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Recall the echo system from the lecture notes in the section "Frequency response of LTI systems: Example D." In this problem, we will consider a different model for how an echo might be generated in a received signal. Specifically, we consider the model below, where T> 0 is a delay and a € (0, 1) is an attenuation factor: f(t). a 1 delay T seconds (a) As an example, suppose a = 0.5 and T = 2. Consider the input signal f(t) as shown below: f(t) 0 Plot the resulting output signal y(t). (b) Let's return to thinking about a and T as generic parameters, not associating them with specific values. In the time domain, we can write the input-output relationship of the echo system as follows: y(t) y(t) = f(t) + ay(t-T). Taking the Fourier transform of both sides of this equation, and using the time delay property of the Fourier transform, we can write Y(w) = F(w) + ???? - Y(w). Fill in the blanks in the equation above. (c) Rearranging terms in the equation above, we can write Y(w)(1-????) = F(w). Dividing Y(w) by F(w), we obtain H(w), the frequency response of the system: H(w)= Using your answer from part (b), find H(w). Y(w) F(w)

Recall the echo system from the lecture notes in the section "Frequency response of LTI systems: Example D." In this problem, we will consider a different model for how an echo might be generated in a received signal. Specifically, we consider the model below, where T> 0 is a delay and a € (0, 1) is an attenuation factor: f(t). a 1 delay T seconds (a) As an example, suppose a = 0.5 and T = 2. Consider the input signal f(t) as shown below: f(t) 0 Plot the resulting output signal y(t). (b) Let's return to thinking about a and T as generic parameters, not associating them with specific values. In the time domain, we can write the input-output relationship of the echo system as follows: y(t) y(t) = f(t) + ay(t-T). Taking the Fourier transform of both sides of this equation, and using the time delay property of the Fourier transform, we can write Y(w) = F(w) + ???? - Y(w). Fill in the blanks in the equation above. (c) Rearranging terms in the equation above, we can write Y(w)(1-????) = F(w). Dividing Y(w) by F(w), we obtain H(w), the frequency response of the system: H(w)= Using your answer from part (b), find H(w). Y(w) F(w)

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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(d) Now let's think about how we could cancel out the echo from a received signal y(t) and recover the original signal f(t). We can actually construct a second LTI system to do this: cancellation system f(t) original echo system α delay T seconds In the frequency domain, we will have y(t) h₂(t) g(t) G(w) = H₂(w)Y(w). If we want G(w) = F(w) (so that g(t) = f(t)), how should we choose H₂(w)? (e) Taking the inverse Fourier transform of H₂(w), we can find the impulse response of the "inverse" (or cancellation) system: h₂(t) = ???? 8(t) + ???? - 8(t – T). Fill in the blanks in the equation above. (f) We can write the input-output relationship of this inverse system as follows: g(t) = ???? - y(t) + ???? - y(t – T). Fill in the blanks in the equation above. (g) Draw a block diagram for the inverse system. (h) Finally, returning to the specific example from part (a) with a = 0.5 and T = 2, show that if you feed the output y(t) from part (a) into the inverse system from part (g), you correctly recover the original signal f(t).
Transcribed Image Text:(d) Now let's think about how we could cancel out the echo from a received signal y(t) and recover the original signal f(t). We can actually construct a second LTI system to do this: cancellation system f(t) original echo system α delay T seconds In the frequency domain, we will have y(t) h₂(t) g(t) G(w) = H₂(w)Y(w). If we want G(w) = F(w) (so that g(t) = f(t)), how should we choose H₂(w)? (e) Taking the inverse Fourier transform of H₂(w), we can find the impulse response of the "inverse" (or cancellation) system: h₂(t) = ???? 8(t) + ???? - 8(t – T). Fill in the blanks in the equation above. (f) We can write the input-output relationship of this inverse system as follows: g(t) = ???? - y(t) + ???? - y(t – T). Fill in the blanks in the equation above. (g) Draw a block diagram for the inverse system. (h) Finally, returning to the specific example from part (a) with a = 0.5 and T = 2, show that if you feed the output y(t) from part (a) into the inverse system from part (g), you correctly recover the original signal f(t).
Recall the echo system from the lecture notes in the section "Frequency response of LTI systems: Example D." In this problem, we will consider a different model for how an echo might be generated in a received signal. Specifically, we consider the model below, where T> 0 is a delay and a € (0, 1) is an attenuation factor: f(t). 0 α 1 delay T seconds (a) As an example, suppose a = 0.5 and T = 2. Consider the input signal f(t) as shown below: f(t) y(t) Plot the resulting output signal y(t). (b) Let's return to thinking about a and T as generic parameters, not associating them with specific values. In the time domain, we can write the input-output relationship of the echo system as follows: t y(t) = f(t) + ay(t-T). Taking the Fourier transform of both sides of this equation, and using the time delay property of the Fourier transform, we can write Y(w) = F(w) + ???? .Y (w). H(w) = Using your answer from part (b), find H(w). Fill in the blanks in the equation above. (c) Rearranging terms in the equation above, we can write Y(w) (1 - ????) = F(w). Dividing Y(w) by F(w), we obtain H(w), the frequency response of the system: Y(w) F(w)
Transcribed Image Text:Recall the echo system from the lecture notes in the section "Frequency response of LTI systems: Example D." In this problem, we will consider a different model for how an echo might be generated in a received signal. Specifically, we consider the model below, where T> 0 is a delay and a € (0, 1) is an attenuation factor: f(t). 0 α 1 delay T seconds (a) As an example, suppose a = 0.5 and T = 2. Consider the input signal f(t) as shown below: f(t) y(t) Plot the resulting output signal y(t). (b) Let's return to thinking about a and T as generic parameters, not associating them with specific values. In the time domain, we can write the input-output relationship of the echo system as follows: t y(t) = f(t) + ay(t-T). Taking the Fourier transform of both sides of this equation, and using the time delay property of the Fourier transform, we can write Y(w) = F(w) + ???? .Y (w). H(w) = Using your answer from part (b), find H(w). Fill in the blanks in the equation above. (c) Rearranging terms in the equation above, we can write Y(w) (1 - ????) = F(w). Dividing Y(w) by F(w), we obtain H(w), the frequency response of the system: Y(w) F(w)
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