Consider a sequence of n dimensional vector

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Unformatted text preview: Model (HMM). A HMM can be either vector or scalar. Example 3: Continuing Examples 1 & 2, let h = [h0 , h1 , · · · , hL−1 ]T and consider the observation random process rn = hT S n + nn (9) where nn is a zero-mean AWGN process. Since S n is a Markov chain, rn is a HMM. Consider a sequence of N -dimensional vector observations of the form r n = f n (S n ) + nn n = 1, 2 , · · · , K (10) where f n (S n ) represents N functions of S n which in general can be time varying, and the nn are AWGN vectors. Let r K = [r 1 , r2 , · · · , rK ] and S K = 1 1 [S 1 , S 2 , · · · , S K ]. The joint PDF of a sequence S K , conditioned on observations 1 r K , is 1 p(rK /S K ) P (S K ) 1 1 1 P (S K /rK ) = . (11) 1 1 K p (r 1 ) Given Eq (10) and AWGN nn , we have that K p(r K /S K ) = 1 1 N =1 p(rn /S n ) . (12) Assuming S n is a Markov chain, we have P (S K /rK ) 1 1 . = K p(rn /S n ) P (S n /S n−1 ) · P (S 0 ) n=1 (13) . where “=” denotes proportional to (i.e. p(r K ) in the denominator of Eq (11) is 1 not a function of S K ). 1 Kevin Buckley - 2007 5 MAP and ML Sequence Estimation for...
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This document was uploaded on 10/12/2009.

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