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For a signal x(n) = { 1, -2} and an impulse response of some LTI system h(n) = {1,1} ↑ ↑ a) Calculate the N=2-point Discrete Fourier Transform (DFT2) of signals x(n) and h(n) i.e. X₂ (k) k=0,..,N-1 and H₂ (k), k=0, ..N-1 b) Calculate linear convolution of the 2 signals: y(n) = x(n) * h(n) c) Let ŷ(n) = IDFT2 {X2 (k) H₂ (k)} k=0,1. (i.e. inverse DFT of the product of X(k) and H(k) for k=0,1) Calculate y(n) for the appropriate number of samples. Is ŷ(n) periodic? If so, what is its' period?

For a signal x(n) = { 1, -2} and an impulse response of some LTI system h(n) = {1,1} ↑ ↑ a) Calculate the N=2-point Discrete Fourier Transform (DFT2) of signals x(n) and h(n) i.e. X₂ (k) k=0,..,N-1 and H₂ (k), k=0, ..N-1 b) Calculate linear convolution of the 2 signals: y(n) = x(n) * h(n) c) Let ŷ(n) = IDFT2 {X2 (k) H₂ (k)} k=0,1. (i.e. inverse DFT of the product of X(k) and H(k) for k=0,1) Calculate y(n) for the appropriate number of samples. Is ŷ(n) periodic? If so, what is its' period?

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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For a signal x(n) = { 1, -2} and an impulse response of some LTI system h(n) = {1,1} ↑ ↑ a) Calculate the N=2-point Discrete Fourier Transform (DFT₂) of signals x(n) and h(n) i.e. X₂ (k) k=0,..,N-1 and H₂(k), k=0, ..N-1 b) Calculate linear convolution of the 2 signals: y(n) = x(n) * h(n) c) Let y(n) = IDFT₂ {X₂(k)H₂(k)} k=0,1. (i.e. inverse DFT of the product of X(k) and H(k) for k=0,1) Calculate y(n) for the appropriate number of samples. Is ŷ(n) periodic? If so, what is its' period? d) Are ŷ (n) in part c) and y(n) in part b) equal? If so, why? Do you expect them to be equal? If they are not equal, briefly explain why? If there is a mismatch between ŷ(n) and y(n), also explain what steps would we need to take to be able to use DFT/IDFT to correctly calculate linear convolution of the signals x(n) and h(n)?
Transcribed Image Text:For a signal x(n) = { 1, -2} and an impulse response of some LTI system h(n) = {1,1} ↑ ↑ a) Calculate the N=2-point Discrete Fourier Transform (DFT₂) of signals x(n) and h(n) i.e. X₂ (k) k=0,..,N-1 and H₂(k), k=0, ..N-1 b) Calculate linear convolution of the 2 signals: y(n) = x(n) * h(n) c) Let y(n) = IDFT₂ {X₂(k)H₂(k)} k=0,1. (i.e. inverse DFT of the product of X(k) and H(k) for k=0,1) Calculate y(n) for the appropriate number of samples. Is ŷ(n) periodic? If so, what is its' period? d) Are ŷ (n) in part c) and y(n) in part b) equal? If so, why? Do you expect them to be equal? If they are not equal, briefly explain why? If there is a mismatch between ŷ(n) and y(n), also explain what steps would we need to take to be able to use DFT/IDFT to correctly calculate linear convolution of the signals x(n) and h(n)?
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