| /* |
| * 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.function.UnaryOperator; |
| import java.util.function.DoubleUnaryOperator; |
| |
| import org.apache.commons.numbers.complex.Complex; |
| import org.apache.commons.numbers.core.ArithmeticUtils; |
| |
| /** |
| * Implements the Fast Sine Transform for transformation of one-dimensional real |
| * data sets. For reference, see James S. Walker, <em>Fast Fourier |
| * Transforms</em>, chapter 3 (ISBN 0849371635). |
| * <p> |
| * There are several variants of the discrete sine transform. The present |
| * implementation corresponds to DST-I, with various normalization conventions, |
| * which are specified by the parameter {@link Norm}. |
| * <strong>It should be noted that regardless to the convention, the first |
| * element of the dataset to be transformed must be zero.</strong> |
| * <p> |
| * DST-I is equivalent to DFT of an <em>odd extension</em> of the data series. |
| * More precisely, if x<sub>0</sub>, …, x<sub>N-1</sub> is the data set |
| * to be sine transformed, the extended data set x<sub>0</sub><sup>#</sup>, |
| * …, x<sub>2N-1</sub><sup>#</sup> is defined as follows |
| * <ul> |
| * <li>x<sub>0</sub><sup>#</sup> = x<sub>0</sub> = 0,</li> |
| * <li>x<sub>k</sub><sup>#</sup> = x<sub>k</sub> if 1 ≤ k < N,</li> |
| * <li>x<sub>N</sub><sup>#</sup> = 0,</li> |
| * <li>x<sub>k</sub><sup>#</sup> = -x<sub>2N-k</sub> if N + 1 ≤ k < |
| * 2N.</li> |
| * </ul> |
| * <p> |
| * Then, the standard DST-I y<sub>0</sub>, …, y<sub>N-1</sub> of the real |
| * data set x<sub>0</sub>, …, x<sub>N-1</sub> is equal to <em>half</em> |
| * of i (the pure imaginary number) times the N first elements of the DFT of the |
| * extended data set x<sub>0</sub><sup>#</sup>, …, |
| * x<sub>2N-1</sub><sup>#</sup> <br> |
| * y<sub>n</sub> = (i / 2) ∑<sub>k=0</sub><sup>2N-1</sup> |
| * x<sub>k</sub><sup>#</sup> exp[-2πi nk / (2N)] |
| * k = 0, …, N-1. |
| * <p> |
| * The present implementation of the discrete sine transform as a fast sine |
| * transform requires the length of the data to be a power of two. Besides, |
| * it implicitly assumes that the sampled function is odd. In particular, the |
| * first element of the data set must be 0, which is enforced in |
| * {@link #apply(DoubleUnaryOperator, double, double, int)}, |
| * after sampling. |
| */ |
| public class FastSineTransform implements RealTransform { |
| /** Operation to be performed. */ |
| private final UnaryOperator<double[]> op; |
| |
| /** |
| * @param normalization Normalization to be applied to the transformed data. |
| * @param inverse Whether to perform the inverse transform. |
| */ |
| public FastSineTransform(final Norm normalization, |
| final boolean inverse) { |
| op = create(normalization, inverse); |
| } |
| |
| /** |
| * @param normalization Normalization to be applied to the |
| * transformed data. |
| */ |
| public FastSineTransform(final Norm normalization) { |
| this(normalization, false); |
| } |
| |
| /** |
| * {@inheritDoc} |
| * |
| * The first element of the specified data set is required to be {@code 0}. |
| * |
| * @throws IllegalArgumentException if the length of the data array is |
| * not a power of two, or the first element of the data array is not zero. |
| */ |
| @Override |
| public double[] apply(final double[] f) { |
| return op.apply(f); |
| } |
| |
| /** |
| * {@inheritDoc} |
| * |
| * The implementation enforces {@code f(x) = 0} at {@code x = 0}. |
| * |
| * @throws IllegalArgumentException if the number of sample points 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 is negative. |
| */ |
| @Override |
| public double[] apply(final DoubleUnaryOperator f, |
| final double min, |
| final double max, |
| final int n) { |
| final double[] data = TransformUtils.sample(f, min, max, n); |
