Wienerfiltering

Wiener filtering is a statistical estimation technique used to estimate a desired signal from a noisy observation by applying a linear filter. It was proposed by Norbert Wiener in 1942 and is optimal in the mean squared error sense for jointly stationary processes with known second-order statistics. It is widely used for denoising, deconvolution, and restoration in signal processing and imaging.

In the standard model, the observed signal is y(t) = s(t) + n(t), where s is the target signal and n is additive noise uncorrelated with s. A linear time-invariant filter with impulse response h(t) produces the estimate û(t) = (h * y)(t). In the frequency domain, the Wiener filter has frequency response H(ω) = S_ss(ω) / [S_ss(ω) + S_nn(ω)], where S_ss and S_nn are the power spectral densities of s and n. Equivalently, H(f) = Φ_sy(f) / Φ_yy(f). The filter minimizes the mean squared error between û and s; note that the Wiener solution is generally noncausal, though causal variants exist for real-time applications.

The approach relies on stationarity and the availability or accurate estimation of second-order statistics. If spectra are unknown, they must be estimated from data or approximated. Adaptations include local or adaptive Wiener filters to handle non-stationary signals. In practice, Wiener filtering is implemented via FFT-based frequency-domain processing or as spatially varying filters in images, where the kernel depends on local signal and noise statistics.

Applications span audio and image denoising, deblurring, communications, and sensor data processing, making Wiener filtering a foundational tool in practical signal restoration.