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.



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:

(1)

Combfunction3.jpg

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


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)

Repfunction3.jpg

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

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) $


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) $ 

Questions and comments

If you have any questions, comments, etc. please post them on this page.


Back to ECE438 slectures, Fall 2014

Alumni Liaison

Prof. Math. Ohio State and Associate Dean
Outstanding Alumnus Purdue Math 2008

Jeff McNeal