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Transforms 71
The main ingredient in expressions (4.3.1) and (4.3.2) is a term of the form cos m1m2
n π,
where m1 and m2 are integers. Due to the periodicity of the cosine function, the entries
of Kn and ˜ Kn may attain the same values for different choices of i,j ∈ N. The
most remarkable case is when n is of the form 2 k for some k ∈ N. In this situation
a lot of coefficients are identical and regularly distributed in the matrix. The poorest
case is when n is a prime number. The entries generated then attain a larger number
of distinct values and substructures are not easily recognized. These observations are
the starting point of an efficient algorithm for the evaluation of the Fourier transform
and its inverse, with a computational cost less than that needed for a matrix-vector
multiplication. This procedure is known as Fast Fourier Transform (abbreviated by
FFT).
The original version of the FFT was officially introduced in cooley and tukey(1965),
although the concept had already been published. Fast Fourier transforms are best
defined in complex space. On the other hand, the transform involving the cosine can be
interpreted as a type of projection into the set of the real numbers. A lot of variants and
improvements have been added by various authors. At the same time, computer pro-
grams have been coded, with the aim of giving a fast and reliable product. High quality
versions of the FFT based on modern parallel architectures are also available. Applica-
tions in the framework of spectral computations for parallel processors are examined in
pelz(1990).
The FFT algorithm is a little bit tricky and needs a certain amount of perseverance
in order to be fully understood. Many books explain, using numerous examples, the
basic ideas of the FFT as well as series of generalizations. Usually, they start with
the complex formulation restricted to the case n = 2 k . Other generalizations (known
as multi-radix FFT algorithms) apply when n is not a power of 2, giving however less
efficient performances. Among the texts we suggest for instance brigham(1974), bini,
capovani, lotti and romani(1981), nussbaumer(1981). Since a full discussion of the
FFT implementation is beyond the scope of this book, we only present an elementary
exposition to bring out its essential features.
The first problem we have in mind is to determine a fast way to compute matrix-