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Due Wednesday September 28, 2011 (in class) | Due Wednesday September 28, 2011 (in class) | ||
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| − | == | + | ==Questions 1-5== |
| − | Pick 5 different continuous-time signals x(t). For each of the signals: | + | 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)|. | + | 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 <math>f_0</math> for the signal (justify your answer). | b) Find the Nyquist rate <math>f_0</math> for the signal (justify your answer). | ||
| − | c) Let | + | c) Let |
| − | + | <math> T = \frac{1}{3 f_0}.</math> | |
| + | Write a mathematical expression for the Fourier transform <math>X_s(f) </math> of | ||
| − | ==Question | + | <math>x_s(t)= comb_T \left( x(t) \right).</math> |
| − | a) What is the relationship between the DT Fourier transform of x[n] and that of y[n]=x[ | + | |
| + | Sketch the graph of <math>|X_s(f)| </math>. | ||
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| + | d) Let | ||
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| + | <math>T = \frac{1}{5 f_0}.</math> | ||
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| + | Write a mathematical expression for the Fourier transform <math>X_d(f) </math> of <math>x_d[n]= x(nT)</math> and sketch the graph of <math>|X_s(f)| </math>. | ||
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| + | ==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 | b) What is the relationship between the DT Fourier transform of x[n] and that of | ||
<math>z[n]=\left\{ \begin{array}{ll} | <math>z[n]=\left\{ \begin{array}{ll} | ||
| − | x[n/ | + | x[n/7],& \text{ if } n \text{ is a multiple of } 7,\\ |
0, & \text{ else}. | 0, & \text{ else}. | ||
\end{array}\right.</math> | \end{array}\right.</math> | ||
(Give the mathematical relation and sketch an example.) | (Give the mathematical relation and sketch an example.) | ||
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== Discussion == | == Discussion == | ||
Revision as of 11:35, 21 September 2011
Contents
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(f) $ of $ x_d[n]= x(nT) $ and sketch the graph of $ |X_s(f)| $.
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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