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# zz-8 - ECE 6010 Lecture 9 Linear Minimum Mean-Square Error...

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ECE 6010 Lecture 9 – Linear Minimum Mean-Square Error Filtering Background Recall that for random variable X and Y with finite variance, the MSE E [( X - h ( Y )) 2 ] is minimized by h ( Y ) = E [ X | Y ] . That is, the best estimate of X using a measured value of Y is to find the conditional average of X . One aspect of this estimate is that: The error is orthogonal to the data. More precisely, the error X - E [ X | Y ] is orthogonal to Y and to every function of Y : E [( X - E [ X | Y ]) g ( Y )] = 0 for all measurable functions g . We will assume that E [ g 2 ( Y )] < . We want to show that h minimizes E [( X - h ( Y )) 2 ] if and only if E [( X - h ( Y )) g ( Y )] = 0 (orthogonality), for all measurable g such that E [ g 2 ( Y )] < . E [( X - E [ X | Y ]) g ( Y )] = E [ E [( X - E [ X | Y ]) | Y ] g ( Y )] = E [( E [ X | Y ] - E [ X | Y ]) g ( Y )] = 0 . Conversely, suppose for some g , E [( X - h ( Y )) g ( Y )] 6 = 0 . Consider the estimate ˆ h ( Y ) = h ( Y ) + αg ( Y ) , where α = E [( X - h ( Y )) g ( Y )] E [ g 2 ( Y )] . Then E [( X - ˆ h ( Y )) 2 ] = E [( X - h ( Y )) 2 ] - ( E [( X - h ( Y )) g ( Y )]) 2 E [ g 2 ( Y )] < E [( X - h ( Y )) 2 ] . Suppose now we are given two random processes { X t } and { Y t } that are statistically related (that is, not independent). Suppose, to begin, that T = R . Suppose we observe Y over the interval [ a, b ] , and based on the information gained we want to estimate X t for some fixed t as a function of { Y t , a t b } . That is, we form ˆ X t = f ( { Y τ , a τ b } ) for some functional f mapping the function to real numbers. If t < b : We say that the operation of the function is smoothing . If t = b : We way that the operation of the function is filtering . If t > b : We way that the operation of the function is prediction . The error in the estimate is X t - ˆ X t . The mean-squared error is E [( X t - ˆ X t ) 2 ] . Fact (built on our previous intuition): The MSE E [( X t - ˆ X t ) 2 ] is minimized by the conditional expectation ˆ X ( t ) = E [ X t | Y τ , a τ b ] . Furthermore, the orthogonality principle applies: X t - E [ X t | Y τ , a τ b ] is orthogonal to every function of { Y τ , a τ b } . While we know the theoretical result, it is difficult in general to compute the desired conditional expectation. Definition 1 Suppose { Y t } is second order. Let H y be the set of all random variables of the form n i =1 a i Y t i + c for n Z and a i , c R and t i [ a, b ] . 2

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ECE 6010: Lecture 9 – Linear Minimum Mean-Square Error Filtering 2 Note that H y may include infinite sequences, so we assume mean-square limits. The set H y contains mean-square derivatives, mean-square integrals, and other linear transfor- mations of { Y t , t [ a, b ] } . (The set H y
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zz-8 - ECE 6010 Lecture 9 Linear Minimum Mean-Square Error...

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