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Homework 3, ECE438, Spring 2017, Prof. Boutin
Hard copy due in class, Wednesday February 1, 2017.
HOMEWORK WITHOUT A PROPER COVER SHEER OR UNSTAPLED WILL NOT BE ACCEPTED. SEE INSTRUCTIONS BELOW.
List all collaborators on your cover page. You can collaborate, but the write up must be your own. Do not copy your collaborator's solution.
The goal of this homework is to to understand the relationship between a signal and a sampling of that signal, viewed in the frequency domain. This time, we are looking at signals beyond pure frequencies.
Question 1
Exercise 1, p. 3.14 of the course notes.
Question 2
Exercise 2, p. 3.14 of the course notes.
Question 3
Consider the signal $ x(t)=\frac{1}{4} \text{sinc } ( \frac{t-2}{5} ). $
a) Obtain the Fourier transform X(f) of the signal and sketch the graph of |X(f)|.
b) What is the Nyquist rate $ f_0 $ for this signal?
c) Let $ T = \frac{1}{4 f_0}. $ Write a mathematical expression for the Fourier transform $ X_s(f) $ of $ x_s(t)= \text{ comb}_T \left( x(t) \right). $ Sketch the graph of $ |X_s(f)| $.
d) Let $ T = \frac{2}{f_0}. $ Write a mathematical expression for the Fourier transform $ {\mathcal X}_d(\omega) $ of $ x_d[n]= x(nT) $ and sketch the graph of $ |{\mathcal X}_d(\omega)| $.
Hand in a hard copy of your solutions. Pay attention to rigor!
Presentation Guidelines
- Write only on one side of the paper.
- Put all problems in order
- Use a "clean" sheet of paper (e.g., not torn out of a spiral book).
- Staple the pages together.
- Include a cover page.
- List all collaborators on your cover page. You can collaborate, but the write up must be your own. Do not copy your collaborator's solution.
- Do not use any other reference besides your course notes. Write a statement to that effect on your cover page, and sign it.
Discussion
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