On Optimal Smoothing in Minimum Statistics Based Noise Tracking

Transcription

1 INTERSPEECH 25 On Optima Smoothing in Minimum Statistics Based Noise Tracking Aeksej Chinaev, Reinhod Haeb-Umbach Department of Communications Engineering, University of Paderborn, 3398 Paderborn, Germany Abstract Noise tracking is an important component of speech enhancement agorithms. Of the many noise trackers proposed, Minimum Statistics ( is a particuary popuar one due to its simpe parameterization and at the same time exceent performance. In this paper we propose to further reduce the number of parameters by giving an aternative derivation of an optima smoothing constant. At the same time the noise tracking performance is improved as is demonstrated by experiments empoying speech degraded by various noise types and at different SNR vaues. Index Terms: speech enhancement, noise tracking, optima smoothing. Introduction In the short time Fourier transform (STFT domain speech is known to have a sparse representation, i.e., the vast majority of energy is concentrated in a few time-frequency (TF sots. It is this property that is expoited in many speech signa extraction agorithms, and it is aso this property that can be expoited for estimating the power spectra density (PSD of the noise. Even in time intervas, where a person is speaking, TF sots exisit that have no or negigibe speech energy, such that the STFT vaues of these sots are governed by the background noise. If the noise is sufficienty stationary, these TF sots suffice to track the noise PSD which can then be used in speech enhancement agorithms. However, a major issue is how to reiaby identify those TF sots in which the noise spectrum can be observed. One way to track the noise PSD without the necessity of an error-prone speech presence probabiity estimation is the famous Minimum Statistics ( approach []. The estimator is deveoped on the observation that the PSD of a noisy speech signa often drops to the noise power eve [2]. To bypass the intermediate time spans with speech activity the agorithm carries out a minimum search on the smoothed spectrum of the noisy speech, using a causa siding window of a certain ength. From the found minima an unbiased noise PSD estimate is provided after a bias compensation. Besides an efficient reaisation of the minimum search and a sophisticated cacuation of the bias compensation factor, a third key component of the approach is an anaytica derivation of the optima smoothing parameter to be used for smoothing the noisy speech spectrum before the minimum search is appied. A recursive averaging of the PSD of the noisy speech is aso done in many other state-of-the-art noise trackers [ 8]. Most of them consider the noise PSD as an unknown but fixed parameter, rather than a random variabe. In this contribution we treat the noise PSD as a random variabe and empoy Bayesian estimation. The update equations for the maximum a posteriori or minimum mean squared error estimates resut in a recursive averaging, thus giving a nice statistica derivation and interpretation of the smoothing operation mentioned above. This derivation aso deivers the optima time-variant smoothing parameter, however ony for the case of a constant noise PSD. It needs to be adjusted for the practica case of a time-variant noise PSD, for which we aso propose a soution. In [] a different derivation of the smoothing parameter is given. The noise PSD is treated as a deterministic parameter, which is estimated in the Maximum Likeihood sense. The optima smoothing parameter is derived anayticay from the statistics of the noisy observations. However, it exhibits some issues, such as a so-caed dead ock for vaues of the a posteriori SNR cose to one and poor performance of the noise tracking in high eves of nonstationary noise. For these issues heuristic soutions are given in []. Here we show that the new derivation based on Bayesian estimation mitigate these issues. The paper is