On Bayesian Estimation of Spectral Components for Broadband Noise Reduction in Audio Signal

On Bayesian Estimation of Spectral Components for Broadband Noise Reduction in Audio SignalsCUED/F-INFENG/TR.404Patrick J.Wolfe and Simon J.GodsillSignal Processing Group,University of CambridgeDepartment of Engineering,Trumpington StreetCB21PZ,Cambridge,UKAugust2001AbstractShort-time spectral attenuation is a common form of audio signal enhancement in which a time-varyingfilter,or suppression rule,is applied to the frequency-domaintransform of a corrupted signal.Here we review suppression rules derived under aGaussian statistical model and interpret them in a Bayesian framework.In particular,the Ephraim and Malah spectral amplitude estimator is both optimal in a minimummean-square error sense and well-known for its associated colourless residual noise.However,its implementation requires the evaluation of computationally expensive ex-ponential and Bessel functions.Here we show that under the same modelling assump-tions,alternative methods of Bayesian estimation lead to suppression rules exhibitingalmost identical behaviour.We derive three such rules and show that,in addition topermitting an efficient implementation,they also yield a more intuitive interpretation.1IntroductionHerein we address broadband noise reduction for audio signals via statistical modelling of their spectral components.We limit our investigation to short-time spectral attenuation,a popular method of broadband noise reduction in which a time-varyingfilter,or suppres-sion rule,is applied to the frequency-domain transform of a corrupted signal.Wefirst review existing suppression rules derived under a Gaussian statistical model and interpret them in a Bayesian framework.We then employ the same model and framework to derive three new suppression rules exhibiting near-optimal behaviour.This report is organised as follows:in the remainder of Section1we introduce the assumed statistical model and estimation framework,and then employ these in an al-ternate derivation of the minimum mean-square error(MMSE)suppression rules due to1y x expxexpxThe mean of the posterior density appearing in(4)follows directly from its Gaussian form:X Yexpif(8)otherwise(9)Y(10)where it is understood that(9)and(10)are defined over the range of in(7)andin(8);X and D denote the respective variances of the th short-time spectral component of the signal and noise.Additionally,definewhere and are interpreted after[7]as the a priori and a posteriori signal-to-noise ratios(SNR),respectively.Under the assumed model,the posterior density Y(following integration w.r.t. the phase term)is Rician[8]with parameters:Y(12)5(13) where is the gamma function[9,eq.8.310.1]and is the confluent hypergeometric function[9,eq.9.210.1].The MMSE solution of Ephraim and Malah is simply thefirst moment of(11);when combined with the optimal phase estimator(the observed phase[2]),it takes the form of a suppression rule:expYexpSince ln is a monotonically increasing function,one may equivalently maximise the natural logarithm of Y.DefineYln constant6cosFrom(15),we have cos;thereforeSolving the above quadratic equation,and substituting(17) Together(17)and(15)define the following suppression rule:ln constant(19) Substituting(12)and(16)into(19)and solving,we arrive at an estimator differing from that of the joint MAP solution only by a factor of two under the square root(owing to the factor(20) Combining(20)with the Ephraim and Malah phase estimator(i.e.,the observed phase )yields the following suppression rule:(22)3Comparison of ApproximationsFigure1shows the Ephraim and Malah suppression rule as a function of instantaneous SNR(defined in[2]as)and a priori SNR.Figures2,3,and4show the gain difference(in dB)between it and each of the three derived suppression rules(note the difference in scale).Table1shows a comparison of the magnitude of gain differences for the three approximations.The MMSE spectral power suppression rule provides the best and most consistent approximation to the Ephraim and Malah suppression rule,with only slightly less suppression in regions of low a priori SNR.The MAP spectral amplitude approximation,although still within5dB of the optimal value over a wide SNR range, is the poorest.While the sign of the deviation of each of these two approximations is constant,that of the joint MAP suppression rule depends on the instantaneous and a priori SNR.Ephraim and Malah[2]show that at high SNR,their derived suppression rule con-verges to the Wiener suppression rule detailed in Section1.3,formulated as a function of :3COMPARISON OF APPROXIMATIONS CUED/F-INFENG/TR.404dB dB Suppression Rule Mean Maximum Range Mean Maximum RangeTable1:Magnitude of deviation from Ephraim and Malah MMSE suppression rule gain This relationship is easily seen from the MMSE spectral power suppression rule given by(22),expanded slightly to the following:(24) As the instantaneous SNR becomes large,(24)may be seen to approach the Wiener sup-pression rule given by(23).As it becomes small,the term in(24)lessens the severity of the attenuation.Capp´e[10]makes the same observation concerning the behaviour of the Ephraim and Malah suppression rule,although the simpler form of the MMSE spectral power estimator shows the influence of the a priori and a posteriori SNR more explicitly.Lastly,we note that the success of the Ephraim and Malah suppression rule is largely due to the authors’decision-directed approach for estimating the a priori SNR[10]. For a given short-time block,the decision-directed a priori SNR estimate is given by a geometric weighting of the SNR in the previous and current blocks:Xarctancos sinsin cosandcos sinexpA solution to the inner integral of(26)is given by[9,p.357,eq.3.338.4]:A solution to(27)is given by[9,p.737,eq.6.631.1]:where is the confluent hypergeometric function[9,eq.9.210.1].Letting15Noting that[9,p.1086,eq.9.212.3]we obtainy which is zero whenx。