Chapter 6

Discrete fourier series and Discrete fourier transform

In the last chapter we studied fourier transform representation of aperiodic signal.Now we consider periodic and finite duration sequences.

Discrete fourier series Representation if a periodic signal

Suppose that

*x*[*n*] is a periodic signal with period N, that is

*x*[*n *+ *N*] = *x*[*n*]As is continues time periodic signal, we would like to represent *x*[*n*] in terms ofdiscrete time complex exponential with period N. These signals are given by

e

j

2*πN kn, k *= 0*,±*1*,±*2*, ... *(6.1)All these signals have frequencies is that are multiples of the some fundamentalfrequency, 2*πN *, and thus harmonically related.These are two important distinction between continuous time and discrete timecomplex exponential. The first one is that harmonically related continuous timecomplex exponential *ej*Ω0*kt *are all distinct for different values of *k*, while thereare only *N *different signals in the set.The reason for this is that discrete time complex exponentials which differ infrequency by integer multiple of 2*π *are identical. Thus

{

ej

2*πN kn} *= *{ej *2*πN *(*k*+*N*)*n}*

So if two values of

*k *differ by multiple of *N*, they represent the same signal.Another difference between continuous time and discrete time complex exponentialis that *{ej*Ω0*kt} *for different *k *have period 2*π*

Ω

0*|k|*

, which changes with

*k*.In discrete time exponential, if *k *and *N *are relative prime than the period is *N*

and not

*N/k*. Thus if *N *is a prime number, all the complex exponentials givenby (6.1) will have period *N*.In a manner analogous to the continuous time, we represent the periodic signal

*x*[*n*] as

*x*[*n*] =1

N

N

−

1 *k*=0 *X*[*k*]*ej *2*πN kn *(6.2)where

*X*[*k*] =

N

−

1 *n−*0 *x*[*n*]*e−j *2*πN kn *(6.3)In equation (6.2) and (6.3) we can sum over any consecutive *N *values. Theequation (6.2) is synthesis equation and equation (6.3) is analysis equation.Some people use the faction 1*/N *in analysis equation. From (6.3) we can seeeasily that

*x*[*k*] = *x*[*k *+ *N*]

1

:D