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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.math4.transform;
import java.util.Random;
import org.apache.commons.math4.analysis.UnivariateFunction;
import org.apache.commons.math4.analysis.function.Sin;
import org.apache.commons.math4.analysis.function.Sinc;
import org.apache.commons.math4.complex.Complex;
import org.apache.commons.math4.exception.MathIllegalArgumentException;
import org.apache.commons.math4.exception.NotStrictlyPositiveException;
import org.apache.commons.math4.exception.NumberIsTooLargeException;
import org.apache.commons.math4.transform.DftNormalization;
import org.apache.commons.math4.transform.FastFourierTransformer;
import org.apache.commons.math4.transform.TransformType;
import org.apache.commons.math4.transform.TransformUtils;
import org.apache.commons.math4.util.FastMath;
import org.junit.Assert;
import org.junit.Test;
/**
* Test case for fast Fourier transformer.
* <p>
* FFT algorithm is exact, the small tolerance number is used only
* to account for round-off errors.
*
*/
public final class FastFourierTransformerTest {
/** The common seed of all random number generators used in this test. */
private final static long SEED = 20110111L;
/*
* Precondition checks.
*/
@Test
public void testTransformComplexSizeNotAPowerOfTwo() {
final int n = 127;
final Complex[] x = createComplexData(n);
final DftNormalization[] norm;
norm = DftNormalization.values();
final TransformType[] type;
type = TransformType.values();
for (int i = 0; i < norm.length; i++) {
for (int j = 0; j < type.length; j++) {
final FastFourierTransformer fft;
fft = new FastFourierTransformer(norm[i]);
try {
fft.transform(x, type[j]);
Assert.fail(norm[i] + ", " + type[j] +
": MathIllegalArgumentException was expected");
} catch (MathIllegalArgumentException e) {
// Expected behaviour
}
}
}
}
@Test
public void testTransformRealSizeNotAPowerOfTwo() {
final int n = 127;
final double[] x = createRealData(n);
final DftNormalization[] norm;
norm = DftNormalization.values();
final TransformType[] type;
type = TransformType.values();
for (int i = 0; i < norm.length; i++) {
for (int j = 0; j < type.length; j++) {
final FastFourierTransformer fft;
fft = new FastFourierTransformer(norm[i]);
try {
fft.transform(x, type[j]);
Assert.fail(norm[i] + ", " + type[j] +
": MathIllegalArgumentException was expected");
} catch (MathIllegalArgumentException e) {
// Expected behaviour
}
}
}
}
@Test
public void testTransformFunctionSizeNotAPowerOfTwo() {
final int n = 127;
final UnivariateFunction f = new Sin();
final DftNormalization[] norm;
norm = DftNormalization.values();
final TransformType[] type;
type = TransformType.values();
for (int i = 0; i < norm.length; i++) {
for (int j = 0; j < type.length; j++) {
final FastFourierTransformer fft;
fft = new FastFourierTransformer(norm[i]);
try {
fft.transform(f, 0.0, Math.PI, n, type[j]);
Assert.fail(norm[i] + ", " + type[j] +
": MathIllegalArgumentException was expected");
} catch (MathIllegalArgumentException e) {
// Expected behaviour
}
}
}
}
@Test
public void testTransformFunctionNotStrictlyPositiveNumberOfSamples() {
final int n = -128;
final UnivariateFunction f = new Sin();
final DftNormalization[] norm;
norm = DftNormalization.values();
final TransformType[] type;
type = TransformType.values();
for (int i = 0; i < norm.length; i++) {
for (int j = 0; j < type.length; j++) {
final FastFourierTransformer fft;
fft = new FastFourierTransformer(norm[i]);
try {
fft.transform(f, 0.0, Math.PI, n, type[j]);
fft.transform(f, 0.0, Math.PI, n, type[j]);
Assert.fail(norm[i] + ", " + type[j] +
": NotStrictlyPositiveException was expected");
} catch (NotStrictlyPositiveException e) {
// Expected behaviour
}
}
}
}
@Test
public void testTransformFunctionInvalidBounds() {
final int n = 128;
final UnivariateFunction f = new Sin();
final DftNormalization[] norm;
norm = DftNormalization.values();
final TransformType[] type;
type = TransformType.values();
for (int i = 0; i < norm.length; i++) {
for (int j = 0; j < type.length; j++) {
final FastFourierTransformer fft;
fft = new FastFourierTransformer(norm[i]);
try {
fft.transform(f, Math.PI, 0.0, n, type[j]);
Assert.fail(norm[i] + ", " + type[j] +
": NumberIsTooLargeException was expected");
} catch (NumberIsTooLargeException e) {
// Expected behaviour
}
}
}
}
/*
* Utility methods for checking (successful) transforms.
