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Compute the DFT of the following signals | Compute the DFT of the following signals | ||
| − | + | a) <math class="inline"> | |
x_1[n] = \left\{ | x_1[n] = \left\{ | ||
\begin{array}{ll} | \begin{array}{ll} | ||
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</math> | </math> | ||
| − | + | b) <math class="inline">x_2[n]= e^{j \frac{\pi}{3} n } \cos ( \frac{\pi}{6} n )</math> | |
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*Note: When asked to compute DFT of a periodic signal x[n], just use the fundamental period of x[n] as N. Same thing for the inverse DFT. -pm | *Note: When asked to compute DFT of a periodic signal x[n], just use the fundamental period of x[n] as N. Same thing for the inverse DFT. -pm | ||
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| + | For Q1,b and c, can we just list the complex exponential and say "by comparing to the DFT pairs we can get the answer X[k]=blah" ? | ||
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[[2011_Fall_ECE_438_Boutin|Back to ECE438, Fall 2011, Prof. Boutin]] | [[2011_Fall_ECE_438_Boutin|Back to ECE438, Fall 2011, Prof. Boutin]] | ||
Revision as of 15:04, 3 October 2011
Contents
Homework 4, ECE438, Fall 2011, Prof. Boutin
Due Wednesday October 5, 2011 (in class)
Questions 1
Compute the DFT of the following signals
a) $ x_1[n] = \left\{ \begin{array}{ll} 1, & n \text{ multiple of } N\\ 0, & \text{ else}. \end{array} \right. $
b) $ x_2[n]= e^{j \frac{\pi}{3} n } \cos ( \frac{\pi}{6} n ) $
c) $ x_3[n] =(\frac{1}{\sqrt{2}}+j \frac{1}{\sqrt{2}})^n $
Question 2
Compute the inverse DFT of $ X[k]= e^{j \pi k }+e^{-j \frac{\pi}{2} k} $.
Question 3
Under which circumstances can one explicitly reconstruct the DTFT of a finite duration signal from its DFT? Justify your answer mathematically.
Question 4
Prove the time shifting property of the DFT.
Discussion
Write your questions/comments here
- Note: When asked to compute DFT of a periodic signal x[n], just use the fundamental period of x[n] as N. Same thing for the inverse DFT. -pm
For Q1,b and c, can we just list the complex exponential and say "by comparing to the DFT pairs we can get the answer X[k]=blah" ?
