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d) <math>x[n]= e^{j \frac{2}{\sqrt{3}} \pi n};</math> | d) <math>x[n]= e^{j \frac{2}{\sqrt{3}} \pi n};</math> | ||
| − | e) <math class="inline"> | + | e) <math class="inline">x[n]= e^{j \frac{\pi}{3} n } \cos ( \frac{\pi}{6} n )</math> |
| − | f) <math> | + | f) <math>x[n]= (-j)^n .</math> |
| − | g) <math class="inline"> | + | g) <math class="inline">x[n] =(\frac{1}{\sqrt{2}}+j \frac{1}{\sqrt{2}})^n </math> |
Note: All of these DFTs are VERY simple to compute. If your computation looks like a monster, please find a simpler approach! | Note: All of these DFTs are VERY simple to compute. If your computation looks like a monster, please find a simpler approach! | ||
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== Question 4 == | == Question 4 == | ||
| − | Under which circumstances can one | + | 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!) | (Yes, this is straight from the notes!) | ||
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*Write question/comment here. | *Write question/comment here. | ||
**answer will go here | **answer will go here | ||
| + | |||
| + | * What is the significance of the subscripts on <math>x[n]</math> on parts e, f, and g of Problem 1? Is it supposed to be the period of <math>x[n]</math>? | ||
| + | **I removed the indices. Just take the fundamental period of the signal as N. -pm | ||
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[[2015_Fall_ECE_438_Boutin|Back to ECE438, Fall 2015, Prof. Boutin]] | [[2015_Fall_ECE_438_Boutin|Back to ECE438, Fall 2015, Prof. Boutin]] | ||
Latest revision as of 10:36, 5 October 2015
Contents
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
