Wiener Filter

CORE DEFINITION

The Wiener filter is a linear filtering technique proposed by American mathematician Norbert Wiener in the 1940s and first formally published in 1949. Its design goal is to recover the original signal from noise-corrupted observations under the criterion of minimum mean square error. The core assumptions of this filter include that both the signal and noise are stationary random processes and that their statistical properties (such as correlation functions and power spectra) are known. Based on these assumptions, the Wiener filter uses statistical optimization to minimize the average squared difference between the filtered signal and the true signal. Unlike ideal pass-stop filters (such as low-pass, high-pass, band-pass, or band-stop), the Wiener filter dynamically adjusts the degree of retention or suppression for each frequency band based on the statistical characteristics of the signal and noise. It has two forms: Non-causal version: can utilize future information to achieve optimal denoising, but is not suitable for real-time systems. Causal version: uses only current and past data to meet real-time processing needs, but is more complex to derive.