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
2πN kn, k = 0,±1,±2, ... (6.1)All these signals have frequencies is that are multiples of the some fundamentalfrequency, 2πN , 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 ejΩ0kt 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 2π are identical. Thus
{
ej
2πN kn} = {ej 2πN (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 {ejΩ0kt} for different k have period 2π
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 2πN kn (6.2)where
X[k] =
N
−
1 n−0 x[n]e−j 2πN 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