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I have been told (Wikipedia agrees) that the Wiener filter is optimal when signal and (additive) noise are WSS. Optimal in the sense that it minimizes the mean-square error.

The Cramér–Rao bound is the lower bound on the variance of an unbiased estimator of a deterministic parameter.

Does that mean that the Wiener filter operating at the Cramér–Rao bound?

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  • $\begingroup$ Here's few page paper related to Wiener & Cramér–Rao bounds - tinyurl.com/y4njfzmv $\endgroup$
    – Juha P
    Commented Jan 28, 2021 at 10:39

2 Answers 2

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While @Gideon Genadi Kogan answer touches some of the issues the main thing is not whether we can define all signal as a set of parameters or not.

The Cramer Rao Lower Bound is a lower bound for a parametric model.
The Wiener Filter is a Bayesian Method with its loss function defined as the MMSE.

So basically they don't have the same modeling of the data.
For Cramer Rao the parameter is fixed but unknown, for the Wiener Filter the data is stochastic.

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Cramér–Rao is a lower bound for variance of parameter estimation. Wiener filter is used for the estimation of a signal rather than the variance of parameters.

In some formulations, the Cramér–Rao can be used for estimation of the lower bound of multiple parameters. Nevertheless, treating a signal as a set of parameters seems like a little bit of stretching the definition.

So main differences are:

  1. The number of the parameters - multiple VS signal length.
  2. Estimation of variance VS estimation of values.
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