ECE438 Week13 Quiz - Rhea

Quiz Questions Pool for Week 13

• Under construction --Zhao

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

• Solution.

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]. $

• Solution.

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

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.

• Solution.

Q5.

• Solution.

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