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Unformatted text preview: ECE 2704 ‐ Homework Number 10 1. Using Properties of the Fourier transform, show that if where indicates convolution Solution Since convolution in the time‐domain is a product in the frequency‐domain, . You will need to convince yourself that . 2. Suppose is the input to a system and is the output. The output is the same as the input . Suppose , , then show that except delayed by some unknown amount . That is, , for all , and that Solution Since seconds. 3. Problem 7.5‐3 (page 766) (a, b, c) Solution . Cannot be realized since (a) , so if we add a time delay of can be realized. (b) , . Thus and for all . In other words, is the impulse response of a non‐causal system. The impulse response starts at , then the system is causal and . Corresponds to a noncausal system. Time‐delay does not help since the nonzero response extends to . However, most of the energy of the signal is centered near , so with a modest time‐delay, it is possible to approximate the desired response. (c) . Corresponds to a noncausal system, and a time‐delay won’t help at all. 4. Problem 7.7‐4 (page 766) (a, b) Solution (a) At point a the signal is Fourier transform of this signal is . . The ‐10,000 0 10,000 ω At point b, the signal is ‐10,000 ‐5,000 0 5,000 10,000 ω At point c, the signal this shifts the frequency spectrum in part(b) to . Using the same analysis in part(a), . ‐30,000 ‐20,000 0 20,000 30,000 ω (b) The spectrum is from approximately rads/sec. to rads/sec, so the bandwidth is 30,000 ...
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- Fall '08