*/
private static Complex[] createComplexData(final int n) {
final Random random = new Random(SEED);
final Complex[] data = new Complex[n];
for (int i = 0; i < n; i++) {
final double re = 2.0 * random.nextDouble() - 1.0;
final double im = 2.0 * random.nextDouble() - 1.0;
data[i] = new Complex(re, im);
}
return data;
}
private static double[] createRealData(final int n) {
final Random random = new Random(SEED);
final double[] data = new double[n];
for (int i = 0; i < n; i++) {
data[i] = 2.0 * random.nextDouble() - 1.0;
}
return data;
}
/** Naive implementation of DFT, for reference. */
private static Complex[] dft(final Complex[] x, final int sgn) {
final int n = x.length;
final double[] cos = new double[n];
final double[] sin = new double[n];
final Complex[] y = new Complex[n];
for (int i = 0; i < n; i++) {
final double arg = 2.0 * FastMath.PI * i / n;
cos[i] = FastMath.cos(arg);
sin[i] = FastMath.sin(arg);
}
for (int i = 0; i < n; i++) {
double yr = 0.0;
double yi = 0.0;
for (int j = 0; j < n; j++) {
final int index = (i * j) % n;
final double c = cos[index];
final double s = sin[index];
final double xr = x[j].getReal();
final double xi = x[j].getImaginary();
yr += c * xr - sgn * s * xi;
yi += sgn * s * xr + c * xi;
}
y[i] = new Complex(yr, yi);
}
return y;
}
private static void doTestTransformComplex(final int n, final double tol,
final DftNormalization normalization,
final TransformType type) {
final FastFourierTransformer fft;
fft = new FastFourierTransformer(normalization);
final Complex[] x = createComplexData(n);
final Complex[] expected;
final double s;
if (type==TransformType.FORWARD) {
expected = dft(x, -1);
if (normalization == DftNormalization.STANDARD){
s = 1.0;
} else {
s = 1.0 / FastMath.sqrt(n);
}
} else {
expected = dft(x, 1);
if (normalization == DftNormalization.STANDARD) {
s = 1.0 / n;
} else {
s = 1.0 / FastMath.sqrt(n);
}
}
final Complex[] actual = fft.transform(x, type);
for (int i = 0; i < n; i++) {
final String msg;
msg = String.format("%s, %s, %d, %d", normalization, type, n, i);
final double re = s * expected[i].getReal();
Assert.assertEquals(msg, re, actual[i].getReal(),
tol * FastMath.abs(re));
final double im = s * expected[i].getImaginary();
Assert.assertEquals(msg, im, actual[i].getImaginary(), tol *
FastMath.abs(re));
}
}
private static void doTestTransformReal(final int n, final double tol,
final DftNormalization normalization,
final TransformType type) {
final FastFourierTransformer fft;
fft = new FastFourierTransformer(normalization);
final double[] x = createRealData(n);
final Complex[] xc = new Complex[n];
for (int i = 0; i < n; i++) {
xc[i] = new Complex(x[i], 0.0);
}
final Complex[] expected;
final double s;
if (type == TransformType.FORWARD) {
expected = dft(xc, -1);
if (normalization == DftNormalization.STANDARD) {
s = 1.0;
} else {
s = 1.0 / FastMath.sqrt(n);
}
} else {
expected = dft(xc, 1);
if (normalization == DftNormalization.STANDARD) {
s = 1.0 / n;
} else {
s = 1.0 / FastMath.sqrt(n);
}
}
final Complex[] actual = fft.transform(x, type);
for (int i = 0; i < n; i++) {
final String msg;
msg = String.format("%s, %s, %d, %d", normalization, type, n, i);
final double re = s * expected[i].getReal();
Assert.assertEquals(msg, re, actual[i].getReal(),
tol * FastMath.abs(re));
final double im = s * expected[i].getImaginary();
Assert.assertEquals(msg, im, actual[i].getImaginary(), tol *
