Signals in Natural Domain
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 of a periodic signal

Suppose that is a periodic signal with period N, that is

As is continues time periodic signal, we would like to represent in terms of discrete time complex exponential signals are given by

                                                                                                          (6.1)

All these signals have frequencies is that are multiples of the some fundamental frequency, , and thus harmonically related.

These are two important distinction between continuous time and discrete time complex exponential. The first one is that harmonically related continuous time complex exponential  are all distinct for different values of k , while there are only N different signals in the set.

The reason for this is that discrete time complex exponentials which differ in frequency by integer multiple of are identical. Thus

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 exponential is that for different k have period 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 given by (6.1) will have period N. In a manner analogous to the continuous time, we represent the periodic signal as
                                                                                                       (6.2)

where

                                                                                                          (6.3)

In equation (6.2) and (6.3) we can sum over any consecutive N values. The equation (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 see easily that

Thus discrete Fourier series coefficients are also periodic with the same period N.

Example 1:

So,   and , since the signal is periodic with periodic with period 5, coefficients are also periodic with period 5, and.

 

 

Now we show that substituting equation (6.3) into (6.2) we indeed get.

interchanging the order of summation we get

                                                                                                            (6.4)

Now the sum


if n - m multiple of N

and for ( n - m ) not a multiple of N this is a geometric series, so sum is

As m varies from 0 to N - 1, we have only one value of m namely m = n , for which the inner sum if non-zero. So we set the RHS of (6.4) as.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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