View source for Slecture Relation in frequency Botao Chen - Rhea

[[Category:slecture]]
[[Category:ECE438Fall2014Boutin]] 
[[Category:ECE]]
[[Category:ECE438]]
[[Category:signal processing]]  

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Frequency domain view of the relationship between a signal and a sampling of that signal
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A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student Botao Chen

Partly based on the [[2014_Fall_ECE_438_Boutin|ECE438 Fall 2014 lecture]] material of [[user:mboutin|Prof. Mireille Boutin]]. 
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==Outline==
#Introduction
#Derivation
#Example
#Conclusion

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==Introduction==

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

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

Now we first concentrate on the relationship between <math> X(f) </math> and <math> X_s(f) </math>. 

We know that <math> x_s(t) = x(t) \times P_T(t) </math>, we can derive the relationship between <math> x_s(t) </math> and <math> x(t) </math> in the following way:

<div style="margin-left: 3em;">
<math>
\begin{align}
F(comb_T(x(t)) &= F(x(t) \times P_T(t))\\
&= X(f)*F(P_T(t))\\
&= X(f)*\frac{1}{T}\sum_{n = -\infty}^\infty \delta(f-\frac{n}{T})\\
&= \frac{1}{T}X(f)*P_\frac{1}{T}(f)\\
&= \frac{1}{T}rep_\frac{1}{T}X(f)\\
\end{align}
</math>
</div>
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Show this relationship in graph below:

Then we are going to find the relation between <math> X_s(f) </math> and <math> X_d(\omega) </math>

We know another way to express CTFT of <math> x_s(t) </math>:

<div style="margin-left: 3em;">
<math>
\begin{align}
X_s(f) &= F(\sum_{n = -\infty}^\infty x(nT)\delta(t-nT))\\
&= \sum_{n = -\infty}^\infty x(nT)F(\delta(t-nT))\\
&= \sum_{n = -\infty}^\infty x(nT)e^{-j2\pi fnT}\\
\end{align}
</math>
</div>
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compare it with DTFT of <math> x_d[n] </math>:

<div style="margin-left: 3em;">
<math>
\begin{align}
X_d(\omega) &= \sum_{n = -\infty}^\infty x_d[n]e^{-j\omega n}\\
&= \sum_{n = -\infty}^\infty x(nT)e^{-j\omega n}\\
\end{align}
</math>
</div>
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we can find that:

<div style="margin-left: 3em;">
<math>
\begin{align}
X_d(2\pi Tf) &= X_s(f)\\
\end{align}
</math>
</div>
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if <math> f = \frac[1][T] </math>