organized as foows. In the next section we introduce the statistica modeing and present the smoothing of the observed noise PSD as a consequence of Bayesian estimation of a constant, however, random noise PSD. In Section 3 we consider the practica case of time-variant noise statistics and adjust the smoothing parameter derived in the Section 2 correspondingy. We then compare its behavior with the soution given in []. Experimenta resuts on noise tracking and speech enhancement are presented in Section 4, before we concude the paper in Section Statistica modeing 2.. estimation of noise PSD The STFT coefficients of the cean speech signa S,k and of the noise signa N,k at frame number and frequency bin k are modeed as uncorreated compex-vaued zero-mean normay distributed random processes with statisticay independent rea and imaginary parts. In this case an additive superposition of the two signas resuts in an exponentiay distributed power spectra density of the noisy speech Y,k 2 = S,k 2 + N,k 2.Its probabiity density function (PDF is given by: p Y,k 2(x; λ Y,,k = ( exp λ Y,,k x λ Y,,k, ( where λ Y,,k = E{ Y,k 2 } = λ S,,k + λ N,,k,andwhere λ S,,k = E{ S,k 2 } and λ N,,k = E{ N,k 2 } are the speech and noise variances. The goa of a noise tracker is to estimate λ N,,k from the observed noisy spectrogram Y,k 2.Sincethe PSD estimator treats each frequency component identicay and independenty of the others, we wi drop the frequency bin index k in the foowing. The noise PSD estimation is especiay chaenging in the presence of speech and if the noise is nonstationary. On the other hand, in the absence of speech, i.e., if λ Y, = λ N, and Copyright 25 ISCA 785 September 6-, 25, Dresden, Germany

2 if the noise is stationary λ N, = λ N for a certain number L of consecutive observations, the unbiased Maximum Likeihood ( estimate of the noise PSD is simpy given by the sampe mean ˆλ N =(/L L (ˆλ = Y 2 with var N =2λ 2 N /L. Athough in the estimation the true noise PSD λ N is treated as an unknown but fixed parameter, the estimate ˆλ N is a random variabe, since it is a function of the random observations Y 2. It has a scaed chi-squared (χ 2 spdf: pˆλ N (x; ν, τ 2 = (τ 2 ν/2 ν/2 x ν/2 exp ( x τ 2 ν/2 (2 Γ(ν/2 for x > and zero ese. Here, ν >, τ 2 = /λ N > and Γ( denote the degrees of freedom, the inverse of the noise variance λ N and the gamma function, respectivey. It is worthwhie noting, that due to the signa processing steps in the STFT computation the degrees of freedom parameter shoud be usuay reduced according to ν L/a for the DC and the Nyquist frequency and otherwise ν 2L/a, wherea is a function of L and of the STFT parameters such as the type of anaysis window and the frame ength [2]. ˆλ N Rather than using the estimate a common estimation technique in noise PSD tracking is a first-order recursive averaging over the past noisy PSDs, assuming speech absence: λ N, = α λ N, +( α Y 2 (3 using a smoothing parameter α. Whie the noise PSD estimators in [2 5] appy a constant smoothing parameter α = α, other sophisticated agorithms as in [, 6 8] use a time-variant smoothing parameter. Simiar to the sampe mean ˆλ N,theestimate λ N, in (3 can be aso approximatey modeed to be a χ 2 s distributed random variabe with appropriate degrees of freedom ν dependend on α. In the approach the smoothed noisy speech according to eq. (3 is the input to the minimum search agorithm Bayesian estimation for stationary noise We new treat the sought-after noise PSD λ N as a random variabe which we seek to estimate by means of Bayesian estimation. For the exponentiay distributed observations, see eq. (, the scaed inverse chi-squared (χ 2 si distribution is a conjugate a priori distribution in speech absence. For x>itisgiven by: p λn (x; ν, τ 2 = (τ ( 2 ν/2 ν/2 x ν/2 exp τ 2 ν/2, (4 Γ(ν/2 x where ν> and τ 2 > denote the degrees of freedom parameter and the scae parameter, respectivey. For this conjugate prior the update equations for the hyper