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

Advanced Digital Signal Processing and Noise Reduction, Second Edition.Saeed V. VaseghiCopyright © 2000 John Wiley & Sons LtdISBNs: 0-471-62692-9 (Hardback): 0-470-84162-1 (Electronic)15CHANNEL EQUALIZATION ANDBLIND DECONVOLUTION15.115.215.315.415.515.615.7IntroductionBlind-Deconvolution Using Channel Input Power SpectrumEqualization Based on Linear Prediction ModelsBayesian Blind Deconvolution and EqualizationBlind Equalization for Digital Communication ChannelsEqualization Based on Higher-Order StatisticsSummaryBlind deconvolution is the process of unravelling two unknownsignals that have been convolved. An important application of blinddeconvolution is in blind equalization for restoration of a signaldistorted in transmission through a communication channel.

Blindequalization has a wide range of applications, for example in digitaltelecommunications for removal of intersymbol interference, in speechrecognition for removal of the effects of microphones and channels, indeblurring of distorted images, in dereverberation of acoustic recordings, inseismic data analysis, etc.In practice, blind equalization is only feasible if some useful statisticsof the channel input, and perhaps also of the channel itself, are available.The success of a blind equalization method depends on how much is knownabout the statistics of the channel input, and how useful this knowledge is inthe channel identification and equalization process.

This chapter begins withan introduction to the basic ideas of deconvolution and channel equalization.We study blind equalization based on the channel input power spectrum,equalization through separation of the input signal and channel responsemodels, Bayesian equalization, nonlinear adaptive equalization for digitalcommunication channels, and equalization of maximum-phase channelsusing higher-order statistics.417Introduction15.1 IntroductionIn this chapter we consider the recovery of a signal distorted, intransmission through a channel, by a convolutional process and observed inadditive noise. The process of recovery of a signal convolved with theimpulse response of a communication channel, or a recording medium, isknown as deconvolution or equalization.

Figure 15.1 illustrates a typicalmodel for a distorted and noisy signal, followed by an equalizer. Let x(m),n(m) and y(m) denote the channel input, the channel noise and the observedchannel output respectively. The channel input/output relation can beexpressed asy (m) = h[ x(m)] + n(m)(15.1)where the function h[· ] is the channel distortion. In general, the channelresponse may be time-varying and non-linear.

In this chapter, it is assumedthat the effects of a channel can be modelled using a stationary, or a slowlytime-varying, linear transversal filter. For a linear transversal filter model ofthe channel, Equation (15.1) becomesP −1y (m) = ∑ hk (m) x(m − k ) + n(m)(15.2)k =0where hk(m) are the coefficients of a Pth order linear FIR filter model of thechannel. For a time-invariant channel model, hk(m)=hk.In the frequency domain, Equation (15.2) becomesY ( f ) = X ( f )H ( f )+ N ( f )(15.3)Noise n(m)x(m)Distortiony(m)H(f)f–1Equaliser^x(m)H (f)fFigure 15.1 Illustration of a channel distortion model followed by an equalizer.418Equalization and Deconvolutionwhere Y(f), X(f), H(f) and N(f) are the frequency spectra of the channeloutput, the channel input, the channel response and the additive noiserespectively.

Ignoring the noise term and taking the logarithm of Equation(15.3) yieldsln Y ( f ) = ln X ( f ) + ln H ( f )(15.4)From Equation (15.4), in the log-frequency domain the effect of channeldistortion is the addition of a “tilt” term ln|H(f)| to the signal spectrum.15.1.1 The Ideal Inverse Channel FilterThe ideal inverse-channel filter, or the ideal equalizer, recovers the originalinput from the channel output signal. In the frequency domain, the idealinverse channel filter can be expressed asH ( f ) H inv ( f ) = 1(15.5)invIn Equation (15.5) H ( f ) is used to denote the inverse channel filter. For−1invthe ideal equalizer we have H ( f )= H ( f ) , or, expressed in the loginvfrequency domain ln H ( f )=−ln H ( f ) .

The general form of Equation(15.5) is given by the z-transform relationH ( z ) H inv ( z ) = z − N(15.6)for some value of the delay N that makes the channel inversion processcausal. Taking the inverse Fourier transform of Equation (15.5), we have thefollowing convolutional relation between the impulse responses of theinvchannel {hk} and the ideal inverse channel response { hk }:∑ hkinv hi−k = δ (i)(15.7)kwhere δ(i) is the Kronecker delta function. Assuming the channel output isnoise-free and the channel is invertible, the ideal inverse channel filter canbe used to reproduce the channel input signal with zero error, as follows.419IntroductionThe inverse filter output xˆ (m) , with the distorted signal y(m) as the input, isgiven asxˆ (m) = ∑ hkinv y (m − k )k= ∑ hkinv ∑ h j x(m − k − j )k(15.8)j= ∑ x(m − i )∑ hkinv hi −kikThe last line of Equation (15.8) is derived by a change of variables i=k+j inthe second line and rearrangement of the terms.

