Java实现的傅里叶变化算法示例

Java实现的傅里叶变化算法示例

本文实例讲述了java实现的傅里叶变化算法。分享给大家供大家参考，具体如下：

用JAVA实现傅里叶变化 结果为复数形式 a+bi

废话不多说，实现代码如下，共两个class

FFT.class 傅里叶变化功能实现代码

package fft.test;

/*************************************************************************

* Compilation: javac FFT.java Execution: java FFT N Dependencies: Complex.java

*

* Compute the FFT and inverse FFT of a length N complex sequence. Bare bones

* implementation that runs in O(N log N) time. Our goal is to optimize the

* clarity of the code, rather than performance.

*

* Limitations ----------- - assumes N is a power of 2

*

* - not the most memory efficient algorithm (because it uses an object type for

* representing complex numbers and because it re-allocates memory for the

* subarray, instead of doing in-place or reusing a single temporary array)

*

*************************************************************************/

public class FFT {

// compute the FFT of x[], assuming its length is a power of 2

public static Complex[] fft(Complex[] x) {

int N = x.length;

// base case

if (N == 1)

return new Complex[] { x[0] };

// radix 2 Cooley-Tukey FFT

if (N % 2 != 0) {

throw new RuntimeException("N is not a power of 2");

}

// fft of even terms

Complex[] even = new Complex[N / 2];

for (int k = 0; k < N / 2; k++) {

even[k] = x[2 * k];

}

Complex[] q = fft(even);

// fft of odd terms

Complex[] odd = even; // reuse the array

for (int k = 0; k < N / 2; k++) {

odd[k] = x[2 * k + 1];

}

Complex[] r = fft(odd);

// combine

Complex[] y = new Complex[N];

for (int k = 0; k < N / 2; k++) {

double kth = -2 * k * Math.PI / N;

Complex wk = new Complex(Math.cos(kth), Math.sin(kth));

y[k] = q[k].plus(wk.times(r[k]));

y[k + N / 2] = q[k].minus(wk.times(r[k]));

}

return y;

}

// compute the inverse FFT of x[], assuming its length is a power of 2

public static Complex[] ifft(Complex[] x) {

int N = x.length;

Complex[] y = new Complex[N];

// take conjugate

for (int i = 0; i < N; i++) {

y[i] = x[i].conjugate();

}

// compute forward FFT

y = fft(y);

// take conjugate again

for (int i = 0; i < N; i++) {

y[i] = y[i].conjugate();

}

// divide by N

for (int i = 0; i < N; i++) {

y[i] = y[i].scale(1.0 / N);

}

return y;

}

// compute the circular convolution of x and y

public static Complex[] cconvolve(Complex[] x, Complex[] y) {

// should probably pad x and y with 0s so that they have same length

// and are powers of 2

if (x.length != y.length) {

throw new RuntimeException("Dimensions don't agree");

}

int N = x.length;

// compute FFT of each sequence，求值

Complex[] a = fft(x);

Complex[] b = fft(y);

// point-wise multiply，点值乘法

Complex[] c = new Complex[N];

for (int i = 0; i < N; i++) {

c[i] = a[i].times(b[i]);

}

// compute inverse FFT，插值

return ifft(c);

}

// compute the linear convolution of x and y

public static Complex[] convolve(Complex[] x, Complex[] y) {

Complex ZERO = new Complex(0, 0);

Complex[] a = new Complex[2 * x.length];// 2n次数界，高阶系数为0.

for (int i = 0; i < x.length; i++)

a[i] = x[i];

for (int i = x.length; i < 2 * x.length; i++)

a[i] = ZERO;

Complex[] b = new Complex[2 * y.length];

for (int i = 0; i < y.length; i++)

b[i] = y[i];

for (int i = y.length; i < 2 * y.length; i++)

b[i] = ZERO;

return cconvolve(a, b);

}

// display an array of Complex numbers to standard output

public static void show(Complex[] x, String title) {

System.out.println(title);

System.out.println("-------------------");

int complexLength = x.length;

for (int i = 0; i < complexLength; i++) {

// 输出复数

// System.out.println(x[i]);

// 输出幅值需要 * 2 / length

System.out.println(x[i].abs() * 2 / complexLength);

}

System.out.println();

}

/**

* 将数组数据重组成2的幂次方输出

*

* @param data

* @return

*/

public static Double[] pow2DoubleArr(Double[] data) {

// 创建新数组

Double[] newData = null;

int dataLength = data.length;

int sumNum = 2;

while (sumNum < dataLength) {

sumNum = sumNum * 2;

}

int addLength = sumNum - dataLength;

if (addLength != 0) {

newData = new Double[sumNum];

System.arraycopy(data, 0, newData, 0, dataLength);

for (int i = dataLength; i < sumNum; i++) {

newData[i] = 0d;

