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;
    }
}