(New page: Category:2010 Fall ECE 438 Boutin == Quiz Questions Pool for Week 13 == *Under construction --Zhao ---- Q1. * Solution. ---- Q2. * S...)
 
 
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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 ==
−
 
−
*Under construction --[[User:zhao148|Zhao]]
 
 
----
 
----
−
Q1.  
+
Q1. Show that the DTFT of time-reversal, <math>x[-n]\,\!</math>, is <math>X(-\omega)\,\!</math>
  
−
* [[ECE438_Week12_Quiz_Q1sol|Solution]].
+
* [[ECE438_Week13_Quiz_Q1sol|Solution]].
 
----
 
----
−
Q2.  
+
Q2. Consider the discrete-time signal
  
−
* Same as HW8, Q2 available [[ECE438_HW8_Solution|here]].
+
:<math>x[n]=2\delta[n]+5 \delta[n-1]+\delta[n-1]- \delta[n-2].</math>
−
----
+
−
Q3.  
+
  
−
* [[Media:Qpw12ece438fa10.pdf|Solution]].
+
a) Determine the DTFT <math>X(\omega)</math> of x[n] and the DTFT of <math>Y(\omega)</math> of y[n]=x[-n].
 +
 
 +
b) Using your result from part a), compute
 +
 
 +
:<math>x[n]* y[n]</math>.
 +
 
 +
c) Consider the discrete-time signal
 +
 
 +
:<math>z[n]=\left\{ \begin{array}{ll}x[(-n)\mod 4],& 0\leq n < 3,\\ 0 & \text{else }\end{array} \right. </math>. 
 +
 
 +
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
 +
 
 +
:<math>z[n]=\left\{ \begin{array}{ll}x[(-n)\mod N],& 0\leq n < N-1,\\ 0 & \text{else }\end{array} \right. </math>. 
 +
 
 +
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 [[ECE438_HW8_Solution|here]].
 
----
 
----
−
Q4.  
+
Q3. Consider the discrete-time signal
  
−
* [[ECE438_Week12_Quiz_Q4sol|Solution]].
+
:<math>x[n]=\delta[n]</math>
 +
 
 +
a) Obtain the N-point DFT X[k] of x[n].
 +
 
 +
b) Obtain the signal y[n] whose DFT is <math> (W_N^{k}+W_N^{2k}+W_N^{3k}) X[k]</math>.
 +
 
 +
c) Now fix <math>N=5</math>. Compute 5-point circular convolution between <math>y[n]</math> and the signal
 +
 
 +
:<math>h[n]=\delta[n]+2\delta[n-1]+3\delta[n-2].</math>
 +
 
 +
* [[ECE438_Week13_Quiz_Q3sol|Solution]].
 
----
 
----
−
Q5.  
+
Q4. Consider a 3X3 FIR filter with coefficients h[m,n] <br/>
 +
 
 +
{| class="wikitable" style="text-align:center" border="1" cellpadding="2" cellspacing="2" width="20%"
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|+ m
 +
! n !! -1!! 0 !! 1
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|-
 +
! 1
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| -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.<br/>
 +
b. Find the output image that results when this filter is applied to the input image shown below:<br/>
 +
 
 +
{| cellspacing="1" cellpadding="1" border="0" width="20%"
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|-
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| 0
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| 0
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| 0
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| 0
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| 0
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| 0
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| 0
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| 0
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| 0
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| 0
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| 0
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|-
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| 0
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| 0
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| 0
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| 0
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| 0
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| 1
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| 0
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| 0
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| 0
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| 0
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| 0
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|-
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| 0
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| 0
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| 0
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| 0
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| 1
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| 1
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| 1
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| 0
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| 0
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| 0
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| 0
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|-
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| 0
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| 0
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| 0
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| 1
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| 1
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| 1
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| 1
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| 1
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| 0
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| 0
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| 0
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|-
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| 0
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| 0
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| 1
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| 1
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| 1
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| 1
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| 1
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| 1
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| 0
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| 0
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|-
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| 0
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| 1
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| 1
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| 1
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|-
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| 0
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| 1
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| 0
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|-
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| 0
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| 1
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| 1
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| 1
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| 1
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| 1
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| 0
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|-
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| 0
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| 1
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| 1
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| 1
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| 1
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| 1
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| 1
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| 1
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| 0
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|-
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| 0
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| 1
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| 1
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| 1
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| 1
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| 1
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| 1
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|-
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| 0
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| 0
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| 0
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| 0
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| 0
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| 0
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| 0
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|}
  
−
* [[ECE438_Week12_Quiz_Q5sol|Solution]].
+
c. Find a simple expression for the frequency response H(<math>\mu ,\nu</math>) of this filter.<br/>
  
 +
* [[ECE438_Week13_Quiz_Q4sol|Solution]].
 
----
 
----
  

Latest revision as of 09:43, 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.

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