PS3_sol - EE341, Autumn 2007 Problem Set 3 SOLUTION Reading...

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EE341, Autumn 2007 Problem Set 3 SOLUTION Reading for this problem set: Chapter 10 of Signals, Systems, and Transforms . 1. Show that, for any function g [ n ], g [ n ] * δ [ n ] = g [ n ] solution From the definition of the convolution sum, g [ n ] * δ [ n ] = X k = -∞ g [ k ] δ [ n - k ] however δ [ n - k ] 6 = 0 only at k = n, therefore: = . . . + g k - 1 [ n ] + g k [ n ] + g k +1 [ n ] + . . . ...which is the impulse representation of the discrete time signal: = g [ n ] 2. Show that the convolution of three signals can be performed in any order, by showing that: ( f [ n ] * g [ n ]) * h [ n ] = f [ n ] * ( g [ n ] * h [ n ]) solution for ( f [ n ] * g [ n ]) * h [ n ] : for f [ n ] * ( g [ n ] * h [ n ]) : let P [ n ] = f [ n ] * g [ n ] let g [ n ] * h [ n ] = Q [ n ] P [ n ] = X k = -∞ f [ k ] g [ n - k ] X k = -∞ h [ k ] g [ n - k ] = Q [ n ] now let i = n - k so k = n - i now let i = n - k so k = n - i P [ n ] = X i = -∞ g [ i ] f [ n - i ] X i = -∞ g [ i ] h
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This note was uploaded on 04/10/2008 for the course E E 341 and 23 taught by Professor Various during the Spring '08 term at University of Washington.

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PS3_sol - EE341, Autumn 2007 Problem Set 3 SOLUTION Reading...

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