KISS FFT - A mixed-radix Fast Fourier Transform based up on the principle, 
"Keep It Simple, Stupid."

    There are many great fft libraries already around.  Kiss FFT is not trying
to be better than any of them.  It only attempts to be a reasonably efficient, 
moderately useful FFT that can use fixed or floating data types and can be 
incorporated into someone's C program in a few minutes with trivial licensing.

USAGE:

    The basic usage for 1-d complex FFT is:

        #include "kiss_fft.h"

        void * cfg = kiss_fft_alloc( nfft ,inverse_fft );

        while ...
        
            ... // put kth sample in cx_in[k].r and cx_in[k].i
            
            kiss_fft( cfg , cx_in , cx_out );
            
            ... // transformed. DC is in cx_out[0].r and cx_out[0].i 
            
        free(cfg);

    Note: frequency-domain data is stored from dc up to 2pi.
    so cx_out[0] is the dc bin of the FFT
    and cx_out[nfft/2] is the Nyquist bin (if exists)

    Declarations are in "kiss_fft.h", along with a brief description of the 
functions you'll need to use. 

Code definitions for 1d complex FFTs are in kiss_fft.c.

You can do other cool stuff with the extras you'll find in tools/

    * arbitrary dimension FFTs (complex only currently, apologies to Steve DeKorte -- mebbe next time )
    * real FFTs
    * fast convolution filtering

The core fft and most tools/ code can be compiled to use float, double 
or 16bit short samples. The default is float.


BACKGROUND:

    I started coding this because I couldn't find a fixed point FFT that didn't 
use assembly code.  I started with floating point numbers so I could get the 
theory straight before working on fixed point issues.  In the end, I had a 
little bit of code that could be recompiled easily to do ffts with short, float
or double (other types should be easy too).  

    Once I got my FFT working, I was curious about the speed compared to
a well respected and highly optimized fft library.  I don't want to criticize 
this great library, so let's call it FFT_BRANDX.
During this process, I learned:

    1. FFT_BRANDX has more than 100K lines of code. The core  of kiss_fft is about 500 lines (cpx 1-d).
    2. It took me an embarrassingly long time to get FFT_BRANDX working.
    3. A simple program using FFT_BRANDX is 522KB. A similar program using kiss_fft is 18KB.
    4. FFT_BRANDX is roughly twice as fast as KISS FFT.

    It is wonderful that free, highly optimized libraries like FFT_BRANDX exist.
But such libraries carry a huge burden of complexity necessary to extract every 
last bit of performance.

    Sometimes simpler is better, even if it's not better.

PERFORMANCE:
    (on Athlon XP 2100+, with gcc 2.96, optimization O3, float data type)

    Kiss performed 1000 1024-pt cpx ffts in 100 ms of cpu time.
    For comparison, it took md5sum 160ms cputime to process the same amount of data

    Transforming 5 minutes of CD quality audio takes about 1 second (nfft=1024). 

DO NOT:
    ... use Kiss if you need the Fastest Fourier Transform in the World
    ... ask me to add features that will bloat the code

UNDER THE HOOD:

    Kiss FFT uses a time decimation, mixed-radix, out-of-place FFT. 
No scaling is done.  Optimized butterflies are used for factors 2,3,4, and 5. 

    The real optimization code only works for even length ffts.  It does two half-length
    FFTs in parallel (packed into real&imag) then twiddles.

    The fast convolution filtering uses the overlap-scrap method, slightly 
    modified to put the scrap at the tail.
    

LICENSE:
    BSD, see COPYING for details. Basically, "free to use&change, give credit where due, no guarantees"

TODO:
    *) Add real optimization for odd length FFTs (DST?)
    *) Add real optimization to the n-dimensional FFT
    *) Add simple windowing function, e.g. Hamming : w(i)=.54-.46*cos(2pi*i/(n-1))
    *) Make the fixed point scaling and bit shifts more easily configurable.
    *) Document/revisit the input/output fft scaling
    *) See if the fixed point code can be optimized a little without adding complexity.

AUTHOR:
    Mark Borgerding
    Mark@Borgerding.net