Adaptive polynomial filters and their application to nonlinear acoustic echo cancellation [Elektronische Ressource] / vorgelegt von Fabian Küch

Adaptive Polynomial Filtersand their Application toNonlinear Acoustic Echo CancellationDer Technischen Fakultat derFriedrich-Alexander-Universitat Erlangen-Nurn bergzur Erlangung des GradesDoktor-Ingenieurvorgelegt vonFabian Kuc hErlangen, 2005Als Dissertation genehmigt vonder Technischen Fakultat derFriedrich-Alexander-UniversitatErlangen-Nurn bergTag der Einreichung: 11. Mai 2005Tag der Promotion: 4. November 2005Dekan: Prof. Dr.-Ing. A. LeipertzBerichterstatter: Prof. W. KellermannProf. Dr.-Ing. P. VaryiiiDanksagungZum Gelingen dieser Arbeit hat eine Vielzahl von Personen beigetragen. Ich mochtedaher an dieser Stelle die Gelegenheit nutzen, einigen von ihnen namentlich meinen Dankauszusprechen.An erster Stelle bedanke ich mich bei Herrn Prof. Dr.-Ing. Kellermann, deres mir ermoglic ht hat, unter seiner anregenden Leitung die zugrunde liegenden wis-senschaftlichen Arbeiten am Lehrstuhl fur Multimediakommunikation und Signalverar-beitung durchzufuhren.Herrn Prof. Dr.-Ing. Vary danke ich fur sein besonderes Interesse an meinemForschungsgebiet, welches er durch die Ubernahme des Zweitgutachtens zum Ausdruckgebracht hat.Des Weiteren mochte ich Herrn Alexander Stenger dafur danken, dass er stets gewillt war, seine umfangreichen Erfahrungen in der Audiosignalverarbeitung mit mir zu teilen.
Publié le : dimanche 1 janvier 2006
Lecture(s) : 23
Source : WWW.OPUS.UB.UNI-ERLANGEN.DE/OPUS/VOLLTEXTE/2006/333/PDF/FABIAN_KUECH_DISSERTATION.PDF
Nombre de pages : 201
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Adaptive Polynomial Filters
and their Application to
Nonlinear Acoustic Echo Cancellation
Der Technischen Fakultat der
Friedrich-Alexander-Universitat Erlangen-Nurn berg
zur Erlangung des Grades
Doktor-Ingenieur
vorgelegt von
Fabian Kuc h
Erlangen, 2005Als Dissertation genehmigt von
der Technischen Fakultat der
Friedrich-Alexander-Universitat
Erlangen-Nurn berg
Tag der Einreichung: 11. Mai 2005
Tag der Promotion: 4. November 2005
Dekan: Prof. Dr.-Ing. A. Leipertz
Berichterstatter: Prof. W. Kellermann
Prof. Dr.-Ing. P. Varyiii
Danksagung
Zum Gelingen dieser Arbeit hat eine Vielzahl von Personen beigetragen. Ich mochte
daher an dieser Stelle die Gelegenheit nutzen, einigen von ihnen namentlich meinen Dank
auszusprechen.
An erster Stelle bedanke ich mich bei Herrn Prof. Dr.-Ing. Kellermann, der
es mir ermoglic ht hat, unter seiner anregenden Leitung die zugrunde liegenden wis-
senschaftlichen Arbeiten am Lehrstuhl fur Multimediakommunikation und Signalverar-
beitung durchzufuhren.
Herrn Prof. Dr.-Ing. Vary danke ich fur sein besonderes Interesse an meinem
Forschungsgebiet, welches er durch die Ubernahme des Zweitgutachtens zum Ausdruck
gebracht hat.
Des Weiteren mochte ich Herrn Alexander Stenger dafur danken, dass er stets gewillt
war, seine umfangreichen Erfahrungen in der Audiosignalverarbeitung mit mir zu teilen.
Frau Marion Schabert hat einen gro en Teil zum stilistischen Erscheinungsbild dieser
Arbeit beigetragen, wofur ich ihr ebenfalls herzlich danke.
Ohne jeden einzelnen personlic h aufzuzahlen, moc hte ich allen ehemaligen Kollegen
danken, die mir die Zeit an der Friedrich-Alexander-Universitat in Erlangen so angenehm
gemacht haben. Ich werde sie { die Zeit und die Kollegen { stets in positiver Erinnerung
behalten.
