Smooth noise reduction effect based on winner filter

Https://en.wikipedia.org/wiki/Wiener_filter

Wiener Filter Solutions

The Wiener filter problem have solutions for three possible cases:one where a noncausal filter was acceptable (requiring an Infinite amount of both past and future data), the case where a causal filter was desired (using an infinite amount of pas T data), and the finite impulse response (FIR) case where a finite amount of past data is used. The first C

ASE is simple to solve, but isn't suited for real-time applications. Wiener ' s main accomplishment was solving the case where the causality requirement are in effect, and in an appendix of Wien Er ' s book Levinson gave the FIR solution.

Noncausal Solution

Where is spectra. Provided that's optimal, then the minimum mean-square error equation reduces to

And the solution is the inverse two-sided Laplace transform of.

Causal Solution

where

• Consists of the causal part of (so is, which part of this fraction has a positive time solution under the inverse Lap Lace transform)
• is the causal component of (i.e., the inverse Laplace transform of are non-zero only for)
• is the anti-causal component of (i.e., the inverse Laplace transform of are non-zero only for)

This general formula are complicated and deserves a more detailed explanation. To write down the solution in a specific case, one should follow these steps:^[2]

1. Start with the spectrum in rational form and factor it to causal and anti-causal components:
   where contains All the zeros and Poles in the left half plane (LHP) and contains the zeroes and Poles in the right half plane (RHP). This is called the Wiener–hopf factorization.
2. Divide by and write out the result as a partial fraction expansion.
3. Select only those terms in this expansion has poles in the LHP. Call these terms.
4. Divide by. The result is the desired filter transfer function.

The original file, the environment noise is already very weak

Post-frame real-time wiener filtering

Smooth noise reduction effect based on winner filter