Subband Particle Filtering for Speech Enhancement

14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyright by EURASIP

SUBBAND PARTICLE FILTERING FOR SPEECH ENHANCEMENT

Ying Deng and V. John Mathews

Dept. of Electrical and Computer Eng., University of Utah 50 S. Central Campus Dr., Rm. 3280 MEB, Salt Lake City, UT 84112, USA phone: + (1)(801) 581-6941, fax: + (1)(801) 581-5281, email: [email protected], [email protected]

ABSTRACT as those discussed in [16, 17] and also in this paper, the integrations Particle filters have recently been applied to speech enhancement used to compute the filtering distribution and the integrations em- when the input speech signal is modeled as a time-varying autore- ployed to estimate the clean speech signal and model parameters do gressive process with stochastically evolving parameters. This type not have closed-form analytical solutions. Approximation methods of modeling results in a nonlinear and conditionally Gaussian state- have to be employed for these computations. The approximation space system that is not amenable to analytical solutions. Prior work methods developed so far can be grouped into three classes: (1) an- in this area involved signal processing in the fullband domain and alytic approximations such as the Gaussian sum filter [19] and the assumed white Gaussian noise with known variance. This paper extended Kalman filter [20], (2) numerical approximations which extends such ideas to subband domain particle filters and colored make the continuous integration variable discrete and then replace noise. Experimental results indicate that the subband particle filter each integral by a summation [21], and (3) sampling approaches achieves higher segmental SNR than the fullband algorithm and is such as the unscented Kalman filter [22] which uses a small num- effective in dealing with colored noise without increasing the com- ber of deterministically chosen samples and the particle filter [23] putational complexity. which uses a larger number of random (Monte Carlo simulation) samples for the computations. The analytic approximations are computationally simple but usually fail in complicated situations. 1. INTRODUCTION The numerical approximations are only suited for low-dimensional Speech enhancement has been an active area of research during the state-spaces. In [16, 17], particle filters have been successfully em- past forty years. Speech enhancement algorithms available in the ployed for speech enhancement. The methods developed in [16, 17] literature can be broadly divided into two categories - non-model were in the fullband domain and only white Gaussian noise with based and model based algorithms. Representative approaches known variance was considered. in the non-model based algorithms include spectral subtractive- In this paper, we present a speech enhancement algorithm that type algorithms [1, 2] and signal subspace-based algorithms [3, 4]. employs particle filters in the subband domain. Typically, subband Model based algorithms employ models of speech in the enhance- speech signals have flatter power spectrum as compared to the cor- ment process. Autoregressive (AR) models are widely used to rep- responding fullband signals. We can therefore use lower order TV- resent the vocal tract transfer function. Example of model based PARCOR models for the subband signals. This usually reduces the speech enhancement algorithms include the iterative Wiener filter- computational complexity of the algorithm. We show in this paper ing approaches [6, 7]. Kalman filtering based algorithms [8, 9, 10] that while maintaining similar computational complexity, the sub- form another class of model-based speech enhancement algorithms. band modeling can model the speech power spectrum more accu- Almost all such methods are based on autoregressive modeling of rately and result in better enhancement results. The enhanced full- speech signals and linear Gaussian state-space representation of the band speech signals are obtained by synthesizing the enhanced sub- system. Speech enhancement algorithms that assume specific prob- band speech signals. The particle filter based speech enhancement ability distributions of speech signals and then derive minimum algorithms in [16, 17] assume white Gaussian noise with known mean-square error estimates of the clean speech signals [11, 12] variance, which is unrealistic in practical applications. This work and those that assume composite source models (a composite source extends the algorithm to solve the enhancement problem in col- model is composed of a finite set of statistically independent sub- ored noise environments. In order to accomplish this, we model the sources with each subsource representing a particular class of sta- colored noise by an AR model and augment the state-space model tistically similar speech sounds) such as Hidden Markov Models for the white noise case. Experimental results show that the sub- (HMMs) and use different estimators for different classes of speech band particle filter achieves higher segmental SNR improvement signals [13, 14] also belong to the class of model based methods. than the fullband scheme in white Gaussian noise without increas- AR models of speech used in the iterative Wiener filtering and ing the computational complexity. The subband particle filter is also Kalman filtering approaches [6, 7, 8, 9, 10] assume that the artic- effective in dealing with colored noise. ulatory shape of the vocal tract remains fixed throughout the anal- The rest of this paper is organized as follows. Section 2 de- ysis interval. However, in reality the vocal tract is changing con- scribes the subband system model and the estimation objectives. In tinuously. To better model the non-stationarity of speech signals, Section 3, we present the subband particle filter and the noise esti- time-varying autoregressive (TVAR) models of speech have been mation algorithm. Section 4 provides experimental results. Finally, proposed [15]. In [16], a TVAR model with stochastically evolv- we make our concluding remarks in Section 5. ing parameters was adopted and shown to outperform standard AR models. By transforming between the AR coefficients and the re- 2. THE SUBBAND SYSTEM MODEL flection coefficients using standard Levinson recursion, Fong [17] In the fullband domain, the noisy speech y(t) can be expressed as used a time-varying partial correlation (TV-PARCOR) model and showed that the TV-PARCOR is a better physical representation of y(t) = s(t)+ n(t), (1) audio signals than the TVAR model. We adopt the TV-PARCOR where s(t) and n(t) are the clean speech signal and the additive model in our method. With the TVAR or TV-PARCOR modeling of background noise, respectively. The fullband signal is decomposed speech signals, the system can be represented in a nonlinear condi- into a set of subband signals using an analysis filter bank and the tionally Gaussian state-space form. Analytic solutions for recursive subband signal can be written as Bayesian state estimation exist only for a small number of specific cases [18]. For nonlinear conditionally Gaussian state-space models yi(t) = si(t)+ ni(t), (2) 14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyright by EURASIP

