Array Beamforming with Linear Difference Equations: 20 (Springer Topics in Signal Processing, 20)

175 Pages·2021·6.421 MB·English

by  Jacob Benesty, Israel Cohen, Jingdong Chen

Springer Topics in Signal Processing
Jacob Benesty
Israel Cohen
Jingdong Chen
Array
Beamforming
with Linear
Diff erence
Equations
Springer Topics in Signal Processing
Volume 20
Serieseditors
JacobBenesty,Montreal,Canada
WalterKellermann,Erlangen,Germany
Moreinformationaboutthisseriesathttp://www.springer.com/series/8109
Jacob Benesty • Israel Cohen (cid:129) Jingdong Chen
Array Beamforming with
Linear Difference Equations
JacobBenesty IsraelCohen
INRS-EMT IsraelInstituteofTechnology
UniversityofQuebec Haifa,Israel
Montreal,QC,Canada
JingdongChen
NorthwesternPolytechnicalUniversity
Xi’an,China
ISSN1866-2609 ISSN1866-2617 (electronic)
SpringerTopicsinSignalProcessing
ISBN978-3-030-68272-9 ISBN978-3-030-68273-6 (eBook)
https://doi.org/10.1007/978-3-030-68273-6
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Abstract
Voice communication and human-machine speech interface systems have to face
the problem of sound acquisition in complex acoustic environments, where noise,
reverberation, and competing sources may coexist. To deal with this challenging
issue,microphonearrays,whichconsistofmultiplesensorsarrangedintoacertain
one-,two-,orthree-dimensionaltopology,havetobeused.Thecentralcomponent
ofanarraysystemissignalprocessing,whichoperatesonthesignalsobservedby
the array of sensors to restore a signal of interest while suppressing the unwanted
signal components. Ideally, one would wish to simply improve the signal-to-noise
ratio (SNR) without adding any distortion to either the signal of interest or the
unwanted signals, thereby preserving the fidelity of both the signal and ambience.
However, this seemingly easy objective turns out to be extremely difficult to
achieve. With all the methods and techniques developed in the literature, signal
enhancement is generally achieved by adding some degree of distortion to either
the signal of interest or the ambience or even both at the same time. There is
one exception: differential beamforming, whose root can be traced back to the
principleofdirectionalmicrophonesinventedinthe1940s;thisprinciplewasthen
introduced into the regime of microphone arrays in the 1990s to form differential
microphonearrays(DMAs),whicharedesignedtomeasurethespatialderivatives
of the sound pressure field. Basically, a first-order DMA consists of two closely
spaced omnidirectional pressure microphones, and its output is the subtraction of
one sensor’s output from that of the other. The two omnidirectional microphones
measure the sound pressure field, each from its own viewpoint. Since they are
closely spaced, the difference between their outputs approximates very well the
spatial derivative of the sound pressure field along the axis that connects the two
sensors.So,thissimplesubtractionoperationyieldsadifferentialbeamformerthat
is responsive to the first-order differential of the acoustic pressure field. Similarly,
a general Nth-order DMA can be formed by subtractively combining the outputs
of two DMAs of order N − 1. This simple way of constructing microphone
arrays has laid out the foundation for differential beamforming. The resulting
beamformersfeatureaprominentproperty,thatis,theirbeampatternsarefrequency
independent by theory, and have a great potential for use in high fidelity sound
v
vi Abstract
acquisition and enhancement as they do not add spectral distortion to the source
signals regardless of the source incidence angle. A great deal of effort has been
devoted to the study and design of differential beamformers over the last two
decades.Manymethodshavebeendeveloped,mostofwhichattempttodesignthe
beamformingfilterfromthebeampatternperspective,thatis,theparametersofthe
beamformersaredeterminedinsuchawaythattheresultingbeampatternmatches
well a target frequency-independent beampattern. Interestingly, few works have
attemptedtoaddresstheproblemfromthedifferenceequationsperspective,though
this view is the foundation for differential beamforming. This book is intended to
study the link between differential beamforming and differential equations, which
is missed in the literature. This link enables to study the fundamental theory
and methods of beamforming from a rather different perspective, leading to new
insights into the problem and new methods to solve the problem. The book first
presentsabriefoverviewoftheproblemsandmethodsforbeamformingandsome
performance measures popularly used either to evaluate beamformers or to derive
optimalbeamformers.Then,first-order,second-order,andgeneralhigh-orderlinear
differenceequationsarediscussed,basedonwhichweshowhowtoformulatethe
beamforming problem and derive different beamforming methods, including fixed
and adaptive ones. Furthermore, we show how to apply the theory of difference
equations to the general problem of speech enhancement and deduce a number
of noise reduction filters, including the maximum SNR filter, the Wiener filter,
the MVDR filter, etc. Also covered in the book are the difference equations and
differentialbeamformingfromthespectralgraphperspective.
Contents
1 Introduction .................................................................. 1
1.1 BeamformingandNoiseReduction.................................... 5
1.2 OrganizationoftheWork............................................... 8
