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I am reading about estimation theory, including topics like Bayesian Estimation (e.g. Wiener Filtering).

It seems that we usually define a filter in terms of tis frequency response (e.g. High Pass, Low Pass). On the other hand, Wiener Filter works by filtering out noise without specific reference to frequency domain.

How are the two related? If not, are they just two different approaches to the filtering problem?

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In Wiener filtering, you filter a noisy signal to more closely resemble a desired signal that you have access to. In Bayesian estimation, you take prior knowledge into account to estimate some state given noisy measurements. In frequency filtering, you just remove frequency content from a signal.

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For linear filtering, it can always be interpreted in the frequency domain, thus, it is impossible to discard the noise without affecting the signal in the same frequency position. For the Wiener filtering, it is based on the signal-to-noise ratio, i.e., which one is the larger, and decide to amplify or keep or attenuate the component. For other advanced filtering techniques, which claim to discard noise without affecting the noise, they should be non-linear filtering.

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