Homework 4, ECE438, Fall 2011, Prof. Boutin

Due Wednesday October 5, 2011 (in class)

Questions 1

Compute the DFT of the following signals

b) $x_1[n] = \left\{ \begin{array}{ll} 1, & n \text{ multiple of } N\\ 0, & \text{ else}. \end{array} \right.$

a) $x_2[n]= e^{j \frac{\pi}{3} n } \cos ( \frac{\pi}{6} \pi )$

c) $x_3[n] =(\frac{1}{\sqrt{2}}+j \frac{1}{\sqrt{2}})^n$

Compute the inverse DFT of $X[k]= e^{j \pi k }+e^{-j \frac{\pi}{2} k}$ .

Under which circumstances can one explicitly reconstruct the DTFT of a finite duration signal from its DFT? Justify your answer mathematically.

Prove the time shifting property of the DFT.

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Note: When asked to compute DFT of a periodic signal x[n], just use the fundamental period of x[n] as N. Same thing for the inverse DFT. -pm