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Automatic Control (AC)
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Communicates & Signal Process (CS)
  
Question 3: Optimization
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Question 2: Signal Processing
 
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August 2017
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August 2010
 
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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>.
+
 
<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/>
+
Equation 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>(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 closed-form expression for resulting reconstructed spectrum <math> X_{r}(\omega) </math>. <br/>
(ii)  
+
(ii) <math> X_{N}(k) </math> is computed as a 12-point DFT of x[n] and used in Eqn (1) with N=12. Write a closed-form expression for the resulting reconstructed spectrum <math> X_{r}(\omega) </math>. <br/>
 +
(iii) <math> X_{N}(k) </math> is computed as an 8-point DFT of x[n] ans used in Eqn (1) with N=8. That is, <math> X_{N}(k) </math> is obtained by sampling the DTFT of x[n] at 8 equi-spaced frequencies between 0 and 2<math> \pi </math>. Write a closed-form expression for the resulting reconstructed spectrum <math> X_{r}(\omega) </math>. <br/>
 +
 
 +
(b) Let x[n] be a discrete-time sinewave of length L=12 as defined below. For all sub-parts of part (b), <math> X_{N}(k) </math> is computed as a 12-pt DFT of x[n] and used in Eqn (1) with N=12.
 +
<center><math> x[n]=cos(\frac{\pi}{3} n) \{u[n]-u[n-12]\} </math> </center><br/>
 +
(i) Write a closed-form expression for the resulting reconstructed spectrum <math> X_{r}(\omega) </math>.<br/>
 +
(ii) What is the numerical value of <math> X_{r}(\frac{\pi}{3}) </math>? The answer is a number and you do not need a calculator to determine the value; this also applies to the next 2 parts.<br/>
 +
(iii) What is the numerical value of <math> X_{r}(\frac{5 \pi}{3}) </math>?<br/>
 +
(iv) What is the numerical value of <math> X_{r}(\frac{\pi}{2}) </math>?<br/>
 +
 