where i is the subband index. recursions The component of the clean speech signal si(t) in the ith subband is modeled as a p-th order TVAR process, i.e., p(xi(t + 1)|yi(1 : t)) =

p(xi(t)|yi(1 : t))p(xi(t + 1)|xi(t))dxi(t) (12) p Z si(t) = ai,t (k)si(t − k)+ σs es (t). (3) ∑ i,t i p(xi(t + 1)|yi(1 : t + 1)) = k=1 p(yi(t + 1)|xi(t + 1))p(xi(t + 1)|yi(1 : t)) T (13) Here, ai,t =[ai,t (1),···,ai,t (p)] is time-varying AR coefﬁcients p(yi(t + 1)|yi(1 : t)) vector associated with the ith subband and esi (t) is a white Gaussian excitation with unit variance. The variance of the excitation is σ 2 . Once the distribution function is known, the MMSE estimate of the si,t state vector is given by We assume that the colored noise statistics change sufﬁciently slowly so that they can be approximated as not changing during ˆxi(t)= E{xi(t)|yi(1 : t)} = xi(t)p(xi(t)|yi(1 : t))dxi(t). (14) short time intervals. In such short time intervals, we model the ith Z subband component ni(t) of the colored noise as a q-th order AR process, i.e., From the state-space representation (10) and (11), we can see