| data[0] = 0; |
| return apply(data); |
| } |
| |
| /** |
| * Perform the FST algorithm (including inverse). |
| * The first element of the data set is required to be {@code 0}. |
| * |
| * @param f Data array to be transformed. |
| * @return the transformed array. |
| * @throws IllegalArgumentException if the length of the data array is |
| * not a power of two, or the first element of the data array is not zero. |
| */ |
| private double[] fst(double[] f) { |
| if (!ArithmeticUtils.isPowerOfTwo(f.length)) { |
| throw new TransformException(TransformException.NOT_POWER_OF_TWO, |
| f.length); |
| } |
| if (f[0] != 0) { |
| throw new TransformException(TransformException.FIRST_ELEMENT_NOT_ZERO, |
| f[0]); |
| } |
| |
| final double[] transformed = new double[f.length]; |
| final int n = f.length; |
| if (n == 1) { |
| transformed[0] = 0; |
| return transformed; |
| } |
| |
| // construct a new array and perform FFT on it |
| final double[] x = new double[n]; |
| x[0] = 0; |
| final int nShifted = n >> 1; |
| x[nShifted] = 2 * f[nShifted]; |
| final double piOverN = Math.PI / n; |
| for (int i = 1; i < nShifted; i++) { |
| final int nMi = n - i; |
| final double fi = f[i]; |
| final double fnMi = f[nMi]; |
| final double a = Math.sin(i * piOverN) * (fi + fnMi); |
| final double b = 0.5 * (fi - fnMi); |
| x[i] = a + b; |
| x[nMi] = a - b; |
| } |
| |
| final FastFourierTransform transform = new FastFourierTransform(FastFourierTransform.Norm.STD); |
| final Complex[] y = transform.apply(x); |
| |
| // reconstruct the FST result for the original array |
| transformed[0] = 0; |
| transformed[1] = 0.5 * y[0].getReal(); |
| for (int i = 1; i < nShifted; i++) { |
| final int i2 = 2 * i; |
| transformed[i2] = -y[i].getImaginary(); |
| transformed[i2 + 1] = y[i].getReal() + transformed[i2 - 1]; |
| } |
| |
| return transformed; |
| } |
| |
| /** |
| * Factory method. |
| * |
| * @param normalization Normalization to be applied to the |
| * transformed data. |
| * @param inverse Whether to perform the inverse transform. |
| * @return the transform operator. |
| */ |
| private UnaryOperator<double[]> create(final Norm normalization, |
| final boolean inverse) { |
| if (inverse) { |
| return normalization == Norm.ORTHO ? |
| f -> TransformUtils.scaleInPlace(fst(f), Math.sqrt(2d / f.length)) : |
| f -> TransformUtils.scaleInPlace(fst(f), 2d / f.length); |
| } else { |
| return normalization == Norm.ORTHO ? |
| f -> TransformUtils.scaleInPlace(fst(f), Math.sqrt(2d / f.length)) : |
| f -> fst(f); |
| } |
| } |
| |
| /** |
| * Normalization types. |
| */ |
| public enum Norm { |
| /** |
| * Should be passed to the constructor of {@link FastSineTransform} to |
| * use the <em>standard</em> normalization convention. The standard DST-I |
| * normalization convention is defined as follows |
| * <ul> |
| * <li>forward transform: y<sub>n</sub> = ∑<sub>k=0</sub><sup>N-1</sup> |
| * x<sub>k</sub> sin(π nk / N),</li> |
| * <li>inverse transform: x<sub>k</sub> = (2 / N) |
| * ∑<sub>n=0</sub><sup>N-1</sup> y<sub>n</sub> sin(π nk / N),</li> |
| * </ul> |
| * where N is the size of the data sample, and x<sub>0</sub> = 0. |
| */ |
| STD, |
| |
| /** |
| * Should be passed to the constructor of {@link FastSineTransform} to |
| * use the <em>orthogonal</em> normalization convention. The orthogonal |
| * DCT-I normalization convention is defined as follows |
| * <ul> |
| * <li>Forward transform: y<sub>n</sub> = √(2 / N) |
| * ∑<sub>k=0</sub><sup>N-1</sup> x<sub>k</sub> sin(π nk / N),</li> |
| * <li>Inverse transform: x<sub>k</sub> = √(2 / N) |
| * ∑<sub>n=0</sub><sup>N-1</sup> y<sub>n</sub> sin(π nk / N),</li> |
| * </ul> |
| * which makes the transform orthogonal. N is the size of the data sample, |
| * and x<sub>0</sub> = 0. |
| */ |
| ORTHO |
| } |
| } |
