Channel Equalization and Blind Deconvolution (779799), страница 7

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The relation between the kthorder cumulant spectra of the input and output signals is given byCY (ω1 , ,ω k −1 ) = H (ω1 ) H (ω k −1 ) H * (ω1 + +ω k −1 ) C X (ω1 , ,ω k −1 )(15.131)where H(ω) is the frequency response of the linear system {hk}. Themagnitude of the kth-order spectrum of the output signal is given as458Equalization and DeconvolutionCY (ω1 , ,ω k −1 ) = H (ω1 ) H (ω k −1 ) H (ω1 + + ω k −1 ) C X (ω1 , , ω k −1 )(15.132)thand the phase of the k -order spectrum is ,ωΦ Y (ω1 ,k −1+ Φ) = Φ H (ω1 ) +H +ω(ω k −1 ) − Φ H (ω1 +k −1)+Φ X (ω1 , ,ω k −1 )(15.133)15.6.3 Blind Equalization Based on Higher-Order CepstraIn this section, we consider blind equalization of a maximum-phase channel,based on higher order cepstra.

Assume that the channel can be modelled byan all-zero filter, and that its z-transfer function H(z) can be expressed as theproduct of a maximum-phase polynomial factor and a minimum-phasefactor asH ( z ) = G H min ( z ) H max ( z −1 ) z − D(15.134)P1H min ( z ) =∏ (1 − α i z −1 ), α i < 1(15.135)i =1P2H max ( z −1 ) =∏ (1 − β i z ), β i < 1(15.136)i =1where G is a gain factor, Hmin(z) is a minimum-phase polynomial with all itszeros inside the unit circle, Hmax(z–1) is a maximum-phase polynomial withall its zeros outside the unit circle, and z–D inserts D unit delays in order tomake Equation (15.134) causal. The complex cepstrum of H(z) is defined ashc (m) = Z −1 (ln H ( z ) )(15.137)where Z–1 denotes the inverse z-transform.

At z=ejω, the z-transform is thediscrete Fourier transform (DFT), and the cepstrum of a signal is obtainedby taking the inverse DFT of the logarithm of the signal spectrum. In thefollowing we consider cepstra based on the power spectrum and the higherorder spectra, and show that the higher-order cepstra have the ability toretain maximum-phase information. Assuming that the channel input x(m) is459Equalisation Based on Higher-Order Statisticsa zero-mean uncorrelated process with variance σ 2x , the power spectrum ofthe channel output can be expressed asσ x2PY (ω ) =H (ω ) H * (ω )2π(15.138)The cepstrum of the power spectrum of y(m) is defined asy c ( m) = IDFT (ln PY (ω ) )(()**= IDFT ln σ x2 G 2 / 2π + ln H min (ω ) + H max (−ω ) + ln H min(ω ) + H max(−ω ))(15.139)where IDFT is the inverse discrete Fourier transform.

SubstitutingEquations (15.135) and (15.36) in (15.139), the cepstrum can be expressedasln G 2σ x2 2π ,m=0y c (m) = − A ( m) + B ( m) m,m>0(15.140) ( − m)Am<0+ B ( − m ) m,((()))where A(m) and B(m) are defined asP1A( m) = ∑α im(15.141)B ( m) = ∑ β im(15.142)i =1P2i =1Note from Equation (15.140) that the along the index m, the maximumphase information B(m) and the minimum-phase information A(m) overlapand cannot be separated.Bi-CepstrumThe bi-cepstrum of a signal is defined as the inverse Fourier transform ofthe logarithm of the bi-spectrum:yc (m1 ,m2 ) = IDFT2 [log CY (ω1 ,ω 2 )](15.143)460Equalization and Deconvolutionwhere IDFT2[.] denotes the two-dimensional inverse discrete Fouriertransform.

