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a) <math>x[n]= \left\{ | a) <math>x[n]= \left\{ | ||
\begin{array}{ll} | \begin{array}{ll} | ||
| − | 3, &\text{ if }n= | + | 3, &\text{ if }n=0,\\ |
| − | 0, &\text{ if }n=2,3,4,5,6,7, | + | 0, &\text{ if }n=1,2,3,4,5,6,7, |
\end{array} | \end{array} | ||
\right.</math> | \right.</math> | ||
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e) <math>x[n]= j^n</math> | e) <math>x[n]= j^n</math> | ||
| − | f) <math class="inline">x[n] =(\frac{1}{\sqrt{2}} | + | f) <math class="inline">x[n] =(\frac{1}{\sqrt{2}}-j \frac{1}{\sqrt{2}})^n </math> |
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b) <math class="inline">X[k]= e^{j \pi k }+e^{-j \frac{\pi}{2} k} </math>. | b) <math class="inline">X[k]= e^{j \pi k }+e^{-j \frac{\pi}{2} k} </math>. | ||
| − | + | Note: Again, these are VERY simple problems. Have pity for your grader, and try to use a simple approach! | |
| − | + | ||
| − | Note: Again, | + | |
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==Question 3== | ==Question 3== | ||
| − | Let x[n] be a DT signal of finite duration N and let <math>{\mathcal X}(\omega)</math> be its DTFT. Consider the periodic signal <math>x_M[n]=\sum_{k=-\infty}^infty x[n+Mk]</math> and its M-point DFT <math>X_M[k]</math>. | + | Let x[n] be a DT signal of finite duration N and let <math>{\mathcal X}(\omega)</math> be its DTFT. Consider the periodic signal <math>x_M[n]=\sum_{k=-\infty}^\infty x[n+Mk]</math> and its M-point DFT <math>X_M[k]</math>. |
Can one reconstruct <math>{\mathcal X}(\omega)</math> from the values of <math>X_M[k]</math>? If yes, explain how and give a mathematical proof of your answer. If no, explain why not (mathematically). | Can one reconstruct <math>{\mathcal X}(\omega)</math> from the values of <math>X_M[k]</math>? If yes, explain how and give a mathematical proof of your answer. If no, explain why not (mathematically). | ||
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== Question 4 == | == 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. | |
| − | + | ||
| − | + | ||
| − | + | ||
| − | 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 x[n] be a signal of duration N beginning with n=0 and let M>N. Let <math>y_M[n]</math> be the signal given by | Hint: To answer this question, let x[n] be a signal of duration N beginning with n=0 and let M>N. Let <math>y_M[n]</math> be the signal given by | ||
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</math> | </math> | ||
| − | and show that the M-point DFT <math>Y_M[k]={\mathcal X} \left( k\frac{2\pi}{M}\right)</math> | + | and show that the M-point DFT <math>Y_M[k]={\mathcal X} \left( k\frac{2\pi}{M}\right)</math>. |
| − | + | ||
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*Write question/comment here. | *Write question/comment here. | ||
**answer will go here | **answer will go here | ||
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| − | [[ | + | [[2016_Fall_ECE_438_Boutin|Back to ECE438, Fall 2016, Prof. Boutin]] |
Latest revision as of 16:11, 27 September 2016
Contents
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
