View source for Fourier Transform rep com Youqin Liu ECE438 slecture - Rhea

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

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Topic 3:Fourier transform of "rep" and "comb"
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A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student Youqin Liu

Partly based on the [[2014_Fall_ECE_438_Boutin|ECE438 Fall 2014 lecture]] material of [[user:mboutin|Prof. Mireille Boutin]]. 
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==1.INTRODUCTION:==
The topic 3 is the Fourier Transform of the Comb and Rep function. In my selecture, I am going to introduce the definition, the Fourier Transformation and the relationship of Comb function and Rep function.

==2.THEORY:==

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[[Image:Combfunction3.jpg]]

Reference:https://engineering.purdue.edu/~bouman/ece637/notes/pdf/RepComb.pdf


<font size= 4>According to the definition of the comb function: </font size>


<math>comb_T\big(X(t)\big)= x(t)\cdot\ P_T(t)</math> 


<font size= 4> where</font size> <math>P_T(t)= \sum_{n=-\infty}^\infty \delta(t-nT)</math>


<font size= 4>Do the Fourier Transform to the function:</font size>


<math>F\bigg(comb_T\big(x(t)\big)\bigg) = F\big(x(t)\cdot P_T(t)\big)</math>


<font size= 4>
According to the property of Fourier Transformation, the multiplication in the time domain is equal to the convolution in the frequency domain.</font size>

<math>F\bigg(comb_T\big(x(t)\big)\bigg) = F\big(x(t)\big)* F\big(P_T(t)\big)</math>

                  <math>=x(f)*F\big(P_T(t)\big)</math>

<font size= 4>Because </font size>   <math>P_T(t)= \sum_{n=-\infty}^\infty \delta(t-nT)</math>   <font size= 4> is a periodic function , so we can expand it to Fourier series. </font size>


 <math>P_T(t)=\sum_{n=-\infty}^\infty F_n e^{jn\cdot 2\pi t/T} </math>


<math>\Rightarrow F_n = \frac{1}{T}\int\limits_{-T/2}^{T/2}P_T(t)e^{jn\cdot 2\pi t/T}dt</math>

       <math>=\frac{1}{T}</math>

<font size= 4>So, </font size>  <math>P_T(t) = \frac{1}{T}\sum_{n=-\infty}^\infty F_n e^{jn\cdot 2\pi t/T} </math>

          <math>=\sum_{n=-\infty}^\infty \frac{1}{T} F(e^{jn\cdot 2\pi t/T}) </math>

          <math>=\sum_{n=-\infty}^\infty \frac{1}{T} \delta(f-\frac{n}{T})</math>

          <math>= \frac{1}{T}P_{1/T}(f)</math>

<font size= 4>So, </font size>  <math>F\bigg(comb_T\big(x(t)\big)\bigg)=X(f)*\frac{1}{T}P_{1/T}(f)</math>

                     <math>=\frac{1}{T}X(f)*P_{1/T}(f)</math>
   
                     <math>=\frac{1}{T}rep_{1/T}X(f)</math>

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<font size =4>(2)</font size>

[[Image:Repfunction3.jpg]]

Reference:https://engineering.purdue.edu/~bouman/ece637/notes/pdf/RepComb.pdf

<font size= 4>According to the definition of Rep function:</font size>


         <math>rep_T\big(x(t)\big):= x(t)*P_T(t)</math>

                    <math>=x(t)*\sum_{n=-\infty}^\infty \delta(t-nT)</math>


<font size= 4>So, </font size><math>F\bigg(rep_T\big(x(t)\big)\bigg)=F\bigg(x(t)*\sum_{n=-\infty}^\infty \delta(t-nT)\bigg)</math>



<font size= 4>Use the impluse-train we get previously, according to the conclusion we get from Fourier Transformation of it, we know:</font size>

          <math>F\big(P_T(t)\big)=\frac{1}{T}P_{1/T}(f)</math>

<font size= 4>So, </font size> <math>F\bigg(rep_T\big(x(t)\big)\bigg)=x(f)\cdot\frac{1}{T}P_{1/T}(f)</math>

                    <math>=\frac{1}{T}x(f)\cdot P_{1/T}(f)</math> 

==[[slecture_title_of_slecture_review|Questions and comments]]==

If you have any questions, comments, etc. please post them on [[slecture_title_of_slecture_review|this page]].
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