Ganz besonders danke ich meiner Frau Katrin { fur all das, was du mir jeden Tag
schenkst.ivv
Contents
1 Introduction 1
2 Polynomial Filters 5
2.1 Introduction to Polynomial Filters . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Applications for P Filters . . . . . . . . . . . . . . . . . . . . . . 8
3 Nonlinear Acoustic Echo Paths 11
3.1 Nonlinear Audio Components . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Digital-to-Analog and Analog-to-Digital Converters . . . . . . . . . 12
3.1.2 Ampli ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.3 Loudspeakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Models for the Echo Path . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.1 Discrete-Time Modeling of Continuous-Time Systems . . . . . . . . 15
3.2.2 Model for the Echo Path . . . . . . . . . . . . . . . . 16
3.3 Methods for Evaluating Nonlinear Distortion . . . . . . . . . . . . . . . . . 18
3.3.1 Characterization by a Parametric Model . . . . . . . . . . . . . . . 18
3.3.2 Correlation-Based Method . . . . . . . . . . . . . . . . . . . . . . . 20
4 Linear Echo Cancellation for Nonlinear Echo Paths 23
4.1 Basics of Linear Acoustic Echo Cancellation . . . . . . . . . . . . . . . . . 23
4.1.1 Adaptation of the Linear Echo Canceller . . . . . . . . . . . . . . . 24
4.1.2 Control of the Adaptation . . . . . . . . . . . . . . . . . . . . . . . 26
4.1.3 Performance Measures . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 E ect of Nonlinearities on Linear Echo Cancellation . . . . . . . . . . . . . 33
4.2.1 Optimum Adaptive Filter Coe cien ts . . . . . . . . . . . . . . . . . 33
4.2.2 Maximum Achievable Echo Attenuation . . . . . . . . . . . . . . . 35
4.2.3 Adaptation Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Volterra Filters in Cartesian Coordinate Representation 41
5.1 Time-Domain Volterra Filters . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.1.1 Adaptation of Time-Domain Volterra Filters . . . . . . . . . . . . . 44vi Contents
5.1.2 Control of the Adaptation . . . . . . . . . . . . . . . . . . . . . . . 47
5.1.2.1 Second-Order Volterra Filters . . . . . . . . . . . . . . . . 47
5.1.2.2 Extension to Higher-Order Volterra Filters . . . . . . . . . 54
5.1.2.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Frequency-Domain Volterra Filters . . . . . . . . . . . . . . . . . . . . . . 57
5.2.1 Partitioned Block Frequency-Domain Volterra Filters . . . . . . . . 59
5.2.1.1 Symmetry Properties . . . . . . . . . . . . . . . . . . . . . 63
5.2.1.2 Vector-based Representation . . . . . . . . . . . . . . . . . 66
5.2.2 Adaptation of Frequency-Domain Volterra Filters . . . . . . . . . . 68
5.2.2.1 Unconstrained Coe cien t Update . . . . . . . . . . . . . . 69
5.2.2.2 Constrained Coe cien t Update . . . . . . . . . . . . . . . 70
5.2.2.3 Mixed Coe cien t Update . . . . . . . . . . . . . . . . . . 71
5.2.3 Step-Size Normalization and Control . . . . . . . . . . . . . . . . . 72
5.2.3.1 Joint of All Kernels . . . . . . . . . . . . . 72
5.2.3.2 Kernel-Dependent Normalization . . . . . . . . . . . . . . 75
5.2.4 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . 77
5.2.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Application to Real Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 Volterra Filters in Diagonal Coordinate Representation 91
6.1 Application to Cascaded Structures . . . . . . . . . . . . . . . . . . . . . . 94
6.2 Time-Domain Volterra Filters . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.1 Adaptation of Time-Domain Volterra Filters . . . . . . . . . . . . . 97
6.2.2 Decorrelation of the Input Signal . . . . . . . . . . . . . . . . . . . 98
6.2.2.1 Linear Filters . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2.2.2 Volterra Filters . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2.2.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3 Frequency-domain Volterra Filters . . . . . . . . . . . . . . . . . . . . . . . 108
6.3.1 Multidelay V Filters . . . . . . . . . . . . . . . . . . . . . . . 108
6.3.2 Adaptation of Multidelay Volterra Filters . . . . . . . . . . . . . . . 112
6.3.3 Control of the Adaptation . . . . . . . . . . . . . . . . . . . . . . . 113
6.3.3.1 Joint Normalization of All Kernels . . . . . . . . . . . . . 113
6.3.3.2 Kernel-Dependent Normalization . . . . . . . . . . . . . . 114
6.3.4 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . 114
6.3.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4 Application to Real Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7 Power Filters 125
7.1 Application to Cascaded Structures . . . . . . . . . . . . . . . . . . . . . . 127
7.2 Orthogonalized Power Filters . . . . . . . . . . . . . . . . . . . . . . . . . 131Contents vii
7.2.1 Orthogonalization of the Input Signals . . . . . . . . . . . . . . . . 131
7.2.2 Equivalent Orthogonalized Structure . . . . . . . . . . . . . . . . . 135
7.2.2.1 Coe cien ts of the Equivalent Orthogonalized Structure . . 136
7.2.2.2 Coe cien t Adjustment for Signal-Adaptive Orthogonal-
ization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.2.3 Adaptation of Orthogonalized Power Filters . . . . . . . . . . . . . 139
7.2.3.1 Coe cien t Update . . . . . . . . . . . . . . . . . . . . . . 139
7.2.3.2 Adaptation Control . . . . . . . . . . . . . . . . . . . . . 140
7.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.3.1 Power Filters as Model for the Echo Path . . . . . . . . . . . . . . 144
7.3.2 E ect of Input Orthogonalization . . . . . . . . . . . . . . . . . . . 147
7.4 Application to Real Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.4.1 Nonlinear Ampli er . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.4.2 Loudspeaker of a Mobile Phone . . . . . . . . . . . . . . 153
8 Summary and Conclusions 157
A A Useful Property of Spherically Invariant Random Processes 161
B Notations 165
B.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
B.2 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
B.3 Mathematical Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
C Titel, Inhaltsverzeichnis, Einleitung und Zusammenfassung 173
C.1 Titel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
C.2 Inhaltsverzeichnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
C.3 Einleitung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
C.4 Zusammenfassung und Schlussfolgerungen . . . . . . . . . . . . . . . . . . 179
Bibliography 184viii Contents1
1 Introduction
Linear adaptive ltering plays an important role in statistical signal processing and re-
spective theoretical and practical results are well established [23]. In practice, however,
nonlinear adaptive ltering often becomes desirable if the considered systems exhibit non-
linear behaviour. Acoustic echo cancellation represents an important example for such
situations.
