on many fields. In the following subsections,we describe a few areas that routinely use signal enhancement techniques, particularlythose developed in the following chapters of this text. Note that we can only cover a fewapplications, but this should leave the reader with no doubt as to the importance andbreadth of application of signal enhancement techniques.

1.1.1 Speech Enhancement and Noise Reduction

In applications related to speech acquisition, processing, recognition, and communica-tions, the speech signal of interest (generally called the “desired speech”) can never berecorded in a pure form; it is always immersed in noise. The noise can come from verydifferent sources. For example, microphones that we use to convert acoustic pressureinto electronic signals have self-noise, even though the noise floor of popularly usedcapacitor microphones has been dropping significantly over the years. The associ-ated digital signal processing boards, including preamplifiers, analog-to-digital (A/D)converters, and processors for processing the signals, may also generate noise. Mostimportantly, noise comes from ambient sources; the environment where we live is fullof different kinds of sounds. While the sensors’ self and circuit noise is generally whitein spectrum, the noise from sound sources in the surrounding environment can varysignificantly from one application scenario to another.

Commonly, noise from acoustic environments can be divided into the following fourbasic categories depending on how the noise is generated:● Additive noise can come from various sources, such as cooling fans, air conditioners,

slamming doors, and passing traffic.● Echoes occur due to the coupling between loudspeakers and microphones.● Reverberation is the result of multipath propagation and is introduced by reflections

from enclosure surfaces.● Interference comes from concurrent sound sources. In some communication appli-

cations, such as teleconferencing, it is possible that each communication site hasmultiple participants and loudspeakers, so there can be multiple competing soundsources.

Combating these four categories of noise has led to the development of diverse acousticsignal processing techniques. They include noise reduction (or speech enhancement),echo cancellation and suppression, speech dereverberation, and source separation, eachof which is a rich subject of research [–]. This text presents many methods, algo-rithms, and techniques that are useful in dealing with additive noise, reverberation, andinterference while its major focus, particularly the signal enhancement part fromChapter to Chapter , is on reducing additive noise.

Additive noise and the desired speech signal are in general statistically independent.While the noise does not modify the speech characteristics directly, the characteristicsof the observation signal are very different from those of the desired speech since itis a mixture of the desired speech signal and noise. Figure . plots a speech signalrecorded in an anechoic (quiet and non-reflective) environment and the same speech

Introduction 3

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Figure 1.1 (a) A speech signal recorded by a microphone in an anechoic environment and (b) thesame speech signal recorded by the same microphone but in a conference room.

signal but recorded in a conference room. The spectrograms of these two signals areshown in Figure .. As can be seen, both the waveform and the spectrogram of the noisysignal are dramatically different from those of the clean speech. The effect of noise maydramatically affect the listener’s perception and also machine processing of the observedspeech. It is therefore generally required to “clean” the observation signal before it isstored, transmitted, or played (through a loudspeaker, for example). This problem isgenerally referred to as either noise reduction or speech enhancement.

1.1.2 Underwater Acoustic Signal Enhancement

Over the last few decades, ocean exploration activity for both military and civilianinterests has been steadily increasing. As a result, there has been growing demandfor underwater communication and signal detection and estimation technologies.Electromagnetic and light waves do not propagate over long distances under water(particularly sea water). In contrast, acoustic waves may propagate across tens or evenhundreds of miles under the sea. Therefore, acoustic waves have played an importantrole in underwater communication and signal detection and estimation. For example,passive sonar systems can detect a submarine from tens of miles away by listeningto the sound produced by the submarine, such as from the propellers, engine, andpumps; active sonars transmit sound pulses into the water and listen to the echoes,thereby detecting underwater features such as the location of fish, sunken objects,vessels, and submarines. Underwater wireless communication systems modulate usefulinformation on acoustic carriers with frequencies between a few kilohertz and a fewtens of kilohertz and transmit the modulated signal from one end to another throughunderwater acoustic channels.

4 Fundamentals of Signal Enhancement and Array Signal Processing

(a)

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Figure 1.2 (a) The spectrogram of the speech signal in Figure 1.1a; (b) the spectrogram of the speechsignal in Figure 1.1b.

