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);
   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
*
******************************************************************************/
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
 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);
 }
 // See Section 3.3.
 public int hashCode() {
   return Objects.hash(re, 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());
 }
}

希望本文所述对大家java程序设计有所帮助。

来源：https://blog.csdn.net/ffj0721/article/details/78521821