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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> | + | e) <math class="inline">x_2[n]= e^{j \frac{\pi}{3} n } \cos ( \frac{\pi}{6} n )</math> |

− | f) <math | + | f) <math>x_8[n]= (-j)^n .</math> |

− | g | + | g) <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, 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 3 == | == Question 3 == | ||

Prove the time shifting property of the DFT. | Prove the time shifting property of the DFT. | ||

+ | |||

+ | ---- | ||

+ | == Question 4 == | ||

+ | Under which circumstances can one explicitly reconstruct the DTFT of a finite duration signal from its DFT? Justify your answer mathematically. | ||

+ | |||

+ | (Yes, this is straight from the notes!) | ||

+ | ---- | ||

---- | ---- | ||

---- | ---- |

## Revision as of 14:51, 2 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_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 explicitly reconstruct the DTFT of a finite duration signal from its DFT? 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