# Homework 6, ECE438, Fall 2015, Prof. Boutin

Hard copy due in class, Wednesday October 8, 2015.

## Questions 1

Compute the DFT of the following signals x[n] (if possible). How does your answer relate to the Fourier series coefficients of x[n]?

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

b) $x[n]= e^{j \frac{2}{5} \pi n};$

c) $x[n]= e^{-j \frac{2}{5} \pi n};$

d) $x[n]= e^{j \frac{2}{\sqrt{3}} \pi n};$

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

f) $x[n]= (-j)^n .$

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

Note: All of these DFTs are VERY simple to compute. If your computation looks like a monster, please find a simpler approach!

## Question 2

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

Note: Again, this is a VERY simple problem. Have pity for your grader, and try to use a simple approach!

## Question 3

Prove the time shifting property of the DFT.

## Question 4

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

(Yes, this is straight from the notes!)

Hand in a hard copy of your solutions. Pay attention to rigor!

## Presentation Guidelines

• Write only on one side of the paper.
• Use a "clean" sheet of paper (e.g., not torn out of a spiral book).
• Staple the pages together.
• Include a cover page.
• What is the significance of the subscripts on $x[n]$ on parts e, f, and g of Problem 1? Is it supposed to be the period of $x[n]$?