FastFourierTransform.java
/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math4.transform;
import java.util.Arrays;
import java.util.function.DoubleUnaryOperator;
import org.apache.commons.numbers.core.ArithmeticUtils;
import org.apache.commons.numbers.complex.Complex;
/**
* Implements the Fast Fourier Transform for transformation of one-dimensional
* real or complex data sets. For reference, see <em>Applied Numerical Linear
* Algebra</em>, ISBN 0898713897, chapter 6.
* <p>
* There are several variants of the discrete Fourier transform, with various
* normalization conventions, which are specified by the parameter
* {@link Norm}.
* <p>
* The current implementation of the discrete Fourier transform as a fast
* Fourier transform requires the length of the data set to be a power of 2.
* This greatly simplifies and speeds up the code. Users can pad the data with
* zeros to meet this requirement. There are other flavors of FFT, for
* reference, see S. Winograd,
* <i>On computing the discrete Fourier transform</i>, Mathematics of
* Computation, 32 (1978), 175 - 199.
*/
public class FastFourierTransform implements ComplexTransform {
/** Number of array slots: 1 for "real" parts 1 for "imaginary" parts. */
private static final int NUM_PARTS = 2;
/**
* {@code W_SUB_N_R[i]} is the real part of
* {@code exp(- 2 * i * pi / n)}:
* {@code W_SUB_N_R[i] = cos(2 * pi/ n)}, where {@code n = 2^i}.
*/
private static final double[] W_SUB_N_R = {
0x1.0p0, -0x1.0p0, 0x1.1a62633145c07p-54, 0x1.6a09e667f3bcdp-1,
0x1.d906bcf328d46p-1, 0x1.f6297cff75cbp-1, 0x1.fd88da3d12526p-1, 0x1.ff621e3796d7ep-1,
0x1.ffd886084cd0dp-1, 0x1.fff62169b92dbp-1, 0x1.fffd8858e8a92p-1, 0x1.ffff621621d02p-1,
0x1.ffffd88586ee6p-1, 0x1.fffff62161a34p-1, 0x1.fffffd8858675p-1, 0x1.ffffff621619cp-1,
0x1.ffffffd885867p-1, 0x1.fffffff62161ap-1, 0x1.fffffffd88586p-1, 0x1.ffffffff62162p-1,
0x1.ffffffffd8858p-1, 0x1.fffffffff6216p-1, 0x1.fffffffffd886p-1, 0x1.ffffffffff621p-1,
0x1.ffffffffffd88p-1, 0x1.fffffffffff62p-1, 0x1.fffffffffffd9p-1, 0x1.ffffffffffff6p-1,
0x1.ffffffffffffep-1, 0x1.fffffffffffffp-1, 0x1.0p0, 0x1.0p0,
0x1.0p0, 0x1.0p0, 0x1.0p0, 0x1.0p0,
0x1.0p0, 0x1.0p0, 0x1.0p0, 0x1.0p0,
0x1.0p0, 0x1.0p0, 0x1.0p0, 0x1.0p0,
0x1.0p0, 0x1.0p0, 0x1.0p0, 0x1.0p0,
0x1.0p0, 0x1.0p0, 0x1.0p0, 0x1.0p0,
0x1.0p0, 0x1.0p0, 0x1.0p0, 0x1.0p0,
0x1.0p0, 0x1.0p0, 0x1.0p0, 0x1.0p0,
0x1.0p0, 0x1.0p0, 0x1.0p0 };
/**
* {@code W_SUB_N_I[i]} is the imaginary part of
* {@code exp(- 2 * i * pi / n)}:
* {@code W_SUB_N_I[i] = -sin(2 * pi/ n)}, where {@code n = 2^i}.
