Channel Equalization and Blind Deconvolution (Vaseghi - Advanced Digital Signal Processing and Noise Reduction), страница 3

Assuming that the length of eachsegment is long compared with the duration of the channel impulseresponse, we can write an approximate convolutional relation for the ithsignal segment asyi (m) ≈ xi (m)∗hi (m)(15.35)The segments are windowed, using a Hamming or a Hanning window, toreduce the spectral leakage due to end effects at the edges of the segment.Taking the complex logarithm of the Fourier transform of Equation (15.35)yieldsln Yi ( f ) = ln Xi ( f ) + ln Hi ( f )(15.36)430Equalization and DeconvolutionTaking the time averages over N segments of the distorted signal recordyields1 N −11 N −11 N −1YfXf=+ln()ln()(15.37)∑ i∑ i∑ ln H i ( f )N i =0N i =0N i =0Estimation of the channel response from Equation (15.37) requires theaverage log spectrum of the undistorted signal X(f).

In Stockham's methodfor restoration of acoustic records, the expectation of the signal spectrum isobtained from a modern recording of the same musical material as that ofthe acoustic recording. From Equation (15.37), the estimate of the logarithmof the channel is given by1ln Hˆ ( f ) =NN −1∑ ln Yi ( f ) −i =01NN −1∑ ln X iM ( f )(15.38)i =0where XM(f) is the spectrum of a modern recording. The equalizer can thenbe defined as− ln Hˆ ( f ),200 Hz ≤ f ≤ 4000 Hzln H inv ( f ) = (15.39)otherwise− 40 dB,In Equation (15.39), the inverse acoustic channel is implemented in therange between 200 and 4000 Hz, where the channel is assumed to beinvertible.

Outside this range, the signal is dominated by noise, and theinverse filter is designed to attenuate the noisy signal.15.2.2 Homomorphic Equalization Using a Bank of High-PassFiltersIn the log-frequency domain, channel distortion may be eliminated using abank of high-pass filters. Consider a time sequence of log-spectra of theoutput of a channel described aslnYt ( f ) = ln X t ( f ) + ln H t ( f )(15.40)where Yt(f) and Xt(f) are the channel input and output derived from a Fouriertransform of the tth signal segment.

From Equation (15.40), the effect of aEqualization Based on Linear Prediction Models431time-invariant channel is to add a constant term ln H(f) to each frequencycomponent of the channel input Xt(f), and the overall result is a timeinvariant tilt of the log-frequency spectrum of the original signal. Thisobservation suggests the use of a bank of narrowband high-pass notch filtersfor the removal of the additive distortion term lnH(f). A simple first-orderrecursive digital filter with its notch at zero frequency is given byln Xˆ t ( f ) =α ln Xˆ t −1 ( f ) + ln Yt ( f ) − ln Yt −1 ( f )(15.41)where the parameter α controls the bandwidth of the notch at zerofrequency. Note that the filter bank also removes any dc component of thesignal ln X(f); for some applications, such as speech recognition, this isacceptable.15.3 Equalization Based on Linear Prediction ModelsLinear prediction models, described in Chapter 8, are routinely used inapplications such as seismic signal analysis and speech processing, for themodelling and identification of a minimum-phase channel.

Linear predictiontheory is based on two basic assumptions: that the channel is minimumphase and that the channel input is a random signal. Standard linearprediction analysis can be viewed as a blind deconvolution method, becauseboth the channel response and the channel input are unknown, and the onlyinformation is the channel output and the assumption that the channel inputis random and hence has a flat power spectrum. In this section, we considerblind deconvolution using linear predictive models for the channel and itsinput. The channel input signal is modelled asX ( z ) = E ( z ) A( z )(15.42)where X(z) is the z-transform of the channel input signal, A(z) is the ztransfer function of a linear predictive model of the channel input and E(z) isthe z-transform of a random excitation signal.

