# Homework 4, ECE438, Fall 2011, Prof. Boutin

Due Wednesday October 5, 2011 (in class)

## Questions 1

Compute the DFT of the following signals

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

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

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

## Question 2

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

## Question 3

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

## Question 4

Prove the time shifting property of the DFT.