FastMath.abs(re));
}
}
private static void doTestTransformFunction(final UnivariateFunction f,
final double min, final double max, int n, final double tol,
final DftNormalization normalization,
final TransformType type) {
final FastFourierTransformer fft;
fft = new FastFourierTransformer(normalization);
final Complex[] x = new Complex[n];
for (int i = 0; i < n; i++) {
final double t = min + i * (max - min) / n;
x[i] = new Complex(f.value(t));
}
final Complex[] expected;
final double s;
if (type == TransformType.FORWARD) {
expected = dft(x, -1);
if (normalization == DftNormalization.STANDARD) {
s = 1.0;
} else {
s = 1.0 / FastMath.sqrt(n);
}
} else {
expected = dft(x, 1);
if (normalization == DftNormalization.STANDARD) {
s = 1.0 / n;
} else {
s = 1.0 / FastMath.sqrt(n);
}
}
final Complex[] actual = fft.transform(f, min, max, n, type);
for (int i = 0; i < n; i++) {
final String msg = String.format("%d, %d", n, i);
final double re = s * expected[i].getReal();
Assert.assertEquals(msg, re, actual[i].getReal(),
tol * FastMath.abs(re));
final double im = s * expected[i].getImaginary();
Assert.assertEquals(msg, im, actual[i].getImaginary(), tol *
FastMath.abs(re));
}
}
/*
* Tests of standard transform (when data is valid).
*/
@Test
public void testTransformComplex() {
final DftNormalization[] norm;
norm = DftNormalization.values();
final TransformType[] type;
type = TransformType.values();
for (int i = 0; i < norm.length; i++) {
for (int j = 0; j < type.length; j++) {
doTestTransformComplex(2, 1.0E-15, norm[i], type[j]);
doTestTransformComplex(4, 1.0E-14, norm[i], type[j]);
doTestTransformComplex(8, 1.0E-14, norm[i], type[j]);
doTestTransformComplex(16, 1.0E-13, norm[i], type[j]);
doTestTransformComplex(32, 1.0E-13, norm[i], type[j]);
doTestTransformComplex(64, 1.0E-12, norm[i], type[j]);
doTestTransformComplex(128, 1.0E-12, norm[i], type[j]);
}
}
}
@Test
public void testStandardTransformReal() {
final DftNormalization[] norm;
norm = DftNormalization.values();
final TransformType[] type;
type = TransformType.values();
for (int i = 0; i < norm.length; i++) {
for (int j = 0; j < type.length; j++) {
doTestTransformReal(2, 1.0E-15, norm[i], type[j]);
doTestTransformReal(4, 1.0E-14, norm[i], type[j]);
doTestTransformReal(8, 1.0E-14, norm[i], type[j]);
doTestTransformReal(16, 1.0E-13, norm[i], type[j]);
doTestTransformReal(32, 1.0E-13, norm[i], type[j]);
doTestTransformReal(64, 1.0E-13, norm[i], type[j]);
doTestTransformReal(128, 1.0E-11, norm[i], type[j]);
}
}
}
@Test
public void testStandardTransformFunction() {
final UnivariateFunction f = new Sinc();
final double min = -FastMath.PI;
final double max = FastMath.PI;
final DftNormalization[] norm;
norm = DftNormalization.values();
final TransformType[] type;
type = TransformType.values();
for (int i = 0; i < norm.length; i++) {
for (int j = 0; j < type.length; j++) {
doTestTransformFunction(f, min, max, 2, 1.0E-15, norm[i], type[j]);
doTestTransformFunction(f, min, max, 4, 1.0E-14, norm[i], type[j]);
doTestTransformFunction(f, min, max, 8, 1.0E-14, norm[i], type[j]);
doTestTransformFunction(f, min, max, 16, 1.0E-13, norm[i], type[j]);
doTestTransformFunction(f, min, max, 32, 1.0E-13, norm[i], type[j]);
doTestTransformFunction(f, min, max, 64, 1.0E-12, norm[i], type[j]);
doTestTransformFunction(f, min, max, 128, 1.0E-11, norm[i], type[j]);
}
}
}
/*
* Additional tests for 1D data.
