 X – arbitrary fixed input distribution  holds for scalar and vector signals Notes Mean of Filtering (causal) MMSE = Smoothing (noncausal) MMSE Γ uniformly.

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Presentation transcript:

 X – arbitrary fixed input distribution  holds for scalar and vector signals Notes Mean of Filtering (causal) MMSE = Smoothing (noncausal) MMSE Γ uniformly distributed in [0,snr] Continuous time time

1. Some direct implications on other results: Equivalency with De Bruijn’s identity  which connects the differential entropy h(  ) to the Fisher’s information matrix J(  ) (connected to the CRLB) Derivative of the divergence gets an interesting form  can be used also in multiuser systems: Information Theory Estimation Theory 2. Duality Information Theory  Estimation Theory: New lower/upper bounds on MMSE and mutual info. I(  ;  ) Results from one domain can be applied to the other Results from one domain can be obtained/proven by the other  e.g., linking the filtering and smoothing by a direct expression in continuous time domain and sandwiching relation in discrete time