## Contents

# 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.

## Discussion

Write your questions/comments here

- 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

For Q1,b and c, can we just list the complex exponential and say "by comparing to the DFT pairs we can get the answer X[k]=blah" ?

- Yes, you should! -pm