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Problem 1. [50 pts] <br> | Problem 1. [50 pts] <br> | ||
Equation 1 below is the formula for reconstructing the DTFT, <math> X(\omega) </math>, from <math>N</math> equi-spaced samples of the DTFT over <math> 0 \leq \omega \leq 2\pi </math>. <math> X_{N}(k) = X(\frac{2\pi k}{N},k=0,1,...,N-1) </math> is the N-pt DFT of x[n], which corresponds to N equi-spaced samples of the DTFT of x[n] over <math>0 \leq \omega \leq 2\pi</math>. | Equation 1 below is the formula for reconstructing the DTFT, <math> X(\omega) </math>, from <math>N</math> equi-spaced samples of the DTFT over <math> 0 \leq \omega \leq 2\pi </math>. <math> X_{N}(k) = X(\frac{2\pi k}{N},k=0,1,...,N-1) </math> is the N-pt DFT of x[n], which corresponds to N equi-spaced samples of the DTFT of x[n] over <math>0 \leq \omega \leq 2\pi</math>. | ||
| − | <center><math> X_{r}(\omega)=\sum_{k=0}^{N-1} X_{N}(k) \frac{sin[\frac{N}{2}(\omega - \frac{2 \pi k}{N})]}{N sin[\frac{1}{2} (\omega -\frac{2 \pi k}{N})]} e^{-j\frac{N-1}{2}(\omega - \frac{2 \pi k}{N}) } </math></center> | + | <center><math> X_{r}(\omega)=\sum_{k=0}^{N-1} X_{N}(k) \frac{sin[\frac{N}{2}(\omega - \frac{2 \pi k}{N})]}{N sin[\frac{1}{2} (\omega -\frac{2 \pi k}{N})]} e^{-j\frac{N-1}{2}(\omega - \frac{2 \pi k}{N}) } </math>(1)</center><br/> |
(a) Let x[n] be a discrete-time rectangular pulse of length <math>L=12</math> as defined below: <br/> | (a) Let x[n] be a discrete-time rectangular pulse of length <math>L=12</math> as defined below: <br/> | ||
| − | <center><math> x[n] = {-1,-1,-1,-1,1,1,1,1,1,1,1,1} </math></center> <br/> | + | <center><math> x[n] = \{-1,-1,-1,-1,1,1,1,1,1,1,1,1 \} </math></center> <br/> |
(i) <math> X_{N}(k) </math> is computed as a 16-point DFT of x[n] and used in Eqn (1) with N=16. Write a close-form expression for resulting reconstructed spectrum <math> X_{r}(\omega) </math>. <br/> | (i) <math> X_{N}(k) </math> is computed as a 16-point DFT of x[n] and used in Eqn (1) with N=16. Write a close-form expression for resulting reconstructed spectrum <math> X_{r}(\omega) </math>. <br/> | ||
(ii) | (ii) | ||
Revision as of 00:18, 11 February 2019
Communicates & Signal Process (CS)
Question 2: Signal Processing
August 2017
Problem 1. [50 pts]
Equation 1 below is the formula for reconstructing the DTFT, $ X(\omega) $, from $ N $ equi-spaced samples of the DTFT over $ 0 \leq \omega \leq 2\pi $. $ X_{N}(k) = X(\frac{2\pi k}{N},k=0,1,...,N-1) $ is the N-pt DFT of x[n], which corresponds to N equi-spaced samples of the DTFT of x[n] over $ 0 \leq \omega \leq 2\pi $.
(a) Let x[n] be a discrete-time rectangular pulse of length $ L=12 $ as defined below:
(i) $ X_{N}(k) $ is computed as a 16-point DFT of x[n] and used in Eqn (1) with N=16. Write a close-form expression for resulting reconstructed spectrum $ X_{r}(\omega) $.
(ii)