*/
private static final double[] W_SUB_N_I = {
0x1.1a62633145c07p-52, -0x1.1a62633145c07p-53, -0x1.0p0, -0x1.6a09e667f3bccp-1,
-0x1.87de2a6aea963p-2, -0x1.8f8b83c69a60ap-3, -0x1.917a6bc29b42cp-4, -0x1.91f65f10dd814p-5,
-0x1.92155f7a3667ep-6, -0x1.921d1fcdec784p-7, -0x1.921f0fe670071p-8, -0x1.921f8becca4bap-9,
-0x1.921faaee6472dp-10, -0x1.921fb2aecb36p-11, -0x1.921fb49ee4ea6p-12, -0x1.921fb51aeb57bp-13,
-0x1.921fb539ecf31p-14, -0x1.921fb541ad59ep-15, -0x1.921fb5439d73ap-16, -0x1.921fb544197ap-17,
-0x1.921fb544387bap-18, -0x1.921fb544403c1p-19, -0x1.921fb544422c2p-20, -0x1.921fb54442a83p-21,
-0x1.921fb54442c73p-22, -0x1.921fb54442cefp-23, -0x1.921fb54442d0ep-24, -0x1.921fb54442d15p-25,
-0x1.921fb54442d17p-26, -0x1.921fb54442d18p-27, -0x1.921fb54442d18p-28, -0x1.921fb54442d18p-29,
-0x1.921fb54442d18p-30, -0x1.921fb54442d18p-31, -0x1.921fb54442d18p-32, -0x1.921fb54442d18p-33,
-0x1.921fb54442d18p-34, -0x1.921fb54442d18p-35, -0x1.921fb54442d18p-36, -0x1.921fb54442d18p-37,
-0x1.921fb54442d18p-38, -0x1.921fb54442d18p-39, -0x1.921fb54442d18p-40, -0x1.921fb54442d18p-41,
-0x1.921fb54442d18p-42, -0x1.921fb54442d18p-43, -0x1.921fb54442d18p-44, -0x1.921fb54442d18p-45,
-0x1.921fb54442d18p-46, -0x1.921fb54442d18p-47, -0x1.921fb54442d18p-48, -0x1.921fb54442d18p-49,
-0x1.921fb54442d18p-50, -0x1.921fb54442d18p-51, -0x1.921fb54442d18p-52, -0x1.921fb54442d18p-53,
-0x1.921fb54442d18p-54, -0x1.921fb54442d18p-55, -0x1.921fb54442d18p-56, -0x1.921fb54442d18p-57,
-0x1.921fb54442d18p-58, -0x1.921fb54442d18p-59, -0x1.921fb54442d18p-60 };
/** Type of DFT. */
private final Norm normalization;
/** Inverse or forward. */
private final boolean inverse;
/**
* @param normalization Normalization to be applied to the
* transformed data.
* @param inverse Whether to perform the inverse transform.
*/
public FastFourierTransform(final Norm normalization,
final boolean inverse) {
this.normalization = normalization;
this.inverse = inverse;
}
/**
* @param normalization Normalization to be applied to the
* transformed data.
*/
public FastFourierTransform(final Norm normalization) {
this(normalization, false);
}
/**
* Computes the standard transform of the data.
* Computation is done in place.
* Assumed layout of the input data:
* <ul>
* <li>{@code dataRI[0][i]}: Real part of the {@code i}-th data point,</li>
* <li>{@code dataRI[1][i]}: Imaginary part of the {@code i}-th data point.</li>
* </ul>
*
* @param dataRI Two-dimensional array of real and imaginary parts of the data.
* @throws IllegalArgumentException if the number of data points is not
* a power of two, if the number of rows of the specified array is not two,
* or the array is not rectangular.
*/
public void transformInPlace(final double[][] dataRI) {
if (dataRI.length != NUM_PARTS) {
throw new TransformException(TransformException.SIZE_MISMATCH,
dataRI.length, NUM_PARTS);
}
final double[] dataR = dataRI[0];
final double[] dataI = dataRI[1];
if (dataR.length != dataI.length) {
throw new TransformException(TransformException.SIZE_MISMATCH,
dataI.length, dataR.length);
}
final int n = dataR.length;
if (!ArithmeticUtils.isPowerOfTwo(n)) {
throw new TransformException(TransformException.NOT_POWER_OF_TWO,
Integer.valueOf(n));
}
if (n == 1) {
return;
} else if (n == 2) {
final double srcR0 = dataR[0];
final double srcI0 = dataI[0];
final double srcR1 = dataR[1];
final double srcI1 = dataI[1];
// X_0 = x_0 + x_1
dataR[0] = srcR0 + srcR1;
dataI[0] = srcI0 + srcI1;
// X_1 = x_0 - x_1
dataR[1] = srcR0 - srcR1;
dataI[1] = srcI0 - srcI1;
normalizeTransformedData(dataRI);
return;
}
bitReversalShuffle2(dataR, dataI);
// Do 4-term DFT.