Acoustic echoes arise from acoustic coupling between the receive path and the transmit
path of telecommunication systems. Such feedback occurs, e.g., with hands-free telecom-
munication terminals or mobile phones. The general setup of the echo cancellation prob-
lem is illustrated in Fig. 1.1. The received signal of the far-end subscriber is output at
receive path (from far-end)
PSfrag replacements
acoustic echoAEC
at near-end
estimate of echo
transmit path (to far-end)
Figure 1.1: General set-up of the acoustic echo cancellation problem.
the near-end loudspeaker. In case of hands-free communication systems, the loudspeaker
signal is re ected at the enclosure and picked up by the microphone. In mobile phones,
the loudspeaker and the microphone are mounted very closely. Thus, the signal leakage
via their shell and the acoustic propagation path along the head of the user have to be
taken into account, too. Due to this acoustic feedback, the loudspeaker signal reaches the
transmit path of the telecommunication device and the far-end subscriber notices a de-
layed version of his own speech. This echo signal represents a very distracting disturbance
for the far-end subscriber and can even inhibit interactive, full-duplex communication.
The most common method to cope with these echoes is to place an acoustic echo
canceller (AEC) in parallel to the propagation path of the echo signal. In the AEC, a
digital replica of the echo signal is estimated which is then subtracted from the observed
microphone signal. The AEC is usually realized on a digital signal processor (DSP) which

2 1. Introduction
implies digital-to-analog conversion of the received far-end signal and analog-to-digital
conversion of the microphone signal. Note that the corresponding signal converters are
not explicitly shown in Fig. 1.1.
Standard approaches for the cancellation of acoustic echoes rely on the assumption
that the echo path can be modeled by a linear system [7]. Accordingly, the AEC is im-
plemented as a linear lter. Since the echo path is unknown and, moreover, can change
during the operation time, the linear lter has to be realized adaptively. Unfortunately,
the simple assumption of a linear echo path does not always hold in practice, as it does
not include the behaviour of nonlinear audio hardware. The nonlinearly distorted compo-
nents of the echo signal can not be captured by a linear AEC and are, thus, transmitted
back to the far-end. Consequently, any non-negligible nonlinear distortion of the echo
signal leads to a reduction of the achievable echo attenuation provided by purely linear
approaches and, thus, impairs the quality of speech communication systems. Possible
sources for nonlinear distortion in the echo path are, e.g., small loudspeakers driven at
high volume or overloaded ampli ers [67]. The problem of nonlinearly distorted echoes is
especially present in mobile communication devices, where high sound levels are desired
while having only low battery voltage available. To give an example: In case of mobile
phones operated in their hands-free mode, consumers usually prefer a nonlinear distor-
tion of the loudspeaker signal over reduced output levels. Nonlinear echoes also occur
in hands-free teleconferencing systems that include small-sized loudspeakers. If the con-
sumer sets the loudspeaker system to its maximum volume, a linear behaviour of small
and/or cheap loudspeakers can not be expected anymore. The listening tests presented
in [67] show that the accepted level of nonlinear distortion of speech is su cien tly high
to cause annoying nonlinear echoes which can not be compensated by linear AECs.
To surpass echo cancellation performance of purely linear approaches, nonlinear meth-
ods have to be taken into consideration, where basically twohes can be applied
[67]:
nonlinear preprocessing of the loudspeaker signal,
nonlinear adaptive ltering in the AEC.
The rst approach aims at a linearization of the audio hardware components via nonlinear
preprocessing of the received far-end signal. Then, the overall echo path to be modeled
by the AEC consists of the acoustic echo path which is extended by the nonlinear prepro-
cessing stage. In case of an ideal preprocessing of the loudspeaker signal, this overall echo
path is linear and, thus, the AEC can also be realized as a linear lter. This approach
can include methods known from the linearization of loudspeakers [15] and/or techniques
that are used to compensate for the nonlinear distortion introduced by overloaded power
ampli ers in digital communication systems [40]. Another method is to intentionally
limit the excitation signal of the loudspeaker in order to avoid nonlinear behaviour of the

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