FFTW for MATLAB
			 http://www.fftw.org

This directory contains files that allow you to call FFTW from MATLAB
(instead of MATLAB's own FFT functions).  This is accomplished by
means of a "MEX" program--a MATLAB external function--that wraps
around the FFTW library.

NOTE: you must have MATLAB 5.0 or later to use these routines.

Once you have compiled and installed the MEX (see below), using FFTW
from within MATLAB is simple:

The forward transform:
	b = fftw(a,-1)

The backwards transform:
	c = fftw(b,+1)

Note that FFTW computes the unnormalized DFT, so "c" in the above code
is a scaled version of the original "a".  (To get back the original
"a", you would compute: c / prod(size(c)).)

To get help on using FFTW in MATLAB, simply type "help fftw" at the
MATLAB prompt.

There are a few points that you should be aware of:

* The first call is expensive:

The first time you call FFTW from within MATLAB, it performs
expensive one-time computations.  (It is figuring out a "plan"--see
the FFTW manual for more information on what is happening.)  So, the
first FFT you compute is slow (it probably takes several seconds).
However, subsequent transforms of the same size will reuse the initial
computations, and will be quite fast (often 2-3 times as fast as
MATLAB's built-in FFT).  So, you should use FFTW within MATLAB when
you are computing many FFTs of the same size and the initial cost is
unimportant.  If you just need a single FFT, use MATLAB's built-in
routines.

To reduce the startup cost, at some slight penalty in performance,
replace FFTW_MEASURE in fftw.c with FFTW_ESTIMATE.

* Small transforms are inefficient:

There is a certain amount of overhead involved in calling FFTW from
MATLAB, and this makes small transforms relatively inefficient.  So,
if you are doing very small transforms in MATLAB, you might be better
off with the built-in routines.  (The exact point at which FFTW begins
to win will depend upon your machine.  It is simple for you to use
MATLAB's timing routines to find out what is best in your
application.)

(One of the major costs is in translating the array from MATLAB's
representation, in which real and imaginary parts are stored
separately, to FFTW's representation, in which complex numbers are
stored as adjacent real/imaginary pairs.)

* FFTW computes multi-dimensional transforms:

The FFTW call in MATLAB computes a transform of the same
dimensionality as the matrix that you give it.  Thus, it is analogous
to the "fftn" routine in MATLAB, rather than the "fft" routine.

* All transforms are out-of-place:

Although the FFTW library is capable of performing in-place
multi-dimensional transforms, the MATLAB routine is out-of-place.
This is simply a restriction of the environment--as far as we can
tell, we are not allowed to modify the inputs that are passed to us,
and must return our results in a separate array.

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			     Installation

Installation of the FFTW MEX routines is straightforward.  First, you
have to compile the FFTW library (see the FFTW manual).  Then, you must
compile the file fftw.c in this directory using the MEX compilation
procedure on your machine.  Finally, you take the MEX file that is
produced, along with the fftw.m file in this directory, and install
them wherever you typically put your MATLAB scripts.

The method for compiling MEX files should be described in your MATLAB
manual.  (You will need to link with the FFTW library that you had
compiled earlier.)  On UNIX systems, you can simply type "make", and
the Makefile in this directory should do the right thing.