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Now we first concentrate on the relationship between <math> X(f) </math> and <math> X_s(f) </math>.  
 
Now we first concentrate on the relationship between <math> X(f) </math> and <math> X_s(f) </math>.  
 +
 
As I say just now, <math> x_s(t) = x(t)*P_T(t) </math>
 
As I say just now, <math> x_s(t) = x(t)*P_T(t) </math>
 +
Given that <math> \mathcal{F}(x(t)) = X(f) </math>, we can find <math> X_s(f) </math> using the convolution property.
 +
 +
<div style="margin-left: 3em;">
 +
<math>
 +
\begin{align}
 +
X_s(f) &= X(f)*\mathcal{F}(p_{\frac{1}{f_s}})\\
 +
&= X(f)*\mathcal{F}(\sum_{k = -\infty}^\infty \delta(t-\frac{k}{f_s}))\\
 +
&= X(f)*f_s\sum_{k = -\infty}^\infty \delta(f-kf_s)\\
 +
&= f_s\sum_{k = -\infty}^\infty X(f)*\delta(t-\frac{k}{f_s})\\
 +
&= f_s\sum_{k = -\infty}^\infty X(f-kf_s)\\
 +
\end{align}
 +
</math>
 +
</div>
 +
<font size>

Revision as of 21:46, 5 October 2014


Frequency domain view of the relationship between a signal and a sampling of that signal

A slecture by ECE student Botao Chen

Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.


Outline

  1. Introduction
  2. Derivation
  3. Example
  4. Conclusion

Introduction

In this slecture I will discuss about the relations between the original signal $ X(f) $ (the CTFT of $ x(t) $ ), sampling continuous time signal $ X_s(f) $ (the CTFT of $ x_s(t) $ ) and sampling discrete time signal $ X_d(\omega) $ (the DTFT of $ x_d[n] $ ) in frequency domain and give a specific example showing the relations.


Derivation

The first thing which need to be clarified is that there two different types of sampling signal: $ x_s(t) $ and $ x_d[n] $. $ x_s(t) $ is created by multiplying a impulse train $ P_T(t) $ with the original signal $ x(t) $ and actually $ x_s(t) $ is $ comb_T(x(t)) $ where T is the sampling period. However the $ x_d[n] $ is $ x(nT) $ where T is the sampling period.

Now we first concentrate on the relationship between $ X(f) $ and $ X_s(f) $.

As I say just now, $ x_s(t) = x(t)*P_T(t) $ Given that $ \mathcal{F}(x(t)) = X(f) $, we can find $ X_s(f) $ using the convolution property.

$ \begin{align} X_s(f) &= X(f)*\mathcal{F}(p_{\frac{1}{f_s}})\\ &= X(f)*\mathcal{F}(\sum_{k = -\infty}^\infty \delta(t-\frac{k}{f_s}))\\ &= X(f)*f_s\sum_{k = -\infty}^\infty \delta(f-kf_s)\\ &= f_s\sum_{k = -\infty}^\infty X(f)*\delta(t-\frac{k}{f_s})\\ &= f_s\sum_{k = -\infty}^\infty X(f-kf_s)\\ \end{align} $

Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood