**FFT Overview**

**FFT and IFFT**

SuperCollider implements a number of UGens supporting FFT based processing. The most basic of these are FFT and IFFT** **which convert data between the time and frequency domains:

**FFT(buffer, input)**

** IFFT(buffer)**

FFT stores spectral data in a local buffer (see Buffer) in the following order: DC, nyquist, real 1f, imag 1f, real 2f, imag 2f, ... real (N-1)f, imag (N-1)f, where f is the frequency corresponding to the window size, and N is the window size / 2.

The buffer's size must correspond to a power of 2. The window size is equivalent to the buffer size, and the window overlap is fixed at 2. Both FFT and IFFT use a Welch window, the combination of which (i.e. Welch^{2}) is a Hanning window.

**Phase Vocoder UGens and Spectral Processing**

In between an FFT and an IFFT one can chain together a number of Phase Vocoder UGens (i.e. 'PV_...') to manipulate blocks of spectral data before reconversion. The process of buffering the appropriate amount of audio, windowing, conversion, overlap-add, etc. is handled for you automatically.

s = Server.local.boot;

b = Buffer.alloc(s,2048,1);

(

{ var in, chain;

in = WhiteNoise.ar(0.8);

chain = FFT(b.bufnum, in);

chain = PV_RandComb(chain, 0.95, Impulse.kr(0.4));

IFFT(chain);

}.play(s);

)

b.free;

PV Ugens write their output data *in place*, i.e. back into the same buffer from which they read. PV UGens which require two buffers write their data into the first buffer, usually called 'bufferA'.

(

b = Buffer.alloc(s,2048,1);

c = Buffer.alloc(s,2048,1);

d = Buffer.read(s,"sounds/a11wlk01.wav");

)

(

{ var inA, chainA, inB, chainB, chain;

inA = LFSaw.ar([100, 150], 0, 0.2);

inB = PlayBuf.ar(1, d.bufnum, BufRateScale.kr(d.bufnum), loop: 1);

chainA = FFT(b.bufnum, inA);

chainB = FFT(c.bufnum, inB);

chain = PV_MagMul(chainA, chainB); // writes into bufferA

0.1 * IFFT(chain);

}.play(s);

)

[b, c, d].do(_.free);

Because each PV UGen overwrites the output of the previous one, it is necessary to copy the data to an additional buffer at the desired point in the chain in order to do parallel processing of input without using multiple FFT UGens. PV_Copy allows for this.

(

b = Buffer.alloc(s,2048,1);

c = Buffer.alloc(s,2048,1);

)

//// proof of concept

(

x = { var inA, chainA, inB, chainB, chain;

inA = LFClipNoise.ar(100);

chainA = FFT(b.bufnum, inA);

chainB = PV_Copy(chainA, c.bufnum);

IFFT(chainA) - IFFT(chainB); // cancels to zero so silent!

}.play(s);

)

x.free;

// IFFTed frames contain the same windowed output data

b.plot(\b, Rect(200, 430, 700, 300)); c.plot(\c, Rect(200, 100, 700, 300));

[b, c].do(_.free);

Note that PV UGens convert as needed between cartesian (complex) and polar representations, therefore when using multiple PV UGens it may be impossible to know in which form the values will be at any given time. FFT produces complex output (see above), so while the following produces a reliable magnitude plot:

b = Buffer.alloc(s,1024);

a = { FFT(b.bufnum, LFSaw.ar(4000)); 0.0 }.play;

(

b.getn(0, 1024, { arg buf;

var z, x;

z = buf.clump(2).flop;

z = [Signal.newFrom(z[0]), Signal.newFrom(z[1])];

x = Complex(z[0], z[1]);

{x.magnitude.plot}.defer

})

)

a.free; b.free;

any Synth using PV UGens might not.

**Multichannel Expansion with FFT UGens**

Care must be taken when using multichannel expansion with FFT UGens, as they require separate buffers. Code such as this can be deceptive:

chain = FFT(bufnum, {WhiteNoise.ar(0.2)}.dup);

The above may seem to work, but does not. It does result in two FFT UGens, but as they both write to the same buffer, the second simply overwrites the data from the first, thus wasting CPU and accomplishing nothing.

When using multichannel expansion with FFT UGens it is necessary to ensure that each one writes to a different buffer. Here's an example of one way to do this:

b = {Buffer.alloc(s,2048,1)}.dup;

(

SynthDef("help-multichannel FFT", { arg out=0, bufnum= #[0, 1]; // bufnum is an array

var in, chain;

in = [SinOsc.ar(0.2), WhiteNoise.ar(0.2)];

chain = FFT(bufnum, in); // each FFT has a different buffer

// now we can multichannel expand as normal

chain = PV_BrickWall(chain, SinOsc.kr(0.1));

Out.ar(out, IFFT(chain));

}).play(s,[\out, 0, \bufnum, b.collect(_.bufnum)]);

)

Note that dup on a UGen just makes a reference to that UGen, because UGen defines -copy to simply return the receiver. (See UGen for more detail.)

a = SinOsc.ar;

a.dup[1] === a

true

Code like IFFT(chain).dup is found throughout the PV help files , and is just a convenient way to copy a mono signal to stereo, without further computation.

See also MultiChannel.

**PV and FFT UGens in the Standard Library**

The following PV UGens are included in the standard SC distribution:

FFT Fast Fourier Transform

IFFT Inverse Fast Fourier Transform

PV_Add complex addition

PV_BinScramble scramble bins

PV_BinShift shift and stretch bin position

PV_BinWipe combine low and high bins from two inputs

PV_BrickWall zero bins

PV_ConformalMap complex plane attack

PV_Copy copy an FFT buffer

PV_CopyPhase copy magnitudes and phases

PV_Diffuser random phase shifting

PV_LocalMax pass bins which are a local maximum

PV_MagAbove pass bins above a threshold

PV_MagBelow pass bins below a threshold

PV_MagClip clip bins to a threshold

PV_MagFreeze freeze magnitudes

PV_MagMul multiply magnitudes

PV_MagDiv division of magnitudes

PV_MagNoise multiply magnitudes by noise

PV_MagShift shift and stretch magnitude bin position

PV_MagSmear average magnitudes across bins

PV_MagSquared square magnitudes

PV_Max maximum magnitude

PV_Min minimum magnitude

PV_Mul complex multiply

PV_PhaseShift shift phase of all bins

PV_PhaseShift270 shift phase by 270 degrees

PV_PhaseShift90 shift phase by 90 degrees

PV_RandComb pass random bins

PV_RandWipe crossfade in random bin order

PV_RectComb make gaps in spectrum

PV_RectComb2 make gaps in spectrum