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/* * 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.math3.transform; import java.io.Serializable; import org.apache.commons.math3.analysis.FunctionUtils; import org.apache.commons.math3.analysis.UnivariateFunction; import org.apache.commons.math3.complex.Complex; import org.apache.commons.math3.exception.MathIllegalArgumentException; import org.apache.commons.math3.exception.util.LocalizedFormats; import org.apache.commons.math3.util.ArithmeticUtils; import org.apache.commons.math3.util.FastMath; /** * Implements the Fast Cosine 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 cosine transform. The present * implementation corresponds to DCT-I, with various normalization conventions, * which are specified by the parameter {@link DctNormalization}. * <p> * DCT-I is equivalent to DFT of an <em>even extension</em> of the data series. * More precisely, if x<sub>0</sub>, …, x<sub>N-1</sub> is the data set * to be cosine transformed, the extended data set * x<sub>0</sub><sup>#</sup>, …, x<sub>2N-3</sub><sup>#</sup> * is defined as follows * <ul> * <li>x<sub>k</sub><sup>#</sup> = x<sub>k</sub> if 0 ≤ k < N,</li> * <li>x<sub>k</sub><sup>#</sup> = x<sub>2N-2-k</sub> * if N ≤ k < 2N - 2.</li> * </ul> * <p> * Then, the standard DCT-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 the N first elements of the DFT of the extended data set * x<sub>0</sub><sup>#</sup>, …, x<sub>2N-3</sub><sup>#</sup> * <br/> * y<sub>n</sub> = (1 / 2) ∑<sub>k=0</sub><sup>2N-3</sup> * x<sub>k</sub><sup>#</sup> exp[-2πi nk / (2N - 2)] * k = 0, …, N-1. * <p> * The present implementation of the discrete cosine transform as a fast cosine * transform requires the length of the data set to be a power of two plus one * (N = 2<sup>n</sup> + 1). Besides, it implicitly assumes * that the sampled function is even. * * @version $Id: FastCosineTransformer.java 1385310 2012-09-16 16:32:10Z tn $ * @since 1.2 */ public class FastCosineTransformer implements RealTransformer, Serializable { /** Serializable version identifier. */ static final long serialVersionUID = 20120212L; /** The type of DCT to be performed. */ private final DctNormalization 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 FastCosineTransformer(final DctNormalization normalization) { this.normalization = normalization; } /** * {@inheritDoc} * * @throws MathIllegalArgumentException if the length of the data array is * not a power of two plus one */ public double[] transform(final double[] f, final TransformType type) throws MathIllegalArgumentException { if (type == TransformType.FORWARD) { if (normalization == DctNormalization.ORTHOGONAL_DCT_I) { final double s = FastMath.sqrt(2.0 / (f.length - 1)); return TransformUtils.scaleArray(fct(f), s); } return fct(f); } final double s2 = 2.0 / (f.length - 1); final double s1; if (normalization == DctNormalization.ORTHOGONAL_DCT_I) { s1 = FastMath.sqrt(s2); } else { s1 = s2; } return TransformUtils.scaleArray(fct(f), s1); } /** * {@inheritDoc} * * @throws org.apache.commons.math3.exception.NonMonotonicSequenceException * if the lower bound is greater than, or equal to the upper bound * @throws org.apache.commons.math3.exception.NotStrictlyPositiveException * if the number of sample points is negative * @throws MathIllegalArgumentException if the number of sample points is * not a power of two plus one */ public double[] transform(final UnivariateFunction f, final double min, final double max, final int n, final TransformType type) throws MathIllegalArgumentException { final double[] data = FunctionUtils.sample(f, min, max, n); return transform(data, type); } /** * Perform the FCT algorithm (including inverse). * * @param f the real data array to be transformed * @return the real transformed array * @throws MathIllegalArgumentException if the length of the data array is * not a power of two plus one */ protected double[] fct(double[] f) throws MathIllegalArgumentException { final double[] transformed = new double[f.length]; final int n = f.length - 1; if (!ArithmeticUtils.isPowerOfTwo(n)) { throw new MathIllegalArgumentException(LocalizedFormats.NOT_POWER_OF_TWO_PLUS_ONE, Integer.valueOf(f.length)); } if (n == 1) { // trivial case transformed[0] = 0.5 * (f[0] + f[1]); transformed[1] = 0.5 * (f[0] - f[1]); return transformed; } // construct a new array and perform FFT on it final double[] x = new double[n]; x[0] = 0.5 * (f[0] + f[n]); x[n >> 1] = f[n >> 1]; // temporary variable for transformed[1] double t1 = 0.5 * (f[0] - f[n]); for (int i = 1; i < (n >> 1); i++) { final double a = 0.5 * (f[i] + f[n - i]); final double b = FastMath.sin(i * FastMath.PI / n) * (f[i] - f[n - i]); final double c = FastMath.cos(i * FastMath.PI / n) * (f[i] - f[n - i]); x[i] = a - b; x[n - i] = a + b; t1 += c; } FastFourierTransformer transformer; transformer = new FastFourierTransformer(DftNormalization.STANDARD); Complex[] y = transformer.transform(x, TransformType.FORWARD); // reconstruct the FCT result for the original array transformed[0] = y[0].getReal(); transformed[1] = t1; for (int i = 1; i < (n >> 1); i++) { transformed[2 * i] = y[i].getReal(); transformed[2 * i + 1] = transformed[2 * i - 1] - y[i].getImaginary(); } transformed[n] = y[n >> 1].getReal(); return transformed; } }