that if the model parameters ai,t , σsi,t , bi and σni are known, the q estimation problem can be solved using a Kalman filter. However, ni(t) = ∑ bi(k)si(t − k)+ σni eni (t). (4) the parameters are unknown and have to be jointly estimated with k=1 the state vector xi(t). This results in a conditionally Gaussian state- space system and has no closed form solution for the computation T Here, bi =[bi(1),···,bi(q)] is the i-th subband AR coefficients of the filtering distribution and the state estimation. Particle filter vector and eni (t) is a white Gaussian excitation with unit variance. as an approximation method is then adopted to solve the estimation 2 problem in this paper. The variance of the excitation is σni and is not known a priori. Given the clean speech and noise models in the subbands, we Let us define a speech parameter vector θi(t) = , , , 2 T can develop a state-space system model in the following manner. [ai,t (1) ··· ai,t (p) logσsi,t ] that is to be estimated with the Let us define xi(t)=[si(t),···,si(t − p + 1),ni(t),···,ni(t − q + state vector xi(t). The noise parameters will be estimated sep- T T T 1)] , ei(t)=[esi (t)eni (t)] and yi(t)=[yi(t)] and the system ma- arately during intervals where speech is absent from the signal. trices To facilitate a particle filter solution, we further assume a TV- s Ai,t 0p×q PARCOR model [17] for the time-varying AR coefficients of the Ai,t = n (5) speech signal. That is, the time-varying AR coefficients are first · 0q×p Ai,t ¸ (p+q)×(p+q) transformed to a set of time-varying reflection coefficients using where Levinson recursion. The corresponding reflection coefficients is applied a constrained Gaussian random walk model. The ai,t (1) ai,t (2) ··· ai,t (p − 1) ai,t (p) constraints imposed are such that stability of the model is ensured. The constrained random walk model for the reflection coefficients  1 0 ··· 0 0  s 0 1 ··· 0 0 is Ai,t = (6)  . . . . .  2  ......  N(ρi,t−1,δρ I); if maxk |ρi,t (k)| < 1   p(ρi,t |ρi,t−1)= (15)  0 0 ··· 1 0  ½ 0; otherwise  p×p Here, ρ =[ρ (1),···,ρ (p)]T is the set of reflection coefficients and i,t i,t i,t associated with the speech signal at time t. The logarithm of speech excitation variance also follows a Gaussian random walk model, b (1) b (2) ··· b (q − 1) b (q) i i i i i.e., we assume that  1 0 ··· 0 0  0 1 0 0 2 2 2 2 An = ··· . (7) p(logσ |logσ )= N(logσ ,δ ). (16) i,t  . . . . .  si,t si,t−1 si,t−1 es  ......    The estimation objectives then become the computation of the  0 0 ··· 1 0   q×q joint distribution p(xi(t),θi(t)|yi(1 : t)) and the MMSE estimates E{xi(t),θi(t)|yi(1 : t))}. Let σsi,t 0 3. SUBBAND PARTICLE FILTERING AND NOISE 0(p−1)×1 0(p−1)×1 ESTIMATION Bi,t =   (8) 0 σni  0 0  The subband particle filter based speech enhancement algorithm is  (q−1)×1 (q−1)×1 (p+q)×2 illustrated in Figure 1. The algorithm first decomposes the input and signal into subband components, performs enhancement in the sub- p−1 q−1 band domain, and then reconstructs the enhanced fullband signal using a synthesis filter bank. In subsection 3.1, a sequential Monte Ci,t = [10···010···0]1×(p+q). (9) Carlo method for estimating the state and speech parameter vector z }| { z }| { from the observed noisy signal is presented. In subsection 3.2, we Then, we can rewrite (2), (3) and (4) in state-space form as will discuss the method used to estimate the noise parameters.

xi(t) = Ai,t xi(t − 1)+ Bi,t ei(t) (10) 3.1 Subband particle filter yi(t) = Ci,t xi(t). (11) The subband particle filter adopted in this paper is the Rao- Blackwellized particle filter similar to those developed in [16, 17]. In order for a sequential minimum mean square error (MMSE) For a tutorial discussion of particle filtering, refer to [23]. We estimation of the state vector, we have to know the distribution func- present the algorithm according for our state-space model in what tion p(xi(t)|yi(1 : t)). This distribution can be obtained using the follows. 14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyright by EURASIP

The importance weights can also be recursively evaluated as

m m m p(xi (1 : t − 1),θi (1 : t − 1)|yi(1 : t − 1)) ω1:t ∝ m m × π(xi (1 : t − 1),θi (1 : t − 1)|yi(1 : t − 1)) m m m m p(xi (t),θi (t)|xi (1 : t − 1),θi (1 : t − 1),yi(1 : t)) m m m m π(xi (t),θi (t)|xi (1 : t − 1),θi (1 : t − 1),yi(1 : t)) m m m m m p(xi (t),θi (t)|xi (1 : t − 1),θi (1 : t − 1),yi(1 : t)) = ω1:t−1 m m m m π(xi (t),θi (t)|xi (1 : t − 1),θi (1 : t − 1),yi(1 : t)) (22)

