| Line 11: | Line 11: | ||
Q2. Consider the discrete-time signal | Q2. Consider the discrete-time signal | ||
| − | <math>x[n]= | + | :<math>x[n]=\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]. | ||
| Line 17: | Line 17: | ||
b) Using your result from part a), compute | b) Using your result from part a), compute | ||
| − | <math>x[n]* y[n]</math>. | + | :<math>x[n]* y[n]</math>. |
c) Consider the discrete-time signal | 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>. | + | :<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]. | Obtain the 4-point circular convolution of x[n] and z[n]. | ||
| Line 27: | Line 27: | ||
d) When computing the N-point circular convolution of x[n] and the signal | 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>. | + | :<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]? | how should N be chosen to make sure that the result is the same as the usual convolution between x[n] and z[n]? | ||
| Line 33: | Line 33: | ||
* Same as HW8 Q3 available [[ECE438_HW8_Solution|here]]. | * Same as HW8 Q3 available [[ECE438_HW8_Solution|here]]. | ||
---- | ---- | ||
| − | Q3. | + | Q3. Consider the discrete-time signal |
| + | |||
| + | :<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=4</math>. Compute 4-point circular convolution between x[n] and the signal | ||
| + | |||
| + | :<math>h[n]=\delta[n]+\delta[n-1]+\delta[n-2].</math> | ||
* [[ECE438_Week13_Quiz_Q3sol|Solution]]. | * [[ECE438_Week13_Quiz_Q3sol|Solution]]. | ||
Revision as of 11:41, 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=4 $. Compute 4-point circular convolution between x[n] and the signal
- $ h[n]=\delta[n]+\delta[n-1]+\delta[n-2]. $
Q4.
Q5.
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