基于winner 滤波平稳降噪效果-白红宇

基于winner 滤波平稳降噪效果

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发布时间：2019-06-19

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

Wiener filter solutions

The Wiener filter problem has solutions for three possible cases: one where a noncausal filter is acceptable (requiring an infinite amount of both past and future data), the case where a filter is desired (using an infinite amount of past data), and the (FIR) case where a finite amount of past data is used. The first c

ase is simple to solve but is not suited for real-time applications. Wiener's main accomplishment was solving the case where the causality requirement is in effect, and in an appendix of Wiener's book gave the FIR solution.

Noncausal solution

G(s) = \frac{S_{x,s}(s)}{S_x(s)}e^{\alpha s}.

Where $S$ are spectra. Provided that $g(t)$ is optimal, then the equation reduces to

E(e^2) = R_s(0) - \int_{-\infty}^{\infty}{g(\tau)R_{x,s}(\tau + \alpha)\,d\tau},

and the solution $g(t)$ is the inverse two-sided of $G(s)$ .

Causal solution

G(s) = \frac{H(s)}{S_x^{+}(s)},

where

$H(s)$ consists of the causal part of $\frac{S_{x,s}(s)}{S_x^{-}(s)}e^{\alpha s}$ (that is, that part of this fraction having a positive time solution under the inverse Laplace transform)

$S_x^{+}(s)$ is the causal component of $S_x(s)$ (i.e., the inverse Laplace transform of $S_x^{+}(s)$ is non-zero only for $t \,\ge\, 0$ )

$S_x^{-}(s)$ is the anti-causal component of $S_x(s)$ (i.e., the inverse Laplace transform of $S_x^{-}(s)$ is non-zero only for $t < 0$ )

This general formula is complicated and deserves a more detailed explanation. To write down the solution $G(s)$ in a specific case, one should follow these steps:

Start with the spectrum $S_x(s)$ in rational form and factor it into causal and anti-causal components:

$S_x(s) = S_x^{+}(s) S_x^{-}(s)$

where $S^{+}$ contains all the zeros and poles in the left half plane (LHP) and $S^{-}$ contains the zeroes and poles in the right half plane (RHP). This is called the .

Divide $S_{x,s}(s)e^{\alpha s}$ by $S_x^{-}(s)$ and write out the result as a partial fraction expansion.

Select only those terms in this expansion having poles in the LHP. Call these terms $H(s)$ .

Divide $H(s)$ by $S_x^{+}(s)$ . The result is the desired filter transfer function $G(s)$ .

原始文件，环境噪音已经很弱了

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