However, processing underwater acoustic signals is by no means an easy task. Firstof all, underwater acoustic channels are generally known as one of the most difficultcommunication media in use today. Underwater acoustic propagation suffers from thetime-varying multipath effect (due to sound reflection at the surface, bottom, and anyobjects in the vicinity, and also sound refraction in the water), frequency-dependentattenuation (due to absorption and signal spreading loss), and a severe Doppler effect(due to the low speed of sound and motion of the transmitter or receiver or the objectsto be detected). Secondly, the ocean is filled with sounds, which interfere with theacoustic signal we are interested in. Underwater sounds are generated by both naturalsources, such as marine animals, breaking waves, rain, cracking sea ice, and underseaearthquakes, as well as man-made sources, such as ships, submarines, and militarysonars.

Marine animals use sound to obtain detailed information about their surroundings.They rely on sound to communicate, navigate, and feed. For example, dolphins candetect individual prey and navigate around objects underwater by emitting short pulsesof sound and listening to the echo. Marine mammal calls can increase ambient noiselevels by – dB in some locations at certain times of year. Blue and fin whalesproduce low-frequency moans at frequencies of – Hz, with estimated source levelsof up to dB at m. Sounds generated by human activities are also an importantpart of the total ocean noise. Undersea sound is used for many valuable purposes,including communication, navigation, defense, research and exploration, and fishing.

Introduction 5

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Figure 1.3 A linear frequency modulated signal: (a) emitted by a transmitter of an underwateracoustic communication system and (b) received by a hydrophone six miles away from the transmitterin an underwater environment.

Sounds generated by human activities cover a wide range of frequencies, from a fewhertz up to several hundred kilohertz, and a wide range of source levels.

The underwater channel condition and noise sets the ultimate limit on the minimumdetectable signal in detection and communication systems. To illustrate how challeng-ing it is to process underwater signals for extracting the useful information, Figure .plots a linear frequency modulation chirp signal transmitted by an acoustic antennaand the signal received by a hydrophone placed six miles away from the transmitter. Themagnitude spectra of these two signals are plotted in Figure .. As seen, the transmittedsignal is dramatically distorted by the acoustic channel and noise. Sophisticated signalenhancement techniques, such as those developed in this text, are needed to extractthe important parameters or information embedded in the transmitted signal from thereceived signal.

1.1.3 Signal Enhancement in Radar Systems

A radar system has a transmitter that emits electromagnetic waves (called radar signals)in look directions. When these waves come into contact with an object, they are usuallyreflected or scattered in many directions. Receivers (usually, but not always, in the samelocation as the transmitter) are then used to receive the echoes. Through processing theechoes, the radar can determine the range, angle, or velocity of the objects of interest.

The invention of the radar dates back to the late th century. Such systems are nowused in a broad range of applications, including air defense, traffic control, aircraftanticollision, ocean surveillance, geological observations, meteorological precipitationmonitoring, and autonomous cars. In order to estimate the range, angle, or velocity of

6 Fundamentals of Signal Enhancement and Array Signal Processing

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Figure 1.4 The power spectrum of the signal in Figure 1.3a; (b) the power spectrum of the signal inFigure 1.3b.

the objects of interest, radar systems must overcome unwanted signals, which can bedivided into the following three categories.● Additive noise is generated by both internal sources (electronics) and external sources

(the natural thermal radiation of the background surrounding the target of interest).In modern radar systems, the internal noise is generally lower than the external noise.

● Clutter is a term used for echoes returned from targets that are not useful to the radarsystem user. Clutters can be generated by irrelevant targets, natural objects such as theground, sea, atmospheric turbulence, ionospheric reflections, and man-made objectssuch as buildings, as illustrated in Figure ..

● Jamming refers to signals received by the radar on its own frequency band but emittedfrom outside sources. Jamming may be intentional, as with an electronic warfaretactic, or unintentional, as with friendly forces’ using equipment that transmits usingthe same frequency range. It is problematic to radar since the jamming signal onlyneeds to travel one way – from the jammer to the radar receiver – whereas the radarechoes travel two ways – from radar to target and to radar – and are thereforesignificantly reduced in power by the time they return to the radar receiver. Therefore,jammers can effectively mask targets along the line of sight from the jammer to theradar, even when they are much less powerful than the jammed radars.

Introduction 7

Wanted echo from aircraft

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Unwanted echo from ground targets

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Unwanted echo from ground targets

Figure 1.5 An illustration of a radar system and its environments.

