Bayesian Enhancement of Speech and Audio Signals which can be Modelled as ARMA Processes

29 Pages·2007·0.39 MB·English

Bayesian Enhancement of Speech
and Audio Signals which can be
Modelled as ARMA Processes
By Simon J. Godsill
email: [email protected]
Signal Processing and Communications Group
University of Cambridge
1st version July 1996, revised October 1996.
Appeared in the International Statistical Review, 65(1), 1-21
Abstract
Inapplicationareaswhichinvolvedigitisedspeechandaudiosignals,
such as coding, digital remastering of old recordings and recognition of
speech, it is often desirable to reduce the e(cid:11)ects of noise with the aim
of enhancing intelligibility and perceived sound quality. We consider
the case where noise sources contain non-Gaussian, impulsive elements
superimposed upon a continuous Gaussian background. Such a situ-
ation arises in areas such as communications channels, telephony and
gramophonerecordingswhereimpulsivee(cid:11)ectsmightbecausedbyelec-
tromagneticinterference(lightningstrikes), electricalswitchingnoiseor
defectsinrecordingmedia,whileelectricalcircuitnoiseorthecombined
e(cid:11)ectofmanydistantatmosphericeventsleadtoacontinuousGaussian
component.
In this paper we discuss the background to this type of noise de-
gradation and describe brie(cid:13)y some existing statistical techniques for
noisereduction. We proposenewmethodsforenhancement basedupon
Markov chain Monte Carlo (MCMC) simulation. Signals are mod-
elled as autoregressive moving-average (ARMA), while noise sources
are treated as discrete and continuous mixtures of Gaussian distribu-
tions. Resultsarepresentedforbothrealandarti(cid:12)ciallycorrupteddata
sequences, illustrating the potential of the new methods.
Danslesapplicationsdedigitalisationdelaparoleetdessignauxau-
dios,telsquelecodage,larestaurationdigitaledevieuxenregistrements
1
sonores et la reconnaissance de la parole, il est souvent d(cid:19)esirable de
r(cid:19)eduirelese(cid:11)ets du^s(cid:18)a lapr(cid:19)esence de bruitpouram(cid:19)eliorerla clart(cid:19)e et la
qualit(cid:19)eduson. Nousconsid(cid:19)eronslecasou(cid:18)lasourcedebruitcontientdes
(cid:19)el(cid:19)ements impulsifsnon-Gaussien,superpos(cid:19)esaubruitGaussiendefond.
Ce genre de situation se manifeste notamment en t(cid:19)el(cid:19)ephonie mobile, en
communication par c^able et sur des vieux enregistrements gramopho-
niques. Le bruit de nature impulsive peut ^etre du^ aux interf(cid:19)erences
(cid:19)electromagn(cid:19)etiques, aux commutations (cid:19)electriques environnantes, ou (cid:18)a
des d(cid:19)efauts li(cid:19)es au support sonore. Par contre, le bruit provenant de
circuits(cid:19)electriques ou de l'e(cid:11)et combin(cid:19)e de plusieurs(cid:19)ev(cid:19)enements atmo-
sph(cid:19)eriques lointains se traduit par du bruit de nature Gaussienne.
Nouspr(cid:19)esentonsdanscetarticlelesprincipesdecegenreded(cid:19)egradation
etfaisonsunedescriptionsommairedequelquesm(cid:19)ethodesd(cid:19)eja(cid:18)existantes
pour la r(cid:19)eduction du bruit. Nous proposons des approches nouvelles
pour l'am(cid:19)elioration des signaux, bas(cid:19)ees sur la m(cid:19)ethode Monte Carlo-
Chaines de Markov (MCMC). Les signaux sont mod(cid:19)elis(cid:19)es comme des
proc(cid:19)ed(cid:19)es autor(cid:19)egressifs (cid:18)a moyenne coulissante (ARMA), tandis que les
sources de bruit sont consid(cid:19)er(cid:19)ees comme des combinaisons lin(cid:19)eaires,
discr(cid:18)etes et continues, de distributions Gaussiennes. Nous pr(cid:19)esentons
desr(cid:19)esultatssurdess(cid:19)equencessynth(cid:19)etiquesetr(cid:19)eellesentach(cid:19)ees debruit
pour illustrer les capacit(cid:19)es de ces m(cid:19)ethodes.
