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I'm confused right off the bat in the introduction to Wiener filters, because how can you know what the "Desired response" is? Where does it come from? And if you know it already, then why build a filter to process an input signal in the first place? Can you provide an example of a typical input $u(n)$ and desired response $d(n)$ (such that that the error $e(n) = d(n) - y(n))$?

I'm reading Haykin's Adaptive Filter Theory, start of Chapter 2.

$u(n)$ = filter input $y(n)$ = linear filter output $d(n)$ = desired signal $e(n)$ = error signal: $d(n) - y(n)$

thanks, js.

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In many applications a Wiener filter is used to decorrelate the desired signal from the input to the filter. The desired system output is then the error signal, which is decorrelated from the filter's input signal.

As an example, in adaptive noise cancellation we assume that we have some noise reference signal (e.g., noise picked up by a microphone). And we have a noisy signal which we want to clean up. The noise in the signal is assumed to be correlated with the noise reference. In terms of Wiener filtering, the noisy signal is the desired signal, the noise reference is the input to the filter, the filter output is hopefully a good estimate of the noise in the desired signal, and the error signal is the actual output of the noise canceler, which is a cleaner version of the originally noisy signal. The figure below shows the set-up (from S.J.Orfanidis: Optimum Signal Processing):

enter image description here

There are many other applications of (adaptive) Wiener filters, such as adaptive equalizers in communication systems (see this answer), and echo cancelation (see this answer).

In sum, the "desired signal" is often a signal contaminated with noise or interference, and the output of the Wiener filter is an estimate of that noise or interference, which is subtracted from the "desired signal". This is possible if at the input of the filter there is some reference of the contaminating signal. The "error signal" is then the actually desired output, which is the noisy "desired signal" minus the noise-estimate at the output of the filter.

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  • $\begingroup$ Thanks! Your response: 1) Uses a different diagram than the basic Wiener filter diagram. 2) Uses an "adaptive" filter, instead of the simple "linear" filter in the basic Wiener filter diagram. 3) In the basic Wiener filter diagram, the estimation error signal and desired response are different signals, yet you combined those into the same signal when you wrote, "The desired system output is then the error signal." Is there not an available example that uses the basic Wiener filter block diagram (just linear filter and sum of output and desired response? $\endgroup$ – j03y_ May 25 at 15:51
  • $\begingroup$ Can you point me to an explanation of the difference between an Optimum Filter and an Adaptive Filter? I googled this earlier, but didn't find a clear explanation. My understanding is that a Wiener filter is "Optimum" but not "Adaptive". Is noise cancellation theoretically the same as recovering a signal from noise, i.e. in receiving radar signals? $\endgroup$ – j03y_ May 25 at 15:53
  • $\begingroup$ @j03y_: It is exactly the same, just a specific application, as you were asking for. Forget about the adaptive part for the moment, the adaptation will stop if everything is stationary (which is an assumption in basic Wiener filtering). You have an input ($y$, which you call $u$), an output $\hat{x}$ (which you call $y$), and an error signal ($e$ in both cases). And the desired signal $d$ is called $x$ in my figure. So there's absolutely no difference to a standard Wiener filter, if you stop the adaptation. $\endgroup$ – Matt L. May 25 at 15:55
  • $\begingroup$ @j03y_: The adaptation of the filter coefficients just tries to keep the filter optimal in time-varying circumstances. If everything is stationary, there's no need for adaptation, but that's just not what usually happens in the real world. $\endgroup$ – Matt L. May 25 at 15:57
  • $\begingroup$ @j03y_: You misunderstood if you think I combined the desired and the error signals. It's just that in noise cancelation, the output of the whole system is the error signal, which equals the actually desired signal, which is different from what you call "desired". Here, "desired" means the cleaned-up signal, which - in terms of Wiener filtering - is just the error signal. $\endgroup$ – Matt L. May 25 at 15:59

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