View source for Slecture Topic 7 - Rhea

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<font size="4">Frequency domain view of the relationship between a signal and a sampling of that signal </font> 

A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student Evan Stockrahm 

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  ==

This slecture will discuss the frequency domain view of the relationship between a signal, and a sampling of that signal. Essentially, given a signal x(t), we are going to take a look at the similarities and differences in X(f) and X<sub>s</sub>(f). X<sub>s</sub>(f) is the Fourier Transform of the sampling, x<sub>s</sub>(t), of x(t). 

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== Derivation  ==

Given an arbitrary signal x(t), its Fourier Transform is X(f) 

The sampling of signal x(t), is the comb of x(t), which is equivalent to multiplying x(t) by the impulse train p<sub>T</sub>(t). 

So, x<sub>s</sub>(t) = x(t) x p<sub>T</sub>(t) = comb<sub>T</sub>(x(t)) 

F(x<sub>s</sub>(t)) = X<sub>s</sub>(f) = (1/T) * rep<sub>1/T</sub>(X(f)) 

Analytically, we can see that the frequency domain view of the sampling's amplitude is scaled by a factor of 1/T. The shape of the frequency responses are the same, but it is repeated every 1/T in the sampling's frequency response. In the following example we will see the relationship graphically.<br>
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== Example  ==

x(t) = cos(w<sub>o</sub>t) &nbsp; &nbsp;(pictured below)

[[Image:XtEvan.JPG]]





&nbsp;X(f) = pi*(delta(w-w<sub>o</sub>) + delta(w+w<sub>o</sub>))<br>&nbsp; &nbsp; &nbsp; &nbsp; = 1/2*(delta(f - f<sub>o</sub>) + delta(f + f<sub>o</sub>)) &nbsp; &nbsp; (pictured below)

[[Image:Xf.JPG]]<br> 

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x<sub>s</sub>(t) = cos(w<sub>o</sub>t) * p<sub>T</sub>(t)<br>&nbsp; &nbsp; &nbsp; &nbsp; = combT(cos(w<sub>o</sub>t)) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(pictured below) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;

[[Image:XstEvan.JPG]]



X<sub>s</sub>(f) = (1/T)*rep1/T(1/2*(delta(f - f<sub>o</sub>) + delta(f + f<sub>o</sub>))) &nbsp; &nbsp; &nbsp; (pictured below)<br>

[[Image:Xsf.JPG]]

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== Conclusion  ==

In conclusion, the frequency domain view of a sampled signal is a scaled and repeated version of the original continuous time signal. The amplitude is scaled by 1/T and the repitition occurs every 1/T.

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If you have any questions, comments, etc. please post them [[Slecture_Topic_7_Evancomments|right here]].
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