/*
* Copyright (c) 2007 by Damien Di Fede <ddf@compartmental.net>
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU Library General Public License as published
* by the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU Library General Public License for more details.
*
* You should have received a copy of the GNU Library General Public
* License along with this program; if not, write to the Free Software
* Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
*/
package ddf.minim.analysis;
import ddf.minim.AudioBuffer;
import ddf.minim.Minim;
/**
* 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 <a
* href="http://en.wikipedia.org/wiki/Nyquist_frequency">Nyquist frequency</a>.
* In other words, given a signal of length <code>N</code>, there will be
* <code>N/2</code> frequency bands in the spectrum.
* <p>
* As an example, if you construct a FourierTransform with a
* <code>timeSize</code> of 1024 and and a <code>sampleRate</code> 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
* <code>5/1024 * 44100 = 0.0048828125 * 44100 = 215 Hz</code>. The width of
* that frequency band is equal to <code>2/1024</code>, 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 22100, 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 <code>getFreq()</code> allows you to query the spectrum with a
* frequency in Hz and the function <code>getBandWidth()</code> will return
* the bandwidth in Hz of each frequency band in the spectrum.
* <p>
* <b>Usage</b>
* <p>
* 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
* <code>audio</code> is an AudioSource and <code>fft</code> is an FFT (one
* of the derived classes of FourierTransform).
*
* <pre>
* 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);
* }
* </pre>
*
* <b>Windowing</b>
* <p>
* Windowing is the process of shaping the audio samples before transforming them
* to the frequency domain. If you call the <code>window()</code> function
* with an appropriate constant, such as FourierTransform.HAMMING, 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 noise in the spectrum somewhat.
* <p>
* <b>Averages</b>
* <p>
* 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.
* <p>
* <code>linAverages()</code> 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.
* <p>
* <code>logAverages()</code> 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 <code>sampleRate/4 - sampleRate/2</code>, 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.
* <p>
* <code>calcAvg()</code> 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.
* <p>
* If you don't want any averages calculated, then you can call
* <code>noAverages()</code>. This will not impact your ability to use
* <code>calcAvg()</code>, it will merely prevent the object from calculating
* an average array every time you use <code>forward()</code>.
* <p>
* <b>Inverse Transform</b>
* <p>
* 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 <code>timeSize()</code> long. The <code>set</code> and
* <code>scale</code> 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
* @see <a href="http://www.dspguide.com/ch8.htm">The Discrete Fourier Transform</a>
*/
public abstract class FourierTransform
{
/** A constant indicating no window should be used on sample buffers. */
public static final int NONE = 0;
/** A constant indicating a Hamming window should be used on sample buffers. */
public static final int HAMMING = 1;
protected static final int LINAVG = 2;
protected static final int LOGAVG = 3;
protected static final int NOAVG = 4;
protected static final float TWO_PI = (float) (2 * Math.PI);
protected int timeSize;
protected int sampleRate;
protected int whichWindow;
protected float[] real;
protected float[] imag;
protected float[] spectrum;
protected float[] averages;
protected int whichAverage;
protected int octaves;
protected int avgPerOctave;
/**
* Construct a FourierTransform that will analyze sample buffers that are
* <code>ts</code> samples long and contain samples with a <code>sr</code>
* sample rate.
