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[[Category:2010 Fall ECE 438 Boutin]]
 
[[Category:2010 Fall ECE 438 Boutin]]
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[[Category:Problem_solving]]
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[[Category:ECE438]]
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[[Category:digital signal processing]]
  
 
== Quiz Questions Pool for Week 13 ==
 
== Quiz Questions Pool for Week 13 ==
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*Under construction --[[User:zhao148|Zhao]]
 
 
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Q1. Show that the DTFT of time-reversal, <math>x[-n]\,\!</math>, is <math>X(-\omega)\,\!</math>  
 
Q1. Show that the DTFT of time-reversal, <math>x[-n]\,\!</math>, is <math>X(-\omega)\,\!</math>  
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Q2. Consider the discrete-time signal
 
Q2. Consider the discrete-time signal
  
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:<math>x[n]=\delta[n]+5 \delta[n-1]+\delta[n-1]- \delta[n-2].</math>
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:<math>x[n]=2\delta[n]+5 \delta[n-1]+\delta[n-1]- \delta[n-2].</math>
  
 
a) Determine the DTFT <math>X(\omega)</math> of x[n] and the DTFT of <math>Y(\omega)</math> of y[n]=x[-n].
 
a) Determine the DTFT <math>X(\omega)</math> of x[n] and the DTFT of <math>Y(\omega)</math> of y[n]=x[-n].
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b) Obtain the signal y[n] whose DFT is <math> (W_N^{k}+W_N^{2k}+W_N^{3k}) X[k]</math>.
 
b) Obtain the signal y[n] whose DFT is <math> (W_N^{k}+W_N^{2k}+W_N^{3k}) X[k]</math>.
  
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c) Now fix <math>N=4</math>. Compute 4-point circular convolution between x[n] and the signal
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c) Now fix <math>N=5</math>. Compute 5-point circular convolution between <math>y[n]</math> and the signal
  
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:<math>h[n]=\delta[n]+\delta[n-1]+\delta[n-2].</math>
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:<math>h[n]=\delta[n]+2\delta[n-1]+3\delta[n-2].</math>
  
 
* [[ECE438_Week13_Quiz_Q3sol|Solution]].
 
* [[ECE438_Week13_Quiz_Q3sol|Solution]].
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b. Find the output image that results when this filter is applied to the input image shown below:<br/>
 
b. Find the output image that results when this filter is applied to the input image shown below:<br/>
  
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0 0 0 0 0 0 0 0 0 0 0 <br/>
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{| cellspacing="1" cellpadding="1" border="0" width="20%"
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0 0 0 0 0 1 0 0 0 0 0 <br/>
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|-
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0 0 0 0 1 1 1 0 0 0 0 <br/>
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| 0  
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0 0 0 1 1 1 1 1 0 0 0 <br/>
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| 0  
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0 0 1 1 1 1 1 1 1 0 0 <br/>
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| 0  
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0 1 1 1 1 1 1 1 1 1 0 <br/>
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| 0  
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0 1 1 1 1 1 1 1 1 1 0 <br/>
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| 0  
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0 1 1 1 1 1 1 1 1 1 0 <br/>
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| 0  
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0 1 1 1 1 1 1 1 1 1 0 <br/>
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| 0  
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0 1 1 1 1 1 1 1 1 1 0 <br/>
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0 0 0 0 0 0 0 0 0 0 0 <br/>
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|}
  
 
c. Find a simple expression for the frequency response H(<math>\mu ,\nu</math>) of this filter.<br/>
 
c. Find a simple expression for the frequency response H(<math>\mu ,\nu</math>) of this filter.<br/>
  
 
* [[ECE438_Week13_Quiz_Q4sol|Solution]].
 
* [[ECE438_Week13_Quiz_Q4sol|Solution]].
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Q5.
 
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* [[ECE438_Week13_Quiz_Q5sol|Solution]].
 
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Latest revision as of 10:43, 11 November 2011


Quiz Questions Pool for Week 13

Q1. Show that the DTFT of time-reversal, $ x[-n]\,\! $, is $ X(-\omega)\,\! $

Q2. Consider the discrete-time signal

$ x[n]=2\delta[n]+5 \delta[n-1]+\delta[n-1]- \delta[n-2]. $

a) Determine the DTFT $ X(\omega) $ of x[n] and the DTFT of $ Y(\omega) $ of y[n]=x[-n].

b) Using your result from part a), compute

$ x[n]* y[n] $.

c) Consider the discrete-time signal

$ z[n]=\left\{ \begin{array}{ll}x[(-n)\mod 4],& 0\leq n < 3,\\ 0 & \text{else }\end{array} \right. $.

Obtain the 4-point circular convolution of x[n] and z[n].

d) When computing the N-point circular convolution of x[n] and the signal

$ z[n]=\left\{ \begin{array}{ll}x[(-n)\mod N],& 0\leq n < N-1,\\ 0 & \text{else }\end{array} \right. $.

how should N be chosen to make sure that the result is the same as the usual convolution between x[n] and z[n]?

  • Same as HW8 Q3 available here.

Q3. Consider the discrete-time signal

$ x[n]=\delta[n] $

a) Obtain the N-point DFT X[k] of x[n].

b) Obtain the signal y[n] whose DFT is $ (W_N^{k}+W_N^{2k}+W_N^{3k}) X[k] $.

c) Now fix $ N=5 $. Compute 5-point circular convolution between $ y[n] $ and the signal

$ h[n]=\delta[n]+2\delta[n-1]+3\delta[n-2]. $

Q4. Consider a 3X3 FIR filter with coefficients h[m,n]

m
n -1 0 1
1 -0.5 0 0.5
0 0 1 0
-1 0.5 0 -0.5

a. Find a difference equation that can be used to implement this filter.
b. Find the output image that results when this filter is applied to the input image shown below:

0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 1 1 1 0 0 0 0
0 0 0 1 1 1 1 1 0 0 0
0 0 1 1 1 1 1 1 1 0 0
0 1 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 1 0
0 0 0 0 0 0 0 0 0 0 0

c. Find a simple expression for the frequency response H($ \mu ,\nu $) of this filter.

Back to ECE 438 Fall 2010 Lab Wiki Page

Back to ECE 438 Fall 2010

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Sees the importance of signal filtering in medical imaging

Dhruv Lamba, BSEE2010