if (inverse) {
for (int i0 = 0; i0 < n; i0 += 4) {
final int i1 = i0 + 1;
final int i2 = i0 + 2;
final int i3 = i0 + 3;
final double srcR0 = dataR[i0];
final double srcI0 = dataI[i0];
final double srcR1 = dataR[i2];
final double srcI1 = dataI[i2];
final double srcR2 = dataR[i1];
final double srcI2 = dataI[i1];
final double srcR3 = dataR[i3];
final double srcI3 = dataI[i3];
// 4-term DFT
// X_0 = x_0 + x_1 + x_2 + x_3
dataR[i0] = srcR0 + srcR1 + srcR2 + srcR3;
dataI[i0] = srcI0 + srcI1 + srcI2 + srcI3;
// X_1 = x_0 - x_2 + j * (x_3 - x_1)
dataR[i1] = srcR0 - srcR2 + (srcI3 - srcI1);
dataI[i1] = srcI0 - srcI2 + (srcR1 - srcR3);
// X_2 = x_0 - x_1 + x_2 - x_3
dataR[i2] = srcR0 - srcR1 + srcR2 - srcR3;
dataI[i2] = srcI0 - srcI1 + srcI2 - srcI3;
// X_3 = x_0 - x_2 + j * (x_1 - x_3)
dataR[i3] = srcR0 - srcR2 + (srcI1 - srcI3);
dataI[i3] = srcI0 - srcI2 + (srcR3 - srcR1);
}
} else {
for (int i0 = 0; i0 < n; i0 += 4) {
final int i1 = i0 + 1;
final int i2 = i0 + 2;
final int i3 = i0 + 3;
final double srcR0 = dataR[i0];
final double srcI0 = dataI[i0];
final double srcR1 = dataR[i2];
final double srcI1 = dataI[i2];
final double srcR2 = dataR[i1];
final double srcI2 = dataI[i1];
final double srcR3 = dataR[i3];
final double srcI3 = dataI[i3];
// 4-term DFT
// X_0 = x_0 + x_1 + x_2 + x_3
dataR[i0] = srcR0 + srcR1 + srcR2 + srcR3;
dataI[i0] = srcI0 + srcI1 + srcI2 + srcI3;
// X_1 = x_0 - x_2 + j * (x_3 - x_1)
dataR[i1] = srcR0 - srcR2 + (srcI1 - srcI3);
dataI[i1] = srcI0 - srcI2 + (srcR3 - srcR1);
// X_2 = x_0 - x_1 + x_2 - x_3
dataR[i2] = srcR0 - srcR1 + srcR2 - srcR3;
dataI[i2] = srcI0 - srcI1 + srcI2 - srcI3;
// X_3 = x_0 - x_2 + j * (x_1 - x_3)
dataR[i3] = srcR0 - srcR2 + (srcI3 - srcI1);
dataI[i3] = srcI0 - srcI2 + (srcR1 - srcR3);
}
}
int lastN0 = 4;
int lastLogN0 = 2;
while (lastN0 < n) {
final int n0 = lastN0 << 1;
final int logN0 = lastLogN0 + 1;
final double wSubN0R = W_SUB_N_R[logN0];
double wSubN0I = W_SUB_N_I[logN0];
if (inverse) {
wSubN0I = -wSubN0I;
}
// Combine even/odd transforms of size lastN0 into a transform of size N0 (lastN0 * 2).
for (int destEvenStartIndex = 0; destEvenStartIndex < n; destEvenStartIndex += n0) {
final int destOddStartIndex = destEvenStartIndex + lastN0;
double wSubN0ToRR = 1;
double wSubN0ToRI = 0;
for (int r = 0; r < lastN0; r++) {
final int destEvenStartIndexPlusR = destEvenStartIndex + r;
final int destOddStartIndexPlusR = destOddStartIndex + r;
final double grR = dataR[destEvenStartIndexPlusR];
final double grI = dataI[destEvenStartIndexPlusR];
final double hrR = dataR[destOddStartIndexPlusR];
final double hrI = dataI[destOddStartIndexPlusR];
final double a = wSubN0ToRR * hrR - wSubN0ToRI * hrI;
final double b = wSubN0ToRR * hrI + wSubN0ToRI * hrR;
// dest[destEvenStartIndex + r] = Gr + WsubN0ToR * Hr
dataR[destEvenStartIndexPlusR] = grR + a;
dataI[destEvenStartIndexPlusR] = grI + b;
// dest[destOddStartIndex + r] = Gr - WsubN0ToR * Hr
dataR[destOddStartIndexPlusR] = grR - a;
dataI[destOddStartIndexPlusR] = grI - b;
// WsubN0ToR *= WsubN0R
final double nextWsubN0ToRR = wSubN0ToRR * wSubN0R - wSubN0ToRI * wSubN0I;
final double nextWsubN0ToRI = wSubN0ToRR * wSubN0I + wSubN0ToRI * wSubN0R;
wSubN0ToRR = nextWsubN0ToRR;
wSubN0ToRI = nextWsubN0ToRI;
}
}
lastN0 = n0;
lastLogN0 = logN0;
}
normalizeTransformedData(dataRI);
}
/**
* {@inheritDoc}
*
* @throws IllegalArgumentException if the length of the data array is not a power of two.
