FastSineTransform.java
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* The ASF licenses this file to You under the Apache License, Version 2.0
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* http://www.apache.org/licenses/LICENSE-2.0
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* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
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* See the License for the specific language governing permissions and
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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
}
}