Table Of ContentSpringer Topics in Signal Processing
Jacob Benesty
Jingdong Chen
Israel Cohen
Design of Circular
Differential
Microphone Arrays
Springer Topics in Signal Processing
Volume 12
Series editors
Jacob Benesty, Montreal, Canada
Walter Kellermann, Erlangen, Germany
More information about this series at http://www.springer.com/series/8109
Jacob Benesty Jingdong Chen
(cid:129)
Israel Cohen
Design of Circular
Differential Microphone
Arrays
123
JacobBenesty Israel Cohen
INRS-EMT Department of ElectricalEngineering
Universityof Quebec Technion, Israel Instituteof Technology
Montreal, QC Haifa
Canada Israel
Jingdong Chen
Northwestern Polytechnical University
Xi’an,Shaanxi
China
ISSN 1866-2609 ISSN 1866-2617 (electronic)
Springer TopicsinSignal Processing
ISBN 978-3-319-14841-0 ISBN 978-3-319-14842-7 (eBook)
DOI 10.1007/978-3-319-14842-7
LibraryofCongressControlNumber:2015930029
SpringerChamHeidelbergNewYorkDordrechtLondon
©SpringerInternationalPublishingSwitzerland2015
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
or information storage and retrieval, electronic adaptation, computer software, or by similar or
dissimilarmethodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt
fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained
hereinorforanyerrorsoromissionsthatmayhavebeenmade.
Printedonacid-freepaper
Springer International Publishing AG Switzerland is part of Springer Science+Business Media
(www.springer.com)
Abstract
Recently,weproposedacompletelynovelandefficientwaytodesigndifferen-
tial beamforming algorithms with linear microphone arrays. Thanks to this
veryflexibleapproach,anyorderofdifferentialarrayscanbedesigned.More-
over,theycanbemaderobustagainstwhitenoiseamplification,whichisthe
main inconvenience in these types of arrays. The other well-known problem
with linear arrays is that electronic steering is not really feasible.
Inthisbook,weextendallthesefundamentalideastocircularmicrophone
arrays and show that we can design small and compact differential arrays of
any order, which can electronically steer in many different directions, with
a good control of the white noise amplification problem, high directional
gains, and frequency-independent responses. We give many practical exam-
ples, proving that differential beamforming with circular microphone arrays
islikelyoneofthebestpossiblechoicesforspeechenhancement(i.e.,noisere-
duction anddereverberation).Almost, allthe materialpresentedin this text
is new and will be of great interest to engineers, students, and researchers
working in microphone arrays and their applications to all types of telecom-
munications, security, and surveillance.
v
Contents
1 Introduction.............................................. 1
1.1 Introduction ........................................... 1
1.2 General Principles of Microphone Arrays .................. 2
1.3 From Linear Differential Arrays to Circular Differential Arrays 10
1.4 Organizationof the Book................................ 11
References ................................................. 12
2 Problem Formulation ..................................... 15
2.1 Signal Model........................................... 15
2.2 Beampattern........................................... 19
2.3 Gain in Signal-to-Noise Ratio (SNR)...................... 26
References ................................................. 30
3 Design of First-Order Circular Differential Arrays ........ 33
3.1 Principle .............................................. 33
3.2 Design Examples ....................................... 36
3.2.1 First-Order Dipole................................ 36
3.2.2 First-Order Cardioid.............................. 39
3.2.3 General First-Order Circular Differential Array ...... 42
References ................................................. 52
4 Design of Second-Order Circular Differential Arrays ...... 53
4.1 Principle and Design Examples with Four Microphones...... 53
4.1.1 Second-Order Dipole.............................. 54
4.1.2 Second-Order Cardioid............................ 57
4.1.3 Second-Order Hypercardioid....................... 60
4.1.4 Second-Order Supercardioid ....................... 63
4.2 Principle and Design Examples with Five Microphones ...... 66
4.2.1 Second-Order Dipole.............................. 67
4.2.2 Second-Order Cardioid............................ 71
4.2.3 Second-Order Hypercardioid....................... 74
vii
viii Contents
4.2.4 Second-Order Supercardioid ....................... 77
References ................................................. 80
5 Design of Third-Order Circular Differential Arrays ....... 81
