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| + | [[Category:optimization]] | ||
| − | = | + | <center> |
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| + | [[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] | ||
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| + | Automatic Control (AC) | ||
| + | Question 3: Optimization | ||
| + | </font size> | ||
| − | + | August 2017 | |
| + | </center> | ||
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| + | <!--哈哈我是注释，:Student answers and discussions for [[QE2013_AC-3_ECE580-1|Part 1]],[[QE2013_AC-3_ECE580-2|2]],[[QE2013_AC-3_ECE580-3|3]],[[QE2013_AC-3_ECE580-4|4]],[[QE2013_AC-3_ECE580-5|5]]不会在浏览器中显示。--> | ||
| + | ---- | ||
| + | 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>. | ||
| + | <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>,(1)<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/> | ||
| + | (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) | ||
| + | ---- | ||
| + | ---- | ||
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Revision as of 00:16, 28 January 2019
Automatic Control (AC)
Question 3: Optimization
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)