 
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 +
Solution :
 +
(a) <br/>
 +
(i)
 +
<math> x[n] = (u[n]-u[n+2]) - 2(u[n]-u[n-4]) </math><br/>
 +
<math> X(w) = e^{-j \frac{12-1}{2}w } \frac{sin(\frac{12}{2}w)}{sin(\frac{1}{2}w)} - 2e^{-j \frac{4-1}{2}w } \frac{sin(\frac{4}{2}w)}{\frac{1}{2}w} </math><br/>
 +
<math> X_{r}(w) = X(w)</math><br/>
 +
(ii)
 +
same,
 +
<math> X_{r}(w)=X(w) </math><br/>
 +
(iii)
 +
<math> x_{r}[n] = \{0,0,0,0,1,1,1,1 \} = (u[n] - u[n-4])* \delta[n-4] </math><br/>
 +
<math> X_{r}(w) = e^{-j \frac{4-1}{2}w} \frac{sin(\frac{4}{2}w)}{\frac{1}{2}w} e^{-j4w}  </math><br/>
 +
 +
(b) <br/>
 +
(i)
 +
<math> X(w)=\frac{1}{2} e^{-j \frac{11}{2}(w- \frac{\pi}{3})} \frac{sin(6(w-\frac{\pi}{3}))}{sin(\frac{1}{2}(w-\frac{\pi}{3}))}  + \frac{1}{2} e^{-j \frac{11}{2} (w+\frac{\pi}{3}) } \frac{sin(6(w+\frac{\pi}{3}))}{sin(\frac{1}{2}(w-\frac{\pi}{3}))} </math><br/>
 +
(ii)
 +
<math> X_{r}(\frac{\pi}{3})=\frac{1}{2} \times  12 = 6 </math><br/>
 +
(iii)
 +
<math> X_{r}(\frac{5}{3} \pi) = \frac{1}{2} \times 12 = 6 </math><br/>
 +
(iv)
 +
<math> X_{r}(\frac{\pi}{2}) = \frac{1}{2} \times 0 = 0 </math><br/>
 +
 +
 +
----
 +
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 +
 +
Problem 2. [50 pts] <br>
 +
Consider a finite-length sinewave of the form below where <math> k_{o} </math> is an interger in the range <math> 0 \leq k_{o} \leq N-1 </math>.<br/>
 +
<center><math> x[n] = e^{j 2 \pi \frac{k_{o}}{N} n} \{ u[n] - u[n-N] \} </math>  (2)</center><br/>
 +
In addition, h[n] is a causal FIR filter of length L, where L < N. In this problem <math> y[n]=x[n] \star h[n] </math> is the linear convolution of the causal sinewave of length N in Equation (1) with a causal FIR filter of length L, where L < N.<br/>
 +
<center><math> y[n]=x[n] \star h[n] </math></center><br/>
 +
(a) The region <math> 0 \leq n \leq L-1 </math> corresponds to partial overlap. The covolution sum can be written as:<br/>
 +
<center><math> y[n]=\sum_{k=??}^{??} h[k] x[n-k] \ \ partial \ \ overlap:\ 0 \leq n \leq L-1  </math>  (3) </center><br/>
 +
Determine the upper and lower limits in the convolution sum above for <math> 0 \leq n \leq L-1 </math><br/>
 +
(b) The region <math> L \leq n \leq N-1 </math> corresponds to full overlap. The convolution sum is: <br/>
 +
<center><math> y[n]=\sum_{k=??}^{??} h[k]x[n-k] \ \ full \ \ overlap: \ L \leq n \leq N-1 </math>  (4)</center><br/>
 +
(i) Determine the upper and lower limits in the convolution sum for <math> L \leq n \leq N-1 </math>.<br/>
 +
(ii) Substituting x[n] in Eqn (1), show that for this range y[n] simplifies to:<br/>
 +
<center><math> y[n]=H_{N}(k_{o}) e^{j 2 \pi \frac{k_{o}}{N} n} \ \ for L \leq n \leq N-1  </math>(5)</center>  <br/>
 +
where <math> H_{N}(k) </math> is the N-point DFT of h[n] evaluated at <math> k = k_{o} </math>. To get the points, you must show all work and explain all details. <br/>
