Revision as of 14:08, 6 September 2016

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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