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Figure 1.6 Normalized magnitude of an echo received by a pulse radar.

Figure . plots the magnitude of a signal received by a pulse radar where the transmittedsignal is a short pulse. The received signal is composed of an echo returned from atarget of interest, two unwanted echoes, and some noise. Figure . shows a radarimage directly mapped from the received signals without using any signal enhancementtechniques. Without signal enhancement, it is difficult to determine the position ofone target, let alone track multiple targets with high resolution. Therefore, signal

8 Fundamentals of Signal Enhancement and Array Signal Processing

Figure 1.7 Illustration of an image displayed in a radar screen without using signal enhancement.

enhancement techniques, particularly those developed in Chapters –, are neededto deal with additive noise, clutter, and jamming in radar systems. This is done byestimating the important parameters embedded in the radar echo signals.

1.1.4 Signal Enhancement in Ultrasound Systems

Ultrasound refers to sound waves with frequencies greater than kHz, a level which iscommonly accepted to be the upper limit of human hearing. This type of high-frequencysound wave is used in many fields for a wide range of applications, such as non-intrusivetesting of products and structures, invisible flaw detection, distance measurement, andmedical diagnosis, to name just a few. One of the best known ultrasound systems is thesonography instrument that is used in medicine to examine many of the body’s internalorgans, including – but not limited to – the heart and blood vessels, liver, gallbladder,pancreas, kidneys, and uterus, as well as unborn children (fetus) in pregnant patients.

Typically, a sonography device consists of an array of transmitters, which sendshort, high-frequency (generally between and MHz) sound pulses into the body.Beamforming is applied to the transmitted pulses so that the ultrasound waves arefocused towards a particular point. As the beamformed waves travel toward the desiredfocal point, they propagate through materials with different densities. With each changein density, reflected waves are produced, some of which propagate back. They are then

Introduction 9

collected by an array of receivers (the transmitters typically become sensors to receivesignals once they have finished generating their respective sound waves). The signalreceived by each receiver is composed of the wanted echoes and noise. Commonly, noisein sonography is one of two major types:● Additive noise is generated from the sensors, amplifiers, A/D converters, and other

electronic system components. It can also come from sources such as backgroundtissues, other organs and anatomical influences, and breathing motion. Generally, thistype of noise is independent (or weakly dependent) on the echo signals and is oftenmodeled mathematically as a white Gaussian noise process.

● Speckle is the result of three sound scattering effects: specular, diffusive, and diffrac-tive. Specular scattering occurs when the scattering object is large compared tothe sound wavelength; diffusive scattering happens when the scattering object issmall relative to the wavelength; diffractive scattering occurs mostly for medium-sizescattering objects. Unlike additive noise, speckles are generally correlated with thewanted echo signals.

To deal with additive noise and speckles in sonography, beamforming, noise reduction,speckle reduction, and many other enhancement processes are applied to the receivedsignals before high-resolution two-dimensional images are formed to display the dis-tances and intensities of the echoes on the screen.

1.2 Approaches to Signal Enhancement

Signal enhancement is one of the most interesting and appealing yet challenging areasof signal processing. Its objective is generally problem oriented, ranging from simplyimproving the SNR, boosting relevant information, restoring the signal of interest, toimproving measures of which only human subjects can judge the quality. As a result,there is no general rule as what method is optimal and it is quite common that a methodthat produces the best enhancement result for one application may not be very usefulfor another. In general, signal enhancement techniques can be classified into one of fourbroad categories, depending on how the information embedded in the signal and noiseare used:● time-domain methods, which directly use temporal information● frequency-domain approaches, which operate on spectra (obtained using the Fourier

transform or other time-to-frequency-domain transformations) of a signal● spatial-domain techniques, which acquire and process a signal of interest using an

array of sensors● combinational methods, which use temporal, spectral, and spatial information.

Time-domain methods typically achieve signal enhancement by applying a finite-impulse-response filter to the noisy signal that is observed at a sensor. So, the core prob-lem of enhancement is converted to one of designing an optimal filter that can attenuatenoise as much as possible while keeping the signal of interest relatively unchanged.The history of this class of methods dates back to the seminal work by Wiener [], inwhich the optimal filter from a second-order-statistics viewpoint is achieved throughthe optimization of the classical mean-squared error (MSE) criterion. The Wiener filter

10 Fundamentals of Signal Enhancement and Array Signal Processing

is well known, and has been intensively investigated for signal enhancement [–].However, while it is optimal from the MSE point of view, it introduces signal distortionif the signal to be enhanced is broadband. The amount of signal distortion may notbe acceptable for some applications. If this is the case, one may consider using somesuboptimal filters that minimize the MSE criterion under certain constraints [, ].