 +
(c) The region <math> N \leq n \leq N+L-2 </math> corresponds to partial overlap. The convolution sum: <br/>
 +
<center><math> y[n] = \sum_{k=??}^{??} h[k]x[n-k] \ \ partial \ \ overlap: \ N \leq n \leq N+L-2 </math>(6)</center>  <br/>
 +
Determine the upper and lower limits in the convolution sum for <math> N \leq n \leq N+L-2 </math>. <br/>
 +
(d) Add the two regions of partial overlap at the beginning and end to form: <br/>
 +
<center><math> z[n] =y[n]+y[n+N]=\sum_{k=??}^{??}h[n]x[n-k] \ \ for: \ 0 \leq n \leq L-1 </math>(7)</center> <br/>
 +
(i) Determine the upper and lower limits in the convolution sum above. <br/>
 +
(ii) Substituting x[n] in Eqn (1), show that for this range z[n] simplifies to: <br/>
 +
<center> <math> z[n] = y[n]+y[n+N]=H_{N}(k_{o})e^{j2 \pi \frac{k_{o}}{N} n} \ \ for \ 0 \leq n \leq L-1 </math>  (8)</center><br/>
 +
where <math> H_{N}(k) </math> is the N-point DFT of h[n] evaluated at <math> k = k_{o}</math> as defined previously. <br/>
 +
(e) <math> y_{N}[n] </math> is formed by computing <math> X_{N}(k) </math> as an N-pt DFT of x[n] in Enq (2), <math> H_{N}(k) </math> as an N-pt DFT of h[n], and then <math> y_{N}[n] </math> as the N-pt inverse DFT of <math> Y_{N}(k) = X_{N}(K)H_{N}(k) </math>. Write a simple, closed-form expression for <math> y_{N}(k) </math>. Is <math> z[n]=y_{N}[n] + y[n+N] \ \ for \ 0 \leq n \leq N-1 </math>? <br/>
 +
 +
 +
Solution: <br/>
 +
(a) <math> y[n] = \sum_{k=0}^{n} h[k] \cdot x[n-k] </math><br/>
 +
So, the lower and upper limits are 0 ~ n. <br/>
 +
(b) <math> y[n] = \sum_{k=0}^{L-1} h[k] \cdot x[n-k] </math><br/>
 +
So, the lower and upper limits are 0 ~ L-1. <br/>
 +
<math> x[n] = e^{2 \pi j (\frac{K_{0}}{N}) n } (u[n] - u[n-N]) </math> <br/>
 +
<math> y[n] = \sum_{k=-\infty}^{+ \infty}  h_{N}[k] \cdot x[n-k] </math><br/>
 +
<math> =  \sum_{k=-\infty}^{+\infty} h_{N}[k] \cdot e^{2 \pi j (\frac{k_{0}}{N})(n-k)} (u[n-k]-u[n-N-k])</math><br/>
 +
<math> = e^{2 \pi jn \frac{k_{0}}{N}} \sum_{k=0}^{L-1} h_{N}[k] e^{-2 \pi j \frac{k_{0}}{N}} k </math><br/>
 +
<math> = e^{2 \pi jn \frac{k_{0}}{N}} \cdot H_{N}(k_{0}) </math><br/>
 +
(c)
 +
<math> y[n] = \sum_{k=n-N+1}^{L-1} h[k] \cdot x[n-k] </math><br/>
 +
(d)
 +
upper and lower limits are 0 ~ L-1. <br/>
 +
<math> z[n] = \sum_{k=-\infty}^{+ \infty}  h_{N}[k] \cdot x[n-k] </math><br/>
 +
<math> =  \sum_{k=-\infty}^{+\infty} h_{N}[k] \cdot e^{2 \pi j (\frac{k_{0}}{N})(n-k)} (u[n-k]-u[n-N-k])</math><br/>
 +
<math> = e^{2 \pi jn \frac{k_{0}}{N}} \sum_{k=0}^{L-1} h_{N}[k] e^{-2 \pi j \frac{k_{0}}{N}} k </math><br/>
 +
<math> = e^{2 \pi jn \frac{k_{0}}{N}} \cdot H_{N}(k_{0}) </math><br/>
 +
(e)
 +
<math> y_{N}[n] = H_{N}(k_{0}) e^{j 2\pi \frac{k_{0}}{N}} n </math><br/>.
 +
Yes, the equation is correct.<br/>
 +
 
