Wiener Filters (779825), страница 3
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Using this relation, and the Fouriertransform property that convolution in time is equivalent to multiplicationin frequency, it is easy to show that the Wiener filter is given by Equation(6.43).6.6 Some Applications of Wiener FiltersIn this section, we consider some applications of the Wiener filter inreducing broadband additive noise, in time-alignment of signals in multichannel or multisensor systems, and in channel equalisation.193Some Applications of Wiener Filters2020 log W(f)0-20-40-60-80-100-60-40-200204060SNR (dB)Figure 6.4 Variation of the gain of Wiener filter frequency response with SNR.6.6.1 Wiener Filter for Additive Noise ReductionConsider a signal x(m) observed in a broadband additive noise n(m)., andmodel asy(m) = x(m) + n(m)(6.45)Assuming that the signal and the noise are uncorrelated, it follows that theautocorrelation matrix of the noisy signal is the sum of the autocorrelationmatrix of the signal x(m) and the noise n(m):Ryy = Rxx + Rnn(6.46)rxy = rxx(6.47)and we can also writewhere Ryy, Rxx and Rnn are the autocorrelation matrices of the noisy signal,the noise-free signal and the noise respectively, and rxy is the crosscorrelation vector of the noisy signal and the noise-free signal.
Substitutionof Equations (6.46) and (6.47) in the Wiener filter, Equation (6.10), yieldsw = ( R xx + Rnn )−1 r xx(6.48)Equation (6.48) is the optimal linear filter for the removal of additive noise.In the following, a study of the frequency response of the Wiener filterprovides useful insight into the operation of the Wiener filter. In thefrequency domain, the noisy signal Y(f) is given by194SignalNoiseWiener filter1.0Wiener filter magnitude W(f)Signal and noise magnitude spectrumWiener Filters0.0Frequency(f)Figure 6.5 Illustration of the variation of Wiener frequency response with signalspectrum for additive white noise. The Wiener filter response broadly follows thesignal spectrum.Y ( f ) = X ( f )+ N ( f )(6.49)where X(f) and N(f) are the signal and noise spectra.
For a signal observedin additive random noise, the frequency-domain Wiener filter is obtained asW( f ) =PXX ( f )PXX ( f ) + PNN ( f )(6.50)where PXX(f) and PNN(f) are the signal and noise power spectra. Dividingthe numerator and the denominator of Equation (6.50) by the noise powerspectra PNN(f) and substituting the variable SNR(f)=PXX(f)/PNN(f) yieldsW( f ) =SNR ( f )SNR ( f ) + 1(6.51)where SNR is a signal-to-noise ratio measure.
Note that the variable, SNR(f)is expressed in terms of the power-spectral ratio, and not in the more usualterms of log power ratio. Therefore SNR(f)=0 corresponds to − ∞ dB.From Equation (6.51), the following interpretation of the Wiener filterfrequency response W(f) in terms of the signal-to-noise ratio can be195Some Applications of Wiener FiltersMagnitudeSignalNoiseSeparable spectraMagnitude(a)Overlapped spectraFrequency(b)FrequencyFigure 6.6 Illustration of separability: (a) The signal and noise spectra do notoverlap, and the signal can be recovered by a low-pass filter; (b) the signal andnoise spectra overlap, and the noise can be reduced but not completely removed.deduced.
For additive noise, the Wiener filter frequency response is a realpositive number in the range 0 ≤ W ( f ) ≤ 1 . Now consider the two limitingcases of (a) a noise-free signal SNR ( f ) = ∞ and (b) an extremely noisysignal SNR(f)=0. At very high SNR, W ( f ) ≈1 , and the filter applies little orno attenuation to the noise-free frequency component.
At the other extreme,when SNR(f)=0, W(f)=0. Therefore, for additive noise, the Wiener filterattenuates each frequency component in proportion to an estimate of thesignal to noise ratio. Figure 6.4 shows the variation of the Wiener filterresponse W(f), with the signal-to-noise ratio SNR(f).An alternative illustration of the variations of the Wiener filterfrequency response with SNR(f) is shown in Figure 6.5. It illustrates thesimilarity between the Wiener filter frequency response and the signalspectrum for the case of an additive white noise disturbance. Note that at aspectral peak of the signal spectrum, where the SNR(f) is relatively high, theWiener filter frequency response is also high, and the filter applies littleattenuation.
At a signal trough, the signal-to-noise ratio is low, and so is theWiener filter response. Hence, for additive white noise, the Wiener filterresponse broadly follows the signal spectrum.6.6.2 Wiener Filter and the Separability of Signal and NoiseA signal is completely recoverable from noise if the spectra of the signaland the noise do not overlap. An example of a noisy signal with separablesignal and noise spectra is shown in Figure 6.6(a). In this case, the signal196Wiener Filtersand the noise occupy different parts of the frequency spectrum, and can beseparated with a low-pass, or a high-pass, filter.
