FastFourierTransformer.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
*
* https://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.
*/
/*
* This is not the original file distributed by the Apache Software Foundation
* It has been modified by the Hipparchus project
*/
package org.hipparchus.transform;
import java.io.Serializable;
import org.hipparchus.analysis.FunctionUtils;
import org.hipparchus.analysis.UnivariateFunction;
import org.hipparchus.complex.Complex;
import org.hipparchus.exception.MathIllegalArgumentException;
import org.hipparchus.exception.MathRuntimeException;
import org.hipparchus.util.ArithmeticUtils;
import org.hipparchus.util.FastMath;
import org.hipparchus.util.MathArrays;
import org.hipparchus.util.MathUtils;
/**
* 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 DftNormalization}.
* <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.
*
* @see DftNormalization
*/
public class FastFourierTransformer implements Serializable {
/** Serializable version identifier. */
private static final long serialVersionUID = 20120210L;
/**
* {@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 };
/** The type of DFT to be performed. */
private final DftNormalization normalization;
/**
* Creates a new instance of this class, with various normalization
* conventions.
*
* @param normalization the type of normalization to be applied to the
* transformed data
*/
public FastFourierTransformer(final DftNormalization normalization) {
this.normalization = normalization;
}
/**
* 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 the first array to be shuffled
* @param b the second array to be shuffled
*/
private static void bitReversalShuffle2(double[] a, double[] b) {
final int n = a.length;
assert b.length == n;
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;
}
}
/**
* Applies the proper normalization to the specified transformed data.
*
* @param dataRI the unscaled transformed data
* @param normalization the normalization to be applied
* @param type the type of transform (forward, inverse) which resulted in the specified data
*/
private static void normalizeTransformedData(final double[][] dataRI,
final DftNormalization normalization, final TransformType type) {
final double[] dataR = dataRI[0];
final double[] dataI = dataRI[1];
final int n = dataR.length;
assert dataI.length == n;
switch (normalization) {
case STANDARD:
if (type == TransformType.INVERSE) {
final double scaleFactor = 1.0 / n;
for (int i = 0; i < n; i++) {
dataR[i] *= scaleFactor;
dataI[i] *= scaleFactor;
}
}
break;
case UNITARY:
final double scaleFactor = 1.0 / FastMath.sqrt(n);
for (int i = 0; i < n; i++) {
dataR[i] *= scaleFactor;
dataI[i] *= scaleFactor;
}
break;
default:
// This should never occur in normal conditions. However this
// clause has been added as a safeguard if other types of
// normalizations are ever implemented, and the corresponding
// test is forgotten in the present switch.
throw MathRuntimeException.createInternalError();
}
}
/**
* Computes the standard transform of the specified complex data. The
* computation is done in place. The input data is laid out as follows
* <ul>
* <li>{@code dataRI[0][i]} is the real part of the {@code i}-th data point,</li>
* <li>{@code dataRI[1][i]} is the imaginary part of the {@code i}-th data point.</li>
* </ul>
*
* @param dataRI the two dimensional array of real and imaginary parts of the data
* @param normalization the normalization to be applied to the transformed data
* @param type the type of transform (forward, inverse) to be performed
* @throws MathIllegalArgumentException if the number of rows of the specified
* array is not two, or the array is not rectangular
* @throws MathIllegalArgumentException if the number of data points is not
* a power of two
*/
public static void transformInPlace(final double[][] dataRI,
final DftNormalization normalization, final TransformType type) {
MathUtils.checkDimension(dataRI.length, 2);
final double[] dataR = dataRI[0];
final double[] dataI = dataRI[1];
MathArrays.checkEqualLength(dataR, dataI);
final int n = dataR.length;
if (!ArithmeticUtils.isPowerOfTwo(n)) {
throw new MathIllegalArgumentException(LocalizedFFTFormats.NOT_POWER_OF_TWO_CONSIDER_PADDING,
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, normalization, type);
return;
}
bitReversalShuffle2(dataR, dataI);
// Do 4-term DFT.