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Latest revision as of 15:37, 24 February 2019


ECE Ph.D. Qualifying Exam

Communicates & Signal Process (CS)

Question 2: Signal Processing

August 2010




Problem 1. [50 pts]

Equation 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 $.

$ 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}) } $(1)

(a) Let x[n] be a discrete-time rectangular pulse of length $ L=12 $ as defined below:

$ x[n] = \{-1,-1,-1,-1,1,1,1,1,1,1,1,1 \} $

(i) $ X_{N}(k) $ is computed as a 16-point DFT of x[n] and used in Eqn (1) with N=16. Write a closed-form expression for resulting reconstructed spectrum $ X_{r}(\omega) $.
(ii) $ X_{N}(k) $ is computed as a 12-point DFT of x[n] and used in Eqn (1) with N=12. Write a closed-form expression for the resulting reconstructed spectrum $ X_{r}(\omega) $.
(iii) $ X_{N}(k) $ is computed as an 8-point DFT of x[n] ans used in Eqn (1) with N=8. That is, $ X_{N}(k) $ is obtained by sampling the DTFT of x[n] at 8 equi-spaced frequencies between 0 and 2$ \pi $. Write a closed-form expression for the resulting reconstructed spectrum $ X_{r}(\omega) $.

(b) Let x[n] be a discrete-time sinewave of length L=12 as defined below. For all sub-parts of part (b), $ X_{N}(k) $ is computed as a 12-pt DFT of x[n] and used in Eqn (1) with N=12.

$ x[n]=cos(\frac{\pi}{3} n) \{u[n]-u[n-12]\} $

(i) Write a closed-form expression for the resulting reconstructed spectrum $ X_{r}(\omega) $.
(ii) What is the numerical value of $ X_{r}(\frac{\pi}{3}) $? The answer is a number and you do not need a calculator to determine the value; this also applies to the next 2 parts.
(iii) What is the numerical value of $ X_{r}(\frac{5 \pi}{3}) $?
(iv) What is the numerical value of $ X_{r}(\frac{\pi}{2}) $?


Solution : (a)
(i) $ x[n] = (u[n]-u[n+2]) - 2(u[n]-u[n-4]) $
$ X(w) = e^{-j \frac{12-1}{2}w } \frac{sin(\frac{12}{2}w)}{sin(\frac{1}{2}w)} - 2e^{-j \frac{4-1}{2}w } \frac{sin(\frac{4}{2}w)}{\frac{1}{2}w} $
$ X_{r}(w) = X(w) $
(ii) same, $ X_{r}(w)=X(w) $
(iii) $ x_{r}[n] = \{0,0,0,0,1,1,1,1 \} = (u[n] - u[n-4])* \delta[n-4] $
$ X_{r}(w) = e^{-j \frac{4-1}{2}w} \frac{sin(\frac{4}{2}w)}{\frac{1}{2}w} e^{-j4w} $

(b)
(i) $ X(w)=\frac{1}{2} e^{-j \frac{11}{2}(w- \frac{\pi}{3})} \frac{sin(6(w-\frac{\pi}{3}))}{sin(\frac{1}{2}(w-\frac{\pi}{3}))} + \frac{1}{2} e^{-j \frac{11}{2} (w+\frac{\pi}{3}) } \frac{sin(6(w+\frac{\pi}{3}))}{sin(\frac{1}{2}(w-\frac{\pi}{3}))} $
(ii) $ X_{r}(\frac{\pi}{3})=\frac{1}{2} \times 12 = 6 $
(iii) $ X_{r}(\frac{5}{3} \pi) = \frac{1}{2} \times 12 = 6 $
(iv) $ X_{r}(\frac{\pi}{2}) = \frac{1}{2} \times 0 = 0 $




Problem 2. [50 pts]
Consider a finite-length sinewave of the form below where $ k_{o} $ is an interger in the range $ 0 \leq k_{o} \leq N-1 $.

$ x[n] = e^{j 2 \pi \frac{k_{o}}{N} n} \{ u[n] - u[n-N] \} $ (2)

In addition, h[n] is a causal FIR filter of length L, where L < N. In this problem $ y[n]=x[n] \star h[n] $ is the linear convolution of the causal sinewave of length N in Equation (1) with a causal FIR filter of length L, where L < N.

$ y[n]=x[n] \star h[n] $

(a) The region $ 0 \leq n \leq L-1 $ corresponds to partial overlap. The covolution sum can be written as:

$ y[n]=\sum_{k=??}^{??} h[k] x[n-k] \ \ partial \ \ overlap:\ 0 \leq n \leq L-1 $ (3)

Determine the upper and lower limits in the convolution sum above for $ 0 \leq n \leq L-1 $
(b) The region $ L \leq n \leq N-1 $ corresponds to full overlap. The convolution sum is:

$ y[n]=\sum_{k=??}^{??} h[k]x[n-k] \ \ full \ \ overlap: \ L \leq n \leq N-1 $ (4)

(i) Determine the upper and lower limits in the convolution sum for $ L \leq n \leq N-1 $.
(ii) Substituting x[n] in Eqn (1), show that for this range y[n] simplifies to:

