State-Space Realizations
The above frequency-domain solutions can be realized in state-space forms. Discrete-time and continuous-time formulations are described in and, respectively. The causal Wiener solution is equivalent to the minimum-variance Kalman filter. The Wiener filter and Kalman filter equivalence is a consequence of the Kalman–Yakubovich–Popov lemma which is also known as the Positive Real Lemma.
The non-causal Wiener solution is known as the minimum-variance smoother. This smoother can attain the best-possible error performance, provided that the model parameters and noise statistics are known precisely.
See the Kalman Filtering Section and the references for examples of state-space Wiener smoother recursions.
Read more about this topic: Wiener Filter
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