if (type == TransformType.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) {
int n0 = lastN0 << 1;
int logN0 = lastLogN0 + 1;
double wSubN0R = W_SUB_N_R[logN0];
double wSubN0I = W_SUB_N_I[logN0];
if (type == TransformType.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) {
int destOddStartIndex = destEvenStartIndex + lastN0;
double wSubN0ToRR = 1;
double wSubN0ToRI = 0;
for (int r = 0; r < lastN0; r++) {
double grR = dataR[destEvenStartIndex + r];
double grI = dataI[destEvenStartIndex + r];
double hrR = dataR[destOddStartIndex + r];
double hrI = dataI[destOddStartIndex + r];
// dest[destEvenStartIndex + r] = Gr + WsubN0ToR * Hr
dataR[destEvenStartIndex + r] = grR + wSubN0ToRR * hrR - wSubN0ToRI * hrI;
dataI[destEvenStartIndex + r] = grI + wSubN0ToRR * hrI + wSubN0ToRI * hrR;
// dest[destOddStartIndex + r] = Gr - WsubN0ToR * Hr
dataR[destOddStartIndex + r] = grR - (wSubN0ToRR * hrR - wSubN0ToRI * hrI);
dataI[destOddStartIndex + r] = grI - (wSubN0ToRR * hrI + wSubN0ToRI * hrR);
// WsubN0ToR *= WsubN0R
double nextWsubN0ToRR = wSubN0ToRR * wSubN0R - wSubN0ToRI * wSubN0I;
double nextWsubN0ToRI = wSubN0ToRR * wSubN0I + wSubN0ToRI * wSubN0R;
wSubN0ToRR = nextWsubN0ToRR;
wSubN0ToRI = nextWsubN0ToRI;
}
}
lastN0 = n0;
lastLogN0 = logN0;
}
normalizeTransformedData(dataRI, normalization, type);
}
/**
* Returns the (forward, inverse) transform of the specified real data set.
*
* @param f the real data array to be transformed
* @param type the type of transform (forward, inverse) to be performed
* @return the complex transformed array
* @throws MathIllegalArgumentException if the length of the data array is not a power of two
*/
public Complex[] transform(final double[] f, final TransformType type) {
final double[][] dataRI = { f.clone(), new double[f.length] };
transformInPlace(dataRI, normalization, type);
return TransformUtils.createComplexArray(dataRI);
}
/**
* Returns the (forward, inverse) transform of the specified real function,
* sampled on the specified interval.
*
* @param f the function to be sampled and transformed
* @param min the (inclusive) lower bound for the interval
* @param max the (exclusive) upper bound for the interval
* @param n the number of sample points
* @param type the type of transform (forward, inverse) to be performed
* @return the complex transformed array
* @throws org.hipparchus.exception.MathIllegalArgumentException
* if the lower bound is greater than, or equal to the upper bound
* @throws org.hipparchus.exception.MathIllegalArgumentException
* if the number of sample points {@code n} is negative
* @throws MathIllegalArgumentException if the number of sample points
* {@code n} is not a power of two
*/
public Complex[] transform(final UnivariateFunction f,
final double min, final double max, final int n,
final TransformType type) {
final double[] data = FunctionUtils.sample(f, min, max, n);
return transform(data, type);
}
/**
* Returns the (forward, inverse) transform of the specified complex data set.
*
* @param f the complex data array to be transformed
* @param type the type of transform (forward, inverse) to be performed
* @return the complex transformed array
* @throws MathIllegalArgumentException if the length of the data array is not a power of two
*/
public Complex[] transform(final Complex[] f, final TransformType type) {
final double[][] dataRI = TransformUtils.createRealImaginaryArray(f);
transformInPlace(dataRI, normalization, type);
return TransformUtils.createComplexArray(dataRI);
}
}