parameters degrees of freedom and scae are given by: ν = ν +2, (5 τ 2 = 2 ν Y 2 + ν ν τ 2, (6 where {ν ; τ 2 } and {ν ; τ 2 } are the parameters of the a priori and a posteriori distributions, respectivey. To get a desired recursive equation we need to condense the posterior PDF to a point estimate. The most popuar are the mode and the mean of the posterior resuting in the maximum a-posteriori (MAP and minimum mean squared error (ME estimates: N, = τ 2 ν ν +2 (7 λme N, = τ 2 ν ν 2. (8 We summarize the two equations in a singe one via λ N, = ν ν +Δν τ 2, (9 which for Δν =2gives the MAP estimate and for Δν = 2 the ME estimate of the noise PSD. Using (5 and (9 in (6 resuts in the recursive equation: λ N, = ν +Δν ν +Δν +2 λ N, + 2 ν +Δν +2 Y 2. ( By comparison of (3 and ( the smoothing parameter α of Bayesian recursion turns out to be a function of ν and Δν: α = ν +Δν ν +Δν +2. ( According to ( α (; is guaranteed for a ν >,if Δν >. ForΔν <, it shoud be ensured that ν Δν. In the Bayesian estimation a sma vaue of α means, that the estimator does not rey on its a-priori knowedge and that the current observation Y 2 is more emphasized. Otherwise, if α is cose to, the estimator is confident in its a-priori knowedge and the observation Y 2 is deepy distrusted. Athough the two equations (3 and ( appy the same recursive smoothing and λ N, is in both cases a χ 2 s distributed random variabes, the modeing of λ N, is quite different. Whie in (3 λ N, is treated as an unknown fixed parameter, is it a χ 2 si distributed random variabe in (. The identity of (3 and ( is a strong anaytica argument for the choice of the χ 2 si distribution as a distribution of λ N, and its practica reevance. It shoud be mentioned that the update equation for the degrees of freedom parameter in (5 is an appropriate choice for stationary noise ony, and it shoud be modified for nonstationary noise tracking. For instance in [9] we used a constant degrees of freedom ν = ν, that is simiar to recursive smoothing with a constant smoothing parameter ike in [2 5]. In this contribution we aso drop the update equation ν = ν 2 +2 and derive an aternative in the next section. 3. Smoothing for nonstationary noise As mentioned above in the approach, λ N, is treated to be a deterministic parameter. An optima smoothing parameter is anayticay derived in [] for observations containing speech pauses by minimizing the mean squared error: α -opt = argmin α E [ ( λn, λ N, 2 λn, ]. (2 This resuts in a smoothing constant that is a function of the smoothed a posteriori SNR ˆγ : α -opt = +(ˆγ, (3 2 ˆγ = λ N,, (4 λ N, where in a practica impementation λ N, = ˆλ N, is used, empoying the noise PSD estimate ˆλ N, obtained from the conventiona approach []. Note, that α -opt may take a very sma vaues, so that λ N, is abe to foow the noisy speech spectrum Y 2 even in speech presence, where ˆγ. 786

3 α (a α -opt α α ME Noise PSD [db] 8-8 (b ˆγ < ˆγ ˆγ ˆλ N, N 2 λ-opt N, λ N, N, λ ME N,.2 α α -opt α α ME. ˆγ 264 Frame number 288 Figure : (a Smoothing parameter α as a function of the smoothed a posteriori SNR ˆγ over ogarithmic scae; (b, top figure sampe noise trajectory for frequency bin 78, 25 Hz (back soid ine and trajectories of smoothed PSD of microphone signa according to eq. (3 for different smoothing parameters (coored ines. The dahsed ine is the bias compensated output of the agorithm, which is used by a smoothing parameter estimators to compute ˆγ ; (b, bottom figure corresponding trajectories of the smoothing parameters. In Fig.