The Discrete Time Fourier Transform

Consider a discrete time signal x[n] expressed in terms of its spectrum X(e) as ,

$x[n] = \frac{1}{2\pi} \int_{2\pi} X(e^{j\omega}) e^{j\omega n} d\omega $ …(i)

and , the spectrum X(e) is expressed in terms of x[n] as ,

$X(e^{j\omega}) = \Sigma_{n=-\infty}^{\infty}x[n] e^{-j\omega n}$ …(ii)

X(e) is also referred as the Fourier Transform and this pair of equation is called the Discrete Time Fourier Transform pair .
Equation (i) is known as Synthesis Equation whereas Equation (ii) is known as Analysis Equation .

The Equation (ii) will converge either if x[n] is absolutely summable , i.e.

$\Sigma_{n=-\infty}^{\infty}|x[n]| < \infty $

or , if the sequence has finite energy , i.e.

$\Sigma_{n=-\infty}^{\infty}|x[n]|^2 < \infty $

Fourier Transform of Periodic Signals :

Consider a Periodic sequence x[n] with period N and with the Fourier series representation

$x[n] = \Sigma_{K =} a_k e^{jk (\frac{2\pi}{N})n}$

and , the Fourier Transform is ,

$x(e^{j\omega}) = \Sigma_{k=-\infty}^{\infty} 2\pi a_k \delta(\omega – \frac{2\pi k}{N}) $

Properties of Discrete Time Fourier Transform :

Here , it will be convenient to adopt notation similar to that used to indicate the property of a signal and its Transform , i.e.

$x(e^{j\omega}) = \zeta (x[n])$

$ x[n] = \zeta^{-1} (x(e^{j\omega}) $

$ x[n] \longleftrightarrow x(e^{j\omega}) $