001    /*
002     * Licensed to the Apache Software Foundation (ASF) under one or more
003     * contributor license agreements.  See the NOTICE file distributed with
004     * this work for additional information regarding copyright ownership.
005     * The ASF licenses this file to You under the Apache License, Version 2.0
006     * (the "License"); you may not use this file except in compliance with
007     * the License.  You may obtain a copy of the License at
008     *
009     *      http://www.apache.org/licenses/LICENSE-2.0
010     *
011     * Unless required by applicable law or agreed to in writing, software
012     * distributed under the License is distributed on an "AS IS" BASIS,
013     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014     * See the License for the specific language governing permissions and
015     * limitations under the License.
016     */
017    package org.apache.commons.math3.transform;
018    
019    import java.io.Serializable;
020    
021    import org.apache.commons.math3.analysis.FunctionUtils;
022    import org.apache.commons.math3.analysis.UnivariateFunction;
023    import org.apache.commons.math3.complex.Complex;
024    import org.apache.commons.math3.exception.MathIllegalArgumentException;
025    import org.apache.commons.math3.exception.util.LocalizedFormats;
026    import org.apache.commons.math3.util.ArithmeticUtils;
027    import org.apache.commons.math3.util.FastMath;
028    
029    /**
030     * Implements the Fast Sine Transform for transformation of one-dimensional real
031     * data sets. For reference, see James S. Walker, <em>Fast Fourier
032     * Transforms</em>, chapter 3 (ISBN 0849371635).
033     * <p>
034     * There are several variants of the discrete sine transform. The present
035     * implementation corresponds to DST-I, with various normalization conventions,
036     * which are specified by the parameter {@link DstNormalization}.
037     * <strong>It should be noted that regardless to the convention, the first
038     * element of the dataset to be transformed must be zero.</strong>
039     * <p>
040     * DST-I is equivalent to DFT of an <em>odd extension</em> of the data series.
041     * More precisely, if x<sub>0</sub>, &hellip;, x<sub>N-1</sub> is the data set
042     * to be sine transformed, the extended data set x<sub>0</sub><sup>&#35;</sup>,
043     * &hellip;, x<sub>2N-1</sub><sup>&#35;</sup> is defined as follows
044     * <ul>
045     * <li>x<sub>0</sub><sup>&#35;</sup> = x<sub>0</sub> = 0,</li>
046     * <li>x<sub>k</sub><sup>&#35;</sup> = x<sub>k</sub> if 1 &le; k &lt; N,</li>
047     * <li>x<sub>N</sub><sup>&#35;</sup> = 0,</li>
048     * <li>x<sub>k</sub><sup>&#35;</sup> = -x<sub>2N-k</sub> if N + 1 &le; k &lt;
049     * 2N.</li>
050     * </ul>
051     * <p>
052     * Then, the standard DST-I y<sub>0</sub>, &hellip;, y<sub>N-1</sub> of the real
053     * data set x<sub>0</sub>, &hellip;, x<sub>N-1</sub> is equal to <em>half</em>
054     * of i (the pure imaginary number) times the N first elements of the DFT of the
055     * extended data set x<sub>0</sub><sup>&#35;</sup>, &hellip;,
056     * x<sub>2N-1</sub><sup>&#35;</sup> <br />
057     * y<sub>n</sub> = (i / 2) &sum;<sub>k=0</sub><sup>2N-1</sup>
058     * x<sub>k</sub><sup>&#35;</sup> exp[-2&pi;i nk / (2N)]
059     * &nbsp;&nbsp;&nbsp;&nbsp;k = 0, &hellip;, N-1.
060     * <p>
061     * The present implementation of the discrete sine transform as a fast sine
062     * transform requires the length of the data to be a power of two. Besides,
063     * it implicitly assumes that the sampled function is odd. In particular, the
064     * first element of the data set must be 0, which is enforced in
065     * {@link #transform(UnivariateFunction, double, double, int, TransformType)},
066     * after sampling.
