## ddf.minim.analysis Class FourierTransform

```java.lang.Object
ddf.minim.analysis.FourierTransform
```
Direct Known Subclasses:
DFT, FFT

`public abstract class FourierTransformextends java.lang.Object`

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 a FourierTransform 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 a FourierTransform 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 (one of the derived classes of FourierTransform).

``` 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

FourierTransform 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 60-120 Hz in half, you will get 60-90Hz and 90-120Hz. 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 11025-22010 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 60-500Hz 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

FourierTransform 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.

Author:
Damien Di Fede
The Discrete Fourier Transform

Field Summary
`protected  float[]` `averages`

`protected  int` `avgPerOctave`

`protected  float` `bandWidth`

`static WindowFunction` `BARTLETT`
A constant indicating a Bartlett window should be used on sample buffers.
`static WindowFunction` `BARTLETTHANN`
A constant indicating a Bartlett-Hann window should be used on sample buffers.
`static WindowFunction` `COSINE`
A constant indicating a Cosine window should be used on sample buffers.
`static WindowFunction` `HAMMING`
A constant indicating a Hamming window should be used on sample buffers.
`static WindowFunction` `HANN`
A constant indicating a Hann window should be used on sample buffers.
`protected  float[]` `imag`

`static WindowFunction` `LANCZOS`
A constant indicating a Lanczos window should be used on sample buffers.
`protected static int` `LINAVG`

`protected static int` `LOGAVG`

`protected static int` `NOAVG`

`static WindowFunction` `NONE`
A constant indicating no window should be used on sample buffers.
`protected  int` `octaves`

`protected  float[]` `real`

`protected  int` `sampleRate`

`protected  float[]` `spectrum`

`protected  int` `timeSize`

`static WindowFunction` `TRIANGULAR`
A constant indicating a Triangular window should be used on sample buffers.
`protected static float` `TWO_PI`

`protected  int` `whichAverage`

`protected  WindowFunction` `windowFunction`

Method Summary
`protected abstract  void` `allocateArrays()`

` int` `avgSize()`
Returns the number of averages currently being calculated.
` float` ```calcAvg(float lowFreq, float hiFreq)```
Calculate the average amplitude of the frequency band bounded by `lowFreq` and `hiFreq`, inclusive.
`protected  void` `doWindow(float[] samples)`

`protected  void` `fillSpectrum()`

` void` `forward(AudioBuffer buffer)`
Performs a forward transform on `buffer`.
` void` ```forward(AudioBuffer buffer, int startAt)```
Performs a forward transform on `buffer`.
`abstract  void` `forward(float[] buffer)`
Performs a forward transform on `buffer`.
` void` ```forward(float[] buffer, int startAt)```
Performs a forward transform on values in `buffer`.
` int` `freqToIndex(float freq)`
Returns the index of the frequency band that contains the requested frequency.
` float` `getAverageCenterFrequency(int i)`
Returns the center frequency of the ith average band.
` float` `getAvg(int i)`
Gets the value of the `ith` average.
` float` `getBand(int i)`
Returns the amplitude of the requested frequency band.
` float` `getBandWidth()`
Returns the width of each frequency band in the spectrum (in Hz).
` float` `getFreq(float freq)`
Gets the amplitude of the requested frequency in the spectrum.
` float` `indexToFreq(int i)`
Returns the middle frequency of the ith band.
` void` `inverse(AudioBuffer buffer)`
Performs an inverse transform of the frequency spectrum and places the result in `buffer`.
`abstract  void` `inverse(float[] buffer)`
Performs an inverse transform of the frequency spectrum and places the result in `buffer`.
` void` ```inverse(float[] freqReal, float[] freqImag, float[] buffer)```
Performs an inverse transform of the frequency spectrum represented by freqReal and freqImag and places the result in buffer.
` void` `linAverages(int numAvg)`
Sets the number of averages used when computing the spectrum and spaces the averages in a linear manner.
` void` ```logAverages(int minBandwidth, int bandsPerOctave)```
Sets the number of averages used when computing the spectrum based on the minimum bandwidth for an octave and the number of bands per octave.
` void` `noAverages()`
Sets the object to not compute averages.
`abstract  void` ```scaleBand(int i, float s)```
Scales the amplitude of the `ith` frequency band by `s`.
` void` ```scaleFreq(float freq, float s)```
Scales the amplitude of the requested frequency by `a`.
`abstract  void` ```setBand(int i, float a)```
Sets the amplitude of the `ith` frequency band to `a`.
`protected  void` ```setComplex(float[] r, float[] i)```

` void` ```setFreq(float freq, float a)```
Sets the amplitude of the requested frequency in the spectrum to `a`.
` int` `specSize()`
Returns the size of the spectrum created by this transform.
` int` `timeSize()`
Returns the length of the time domain signal expected by this transform.
` void` `window(WindowFunction windowFunction)`
Sets the window to use on the samples before taking the forward transform.

Methods inherited from class java.lang.Object
`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

Field Detail

### LINAVG

`protected static final int LINAVG`
Constant Field Values

### LOGAVG

`protected static final int LOGAVG`
Constant Field Values

### NOAVG

`protected static final int NOAVG`
Constant Field Values

### NONE

`public static final WindowFunction NONE`
A constant indicating no window should be used on sample buffers. Also referred as a Rectangular window.

### HAMMING

`public static final WindowFunction HAMMING`
A constant indicating a Hamming window should be used on sample buffers.

### HANN

`public static final WindowFunction HANN`
A constant indicating a Hann window should be used on sample buffers.

### COSINE

`public static final WindowFunction COSINE`
A constant indicating a Cosine window should be used on sample buffers.

### TRIANGULAR

`public static final WindowFunction TRIANGULAR`
A constant indicating a Triangular window should be used on sample buffers.

### BARTLETT

`public static final WindowFunction BARTLETT`
A constant indicating a Bartlett window should be used on sample buffers.

### BARTLETTHANN

`public static final WindowFunction BARTLETTHANN`
A constant indicating a Bartlett-Hann window should be used on sample buffers.

### LANCZOS

`public static final WindowFunction LANCZOS`
A constant indicating a Lanczos window should be used on sample buffers.

### TWO_PI

`protected static final float TWO_PI`
Constant Field Values

### timeSize

`protected int timeSize`

### sampleRate

`protected int sampleRate`

### bandWidth

`protected float bandWidth`

### windowFunction

`protected WindowFunction windowFunction`

### real

`protected float[] real`

### imag

`protected float[] imag`

### spectrum

`protected float[] spectrum`

### averages

`protected float[] averages`

### whichAverage

`protected int whichAverage`

### octaves

`protected int octaves`

### avgPerOctave

`protected int avgPerOctave`
Method Detail

### allocateArrays

`protected abstract void allocateArrays()`

### setComplex

