Line 49: Line 49:
 
----
 
----
 
== Question 4 ==
 
== Question 4 ==
Under which circumstances can one explicitly reconstruct the DTFT of a finite duration signal from its DFT? Justify your answer mathematically.
+
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!)

Revision as of 13:52, 2 October 2015


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_2[n]= e^{j \frac{\pi}{3} n } \cos ( \frac{\pi}{6} n ) $

f) $ x_8[n]= (-j)^n . $

g) $ 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, 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

Back to ECE438, Fall 2015, Prof. Boutin

Alumni Liaison

BSEE 2004, current Ph.D. student researching signal and image processing.

Landis Huffman