References..................................................................... 9
2 ABriefOverviewofConventionalBeamforming ........................ 13
2.1 SignalModel............................................................ 13
2.2 LinearBeamforming.................................................... 15
2.3 PerformanceMeasures.................................................. 16
2.4 ExamplesofOptimalBeamformers.................................... 18
References..................................................................... 21
3 BeamformingwithFirst-OrderLinearDifferenceEquations.......... 23
3.1 SignalModel............................................................ 23
3.2 First-OrderLinearDifferenceBeamforming .......................... 25
3.3 PerformanceMeasures.................................................. 26
3.4 ExamplesofOptimalFixedBeamformers............................. 30
3.5 ExamplesofOptimalAdaptiveBeamformers......................... 51
References..................................................................... 65
4 BeamformingwithSecond-OrderLinearDifferenceEquations ....... 67
4.1 SignalModel............................................................ 67
4.2 Second-OrderLinearDifferenceBeamforming....................... 69
4.3 PerformanceMeasures.................................................. 70
4.4 ExamplesofOptimalFixedBeamformers............................. 74
4.5 ExamplesofOptimalAdaptiveBeamformers......................... 96
References..................................................................... 108
5 BeamformingwithHigher-OrderLinearDifferenceEquations........ 109
5.1 SignalModel............................................................ 109
5.2 Higher-OrderLinearDifferenceBeamforming........................ 111
5.3 PerformanceMeasures.................................................. 112
5.4 ExamplesofOptimalFixedBeamformers............................. 115
vii
viii Contents
5.5 ExamplesofOptimalAdaptiveBeamformers......................... 127
References..................................................................... 140
6 ApplicationtoSpeechEnhancement....................................... 141
6.1 SignalModel............................................................ 141
6.2 LinearDifferenceFiltering ............................................. 144
6.3 PerformanceMeasures.................................................. 145
6.4 ExamplesofOptimalSpeechEnhancementFilters ................... 148
References..................................................................... 155
7 ASpectralGraphPerspective.............................................. 157
7.1 MathematicalBackground.............................................. 157
7.2 BeamformingwithLinearDifferenceEquationsonGraphs.......... 160
References..................................................................... 166
Index............................................................................... 167
Chapter 1
Introduction
Ever since the invention of the telephone, sound signal acquisition and processing
hasbecomeacenterofacousticandspeechprocessing.Overthelasthalfcentury,
thetechnologicalregimeofsoundacquisitionandprocessinghasgonethroughsig-
nificantchanges.Broadly,thetechnologicaldevelopmentcanbedividedintothree
major phases. Before the 1960s, most sound acquisition systems were equipped
with a single microphone sensor. In such systems, close-talking sensors with low
sensitivity were generally used and analog pre-amplifiers and pre-filtering circuits
hadtobeproperlydesignedandusedtoensureacquisitionquality,i.e.,topickupthe
soundfromaclose-talkingsourcewhileavoidingambientnoiseand/orinterference
entering into the system. The invention of the analog/digital (A/D) converters and
microprocessorsinthe1960sopenedaneweraforacousticsignalprocessing.Still
withasinglemicrophonesensor,nowthesystemswereabletocovertthesensor’s
outputintothedigitaldomainformoreadvancedandsophisticatedprocessingsuch
asspectralanalysis,statisticalestimation,filtering,etc.Thoseadvancedfilteringand
estimationprocessingcanhelpcleantheacousticsignalofinterestfromitsobser-
vationscorruptedbynoiseandecho,therebyimprovingsignificantlythequalityof
full-duplexvoicecommunication.Theconfluenceofhands-freecommunicationand
human-machinespeechinterfacesoverrecentdecadeshaspushedsoundacquisition
andprocessingintoitscurrentphase.Themajordifferencebetweentheapplications
in the current phase and those in the previous ones is that the sound sources
of interest are no longer close to the microphone sensors anymore. As sound
propagation is governed by the inverse square law, the sound intensity decreases
dramaticallyasthedistancebetweenthesourceandthemicrophoneincreasesand,
asaresult,sensorswithhighsensitivity(forsomesensors,preamplificationcanbe
usedtoincreasethesystemsensitivity)havetobeusedfordistantsoundacquisition.
Asideeffectofthisisthatnoise,reverberation,echo,andinterferingsoundsignals
from the surrounding environments are also picked up by the high-sensitivity
sensorsleadingtolowsignal-to-noiseratio(SNR)and/orlowsignal-to-interference
ratio(SIR),whichmakestheextractionofthesignalofinterestverychallenging.
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J.Benestyetal.,ArrayBeamformingwithLinearDifferenceEquations,
SpringerTopicsinSignalProcessing20,https://doi.org/10.1007/978-3-030-68273-6_1