The.Algorithm.Design.Manual.Springer-Verlag.1998

Discrete Fourier Transform
an algorithm that computes the discrete Fourier transform in . This is arguably the most
important algorithm known, as measured by practical impact, for it opened the door to modern image
processing. There are several different algorithms that call themselves FFTs, all of which are based on a
divide-and-conquer approach. Essentially, the problem of computing the discrete Fourier transform on n
points is reduced to computing two transforms on n/2 points each and is then applied recursively
The FFT usually assumes that n is a power of two. If this is not the case for your data, you are usually
better off padding your data with zeros to create elements rather than hunting for a more general
code.
Since many image processing systems have strong real-time constraints, FFTs are often implemented in
hardware, or at least in assembly language tuned to the particular machine. Be aware of this possibility if
the codes below prove too slow.
Implementations: FFTPACK is a package of Fortran subprograms for the fast Fourier transform of
periodic and other symmetric sequences, written by P. Swartzrauber. It includes complex, real, sine,
cosine, and quarter-wave transforms. A C language translation of the main routines is also provided.
FFTPACK resides on Netlib (see Section ) at http://www.netlib.org/fftpack.
Algorithm 545 [Fra79] of the Collected Algorithms of the ACM is an implementation of the fast Fourier
transform optimizing virtual memory performance and written in Fortran. See Section for further
information.
XTango (see Section ) is an algorithm animation system for UNIX and X-windows, which includes
an interesting animation of the fast Fourier transform.
A Pascal implementation of the fast Fourier transform for points appears in [MS91]. For more
details, see Section . Sedgewick [Sed92] provides a bare bones implementation of the fast Fourier
transform in C++. See Section for details.
Notes: Brigham [Bri74] is an excellent introduction to Fourier transforms and the FFT and is strongly
recommended, as is the exposition in [PFTV86]. Expositions in algorithms texts on the fast Fourier
transform include [AHU74, Baa88, CLR90, Man89].
Credit for inventing the fast Fourier transform is usually given to Cooley and Tukey [CT65], although it
is not completely deserved. See [Bri74] for a complete history.
An interesting divide-and-conquer algorithm for polynomial multiplication [KO63] does the job in
file:///E|/BOOK/BOOK4/NODE146.HTM (3 of 4) [19/1/2003 1:30:34]