SPEECH ENHANCEMENT USING AN MMSE SPECTRAL AMPLITUDE ESTIMATOR BASED ON A MODULATION DOMAIN KALMAN FILTER WITH A GAMMA PRIOR

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1 SPEECH ENHANCEMENT USING AN MMSE SPECTRAL AMPLITUDE ESTIMATOR BASED ON A MODULATION DOMAIN KALMAN FILTER WITH A GAMMA PRIOR Yu Wag? Mike Brookes? Egieerig Departmet Uiversity of Cambridge Uited Kigdom Departmet of Electrical ad Electroic Egieerig Imperial Collge Lodo Uited Kigdom yw96@cam.ac.uk mike.brookes@imperial.ac.uk ABSTRACT I this paper we propose a miimum mea square error spectral estimator for clea speech spectral amplitudes that uses a Kalma filter to model the temporal dyamics of the spectral amplitudes i the modulatio domai. Usig a two-parameter Gamma distributio to model the prior distributio of the speech spectral amplitudes we derive closed form expressios for the posterior mea ad variace of the spectral amplitudes as well as for the associated update step of the Kalma filter. The performace of the proposed algorithm is evaluated o the TIMIT core test set usig the perceptual evaluatio of speech quality ) measure ad segmetal SNR measure ad is show to give a cosistet improvemet over a wide rage of SNRs whe compared to competitive algorithms. Idex Terms speech ehacemet modulatio domai Kalma filter miimum mea-square error MMSE) estimator. INTRODUCTION Over several decades umerous speech ehacemet algorithms have bee proposed. Amog the most popular are those such as [ ] which apply a variable gai i the short time Fourier trasform STFT) domai to estimate the spectral amplitudes of the clea speech. Although these STFT-domai ehacemet algorithms ofte improve the sigal-to-oise ratio SNR) dramatically the temporal dyamics of the speech spectral amplitudes are ot icorporated ito the derivatio of the estimator. There is evidece however that sigificat iformatio i speech is carried by the modulatio of spectral evelopes i additio to the evelopes themselves [ 5]. Spectral modulatio-domai processig has bee used i speech recogitio [6 7] i speech itelligibility metrics [8 9] ad i speech ehacemet [ ]. I oe such ehacemet algorithm [] the temporal evelope of the amplitude spectrum of the oisy speech is processed separately i each subbad by a Kalma filter KF) i order to obtai the spectral amplitudes of the ehaced speech. This modulatio-domai KF combies the estimated dyamics of the speech spectral amplitudes with the observed oisy speech amplitudes to give a miimum mea square error MMSE) estimate of the amplitude spectrum of the clea speech uder the assumptio that the spectral amplitudes of both the clea speech ad the oise are Gaussia distributed. I this paper we propose a MMSE spectral amplitude estimator uder the assumptio that the speech amplitudes follow a geeralized Gamma distributio []. The advatage of the proposed Yu Wag was a PhD studet at Imperial College Lodo durig the course of this work. estimator over previously proposed spectral amplitude estimators [ ] is that it icorporates temporal cotiuity ito the MMSE estimator by the use of the KF ad that it uses a Gamma prior which is a more appropriate model for the speech spectral amplitudes tha a Gaussia prior [].. SIGNAL MODEL AND KALMAN FILTER We assume a additive model i the STFT domai i which for frequecy bi k of frame Y k X k + W k ) where X ad W deote the complex-valued STFT coefficiets of the clea speech ad the oise respectively. Sice each frequecy bi is processed idepedetly withi our algorithm we omit the frequecy idex k i the remaider of this paper. We deote the spectral amplitudes as: X A Y R ad W N. The predictio model we assume for the clea speech spectral amplitudes is a F a + v ) where a [A A p+] T is the p-dimesioal state vector ad v deotes the zero-mea predictio residual with covariace apple matrix Q. The p p) trasitio matrix has the form b F T where b I [b b p] T is the vector of liear predictio LPC) coefficiets for the speech spectral amplitudes i frame. Our model differs from that used i [] i two respects: we treat the oise ad speech as additive i the complex STFT domai rather tha i the spectral amplitude domai ad we use a geeralized Gamma prior for the speech amplitudes rather tha a Gaussia prior.. PROPOSED ESTIMATOR DESCRIPTION A block diagram of the proposed algorithm is show i Fig.. The oisy speech yt) is coverted to the time-frequecy domai R k e j k usig the STFT [5]. I order to perform LPC modellig i the modulatio domai the oise power spectrum is estimated usig for example [6] or [7] ad the speech is passed through a covetioal MMSE ehacer [] to reduce the effects of the oise o the modellig. Followig this the sequece of spectral amplitudes i each frequecy bi is divided ito overlappig modulatio frames. Autocorrelatio LPC [8] is performed o each modulatio frame to determie the coefficiets b ad thece the trasitio matrix F i ).

