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Homework 3, ECE438, Fall 2016, Prof. Boutin
Hard copy due in class, Wednesday September 14, 2016.
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
Consider the signal $ x(t)=4 \text{sinc } ( \frac{t-3}{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)| $.
Question 2
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
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