067     *
068     * @version $Id: FastSineTransformer.java 1385310 2012-09-16 16:32:10Z tn $
069     * @since 1.2
070     */
071    public class FastSineTransformer implements RealTransformer, Serializable {
072    
073        /** Serializable version identifier. */
074        static final long serialVersionUID = 20120211L;
075    
076        /** The type of DST to be performed. */
077        private final DstNormalization normalization;
078    
079        /**
080         * Creates a new instance of this class, with various normalization conventions.
081         *
082         * @param normalization the type of normalization to be applied to the transformed data
083         */
084        public FastSineTransformer(final DstNormalization normalization) {
085            this.normalization = normalization;
086        }
087    
088        /**
089         * {@inheritDoc}
090         *
091         * The first element of the specified data set is required to be {@code 0}.
092         *
093         * @throws MathIllegalArgumentException if the length of the data array is
094         *   not a power of two, or the first element of the data array is not zero
095         */
096        public double[] transform(final double[] f, final TransformType type) {
097            if (normalization == DstNormalization.ORTHOGONAL_DST_I) {
098                final double s = FastMath.sqrt(2.0 / f.length);
099                return TransformUtils.scaleArray(fst(f), s);
100            }
101            if (type == TransformType.FORWARD) {
102                return fst(f);
103            }
104            final double s = 2.0 / f.length;
105            return TransformUtils.scaleArray(fst(f), s);
106        }
107    
108        /**
109         * {@inheritDoc}
110         *
111         * This implementation enforces {@code f(x) = 0.0} at {@code x = 0.0}.
112         *
113         * @throws org.apache.commons.math3.exception.NonMonotonicSequenceException
114         *   if the lower bound is greater than, or equal to the upper bound
115         * @throws org.apache.commons.math3.exception.NotStrictlyPositiveException
116         *   if the number of sample points is negative
117         * @throws MathIllegalArgumentException if the number of sample points is not a power of two
118         */
119        public double[] transform(final UnivariateFunction f,
120            final double min, final double max, final int n,
121            final TransformType type) {
122    
123            final double[] data = FunctionUtils.sample(f, min, max, n);
124            data[0] = 0.0;
125            return transform(data, type);
126        }
127    
128        /**
129         * Perform the FST algorithm (including inverse). The first element of the
130         * data set is required to be {@code 0}.
131         *
132         * @param f the real data array to be transformed
133         * @return the real transformed array
134         * @throws MathIllegalArgumentException if the length of the data array is
135         *   not a power of two, or the first element of the data array is not zero
136         */
137        protected double[] fst(double[] f) throws MathIllegalArgumentException {
138    
139            final double[] transformed = new double[f.length];
140    
141            if (!ArithmeticUtils.isPowerOfTwo(f.length)) {
142                throw new MathIllegalArgumentException(
143                        LocalizedFormats.NOT_POWER_OF_TWO_CONSIDER_PADDING,
144                        Integer.valueOf(f.length));
145            }
146            if (f[0] != 0.0) {
147                throw new MathIllegalArgumentException(
148                        LocalizedFormats.FIRST_ELEMENT_NOT_ZERO,
149                        Double.valueOf(f[0]));
150            }
151            final int n = f.length;
152            if (n == 1) {       // trivial case
153                transformed[0] = 0.0;
154                return transformed;
155            }
156    
157            // construct a new array and perform FFT on it
158            final double[] x = new double[n];
159            x[0] = 0.0;
160            x[n >> 1] = 2.0 * f[n >> 1];
161            for (int i = 1; i < (n >> 1); i++) {
162                final double a = FastMath.sin(i * FastMath.PI / n) * (f[i] + f[n - i]);
163                final double b = 0.5 * (f[i] - f[n - i]);
164                x[i]     = a + b;
165                x[n - i] = a - b;
166            }
167            FastFourierTransformer transformer;
168            transformer = new FastFourierTransformer(DftNormalization.STANDARD);
169            Complex[] y = transformer.transform(x, TransformType.FORWARD);
170    
171            // reconstruct the FST result for the original array
172            transformed[0] = 0.0;
173            transformed[1] = 0.5 * y[0].getReal();
174            for (int i = 1; i < (n >> 1); i++) {
175                transformed[2 * i]     = -y[i].getImaginary();
176                transformed[2 * i + 1] = y[i].getReal() + transformed[2 * i - 1];
177            }
178    
179            return transformed;
180        }
181    }