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Homework 3, ECE438, Fall 2011, Prof. Boutin
Due Wednesday September 28, 2011 (in class)
Questions 1-5
Pick 5 different continuous-time signals x(t) (at least three of which should be band-limited, and at least one should be a pure frequency). For each of the signals:
a) Obtain the Fourier transform X(f) of the signal and sketch the graph of |X(f)|. (Do not simply obtain the Fourier transform from a table; either use the definition of the Fourier transform or use some other way to fully justify your answer.)
b) Find the Nyquist rate $ f_0 $ for the signal (justify your answer).
c) Let
$ T = \frac{1}{3 f_0}. $
Write a mathematical expression for the Fourier transform $ X_s(f) $ of
$ x_s(t)= comb_T \left( x(t) \right). $
Sketch the graph of $ |X_s(f)| $.
d) Let
$ T = \frac{1}{5 f_0}. $
Write a mathematical expression for the Fourier transform $ X_d(\omega) $ of $ x_d[n]= x(nT) $ and sketch the graph of $ |X_d(\omega)| $.
- Note: For help coming up with band-limited signals, see the following practice problem
- Opps! I actually should have written $ X_d( \omega ) $ : (. pm
Question 6
a) What is the relationship between the DT Fourier transform of x[n] and that of y[n]=x[5n]? (Give the mathematical relation and sketch an example.)
b) What is the relationship between the DT Fourier transform of x[n] and that of
$ z[n]=\left\{ \begin{array}{ll} x[n/7],& \text{ if } n \text{ is a multiple of } 7,\\ 0, & \text{ else}. \end{array}\right. $
(Give the mathematical relation and sketch an example.)
Discussion
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