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[[Category:2010 Fall ECE 438 Boutin]] | [[Category:2010 Fall ECE 438 Boutin]] | ||
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| + | [[Category:digital signal processing]] | ||
== Quiz Questions Pool for Week 13 == | == Quiz Questions Pool for Week 13 == | ||
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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 | ||
| − | :<math>x[n]=\delta[n]+5 \delta[n-1]+\delta[n-1]- \delta[n-2].</math> | + | :<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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* [[ECE438_Week13_Quiz_Q4sol|Solution]]. | * [[ECE438_Week13_Quiz_Q4sol|Solution]]. | ||
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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]
| 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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