Non-Stationary Bayesian Modelling and Enhan ement ofSpee h
This paper applies time-varying AR (TVAR) models with sto hasti ally evolving parameters to the problem of spee h modelling and enhan ement. The sto hasti evolution models for the TVAR parameters are Markovian di usion pro esses. The problem and estimation obje tives are formulated within a Bayesian framework. Two eÆ ient iterative algorithms are developed to a hieve these obje tives. The rst is a Markov hain Monte Carlo (MCMC) algorithm whi h generates samples from the posterior distribution based on whi h the minimum mean square error (MMSE) estimates of the TVAR parameters and lean spee h an be omputed. The se ond is a sto hasti optimisation algorithm whi h omputes the marginal maximum a posteriori (MMAP) estimate of the TVAR parameters. The lean spee h an then be obtained by running a xed-interval Kalman smoother with this estimate of the TVAR parameters. Contrary to the EM-type algorithms, the estimation s hemes work without introdu ing a set of \missing data" (the lean spee h in this ase). Nevertheless, at ea h iteration the omputational omplexity of the algorithms is still linear in the number of samples in the analysis window. Performan e measures based on predi tive distributions are used in simulation studies to ompare the modelling and signal reonstru tion performan e of the TVAR model to that of the standard xed-parameter AR model on both syntheti and real spee h data sets. SP Edi s: 1-ENHA 1 I Introdu tion A widely used and popular model for spee h is the autoregressive (AR) pro ess [16℄. In pra ti al appli ations, however, lean spee h signals are rarely available, the spee h being ontaminated by some ba kground or appli ation-spe i noise pro ess. For the subsequent pro essing to be su essful, these noise pro esses need to be expli itly a ounted for within the hosen modelling framework. Fortunately, the bulk of these an be adequately modelled as slowly time-varying additive white Gaussian or Gaussian mixture pro esses. This basi assumption led to the development of many eÆ ient spee h enhan ement and signal re onstru tion algorithms (see [8, 13℄ for some examples), and is hen e also adopted here. Asso iated with the AR spee h produ tion model is a stylised arti ulatory on guration whi h remains xed throughout the analysis interval. In reality, however, the vo al tra t is ontinually hanging, sometimes slowly, sometimes rapidly. To partially re on ile the time-varying nature of the vo al tra t with the time invarian e of the model, the spee h is usually pro essed in short segments, during ea h of whi h the signal is assumed to be stationary. However, sin e the framing is de ned a priori with no relation to the phoneti information, non-stationary frames are still likely to o ur, even for very short analysis intervals. In these ir umstan es non-stationary models may provide more true-to-life approximations of the behaviour of the vo al tra t. One su h model is the time-varying AR (TVAR) pro ess. It is a simple extension of the standard AR pro ess where the AR oeÆ ients are allowed to vary with time. TVAR models were proposed in the ontext of spee h modelling in e.g. [5, 10, 14℄. Sin e it allows for the ontinually hanging behaviour of the spee h signal, the TVAR model leads to more a urate signal representations. In addition, analysis over longer data frames is possible. This paper onsiders a generalisation of the TVAR model with sto hasti ally evolving parameters for non-stationary spee h modelling and enhan ement. Random walk sto hasti evolution models are adopted for both the TVAR oeÆ ients and the ex itation and observation noise varian es, the former enfor ing stability by restri ting the system poles to lie stri tly within the unit ir le. The problem and estimation strategy are formulated within a Bayesian framework. The resulting inferen e pro edures are analyti ally intra table, requiring numeri al methods for their solution. 2 Two lasses of iterative algorithms that have been su essfully applied before in the ontext of numeri al Bayesian inferen e are the sto hasti Markov hain Monte Carlo (MCMC) methods [17, 18℄ and simulated annealing optimisation algorithms [19℄. Here one MCMC and one simulated annealing optimisation algorithm are developed to perform minimum mean square error (MMSE) estimation of the TVAR model parameters and lean spee h, and marginal maximum a posteriori (MMAP) estimation of the TVAR model parameters, respe tively. In ontrast to the EM-type methods, these algorithms avoid the introdu tion of any \missing data" sets, while keeping a omputational ost that is linear in the number of samples in the analysis window. The Bayesian formulation also fa ilitates dire t model omparison between di erent models by de ning performan e measures based on the orresponding predi tive distributions. The analyti evaluation of these measures is intra table in general, but a urate Monte Carlo approximations an be obtained by using the samples from the MCMC