```protected void setComplex(float[] r,
float[] i)```

### fillSpectrum

`protected void fillSpectrum()`

### noAverages

`public void noAverages()`
Sets the object to not compute averages.

### linAverages

`public void linAverages(int numAvg)`
Sets the number of averages used when computing the spectrum and spaces the averages in a linear manner. In other words, each average band will be `specSize() / numAvg` bands wide.

Parameters:
`numAvg` - how many averages to compute

### logAverages

```public void logAverages(int minBandwidth,
int bandsPerOctave)```
Sets the number of averages used when computing the spectrum based on the minimum bandwidth for an octave and the number of bands per octave. For example, with audio that has a sample rate of 44100 Hz, `logAverages(11, 1)` will result in 12 averages, each corresponding to an octave, the first spanning 0 to 11 Hz. To ensure that each octave band is a full octave, the number of octaves is computed by dividing the Nyquist frequency by two, and then the result of that by two, and so on. This means that the actual bandwidth of the lowest octave may not be exactly the value specified.

Parameters:
`minBandwidth` - the minimum bandwidth used for an octave
`bandsPerOctave` - how many bands to split each octave into

### window

`public void window(WindowFunction windowFunction)`
Sets the window to use on the samples before taking the forward transform. If an invalid window is asked for, an error will be reported and the current window will not be changed.

Parameters:
`windowFunction` -

### doWindow

`protected void doWindow(float[] samples)`

### timeSize

`public int timeSize()`
Returns the length of the time domain signal expected by this transform.

Returns:
the length of the time domain signal expected by this transform

### specSize

`public int specSize()`
Returns the size of the spectrum created by this transform. In other words, the number of frequency bands produced by this transform. This is typically equal to `timeSize()/2 + 1`, see above for an explanation.

Returns:
the size of the spectrum

### getBand

`public float getBand(int i)`
Returns the amplitude of the requested frequency band.

Parameters:
`i` - the index of a frequency band
Returns:
the amplitude of the requested frequency band

### getBandWidth

`public float getBandWidth()`
Returns the width of each frequency band in the spectrum (in Hz). It should be noted that the bandwidth of the first and last frequency bands is half as large as the value returned by this function.

Returns:
the width of each frequency band in Hz.

### setBand

