solnsmid1_600

solnsmid1_600 - ECE-600 Introduction to Digital Signal...

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Unformatted text preview: ECE-600 Introduction to Digital Signal Processing Au10 Midterm #1 Oct. 29, 2010 MIDTERM #1 SOLUTIONS 1. (a) Recall that the IDTFT of a unit-height rectangle spanning the interval [ ω ,ω ) equals sin( ω n ) / ( πn ), which has height ω /π and nulls at kπ/ω for integers k negationslash = 0. Here, ω = π/ 3, giving something like the figure below. x [ n ] n 1 / 3 3 6 9 − 3 − 6 − 9 (b) We can derive W ( e jω ) = ∑ ∞ n = −∞ ( x [ − n ] e j π 3 n ) e − jωn = ∑ ∞ m = −∞ ( x [ m ] e − j π 3 m ) e jωn = ∑ ∞ m = −∞ x [ m ] e − j ( π 3 − ω ) n = X ( e j ( π 3 − ω ) ). Note also that, since X ( e − jω ) = X ( e jω ) due to symmetry, we can also write W ( e jω ) = X ( e j ( ω − π 3 ) ). Both yield the plot below. W ( e jω ) ω 1 2 π 3 π − π (c) Since X ( e jω ) is conjugate symmetric (actually, it is real-valued symmetric, which is a special case of conjugate symmetric), we know x [ n ] ∈ R , so that | x [ n ] | 2 = x [ n ] 2 ....
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This note was uploaded on 03/05/2011 for the course ECE 600 taught by Professor Clymer,b during the Fall '08 term at Ohio State.

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solnsmid1_600 - ECE-600 Introduction to Digital Signal...

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