Fourier transform - 2

[logistic_guy]

Any function \(\displaystyle g(t)\) can be split unambiguously into an even part and odd part, as shown by

\(\displaystyle g(t) = g_e(t) + g_o(t)\)

The even part is defined by

\(\displaystyle g_e(t) = \frac{1}{2}\bigg[g(t) + g(-t)\bigg]\)

and the odd part is defined by

\(\displaystyle g_o(t) = \frac{1}{2}\bigg[g(t) - g(-t)\bigg]\)

\(\displaystyle \bold{(a)}\) Evaluate the even and odd parts of a rectangular pulse defined by

\(\displaystyle g(t) = A \ \text{rect}\left(\frac{t}{T} - \frac{1}{2}\right)\)

\(\displaystyle \bold{(b)}\) What are the Fourier transforms of these two parts of the pulse?

 

[logistic_guy]

\(\displaystyle \bold{(a)}\) Evaluate the even and odd parts of a rectangular pulse defined by

\(\displaystyle \textcolor{blue}{\bold{Even \ Part.}}\)

\(\displaystyle g_e(t) = \frac{1}{2}\bigg[g(t) + g(-t)\bigg] = \frac{1}{2}\bigg[A \ \text{rect}\left(\frac{t}{T} - \frac{1}{2}\right) + A \ \text{rect}\left(\frac{-t}{T} - \frac{1}{2}\right)\bigg]\)


\(\displaystyle = \frac{1}{2}\bigg[A \ \text{rect}\left(\frac{t}{T} - \frac{1}{2}\right) + A \ \text{rect}\left(-\left[\frac{t}{T} + \frac{1}{2}\right]\right)\bigg]\)


\(\displaystyle = \frac{A}{2}\bigg[\ \text{rect}\left(\frac{t}{T} - \frac{1}{2}\right) + \ \text{rect}\left(\frac{t}{T} + \frac{1}{2}\right)\bigg] \ \ \ \ \textcolor{red}{\text{Even function}} \rightarrow \textcolor{blue}{\text{rect}(-t) = \text{rect}(t)}\)

 

[logistic_guy]

\(\displaystyle \bold{(a)}\) Evaluate the even and odd parts of a rectangular pulse defined by

\(\displaystyle \textcolor{blue}{\bold{Odd \ Part.}}\)

\(\displaystyle g_o(t) = \frac{1}{2}\bigg[g(t) - g(-t)\bigg] = \frac{1}{2}\bigg[A \ \text{rect}\left(\frac{t}{T} - \frac{1}{2}\right) - A \ \text{rect}\left(\frac{-t}{T} - \frac{1}{2}\right)\bigg]\)


\(\displaystyle = \frac{1}{2}\bigg[A \ \text{rect}\left(\frac{t}{T} - \frac{1}{2}\right) - A \ \text{rect}\left(-\left[\frac{t}{T} + \frac{1}{2}\right]\right)\bigg]\)


\(\displaystyle = \frac{A}{2}\bigg[\ \text{rect}\left(\frac{t}{T} - \frac{1}{2}\right) - \ \text{rect}\left(\frac{t}{T} + \frac{1}{2}\right)\bigg] \ \ \ \ \textcolor{red}{\text{Even function}} \rightarrow \textcolor{blue}{\text{rect}(-t) = \text{rect}(t)}\)

 

[logistic_guy]

\(\displaystyle \bold{(b)}\) What are the Fourier transforms of these two parts of the pulse?

\(\displaystyle \textcolor{blue}{\bold{Even \ Part.}}\)

\(\displaystyle g_e(t) = \frac{A}{2}\bigg[\ \text{rect}\left(\frac{t}{T} - \frac{1}{2}\right) + \ \text{rect}\left(\frac{t}{T} + \frac{1}{2}\right)\bigg]\)

We know that the Fourier transform of \(\displaystyle \text{rect}\left(\frac{t}{T}\right)\) is \(\displaystyle T\text{sinc}(fT)\),

then by using time-shifting property we have:

\(\displaystyle \mathcal{F}\left\{\text{rect}\left(\frac{t}{T} - \frac{1}{2}\right)\right\} = T\text{sinc}(fT) e^{-j\pi fT}\)

And

\(\displaystyle \mathcal{F}\left\{\text{rect}\left(\frac{t}{T} + \frac{1}{2}\right)\right\} = T\text{sinc}(fT) e^{j\pi fT}\)

Then,

\(\displaystyle \mathcal{F}\left\{g_e(t)\right\} = \frac{A}{2}\bigg[\ T\text{sinc}(fT) e^{-j\pi fT} + \ T\text{sinc}(fT) e^{j\pi fT} \ \bigg]\)

With a little simplification, we get:

\(\displaystyle \mathcal{F}\left\{g_e(t)\right\} = \textcolor{blue}{AT\text{sinc}(fT)\cos(\pi fT)}\)