```public abstract void setBand(int i,
float a)```
Sets the amplitude of the `ith` frequency band to `a`. You can use this to shape the spectrum before using `inverse()`.

Parameters:
`i` - the frequency band to modify
`a` - the new amplitude

### scaleBand

```public abstract void scaleBand(int i,
float s)```
Scales the amplitude of the `ith` frequency band by `s`. You can use this to shape the spectrum before using `inverse()`.

Parameters:
`i` - the frequency band to modify
`s` - the scaling factor

### freqToIndex

`public int freqToIndex(float freq)`
Returns the index of the frequency band that contains the requested frequency.

Parameters:
`freq` - the frequency you want the index for (in Hz)
Returns:
the index of the frequency band that contains freq

### indexToFreq

`public float indexToFreq(int i)`
Returns the middle frequency of the ith band.

Parameters:
`i` - the index of the band you want to middle frequency of

### getAverageCenterFrequency

`public float getAverageCenterFrequency(int i)`
Returns the center frequency of the ith average band.

Parameters:
`i` - which average band you want the center frequency of.

### getFreq

`public float getFreq(float freq)`
Gets the amplitude of the requested frequency in the spectrum.

Parameters:
`freq` - the frequency in Hz
Returns:
the amplitude of the frequency in the spectrum

### setFreq

```public void setFreq(float freq,
float a)```
Sets the amplitude of the requested frequency in the spectrum to `a`.

Parameters:
`freq` - the frequency in Hz
`a` - the new amplitude

### scaleFreq

```public void scaleFreq(float freq,
float s)```
Scales the amplitude of the requested frequency by `a`.

Parameters:
`freq` - the frequency in Hz
`s` - the scaling factor

### avgSize

`public int avgSize()`
Returns the number of averages currently being calculated.

Returns:
the length of the averages array

### getAvg

`public float getAvg(int i)`
Gets the value of the `ith` average.

Parameters:
`i` - the average you want the value of
Returns:
the value of the requested average band

### calcAvg

```public float calcAvg(float lowFreq,
float hiFreq)```
Calculate the average amplitude of the frequency band bounded by `lowFreq` and `hiFreq`, inclusive.

Parameters:
`lowFreq` - the lower bound of the band
`hiFreq` - the upper bound of the band
Returns:
the average of all spectrum values within the bounds

### forward

`public abstract void forward(float[] buffer)`
Performs a forward transform on `buffer`.

Parameters:
`buffer` - the buffer to analyze

### forward

```public void forward(float[] buffer,
int startAt)```
Performs a forward transform on values in `buffer`.

Parameters:
`buffer` - the buffer of samples
`startAt` - 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.

### forward

`public void forward(AudioBuffer buffer)`
Performs a forward transform on `buffer`.

Parameters:
`buffer` - the buffer to analyze

### forward

```public void forward(AudioBuffer buffer,
int startAt)```
Performs a forward transform on `buffer`.

Parameters:
`buffer` - the buffer of samples
`startAt` - 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.

### inverse

`public abstract void inverse(float[] buffer)`
Performs an inverse transform of the frequency spectrum and places the result in `buffer`.

Parameters:
`buffer` - the buffer to place the result of the inverse transform in

### inverse

`public void inverse(AudioBuffer buffer)`
Performs an inverse transform of the frequency spectrum and places the result in `buffer`.

Parameters:
`buffer` - the buffer to place the result of the inverse transform in

### inverse

```public void inverse(float[] freqReal,
float[] freqImag,
float[] buffer)```
Performs an inverse transform of the frequency spectrum represented by freqReal and freqImag and places the result in buffer.

Parameters:
`freqReal` - the real part of the frequency spectrum
`freqImag` - the imaginary part the frequency spectrum
`buffer` - the buffer to place the inverse transform in