m m ω1:t and the normalized importance weights are ω¯1:t = N m . ∑m=1 ω1:t Figure 1: Subband speech enhancement system. Thus, given an estimate of p(xi(1 : t − 1),θi(1 : t − 1)|yi(1 : t − 1)), the estimate of p(xi(1 : t),θi(1 : t)|yi(1 : t)) is obtained by m m m m augmenting (xi (1 : t − 1),θi (1 : t − 1)) with (xi (t),θi (t)),m = 1,...,N and recursively updating the importance weights ac- m m cording to (22). (xi (t),θi (t)),m = 1,...,N are sampled from 3.1.1 Sequential Bayesian importance sampling m m π(xi(t),θi(t)|x (1 : t − 1),θ (1 : t − 1),yi(1 : t)). The marginal N xm t , m i i Suppose that it is possible to sample particles { i (1 : ) θi (1 : distribution p(xi(t),θi(t)|yi(1 : t)) is estimated as t);m = 1,...,N} according to p(xi(1 : t),θi(1 : t)|yi(1 : t)). An empirical estimate of this distribution is N m m m pˆN (xi(t),θi(t)|yi(1 : t)) = ω¯ δ x (t),θ (t)). (23) N ∑ 1:t ( i i 1 m=1 pN (xi(1 : t),θi(1 : t)|yi(1 : t)) = δ xm t , m t , (17) N ∑ ( i (1: ) θi (1: )) m=1 3.1.2 Rao-Blackwellization where δ(·) is the Dirac delta function. Using this empirical distri- Recall that p(xi(t),θi(1 : t)|yi(1 : t)) = p(xi(t)|θi(1 : t),yi(1 : bution, the MMSE state and speech parameters estimates can be t))p(θi(1 : t)|yi(1 : t)) and that p(xi(t)|θi(1 : t),yi(1 : t)) is a Gaus- obtained as sian distribution that can be analytically evaluated using a Kalman filter. We assume that the noise parameters are already estimated (ˆxi(1 : t),θî(1 : t)) and known. Then from the Rao-Blackwell theorem [25], we can re- duce the estimation variance by only sampling p(θi(1 : t)|yi(1 : t)) m = (xi(1 : t),θi(1 : t))pN (dxi(1 : t),dθi(1 : t)|yi(1 : t)) and analytically evaluating p x t θ 1 : t ,y 1 : t to obtain an Z ( i( )| i ( ) i( )) estimate of p(xi(t),θi(1 : t)|yi(1 : t)). We can summarize the Rao- 1 N Blackwellized particle filter as follows. = (xm(1 : t),θ m(1 : t)) (18) ∑ i i • m N m 1 Sample π(θi(t)|θi(1 : t − 1),yi(1 : t)) for θi (t),m = 1,...,N = m m m and θi (1 : t)=(θi (1 : t − 1),θi (t)). According to the strong law of large numbers, this estimate con- • For m = 1,...,N, evaluate the importance weights up to a nor- m m verges to the true estimate as N goes to infinity [23]. m m p(θi (t)|θi (1:t−1),yi(1:t)) malizing constant ω1:t ∝ ω1:t 1 m m Unfortunately, p(xi(1 : t),θi(1 : t)|yi(1 : t)) is usually too com- − π(θi (t)|θi (1:t−1),yi(1:t)) plicated to sample directly. Instead, a simpler distribution π(xi(1 : • p(θi(1 : t)|yi(1 : t)) can then be approximated byp ˆN (θi(1 : N m t),θi(1 : t)|yi(1 : t)) which can be easily sampled from and whose t)|y (1 : t)) = ∑ ω¯ δ m . i m=1 1:t θi (1:t) support includes that of p(xi(1 : t),θi(1 : t)|yi(1 : t)) is employed. • The estimates of the speech parameters and the state vector can This method is called Bayesian importance sampling (BIS) [24]. be expressed as An empirical estimate of p(xi(1 : t),θi(1 : t)|yi(1 : t)) using BIS is given by N θˆ(t) = ω¯ m θ m(t) (24) N i ∑ 1:t i m m=1 pˆN (xi(1 : t),θi(1 : t)|yi(1 : t)) = ω¯ δ xm 1:t ,θ m 1:t , (19) ∑ 1:t ( i ( ) i ( )) N m=1 m m ˆxi(t) = ω¯1:t E{xi(t)|θi (1 : t),yi(1 : t)}, (25) m ∑ m ω1:t m=1 where, the normalized importance weights ω¯1:t = N m and the ∑m=1 ω1:t p(xm(1:t),θ m(1:t)|y (1:t)) m m i i i where, E{xi(t)|θi (1 : t),yi(1 : t)} can be computed using a importance weights ω1:t ∝ m m . With the BIS, the π(xi (1:t),θi (1:t)|yi(1:t)) Kalman filter. For details of Kalman filtering, please refer to MMSE state and speech parameters estimates can be obtained as [26]. N ˆ m m m 3.1.3 Resampling (ˆxi(1 : t),θi(1 : t)) = ∑ ω¯1:t (xi (1 : t),θi (1 : t)). (20) m=1 One problem with the sequential BIS is that after several time steps, many importance weights will have insignificant values. This will In order to estimate p(xi(1 : t),θi(1 : t)|yi(1 : t)) at any time m cause large estimation variances. In order to alleviate this problem, t without changing the past simulated trajectories (xi (1 : t − m many resampling schemes have been proposed such as sampling 1),θi (1 : t −1)),m = 1,...,N, we employ a sequential BIS scheme. importance resampling [27], residual resampling [28] and strati- The basic idea of sequential BIS is that π(xi(1 : t − 1),θi(1 : t − fied resampling [29]. The generic stratified resampling scheme is 1)|yi(1 : t − 1)) is a factor of π(xi(1 : t),θi(1 : t)|yi(1 : t)), i.e., adopted in this paper. For details of the algorithm, please refer to [29]. π(xi(1 : t),θi(1 : t)|yi(1 : t)) = π(xi(1 : t − 1),θi(1 : t − 1)|yi(1 : t − 1)) × 3.2 Noise parameter estimation π(xi(t),θi(t)|xi(1 : t − 1),θi(1 : t − 1),yi(1 : t)). For noise parameter estimation, we first design a voice activity de- (21) tector in each subband. Then we collect all the noise only segments 14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyright by EURASIP

