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==Questions 1== | ==Questions 1== | ||
| − | Compute the DFT of the following signals | + | 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) <math class="inline"> | a) <math class="inline"> | ||
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</math> | </math> | ||
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| + | b) <math>x_1[n]= e^{j \frac{2}{3} \pi n};</math> | ||
| + | c) <math>x_5[n]= e^{-j \frac{2}{1000} \pi n};</math> | ||
| − | + | d) <math>x_2[n]= e^{j \frac{2}{\sqrt{3}} \pi n};</math> | |
| + | e) <math>x_6[n]= \cos\left( \frac{2}{1000} \pi n\right) ;</math> | ||
| + | |||
| + | f) <math class="inline">x_2[n]= e^{j \frac{\pi}{3} n } \cos ( \frac{\pi}{6} n )</math> | ||
| + | |||
| + | g) <math>x_8[n]= (-j)^n .</math> | ||
| + | |||
| + | h) <math class="inline">x_3[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, look for a simpler approach! | ||
| + | ---- | ||
==Question 2 == | ==Question 2 == | ||
Compute the inverse DFT of <math class="inline">X[k]= e^{j \pi k }+e^{-j \frac{\pi}{2} k} </math>. | Compute the inverse DFT of <math class="inline">X[k]= e^{j \pi k }+e^{-j \frac{\pi}{2} k} </math>. | ||
| − | + | Note: Again, this is a VERY simple problem. Have pity for your grader, and try to use a simple approach! | |
| + | ---- | ||
== Question 3 == | == Question 3 == | ||
Prove the time shifting property of the DFT. | Prove the time shifting property of the DFT. | ||
Latest revision as of 06:30, 29 September 2014
Contents
Homework 5, ECE438, Fall 2014, Prof. Boutin
Hard copy due in class, Monday October 6, 2014.
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.
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_1[n] = \left\{ \begin{array}{ll} 1, & n \text{ multiple of } N\\ 0, & \text{ else}. \end{array} \right. $
b) $ x_1[n]= e^{j \frac{2}{3} \pi n}; $
c) $ x_5[n]= e^{-j \frac{2}{1000} \pi n}; $
d) $ x_2[n]= e^{j \frac{2}{\sqrt{3}} \pi n}; $
e) $ x_6[n]= \cos\left( \frac{2}{1000} \pi n\right) ; $
f) $ x_2[n]= e^{j \frac{\pi}{3} n } \cos ( \frac{\pi}{6} n ) $
g) $ x_8[n]= (-j)^n . $
h) $ x_3[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, look for 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.
Discussion
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