MT(483)S09_Solns

MT(483)S09_Solns - EE 483 Introduction to Digital Signal...

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1 EE 483 – Introduction to Digital Signal Processing Spring 2009 Midterm Examination – February 25, 2009 1. (10 Points) Consider the following finite-length sequences: (i) h [ n ], - M n N , and (ii) g [ n ], K n N , where M , N and K are positive integers with K < N . Define (a) y 1 [ n ] = h [ n ]O * h [ n ], (b) y 2 [ n ] = h [ n ]O * g [ n ], (c) y 3 [ n ] = g [ n ]O * g [ n ]. What are the lengths of each of the above convolved sequences? What are the ranges of the index n for which each of the above convolved sequences are defined? (a) Length: 2( N + M ) + 1. Range: - 2 M n 2 N . (b) Length 2 N + M - K + 1. Range: - M + K n 2 N . (c) Length: 2( N - K ) + 1. Range: 2 K n 2 N . 2. (15 Points) (a) ( - 1) n x [ n ] e - j ϖ n n =-∞ = x [ n ] e - j π n e - j n n =-∞ = x [ n ] e - j ( + ) n n =-∞ = X ( e j ( + ) ). (b) From part (a), ( - 1) n n =-∞ x [ n ] = X ( e j ). (c) ( 29 ( 29 n j n n n n j n j j e n x e e n x e Y ϖ - -∞ = -∞ = π - ϖ - ϖ - ϖ - + = + = ) 1 ( 1 ] [ ] [ ) ( ) ( = 2 x [2 m ] e - j 2 m m =-∞ . Hence, Y ( e j ) depends only on the even-indexed samples x [2 m ] and is independent of the odd- indexed samples x [2 m + 1]. 3.
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MT(483)S09_Solns - EE 483 Introduction to Digital Signal...

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