and construct a sequence of noise samples. We can then estimate Input SegSNR improvement(dB) the noise parameters using the Yule-Walker method from the noise SegSNR(dB) Fullband Subband only sequence. -5 6.90 12.37 The voice activity detector we adopt here is based on the min- 0 5.15 10.25 imum controlled recursive averaging noise spectrum estimation 5 3.42 9.61 method [12]. We summarize the algorithm as follows. For each 10 2.44 7.52 subband noisy signal yi(t), we first estimate the energy recursively as, Table 1: Comparison of SegSNR improvement in white Gaussian 2 Si(t)= αsSi(t − 1)+(1 − αs)yi (t), (26) Noise. where 0 < αs < 1 is a forgetting factor. Then we track the minimum value of Si(t) denoted by Si,min(t). A samplewise comparison of Input SegSNR improvement(dB) the smoothed energy and the corresponding variable in the previous SegSNR(dB) Cohen [33] Subband PF frame allows the following update for the minimum value Traffic F-16 Traffic F-16 S (t)= min{S (t − 1),S (t)}. (27) -5 7.64 5.98 7.82 8.12 i,min i,min i 0 5.67 4.07 5.30 6.77 Finally, we compute the ratio of the smoothed energy to its mini- 5 3.30 2.59 3.14 4.75 mum value and compare the ratio with a threshold T. If the ratio 10 1.23 0.83 1.11 2.89 is larger than the threshold, we consider it speech active. Other- wise, we consider it noise only. Once the noise only sequence is Table 2: Comparison of SegSNR improvement in colored noise. obtained, a standard Yule-Walker autoregressive parameter estimation algorithm [30] is applied to get the noise parameters.

