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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 Sine Transform for transformation of one-dimensional real
data sets. For reference, see James S. Walker, Fast Fourier
Transforms, chapter 3 (ISBN 0849371635).
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 DstNormalization. It should be noted that regardless to the convention, the first
element of the dataset to be transformed must be zero.
DST-I is equivalent to DFT of an odd extension of the data series.
More precisely, if x0, …, xN-1 is the data set
to be sine transformed, the extended data set x0#,
…, x2N-1# is defined as follows
- x0# = x0 = 0,
- xk# = xk if 1 ≤ k < N,
- xN# = 0,
- xk# = -x2N-k if N + 1 ≤ k <
2N.
Then, the standard DST-I y0, …, yN-1 of the real
data set x0, …, xN-1 is equal to half
of i (the pure imaginary number) times the N first elements of the DFT of the
extended data set x0#, …,
x2N-1#
yn = (i / 2) ∑k=02N-1
xk# exp[-2πi nk / (2N)]
k = 0, …, N-1.
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 transform(UnivariateFunction, double, double, int, TransformType), after sampling.
Since: 1.2
/**
* 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 DstNormalization}.
* <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 #transform(UnivariateFunction, double, double, int, TransformType)},
* after sampling.
*
* @since 1.2
*/
public class FastSineTransformer implements RealTransformer, Serializable {
Serializable version identifier. /** Serializable version identifier. */
static final long serialVersionUID = 20120211L;
The type of DST to be performed. /** The type of DST to be performed. */
private final DstNormalization normalization;
Creates a new instance of this class, with various normalization conventions.
Params: - normalization – the type of normalization to be applied to the transformed data
/**
* 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 FastSineTransformer(final DstNormalization normalization) {
this.normalization = normalization;
}
{@inheritDoc} The first element of the specified data set is required to be 0. Throws: - MathIllegalArgumentException – if the length of the data array is
not a power of two, or the first element of the data array is not zero
/**
* {@inheritDoc}
*
* The first element of the specified data set is required to be {@code 0}.
*
* @throws MathIllegalArgumentException if the length of the data array is
* not a power of two, or the first element of the data array is not zero
*/
public double[] transform(final double[] f, final TransformType type) {
if (normalization == DstNormalization.ORTHOGONAL_DST_I) {
final double s = FastMath.sqrt(2.0 / f.length);
return TransformUtils.scaleArray(fst(f), s);
}
if (type == TransformType.FORWARD) {
return fst(f);
}
final double s = 2.0 / f.length;
return TransformUtils.scaleArray(fst(f), s);
}
{@inheritDoc} This implementation enforces f(x) = 0.0 at x = 0.0. Throws: - NonMonotonicSequenceException –
if the lower bound is greater than, or equal to the upper bound
- NotStrictlyPositiveException –
if the number of sample points is negative
- MathIllegalArgumentException – if the number of sample points is not a power of two
/**
* {@inheritDoc}
*
* This implementation enforces {@code f(x) = 0.0} at {@code x = 0.0}.
*
* @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
*/
public double[] transform(final UnivariateFunction f,
final double min, final double max, final int n,
final TransformType type) {
final double[] data = FunctionUtils.sample(f, min, max, n);
data[0] = 0.0;
return transform(data, type);
}
Perform the FST algorithm (including inverse). The first element of the data set is required to be 0. Params: - f – the real data array to be transformed
Throws: - MathIllegalArgumentException – if the length of the data array is
not a power of two, or the first element of the data array is not zero
Returns: the real transformed array
/**
* Perform the FST algorithm (including inverse). The first element of the
* data set is required to be {@code 0}.
*
* @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, or the first element of the data array is not zero
*/
protected double[] fst(double[] f) throws MathIllegalArgumentException {
final double[] transformed = new double[f.length];
if (!ArithmeticUtils.isPowerOfTwo(f.length)) {
throw new MathIllegalArgumentException(
LocalizedFormats.NOT_POWER_OF_TWO_CONSIDER_PADDING,
Integer.valueOf(f.length));
}
if (f[0] != 0.0) {
throw new MathIllegalArgumentException(
LocalizedFormats.FIRST_ELEMENT_NOT_ZERO,
Double.valueOf(f[0]));
}
final int n = f.length;
if (n == 1) { // trivial case
transformed[0] = 0.0;
return transformed;
}
// construct a new array and perform FFT on it
final double[] x = new double[n];
x[0] = 0.0;
x[n >> 1] = 2.0 * f[n >> 1];
for (int i = 1; i < (n >> 1); i++) {
final double a = FastMath.sin(i * FastMath.PI / n) * (f[i] + f[n - i]);
final double b = 0.5 * (f[i] - f[n - i]);
x[i] = a + b;
x[n - i] = a - b;
}
FastFourierTransformer transformer;
transformer = new FastFourierTransformer(DftNormalization.STANDARD);
Complex[] y = transformer.transform(x, TransformType.FORWARD);
// reconstruct the FST result for the original array
transformed[0] = 0.0;
transformed[1] = 0.5 * y[0].getReal();
for (int i = 1; i < (n >> 1); i++) {
transformed[2 * i] = -y[i].getImaginary();
transformed[2 * i + 1] = y[i].getReal() + transformed[2 * i - 1];
}
return transformed;
}
}