Channel Equalization and Blind Deconvolution (Vaseghi - Advanced Digital Signal Processing and Noise Reduction), страница 4

Theseclusters can be identified and the centre of each cluster taken as a pole of thechannel model This method assumes that the poles of the time-varyingsignal are well separated in space from the poles of the time-invariantsignal.15.4 Bayesian Blind Deconvolution and EqualizationThe Bayesian inference method, described in Chapter 4, provides a generalframework for inclusion of statistical models of the channel input and thechannel response. In this section we consider the Bayesian equalizationmethod, and study the case where the channel input is modelled by a set ofhidden Markov models.

The Bayesian risk for a channel estimate hˆ isdefined asR (hˆ | y ) = ∫ ∫ C (hˆ , h) f X ,H |Y ( x , h y ) d x d hHX1C (hˆ , h) f Y | H ( y h) f H (h) d h=f Y ( y) H∫(15.55)where C( hˆ, h) is the cost of estimating the channel h as hˆ , f X , H|Y ( x, h y ) isthe joint posterior density of the channel h and the channel input x,f Y|H ( y h) is the observation likelihood, and fH(h) is the prior pdf of thechannel. The Bayesian estimate is obtained by minimisation of the riskfunction R (hˆ | y ) . There are a variety of Bayesian-type solutions dependingon the choice of the cost function and the prior knowledge, as described inChapter 4.In this section, it is assumed that the convolutional channel distortion istransformed into an additive distortion through transformation of thechannel output into log-spectral or cepstral variables.

Ignoring the channel436Equalization and Deconvolutionnoise, the relation between the cepstra of the channel input and outputsignals is given byy ( m) = x ( m) + h(15.56)where the cepstral vectors x(m), y(m) and h are the channel input, thechannel output and the channel respectively.15.4.1 Conditional Mean Channel EstimationA commonly used cost function in the Bayesian risk of Equation (15.55) is2the mean square error C (h − hˆ )=| h − hˆ | , which results in the conditionalmean (CM) estimate defined ashˆ CM = ∫ h f H |Y (h | y ) d h(15.57)HThe posterior density of the channel input signal may be conditioned on anestimate of the channel vector hˆ and expressed as f X |Y , H ( x | y , hˆ ) . Theconditional mean of the channel input signal given the channel output y andan estimate of the channel ĥ isxˆ CM =E[ x | y , hˆ ]= ∫ x f X |Y , H ( x | y , hˆ ) dx(15.58)XEquations (15.57) and (15.58) suggest a two-stage iterative method forchannel estimation and the recovery of the channel input signal.15.4.2 Maximum-Likelihood Channel EstimationThe ML channel estimate is equivalent to the case when the Bayes costfunction and the channel prior are uniform.

Assuming that the channel inputsignal has a Gaussian distribution with mean vector µx and covarianceBayesian Blind Deconvolution and Equalization437matrix Σxx, the likelihood of a sequence of N P-dimensional channel outputvectors {y(m)} given a channel input vector h isf Y | H ( y (0 ) , , y( N − 1) h) = ∏ fN −1X( y ( m) − h)m=0N −1=∏1m =0 (2π )P/2Σ xx1/ 2{−1exp [ y ( m) − h − µ x ] T Σ xx[ y (m) − h − µ x ]}(15.59)To obtain the ML estimate of the channel h, the derivative of the loglikelihood function ln fY(y|h) with respect to h is set to zero to yield1 N −1hˆ ML =∑ (y(m) − µ x )N m= 0(15.60)15.4.3 Maximum A Posteriori Channel EstimationThe MAP estimate, like the ML estimate, is equivalent to a Bayesianestimator with a uniform cost function. However, the MAP estimateincludes the prior pdf of the channel.

The prior pdf can be used to confinethe channel estimate within a desired subspace of the parameter space.Assuming that the channel input vectors are statistically independent, theposterior pdf of the channel given the observation sequence Y={y(0), ...,y(N–1)} isf H |Y (h y (0) , , y ( N − 1) ) ==N −1∏m =0N −1∏m =01f Y | H ( y (m) h) f H (h)f Y ( y (m))1f X ( y (m) − h) f H (h)f Y ( y (m))(15.61)Assuming that the channel input x(m) is Gaussian, fX(x(m))=N (x, µx, Σxx),with mean vector µx and covariance matrix Σxx, and that the channel h isalso Gaussian, fH(h)=N (h, µh, Σhh), with mean vector µh and covariancematrix Σhh, the logarithm of the posterior pdf is438Equalization and Deconvolutionln f H |Y (h y (0) ,−N −1 , y( N − 1)) = − ∑ ln f ( y(m)) − NP ln(2π ) − 12 ln( ΣN −1xxΣ hh )m =0∑ 2 {[ y(m) − h − µ x ]T Σ xx−1 [ y (m) − h − µ x ] + (h − µ h ) T Σ hh−1 (h − µ h )}1m =0(15.62)The MAP channel estimate, obtained by setting the derivative of the logposterior function ln fH|Y(h|y) to zero, is−1−1hˆ MAP = (Σ xx + Σ hh ) Σ hh ( y − µ x )+ (Σ xx + Σ hh ) Σ xx µ h(15.63)wherey=1 N −1∑ y ( m)N m =0(15.64)is the time-averaged estimate of the mean of observation vector.