Keywords: ARMA model, Markov chain Monte Carlo, noise reduction,
impulsive noise, outliers, robust
1 Introduction
This paper is concerned with the reconstruction of acoustically recorded sig-
nals, such as speech and music, which are degraded both by impulsive non-
Gaussian noise sources (`outliers') and continuous Gaussian background noise.
Noise degradation of this type is quite common in a number of physical meas-
urement systems. In the case of gramophone recordings there are several
mechanisms for impulsive defects in the physical storage medium, including
natural irregularities, scratches, cracks and dust particles. These all give rise
to localized noise artefacts in the recorded sound which are perceived as the
characteristic `click' and `crackle' noise associated with old recordings. In
analogue communications channels impulsive noise occurs as a result of elec-
tromagnetic interference, switching noise and atmospheric noise. Background
noise, which can occur for example as thermal noise in electronic circuitry or
the combinationof many distant interference e(cid:11)ects, willbe present to varying
degrees in all forms of recorded sound, and is often a signi(cid:12)cant perceptual
factor even when high energy impulses are also present.
In the (cid:12)rst part of this paper some existing methods for treatment of im-
pulsive and background noise are discussed. In the second part of the paper
new techniques are proposed for enhancement of acoustical signals which can
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be modelled as ARMA processes. These techniques apply Bayesian method-
ology to the problem of removing both impulsive and background noise in
one single procedure. Results of processing are presented for both arti(cid:12)cially
generated data and real data, illustrating the potential of Bayesian numerical
methods in this important area.
2 Background
2.1 The noisy recording system
Figure 1 shows in schematic form the noisy recording and playback system.
We describe the system inrelationto an electricallyrecorded gramophone disk
system, although a similar model can be applied for mechanically recorded
disks, tape recordings and transmission of signals in noisy communication
channels.
x(t) represents the input signal to the recording system, usually the sound
pressure levels at the input to a microphone or recording horn. The recording
system, which willincludethe microphoneresponse andany subsequent equal-
ization circuitry through which the signal passes, is represented by a linear
time-invariant(LTI)system withfrequency response Hr(!). Non-linearitywill
ofcourse be present inthis system, particularlyat very highinput levels which
saturate the electronics, but we do not consider such e(cid:11)ects here (see God-
sill, Rayner and Capp(cid:19)e (1996) for some possible approaches to non-linearity
problems in audio).
The output from the recording system is the signal recorded directly onto
the physical recording medium, for example the groove wall modulation on
a gramophone disk. Here an additive noise source v(t) is superimposed. v(t)
represents thecombinede(cid:11)ectofirregularitiesinthephysicalmedium,damage
due to wear and tear, excessive playback and dust and dirt particles adhering
to the medium. Such noise will be largely additive in nature, although the
occasional deep gouge or breakage in the medium may lead to `replacement'
noise in which the underlying signal is lost completely for a short interval. A
typical example of the noise signal v(t), obtained from a 78rpm recording and
sampled at a rate of 44,100 Hz with 16-bit resolution, is shown in (cid:12)gure 2.
Finally, the noisy signal passes through the playback system Hp(!), which
includesthemechanicalresponse ofthestylus, cartridgeandtone-armsetup as
well as any post-equalization circuitry applied to the signal, leading to a noisy
output signal y(t) which may be digitally sampled and stored on a computer.
Anexampleofanoutputsignaly(t)sampledfromanorchestralrecordingwith
very little noise content is shown in (cid:12)gure 3. A noisy output signal is shown
in (cid:12)gure 9.
We have already stated that the system as described is somewhat idealized
in the assumptions of linearityfor Hr and Hp as well as the additivity assump-
tion for the noise. A further elaboration to the system might include extra
3
noise sources at the input and output stages. The input noise, which could in-
cludebackground noiseinthe recordingenvironment andmicrophone/thermal
noise in the circuitry, is usually the more signi(cid:12)cant e(cid:11)ect, since output noise
can be controlled by suitable choice of playback apparatus. Here we assume
that both of these sources of noise are negligible compared with the noise in
the physical recording medium, which is usually the case for gramophone disk
recordings.
