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Unformatted text preview: ENEE 630 Fall 2009 Homework #9 Take-home Exercise. Solution will be posted on the course website Problem 1 Find the variance of the unbiased ACF estimator ˆ r ′ xx [ k ] = 1 N − k ∑ N − 1 − k n =0 x [ n ] x [ n + k ] 0 ≤ k ≤ N- 1 for real data which is a zero-mean white Gaussian process with variance σ 2 x . What happens as the lag k increases? Repeat the problem for the biased ACF estimator and explain your results: ˆ r xx [ k ] = 1 N N − 1 − k summationdisplay n =0 x [ n ] x [ n + k ] Hint: First prove that for any real zero-mean Gaussian process the variance of the unbiased ACF estimator is var [ˆ r ′ xx [ k ]] = 1 N- k N − 1 − k summationdisplay j = − ( N − 1 − k ) parenleftbigg 1- | j | N- k parenrightbigg ( r 2 xx [ j ] + r xx [ j + k ] r xx [ j- k ]) . Problem 2 Show that the exponential signal x [ n ] = p summationdisplay i =1 A c i z n i may be generated by the recursive difference equation x [ n ] =- p summationdisplay k =1 a [ k ] x [ n- k ]...
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This note was uploaded on 10/31/2010 for the course EE 630 taught by Professor Wu during the Spring '10 term at Aarhus Universitet, Aarhus.
- Spring '10