[RP]Lecture Note VI - Kyung Hee University Department of...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) VI-1 C1002900 RP Lecture Handout VI: Mean Square Estimation Reading: Chapter 7.3 The estimation problem is fundamental in the applications of probability. Let X be a random variable with some known distribution. Suppose X is not observed but we wish to estimate X . (Optimality Criterion) Minimizing the mean square value of estimation error: minimum mean square error (MMSE) estimation. In many applications, a random variable X is estimated based on observation of the other random variable Y . Thus, an estimator is a function of Y . The two most frequently considered classes are essentially: all functions, leading to the best unconstrained estimator all linear functions, leading to the best linear estimator. 6.1. MS Estimation without Information If we use a constant c to estimate X , then the estimation error is eXc  . What is the best guess c of the value of X ?    2 argmin . c cX c MS error:    X eX c xc f xd x   2 2 X de xcf xd x dc   20   . X cx f x d xX    Solution:   Optimal (least) MS error: Var .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) VI-2 6.2. MS Estimation with Information: Nonlinear MS Estimation Now, we observe that Yy . Then,    XcY y  2 is minimized by   . cX Y g y y  View the estimator as a function gy .   XY minimizes {( ( )) } Xg Y 2 over all estimators g . We wish to estimate X by a function gY of the random variable Y . Our problem now is to find the optimal (unconstrained) that minimizes the MS error      , ,. eX g Y x g y f x y dxdy        2 2 Since   , , Y fx y y f y , is minimized if it is minimized for every . ey Y ef y x g y f x y d x d y   2    XY y If Xh Y , then      . hY Y y hy  Optimal estimator . gy hy e 0 If X and Y are independent, then     . X Knowledge of Y have no effect on the estimate of X . Optimal (least) MS estimator:  
Background image of page 2
Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) VI-3 6.3. Linear MS Estimation Linear estimation problem: estimate X in term of a linear function aY b of Y .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 06/10/2010 for the course ELECTRONIC C1002900 taught by Professor Hyungdongshin during the Spring '10 term at Kyung Hee.

Page1 / 9

[RP]Lecture Note VI - Kyung Hee University Department of...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online