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Unformatted text preview: ECE 3250 HOMEWORK ASSIGNMENT VI Fall 2009 1. Show that every continuous-time system with input space X consisting of all decent signals and and system mapping of the form S ( x ) = Conv h ( x ) + N X k =1 d k Shift t k ( x ) , where h is decent and has finite duration, has a frequency response. 2. Show that every (causal) BIBO stable continuous-time system has a frequency re- sponse. Also, find the impulse response h of some causal continuous-time system that doesn’t have a frequency response. 3. A certain LTI system has frequency response b H with specification b H (Ω) = 1 3 + j Ω for all Ω ∈ R . Let x be the signal with Fourier series x ( t ) = ∞ X k =-∞ 1 k 2 + 1 e jk 17 t . Find a Fourier series for the system’s output S ( x ) in response to input x . Why might one say that the system acts as a crude low-pass filter? 4. Consider the averager system whose input space X is the set of all decent F-valued signals and whose system mapping is S ( x )( t ) = 1 T o Z t t- T o x...
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- Digital Signal Processing, LTI system theory