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=Homework 3, [[ECE438]], Fall 2011, [[user:mboutin|Prof. Boutin]]=
 
=Homework 3, [[ECE438]], Fall 2011, [[user:mboutin|Prof. Boutin]]=
 
Due Wednesday September 28, 2011 (in class)
 
Due Wednesday September 28, 2011 (in class)
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<math>T = \frac{1}{5 f_0}.</math>
 
<math>T = \frac{1}{5 f_0}.</math>
  
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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Write a mathematical expression for the Fourier transform <math>X_d(\omega) </math> of  <math>x_d[n]= x(nT)</math> and sketch the graph of <math>|X_d(\omega)| </math>.
  
 
*Note: For help coming up with band-limited signals, see the following [[Practice_problem_finding_band_limited_signals_ECE438F11|practice problem]]
 
*Note: For help coming up with band-limited signals, see the following [[Practice_problem_finding_band_limited_signals_ECE438F11|practice problem]]
 +
*Opps! I actually should have written <math>X_d( \omega ) </math> : (. pm
 
==Question 6==
 
==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.)
 
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.)

Latest revision as of 02:54, 31 August 2013


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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