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Homework 6, ECE438, Fall 2015, Prof. Boutin

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


Question 1

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.
  • Do not let your dog play with your homework.

Discussion

  • Write question/comment here.
    • answer will go here
  • 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] $?
    • I removed the indices. Just take the fundamental period of the signal as N. -pm

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