Similarly, the channel outputcan be modelled by a linear predictive model H(z) with input X(z) andoutput Y(z) asY ( z) = X ( z)H ( z)(15.43)432Equalization and DeconvolutionFigure 15.7 illustrates a cascade linear prediction model for a channel inputprocess X(z) and a channel response H(z). The channel output can beexpressed asY ( z ) = E ( z ) A( z ) H ( z )= E ( z ) D( z )(15.44)where(15.45)D( z ) = A( z ) H ( z )The z-transfer function of the linear prediction models of the channel inputsignal and the channel can be expanded asA( z ) =G1P1−∑ a k z=−kk =1H ( z) =(15.46)∏ (1 − α−1kz )k =1G2QG1P1−∑ bk z − k=k =1G2(15.47)Q∏ (1 − βkz −1 )k =1where {ak,αk} and {bk,βk} are the coefficients and the poles of the linearprediction models for the channel input signal and the channel respectively.Substitution of Equations (15.46) and (15.47) in Equation (15.45) yields thecombined input-channel model asD( z ) =GP +Q1− ∑ d k zk =1=−kGP +Q∏ (1 − γ k z(15.48)−1)k =1The total number of poles of the combined model for the input signal andthe channel is the sum of the poles of the input signal model and the channelmodel.433Equalization Based on Linear Prediction Modelse(m)Channel inputsignal modelx(m)Channel modely(m)H(z)A(z)D(z) = A(z) H(z)Figure 15.7 A distorted signal modelled as cascade of a signal model and achannel model.15.3.1 Blind Equalization Through Model FactorisationA model-based approach to blind equalization is to factorise the channeloutput model D(z)=A(z)H(z) into a channel input signal model A(z) and achannel model H(z).

If the channel input model A(z) and the channel modelH(z) are non-factorable then the only factors of D(z) are A(z) and H(z).However, z-transfer functions are factorable into the roots, the so-calledpoles and zeros, of the models. One approach to model-based deconvolutionis to factorize the model for the convolved signal into its poles and zeros,and classify the poles and zeros as either belonging to the signal orbelonging to the channel.Spencer and Rayner (1990) developed a method for blind deconvolutionthrough factorization of linear prediction models, based on the assumptionthat the channel is stationary with time-invariant poles whereas the inputsignal is non-stationary with time-varying poles. As an application, theyconsidered the restoration of old acoustic recordings where a time-varyingaudio signal is distorted by the time-invariant frequency response of therecording equipment.

For a simple example, consider the case when thesignal and the channel are each modelled by a second-order linear predictivemodel. Let the time-varying second-order linear predictive model for thechannel input signal x(m) bex(m) = a1 (m) x(m − 1)+a 2 (m) x(m − 2)+G1 (m)e(m)(15.49)where a1(m) and a2(m) are the time-varying coefficients of the linearpredictor model, G1(m) is the input gain factor and e(m) is a zero-mean, unitvariance, random signal.

Now let α1(m) and α2(m) denote the time-varying434Equalization and Deconvolutionpoles of the predictor model of Equation (15.49); these poles are the roots ofthe polynomial[][]1 − a1 (m) z −1 − a 2 (m) z −2 = 1 − z −1α1 (m) 1 − z −1α 2 (m) = 0(15.50)Similarly, assume that the channel can be modelled by a second-orderstationary linear predictive model asy (m) = h1 y (m − 1)+ h2 y (m − 2)+G2 x(m)(15.51)where h1 and h2 are the time-invariant predictor coefficients and G2 is thechannel gain. Let β1 and β2 denote the poles of the channel model; these arethe roots of the polynomial1 − h1 z −1 − h2 z −2 = (1 − z −1 β1 )(1 − z −1 β 2 ) = 0(15.52)The combined cascade of the two second-order models of Equations (15.49)and (15.51) can be written as a fourth-order linear predictive model withinput e(m) and output y(m):y (m) = d1 (m) y (m − 1)+d 2 (m) y (m − 2)+d 3 (m) y (m − 3)+ d 4 (m) y (m − 4)+ Ge(m)(15.53)where the combined gain G=G1G2. The poles of the fourth order predictormodel of Equation (15.53) are the roots of the following polynomial:1 − d1 (m) z −1 −d 2 (m) z −2 −d 3 (m) z −3 − d 4 (m) z −4 =[][][][]= 1 − z −1α1 (m) 1 − z −1α 2 (m) 1 − z −1 β1 1 − z −1 β 2 = 0(15.54)In Equation (15.54) the poles of the fourth order predictor are α1(m) , α2(m),β1 and β2 .

The above argument on factorisation of the poles of time-varyingand stationary models can be generalised to a signal model of order P and achannel model of order Q.In Spencer and Rayner, the separation of the stationary poles of thechannel from the time-varying poles of the channel input is achievedthrough a clustering process. The signal record is divided into N segmentsand each segment is modelled by an all-pole model of order P+Q where Pand Q are the assumed model orders for the channel input and the channelBayesian Blind Deconvolution and Equalization435respectively. In all, there are N(P+Q) values which are clustered to formP+Q clusters. Even if both the signal and the channel were stationary, thepoles extracted from different segments would have variations due to therandom character of the signals from which the poles are extracted.Assuming that the variances of the estimates of the stationary poles aresmall compared with the variations of the time-varying poles, it is expectedthat, for each stationary pole of the channel, the N values extracted from Nsegments will form an N-point cluster of a relatively small variance.