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The Fast Fourier Transform 187
5.6 THE FAST FOURIER TRANSFORM
The DFT (5.24) introduced earlier is the only transform that is discrete in
both the time and the frequency domains, and is defined for finite-duration
sequences. Although it is a computable transform, the straightforward
implementation of (5.24) is very inefficient, especially when the sequence
length N is large. In 1965 Cooley and Tukey [1] showed a procedure to
substantially reduce the amount of computations involved in the DFT.
This led to the explosion of applications of the DFT, including in the
digital signal processing area. Furthermore, it also led to the development
of other efficient algorithms. All these efficient algorithms are collectively
known as fast Fourier transform (FFT) algorithms.
Consider an N-point sequence x(n). Its N-point DFT is given by
(5.24) and reproduced here
X(k) =
N−1
∑
n=0
x(n)W nk
N , 0 ≤ k ≤ N − 1 (5.46)
where W N = e −j2π/N .Toobtain one sample of X(k), we need N complex
multiplications and (N −1) complex additions. Hence to obtain a complete
set of DFT coefficients, we need N 2 complex multiplications and N(N −1)
≃ N 2 complex additions. Also one has to store N 2 complex coefficients
{
W
nk
N
}
(or generate internally at an extra cost). Clearly, the number of
DFT computations for an N-point sequence depends quadratically on N,
which will be denoted by the notation
C N = o ( N 2)
For large N, o ( N 2) is unacceptable in practice. Generally, the processing
time for one addition is much less than that for one multiplication.
Hence from now on we will concentrate on the number of complex multiplications,
which itself requires 4 real multiplications and 2 real additions.
Goal of an Efficient Computation In an efficiently designed algorithm
the number of computations should be constant per data sample,
and therefore the total number of computations should be linear with
respect to N.
The quadratic dependence on N can be reduced by realizing that most
of the computations (which are done again and again) can be eliminated
using the periodicity property
W kn
N
= W k(n+N)
N
= W (k+N)n
N
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