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 is:

        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 
        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 even length FFT)

    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.
with sample usage code.  For more functionality, like 2d FFTs you may need to add 
other source files to your project.

    The 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 wanted to get some performance numbers against 
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. KISS has less than 1k.
    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 ffts in 100 ms of cpu time.
    For comparison, it took md5sum 160ms cputime to process the same amount of data

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. 

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

TODO:
    *) Add 2d real optimized FFT
    *) Make a better self-test program(s).  Should report snr & timing for short,float, 
        or double.
    *) 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