java 实现傅立叶变换算法 及复数的运算
最近项目需求，需要把python中的算法移植到java上，其中有一部分需要用到复数的运算和傅立叶变换算法，废话不多说 如下：
package qrs;
/**
* 复数的运算
*
*/
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
public 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(im), 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);
}
// sample client 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());
}
}
傅立叶变换部分需要依赖复数类
package qrs;
/*************************************************************************
* 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);
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
*********************************************************************/
