Channel Equalization and Blind Deconvolution (Vaseghi - Advanced Digital Signal Processing and Noise Reduction), страница 2

Figure 15.3 illustrates thefrequency response of a channel that has one invertible andtwo non-invertible regions. In the non-invertible regions, thesignal frequencies are heavily attenuated and lost to channelnoise. In the invertible region, the signal is distorted butrecoverable. This example illustrates that the inverse filtermust be implemented with care in order to avoid undesirableresults such as noise amplification at frequencies with lowSNR.15.1.4 Minimum- and Maximum-Phase ChannelsFor stability, all the poles of the transfer function of a channel must lieinside the unit circle.

If all the zeros of the transfer function are also insidethe unit circle then the channel is said to be a minimum-phase channel. Ifsome of the zeros are outside the unit circle then the channel is said to be amaximum-phase channel. The inverse of a minimum-phase channel has allits poles inside the unit circle, and is therefore stable. The inverse of amaximum-phase channel has some of its poles outside the unit circle;therefore it has an exponentially growing impulse response and is unstable.However, a stable approximation of the inverse of a maximum-phaseInputChannel distortionX(f)H(f)Noninvertible InvertibleOutputY(f)=X(f)H(f)NoninvertibleChannelnoiseffFigure 15.3 Illustration of the invertible and noninvertible regions of a channel.f424Equalization and DeconvolutionMinimum-phaseMaximum-phasehmin(k)hmax(k)kkFigure 15.4 Illustration of the zero diagram and impulse response of fourth ordermaximum-phase and minimum-phase FIR filters.channel may be obtained by truncating the impulse response of the inversefilter.

Figure 15.3 illustrates examples of maximum-phase and minimumphase fourth-order FIR filters.When both the channel input and output signals are available, in thecorrect synchrony, it is possible to estimate the channel magnitude andphase response using the conventional least square error criterion. In blinddeconvolution, there is no access to the exact instantaneous value or thetiming of the channel input signal. The only information available is thechannel output and some statistics of the channel input. The second orderstatistics of a signal (i.e.

the correlation or the power spectrum) do notinclude the phase information; hence it is not possible to estimate thechannel phase from the second-order statistics. Furthermore, the channelphase cannot be recovered if the input signal is Gaussian, because aGaussian process of known mean is entirely specified by the autocovariancematrix, and autocovariance matrices do not include any phase information.For estimation of the phase of a channel, we can either use a non-linearestimate of the desired signal to direct the adaptation of a channel equalizeras in Section 15.5, or we can use the higher-order statistics as in Section15.6.425Introduction15.1.5 Wiener EqualizerIn this section, we consider the least squared error Wiener equalization.Note that, in its conventional form, Wiener equalization is not a form ofblind equalization, because the implementation of a Wiener equalizerrequires the cross-correlation of the channel input and output signals, whichare not available in a blind equalization application.

The Wiener filterestimate of the channel input signal is given byP −1xˆ (m) = ∑ hˆkinv y (m − k )(15.16)k =0where hˆk is an FIR Wiener filter estimate of the inverse channel impulseresponse. The equalization error signal v(m) is defined asinvP −1v(m) = x(m) − ∑ hˆkinv y (m − k )(15.17)k =0The Wiener equalizer with input y(m) and desired output x(m) is obtainedfrom Equation (6.10) in Chapter 6 ashˆ inv = R −yy1r xy(15.18)where Ryy is the P × P autocorrelation matrix of the channel output, and rxyis the P-dimensional cross-correlation vector of the channel input and outputsignals. A more expressive form of Equation (15.18) can be obtained bywriting the noisy channel output signal in vector equation form asy = Hx + n(4.19)where y is an N-sample channel output vector, x is an N+P-sample channelinput vector including the P initial samples, H is an N×(N+P) channeldistortion matrix whose elements are composed of the coefficients of thechannel filter, and n is a noise vector.

The autocorrelation matrix of thechannel output can be obtained from Equation (15.19) asR yy = E [ yy T ] = HR xx H T +Rnn(15.20)426Equalization and Deconvolutionwhere E [·] is the expectation operator. The cross-correlation vector rxy ofthe channel input and output signals becomesrxy =E[ xy ] = Hrxx(15.21)Substitution of Equation (15.20) and (15.21) in (15.18) yields the Wienerequalizer as−1hˆ inv = HR xx H T +Rnn Hr xx(15.22)()The derivation of the Wiener equalizer in the frequency domain is asfollows. The Fourier transform of the equalizer output is given byXˆ ( f ) = Hˆ inv ( f )Y ( f )(15.23)invwhere Y(f), the channel output and Hˆ ( f ) is the frequency response of theWiener equalizer. The error signal V(f) is defined asV ( f ) = X ( f ) − Xˆ ( f )= X ( f ) − Hˆ inv ( f )Y ( f )(15.24)As in Section 6.5 minimisation of the expectation of the squared magnitudeof V(f) results in the frequency Wiener equalizer given byP (f)Hˆ inv ( f ) = XYPYY ( f )=PXX ( f ) H * ( f )(15.25)2PXX ( f ) H ( f ) + PNN ( f )where PXX(f) is the channel input power spectrum, PNN(f) is the noise powerspectrum, PXY(f) is the cross-power spectrum of the channel input andoutput signals, and H(f) is the frequency response of the channel.

