Brief explanation about Fourier Transform
for the CT Fourier Transform, there are two important formulas that we have to know which are,
$ x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty}X(\omega)e^{j\omega t} d\omega $
and
$ X(\omega) = \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt $
Above equations are referred to as the Fourier transform pair.
$ X(\omega) $ stands for Fourier Transform and the first equation is inverse Fourier transform equation.
In our textbook(Signal and System,second edition,oppenheim), If we look up p328 and p329, they have very useful Table for Fourier Properties and Basic Fourier Transform Pairs.
for the DT Fourier Transform, it also has two important equations that we have to know which are,
$ x[n]=\frac{1}{2\pi}\int_{2\pi}X(e^{j\omega})e^{j\omega n}d\omega $
$ X(e^{j\omega}) = \sum_{n =-\infty}^{\infty} x[n]e^{-j\omega n} $
$ X(e^{j\omega}) $ is referred to as the discrete time Fourier transform and the pair of equation as the discret time Fourier transform pair.
x[n] equation is the synthesis equation.
p391 and 392 has very useful Tables for DT Fourier Transform.
Example question for FT
Q.Determine the inverse Fourier transforms $ X(\omega) = 4\pi\delta(\omega) + 6\pi\delta(\omega + 2\pi) $
A.$ x(t) = (\frac{1}{2\pi})\int_{-\infty}^{\infty} [4\pi\delta(\omega) + 6\pi\delta(\omega + 2\pi)]e^{j\omega t}d\omega $
$ =(\frac{1}{2\pi})[4\pi e^{j0t} + 6\pi e^{-j2\pi t}] $
$ =1 + 3e^{-j2\pi t}\! $
Q.compute the F.T of $ x[n] = a^n u[n] $
A.$ X(\omega) = \sum^{\infty}_{n=-\infty}x[n]e^{-j\omega n} $
$ =\sum_{\infty}^{n=-\infty}a^{n}u[n]e^{-j\omega n} $
$ =\sum_{n=0}^{\infty}a^{n}e^{-j\omega n} $
$ =\sum_{n=0}^{\infty}(a e^{-jw})^n $
$ =\frac{1}{1-a e^{-j\omega}} $