Note thatfor a Gaussian process the MAP and conditional mean estimates areidentical.15.4.4 Channel Equalization Based on Hidden Markov ModelsThis section considers blind deconvolution in applications where thestatistics of the channel input are modelled by a set of hidden Markovmodels. An application of this method, illustrated in Figure 15.8, is inrecognition of speech distorted by a communication channel or amicrophone.

A hidden Markov model (HMM) is a finite-state Bayesianmodel, with a Markovian state prior and a Gaussian observation likelihood(see chapter 5). An N-state HMM can be used to model a non-stationaryprocess, such as speech, as a chain of N stationary states connected by a setof Markovian state transitions. The likelihood of an HMM Mi and asequence of N P-dimensional channel input vectors X=[x(0), ..., x(N–1)] canbe expressed in terms of the state transition and the observation pdfs of Miasf X |M ( X | M i ) = ∑ f X |M , S ( X | M i , s) PS |M ( s | M i )(15.65)s439Bayesian Blind Deconvolution and EqualizationHMMs of the channel inputa11s1a12a22s2aNN...sM1Nn......a11s1a12xa22s2ChannelhaNNs...NMy+BayesianEstimatorVFigure 15.8 Illustration of a channel with the input modelled by a set of HMMs.where f X |M , S ( X | M i , s ) is the likelihood that the sequence X=[x(0), ...,x(N–1)] was generated by the state sequence s=[s(0), ..., s(N–1)] of themodel Mi, and Ps|M ( s | M i ) is the Markovian prior pmf of the state sequences.

The Markovian prior entails that the probability of a transition to the statei at time m depends only on the state at time m–1 and is independent of theprevious states. The transition probability of a Markov process is defined asa ij = P(s (m) = j | s (m − 1) = i )(15.66)where aij is the probability of making a transition from state i to state j. TheHMM state observation probability is often modelled by a multivariateGaussian pdf asf X |M ,S ( x | M i , s ) =1(2π )P/2| Σ xx ,s |1/ 21−1exp − [ x − µ x,s ]T Σ xx,s [ x − µ x, s ] 2(15.67)where µx,s and Σxx,s are the mean vector and the covariance matrix of theGaussian observation pdf of the HMM state s of the model Mi.The HMM-based channel equalization problem can be stated asfollows: Given a sequence of N P-dimensional channel output vectorsY=[y(0), ..., y(N–1)], and the prior knowledge that the channel input440Equalization and Deconvolutionsequence is drawn from a set of V HMMs M={Mi i=1, ..., V}, estimate thechannel response and the channel input.The joint posterior pdf of an input word Mi and the channel vector hcan be expressed asf M , H |Y (M i , h | Y ) = PM | H ,Y (M i h, Y ) f H |Y (h | Y )(15.68)Simultaneous joint estimation of the channel vector h and classification ofthe unknown input word Mi is a non-trivial exercise.

The problem is usuallyapproached iteratively by making an estimate of the channel response, andthen using this estimate to obtain the channel input as follows. From Bayes’rule, the posterior pdf of the channel h conditioned on the assumption thatthe input model is Mi and given the observation sequence Y can beexpressed asf H |M ,Y (h M i , Y ) =1f Y |M , H (Y M i , h) f H |M (h | M i )f Y |M (Y | M i )(15.69)The likelihood of the observation sequence, given the channel and the inputword model, can be expressed asf Y |M , H (Y | M i , h) = f X |M (Y − h | M i )(15.70)where it is assumed that the channel output is transformed into cepstralvariables so that the channel distortion is additive.