*
* @param ts
* the length of the buffers that will be analyzed
* @param sr
* the sample rate of the samples that will be analyzed
*/
FourierTransform(int ts, float sr)
{
timeSize = ts;
sampleRate = (int)sr;
noAverages();
allocateArrays();
whichWindow = NONE;
}
// allocating real, imag, and spectrum are the responsibility of derived
// classes
// because the size of the arrays will depend on the implementation being used
// this enforces that responsibility
protected abstract void allocateArrays();
protected void setComplex(float[] r, float[] i)
{
if (real.length != r.length && imag.length != i.length)
{
Minim
.error("FourierTransform.setComplex: the two arrays must be the same length as their member counterparts.");
}
else
{
System.arraycopy(r, 0, real, 0, r.length);
System.arraycopy(i, 0, imag, 0, i.length);
}
}
// fill the spectrum array with the amps of the data in real and imag
// used so that this class can handle creating the average array
// and also do spectrum shaping if necessary
protected void fillSpectrum()
{
for (int i = 0; i < spectrum.length; i++)
{
spectrum[i] = (float) Math.sqrt(real[i] * real[i] + imag[i] * imag[i]);
}
if (whichAverage == LINAVG)
{
int avgWidth = (int) spectrum.length / averages.length;
for (int i = 0; i < averages.length; i++)
{
float avg = 0;
int j;
for (j = 0; j < avgWidth; j++)
{
int offset = j + i * avgWidth;
if (offset < spectrum.length)
{
avg += spectrum[offset];
}
else
break;
}
avg /= j;
averages[i] = avg;
}
}
else if (whichAverage == LOGAVG)
{
for (int i = 0; i < octaves; i++)
{
float lowFreq, hiFreq, freqStep;
if (i == 0)
lowFreq = 0;
else
lowFreq = (sampleRate / 2) / (float) Math.pow(2, octaves - i);
hiFreq = (sampleRate / 2) / (float) Math.pow(2, octaves - i - 1);
freqStep = (hiFreq - lowFreq) / avgPerOctave;
float f = lowFreq;
for (int j = 0; j < avgPerOctave; j++)
{
int offset = j + i * avgPerOctave;
averages[offset] = calcAvg(f, f + freqStep);
f += freqStep;
}
}
}
}
/**
* Sets the object to not compute averages.
*
*/
public void noAverages()
{
averages = new float[0];
whichAverage = NOAVG;
}
/**
* 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
* <code>specSize() / numAvg</code> bands wide.
*
* @param numAvg
* how many averages to compute
*/
public void linAverages(int numAvg)
{
if (numAvg > spectrum.length / 2)
{
Minim.error("The number of averages for this transform can be at most "
+ spectrum.length / 2 + ".");
}
else
{
averages = new float[numAvg];
}
whichAverage = LINAVG;
}
/**
* 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,
* <code>logAverages(11, 1)</code> 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.
*
* @param minBandwidth
* the minimum bandwidth used for an octave
* @param bandsPerOctave
* how many bands to split each octave into
*/
public void logAverages(int minBandwidth, int bandsPerOctave)
{
float nyq = (float) sampleRate / 2f;
octaves = 1;
while ((nyq /= 2) > minBandwidth)
octaves++;
Minim.debug("Number of octaves = " + octaves);
avgPerOctave = bandsPerOctave;
averages = new float[octaves * bandsPerOctave];
whichAverage = LOGAVG;
}
/**
* 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.
*
* @param which
* FourierTransform.HAMMING or FourierTransform.NONE
*/
public void window(int which)
{
if (which < 0 || which > 1)
{
Minim.error("Invalid window type.");
}
else
{
whichWindow = which;
}
}
protected void doWindow(float[] samples)
{
switch (whichWindow)
{
case HAMMING:
hamming(samples);
break;
}
}
// windows the data in samples with a Hamming window
protected void hamming(float[] samples)
{
for (int i = 0; i < samples.length; i++)
{
samples[i] *= (0.54f - 0.46f * Math
.cos(TWO_PI * i / (samples.length - 1)));
}
}
/**
* Returns the length of the time domain signal expected by this transform.
*
* @return the length of the time domain signal expected by this transform
*/
public int timeSize()
{
return timeSize;
}
/**
* 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 <code>timeSize()/2 + 1</code>, see above for an explanation.
*
* @return the size of the spectrum
*/
public int specSize()
{
return spectrum.length;
}
/**
* Returns the amplitude of the requested frequency band.
*
* @param i
* the index of a frequency band
* @return the amplitude of the requested frequency band
*/
public float getBand(int i)
{
if (i < 0) i = 0;
if (i > spectrum.length - 1) i = spectrum.length - 1;
return spectrum[i];
}
/**
* 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.
*
* @return the width of each frequency band in Hz.
*/
public float getBandWidth()
{
return (2f / (float) timeSize) * sampleRate;
}
/**
* Sets the amplitude of the <code>i<sup>th</sup></code> frequency band to
* <code>a</code>. You can use this to shape the spectrum before using
* <code>inverse()</code>.