$ y[n]=H_{N}(k_{o}) e^{j 2 \pi \frac{k_{o}}{N} n} \ \ for L \leq n \leq N-1 $(5)

where $ H_{N}(k) $ is the N-point DFT of h[n] evaluated at $ k = k_{o} $. To get the points, you must show all work and explain all details.
(c) The region $ N \leq n \leq N+L-2 $ corresponds to partial overlap. The convolution sum:

$ y[n] = \sum_{k=??}^{??} h[k]x[n-k] \ \ partial \ \ overlap: \ N \leq n \leq N+L-2 $(6)

Determine the upper and lower limits in the convolution sum for $ N \leq n \leq N+L-2 $.
(d) Add the two regions of partial overlap at the beginning and end to form:

$ z[n] =y[n]+y[n+N]=\sum_{k=??}^{??}h[n]x[n-k] \ \ for: \ 0 \leq n \leq L-1 $(7)

(i) Determine the upper and lower limits in the convolution sum above.
(ii) Substituting x[n] in Eqn (1), show that for this range z[n] simplifies to:

$ z[n] = y[n]+y[n+N]=H_{N}(k_{o})e^{j2 \pi \frac{k_{o}}{N} n} \ \ for \ 0 \leq n \leq L-1 $ (8)

where $ H_{N}(k) $ is the N-point DFT of h[n] evaluated at $ k = k_{o} $ as defined previously.
(e) $ y_{N}[n] $ is formed by computing $ X_{N}(k) $ as an N-pt DFT of x[n] in Enq (2), $ H_{N}(k) $ as an N-pt DFT of h[n], and then $ y_{N}[n] $ as the N-pt inverse DFT of $ Y_{N}(k) = X_{N}(K)H_{N}(k) $. Write a simple, closed-form expression for $ y_{N}(k) $. Is $ z[n]=y_{N}[n] + y[n+N] \ \ for \ 0 \leq n \leq N-1 $?


Solution:
(a) $ y[n] = \sum_{k=0}^{n} h[k] \cdot x[n-k] $
So, the lower and upper limits are 0 ~ n.
(b) $ y[n] = \sum_{k=0}^{L-1} h[k] \cdot x[n-k] $
So, the lower and upper limits are 0 ~ L-1.
$ x[n] = e^{2 \pi j (\frac{K_{0}}{N}) n } (u[n] - u[n-N]) $
$ y[n] = \sum_{k=-\infty}^{+ \infty} h_{N}[k] \cdot x[n-k] $
$ = \sum_{k=-\infty}^{+\infty} h_{N}[k] \cdot e^{2 \pi j (\frac{k_{0}}{N})(n-k)} (u[n-k]-u[n-N-k]) $
$ = e^{2 \pi jn \frac{k_{0}}{N}} \sum_{k=0}^{L-1} h_{N}[k] e^{-2 \pi j \frac{k_{0}}{N}} k $
$ = e^{2 \pi jn \frac{k_{0}}{N}} \cdot H_{N}(k_{0}) $
(c) $ y[n] = \sum_{k=n-N+1}^{L-1} h[k] \cdot x[n-k] $
(d) upper and lower limits are 0 ~ L-1.
$ z[n] = \sum_{k=-\infty}^{+ \infty} h_{N}[k] \cdot x[n-k] $
$ = \sum_{k=-\infty}^{+\infty} h_{N}[k] \cdot e^{2 \pi j (\frac{k_{0}}{N})(n-k)} (u[n-k]-u[n-N-k]) $
$ = e^{2 \pi jn \frac{k_{0}}{N}} \sum_{k=0}^{L-1} h_{N}[k] e^{-2 \pi j \frac{k_{0}}{N}} k $
$ = e^{2 \pi jn \frac{k_{0}}{N}} \cdot H_{N}(k_{0}) $
(e) $ y_{N}[n] = H_{N}(k_{0}) e^{j 2\pi \frac{k_{0}}{N}} n $
. Yes, the equation is correct.



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