5.1 Principle and Design Example with Six Microphones........ 81
5.2 Principle and Design Example with Seven Microphones ..... 85
5.3 Generalization ......................................... 89
References ................................................. 90
6 Superdirective Beamforming with Circular Arrays........ 91
6.1 Superdirective Beamforming with ULAs ................... 91
6.2 Superdirective Beamforming with UCAs................... 93
6.2.1 Without the Symmetry Constraint ................. 94
6.2.2 With the Symmetry Constraint .................... 96
6.2.3 With the Maximum of Nulls ....................... 98
6.3 Design Examples ....................................... 98
6.3.1 Superdirective Beamforming with a ULA............ 98
6.3.2 Superdirective Beamforming with a UCA – without
the Symmetry Constraint ......................... 99
6.3.3 Superdirective Beamforming with a UCA – with the
Symmetry Constraint............................. 105
6.3.4 Superdirective Beamforming with a UCA – with the
Maximum of Nulls................................ 107
References ................................................. 111
7 Minimum-NormSolution for Robust Circular Differential
Arrays.................................................... 113
7.1 Adaptive Beamforming versus Circular Differential Arrays ... 113
7.2 Minimum-Norm Filter .................................. 116
7.3 Design Examples ....................................... 117
7.3.1 First-Order Cardioid.............................. 118
7.3.2 Second-Order Cardioid............................ 124
7.3.3 Third-Order Cardioid............................. 130
7.3.4 Third-Order Pattern with Three Distinct Nulls....... 136
References ................................................. 141
8 Design of Circular Differential Arrays with the
Jacobi-Anger Expansion .................................. 143
8.1 Beampattern Definitions Revisited........................ 143
8.2 Design Principle........................................ 146
8.3 First-Order Differential Arrays ........................... 147
8.4 Second-Order Differential Arrays ......................... 149
8.5 Third-Order Differential Arrays .......................... 151
8.6 Adaptive Beamforming versus Differential Arrays........... 159
8.7 Design of the Ideal Beampattern ......................... 163
References ................................................. 164
Contents ix
Index......................................................... 165
Chapter 1
Introduction
1.1 Introduction
In general, a microphone array refers to a sound acquisition system that
uses multiple microphones to sample the sound field with spatial diversity.
These microphones are arranged into a particular geometry in which each
sensor’spositionrelativetoareferencepointisknowntothesubsequentpro-
cessors. This kind of arrays, when combined with proper signal processing
algorithms that can take advantage of the spatial, temporal, and frequency
information embedded in the observations, can be used to solve many im-
portant room acoustic problems such as source localization/tracking, noise
reduction/speech enhancement, source separation, dereverberation, spatial
sound recording, etc. Consequently, the design of such microphone arrays
and the associated processing algorithms has attracted a significant amount
of researchand engineering interest over several decades.
Someofthefundamentalprinciplesofmicrophonearrayprocessingcanbe
tracedbacktothe1930’swhenthedirectionalmicrophoneswereinvented[1],
[2]. The formal terminology of “microphone array” however, appears in the
1960’s.Early works in this area werestrongly influenced by the sensor array
theorydevelopedinthefieldofradarandsonarprocessing.Itwassoonfound
that the problems encountered in microphone arrays are very different from
those in radar and sonar.The causes that lead to the difference aremultiple
andthemajoronesareasfollows.1)Microphonearraysdealwithspeechand
audiosignals,whicharebroadbandinnatureandhaveafrequencyrangefrom
a few Hz to 20 kHz. It is extremely challenging,if not impossible, to develop
algorithms that can perform uniformly over such a largefrequency range.2)
Speechandaudiosignalsarehighlynonstationary;asaresult,theestimation
of their statistics is very difficult. 3) The environments where microphone
arraysaresupposedtoworkaregenerallyreverberantanditisnotuncommon
tohavethousandsofreflections.Itiswellknownthatreverberationcanhave
a very negative impact on the performance of microphone array processing
(cid:2)SpringerInternationalPublishingSwitzerland2015 1
J.Benestyetal., DesignofCircularDifferentialMicrophoneArrays,
SpringerTopicsinSignalProcessing12,DOI10.1007/978-3-319-14842-7_1