Note thatin the absence of noise, PNN(f)=0 and the Wiener inverse filter becomesH inv ( f ) = H −1 ( f ) .Blind Equalization Using Channel Input Power Spectrum42715.2 Blind Equalization Using Channel Input Power SpectrumOne of the early papers on blind deconvolution was by Stockham et al.(1975) on dereverberation of old acoustic recordings. Acoustic recorders, asillustrated in Figure 15.5, had a bandwidth of about 200 Hz to 4 kHz.However, the limited bandwidth, or even the additive noise or scratch noisepulses, are not considered as the major causes of distortions of acousticrecordings.

The main distortion on acoustic recordings is due toreverberations of the recording horn instrument. An acoustic recording canbe modelled as the convolution of the input audio signal x(m) and theimpulse response of a linear filter model of the recording instrument {hk}, asin Equation (15.2), reproduced here for convenienceP −1y (m) = ∑ hk x(m − k )+ n(m)(15.26)k =0or in the frequency domain asY( f )= X ( f )H( f )+ N( f )(15.27)where H(f) is the frequency response of a linear time-invariant model of theacoustic recording instrument, and N(f) is an additive noise.

MultiplyingFigure 15.5 Illustration of the early acoustic recording process on a wax disc.Acoustic recordings were made by focusing the sound energy, through a hornvia a sound box, diaphragm and stylus mechanism, onto a wax disc. Thesound was distorted by reverberations of the horn.428Equalization and Deconvolutionboth sides of Equation (15.27) with their complex conjugates, and taking theexpectation, we obtainE [Y ( f )Y ∗ ( f )] = E [(X ( f ) H ( f ) + N ( f ) )(X ( f ) H ( f ) + N ( f ) )∗ ] (15.28)Assuming the signal X(f) and the noise N(f) are uncorrelated Equation(15.28) becomes2PYY ( f ) = PXX ( f ) H ( f ) + PNN ( f )(15.29)where PYY(f), PXX(f) and PNN(f) are the power spectra of the distorted signal,the original signal and the noise respectively. From Equation (15.29) anestimate of the spectrum of the channel response can be obtained as2H( f ) =PYY ( f )− PNN ( f )PXX ( f )(15.30)In practice, Equation (15.30) is implemented using time-averaged estimatesof the of the power spectra.15.2.1 Homomorphic EqualizationIn homomorphic equalization, the convolutional distortion is transformed,first into a multiplicative distortion through a Fourier transform of thedistorted signal, and then into an additive distortion by taking the logarithmof the spectrum of the distorted signal.

A further inverse Fourier transformoperation converts the log-frequency variables into cepstral variables asillustrated in Figure 15.6. Through homomorphic transformationconvolution becomes addition, and equalization becomes subtraction.Ignoring the additive noise term and transforming both sides ofEquation (15.27) into log-spectral variables yieldsln Y ( f ) = ln X ( f ) + ln H ( f )(15.31)Note that in the log-frequency domain, the effect of channel distortion is theaddition of a tilt to the spectrum of the channel input. Taking theexpectation of Equation (15.31) yields429Blind Equalization Using Input Power Spectrumln |X(f)|+ln|X(f)|X(f)H(f)y(m)=x(m)*h(m)Fouriertransformln|.|Inverse Fouriertransformxc (m)+hc (m)Homomorphic analysisFigure 15.6 Illustration of homomorphic analysis in deconvolution.E [ln Y ( f )] = E[ln X ( f )]+ ln H ( f )(15.32)In Equation (15.32), it is assumed that the channel is time-invariant; henceE [ln H ( f )] = ln H ( f ) .

Using the relation ln z =ln| z |+ j∠ z , the termE [ln X ( f )] can be expressed asE [ln X ( f )] = E [ln| X ( f ) |]+ jE[∠X ( f )](15.33)The first term on the right-hand side of Equation (15.33), E [ln| X ( f ) |] , isnon-zero, and represents the frequency distribution of the signal power indecibels, whereas the second term E [∠X ( f )] is the expectation of thephase, and can be assumed to be zero. From Equation (15.32), the logfrequency spectrum of the channel can be estimated asln H ( f ) = E [ln Y ( f )]−E [ln X ( f )](15.34)In practice, when only a single record of a signal is available, the signal isdivided into a number of segments, and the average signal spectrum isobtained over time across the segments.