For the ideal inversechannel filter, substitution of Equation (15.7) in Equation (15.8) yieldsxˆ (m) = ∑ δ (i ) x(m − i ) = x(m)(15.9)iwhich is the desired result. In practice, it is not advisable to implementHinv(f) simply as H–1(f) because, in general, a channel response may be noninvertible. Even for invertible channels, a straightforward implementation ofthe inverse channel filter H–1(f) can cause problems. For example, atfrequencies where H(f) is small, its inverse H–1(f) is large, and this can leadto noise amplification if the signal-to-noise ratio is low.15.1.2 Equalization Error, Convolutional NoiseThe equalization error signal, also called the convolutional noise, is definedas the difference between the channel equalizer output and the desiredsignal:v(m) = x(m) − xˆ (m)P −1= x(m) − ∑ hˆkinv y (m − k )(15.10)k =0invwhere hˆk is an estimate of the inverse channel filter.

Assuming that thereinvis an ideal equalizer hk that can recover the channel input signal x(m) fromthe channel output y(m), we have420Equalization and DeconvolutionP −1x(m) = ∑ hkinv y (m − k )(15.11)k =0Substitution of Equation (15.11) in Equation (15.10) yieldsP −1P −1k =0P −1k =0v(m) = ∑ hkinv y (m − k ) − ∑ hˆkinv y (m − k )~= ∑ hkinv y (m − k )(15.12)k =0~ inv invinvwhere hk = hk − hˆk .

The equalization error signal v(m) may be viewed~ invas the output of an error filter hk in response to the input y(m–k), hencethe name “convolutional noise” for v(m). When the equalization process isproceeding well, such that xˆ (m) is a good estimate of the channel inputx(m), then the convolutional noise is relatively small and decorrelated andcan be modelled as a zero mean Gaussian random process.15.1.3 Blind EqualizationThe equalization problem is relatively simple when the channel response isknown and invertible, and when the channel output is not noisy.

However,in most practical cases, the channel response is unknown, time-varying,non-linear, and may also be non-invertible. Furthermore, the channel outputis often observed in additive noise.Digital communication systems provide equalizer-training periods,during which a training pseudo-noise (PN) sequence, also available at thereceiver, is transmitted. A synchronised version of the PN sequence isgenerated at the receiver, where the channel input and output signals areused for the identification of the channel equalizer as illustrated in Figure15.2(a).

The obvious drawback of using training periods for channelequalization is that power, time and bandwidth are consumed for theequalization process.421IntroductionNoise n(m)x(m)ChannelH(z)+y(m)Inverse Channel^x(m)Hinv(z)Adaptationalgorithm(a) Conventional deconvolutionNoise n(m)x(m)ChannelH(z)+y(m)Inverse channelHinv(z)Bayesianestimationalgorithm^x(m)(b) Blind deconvolutionFigure 15.2 A comparative illustration of (a) a conventional equalizer withaccess to channel input and output, and (b) a blind equalizer.It is preferable to have a “blind” equalization scheme that can operatewithout access to the channel input, as illustrated in Figure 15.2(b).Furthermore, in some applications, such as the restoration of acousticrecordings, or blurred images, all that is available is the distorted signal andthe only restoration method applicable is blind equalization.Blind equalization is feasible only if some statistical knowledge of thechannel input, and perhaps that of the channel, is available.

Blindequalization involves two stages of channel identification, anddeconvolution of the input signal and the channel response, as follows:(a) Channel identification. The general form of a channel estimator can beexpressed ashˆ =ψ ( y, M x , M h )(15.13)where ψ is the channel estimator, the vector hˆ is an estimate of thechannel response, y is the channel output, and Mx and Mh are statisticalmodels of the channel input and the channel response respectively.422Equalization and DeconvolutionChannel identification methods rely on utilisation of a knowledge of thefollowing characteristics of the input signal and the channel:(i)The distribution of the channel input signal: for example, indecision-directed channel equalization, described in Section15.5, the knowledge that the input is a binary signal is used ina binary decision device to estimate the channel input and to“direct” the equalizer adaptation process.(ii) the relative durations of the channel input and the channelimpulse response: the duration of a channel impulse responseis usually orders of magnitude smaller than that of the channelinput.

This observation is used in Section 15.3.1 to estimate astationary channel from the long-time averages of the channeloutput.(iii) The stationary, or time-varying characteristics of the inputsignal process and the channel: in Section 15.3.1, a method isdescribed for the recovery of a non-stationary signal convolvedwith the impulse response of a stationary channel.(b) Channel equalization. Assuming that the channel is invertible, thechannel input signal x(m) can be recovered using an inverse channelfilter asP −1xˆ (m) = ∑ hˆkinv y (m − k )(15.14)k =0In the frequency domain, Equation (15.14) becomesXˆ ( f ) = Hˆ inv ( f )Y ( f )(15.15)In practice, perfect recovery of the channel input may not be possible,either because the channel is non-invertible or because the output isobserved in noise. A channel is non-invertible if:(i) The channel transfer function is maximum-phase: the transferfunction of a maximum-phase channel has zeros outside theunit circle, and hence the inverse channel has unstable poles.Maximum-phase channels are considered in the followingsection.423Introduction(ii) The channel transfer function maps many inputs to the sameoutput: in these situations, a stable closed-form equation forthe inverse channel does not exist, and instead an iterativedeconvolution method is used.