[pd] hanning window + fft

[Chuckk Hubbard]

On 9/4/06, Kevin McCoy <km.takewithyou at gmail.com> wrote:
>
>
> > Has anyone made an example patch that explains this?  Something like a
> > simple FFT filter with switchable windows, so you can hear the difference?
>
>
>
> Actually anything (links, explanations, etc) that explains or details FFT as
> it relates to Pd would be great... the example patches are a bit cryptic, or
> sometimes jump a bit too quickly, at least for me.  I want to eventually
> generate visuals based on obtaining frequency data from sound, but I just
> couldn't get my mind around fft~, rfft~ etc. so I am going cheap using
> bandpass filters now which is not really what I wanted.  (Sidenote - does
> anyone know how other artists are accomplishing this?  I assumed they were
> using FFT)
>
> I know this is a fairly advanced topic but I'd really love to figure out how
> to make it work.

The explanation that did it best for me is in R.W. Hamming's 'Digital
Filters', a pretty dense book (not for beginners like me, but cross
referencing this book with Miller Puckette's-
http://www-crca.ucsd.edu/~msp/techniques.htm - helped me make sense of
some of it).
Hamming explains that, if you take any two harmonically related sines
or cosines, or one of each, or one of them and a zero-frequency
component, then multiply the two functions over a full period, it will
integrate to zero; meaning that the sums of all the positive values
minus the sums of the negative values is zero.  His explanation is
more accurate, but confusing nonetheless.  The only time the result
isn't zero is if you are multiplying a sine or cosine or a
zero-frequency part by itself at the same frequency.

This means that, since any function is theoretically the sum of an
infinite number of sines and cosines, if you multiply either a sine or
a cosine of a certain frequency by any function, the integral of the
result will be exactly zero for every harmonically related frequency
except the one you multiplied by, so the integral of this product
tells you how strong the particular sine or cosine is in the total.

I'm probably leaving out a lot, and I might be downright wrong about
something.  But having some sort of intuitive understanding seems to
be the only way I can use any tool without feeling like it's magic.

If you did a FFT with a block size larger than an entire audio file,
as I understand it, you wouldn't need a window.  Looping Martin Luther
King's "I Have a Dream" speech (or the audio from the second season of
CHIPS, or a recording of penguins in their natural habitat) over and
over with a 30-minute pause between each round would technically be a
"periodic function", so analyzing it this way would create an accurate
representation; unfortunately, you would have exactly one value for
each frequency, which tells you nothing about the frequency content at
any point.
To analyze changing frequency content, you have to take smaller
blocks.  Repeating these to make them "periodic functions" means the
beginning and end of any one block of sound will run together.  If
this happens to create a continuous function, great, but the chances
of that happening with every block are pretty low (if a block is 64
samples, at 44100 samples per second, and the second season of CHIPS
is 23 episodes, that's a lot of blocks).
Windowing the input simply forces that part of the function to be
continuous, by forcing beginning and end to the same value and
direction.  To prevent the data from that part of the function from
being completely lost, alternating FFTs can be taken with windows in
opposite phase.

I don't exactly understand what happens with the result, though.
Whether the frequency values are averaged, or whether resynthesis is
done with the same overlapping.  I also don't understand
upsampling/downsampling with the block~ object, but I'm not sure
that's related.

-Chuckk












>
> Glad this came up!!
> Kevin
>
>
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