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Concept
Wiener filter
Summary
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 Wiener filter minimizes the mean square error between the estimated random process and the desired process.
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 that known signal to produce the estimate as an output. For example, the known signal might consist of an unknown signal of interest that has been corrupted by additive noise. The Wiener filter can be used to filter out the noise from the corrupted signal to provide an estimate of the underlying signal of interest. 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:
Assumption: signal and (additive) noise are stationary linear stochastic processes with known spectral characteristics or known autocorrelation and cross-correlation
Requirement: the filter must be physically realizable/causal (this requirement can be dropped, resulting in a non-causal solution)
Performance criterion: minimum mean-square error (MMSE)
This filter is frequently used in the process of deconvolution; for this application, see Wiener deconvolution.
Let be an unknown signal which must be estimated from a measurement signal . Where alpha is a tunable parameter. is known as prediction, is known as filtering, and is known as smoothing (see Wiener filtering chapter of for more details).
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Estimation theory
Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements.
Linear prediction
Linear prediction is a mathematical operation where future values of a discrete-time signal are estimated as a linear function of previous samples. In digital signal processing, linear prediction is often called linear predictive coding (LPC) and can thus be viewed as a subset of filter theory. In system analysis, a subfield of mathematics, linear prediction can be viewed as a part of mathematical modelling or optimization. The most common representation is where is the predicted signal value, the previous observed values, with , and the predictor coefficients.
Noise reduction
Noise reduction is the process of removing noise from a signal. Noise reduction techniques exist for audio and images. Noise reduction algorithms may distort the signal to some degree. Noise rejection is the ability of a circuit to isolate an undesired signal component from the desired signal component, as with common-mode rejection ratio. All signal processing devices, both analog and digital, have traits that make them susceptible to noise.