# HWIII - ECE 3250 HOMEWORK ASSIGNMENT III Fall 2009 1 A...

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ECE 3250 HOMEWORK ASSIGNMENT III Fall 2009 1. A certain LTI system has impulse response h whose specification is h ( n ) = 5 - n if 0 n < 7 0 otherwise . (a) What is D h , the set of all x F Z for which Conv h ( x ) exists? (b) Find the output S ( x ) of the system when the input is the signal x whose specifi- cation is x ( n ) = 3 for all n Z . (c) Find S ( u ), the step response of the system, which is the output of the system when its input is a unit step. What I’d like here are formulas for S ( u )( n ) for all n Z . (d) Suppose x F Z has duration 13. What is the maximum possible duration for S ( x )? 2. Denote by S the system mapping for the causal sliding-window M -fold averager with M = 5. (a) Find a non-constant signal x F Z for which S ( x )( n ) = 11 for every n Z . (b) Show that lim n →∞ S ( x )( n ) = 0 for every left-sided signal x F Z . Show also that lim n →-∞ S ( x )( n ) = 0 for every right-sided signal x F Z . (c) Find a signal x F Z that doesn’t have finite duration but that satisfies lim n →±∞ S ( x )( n ) = 0 .

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