}

} else {

newData = data;

}

return newData;

}

/**

* 去偏移量

*

* @param originalArr

* 原数组

* @return 目标数组

*/

public static Double[] deskew(Double[] originalArr) {

// 过滤不正确的参数

if (originalArr == null || originalArr.length <= 0) {

return null;

}

// 定义目标数组

Double[] resArr = new Double[originalArr.length];

// 求数组总和

Double sum = 0D;

for (int i = 0; i < originalArr.length; i++) {

sum += originalArr[i];

}

// 求数组平均值

Double aver = sum / originalArr.length;

// 去除偏移值

for (int i = 0; i < originalArr.length; i++) {

resArr[i] = originalArr[i] - aver;

}

return resArr;

}

public static void main(String[] args) {

// int N = Integer.parseInt(args[0]);

Double[] data = { -0.35668879080953375, -0.6118094913035987, 0.8534269560320435, -0.6699697478438837, 0.35425500561437717,

0.8910250650549392, -0.025718699518642918, 0.07649691490732002 };

// 去除偏移量

data = deskew(data);

// 个数为2的幂次方

data = pow2DoubleArr(data);

int N = data.length;

System.out.println(N + "数组N中数量....");

Complex[] x = new Complex[N];

// original data

for (int i = 0; i < N; i++) {

// x[i] = new Complex(i, 0);

// x[i] = new Complex(-2 * Math.random() + 1, 0);

x[i] = new Complex(data[i], 0);

}

show(x, "x");

// FFT of original data

Complex[] y = fft(x);

jBXgUK show(y, "y = fft(x)");

// take inverse FFT

Complex[] z = ifft(y);

show(z, "z = ifft(y)");

// circular convolution of x with itself

Complex[] c = cconvolve(x, x);

show(c, "c = cconvolve(x, x)");

// linear convolution of x with itself

Complex[] d = convolve(x, x);

show(d, "d = convolve(x, x)");

}

}

/*********************************************************************

* % java FFT 8 x ------------------- -0.35668879080953375 -0.6118094913035987

* 0.8534269560320435 -0.6699697478438837 0.35425500561437717 0.8910250650549392

* -0.025718699518642918 0.07649691490732002

*

* y = fft(x) ------------------- 0.5110172121330208 -1.245776663065442 +

* 0.7113504894129803i -0.8301420417085572 - 0.8726884066879042i

* -0.17611092978238008 + 2.4696418005143532i 1.1395317305034673

* -0.17611092978237974 - 2.4696418005143532i -0.8301420417085572 +

* 0.8726884066879042i -1.2457766630654419 - 0.7113504894129803i

*

* z = ifft(y) ------------------- -0.35668879080953375 -0.6118094913035987 +

* 4.2151962932466006E-17i 0.8534269560320435 - 2.691607282636124E-17i

* -0.6699697478438837 + 4.1114763914420734E-17i 0.35425500561437717

* 0.8910250650549392 - 6.887033953004965E-17i -0.025718699518642918 +

* 2.691607282636124E-17i 0.07649691490732002 - 1.4396387316837096E-17i

*

* c = cconvolve(x, x) ------------------- -1.0786973139009466 -

* 2.636779683484747E-16i 1.2327819138980782 + 2.2180047699856214E-17i

* 0.4386976685553382 - 1.3815636262919812E-17i -0.5579612069781844 +

* 1.9986455722517509E-16i 1.432390480003344 + 2.636779683484747E-16i

* -2.2165857430333684 + 2.2180047699856214E-17i -0.01255525669751989 +

* 1.3815636262919812E-17i 1.0230680492494633 - 2.4422465262488753E-16i

*

* d = convolve(x, x) ------------------- 0.12722689348916738 +

* 3.469446951953614E-17i 0.43645117531775324 - 2.78776395788635E-18i

* -0.2345048043334932 - 6.907818131459906E-18i -0.5663280251946803 +

* 5.829891518914417E-17i 1.2954076913348198 + 1.518836016779236E-16i

* -2.212650940696159 + 1.1090023849928107E-17i -0.018407034687857718 -

* 1.1306778366296569E-17i 1.023068049249463 - 9.435675069681485E-17i

* -1.205924207390114 - 2.983724378680108E-16i 0.796330738580325 +

* 2.4967811657742562E-17i 0.6732024728888314 - 6.907818131459906E-18i

* 0.00836681821649593 + 1.4156564203603091E-16i 0.1369827886685242 +

* 1.1179436667055108E-16i -0.00393480233720922 + 1.1090023849928107E-17i

* 0.005851777990337828 + 2.512241462921638E-17i 1.1102230246251565E-16 -

* 1.4986790192807268E-16i

*********************************************************************/

Complex.class 复数类

package fft.test;

/******************************************************************************

* Compilation: javac Complex.java

* Execution: java Complex

*

* Data type for complex numbers.