algorithm. In the simulation studies presented later su h a measure is used to ompare the modelling and re onstru tion performan e of the TVAR model to that of the standard xed-parameter AR model. The remainder of the paper is organised as follows. The model spe i ation and estimation obje tives are stated in Se tion II. Se tion III develops an eÆ ient MCMC algorithm to ompute the MMSE estimates of the model parameters and the lean spee h. In Se tion IV a simulated annealing optimisation algorithm is developed to ompute the MMAP estimate of the model parameters with the lean spee h integrated out. Se tion V presents and dis usses simulation results on syntheti and real data sets. In Se tion VI, some on lusions are drawn and a number of suggestions for future investigation are made. Appendix A re alls the Kalman ltering and xedinterval smoothing equations, whi h form an integral part of the estimation algorithms developed here. In Appendix B a method is presented to simulate from a Gaussian mixture distribution to a power greater than one, and nally, the proofs of some of the more important lemmas and propositions used in this paper are grouped together in Appendix C. 3 II Model Spe ifi ation and Estimation Obje tives A Signal Model and State-Spa e Representation The spee h signal at dis rete time t 2 T , f1; : : : ; Tg, T being the T -sample analysis window, is modelled as the output of a k-th order TVAR pro ess, parameterised by a ve tor t 2 Rn , i.e. xt = k Xi=1 ai;t ( t)xt i + et ( t) et; et iid N (0; 1) ; (1) where at ( t) , (a1;t ( t) ; : : : ; ak;t ( t)) are the TVAR oeÆ ients, and 2et ( t) is the varian e of the TVAR innovation sequen e. The signal is assumed to be submerged in additive white Gaussian noise (AWGN), so that the observed value at dis rete time t 2 T be omes yt = xt + nt ( t)nt; nt iid N (0; 1) ; (2) where fntg is a white noise pro ess independent of fetg, and 2nt ( t) is the varian e of the observation noise. Conditionally on f tg the signal model is linear, fa ilitating a onditionally Gaussian statespa e (CGSS) representation. More pre isely, de ning the ve tors t , (xt; : : : ; xt k+1), yt , (yt), vt , (et) and wt , (nt), and the system matri es At ( t) , 2664 aTt ( t) Ik 1 0k 1 13775 Bt ( t) , 2664 et ( t) 0k 1 13775 (3) Ct ( t) = C , 1 01 k 1 Dt ( t) , nt ( t) ; (4) the signal model of (1) and (2) is readily expressed in the CGSS form given by t = At ( t) t 1 +Bt ( t)vt; vt iid N (0nv 1; Inv) (5) yt = Ct ( t) t +Dt ( t)wt; wt iid N (0nw 1; Inw) ; (6) where t 2 Rn is the system state, yt 2 Rny is the observation, and vt 2 Rnv and wt 2 Rnw are the system disturban es at dis rete time t 2 T , respe tively. It is further assumed that Dt ( t)DTt ( t) > 0, for all t 2 T , 0 N (m0 ( 0) ;P0 ( 0)), with P0 ( 0) > 0, and that 0, vt and wt are mutually independent for all t 2 T . 4 B Parameterisation and Evolution Model For the parameterisation adopted here the oeÆ ients of the TVAR model are used without modi ation, and the ex itation and observation noise varian es are parameterised by their orresponding logarithms, i.e. et , log 2et and nt , log 2nt . Reparameterising the varian es in this way leads to a representation where the resulting variables are un onstrained. Thus, the full parameter ve tor for this parameterisation an be expressed as t , at; et ; nt , n = k + 2, with orresponding support , Ak R R, where Ak is the region of stability for the oeÆ ients of a k-th order stationary AR pro ess. The model order k is assumed to be known throughout. Remark 1 at 2 Ak, for all t 0, is a suÆ ient, but not ne essary, ondition for the TVAR pro ess to be stable. Finding the true region of stability for the oeÆ ients of a general TVAR pro ess is diÆ ult, and hen e the simpler ondition will be enfor ed here, as was done for the xed-parameter AR pro ess in e.g. [2℄. The TVAR model parameters are assumed to evolve a priori a ording to a rst-order Markov pro ess. This pro ess is fully spe i ed by its initial state distribution p ( 0) and state transition distribution p ( tj t 1), for all t 2 T . Furthermore, it is assumed that the omponents of the parameter ve tor evolve independently, so that these distributions an be fa torised as p ( 0) , n Y i=1 p ( i;0) ; p ( tj t 1) , n Y i=1 p ( i;tj i;t 1) ; 8t 2 T : (7) This evolution model is similar to the previously de ned \smoothness priors" models for the TVAR oeÆ ients, proposed in [12℄. Here ea h of the omponents of the parameter ve tor is assumed to evolve a ording to the trun ated Gaussian random walk di usion pro ess, de ned by p ( i;0) / N i;0; 0; 2i;0 I ( 0) ; p ( i;tj i;t 1) / N i;t; i;t 1; 2i;1 I ( t) ; 8t > 0; (8) where the parameters 2i;0; 2i;1 are assumed to be known. C Estimation Obje tives Given the observations y1:T , all Bayesian inferen e for the signal model in Se tion II relies on the joint posterior distribution p ( 0:T ; 0:T jy1:T ). Two optimal estimation problems are onsidered here, viz. 5 MMSE estimation of 0:T and 0:T . Compute the optimal (in the mean square sen