2 STFT$ k ISTFT$ Yk x t) KF$Update$$ a P k a P Aframe$delay$$ KF$Predict$$ $$$Noise$$ Esmator$ a P $ ehacer$.6.. F b k µ.5 σ. µ. σ. µ.5 σ. µ. σ. µ8. σ..8 pa) yt) Fig.. Block diagram of the proposed Kalma Filter MMSE estimator... Kalma filter predictio step F a P F T P F ) + Q where a ad P deote respectively the a priori estimates of the amplitude state vector ad of the correspodig covariace matrix at time ad a deotes the a posteriori estimate of the state vector at time. The first elemet of the state vector a correspods to the spectral amplitude i the curret frame A ad so its a priori mea ad variace are give by ) V ara R dt a ) ct P c 6) I this sectio we describe the KF MMSE update step which determies a updated state estimate by combiig the predicted state vector ad covariace the estimated oise ad the observed spectral amplitude. Withi the update step we model the prior speech amplitude A usig a -parameter Gamma distributio a ) exp a φ λ Fig.. The curve of ad < versus where < +.5) ) µ 5).. Kalma Filter MMSE update model ) true fitted + arcta ) < <. We ca elimiate where R represets the observed speech amplitudes up to time ad c [...]T. p a R.5 ) a EA R 8 From the time update model ) we obtai the KF predictio equatios 6 Fig.. Curves of Gamma probability desity fuctio for 7) with variace ad differet meas..5 µ a Modulao$ Domai$LPC$ +.5) ) 8) +.5) ) 9). betwee 8) ad 9) to obtai µ.5) ) µ + ) We eed to solve the o-liear equatio ) to determie from the value of which ca be calculated from µ ad <. Istead of dealig with di ad satisfies < rectly it is coveiet to set arcta ) where lies i the rage < <. The solid lie i Fig. shows the fuctio ). We ca approximate this fuctio well with a low-order polyomial that is costraied to pass through the poits ) ad ) ad i the experimets i Sec.. we use the quartic approximatio 7) where ) is the Gamma fuctio. The distributio is obtaied by settig c i the geeralized Gamma distributio give i [9] ad the two parameters ad are chose to match the mea µ ad variace of the predicted amplitude from 5) ad 6). Examples of the probability desity fuctios from 7) with variace ad meas µ i the rage.5 to 8 are show i Fig. from which it ca be see that the distributio i 7) is sufficietly flexible to model the outcome of the predictio over a wide rage of µ /. At frame the mea ad variace of the Gamma distributio i 7) ca be expressed i terms of ad [9] as ) which is show with asterisks i Fig.. Give we ca use this polyomial to obtai ad thece by the iverse trasform ta )... Derivatio of estimator The MMSE estimate of A is give by the coditioal expectatio µ EA R ) ˆ a pa R )da )

3 Usig Bayes rule the coditioal probability is expressed as p a R )pa y R ) p y a R ) p a p y R ) R ) d ) where is the realizatio of the radom variable which represets the phase of the clea speech. Because Y is coditioally idepedet of R give a ad ) becomes p a R ) p y a ) p a R ) d p y R ) ) Followig [] the observatio oise is assumed to be complex Gaussia distributed with variace EN ) leadig to the observatio prior model py a ) exp y a e j ) Uder the assumptio of the statistical models previously defied ad assumig that the phase compoets ad amplitude compoets ad A are idepedet we ca ow calculate a closed-form expressio for the estimator ) usig [ Eq ] s µ +.5) ) + ) M +.5; ; M ;; where M is the cofluet hypergeometric fuctio [] ad EA Y ) µ + R + R + 5) are the a priori SNR ad a posteriori SNR respectively. The variace of the posterior estimate is give by E A R E A R )) M +;; + ) M ;;.. Update of state vector + R µ +. 6) The fial step is to update the etire state vector ad the associated covariace matrix a ad P. I order to decorrelate the curret observatio from the rest of the state vector we decompose the covariace matrix P as P apple g T g G where g is a p state vector as )-dimesioal vector. We ow trasform the z H a 7) usig the trasformatio matrix H " T # I. The g covariace matrix U of the trasformed state vector z is give by U E z z T H P H T " T G ggt We see that the first elemet of z is equal to µ ad ucorrelated with ay of the other elemets ad is therefore distributed as N µ ). Usig the posterior mea ad variace from 5) ad 6) ad c [... ] T we ca update the trasformed mea vector ad covariace matrix as z z +µ µ )c U U + cc T. Ivertig the trasformatio i 7) we obtai after some algebraic maipulatio the followig update equatios a a + µ µ P c 8) P P + P cc T P. 9) I this sectio we have derived the update equatios for the KF. For each acoustic frame of oisy speech we first use ) ad ) to calculate the a priori state vector a ad the correspodig covariace P ad solve ) to fid. We the use 5) ad 6) to calculate the a posteriori estimate of the amplitude ad the correspodig variace respectively. Fially the KF state vector ad its covariace matrix are updated usig 8) ad 9).. IMPLEMENTATION AND EVALUATION.. Implemetatio of algorithm I this sectio we compare the performace of the proposed KF based MMSE ) estimator with five other algorithms: i) the baselie log-amplitude MMSE ehacer from [ ]; ii) the perceptually motivated MMSE estimator from [ ] usig a weighted Euclidea distortio measure with a power expoet of p ; iii) ModSub the modulatiodomai spectral subtractio from []; iv) the versio of the modulatio-domai Kalma filter from [] that extracts the modulatio-domai LPC coefficiets from ehaced speech usig the algorithm [ ]); v) KFMMSEI a itermediate versio of our proposed algorithm that assumes the speech ad oise add i the STFT amplitude domai rather tha the complex STFT domai i.e. replacig ) with Y k X k + W k ). The parameters of all the algorithms were chose to optimize performace o a subset of the traiig set of the TIMIT database []. We have used a acoustic frame legth of ms with a ms icremet which gives a 5 Hz samplig frequecy i the modulatio domai. The speech LPC models are determied from a modulatio frame of duratio 8 ms acoustic frames) with a 6 ms frame icremet ad the model orders i both the ad algorithms are l. I the experimets we use the core test set from the TIMIT database which cotais 6 male ad 8 female speakers #.

