Channel Equalization and Blind Deconvolution (779799), страница 5
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Thus a channel estimatehˆw is based on the hypothesis that the input word is w. It is expected that abetter channel estimate is obtained from the correctly hypothesised HMM,and a poorer estimate from an incorrectly hypothesised HMM. Thehypothesised-input HMM algorithm is as follows (Figure 15.9):For i =1 to number of words V {step 1 Using each HMM, Mi, make an estimate of the channel, hˆi ,step 2 Using the channel estimate, ĥi , estimate the channel inputxˆ (m)= y (m) − hˆistep 3 Compute a probability score for model Mi, given the estimate[ xˆ ( m)].
}Select the channel estimate associated with the most probable word.Figure 15.10 shows the ML channel estimates of two channels usingunweighted average and hypothesised-input methods.444Equalization and Deconvolutiony=x+hChannelestimate/Mi^hi+^^xi =y–hProbabilityscorefor M iP(M i|y)iFigure 15.9 Hypothesised channel estimation procedure.0.0Frequency kHz. 8.0Actual channel distortionChannel estimate using statistical averages over all modelsdBChannel estimate using the statistics of the most likely modelsActual channel distortiondBChannel estimate using statistical averages over all models.Figure 15.10 Illustration of actual and estimated channel response fortwo channels.Channel estimate using the statistics of the most likely modelsMethod III: Decision-Directed EqualizationBlind adaptive equalizers are often composed of two distinct sections: anadaptive linear equalizer followed by a non-linear estimator to improve theequalizer output.
The output of the non-linear estimator is the final estimateof the channel input, and is used as the desired signal to direct the equalizeradaptation. The use of the output of the non-linear estimator as the desiredsignal assumes that the linear equalization filter removes a large part of thechannel distortion, thereby enabling the non-linear estimator to produce anaccurate estimate of the channel input. A method of ensuring that theequalizer locks into, and cancels a large part of the channel distortion is touse a startup, equalizer training period during which a known signal istransmitted.445Equalization Based on Linear Prediction Modelsy(m)Equalisation filterhinvz(m)=x(m)+v(m)HMMclassifier/estimator^x(m)–++LMS adaptationalgorithme(m) error signalFigure 15.11 A decision-directed equalizer.Figure 15.11 illustrates a blind equalizer incorporating an adaptive linearfilter followed by a hidden Markov model classifier/estimator.
The HMMclassifies the output of the filter as one of a number of likely signals andprovides an enhanced output, which is also used for adaptation of the linearfilter. The output of the equalizer z(m) is expressed as the sum of the inputto the channel x(m) and a so-called convolutional noise term v(m) asz ( m ) = x ( m ) +v ( m )(15.82)The HMM may incorporate state-based Wiener filters for suppression of theconvolutional noise v(m) as described in Section 5.5. Assuming that theLMS adaptation method is employed, the adaptation of the equalizercoefficient vector is governed by the following recursive equation:hˆ inv (m) = hˆ inv (m − 1) + µ e(m) y (m)(15.83)invwhere hˆ (m) is an estimate of the optimal inverse channel filter, µ is anadaptation step size and the error signal e(m) is defined ase(m) = xˆ HMM (m) − z (m)(15.84)HMM(m) is the output of the HMM-based estimator and is used aswhere xˆthe correct estimate of the desired signal to direct the adaptation process.446Equalization and Deconvolution15.5 Blind Equalization for Digital Communication ChannelsHigh speed transmission of digital data over analog channels, such astelephone lines or a radio channels, requires adaptive equalization to reducedecoding errors caused by channel distortions.
In telephone lines, thechannel distortions are due to the non-ideal magnitude response and thenonlinear phase response of the lines. In radio channel environments, thedistortions are due to non-ideal channel response as well as the effects ofmultipath propagation of the radio waves via a multitude of different routeswith different attenuations and delays. In general, the main types ofdistortions suffered by transmitted symbols are amplitude distortion, timedispersion and fading. Of these, time dispersion is perhaps the mostimportant, and has received a great deal of attention. Time dispersion hasthe effect of smearing and elongating the duration of each symbol.
In highspeed communication systems, where the data symbols closely follow eachother, time dispersion results in an overlap of successive symbols, an effectknown as intersymbol interference (ISI), illustrated in Figure 15.12.In a digital communication system, the transmitter modem takes N bitsof binary data at a time, and encodes them into one of 2N analog symbols fortransmission, at the signalling rate, over an analog channel. At the receiverthe analog signal is sampled and decoded into the required digital format.Most digital modems are based on multilevel phase-shift keying, orcombined amplitude and phase shift keying schemes. In this section weconsider multi-level pulse amplitude modulation (M-ary PAM) as aconvenient scheme for the study of adaptive channel equalization.Assume that at the transmitter modem, the kth set of N binary digits ismapped into a pulse of duration Ts seconds and an amplitude a(k).
