23-08-2012, 12:11 PM
WIENER FILTERS
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Formulated by Norbert Wiener, forms the foundation of data-dependent linear least square error filters. Wiener filters play a central role in a wide range of applications such as linear prediction, echo cancellation, signal restoration, channel equalisation and system identification. The coefficients of a Wiener filter are calculated to minimise the average squared distance between the filter output and a desired signal. In its basic form, the Wiener theory assumes that the signals are stationary processes. However, if the filter coefficients are periodically recalculated for every block of N signal samples then the filter adapts itself to the average characteristics of the signals within the blocks and becomes block-adaptive. A block-adaptive (or segment adaptive) filter can be used for signals such as speech and image that may be considered almost stationary over a relatively small block of samples. In this chapter, we study Wiener filter theory, and consider alternative methods of formulation of the Wiener filter problem. We consider the application of Wiener filters in channel equalisation, time-delay estimation and additive noise reduction. A case study of the frequency response of a Wiener filter, for additive noise reduction, provides useful insight into the operation of the filter. We also deal with some implementation issues of Wiener filters.
Wiener Filters: Least Square Error Estimation
Wiener formulated the continuous-time, least mean square error, estimation
problem in his classic work on interpolation, extrapolation and smoothing
of time series (Wiener 1949). The extension of the Wiener theory from
continuous time to discrete time is simple, and of more practical use for
implementation on digital signal processors. A Wiener filter can be an
infinite-duration impulse response (IIR) filter or a finite-duration impulse
response (FIR) filter. In general, the formulation of an IIR Wiener filter
results in a set of non-linear equations, whereas the formulation of an FIR
Wiener filter results in a set of linear equations and has a closed-form
solution. In this chapter, we consider FIR Wiener filters, since they are
relatively simple to compute, inherently stable and more practical. The main
drawback of FIR filters compared with IIR filters is that they may need a
large number of coefficients to approximate a desired response.
Interpretation of Wiener Filters as Projection in Vector Space
In this section, we consider an alternative formulation of Wiener filters
where the least square error estimate is visualized as the perpendicular
minimum distance projection of the desired signal vector onto the vector
space of the input signal. A vector space is the collection of an infinite
number of vectors that can be obtained from linear combinations of a
number of independent vectors.
In order to develop a vector space interpretation of the least square
error estimation problem, we rewrite the matrix Equation (6.11) and express
the filter output vector xˆ as a linear weighted combination of the column
vectors of the input signal matrix as
Implementation of Wiener Filters
The implementation of a Wiener filter for additive noise reduction, using
Equations (6.48)–(6.50), requires the autocorrelation functions, or
equivalently the power spectra, of the signal and noise. The noise power
spectrum can be obtained from the signal-inactive, noise-only, periods. The
assumption is that the noise is quasi-stationary, and that its power spectra
remains relatively stationary between the update periods. This is a
reasonable assumption for many noisy environments such as the noise
inside a car emanating from the engine, aircraft noise, office noise from
computer machines, etc. The main practical problem in the implementation
of a Wiener filter is that the desired signal is often observed in noise, and
that the autocorrelation or power spectra of the desired signal are not readily
available. Figure 6.9 illustrates the block-diagram configuration of a system
for implementation of a Wiener filter for additive noise reduction.