The Wiener technique, by its assumption, can only deal with stationary signals. Onepopularly used approach to extending the Wiener filter to deal with nonstationarysignals is to relax the stationarity assumption to one of short-time stationarity. Then,the Wiener filter is computed using signals within only a short-time, sliding window.In this case, the length of the short-time window plays an important role on the tradeoffbetween the nonstationarity and performance within the short-time window. Thereare, of course, other ways to deal with signal enhancement of nonstationary signals,for example, combining the Kalman filter and the linear-prediction-coding method [].Comprehensive coverage of signal enhancement using temporal information will begiven in Chapter .

Frequency-domain methods, as the name indicates, explicitly operate on the spectraof the signal to be processed. The root of this class of methods can be traced backto the s, when low-pass, high-pass, and band-pass filters were invented to filterout noise that occupies different frequency bands to the signal of interest. Today, thebasic principle of band-pass filtering is still widely used in signal enhancement, butoften in more complicated forms, such as comb filters [] and binary masking [].Band-pass filtering, comb filtering, and binary masking are hard-decision methodsin the sense that, given a narrow frequency band, they either completely remove thesignal component or keep it unchanged. In comparison, a soft decision can be achievedthrough the spectral enhancement method, which was first developed in the s usinganalog circuits []. A digital-domain version of this method, which is called “spectralsubtraction”, was then developed in the late s []. While it is very useful and isoften used as a benchmark against which other techniques are compared, the spec-tral substraction method has no optimality properties associated with it. An optimalspectral enhancement framework was developed in the early s []. This unifieda broad class of enhancement algorithms, including spectral substraction, frequency-domain Wiener filtering, and maximum likelihood envelope estimator. Following thiswork, an optimal spectral amplitude estimator using statistical estimation theory wasdeveloped in the early s. Following this work, many statistical spectral estimatorswere developed, including the minimum mean-squared error (MMSE) estimator [],the MMSE log-spectral amplitude estimator, the maximum-likelihood (ML) spectralamplitude estimator, the ML spectral power estimator, and the maximum a posteriori(MAP) spectral amplitude estimator. Today, there are still tremendous efforts to findbetter spectral amplitude estimators, inspired by the work of McAulay and Malpass[] and Ephraim and Malah []. A broad coverage of frequency-domain methods willbe given in Chapter .

When multiple sensors are used, the spatial information embedded in the sensors’outputs can be exploited to enhance the signal of interest and reduce unwanted noise.This can be done in a straightforward way by extending the single-channel methods ofChapters and to the multichannel cases (see, respectively, Chapters and ). It canalso be done in a different way through array beamforming, which will be discussed inthe next section.

Introduction 11

1.3 Array Signal Processing

An array consists of a set of sensors positioned at known locations with reference toa common point. The sensors collect signals from sources in their own field of viewand the output of each sensor is composed of these source components as well asnoise. By processing the sensors’ outputs, two groups of functionalities can be achieved:estimation of important parameters of sources and enhancement of some signals ofinterest.

The history of array signal processing dates back to World War II. Early efforts in thisfield were mainly focused on parameter estimation: estimating the range, angle, andvelocity of the sources of interest. A wide range of processing methods were developedto this end, including fixed beamforming (or spatial filtering), matched filtering, Capon’sadaptive beamforming, the MUSIC (Multiple SIgnal Classification) method and itsvarieties, and the ESPRIT (Estimation of Signal Parameters by Rotational InvarianceTechniques) algorithm, to name but a few. The reader is referred to the literature for anin-depth consideration of the problem of array parameter estimation and the associatedmethods [–]. In this book, we choose to focus on the signal enhancement problemwith the use of sensor arrays.