Figure 6.6(b) illustrates amore common example of a signal and noise process with overlappingspectra. For this case, it is not possible to completely separate the signalfrom the noise. However, the effects of the noise can be reduced by using aWiener filter that attenuates each noisy signal frequency in proportion to anestimate of the signal-to-noise ratio as described by Equation (6.51).6.6.3 The Square-Root Wiener FilterIn the frequency domain, the Wiener filter output Xˆ ( f ) is the product of theinput frequency X(f) and the filter response W(f) as expressed in Equation(6.38). Taking the expectation of the squared magnitude of both sides ofEquation (6.38) yields the power spectrum of the filtered signal asE[| Xˆ ( f ) | 2 ] = W ( f ) 2 E[| Y ( f ) | 2 ]= W ( f ) 2 PYY ( f )(6.52)Substitution of W(f) from Equation (6.43) in Equation (6.52) yields2PXY(f)2ˆE[| X ( f ) | ] =PYY ( f )(6.53)Now, for a signal observed in an uncorrelated additive noise we havePYY ( f )= PXX ( f )+ PNN ( f )(6.54)PXY ( f )= PXX ( f )(6.55)andSubstitution of Equations (6.54) and (6.55) in Equation (6.53) yieldsE[| Xˆ ( f ) | 2 ] =2PXX(f)PXX ( f ) + PNN ( f )(6.56)Now, in Equation (6.38) if instead of the Wiener filter, the square root ofthe Wiener filter magnitude frequency response is used, the result is197Some Applications of Wiener FiltersXˆ ( f ) = W ( f ) 1 / 2 Y ( f )(6.57)and the power spectrum of the signal, filtered by the square-root Wienerfilter, is given byE [| Xˆ ( f ) | 2 ] = [W ( f ) 1/ 2 ] E[ Y ( f ) 2 ] =2PXY ( f )PYY ( f ) = PXY ( f ) (6.58)PYY ( f )Now, for uncorrelated signal and noise Equation (6.58) becomesE[| Xˆ ( f ) | 2 ] = PXX ( f )(6.59)Thus, for additive noise the power spectrum of the output of the square-rootWiener filter is the same as the power spectrum of the desired signal.6.6.4 Wiener Channel EqualiserCommunication channel distortions may be modelled by a combination of alinear filter and an additive random noise source as shown in Figure 6.7.The input/output signals of a linear time invariant channel can be modelledasP −1y ( m ) = ∑ hk x ( m − k ) + n ( m )(6.60)k =0where x(m) and y(m) are the transmitted and received signals, [hk] is theimpulse response of a linear filter model of the channel, and n(m) modelsthe channel noise.
In the frequency domain Equation (6.60) becomesY ( f )= X ( f )H ( f )+ N ( f )(6.61)where X(f), Y(f), H(f) and N(f) are the signal, noisy signal, channel and noisespectra respectively. To remove the channel distortions, the receiver isfollowed by an equaliser. The equaliser input is the distorted channeloutput, and the desired signal is the channel input. Using Equation (6.43) itis easy to show that the Wiener equaliser in the frequency domain is givenby198Wiener Filtersnoise n(m)x(m)Distortiony(m)H (f)Equaliser^x(m)H –1 (f)ffFigure 6.7 Illustration of a channel model followed by an equaliser.W ( f )=PXX ( f ) H * ( f )PXX ( f ) H ( f ) 2 + PNN ( f )(6.62)where it is assumed that the channel noise and the signal are uncorrelated.In the absence of channel noise, PNN(f)=0, and the Wiener filter is simplythe inverse of the channel filter model W(f)=H–1(f).
The equalisationproblem is treated in detail in Chapter 15.6.6.5 Time-Alignment of Signals in Multichannel/MultisensorSystemsIn multichannel/multisensor signal processing there are a number of noisyand distorted versions of a signal x(m), and the objective is to use all theobservations in estimating x(m), as illustrated in Figure 6.8, where the phaseand frequency characteristics of each channel is modelled by a linear filter.As a simple example, consider the problem of time-alignment of two noisyrecords of a signal given asy1 (m)= x(m)+n1 (m)y 2 ( m)= A x ( m − D )+ n 2 ( m)(6.63)(6.64)where y1(m) and y2(m) are the noisy observations from channels 1 and 2,n1(m) and n2(m) are uncorrelated noise in each channel, D is the time delayof arrival of the two signals, and A is an amplitude scaling factor.
Nowassume that y1(m) is used as the input to a Wiener filter and that, in theabsence of the signal x(m), y2(m) is used as the “desired” signal. The errorsignal is given by199Some Applications of Wiener Filtersn1(m)x(m)y1(m)h1(m)^x(m)w1(m)n2(m)x(m)h2(m)...x(m)^x(m)y2(m)w2(m)...nK(m)hK(m)yK(m)^x(m)wK(m)Figure 6.8 Illustration of a multichannel system where Wiener filters are used totime-align the signals from different channels.P −1e ( m ) = y 2 ( m ) − ∑ wk y1 ( m )k =0P −1 P −1= A x ( m − D ) − ∑ wk x ( m ) + ∑ wk n1 ( m ) + n2 ( m ) k =0k =0(6.65)The Wiener filter strives to minimise the terms shown inside the squarebrackets in Equation (6.65).
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