Revision as of 09:05, 21 September 2011

Homework 3, ECE438, Fall 2011, Prof. Boutin

Due Wednesday September 28, 2011 (in class)

Question 1

Pick 5 different continuous-time signals x(t). For each of the signals:

a) Obtain the Fourier transform X(f) of the signal and sketch the graph of |X(f)|.

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 $ comb_T \left( x(t) \right) $ and sketch the graph of $ |X_s(f)| $.

d) Let T = . Write a mathematical expression for the Fourier transform $ X_d(f) $ of $ x_d[n]= x(nT) $ and sketch the graph of $ |X_s(f)| $.

Question 2

a) What is the relationship between the DT Fourier transform of x[n] and that of y[n]=x[3n]? (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/4],& \text{ if } n \text{ is a multiple of } 4,\\ 0, & \text{ else}. \end{array}\right. $

(Give the mathematical relation and sketch an example.)

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