The basic principle of signal enhancement using an array of sensors can be illustratedin Figure .. Consider a simple example with a uniformly spaced linear array of Msensors and assume that there is a single source in the farfield such that its sphericalwavefront appears planar at the array. If we neglect the propagation attenuation, thesignals received at the M sensors can be written as

Ym( f ) = Xm( f ) + Vm( f ) (.)= X( f )e−𝚥𝜋f (t+𝜏m−) + Vm( f )= X( f )e−𝚥𝜋f 𝜏m− + Vm( f ), m = , ,… ,M,

X ( f )

θ

δ

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wav

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V1 ( f )

Y2 ( f )YM ( f )

VM ( f )

Figure 1.8 Illustration of an array system for signal enhancement.

12 Fundamentals of Signal Enhancement and Array Signal Processing

where f is frequency, t is the propagation time from the source X(f ) to sensor (thereference sensor), X( f ) = X( f )e−𝚥𝜋ft is the signal component at sensor , 𝜏m− isthe relative time delay between the mth sensor and the reference one, and 𝚥 =

√− is

the imaginary unit. It is assumed that X( f ) is uncorrelated with Vm( f ), m = , ,… ,M.With a uniform linear array (ULA) and a farfield source, the delay 𝜏m− can be expressedin the following form according to the geometry shown in Figure .:

𝜏m− = (m − )𝛿 cos 𝜃∕c, m = , ,… ,M, (.)

where 𝛿 is the spacing between two neighboring sensors, c represents velocity of wavepropagation, and 𝜃 is the signal incidence angle.

Now let us consider processing the M signals Ym(f ), m = , ,… ,M, to extract thesource signal X(f ) (up to a delay) and reduce the effect of Vm(f ). A straightforward wayof doing this is to multiply Ym(f ) by e𝚥𝜋f 𝜏m− , and then average the results. This gives anoutput:

Z(f ) = M

M∑m=

Ym(f )e𝚥𝜋f 𝜏m− (.)

= X(f ) +

M

M∑m=

Vm(f )e𝚥𝜋f 𝜏m− .

To check whether the output Z(f ) is less noisy than the input, let us compare the inputand output SNRs. The input SNR, according to the signal model given in (.), is definedas the SNR at the reference sensor:

iSNR(f ) =𝜙X

(f )𝜙V

(f ), (.)

where 𝜙X(f ) = E

[|X(f )|] and 𝜙V(f ) = E

[|V(f )|] are the variances of X(f ) andV(f ) respectively, and E[⋅] denotes mathematical expectation.

The output SNR – the SNR of the Z(f ) signal – is written as

oSNR(f ) =𝜙X

(f )

M E[|||∑M

m= Vm(f )e𝚥𝜋f 𝜏m−|||] . (.)

Now, if all the noise signals Vm(f ), m = , ,… ,M, are uncorrelated with each otherand have the same variance, it is easy to check that

E⎡⎢⎢⎣||||||

M∑m=

Vm(f )e𝚥𝜋f 𝜏m−

||||||⎤⎥⎥⎦ = M × E

[||V(f )||]. (.)

It follows immediately that oSNR(f ) = M × iSNR(f ). Thus, a simple phase shiftingand averaging operation of the sensors’ outputs results in an SNR improvement bya factor of M, the number of sensors. The underlying physical principle behind the

Introduction 13

SNR improvement can be explained as follows. Through appropriate phase shifting, thesignal components from the source of interest have been coherently combined whilethe noise signals from different sensors add only incoherently as they are uncorrelatedwith each other, yielding a gain for the overall output signal compared to the noise.

Of course, the above example is only a particular case. More generally, an estimateof the source signal X(f ) can be obtained through weighted linear combination of thesensors’ outputs:

Z(f ) =M∑

m=H∗

m(f )Ym(f ), (.)

where Hm(f ), m = , ,… ,M, are complex weighting coefficients. The process offinding the appropriate values of Hm(f ) so that Z(f ) is a good estimate of X(f ) iscalled beamforming. Therefore, the coefficients Hm(f ) are also called beamformingcoefficients and the vector that consists of all the coefficients is called the beamformingfilter or beamformer.

Beamforming has been the central problem of array signal processing ever sincesensor arrays were invented, and a large number of algorithms has been developed anddescribed in the literature. By and large, the developed algorithms fall into two majorcategories – fixed or adaptive beamforming – depending on whether the noise or signalstatistics are considered in forming the beamforming filters.