4. EXPERIMENTAL RESULTS dB. From Table 1 and Figure 2, we can see that the subband domain speech enhancement algorithm exhibits much smaller estima- The filter bank we used to obtain our experimental results was tion variance and thus much higher segSNR improvement. This a nonuniform pseudo-QMF bank [31, 32] which achieves critical is because while maintaining similar computation complexity, the band division. The length of the prototype filter is 896 samples. A subband modeling can model the speech power spectrum more ac- 1s long clean speech signal with sampling rate 16kHz was used. For curately and result in better enhancement results. noise parameter estimation, the forgetting factor αs was chosen to be 0.8 and the threshold T was set to 5. For colored noise, we chose a first order AR model with q = 1 to represent the noise signal in Input Noisy Speech Signal each subband. We also employed a first order time-varying autore- 0.4 gressive model p = 1 for the speech signal in each subband. The 0.2 importance distribution was chosen to be the prior distribution, i.e., 0 π(θi(t)|θi(1 : t −1),yi(1 : t)) = p(θi(t)|θi(t −1). With the Gaussian random walk models defined in (15) and (16), it is easy to sample −0.2 −0.4 this prior distribution. The number of particles N was selected as 0 100 200 300 400 500 600 700 800 900 1000 100 in all the experiments. The variances of the Gaussian random Fullband RBPF Estimates 0.4 walk models in (15) and (16) were set to δ 2 = 0.001 and δ 2 = 0.01. clean speech ρ es 0.2 estimated For the first set of experiments, we compare the algorithm of this paper with the fullband Rao-Blackwellized particle filter [17] in 0 white Gaussian noise to show that our subband particle filter algo- −0.2 rithm achieves higer segmental SNR improvement and at the same −0.4 0 100 200 300 400 500 600 700 800 900 1000 time takes similar CPU time per iteration as compared to the full- Subband RBPF Estimates 0.4 band particle filter algorithm. The segmental SNR is a widely used clean speech objective measure for speech enhancement systems and is defined 0.2 estimated as 0 SegSNR −0.2 −0.4 L−1 0 100 200 300 400 500 600 700 800 900 1000 ∑ |s(n + mL)|2 t 1 M−1   = 10log n=0 , M ∑ 10  L−1  Figure 2: Comparison of the estimated clean speech at input m=0  2   ∑ |s(n + mL) − sˆ(n + mL)|  SegSNR of 5 dB.    n=0  (28) For the second set of experiments, we compare the SegSNR where s(n) ands ˆ(n) denote the clean speech and the enhanced improvement of our algorithm to the two-state modeling algorithm speech, respectively. Here, M is the number of frames in the speech [33] when the clean speech signal is corrupted by colored noise. segment and L is the number of samples per frame. The segmen- Two colored noise signals - traffic and F-16 noise - were added to tal SNR improvement was estimated by subtracting the SegSNR of a clean speech signal at different Segmental SNRs. Table 2 shows the enhanced speech from the SegSNR associated with the noisy the experimental results. From Table 2, we can see that the subband speech. White Gaussian noise was added to a clean speech signal particle filter performs 2 − 3dB better for the F-16 noise case than at different segmental SNRs. Without optimization, the CPU times the two-state modeling algorithm. The two-state modeling algo- were 0.9080s per iteration for the subband algorithm and 0.8982s rithm performs slightly better or similar to the particle filter for the per iteration for the fullband algorithm on a standard 1.4 GHz PC. traffic noise case. However, the performance difference is less than Table 1 shows the comparison of the SegSNR improvements in this 0.4dB in all cases tabulated in Table 2. Informal listening tests have example. Figure 2 shows the comparison of the estimated clean also shown that the subband particle filter method exhibits lower speech signal for these two approaches at an input SegSNR of 5 residual noise than the two-state modeling approach [33]. 14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyright by EURASIP

5. CONCLUSIONS [16] J. Vermaak, C. Andrieu, A. Doucet and S.J. Godsill, “Parti- cle methods for Bayesian modeling and enhancement of speech This paper presented a subband domain particle filter based speech singlas,” IEEE Trans. Speech and Audio Processing, vol. 10, enhancement system. We have shown through experiments that the pp. 173-185, Mar. 2002. subband domain particle filter performs better in terms of segmental SNR as compared to the corresponding fullband domain algo- [17] W. Fong, S.J. Godsill, A. Doucet and M. West, “Monte rithm. The algorithm is able to deal with colored noise, whereas Carlo smoothing with application to audio signal enhance- only white Gaussian noise with known variance was considered ment,” IEEE Trans. Signal Processing, vol. 50, pp. 438-449, in previous application of particle filters to speech enhancement Feb. 2002. [16, 17]. We compared our speech enhancement results in colored [18] B. Ristic, S. Arulampalam and N. 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