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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 the following signals, where N is the fundamental period of the signal:
a) $ x[n]= \left\{ \begin{array}{ll} 3, if n=1\\ 0, if n=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} $.
c)
Note: Again, this is a VERY simple problem. 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
Prove the time-shifting property of the DFT.
Question 5
What is the effect of padding a finite duration signal with zeros (up to length M) before taking the M-point DTF of its periodic repetition (with period M)? How will this affect the DFT?
Hint: To answer this question, let tet x[n] be a signal of duration N beginning with n=0. Let y[n] be given by
$ y[n]=\left\{ \begin{array}{ll} x[n],&\\ 0,& \end{array} \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
