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A continuous-time LTI system has impulse response et, -{ h(t) = 1

A continuous-time LTI system has impulse response et, -{ h(t) = 1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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5. A continuous-time LTI system has impulse response 1 < t < 2, 0, otherwise. h(t) = (a) Using convolution in the time domain, find the response to the input f(t) = u(t). (b) Find the frequency response H(w). (c) Using the frequency response, find the response to the input f(t) = 3 cos (t+). To do this, follow these steps: • begin by using Euler's rule to express f(t) as the sum of two complex exponential signals • using the frequency response expression you derived in part (b), determine the system's response to each of these two complex exponentials use Euler's rule to re-combine the outputs into one real-valued cosine expression (d) Now, we can take advantage of a shortcut to solve problems like part (c). In class, we discussed the following useful fact: for an LTI system with a real-valued impulsc response h(t), if the input f(t) = A cos(wot + p) for some amplitude A, frequency wo, and phase o, then the output will be y(t) = A|H(wo)| cos(wot + + 2H(wo)). Use this fact to check your answer from part (c), showing your work.
Transcribed Image Text:5. A continuous-time LTI system has impulse response 1 < t < 2, 0, otherwise. h(t) = (a) Using convolution in the time domain, find the response to the input f(t) = u(t). (b) Find the frequency response H(w). (c) Using the frequency response, find the response to the input f(t) = 3 cos (t+). To do this, follow these steps: • begin by using Euler's rule to express f(t) as the sum of two complex exponential signals • using the frequency response expression you derived in part (b), determine the system's response to each of these two complex exponentials use Euler's rule to re-combine the outputs into one real-valued cosine expression (d) Now, we can take advantage of a shortcut to solve problems like part (c). In class, we discussed the following useful fact: for an LTI system with a real-valued impulsc response h(t), if the input f(t) = A cos(wot + p) for some amplitude A, frequency wo, and phase o, then the output will be y(t) = A|H(wo)| cos(wot + + 2H(wo)). Use this fact to check your answer from part (c), showing your work.
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