Thus themodulator output signal, which is the input to the communication channel,is given asx(t ) = ∑ a (k )r (t − kTs )(15.85)kTransmitted waveform11010Received waveform1timetimeFigure 15.12 Illustration of intersymbol interference in a binary pulse amplitudemodulation system.Blind Equalization for Digital Communication Channels447where r(t) is a pulse of duration Ts and with an amplitude a(k) that canassume one of M=2N distinct levels. Assuming that the channel is linear, thechannel output can be modelled as the convolution of the input signal andchannel response:∞y (t ) = ∫ h(τ ) x(t − τ )dτ(15.86)−∞where h(t) is the channel impulse response. The sampled version of thechannel output is given by the following discrete-time equation:y (m) = ∑ hk x(m − k )(15.87)kTo remove the channel distortion, the sampled channel output y(m) is passedinvto an equalizer with an impulse response hˆk .
The equalizer output z(m) isgiven asz (m) = ∑ hˆkinv y (m − k )k= ∑ x(m − j )∑ hˆkinv h j − kj(15.88)kwhere Equation (15.87) is used to obtain the second line of Equation(15.88). The ideal equalizer output is z(m)=x(m–D)=a(m–D) for some delayD that depends on the channel response and the length of the equalizer.From Equation (15.88), the channel distortion would be cancelled ifhmc =hm ∗hˆminv =δ (m − D)(15.89)where hmc is the combined impulse response of the cascade of the channeland the equalizer. A particular form of channel equalizer, for the eliminationof ISI, is the Nyquist zero-forcing filter, where the impulse response of thecombined channel and equalizer is defined as1, k = 0h c (kTs + D) = 0, k ≠ 0(15.90)448Equalization and DeconvolutionNote that in Equation (15.90), at the sampling instants the channel distortionis cancelled, and hence there is no ISI at the sampling instants. A functionthat satisfies Equation (15.90) is the sinc function hc(t)=sin(πfst)/πfst, wherefs=1/Ts.
Zero-forcing methods are sensitive to deviations of hc(t) from therequirement of Equation (15.90), and also to jitters in the synchronisationand the sampling process.15.5.1 LMS Blind EqualizationIn this section, we consider the more general form of the LMS-basedadaptive equalizer followed by a nonlinear estimator. In a conventionalsample-adaptive filter, the filter coefficients are adjusted to minimise themean squared distance between the filter output and the desired signal. Inblind equalization, the desired signal (which is the channel input) is notavailable. The use of an adaptive filter for blind equalization, requires aninternally generated desired signal as illustrated in Figure 15.13.
Digitalblind equalizers are composed of two distinct sections: an adaptive equalizerthat removes a large part of the channel distortion, followed by a non-linearestimator for an improved estimate of the channel input. The output of thenon-linear estimator is the final estimate of the channel input, and is used asthe desired signal to direct the equalizer adaptation. A method of ensuringthat the equalizer removes a large part of the channel distortion is to use astart-up, equalizer training, period during which a known signal istransmitted.Assuming that the LMS adaptation method is employed, the adaptationof the equalizer coefficient vector is governed by the following recursiveequation:hˆ inv (m) = hˆ inv (m − 1) + µe(m) y (m)(15.91)invinvwhere hˆ (m) is an estimate of the optimal inverse channel filter h , thescalar µ is the adaptation step size, and the error signal e(m) is defined ase(m) = ψ ( z(m)) − z(m)= xˆ (m ) − z(m)(15.92)where xˆ (m) = ψ ( z(m) ) is a non-linear estimate of the channel input.
Forexample, in a binary communication system with an input alphabet {±a} we449Blind Equalization for Digital Communication Channelsy(m)Equalisation filterhinvz(m)=x(m)+v(m)Decision device^x(m)M-level quantiser–++LMS adaptationalgorithme(m) error signalFigure 15.13 Configuration of an adaptive channel equalizer with an estimate ofthe channel input used as an “internally” generated desired signalcan use a signum non-linearity such that xˆ (m) = a. sgn( z(m)) where thefunction sgn(·) gives the sign of the argument. In the following, we use aBayesian framework to formulate the nonlinear estimator ψ().Assuming that the channel input is an uncorrelated process and theequalizer removes a large part of the channel distortion, the equalizer outputcan be expressed as the sum of the desired signal (the channel input) plus anuncorrelated additive noise term:z( m) = x (m) + v( m)(15.93)where v(m) is the so-called convolutional noise defined asv(m) = x(m) − ∑ hˆkinv y (m − k )=∑k(hkinv − hˆkinv ) y (m − k )(15.94)kIn the following, we assume that the non-linear estimates of the channelinput are correct, and hence the error signals e(m) and v(m) are identical.Owing to the averaging effect of the channel and the equalizer, each sampleof convolutional noise is affected by many samples of the input process.From the central limit theorem, the convolutional noise e(m) can bemodelled by a zero-mean Gaussian process as450Equalization and Deconvolution e 2 ( m) 1f E (e(m) ) =exp −2 2π σ eσ2e (15.95)where σ e2 , the noise variance, can be estimated using the recursive timeupdate equationσ e2 (m) = ρσ e2 (m − 1) + (1 − ρ )e 2 (m)(15.96)where ρ < 1 is the adaptation factor.
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