The aim of an enhancement or restoration system will be to reconstruct
the noise-free input signal x(t) given only the noisy output signal y(t). When
x(t) is a sample function from a stochastic process this will be a statistical
estimation task.
v(t)
x(t) y(t)
Recording System
Playback System
+
H (ω ) H (ω )
r p
Figure 1: Schematic of the noisy recording/playback system
2.2 Statistical Enhancement Methods
Itcanbeseen from(cid:12)gure2thatthe noisesignalsfromagramophonerecording
are highly non-Gaussian and that any e(cid:11)ective restoration system will have
to take this non-Gaussianity into account. Most existing systems treat the
high amplitude `impulsive' noise, which corresponds to large defects on the
recording medium, and the lower amplitude `background' noise in separate
procedures. The reasoning behind such schemes is that most of the impulsive
noise can be eliminated using simple threshold-based detectors followed by
interpolation,leavingbehindabackground noisecomponentwhichisGaussian
4
1500
1000
500
e
d
u
plit
m
A
0
−500
−1000
0 100 200 300 400 500 600 700 800 900 1000
Sample Number
Figure 2: Typical noise signal from a 78rpm gramophone recording
(to a good approximation) and for which linear restoration methods will be
appropriate. We (cid:12)rst consider brie(cid:13)y some of these existing techniques before
formulatingournewmethod,whichtreatsbothimpulsesandbackgroundnoise
inonesingleprocedure. The focus hereisonstatisticallybased methodswhich
are related to the proposed scheme, and hence the coverage is by no means
complete. A more complete review of audio enhancement techniques can be
found in Godsill et al. (1996).
Note that the methods described here assume `(cid:13)at' recording/playback
characteristics (i.e. Hr(!) = Hp(!) = 1). Furthermore, it is assumed that
the data have been sampled at a uniform rate so that the noisy signal may be
modelled as a discrete time series:
(1) yt = xt +vt; t = (cid:0)1;::: ;(cid:0)1;0;1;:::1
2.2.1 ARMA modelling
One suitable model for the underlying signal xt is the Autoregressive Moving
Average (ARMA) model:
(cid:18)(L)
(2) xt = et
(cid:30)(L)
q
where L is the unit delay, (cid:18)(L) = 1(cid:0)(cid:18)1L(cid:0):::(cid:0)(cid:18)qL is the MA polynomial
p
and (cid:30)(L) = 1 (cid:0) (cid:30)1L (cid:0) ::: (cid:0) (cid:30)pL is the AR polynomial (q < p). et is the
`error' or `excitation' sequence. When (cid:18)(L) = 1 we have a pure AR model,
while (cid:30)(L) = 1 gives an MA model.
5
4000
3000
2000
1000
e
d
u
plit 0
m
A
−1000
−2000
−3000
−4000
0 100 200 300 400 500 600 700 800 900 1000
Sample Number
Figure 3: Noise-free music signal from a 78rpm gramophone recording
Such a model has an appealing physical source-(cid:12)lter interpretation for
sound signals, in particular speech. The excitation signal et can be associ-
(cid:18)(L)
ated with the glottal waveform which drives a `(cid:12)lter' , corresponding to
(cid:30)(L)
the transfer function of the vocal tract (Proakis, Deller and Hansen, 1993).
A similar argument can be applied to certain musical instruments and acous-
tical sounds. Of course, if several sounds, each of which can be modelled as
ARMA processes, are added together (as in the case of an orchestra, for ex-
ample) the resulting process can still be modelled as ARMA (Box, Jenkins
and Reinsel, 1994, section A4.3).
The ARMA (cid:12)lter characteristics willvary with time as di(cid:11)erent sounds are
produced. Thisisoftenaccounted forbysegmentingthedataintosmallblocks
of perhaps 1/40 second, in which the vocal tract response is approximately
time-invariant; processing is then applied independently to each block in turn.
2.2.2 Removal of impulses
The detection of additive impulses in an acoustical signal is very much akin
to the detection of outliers in statistical data. Simple methods rely on the
fact that acoustical signals are usually `smooth' compared with impulsive dis-
tortions. Hence operations which emphasize the `roughness' of a signal, such
as high-pass (cid:12)ltering or di(cid:11)erentiation, will accentuate the impulses, allowing
for detection based on ouptut amplitudes. Samples corresponding to detected
impulses can then be replaced with values obtained from some suitable inter-
polation scheme (see Nieminen, Heinonen and Neuvo (1987), Vali(cid:18)ere (1991)
and Kasparis and Lane (1993) for some of the many approaches which can be
6
applied to this problem).
