Wiener filter

In signal processing, the Wiener filter is a filter used to produce an estimate of a desired or target random process by linear time-invariant (LTI) filtering of an observed noisy process, assuming known stationary signal and noise spectra, and additive noise.

The goal of the wiener filter is to compute a statistical estimate of an unknown signal using a related signal as an input and filtering it to produce the estimate.

The Wiener filter is based on a statistical approach, and a more statistical account of the theory is given in the minimum mean square error (MMSE) estimator article.

Typical deterministic filters are designed for a desired frequency response.

However, the design of the Wiener filter takes a different approach.

One is assumed to have knowledge of the spectral properties of the original signal and the noise, and one seeks the linear time-invariant filter whose output would come as close to the original signal as possible.

Wiener filters are characterized by the following:[1] This filter is frequently used in the process of deconvolution; for this application, see Wiener deconvolution.

is known as smoothing (see Wiener filtering chapter of [1] for more details).

The Wiener filter problem has solutions for three possible cases: one where a noncausal filter is acceptable (requiring an infinite amount of both past and future data), the case where a causal filter is desired (using an infinite amount of past data), and the finite impulse response (FIR) case where only input data is used (i.e. the result or output is not fed back into the filter as in the IIR case).

The first case is simple to solve but is not suited for real-time applications.

Wiener's main accomplishment was solving the case where the causality requirement is in effect; Norman Levinson gave the FIR solution in an appendix of Wiener's book.

is optimal, then the minimum mean-square error equation reduces to and the solution

where This general formula is complicated and deserves a more detailed explanation.

in a specific case, one should follow these steps:[2] The causal finite impulse response (FIR) Wiener filter, instead of using some given data matrix X and output vector Y, finds optimal tap weights by using the statistics of the input and output signals.

In order to derive the coefficients of the Wiener filter, consider the signal w[n] being fed to a Wiener filter of order (number of past taps) N and with coefficients

The Wiener filter is designed so as to minimize the mean square error (MMSE criteria) which can be stated concisely as follows: where

For simplicity, the following considers only the case where all these quantities are real.

The mean square error (MSE) may be rewritten as: To find the vector

which minimizes the expression above, calculate its derivative with respect to each

known respectively as the autocorrelation of w[n] and the cross-correlation between w[n] and s[n] can be defined as follows: The derivative of the MSE may therefore be rewritten as: Note that for real

, these matrices are known to be positive definite and therefore non-singular yielding a unique solution to the determination of the Wiener filter coefficient vector,

In some articles, the cross correlation function is defined in the opposite way:

The realization of the causal Wiener filter looks a lot like the solution to the least squares estimate, except in the signal processing domain.

is The FIR Wiener filter is related to the least mean squares filter, but minimizing the error criterion of the latter does not rely on cross-correlations or auto-correlations.

This involves computing partial derivatives with respect to both the real and imaginary parts of

The resulting Wiener-Hopf equations are: which can be rewritten in matrix form: Note here that:

The Wiener filter has a variety of applications in signal processing, image processing,[3] control systems, and digital communications.

These applications generally fall into one of four main categories: For example, the Wiener filter can be used in image processing to remove noise from a picture.

The filter was proposed by Norbert Wiener during the 1940s and published in 1949.

[4][5] The discrete-time equivalent of Wiener's work was derived independently by Andrey Kolmogorov and published in 1941.