Wienersuodatus

Wienersuodatus, known in English as Wiener filtering, is a signal processing technique used to estimate a desired signal from a noisy observation by applying a linear time-invariant filter that minimizes the mean squared error between the estimate and the true signal. Named after Norbert Wiener, the method relies on assumed known statistics of the signal and the noise and yields an optimal linear estimator under the minimum mean square error criterion.

In the frequency domain, the Wiener filter H(ω) is defined as the ratio of the cross-power spectrum between the desired signal and the observed signal to the power spectrum of the observed signal: H(ω) = S_sd(ω)/S_xx(ω). In the time domain, the filter impulse response h[k] is obtained from the Wiener–Hopf equations: R_xx h = r_xd, where R_xx is the autocorrelation of the observed signal and r_xd is the cross-correlation between the observed signal and the desired signal. The filter can be causal if only past and present inputs are used.

In practice, Wiener filtering is often implemented in adaptive forms when statistics are unknown or non-stationary. Adaptive Wiener filters use algorithms such as least-mean-square (LMS) or recursive-least-squares (RLS) to update the filter coefficients in real time as signal properties change.

Applications of Wiener filtering span audio and speech denoising, image deblurring, wireless communications, radar and sonar signal processing, and preprocessing for deconvolution or compression. Its main advantage is optimality under the MMSE criterion for stationary signals, while limitations include dependence on accurate statistical models and diminished performance if those statistics are incorrect or if the signal is highly non-stationary.