Homework 6, ECE438, Fall 2015, Prof. Boutin

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

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

f) $$ x_8[n]= (-j)^n . $$

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

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

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!

Prove the time shifting property of the DFT.

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

(Yes, this is straight from the notes!)

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