Lect12 - Stochastic Process 12/22/2006 Lecture 12...

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Stochastic Process 12/22/2006 Lecture 12 Prediction and Kalman Filtering NCTUEE Summary In this lecture, I will discuss: MMSE Prediction with Linear State Variable Model Kalman Filter Notation We will use the following notation rules, unless otherwise noted, to represent symbols during this course. Boldface upper case letter to represent MATRIX Boldface lower case letter to represent vector Superscript ( · ) T and ( · ) H to denote transpose and hermitian (conjugate transpose), respectively Upper case italic letter to represent RANDOM VARIABLE 12-1
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1 Prediction (1) Review: (a) MMSE: (Lecture 13) We want to estimate a random vector x based on observation y using the minimum mean-squared error criterion ˆ g mmse ( y ) = arg min g ( y ) E h x - g ( y ) 2 i = E [ x | y ] , ˆ x mmse (b) LMMSE: (Lecture 13) We want the rule g ( y ) to be constrained by g ( y ) = A · y + b . ˆ x lmmse = m x + K xy K y - 1 ( y - m y ) . (c) Recall from HW 2. Let x , y and z be collectively jointly Gaussian random vectors. — If y and z are statistically independent, then E [ x | y , z ] = E [ x | y ] + E [ x | z ] - m x , where m x = E [ x ] . — If y and z are not necessarily statistically independent, then E [ x | y , z ] = E [ x | y , ˜ z ] = E [ x | y ] + E [ x | ˜ z ] - m x , where ˜ z = z - E [ z | y ] . 12-2
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(2) (Vector random sequence) A vector random sequence is a mapping from a probability space Ω into the space of vector-valued sequence. In other words, it’s a sequence of random vectors. (3) Basic state variable model [1] — Often used to study a time-varying (dynamic) phenomena em- bedded in observations or measurements, e.g. time-varying chan- nels appeared in the received signals of a wireless link — Characterized by state vector x [ k ] and measurement vector z [ k ] . State vector: x [ k + 1] = A [ k + 1 ,k ] · x [ k ] + B [ k + 1 ,k ] · w [ k ] + C [ k + 1 ,k ] · u [ k ] Measurement vector: z [ k + 1] = H [ k + 1] · x [ k + 1] + v [ k + 1] . w
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This note was uploaded on 11/28/2010 for the course EE 301 taught by Professor Gfung during the Winter '10 term at National Chiao Tung University.

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Lect12 - Stochastic Process 12/22/2006 Lecture 12...

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