(a the green dash-dotted ine depicts the optima α -opt as a function of ˆγ from eq. (3. However, there are two issues with this smoothing parameter []. The first is the so caed dead ock probem for ˆγ : if ˆγ, then α -opt and the noise PSD estimate according to eq. (3 wi hardy change and can get stuck at its current vaue. A second issue is the imited tracking performance in high eves of nonstationary noise: for arge vaues of ˆγ, α -opt wi become very sma, λ N, wi amost instanty foow the observations and thus a too arge estimator variance is induced. A third, equay important issue is a weak tracking behavior of α -opt for < ˆγ < with the theoretica ower bound α -opt =.5for ˆγ =. This resuts in a imited abiity of the recursive averaging to foow the noisy speech spectrum for sma power eve λ N, < ˆλ N,. Bins with such a ow power eve wi most ikey contain noise ony. Tracking of these vaues is crucia, since the recursive estimates λ N, are used for the minimum search in the approach, as mentioned above. Therefore in a practica impementation Martin proposed to use the suboptima smoothing parameter α according to: ( α =max α max α -opt, α min(snr (5 with an upper bound α max =.96 and a SNR dependent ower bound α min(snr cacuated for positive vaues of the overa signa-to-noise ratios SNR > measured in db as per: α min(snr =min (.3, SNR R.64 f S, (6 where R and f S are the frame shift (in number of sampes and the samping frequency, respectivey []. (6 aows α min to be smaer than.3 for arge SNR vaues. In the Fig.(a α is depicted by the soid bue curve for α min =.3. Additionay α is adjusted according to an error monitoring, that does not need any parameter and therefore is not taken into account in the Fig.(a. It is a heuristic adjustment of the theoretica resut to sove ony the first two issues discussed above. To overcome a mentioned issues with α -opt we suggest to contro the degrees of freedom parameter ν of ( and thus the smoothing parameter α by a measure of the instantaneous degree of nonstationarity (DN d according to { /d for Δν, ν = (7 Δν +/d for Δν <. where DN is defined as foows: d = n(ˆγ, (8 which is simiar to the measure of degree of nonstationary defined in []. In comparision to [] our definition of DN is more intuitive and needs no additiona parameter. Inserting (8 and (7 in ( eads to the proposed smoothing parameter: = ( + 2 n ˆγ for Δν, ( + 2 n ˆγ for Δν <. (9 As a function of ˆγ, has a singe parameter Δν, which determines, which point estimate is used in (9 either the MAP or the ME estimate. The curves: = +2 n ˆγ +4 n ˆγ, (2 αme = +2 n ˆγ (2 are depicted in the Fig.(a as dashed ines. It is obvious that = f(ˆγ avoids the dead ock probem, since its first derivative / ˆγ for ˆγ =. Its minimum vaue is cacuated to min =Δν/(Δν +2 for Δν and min =ese. Moreover the proposed is symmetric with respect to the axis ˆγ =and is abe to foow the noisy spectrogram rapidy not ony for arge ˆγ > but aso for sma vaues ˆγ <. To justify the choice of the proposed function we first consider stationary noise in speech absence. In this case the smoothed PSD of the microphone signa is approximatey equa to the noise PSD estimate deivered by the approach, λ N, ˆλ N,, resuting in ˆγ according to eq. (4. Consequenty d, therefore ν and the smoothing parameter assumes a vaue cose to, which is desired in 787

4 SDM LEV -db -3dB 4dB db 8dB 25dB (a 2 white destr.-engine factory- babbe 36.3% 3.9% 35.% 36.8% ME ME ME ME (b white destr.-engine factory- babbe 3 2.% % % % ME ME ME ME Figure 2: Noise tracking performance of the approach with α -based and with α ME -based smoothing in terms of (a spectra distance measure (SDM and (b ogarithmic error variance (LEV for 4 noise types of the NOISEX-92 database. stationary noise. However this is vaid for quite a sma range of ˆγ ony. For ˆγ, quicky decreases, and the smoothed PSD is abe to foow changes in the noise spectrum, both for speech presence and for decaying noise power eves. Further decreases for ˆγ > not as rapidy as α -opt.sowedo not expect a too arge estimator variance and consequenty do not need any ower bound even for = α ME. In Fig.