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Problem 3 For two signals x₁ (n) = {2, 1} and x₂ (n) = {-1, 2}, 个 个 a) Calculate 2-point Discrete Fourier Transforms X2,1(k) = DFT₂{x₁ (n)} and X2,2(k) = DFT₂{x₂ (n)} of x₁ (n) and x₂ (n), respectively (k=0,1) b) Using part a), find x3 (n) = IDFT2{X3(k)},where X3(k) = X₁(k)X₂(k), k = 0,1 c) Find linear convolution of x₁ (n) and x₂ (n) (without using DFT). d) Explain briefly why are the results of b) and c) different. What is the minimum number of points DFT required to make the result of b) equal to the result of c)? e) Using the number of points N determined in d), calculate N-point DFT of x₁ (n) and x₂ (n) i.e. X₁,N(K), X2,N(K), k = 0,1,.., N-1 f) Using the number of points N determined in d), calculate xp(n) = IDFT{Xp(k)},where Xp(k) = X₁,N(K)X2,N(K), k = 0,1,.., N-1

Problem 3 For two signals x₁ (n) = {2, 1} and x₂ (n) = {-1, 2}, 个 个 a) Calculate 2-point Discrete Fourier Transforms X2,1(k) = DFT₂{x₁ (n)} and X2,2(k) = DFT₂{x₂ (n)} of x₁ (n) and x₂ (n), respectively (k=0,1) b) Using part a), find x3 (n) = IDFT2{X3(k)},where X3(k) = X₁(k)X₂(k), k = 0,1 c) Find linear convolution of x₁ (n) and x₂ (n) (without using DFT). d) Explain briefly why are the results of b) and c) different. What is the minimum number of points DFT required to make the result of b) equal to the result of c)? e) Using the number of points N determined in d), calculate N-point DFT of x₁ (n) and x₂ (n) i.e. X₁,N(K), X2,N(K), k = 0,1,.., N-1 f) Using the number of points N determined in d), calculate xp(n) = IDFT{Xp(k)},where Xp(k) = X₁,N(K)X2,N(K), k = 0,1,.., N-1

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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Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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Problem 3 For two signals x₁ (n) = {2, 1} and x₂ (n) = {-1, 2}, 个 个 a) Calculate 2-point Discrete Fourier Transforms X2,1(k) = DFT₂{x₁(n)} and X2,2 (k) = DFT₂{x₂ (n)} of x₁ (n) and x₂ (n), respectively (k=0,1) b) Using part a), find x3 (n) = IDFT2{X3(k)},where X3(k) = X₁(k)X₂(k), k = 0,1 c) Find linear convolution of x₁ (n) and x₂ (n) (without using DFT). d) Explain briefly why are the results of b) and c) different. What is the minimum number of points DFT required to make the result of b) equal to the result of c)? e) Using the number of points N determined in d), calculate N-point DFT of x₁ (n) and x₂ (n) i.e. X₁,N(K), X2,N(K), k = 0,1,.., N-1 f) Using the number of points N determined in d), calculate xp (n) = IDFTN{Xp (k)},where Xp (k) = X₁,N(K)X2,N(k), k = 0,1,.., N - 1 g) Would you be able to achieve the same result as in f) if you used DFT using N with twice as many points as you determined in d) ?
Transcribed Image Text:Problem 3 For two signals x₁ (n) = {2, 1} and x₂ (n) = {-1, 2}, 个 个 a) Calculate 2-point Discrete Fourier Transforms X2,1(k) = DFT₂{x₁(n)} and X2,2 (k) = DFT₂{x₂ (n)} of x₁ (n) and x₂ (n), respectively (k=0,1) b) Using part a), find x3 (n) = IDFT2{X3(k)},where X3(k) = X₁(k)X₂(k), k = 0,1 c) Find linear convolution of x₁ (n) and x₂ (n) (without using DFT). d) Explain briefly why are the results of b) and c) different. What is the minimum number of points DFT required to make the result of b) equal to the result of c)? e) Using the number of points N determined in d), calculate N-point DFT of x₁ (n) and x₂ (n) i.e. X₁,N(K), X2,N(K), k = 0,1,.., N-1 f) Using the number of points N determined in d), calculate xp (n) = IDFTN{Xp (k)},where Xp (k) = X₁,N(K)X2,N(k), k = 0,1,.., N - 1 g) Would you be able to achieve the same result as in f) if you used DFT using N with twice as many points as you determined in d) ?
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