In fixed beamforming, the beamforming filters are designed explicitly using the arraygeometry information as well as the assumed knowledge of the look direction andnoise statistics. Once computed, the coefficients of the beamforming filters will be fixedregardless of the particular application environment. It is for this reason that this designprocess is called fixed beamforming. The representative algorithms include the delay-and-sum beamformer, the maximum directivity factor beamformer, and the superdi-rective beamformer. The reader may find a discussion of these algorithms in differentcontexts and applications in the literature [, –]. The basic theory and methodsfor designing fixed beamformers from a narrowband perspective will be covered inChapter . While the principles presented in this chapter are rather general, in designingoptimal fixed beamformers, the resulting beamformers may be insufficient to deal withbroadband signals as their beampatterns may vary with frequency. Chapters and are also concerned with fixed beamforming, but with focus on processing broadbandsignals where beampatterns are expected to be the same across a band of frequencies.

In comparison with fixed beamforming, adaptive beamforming algorithms considerusing either the noise statistics or the statistics of the array observation data to optimizethe beamforming filters. The performance of adaptive beamforming can be more opti-mal than its fixed counterpart as long as the signal statistics are correctly estimated. Therepresentative algorithms in this category include the minimum variance distortionlessresponse (MVDR) beamformer, which is also known as the Capon’s beamformer [],the linearly constrained minimum variance (LCMV) beamformer, which is also calledthe Frost’s beamformer [], and the generalized sidelobe canceller [, ]. Manyapplications of adaptive beamforming can be found in the literature [–, , ].The fundamental theory and methods for adaptive beamforming from a frequency-domain perspective will be covered in Chapter .

14 Fundamentals of Signal Enhancement and Array Signal Processing

While one may see that most discussion on beamforming in both the literature andthis text concerns the frequency domain, it is also possible to formulate this problemin the time domain. Chapter is devoted to a time-domain framework for arraybeamforming, which can be used to design both fixed and adaptive beamformers aswell as narrowband and broadband beamformers.

1.4 Organization of the Book

This book attempts to cover the most basic concepts, fundamental principles, and prac-tical methods of signal enhancement and array beamforming. The material discussedoccupies ten chapters, which are divided into two parts.

The first part, Signal Enhancement, consists of five chapters: from Chapter toChapter . We start to discuss the signal enhancement problem in the time domainwith a single sensor in Chapter . With a single sensor, we show how to exploit thetemporal information so as to reduce the level of the additive noise from the sensor’soutput, thereby enhancing the signal of interest. This chapter presents two fundamentalapproaches: one deals with the problem from the Wiener filtering perspective and theother from a spectral mode perspective based on the joint diagonalization of the corre-lation matrices of the signal of interest and the noise. In both approaches, we present theproblem formulation and performance measures that can be used to evaluate the signalenhancement performance. Different cost functions are also presented, and based onthese we discuss how to derive useful optimal enhancement filters.

Chapter continues the investigation of the single-channel signal enhancementproblem, but in the frequency domain. The frequency-domain approach is equivalentto the spectral method discussed in Chapter in the sense that the observation signalat each frequency band can be processed independently from the others. The advantageof the algorithms in this chapter is that they can all be implemented efficiently thanksto the use of the fast Fourier transform. Again, we start from problem formulationand then discuss how to perform signal enhancement with just simple gains at eachfrequency band. Relevant performance measures are defined and we show how to deriveseveral kinds of enhancement gain, some of which can achieve a compromise betweendistortion of the desired signal and reduction of the additive noise.

Chapter is basically an extension of Chapter . The fundamental difference is thatin this chapter we consider the signal enhancement problem with the use of multiplesensors, which are located at distinct positions in the space. In this case, every sensorpicks up the signal of interest and noise from its own viewpoint. Now, in addition tothe temporal information, the spatial information from the multiple sensors can alsobe exploited to enhance the signal of interest. As a result, either a better enhancementperformance or more flexibility to compromise between noise reduction and desiredsignal distortion can be achieved, as compared to the single-channel scenario. Similarto Chapter , we also discuss two approaches: the Wiener filtering one and the onebased on the joint diagonalization of the correlation matrices of the signal of interestand noise.

Chapter deals with the problem of signal enhancement with multiple sensors in thefrequency domain. As in Chapter , the spatial information embedded in the multiplesensors is exploited to enhance the signal of interest, but in a way that is easier to