One method which has proved e(cid:11)ective employs autoregressive (AR) mod-
elling of the audio signal (Vaseghi and Rayner, 1990; Vaseghi and Frayling-
Cork, 1992). The AR coe(cid:14)cients are estimated using methods which are
robust to outliers (see e.g. Kleiner, Martin and Thomson (1979) and Martin
(1980)). Detection of impulses is then achieved by thresholding the estimated
AR excitation sequence corresponding to the noisy data, while reconstruc-
tion is performed by maximum likelihood interpolation of the corrupted data
valuesusingtheestimatedARmodel(Janssen, VeldhuisandVries,1986). Dis-
advantages of the method include the inability to detect small impulses in the
vicinity of much larger disturbances as well as the introduction of distortion
in the presence of certain signal characteristics.
Godsilland Rayner (1992)and Godsilland Rayner (1995a)have developed
sequential Bayesian methods which improve performance by allowing very ac-
curate detection of audio outliers occurring in arbitrary con(cid:12)gurations and
bursts. These methods assume an AR model for the signal with a switched,
additive Gaussian impulsive noise process and can be related to Bayesian
schemes for treatment of additive outliers in statistical data (see e.g. Box
and Tiao (1968) and Abraham and Box (1979)). Further work in this area
(O(cid:19) Ruanaidh and Fitzgerald, 1994; Godsill and Rayner, 1996b) has applied
Markov chain Monte Carlo methodology to the removal of impulsive noise
from audio signals. The promising results of this work motivate the methods
proposed later in this paper.
2.2.3 Removal of background noise
The area of background noise removal for speech and audio has received a
great deal of research attention over the past twenty years, particularly in the
speech processing community. Detailed review material can be found in Lim
and Oppenheim (1979), Lim (1983), Boll (1992) and Godsill et al. (1996). We
mention here only statistically-based methods relevant to the current exposi-
tion.
LimandOppenheim(1978)studiedtheproblemofenhancingspeech(mod-
elled as an AR process) in additive Gaussian noise. They proposed iterat-
ive maximum likelihood methods for estimation of parameters and enhanced
speech. Musicus and Lim (1979) extended the scheme to speech modelled
as an ARMA process using an Expectation-Maximize (EM)-based algorithm
(Dempster, Laird and Rubin, 1977), while Dembo and Zeitouni (1988) de-
rived EM methods for time-varying ARMA processes in noise. More recent
work along the same lines (Hansen and Clements, 1991; Ephraim, 1992) has
incorporated modelling constraints speci(cid:12)c to speech signals. We prefer to
use generic modelling techniques which are appropriate for a wide range of
acoustical signals, including both speech and music.
7
3 Proposed Methods
3.1 Overview
The methods described above operate in a two stage scheme, the (cid:12)rst of which
eliminates non-Gaussian impulsive disturbances and the second which reduces
the background component. This isclearly sub-optimaland itmay be possible
to obtain greater (cid:12)delity to the originalsource materialif alltypes of noise are
treated within one single procedure. An empiricalBayes procedure for achiev-
ing this goal was described in Godsill (1993) as an extension to the impulse
detection algorithms presented therein and in Godsill and Rayner (1995a).
Nied(cid:19)zwiecki and Cisowski (1996) have recently presented methods for address-
ing the same problem which use the Extended Kalman Filter to perform both
adaptive parameter estimation and signal enhancement. Their methods are
least-squares based andrequireaheuristicpriordetection ofimpulsepositions.
We bypass this requirement by the use of Bayesian probabilistic methods and
an explicit model for the noise generation process.
We present here a fully Bayesian approach to the enhancement of speech
and audio signals, modelled as ARMA processes, in the presence of both im-
pulsive noise and continuous background noise. Bayesian computations are
performed using a Markov chain Monte Carlo (MCMC) simulation technique
based around the Gibbs sampler (Geman and Geman, 1984; Gelfand and
Smith, 1990). The work lies within the broad class of conditionally Gaus-
sian state-space time series processing as discussed by a number of authors
in recent years, including Shephard (1994) and Carter and Kohn (1994). We
use a form of ARMA model which leads to Gaussian conditionals for the AR
and MA coe(cid:14)cients and discuss an e(cid:14)cient scheme for detecting both innov-
ational and observational outliers (impulses), which forms an integral part of
the MCMC simulation. We discuss the relationship of the methods to existing
MCMCtechniques forARMAmodelling(ChibandGreenberg,1994;Marriott,
Ravishanker, Gelfand and Pai, 1996; Barnett, Kohn and Sheather, 1996).
The framework presented is quite general, allowingfor a full ARMA model
withimpulsivenoiseinbothexcitationsequenceandobservationnoise. Special
cases ofthemethods willbe ofuse inlesstesting conditions; inparticularnoise
reduction for pure AR or MA processes is easily achieved, as is noise reduction
for simple Gaussian noise with no impulsive elements.
