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Homework 6, ECE438, Fall 2016, Prof. Boutin

Hard copy due in class, Wednesday October 5, 2016.

Question 1

Compute the N-point DFT of each of the following periodic signals. ( Use the fundamental period of the signal as N):

a) $ x[n]= \left\{ \begin{array}{ll} 3, &\text{ if }n=0,\\ 0, &\text{ if }n=1,2,3,4,5,6,7, \end{array} \right. $ x[n] periodic with period 8.

b) $ x[n]= e^{j \frac{2}{7} \pi n}; $

c) $ x[n]=sin(\frac{\pi}{8} n) $

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

e) $ x[n]= j^n $

f) $ 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 N-point DFT of

a) $ X[k]= e^{j \frac{\pi}{6} k } $.

b) $ X[k]= e^{j \pi k }+e^{-j \frac{\pi}{2} k} $.

Note: Again, these are VERY simple problems. Have pity for your grader, and try to use a simple approach!

Question 3

Let x[n] be a DT signal of finite duration N and let $ {\mathcal X}(\omega) $ be its DTFT. Consider the periodic signal $ x_M[n]=\sum_{k=-\infty}^\infty x[n+Mk] $ and its M-point DFT $ X_M[k] $.

Can one reconstruct $ {\mathcal X}(\omega) $ from the values of $ X_M[k] $? If yes, explain how and give a mathematical proof of your answer. If no, explain why not (mathematically).

Question 4

What is the effect of padding a finite duration signal with zeros (up to length M>N) before taking the M-point DTF of its periodic repetition (with period M)? How will this affect the DFT? State your answer in a few simple words and give a mathematical proof.

Hint: To answer this question, let x[n] be a signal of duration N beginning with n=0 and let M>N. Let $ y_M[n] $ be the signal given by

$ y[n]=\left\{ \begin{array}{ll} x[n],& 0\leq n <N \\ 0,& N\leq n <M \end{array} \right. $

and show that the M-point DFT $ Y_M[k]={\mathcal X} \left( k\frac{2\pi}{M}\right) $.

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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