Fourier Representation of Finite Duration sequence

The Discrete Fourier Transform (DFT)

We now consider the sequence such that and. Thus can be take non-zero values only for. Such sequences are known as finite length sequences, and N is called the length of the sequence. If a sequence has length M, we consider it to be a length N sequence where. In these cases last ( N - M ) sample values are zero. To each finite length sequence of length N we can always associate a periodic sequence defined by

Note that defined by equation (6.16) will always be a periodic sequence with period N, whether is of finite length N or not. But when has finite length N, we can recover the sequence from by defining

This is because of has finite length N , then there is no overlap between terms and for different values of.

Recall that if
n = kN + r, where
then n modulo N = r ,

i.e. we add or subtract multiple of N from n until we get a number lying between 0 to N - 1. We will use ((n))N to denote n modulo N. Then for finite length sequences of length N equation (6.16) can be written as

We can extract from using equation (6.17). Thus there is one-to- one correspondance between finite length sequences of length N , and periodic sequences of period N.

Given a finite length sequence we can associate a periodic sequence with it.
This periodic sequence has discrete Fourier series coefficients which are also periodic with period N. From equations (6.2) and (6.3) we see that we need values of for and for 0 = k = N - 1. Thus we define discrete Fourier transform of finite length sequence as

where is DFS coefficient of associated periodic sequence. From we can get by the relation.

then from this we can get using synthesis equation (6.2) and finally using equation (6.17). In equations (6.2) and (6.3) summation interval is 0 to N - 1, we can write X [k ] directly in terms of x[n], and x[n] directly in terms of X[k] as

For convenience of notation, we use the complex quantity

with this notation, DFT analysis and synthesis equations are written a follows

If we use values of k and n outside the interval 0 to N - 1 in equation (6.20) and (6.21), then we will not get values zero, but we will get periodic repetition of and respectively. In defining DFT, we are concerned with values only in interval 0 to N - 1. Since a sequence of length M can also be considered a sequence of length , we also specify the length of the sequence by saying N-point-DFT, of sequence.

Sampling of the Fourier transform:

For sequence of length N, we have two kinds of representations, namely, discrete time Fourier transform and discrete Fourier transform. The DFT values can be considered as samples of

(as x[n] = 0 n < 0, for n < 0, and n > N - 1)

Thus is is obtained by sampling at.