public class FFT extends FourierTransform
timeSize
that is not a power of two, an IllegalArgumentException will be
thrown.
A Fourier Transform is an algorithm that transforms a signal in the time
domain, such as a sample buffer, into a signal in the frequency domain, often
called the spectrum. The spectrum does not represent individual frequencies,
but actually represents frequency bands centered on particular frequencies.
The center frequency of each band is usually expressed as a fraction of the
sampling rate of the time domain signal and is equal to the index of the
frequency band divided by the total number of bands. The total number of
frequency bands is usually equal to the length of the time domain signal, but
access is only provided to frequency bands with indices less than half the
length, because they correspond to frequencies below the Nyquist frequency.
In other words, given a signal of length N
, there will be
N/2
frequency bands in the spectrum.
As an example, if you construct an FFT with a
timeSize
of 1024 and and a sampleRate
of 44100
Hz, then the spectrum will contain values for frequencies below 22010 Hz,
which is the Nyquist frequency (half the sample rate). If you ask for the
value of band number 5, this will correspond to a frequency band centered on
5/1024 * 44100 = 0.0048828125 * 44100 = 215 Hz
. The width of
that frequency band is equal to 2/1024
, expressed as a
fraction of the total bandwidth of the spectrum. The total bandwith of the
spectrum is equal to the Nyquist frequency, which in this case is 22050, so
the bandwidth is equal to about 50 Hz. It is not necessary for you to
remember all of these relationships, though it is good to be aware of them.
The function getFreq()
allows you to query the spectrum with a
frequency in Hz and the function getBandWidth()
will return
the bandwidth in Hz of each frequency band in the spectrum.
Usage
A typical usage of the FFT is to analyze a signal so that the
frequency spectrum may be represented in some way, typically with vertical
lines. You could do this in Processing with the following code, where
audio
is an AudioSource and fft
is an FFT.
fft.forward(audio.left); for (int i = 0; i < fft.specSize(); i++) { // draw the line for frequency band i, scaling it by 4 so we can see it a bit better line(i, height, i, height  fft.getBand(i) * 4); }Windowing
Windowing is the process of shaping the audio samples before transforming them to the frequency domain. The Fourier Transform assumes the sample buffer is is a repetitive signal, if a sample buffer is not truly periodic within the measured interval sharp discontinuities may arise that can introduce spectral leakage. Spectral leakage is the speading of signal energy across multiple FFT bins. This "spreading" can drown out narrow band signals and hinder detection.
A windowing function
attempts to reduce spectral leakage by attenuating the measured sample buffer
at its end points to eliminate discontinuities. If you call the window()
function with an appropriate WindowFunction, such as HammingWindow()
,
the sample buffers passed to the object for analysis will be shaped by the current
window before being transformed. The result of using a window is to reduce
the leakage in the spectrum somewhat.
Averages
FFT also has functions that allow you to request the creation of an average spectrum. An average spectrum is simply a spectrum with fewer bands than the full spectrum where each average band is the average of the amplitudes of some number of contiguous frequency bands in the full spectrum.
linAverages()
allows you to specify the number of averages
that you want and will group frequency bands into groups of equal number. So
if you have a spectrum with 512 frequency bands and you ask for 64 averages,
each average will span 8 bands of the full spectrum.
logAverages()
will group frequency bands by octave and allows
you to specify the size of the smallest octave to use (in Hz) and also how
many bands to split each octave into. So you might ask for the smallest
octave to be 60 Hz and to split each octave into two bands. The result is
that the bandwidth of each average is different. One frequency is an octave
above another when it's frequency is twice that of the lower frequency. So,
120 Hz is an octave above 60 Hz, 240 Hz is an octave above 120 Hz, and so on.
When octaves are split, they are split based on Hz, so if you split the
octave 60120 Hz in half, you will get 6090Hz and 90120Hz. You can see how
these bandwidths increase as your octave sizes grow. For instance, the last
octave will always span sampleRate/4  sampleRate/2
, which in
the case of audio sampled at 44100 Hz is 1102522010 Hz. These
logarithmically spaced averages are usually much more useful than the full
spectrum or the linearly spaced averages because they map more directly to
how humans perceive sound.
calcAvg()
allows you to specify the frequency band you want an
average calculated for. You might ask for 60500Hz and this function will
group together the bands from the full spectrum that fall into that range and
average their amplitudes for you.
If you don't want any averages calculated, then you can call
noAverages()
. This will not impact your ability to use
calcAvg()
, it will merely prevent the object from calculating
an average array every time you use forward()
.
Inverse Transform
FFT also supports taking the inverse transform of a spectrum.
This means that a frequency spectrum will be transformed into a time domain
signal and placed in a provided sample buffer. The length of the time domain
signal will be timeSize()
long. The set
and
scale
functions allow you the ability to shape the spectrum
already stored in the object before taking the inverse transform. You might
use these to filter frequencies in a spectrum or modify it in some other way.
FourierTransform
,
The Fast Fourier Transformaverages, avgPerOctave, bandWidth, BARTLETT, BARTLETTHANN, BLACKMAN, COSINE, currentWindow, GAUSS, HAMMING, HANN, imag, LANCZOS, LINAVG, LOGAVG, NOAVG, NONE, octaves, real, sampleRate, spectrum, timeSize, TRIANGULAR, TWO_PI, whichAverage
Constructor and Description 

