1. EE513 Audio Signals and Systems
Wiener Inverse Filter
Kevin D. Donohue Electrical and Computer Engineering University of Kentucky

2. Weiner Filters
A class of filters, referred to as Wiener filters, exploit correlation information between signal and noise to enhance SNR or reduce distortion. The Wiener filter is the optimal filter for enhancing SNR of a random signal in random noise. The signals and noise are characterized by their PSDs or Acs, and objective metrics are either SNR enhancement or minimization of least-square error. These filters are named after Norbert Weiner:

3. Wiener Filters and Noise
Let s[n] be the original signal and y[n] be the corrupted version. The error signal or noise is given by:
Minimizing the error in the L2 or mean square error (MSE) sense means minimizing the expected value of:
This is equivalent to maximizing the SNR:
Signal Power
Noise Power

4. Wiener Filter Objective
Let w[n] be the filter to maximize SNR or equivalently to minimize the MSE:
Express MSE in terms of the above equation:
where  is a delay parameter to relax a causality constraint and typically improve performance. The first equation can be express more directly in the frequency domain:

5. Wiener Filter and SNR
Assuming the signal and noise are uncorrelated, zero-mean stationary processes, it can be shown that the optimal filter for minimizing MSE is:
Can also be rewritten as:
PSD of Noise
PSD of Signal

6. FIR Inverse (Wiener) Filters
An inverse filter undoes distortions due to frequency selective channels/systems and restore the original transmitted/driving signal. This type of filtering is sometimes referred to as deconvolution. Let h(n) denote the impulse response of the channel/system. The inverse filter, hI(n), is described by:

7. FIR Inverse Filters - Polynomial Division
Assume that for practical purposes the channel/system can be modeled as an all-pole system, therefore the inverse filter is an all-zero system. A direct way of obtaining the impulse response of the inverse filter, hI(n), is to expand rational polynomial through long division and truncate the sequence after M+1 coefficients:
The resulting error becomes:

8. FIR Inverse Filters – Least Squares
Another design can be obtained via a least-squares approach: - h ( n )
FIR Filter {b k }
Minimize Sum of
Squared Errors { b e
where d(n) is the desired response and the error of the filter output is e(n). The error and overall squared error E2 are given by:

9. FIR Inverse Weiner Filter
After minimizing E2 (take the derivatives with respect to each bk and set the result to zero), it can be shown that the optimal set of {bk}’s are the solution to the M equations given by:
where rhh(.) is the autocorrelation for h(n), and rdh(.) is the cross-correlation between h(n) and d(n):

10. FIR Inverse Filters – Least Squares
For the special case where d(n) = (n):
Therefore, the following system of equations can be used to solve for the filter coefficients:
Matrix is Symmetric and Toeplitz, can use Levenson- Durbin algorithm to solve

11. Example
Given desired response (original input to the system) d(n) and the actual response of system h(n) up to length N, design an Mth order FIR (Wiener) inverse filter. Create the following vector and matrix:
Then compute desired filter coefficients by solving the following matrix equation for b:
where

12. Example
Then test for stability (was original system minimum phase?), apply h(n) (plus a little noise, less than -3 dB) to the inverse filter. If the result is garbage (not close to the signal of interest), add a delay to the desired response and repeat (i.e. use):
Insert 0’s to delay desired response. This provides the filter with more degrees of freedom to undo the system response at the expense of delaying the output.
Increasing filter order may also improve performance.