Wiener Filters (Vaseghi - Advanced Digital Signal Processing and Noise Reduction), страница 4

Using the Wiener filter Equation (6.10), wehavew = R −y11y1 r y1 y 2(6.66)= (R xx + Rn 1 n 1 ) −1 Arxx ( D )where rxx(D)=E [x(PD)x(m)]. The frequency-domain equivalent ofEquation (6.65) can be derived asW( f )=PXX ( f )Ae − jωDPXX ( f ) + PN1 N1 ( f )(6.67)Note that in the absence of noise, the Wiener filter becomes a pure phase (ora pure delay) filter with a flat magnitude response.200Wiener FiltersX(f)=Y(f) – N(f)Noisy signalNoisy signalspectrum estimatorW( f ) =Y(f)SilenceDetectorX(f)Y(f)WienercoefficientvectorNoise spectrumestimatorFigure 6.9 Configuration of a system for estimation of frequency Wiener filter.6.6.6 Implementation of Wiener FiltersThe implementation of a Wiener filter for additive noise reduction, usingEquations (6.48)–(6.50), requires the autocorrelation functions, orequivalently the power spectra, of the signal and noise. The noise powerspectrum can be obtained from the signal-inactive, noise-only, periods.

Theassumption is that the noise is quasi-stationary, and that its power spectraremains relatively stationary between the update periods. This is areasonable assumption for many noisy environments such as the noiseinside a car emanating from the engine, aircraft noise, office noise fromcomputer machines, etc. The main practical problem in the implementationof a Wiener filter is that the desired signal is often observed in noise, andthat the autocorrelation or power spectra of the desired signal are not readilyavailable. Figure 6.9 illustrates the block-diagram configuration of a systemfor implementation of a Wiener filter for additive noise reduction.

Anestimate of the desired signal power spectra is obtained by subtracting anestimate of the noise spectra from that of the noisy signal. A filter bankimplementation of the Wiener filter is shown in Figure 6.10, where theincoming signal is divided into N bands of frequencies. A first-orderintegrator, placed at the output of each band-pass filter, gives an estimate ofthe power spectra of the noisy signal.

The power spectrum of the originalsignal is obtained by subtracting an estimate of the noise power spectrumfrom the noisy signal. In a Bayesian implementation of the Wiener filter,prior models of speech and noise, such as hidden Markov models, are usedto obtain the power spectra of speech and noise required for calculation ofthe filter coefficients.201The Choice of Wiener Filter OrderY(f1 )BPF(f1 ).2Y2 (f1 )y(m)ρ..W(f1 ) =X2 (f1 )Y2 (f1 )^x(m)N2 (f1 )Z...X2 (f1 )–1...Y(fN)BPF(fN).2W(fN) =X2 (fN)Y2 (fN)Y2 (fN)ρX2 (fN)..N2 (fN)Z–1Figure 6.10 A filter-bank implementation of a Wiener filter.6.7 The Choice of Wiener Filter OrderThe choice of Wiener filter order affects:(a) the ability of the filter to remove distortions and reduce the noise;(b) the computational complexity of the filter; and(c) the numerical stability of the of the Wiener solution, Equation(6.10).The choice of the filter length also depends on the application and themethod of implementation of the Wiener filter.

For example, in a filter-bankimplementation of the Wiener filter for additive noise reduction, the numberof filter coefficients is equal to the number of filter banks, and typically the202Wiener Filtersnumber of filter banks is between 16 to 64. On the other hand for manyapplications, a direct implementation of the time-domain Wiener filterrequires a larger filter length say between 64 and 256 taps.A reduction in the required length of a time-domain Wiener filter canbe achieved by dividing the time domain signal into N sub-band signals.Each sub-band signal can then be decimated by a factor of N. Thedecimation results in a reduction, by a factor of N, in the required length ofeach sub-band Wiener filter.

In Chapter 14, a subband echo canceller isdescribed.6.8 SummaryA Wiener filter is formulated to map an input signal to an output that is asclose to a desired signal as possible. This chapter began with the derivationof the least square error Wiener filter. In Section 6.2, we derived the blockdata least square error Wiener filter for applications where only finitelength realisations of the input and the desired signals are available. In suchcases, the filter is obtained by minimising a time-averaged squared errorfunction. In Section 6.3, we considered a vector space interpretation of theWiener filters as the perpendicular projection of the desired signal onto thespace of the input signal.In Section 6.4, the least mean square error signal was analysed.

Themean square error is zero only if the input signal is related to the desiredsignal through a linear and invertible filter. For most cases, owing to noiseand/or nonlinear distortions of the input signal, the minimum mean squareerror would be non-zero. In Section 6.5, we derived the Wiener filter in thefrequency domain, and considered the issue of separability of signal andnoise using a linear filter.

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