*/
/**
* Test of transformer for the ad hoc data taken from Mathematica.
*/
@Test
public void testAdHocData() {
FastFourierTransformer transformer;
transformer = new FastFourierTransformer(DftNormalization.STANDARD);
Complex result[]; double tolerance = 1E-12;
double x[] = {1.3, 2.4, 1.7, 4.1, 2.9, 1.7, 5.1, 2.7};
Complex y[] = {
new Complex(21.9, 0.0),
new Complex(-2.09497474683058, 1.91507575950825),
new Complex(-2.6, 2.7),
new Complex(-1.10502525316942, -4.88492424049175),
new Complex(0.1, 0.0),
new Complex(-1.10502525316942, 4.88492424049175),
new Complex(-2.6, -2.7),
new Complex(-2.09497474683058, -1.91507575950825)};
result = transformer.transform(x, TransformType.FORWARD);
for (int i = 0; i < result.length; i++) {
Assert.assertEquals(y[i].getReal(), result[i].getReal(), tolerance);
Assert.assertEquals(y[i].getImaginary(), result[i].getImaginary(), tolerance);
}
result = transformer.transform(y, TransformType.INVERSE);
for (int i = 0; i < result.length; i++) {
Assert.assertEquals(x[i], result[i].getReal(), tolerance);
Assert.assertEquals(0.0, result[i].getImaginary(), tolerance);
}
double x2[] = {10.4, 21.6, 40.8, 13.6, 23.2, 32.8, 13.6, 19.2};
TransformUtils.scaleArray(x2, 1.0 / FastMath.sqrt(x2.length));
Complex y2[] = y;
transformer = new FastFourierTransformer(DftNormalization.UNITARY);
result = transformer.transform(y2, TransformType.FORWARD);
for (int i = 0; i < result.length; i++) {
Assert.assertEquals(x2[i], result[i].getReal(), tolerance);
Assert.assertEquals(0.0, result[i].getImaginary(), tolerance);
}
result = transformer.transform(x2, TransformType.INVERSE);
for (int i = 0; i < result.length; i++) {
Assert.assertEquals(y2[i].getReal(), result[i].getReal(), tolerance);
Assert.assertEquals(y2[i].getImaginary(), result[i].getImaginary(), tolerance);
}
}
/**
* Test of transformer for the sine function.
*/
@Test
public void testSinFunction() {
UnivariateFunction f = new Sin();
FastFourierTransformer transformer;
transformer = new FastFourierTransformer(DftNormalization.STANDARD);
Complex result[]; int N = 1 << 8;
double min, max, tolerance = 1E-12;
min = 0.0; max = 2.0 * FastMath.PI;
result = transformer.transform(f, min, max, N, TransformType.FORWARD);
Assert.assertEquals(0.0, result[1].getReal(), tolerance);
Assert.assertEquals(-(N >> 1), result[1].getImaginary(), tolerance);
Assert.assertEquals(0.0, result[N-1].getReal(), tolerance);
Assert.assertEquals(N >> 1, result[N-1].getImaginary(), tolerance);
for (int i = 0; i < N-1; i += (i == 0 ? 2 : 1)) {
Assert.assertEquals(0.0, result[i].getReal(), tolerance);
Assert.assertEquals(0.0, result[i].getImaginary(), tolerance);
}
min = -FastMath.PI; max = FastMath.PI;
result = transformer.transform(f, min, max, N, TransformType.INVERSE);
Assert.assertEquals(0.0, result[1].getReal(), tolerance);
Assert.assertEquals(-0.5, result[1].getImaginary(), tolerance);
Assert.assertEquals(0.0, result[N-1].getReal(), tolerance);
Assert.assertEquals(0.5, result[N-1].getImaginary(), tolerance);
for (int i = 0; i < N-1; i += (i == 0 ? 2 : 1)) {
Assert.assertEquals(0.0, result[i].getReal(), tolerance);
Assert.assertEquals(0.0, result[i].getImaginary(), tolerance);
}
}
}