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Unformatted text preview: ECE 3250 HOMEWORK VI SOLUTIONS Fall 2009 1. By definition, a LTI system has a frequency response if and only if for all Ω ∈ R the signal t 7→ e j Ω t is an admissible input for the system. A signal x is an admissible input for a system such as the one in the problem if and only if x ∈ D h . Since h is decent and has finite duration, every decent signal x is in D h . Accordingly, the signal t 7→ e j Ω t is in D h and is therefore an admissible input for all Ω ∈ R . The system, in other words, has a frequency response. 2. Suppose a (causal) BIBO stable system has impulse response h . For any Ω o ∈ R , the input signal t 7→ e j Ω o t is bounded. Since the system is BIBO stable, all these inputs are in D h , which means, by definition, that the system has a frequency response. A causal LTI system that’s not BIBO stable is the system with impulse response h with specification h ( t ) = e 3 t u ( t ) for all t ∈ R . For any Ω o ∈ R , the convolution of h with x , where x ( t ) = e j Ω o t for all t ∈ R , does not exist. Hence the system does not have a frequency response. 3. One way of characterizing the frequency response b H of a LTI system is as follows: if the input x has specification x ( t ) = e j Ω o t for t ∈ R , then the output y has specification y ( t ) = b H (Ω o ) e j Ω o t for t ∈ R . In this problem, the input x splits up into the sum of such pure sinusoidal terms. For each k ∈ Z , the output corresponding to input signal t 7→ e jk 17 t is the signal t 7→ b H ( k 17) e jk 17 t = 1 3 + jk 17 e jk 17 t ....
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 '07
 DELCHAMPS
 Fourier Series, Frequency, LTI system theory

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