(b trajectories of the noise PSD and the smoothed PSD of the microphone signa according to eq. (3 are shown for the different options for the smoothing parameter α,aswe as the trajectories of the smoothing paramters themseves. For a smoothed PSDs, the same ˆλ N, is used for cacuating ˆγ. The depicted time span can be divided in 3 distinctive parts: ˆγ <, ˆγ and ˆγ. The main advantage of the proposed smoothed noisy speech PSD trajectories λ N, and λ N, MAP ME is their abiity to foow the changes of the noise power eve for ˆγ <. It is due to the sma vaues of and α ME,which can be seen in the bottom picture. This property can be used to better estimate the noise power foor in the approach. For ˆγ (power push in noise signa a α drop to their minimum vaues and try to foow the rapid changes in noise power. Bacause of the reative high minimum vaue of min =.5, N, performs not we enough in such time segments. In the ast time interva with ˆγ the dead ock probem of α -opt can be observed: hardy foows the noise trajectory. λ -opt N, 4. Experimenta resuts The performance of the proposed smoothing parameter α ME versus conventiona α is experimentay evauated empoying speech degraded by various noise types and at goba SNR vaues varied from - db to 25 db in steps of 7 db. The cean speech signas are generated by concatenating utterances of mae and femae speakers from the TIMIT database [], having a tota ength of 3 minutes. To them the noise signas of ΔSTOI [%] white -db -3dB 4dB db 8dB 25dB destr.-engine factory- babbe eopard m-9 factory-2 destr-oper. f6 hf-channe buccaneer- buccaneer-2 pink Figure 3: Improvement of the short-time objectve inteigibiity (STOI measure ΔSTOI=STOI ME STOI of the enhanced signas obtained by using the approach with α ME -based versus with α -based smoothing for NOISEX-92 database. 3 different noise types, which were taken from the NOISEX- 92 database [2], were artificiay added. A signas are samped at 6 khz. The STFT spectra anaysis used a Hann window of 24 sampes ength with a frame overap of 5%. Since the impementation of the approach of [] was avaiabe, the proposed function α ME from (2 is simpy integrated in the impementation. The ength of the window for minimum search is set to D = U V =96frames devided into U =8subwindows of the ength of V =2frames. The performance of the -based noise tracking with either α - based or α ME -based smoothing of the noisy speech PSD is given in Fig. 2 in terms of the spectra distance measure (SDM [3] and ogarithmic error variance (LEV [4], where the true noise periodogram N,k 2 is used as a reference noise PSD. We observed that both the estimator errors measured by SDM and the estimator variances measured by LEV reduced for α ME - based smoothing versus the α -based smoothing for a SNR vaues and for a considered noise types of the NOISEX-92 database, of which ony 4 representatives are depicted in Fig. 2. The absoute vaues and the reductions given in % of SDM and LEV are the averages over a SNR vaues. Further both noise trackers are combined with the optimay-modified og-spectra ampitude (OM-LSA estimator to obtain the enhanced speech signas [5]. Fig. 3 shows the improvement of the short-time objectve inteigibiity (STOI measure ΔSTOI = STOI ME STOI of enhanced signas [6] obtained by using the approach with α ME -based versus with α -based smoothing. The improvements for the NOISEX-92 database are sma, however consistent over a noise types and SNR vaues. As expected the best over a SNR vaues averaged improvements of 2.5% and 2% are achieved for white and pink noise types, respectivey. 5. Concusions In this contribution we proposed an aternative computation of the smoothing parameter, which is used to smooth the PSD of the microphone signa at the input of the Minimum Statistics based noise tracking. The smoothing parameter is controed by a degree of nonstationary measure. Its singe parameter, Δν, is chosen to reaize an ME estimate of the noisy speech PSD. The experimenta evauation showed that the performance of the -based noise tracking was improved for a noise types of the NOISEX-92 database and a tested SNR vaues, compared to the use of the origina smoothing parameter as proposed in []. 788

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