4 5 5.5 segsnr db) Global SNR of oisy speech db) segsnr db) Global SNR of oisy speech db) Fig.. Average segmetal SNR of ehaced speech speech after processig by four algorithms plotted agaist the global SNR of the iput speech corrupted by additive car oise left) ad street oise right). The algorithm acroyms are defied i the text Global SNR of oisy speech db) Global SNR of oisy speech db) Fig. 5. Average quality of ehaced speech after processig by four algorithms plotted agaist the global SNR of the iput speech corrupted by additive car oise left) ad street ose right). each readig 8 distict seteces totallig 9 seteces) ad the speech is corrupted by the oise from the RSG- database [5] ad the ITU-T test sigals database [6] at 5 5 ad 5 db global SNR. A Hammig widow is used i the STFT aalysis ad sythesis ad the oise power spectrum k is estimated usig the algorithm from [7] as implemeted i []. It is possible for the algorithm to lock up with µ ; to prevet this we impose the costrait >.5 i )... Performace evaluatios The performace of the algorithms is evaluated usig both segmetal SNR segsnr) ad the perceptual evaluatio of speech quality ) measure defied i ITU-T P.86. All the measured val-.5 ModSub I Fig. 7. Box plot showig the differece i score betwee competig algorithms ad the proposed algorithm for 76 speech+oise combiatios. ues show are averages over the 9 seteces i the TIMIT core test set. Figure shows the average segsnr of speech eghaced by the proposed algorithm ) as well as by the ad algorithms. The left ad right plots respectively show results for car oise [5] ad street oise [6]. We see that for car oise which is predomiatly low frequecy gives the best segsnr especially at poor SNRs where it is approximately db better tha the ext best algorithm. For street oise however which has a broader spectrum the situatio is reversed ad the algorithm has the best performace especially at SNRs above 5 db. Figure 5 shows the correspodig average scores for car oise left plot) ad street oise right plot). We see that with this measure the algorithm clearly has the highest performace. For car oise the score from the algorithm is approximately. better tha that of the other algorithms at SNRs below 5 db while for street oise the correspodig figure is.5. These differeces correspod to SNR improvemets of db ad.5 db respectively. To assess the robustess to oise type we have evaluated the algorithms usig twelve differet oise types from [5] with the average SNR for each oise type chose to give a mea score of. for the oisy speech. I Fig. 6 the solid lies show the media the boxes the iterquartile rage ad the whiskers the extreme values for the 98 speech-plus-oise combiatios. Figure 7 shows box plots of the differece i score betwee competig algorithms ad. We see that i all cases the etire box lies below the axis lie; this idicates that results i a improvemet for a overwhelmig majority of speechplus-oise combiatios. The I box plot demostrates the small but cosistet beefit of usig a additive model i the complex STFT domai rather tha the amplitude domai..5.5 ModSub I Fig. 6. Box plot of the scores for oisy speech processed by six ehacemet algorithms. The plots show the media iterquartile rage ad extreme values from 76 speech+oise combiatios. 5. CONCLUSION I this paper we have proposed a MMSE spectral amplitude estimator based o a modulatio domai Kalma filter. The ovel MMSE estimator icorporates a model of the temporal dyamics of spectral amplitudes withi each frequecy bi by usig a Kalma filter. We have show how the parameters of the speech prior model ca be estimated from the predicted state vector from the Kalma filter ad used to calculate the estimator i the update step. The proposed algorithm gives a cosistet improvemet i over all the competitive algorithms demostratig that ca be improved by about. over the baselie ehacer for a wide rage of SNRs.

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