The relationship between the bi-spectra of the input and output ofa linear system isCY (ω1 ,ω 2 ) = H (ω1 ) H (ω 2 ) H * (ω1 +ω 2 ) C X (ω1 ,ω 2 )(15.144)Assuming that the input x(m) of the linear time-invariant system {hk} is anuncorrelated non-Gaussian process, the bi-spectrum of the output can bewritten asCY (ω1 ,ω 2 ) =γ x(3) G 3( 2π ) 2H min (ω1 ) H max ( −ω1 ) H min (ω 2 ) H max ( −ω 2 )(15.145)**× H min(ω1 + ω 2 ) H max(−ω1 − ω 2 )where γ x (2π ) is the third-order cumulant of the uncorrelated randominput process x(m). Taking the logarithm of Equation (15.145) yields(3)2lnC y (ω1 , ω 2 ) = ln| A |+ln H min (ω1 )+ln H max (−ω1 )+ln H min (ω 2 )+ln H max (−ω 2 )**ln H min(ω1 + ω 2 )+ln H max(−ω1 − ω 2 )(15.146)(3) 32where A =γ x G (2π ) .

The bi-cepstrum is obtained through the inverseDiscrete Fourier transform of Equation (15.146) as ln A , ( m1 )m1 ,− A ( m2 )m2 ,− A− B ( − m1 ) m ,1y c (m1 , m2 ) = ( − m2 )m2 ,B ( m2 )m2 ,− B ( − m2 )m2 ,A 0,m1 = m2 = 0m1 > 0, m2 = 0m2 > 0, m1 = 0m1 < 0, m2 = 0m2 < 0, m1 = 0m1 = m2 >0m1 = m2 < 0otherwise(15.147)Equalisation Based on Higher-Order Statistics461Note from Equation (15.147) that the maximum-phase information B(m) andthe minimum-phase information A(m) are separated and appear in differentregions of the bi-cepstrum indices m1 and m2.The higher-order cepstral coefficients can be obtained either from theIDFT of higher-order spectra as in Equation (15.147) or using parametricmethods as follows.

In general, the cepstral and cumulant coefficients can berelated by a convolutional equation. Pan and Nikias (1988) have shown thatthe recursive relation between the bi-cepstrum coefficients and the thirdorder cumulants of a random process isy c (m1 , m2 )∗ [− m1 c y (m1 , m2 )]= − m1 c y (m1 , m2 )(15.148)Substituting Equation (15.147) in Equation (15.148) yields∞∑A(i )[c x (m1 − i, m2 ) − c x (m1 + i, m2 + i )]+ B (i ) [c x (m1 − i, m2 − i ) − c x (m1 + i, m2 )]i =1= − m1c x (m1 , m2 )(15.149)The truncation of the infinite summation in Equation (15.149) provides anapproximate equation asP∑ A(i) [c x (m1 − i, m2 ) − c x (m1 + i, m2 + i)]i =1Q+∑Bi =1(15.150)(i )[c x (m1 − i, m2 − i ) − c x (m1 + i, m2 )]≈ − m1c x (m1 , m2 )Equation (15.150) can be used to solve for the cepstral parameters A(m) andB(m).Tri-CepstrumThe tri-cepstrum of a signal y(m) is defined as the inverse Fourier transformof the tri-spectrum:y c (m1 ,m2 ,m3 ) = IDFT3 [ln CY (ω1 ,ω 2 ,ω 3 )](15.151)462Equalization and Deconvolutionwhere IDFT3[·] denotes the three-dimensional inverse discrete Fouriertransform.