Straightforward extensions to the framework which may be of practical
importance (but are not detailed here) include the incorporation of non-(cid:13)at
recording/playback responses Hr and Hs, as well as the possibility of the ob-
servation noise being a dependent process (i.e. non-white or `coloured'). Each
of these extensions requires the inclusion of an extra linear (cid:12)ltering operator
into the model.
8
3.2 Signal and Noise Models
The underlying signal xt is modelled as an ARMA(p;q) process (see equation
(2)), which is assumed to be minimum phase and stationary so that all poles
(cid:0)1 (cid:0)1
and zeros (i.e. the solutions of (cid:30)(z ) = 0 and (cid:18)(z ) = 0) lie strictly within
the unit circle. This ensures a unique ARMA representation (Box et al., 1994,
section 6.4)
et is an i.i.d. excitation process. Non-Gaussian impulsive elements are
modelled in the excitation in order to allow for transient e(cid:11)ects in the data
such as note or word `attacks' and the impulse-like excitations found in voiced
speech signals.
In order to facilitateMCMC sampling we use a factorization of the ARMA
model into a pure AR process ut cascaded with an MA (cid:12)lter:
1
(3) xt = (cid:18)(L)ut; ut = et
(cid:30)(L)
Thisfactorizationensures GaussianconditionallikelihoodsforboththeAR
and the MA coe(cid:14)cients when the ARMA process is observed inadditive noise,
as noted by Musicus and Lim (1979). A schematic diagram for the model is
given in (cid:12)gure 4.
et ut xt
AR System MA System
θ
φ (L)
1/ (L)
Figure 4: Schematic of ARMA model used
The observation equation (1) and factorized ARMA representation (3) are
readily combined into state-space form as:
9
(4) yt = z(cid:11)t +vt
(5) (cid:11)t = T(cid:11)t(cid:0)1 +wet
where
T
(6) (cid:11)t = [ut;ut(cid:0)1;::: ;ut(cid:0)p+1]
(7) z = [1;(cid:0)(cid:18)1;:::;(cid:0)(cid:18)q;0;::: ;0]
2 3 2 3
(cid:30)1 (cid:30)2 ::: (cid:30)p(cid:0)1 (cid:30)p 1
6 7 6 7
61 0 ::: 0 07 607
6 7 6 7
(8) T = 66 0 1 ::: 0 077; w = 66...77
664 ... ... ... ... ... 775 6640775
0 0 ::: 1 0 0
This state-space form will allow fast Kalman (cid:12)lter-based smoothing oper-
ations to be used in the signal reconstruction step of the Gibbs sampler.
In the same manner as for the excitation noise, impulsive elements are
included in the observation noise, vt. This allows for the characteristic out-
lier e(cid:11)ects which are associated with recording media such as gramophone
disks and with communications channels. Both noise sources are modelled as
Gaussian with switched, time-varying variance components:
2 2 2
(9) vt (cid:24) N(0;(1(cid:0)it)(cid:27)v +itgt(cid:27)v)
2 2 2
(10) et (cid:24) N(0;(1(cid:0)jt)(cid:27)e +jtht(cid:27)e)
Here the it (jt) are indicator variables which switch a noise component
2 2
between a continuous background component with variance (cid:27)v ((cid:27)e) and a
2 2 2 2
volatile impulsive component with variance gt(cid:27)v (ht(cid:27)e) and gt > 1 (ht > 1).
A similarnoise model was proposed by Godsill(1993, chapter 4) and has been
implemented for AR data using the Gibbs sampler in Godsill and Rayner
(1996a). Such a model can be viewed as a logical extension to the purely
impulsive (i.e. no background component) models used by Godsill (1993)
and Godsill and Rayner (1995a) for the processing of audio signals and more
recently implemented using MCMC methods (Godsill and Rayner, 1996b).
This latter work is closely related to MCMC work instatisticaloutlieranalysis
(McCulloch and Tsay, 1994; Barnett et al., 1996).
For computational convenience we choose conditionally conjugate priors
for the noise process variance parameters. Markov dependence is chosen for
the indicator variables in order to model the observed temporal `clustering'
of impulsive noise in sound recordings (see (cid:12)gure 2), corresponding to small
areas of degradation in the recording medium (similar models have been used
to model interference in, for example, telephony (Gilbert, 1960)). The priors
10