FFT(int timeSize,
float sampleRate)
Constructs an FFT that will accept sample buffers that are
timeSize long and have been recorded with a sample rate of
sampleRate . 
Modifier and Type  Method and Description 

protected void 
allocateArrays() 
void 
forward(float[] buffer)
Performs a forward transform on
buffer . 
void 
forward(float[] buffReal,
float[] buffImag)
Performs a forward transform on the passed buffers.

void 
forward(float[] buffer,
int startAt)
Performs a forward transform on values in
buffer . 
void 
inverse(float[] buffer)
Performs an inverse transform of the frequency spectrum and places the
result in
buffer . 
void 
scaleBand(int i,
float s)
Scales the amplitude of the
i^{th} frequency band
by s . 
void 
setBand(int i,
float a)
Sets the amplitude of the
i^{th} frequency band to
a . 
avgSize, calcAvg, doWindow, fillSpectrum, forward, forward, freqToIndex, getAverageBandWidth, getAverageCenterFrequency, getAvg, getBand, getBandWidth, getFreq, getSpectrumImaginary, getSpectrumReal, indexToFreq, inverse, inverse, linAverages, logAverages, noAverages, scaleFreq, setComplex, setFreq, specSize, timeSize, window
public FFT(int timeSize, float sampleRate)
timeSize
long and have been recorded with a sample rate of
sampleRate
. timeSize
must be a
power of two. This will throw an exception if it is not.timeSize
 int: the length of the sample buffers you will be analyzingsampleRate
 float: the sample rate of the audio you will be analyzingprotected void allocateArrays()
allocateArrays
in class FourierTransform
public void scaleBand(int i, float s)
FourierTransform
i^{th}
frequency band
by s
. You can use this to shape the spectrum before using
inverse()
.scaleBand
in class FourierTransform
i
 int: the frequency band to modifys
 float: the scaling factorpublic void setBand(int i, float a)
FourierTransform
i^{th}
frequency band to
a
. You can use this to shape the spectrum before using
inverse()
.setBand
in class FourierTransform
i
 int: the frequency band to modifya
 float: the new amplitudepublic void forward(float[] buffer)
FourierTransform
buffer
.forward
in class FourierTransform
buffer
 float[]: the buffer to analyze, must be the same length as timeSize()public void forward(float[] buffer, int startAt)
FourierTransform
buffer
.forward
in class FourierTransform
buffer
 float[]: the buffer to analyze, must be the same length as timeSize()startAt
 int: the index to start at in the buffer. there must be at least timeSize() samples
between the starting index and the end of the buffer. If there aren't, an
error will be issued and the operation will not be performed.public void forward(float[] buffReal, float[] buffImag)
buffReal
 the real part of the time domain signal to transformbuffImag
 the imaginary part of the time domain signal to transformpublic void inverse(float[] buffer)
FourierTransform
buffer
.inverse
in class FourierTransform
buffer
 float[]: the buffer to place the result of the inverse transform in