The tri-spectra of the input and output of the linear system arerelated byCY (ω1 ,ω 2 ,ω 3 ) = H (ω1 ) H (ω 2 ) H (ω 3 ) H * (ω1 +ω 2 +ω 3 ) C X (ω1 ,ω 2 ,ω 3 )(15.152)Assuming that the channel input x(m) is uncorrelated, Equation (15.152)becomesγ x( 4) G 4CY (ω1 ,ω 2 ,ω 3 ) =(2π )3H (ω1 ) H (ω 2 ) H (ω 3 ) H * (ω1 +ω 2 +ω 3 )(15.153)where γ x (2π ) is the fourth-order cumulant of the input signal. Takingthe logarithm of the tri-spectrum gives( 4)lnCY (ω1 , ω 2 , ω 3 ) =3γ x( 4) G 4(2π ) 3+ln H min (ω1 )+ln H max ( −ω1 )+ln H min (ω 2 )+ln H max ( −ω 2 )**+ln H min (ω 3 )+ln H max ( −ω 3 ) + ln H min(ω1 + ω 2 + ω 3 )+ln H max( −ω1 − ω 2 − ω 3 )(15.154)From Equations (15.151) and (15.154), we have ln A, ( m1 )m1 ,− A ( m2 )m2 ,− A− A ( m3 ) m ,3m()− B 1 m ,1y c (m1 , m2 , m3 ) = ( − m2 )m2 ,B ( − m3 )m3 ,B ( m2 )m2 ,− B A ( m2 ) m ,2 0m1 = m2 = m3 = 0m1 > 0, m2 = m3 = 0m2 > 0, m1 = m3 = 0m3 > 0, m1 = m2 = 0m1 < 0, m2 = m3 = 0m2 < 0, m1 = m3 = 0m3 < 0, m1 = m2 = 0m1 = m2 = m3 > 0m1 = m2 = m3 < 0otherwise(15.155)463Equalisation Based on Higher-Order Statistics( 4) 43where A =γ x G (2π ) .

Note from Equation (15.155) that the maximumphase information B(m) and the minimum-phase information A(m) areseparated and appear in different regions of the tri-cepstrum indices m1, m2and m3.Calculation of Equalizer Coefficients from the Tri-cepstrumAssuming that the channel z-transfer function can be described by Equation(15.134), the inverse channel can be written asH inv ( z ) =11invinv( z ) H max( z −1 )== H min1−H ( z ) H min ( z ) H max ( z )(15.156)where it is assumed that the channel gain G is unity. In the time domainEquation (15.156) becomesinvinvh inv (m) = hmin(m)∗ hmax( m)(15.157)Pan and Nikias (1988) describe an iterative algorithm for estimation of thetruncated impulse response of the maximum-phase and the minimum-phaseinvinvfactors of the inverse channel transfer function.

Let hˆmin(i,m) , hˆmax (i, m)denote the estimates of the mth coefficients of the maximum-phase andminimum-phase parts of the inverse channel at the ith iteration . The Pan andNikias algorithm is the following:(a) Initialisationinvinvhˆmin(i,0) = hˆmax(i,0)=1(15.158)(b) Calculation of the minimum-phase polynomial1 m+1invinvhˆmin(i,m) = ∑ Aˆ ( k −1) hˆmin(i,m − k + 1)m k =2i=1, ..., P1 (15.159)(c) Calculation of the maximum-phase polynomial1 0 ˆ (1−k ) ˆ invinvhˆmax(i,m) =∑ B hmax (i,m − k + 1)m k =m+1i= –1, ..., –P2 (15.160)464Equalization and DeconvolutionThe maximum-phase and minimum-phase components of the inversechannel response are combined in Equation (15.157) to give the inversechannel equalizer.15.7 SummaryIn this chapter, we considered a number of different approaches to channelequalization.

The chapter began with an introduction to models for channeldistortions, the definition of an ideal channel equalizer, and the problemsthat arise in channel equalization due to noise and possible non-invertibilityof the channel. In some problems, such as speech recognition or restorationof distorted audio signals, we are mainly interested in restoring themagnitude spectrum of the signal, and phase restoration is not a primaryobjective.

In other applications, such as digital telecommunication therestoration of both the amplitude and the timing of the transmitted symbolsare of interest, and hence we need to equalise for both the magnitude and thephase distortions.In Section 15.1, we considered the least square error Wienerequalizer. The Wiener equalizer can only be used if we have access to thechannel input or the cross-correlation of the channel input and outputsignals.For cases where a training signal cannot be employed to identify thechannel response, the channel input is recovered through a blindequalization method.

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