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)

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, $F\bigg(rep_T\big(x(t)\big)\bigg)=F\bigg(x(t)*\sum_{n=-\infty}^\infty \delta(t-nT)\bigg)$

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, $F\bigg(rep_T\big(x(t)\big)\bigg)=x(f)\cdot\frac{1}{T}P_{1/T}(f)$

$=\frac{1}{T}x(f)\cdot P_{1/T}(f)$