Wienerfilter

The Wiener filter is a linear filter designed to estimate a desired signal from a noisy observation by minimizing the mean squared error between the estimated and true signals. It is grounded in the statistical properties of the signal and the noise and seeks the filter response that yields the best linear unbiased estimate under those statistics.

In the common model, the observed signal y(t) is the sum of the desired signal x(t) and additive noise n(t). If x and n are wide-sense stationary and uncorrelated, the optimal linear filter in the frequency domain has a transfer function H(f) = S_xx(f) / [S_xx(f) + S_nn(f)], where S_xx and S_nn are the power spectral densities of the signal and noise, respectively. Equivalently, in the time domain the impulse response is found by solving Wiener-Hopf equations. The filter thus balances passing the signal spectrum with suppressing noise, depending on how much signal and noise energy there is at each frequency.

Assumptions include linear time-invariance and knowledge (or good estimates) of second-order statistics. In practice, when statistics are unknown or non-stationary, adaptive versions are used, updating estimates over time (for example via LMS or RLS algorithms). Applications span noise reduction in audio and speech, image and video restoration, and channel equalization in communications, as well as other domains requiring estimation from noisy measurements.

Limitations include sensitivity to incorrect statistics, potential over-smoothing of details, and computational demands for real-time or high-dimensional data. The Wiener filter remains a foundational tool in statistical signal processing, named after Norbert Wiener.