THE WIENER FILTER — a dart

THE WIENER FILTER the optimal blend of signal and noise — recover the clean part in the mean-square sense

Given a signal buried in noise, what is the best linear way to recover the clean part? Norbert Wiener's answer: weight each component by how much of it is signal versus noise. Where the signal-to-noise ratio is high, pass it through; where noise dominates, suppress it — the optimal gain is S/(S+N) in terms of the power spectra. It is the mathematically optimal denoiser under the mean-square criterion, the ancestor of every deconvolution and spectral-subtraction scheme. Reconstruction of the clean signal is caused by knowing the statistics: the structure — the relative power of signal and noise — is what tells the filter how hard to trust each frequency.

THE TECHNIQUE optimal gain S/(S+N); pass signal, suppress noise

An observed value that is signal plus noise, with known variances. The demo computes the Wiener gain and reconstructs the clean estimate: live demo

HISTORY & CREDIT Wiener 1949 · Kolmogorov 1941

“Denoising is a matter of tuning knobs.” — Wiener made it optimal: the gain S/(S+N) minimizes the mean-square error exactly, given the statistics. The best linear recovery, not a heuristic. cited

1941 · Andrey Kolmogorov — the discrete prediction/filtering theory.
1949 · Norbert Wiener — the continuous optimal filter (the Wiener–Kolmogorov theory).
now · image deconvolution, audio denoising, spectral subtraction, the LMMSE estimator.

Weight each part by its signal-to-noise ratio and the mean-square error is minimized. The clean signal recovered because the statistics say how much of it is really there. Wiener-Kolmogorov

RECOMMEND FOR I-13 optimal denoised estimate, computed

On the canonical compiler, signal power 4 and noise power 1 give Wiener gain 0.8; an observed 5 reconstructs to a clean estimate 4:

$ i13 run rec_wiener.i13 # gain S/(S+N), estimate = gain * observed w = 0.8 -- optimal gain: 80% signal here estimate = 4 -- the denoised reconstruction (mean-square optimal)

Recommend: the Wiener filter is reconstruction of the clean signal, weighted by structure, and i13 computes its gain and estimate: with signal power 4 and noise power 1, the optimal gain is 0.8 and the observed 5 denoises to 4. The load-bearing structure is the known statistics: the gain S/(S+N) is derived to minimize mean-square error, so the recovery is optimal, not tuned. It is the frequency-domain sibling of the Kalman filter (dart 306) — where Kalman fuses over time, Wiener fuses over spectrum — and the batch's statement that even noisy recovery is caused by a structure: here, how much of each component is truly signal.