Homework 8, ECE438, Fall 2010, Prof. Boutin
Due in class, Wednesday November 3, 2010.
The discussion page for this homework is here.
Question 1
Consider two discrete-time signals with the same (finite) duration N. Let $ X_1(z) $ be the z-transform of the first signal, and $ X_2[k] $ be the N-point DFT of the second signal. If we assume that
$ X_2[k]=\left. X_1(z) \right|_{z=\frac{1}{2}e^{-j \frac{2 \pi}{N} k}}, \text{ for }k=0,1,\ldots,N-1, $
then what is the relationship between the two signals?
Question 2
Consider the discrete-time signal
$ x[n]=6\delta[n]+5 \delta[n-1]+4 \delta[n-2]+3 \delta[n-3]+2 \delta[n-4]+\delta[n-5]. $
a) Obtain the six-point DFT X[k] of x[n].
b) Obtain the signal y[n] whose DFT is $ W_6^{-2k} X[k] $.
c) Compute six-point circular convolution between x[n] and the signal
$ h[n]=\delta[n]+\delta[n-1]+\delta[n-2]. $
d) If we convolve x[n] with the given h[n] by N-point convolution, how large should N be to insure that the result is the same as the periodic repetition (with period N) of the usual convolution between x[n] and h[n]?
Question 3
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]?
