Continuous-Time Wiener Filters

• Approach

• Example 4-1, Continuous-Time Wiener Filter

• Wiener Filters Mean-Square Error

• MSE of Example 4-1

Discrete-Time Wiener IIR Filters

• Overview

• Finding S_d(z) and S_x(z) from R_x(t)

  • GMP

  • Z transform

• Example 4-2, S_d(z) from R_x(t)

  • GMP or can be approx. as a GMP

  • Z transform

• Example 4-3, Discrete Wiener Filter

• Example 4-4, Digital Wiener Filter

• Example 4-5, Discrete Wiener Filter Simulation 
• Mean-Square Error
• Example 4-6, MSE of Example 4-3
  • Z-Domain Approach
  • Time Domain Approach

Discrete-Time Wiener FIR Filters

• Using the Filter Impulse Response 
• Optimal Discrete Wiener Filter Using R_x & R_x
• R_x  Conditioning
• Selection of Number of R_d terms
• Example 4-7, Optimum Wiener FIR Filter

Prediction & Smoothing

• Prediction
• Smoothing
• Example 4-8, One-Step Discrete Predictor

Complimentary (Matched) Filters

• Overview
• Design & Implementation Steps
• Example 4-9, Complimentary Filter Design

Adaptive Linear Filters

Solved Problems

• Continuous-Time Wiener Filter
• IIR Wiener Filter
• FIR Wiener Filter #1
• FIR Wiener Filter #2
• FIR Wiener Filter #3
• IIR One-Stage Predictor
• Adaptive Linear Filter

Chapter 4 Quiz Review

For Questions and Comments on the Structure or Design of this page ,e-mail Rajesh

Continuous-Time Wiener Filters

References : Discrete : Modern Filters,Haykin,Macmillan

Continuous : Intro. to Random signals and Applied Kalman Filtering, Brown & Hwang, Wiley.

A Wiener filter minimizes the mean-square estimation error,.Wiener filtering is restricted to a class of problems where white noise is filtered from a noise-like (Random) process. A Wiener Filter is a "Linear Filter". A nonlinear or Kalman filter may be better. However, many random signals/processes are Gauss-Markov processes. Hence, Wiener filtering can be use also, when we have and signals and is low frequency noise and   is high frequency noise, a complimentary wiener filter can be used. Wiener filters are widely used in communication instrumentation systems.

Approach

The approach developed by Wiener is based on manipulating the power spectrum of the signal + noise as the noise we want to minimize is spread across the spectrum or in the frequency band of interest. Hence, it can't be easily filtered out using a low pass filter.

The above process is an ideal depiction of what we would like the Wiener filter to be. That is, 

However, the filter is NONCAUSAL (depends on future data, because of the transfer function :, The approach is to first transform so that it looks as much as possible like white noise which is shown as . The reason is that would then be an impulse function and all future values of the input would be uncorrelated with the present and past values. Thus, we would not be ignoring any future information that might lead to a better estimate.

Continuous-Time Wiener Filter

It can be shown that and are:

is a "whitening" filter for an input x = d+v,

and

Prediction and smoothing are sometimes expressed in the above.

The Wiener filter is : 

Note : Subtract out the mean,, before filtering and add the mean of the desired signal,, back at the output of the filter.