Representation of Discrete periodic signal.

A periodic discrete time signal x[n] with period N can be represented as a Fourier series:

 where

Here the summation ranges over any consecutive N integers of  x[n],

where N is the period of the discrete time signal x[n].

Here equation (i) is called the Synthesis Equation and equation (ii) is called the Analysis Equation.

Now since x[n] is periodic with period N; the Fourier series coefficients are related as;