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− | + | Topic 3:Fourier transform of "rep" and "comb" | |
</font size> | </font size> | ||
− | A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student | + | 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]]. | Partly based on the [[2014_Fall_ECE_438_Boutin|ECE438 Fall 2014 lecture]] material of [[user:mboutin|Prof. Mireille Boutin]]. | ||
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<font size= 5> | <font size= 5> | ||
− | '''1. | + | '''1.INTRODUCTION: |
− | 2.THEORY:''' | + | In my selecture, I am going to introduce the definition, the Fourier Transformation and the relationship of Comb function and Rep function. |
+ | |||
+ | '''2.THEORY:''' | ||
</font size> | </font size> | ||
<font size= 3>(1)According to the definition of the comb function: </font size> | <font size= 3>(1)According to the definition of the comb function: </font size> | ||
<math>comb_T\big(X(t)\big)= x(t)\cdot\ P_T(t)</math> | <math>comb_T\big(X(t)\big)= x(t)\cdot\ P_T(t)</math> | ||
+ | |||
<font size= 3> where</font size> <math>P_T(t)= \sum_{n=-\infty}^\infty \delta(t-nT)</math> | <font size= 3> where</font size> <math>P_T(t)= \sum_{n=-\infty}^\infty \delta(t-nT)</math> | ||
+ | |||
<font size= 3>Do the Fourier Transform to the function:</font size> | <font size= 3>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> | <math>F\bigg(comb_T\big(x(t)\big)\bigg) = F\big(x(t)\cdot P_T(t)\big)</math> | ||
+ | |||
<font size= 3> | <font size= 3> | ||
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<math>=\frac{1}{T}rep_{1/T}X(f)</math> | <math>=\frac{1}{T}rep_{1/T}X(f)</math> | ||
+ | |||
<font size= 3>(2)According to the definition of Rep function:</font size> | <font size= 3>(2)According to the definition of Rep function:</font size> | ||
+ | |||
<math>rep_T\big(x(t)\big):= x(t)*P_T(t)</math> | <math>rep_T\big(x(t)\big):= x(t)*P_T(t)</math> | ||
− | <math>=x(t)*\sum_{n=-\infty}^\infty \delta(t-nT)</math> | + | <math>=x(t)*\sum_{n=-\infty}^\infty \delta(t-nT)</math> |
− | <font size= | + | <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= 3>[https://engineering.purdue.edu/~bouman/ece637/notes/pdf/RepComb.pdf (3)Graph of the relationship between comb and rep function]</font size> | ||
<font size= 3>Use the impluse-train we get previously, according to the conclusion we get from Fourier Transformation of it, we know:</font size> | <font size= 3>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> | + | <math>F\big(P_T(t)\big)=\frac{1}{T}P_{1/T}(f)</math> |
<font size= 3>So, </font size> <math>F\bigg(rep_T\big(x(t)\big)\bigg)=x(f)\cdot\frac{1}{T}P_{1/T}(f)</math> | <font size= 3>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> | + | <math>=\frac{1}{T}x(f)\cdot P_{1/T}(f)</math> |
− | + | ||
Revision as of 19:28, 30 September 2014
Topic 3:Fourier transform of "rep" and "comb"
A slecture by ECE student Youqin Liu
Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.
ECE438 SELECTURE
1.INTRODUCTION:
In my selecture, I am going to introduce the definition, the Fourier Transformation and the relationship of Comb function and Rep function.
2.THEORY:
(1)According to the definition of the comb function: $ comb_T\big(X(t)\big)= x(t)\cdot\ P_T(t) $
where $ P_T(t)= \sum_{n=-\infty}^\infty \delta(t-nT) $
Do the Fourier Transform to the function:
$ F\bigg(comb_T\big(x(t)\big)\bigg) = F\big(x(t)\cdot P_T(t)\big) $
According to the property of Fourier Transformation, the multiplication in the time domain is equal to the convolution in the frequency domain.
$ F\bigg(comb_T\big(x(t)\big)\bigg) = F\big(x(t)\big)* F\big(P_T(t)\big) $
$ =x(f)*F\big(P_T(t)\big) $
Because $ P_T(t)= \sum_{n=-\infty}^\infty \delta(t-nT) $ is a periodic function , so we can expand it to Fourier series.
$ P_T(t)\sum_{n=-\infty}^\infty F_n e^{jn\cdot 2\pi t/T} $
$ \Rightarrow F_n = \frac{1}{T}\int\limits_{-T/2}^{T/2}P_T(t)e^{jn\cdot 2\pi t/T}dt $
$ =\frac{1}{T} $
So, $ P_T(t) = \frac{1}{T}\sum_{n=-\infty}^\infty F_n e^{jn\cdot 2\pi t/T} $
$ =\sum_{n=-\infty}^\infty \frac{1}{T} F(e^{jn\cdot 2\pi t/T}) $
$ =\sum_{n=-\infty}^\infty \frac{1}{T} \delta(f-\frac{n}{T}) $
$ = \frac{1}{T}P_{1/T}(f) $
So, $ F\bigg(comb_T\big(x(t)\big)\bigg)=X(f)*\frac{1}{T}P_{1/T}(f) $
$ =\frac{1}{T}X(f)*P_{1/T}(f) $ $ =\frac{1}{T}rep_{1/T}X(f) $
(2)According to the definition of Rep function:
$ rep_T\big(x(t)\big):= x(t)*P_T(t) $
$ =x(t)*\sum_{n=-\infty}^\infty \delta(t-nT) $
So, $ F\bigg(rep_T\big(x(t)\big)\bigg)=F\bigg(x(t)*\sum_{n=-\infty}^\infty \delta(t-nT)\bigg) $
(3)Graph of the relationship between comb and rep function
Use the impluse-train we get previously, according to the conclusion we get from Fourier Transformation of it, we know:
$ F\big(P_T(t)\big)=\frac{1}{T}P_{1/T}(f) $
So, $ F\bigg(rep_T\big(x(t)\big)\bigg)=x(f)\cdot\frac{1}{T}P_{1/T}(f) $
$ =\frac{1}{T}x(f)\cdot P_{1/T}(f) $
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