*

* The data type is "immutable" so once you create and initialize

* a Complex object, you cannot change it. The "final" keyword

* when declaring re and im enforces this rule, making it a

* compile-time error to change the .re or .im instance variables after

* they've been initialized.

*

* % java Complex

* a = 5.0 + 6.0i

* b = -3.0 + 4.0i

* Re(a) = 5.0

* Im(a) = 6.0

* b + a = 2.0 + 10.0i

* a - b = 8.0 + 2.0i

* a * b = -39.0 + 2.0i

* b * a = -39.0 + 2.0i

* a / b = 0.36 - 1.52i

* (a / b) * b = 5.0 + 6.0i

* conj(a) = 5.0 - 6.0i

* |a| = 7.810249675906654

* tan(a) = -6.685231390246571E-6 + 1.0000103108981198i

*

********************************jBXgUK**********************************************/

import java.util.Objects;

public class Complex {

private final double re; // the real part

private final double im; // the imaginary part

// create a new object with the given real and imaginary parts

public Complex(double real, double imag) {

re = real;

im = imag;

}

// return a string representation of the invoking Complex object

public String toString() {

if (im == 0)

return re + "";

if (re == 0)

return im + "i";

if (im < 0)

return re + " - " + (-im) + "i";

return re + " + " + im + "i";

}

// return abs/modulus/magnitude

public double abs() {

return Math.hypot(re, im);

}

// return angle/phase/argument, normalized to be between -pi and pi

public double phase() {

return Math.atan2(im, re);

}

// return a new Complex object whose value is (this + b)

public Complex plus(Complex b) {

Complex a = this; // invoking object

double real = a.re + b.re;

double imag = a.im + b.im;

return new Complex(real, imag);

}

// return a new Complex object whose value is (this - b)

public Complex minus(Complex b) {

Complex a = this;

double real = a.re - b.re;

double imag = a.im - b.im;

return new Complex(real, imag);

}

// return a new Complex object whose value is (this * b)

public Complex times(Complex b) {

Complex a = this;

double real = a.re * b.re - a.im * b.im;

double imag = a.re * b.im + a.im * b.re;

return new Complex(real, imag);

}

// return a new object whose value is (this * alpha)

public Complex scale(double alpha) {

return new Complex(alpha * re, alpha * im);

}

// return a new Complex object whose value is the conjugate of this

public Complex conjugate() {

return new Complex(re, -im);

}

// return a new Complex object whose value is the reciprocal of this

public Complex reciprocal() {

double scale = re * re + im * im;

return new Complex(re / scale, -im / scale);

}

// return the real or imaginary part

publicjBXgUK double re() {

return re;

}

public double im() {

return im;

}

// return a / b

public Complex divides(Complex b) {

Complex a = this;

return a.times(b.reciprocal());

}

// return a new Complex object whose value is the complex exponential of

// this

public Complex exp() {

return new Complex(Math.exp(re) * Math.cos(im), Math.exp(re) * Math.sin(im));

}

// return a new Complex object whose value is the complex sine of this

public Complex sin() {

return new Complex(Math.sin(re) * Math.cosh(jBXgUKim), Math.cos(re) * Math.sinh(im));

}

// return a new Complex object whose value is the complex cosine of this

public Complex cos() {

return new Complex(Math.cos(re) * Math.cosh(im), -Math.sin(re) * Math.sinh(im));

}

// return a new Complex object whose value is the complex tangent of this

public Complex tan() {

return sin().divides(cos());

}

// a static version of plus

public static Complex plus(Complex a, Complex b) {

double real = a.re + b.re;

double imag = a.im + b.im;

Complex sum = new Complex(real, imag);

return sum;

}

// See Section 3.3.

public boolean equals(Object x) {

if (x == null)

return false;

if (this.getClass() != x.getClass())

return false;

Complex that = (Complex) x;

return (this.re == that.re) && (this.im == that.im);

}

// See Section 3.3.

public int hashCode() {

return Objects.hash(re, im);

}

// sample clienthttp:// for testing

public static void main(String[] args) {

Complex a = new Complex(3.0, 4.0);

Complex b = new Complex(-3.0, 4.0);

System.out.println("a = " + a);

System.out.println("b = " + b);

System.out.println("Re(a) = " + a.re());

System.out.println("Im(a) = " + a.im());

System.out.println("b + a = " + b.plus(a));

System.out.println("a - b = " + a.minus(b));

System.out.println("a * b = " + a.times(b));

System.out.println("b * a = " + b.times(a));

System.out.println("a / b = " + a.divides(b));

System.out.println("(a / b) * b = " + a.divides(b).times(b));

System.out.println("conj(a) = " + a.conjugate());

System.out.println("|a| = " + a.abs());

System.out.println("tan(a) = " + a.tan());

}

}

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