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154 Chapter 5 THE DISCRETE FOURIER TRANSFORM
5.3 THE DISCRETE FOURIER TRANSFORM
The discrete Fourier series provides a mechanism for numerically computing
the discrete-time Fourier transform. It also alerted us to a potential
problem of aliasing in the time domain. Mathematics dictates that the
sampling of the discrete-time Fourier transform result in a periodic sequence
˜x(n). But most of the signals in practice are not periodic. They
are likely to be of finite duration. How can we develop a numerically computable
Fourier representation for such signals? Theoretically, we can take
care of this problem by defining a periodic signal whose primary shape is
that of the finite-duration signal and then using the DFS on this periodic
signal. Practically, we define a new transform called the discrete Fourier
transform (DFT), which is the primary period of the DFS. This DFT
is the ultimate numerically computable Fourier transform for arbitrary
finite-duration sequences.
First we define a finite-duration sequence x(n) that has N samples
over 0 ≤ n ≤ N − 1asanN-point sequence. Let ˜x(n) beaperiodic signal
of period N, created using the N-point sequence x(n); that is, from (5.19)
˜x(n) =
∞∑
r=−∞
x(n − rN)
This is a somewhat cumbersome representation. Using the modulo-N
operation on the argument we can simplify it to
˜x(n) =x(n mod N) (5.21)
A simple way to interpret this operation is the following: if the argument
n is between 0 and N − 1, then leave it as it is; otherwise add or subtract
multiples of N from n until the result is between 0 and N − 1. Note
carefully that (5.21) is valid only if the length of x(n)isN or less. Furthermore,
we use the following convenient notation to denote the modulo-N
operation.
x((n)) N
△
= x(n mod N) (5.22)
Then the compact relationships between x(n) and ˜x(n) are
˜x(n) =x((n)) N
x(n) =˜x(n)R N (n)
(Periodic extension)
(Window operation)
(5.23)
The rem(n,N) function in MATLAB determines the remainder after dividing
n by N. This function can be used to implement our modulo-N
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