*
* @param i
* the frequency band to modify
* @param a
* the new amplitude
*/
public abstract void setBand(int i, float a);
/**
* Scales the amplitude of the <code>i<sup>th</sup></code> frequency band
* by <code>s</code>. You can use this to shape the spectrum before using
* <code>inverse()</code>.
*
* @param i
* the frequency band to modify
* @param s
* the scaling factor
*/
public abstract void scaleBand(int i, float s);
/**
* Returns the index of the frequency band that contains the requested
* frequency.
*
* @param freq
* the frequency you want the index for (in Hz)
* @return the index of the frequency band that contains freq
*/
public int freqToIndex(float freq)
{
// special case: freq is lower than the bandwidth of spectrum[0]
if (freq < getBandWidth() / 2) return 0;
// special case: freq is within the bandwidth of spectrum[512]
if (freq > sampleRate / 2 - getBandWidth() / 2) return 512;
// all other cases
float fraction = freq / (float) sampleRate;
int i = Math.round(timeSize * fraction);
return i;
}
/**
* Gets the amplitude of the requested frequency in the spectrum.
*
* @param freq
* the frequency in Hz
* @return the amplitude of the frequency in the spectrum
*/
public float getFreq(float freq)
{
return getBand(freqToIndex(freq));
}
/**
* Sets the amplitude of the requested frequency in the spectrum to
* <code>a</code>.
*
* @param freq
* the frequency in Hz
* @param a
* the new amplitude
*/
public void setFreq(float freq, float a)
{
setBand(freqToIndex(freq), a);
}
/**
* Scales the amplitude of the requested frequency by <code>a</code>.
*
* @param freq
* the frequency in Hz
* @param s
* the scaling factor
*/
public void scaleFreq(float freq, float s)
{
scaleBand(freqToIndex(freq), s);
}
/**
* Returns the number of averages currently being calculated.
*
* @return the length of the averages array
*/
public int avgSize()
{
return averages.length;
}
/**
* Gets the value of the <code>i<sup>th</sup></code> average.
*
* @param i
* the average you want the value of
* @return the value of the requested average
*/
public float getAvg(int i)
{
float ret;
if (averages.length > 0)
ret = averages[i];
else
ret = 0;
return ret;
}
/**
* Calculate the average amplitude of the frequency band bounded by
* <code>lowFreq</code> and <code>hiFreq</code>, inclusive.
*
* @param lowFreq
* the lower bound of the band
* @param hiFreq
* the upper bound of the band
* @return the average of all spectrum values within the bounds
*/
public float calcAvg(float lowFreq, float hiFreq)
{
int lowBound = freqToIndex(lowFreq);
int hiBound = freqToIndex(hiFreq);
float avg = 0;
for (int i = lowBound; i <= hiBound; i++)
{
avg += spectrum[i];
}
avg /= (hiBound - lowBound + 1);
return avg;
}
/**
* Performs a forward transform on <code>buffer</code>.
*
* @param buffer
* the buffer to analyze
*/
public abstract void forward(float[] buffer);
/**
* Performs a forward transform on <code>buffer</code>.
*
* @param buffer
* the buffer to analyze
*/
public void forward(AudioBuffer buffer)
{
forward(buffer.toArray());
}
/**
* Performs an inverse transform of the frequency spectrum and places the
* result in <code>buffer</code>.
*
* @param buffer
* the buffer to place the result of the inverse transform in
*/
public abstract void inverse(float[] buffer);
/**
* Performs an inverse transform of the frequency spectrum and places the
* result in <code>buffer</code>.
*
* @param buffer
* the buffer to place the result of the inverse transform in
*/
public void inverse(AudioBuffer buffer)
{
inverse(buffer.toArray());
}
/**
* Performs an inverse transform of the frequency spectrum represented by
* freqReal and freqImag and places the result in buffer.
*
* @param freqReal
* the real part of the frequency spectrum
* @param freqImag
* the imaginary part the frequency spectrum
* @param buffer
* the buffer to place the inverse transform in
*/
public void inverse(float[] freqReal, float[] freqImag, float[] buffer)
{
setComplex(freqReal, freqImag);
inverse(buffer);
}
}