For a given input modelMi, and state sequence s=[s(0), s(1), ..., s(N–1)], the pdf of a sequence of Nindependent observation vectors Y=[y(0), y(1), ..., y(N–1)] isN −1f Y | H ,S,M (Y | h, s, M i ) =N −1=1∏ (2π ) P/2 | Σm =0∏ f X |S ,M ( y(m ) − h | s(m ),M i )m =0xx , s ( m )1/ 2|1−1exp − [y (m) − h − µ x , s ( m) ]T Σ xx, s ( m ) [y ( m) − h − µ x , s ( m ) ] 2(15.71)Taking the derivative of the log-likelihood of Equation (15.71) with respectto the channel vector h yields a maximum likelihood channel estimate asBayesian Blind Deconvolution and Equalizationhˆ ML (Y,s) =N −1∑m= 0441−1 N∑−1 Σ −1  Σ −1xx ,s( m) (y(m) − µ x,s( m) ) k = 0 xx ,s( k ) (15.72)Note that when all the state observation covariance matrices are identical thechannel estimate becomes1hˆ ML (Y , s) =NN −1∑ ( y ( m) − µ x , s ( m ) )(15.73)m=0The ML estimate of Equation (15.73) is based on the ML state sequence s ofMi.

In the following section we consider the conditional mean estimate overall state sequences of a model.15.4.5 MAP Channel Estimate Based on HMMsThe conditional pdf of a channel h averaged over all HMMs can beexpressed asVf H |Y (h | Y ) = ∑∑ f H |Y , S ,M (h Y , s, M i )PS M ( s M i ) PM (M i )(15.74)i =1 swhere PM (M i ) is the prior pmf of the input words. Given a sequence of NP-dimensional observation vectors Y=[y(0), ..., y(N–1)], the posterior pdf ofthe channel h along a state sequence s of an HMM Mi is defined asfY | H ,S ,M (h Y , s, Mi ) ==1 N −1∏fY (Y ) m = 0 ( 2π ) P Σ1fY | H ,S,M (Y h, s, M i ) f H (h)fY (Y )11/ 2xx , s ( m )Σ hh1/ 2 1−1exp − [y ( m) − h − µ x , s ( m) ]T Σ xx, s ( m ) [y ( m) − h − µ x , s ( m ) ] 2× exp − (h − µ h )T Σ h−h1 (h − µ h ) 21(15.75)where it is assumed that each state of the HMM has a Gaussian distributionwith mean vector µx,s(m) and covariance matrix Σxx,s(m), and that the channelh is also Gaussian-distributed, with mean vector µh and covariance matrix442Equalization and DeconvolutionΣhh.

The MAP estimate along state s, on the left-hand side of Equation(15.75), can be obtained asN −1()N −1−1−1 hˆ MAP (Y , s, M i ) = ∑  ∑ Σ xx, s ( k ) + Σ hh k 0m =0 =−1−1Σ xx, s ( m ) [y ( m) − µ x , s ( m ) ]−1−1−1 −1+  ∑ (Σ xx, s ( k ) + Σ hh ) Σ hh µ hN −1 k =0(15.76)The MAP estimate of the channel over all state sequences of all HMMs canbe obtained asVhˆ (Y ) = ∑ ∑ hˆ MAP (Y , s, M i ) PS|M (s | M i ) PM (M i )(15.77)i=1 S15.4.6 Implementations of HMM-Based DeconvolutionIn this section, we consider three implementation methods for HMM-basedchannel equalization.Method I: Use of the Statistical Averages Taken Over All HMMsA simple approach to blind equalization, similar to that proposed byStockham, is to use as the channel input statistics the average of the meanvectors and the covariance matrices, taken over all the states of all theHMMs asµx =1V NsV Ns∑∑ µ Mi , j , Σ xx =i =1 j =11V NsV Ns∑∑ Σ M , ji =1 j =1i(15.78)where µ M i , j and Σ M i , j are the mean and the covariance of the jth state of theith HMM, V and Ns denote the number of models and number of states permodel respectively.

The maximum likelihood estimate of the channel, hˆ ML ,is defined as443Bayesian Blind Deconvolution and Equalizationhˆ ML = ( y − µ x )(15.79)where y is the time-averaged channel output. The estimate of the channelinput isxˆ (m) = y(m) − hˆ ML(15.80)Using the averages over all states and models, the MAP channel estimatebecomeshˆ MAP (Y ) =N −1∑(Σ xx + Σ hh )−1 Σ hh (y(m) − µ x ) + (Σ xx + Σ hh )−1 Σ xx µ hm= 0(15.81)Method II: Hypothesised-Input HMM EqualizationIn this method, for each candidate HMM in the input vocabulary, a channelestimate is obtained and then used to equalise the channel output, prior tothe computation of a likelihood score for the HMM.