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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>. | ||
| − | c) Now fix <math>N= | + | 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]+\delta[n-1]+\delta[n-2].</math> | + | :<math>h[n]=\delta[n]+2\delta[n-1]+3\delta[n-2].</math> |
* [[ECE438_Week13_Quiz_Q3sol|Solution]]. | * [[ECE438_Week13_Quiz_Q3sol|Solution]]. | ||
Revision as of 10:59, 17 November 2010
Quiz Questions Pool for Week 13
- Under construction --Zhao
Q1. Show that the DTFT of time-reversal, $ x[-n]\,\! $, is $ X(-\omega)\,\! $
Q2. Consider the discrete-time signal
- $ x[n]=\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.
Q5.
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