*/
@Override
public Complex[] apply(final double[] f) {
final double[][] dataRI = {
Arrays.copyOf(f, f.length),
new double[f.length]
};
transformInPlace(dataRI);
return TransformUtils.createComplex(dataRI);
}
/**
* {@inheritDoc}
*
* @throws IllegalArgumentException if the number of sample points
* {@code n} is not a power of two, if the lower bound is greater than,
* or equal to the upper bound, if the number of sample points {@code n}
* is negative
*/
@Override
public Complex[] apply(final DoubleUnaryOperator f,
final double min,
final double max,
final int n) {
return apply(TransformUtils.sample(f, min, max, n));
}
/**
* {@inheritDoc}
*
* @throws IllegalArgumentException if the length of the data array is
* not a power of two.
*/
@Override
public Complex[] apply(final Complex[] f) {
final double[][] dataRI = TransformUtils.createRealImaginary(f);
transformInPlace(dataRI);
return TransformUtils.createComplex(dataRI);
}
/**
* Applies normalization to the transformed data.
*
* @param dataRI Unscaled transformed data.
*/
private void normalizeTransformedData(final double[][] dataRI) {
final double[] dataR = dataRI[0];
final double[] dataI = dataRI[1];
final int n = dataR.length;
switch (normalization) {
case STD:
if (inverse) {
final double scaleFactor = 1d / n;
for (int i = 0; i < n; i++) {
dataR[i] *= scaleFactor;
dataI[i] *= scaleFactor;
}
}
break;
case UNIT:
final double scaleFactor = 1d / Math.sqrt(n);
for (int i = 0; i < n; i++) {
dataR[i] *= scaleFactor;
dataI[i] *= scaleFactor;
}
break;
default:
throw new IllegalStateException(); // Should never happen.
}
}
/**
* Performs identical index bit reversal shuffles on two arrays of
* identical size.
* Each element in the array is swapped with another element based
* on the bit-reversal of the index.
* For example, in an array with length 16, item at binary index 0011
* (decimal 3) would be swapped with the item at binary index 1100
* (decimal 12).
*
* @param a Array to be shuffled.
* @param b Array to be shuffled.
*/
private static void bitReversalShuffle2(double[] a,
double[] b) {
final int n = a.length;
final int halfOfN = n >> 1;
int j = 0;
for (int i = 0; i < n; i++) {
if (i < j) {
// swap indices i & j
double temp = a[i];
a[i] = a[j];
a[j] = temp;
temp = b[i];
b[i] = b[j];
b[j] = temp;
}
int k = halfOfN;
while (k <= j && k > 0) {
j -= k;
k >>= 1;
}
j += k;
}
}
/**
* Normalization types.
*/
public enum Norm {
/**
* Should be passed to the constructor of {@link FastFourierTransform}
* to use the <em>standard</em> normalization convention. This normalization
* convention is defined as follows
* <ul>
* <li>forward transform: \( y_n = \sum_{k = 0}^{N - 1} x_k e^{-2 \pi i n k / N} \),</li>
* <li>inverse transform: \( x_k = \frac{1}{N} \sum_{n = 0}^{N - 1} y_n e^{2 \pi i n k / N} \),</li>
* </ul>
* where \( N \) is the size of the data sample.
*/
STD,
/**
* Should be passed to the constructor of {@link FastFourierTransform}
* to use the <em>unitary</em> normalization convention. This normalization
* convention is defined as follows
* <ul>
* <li>forward transform: \( y_n = \frac{1}{\sqrt{N}} \sum_{k = 0}^{N - 1} x_k e^{-2 \pi i n k / N} \),</li>
* <li>inverse transform: \( x_k = \frac{1}{\sqrt{N}} \sum_{n = 0}^{N - 1} y_n e^{2 \pi i n k / N} \),</li>
* </ul>
* where \( N \) is the size of the data sample.
*/
UNIT;
}
}