Java实现的傅里叶变化算法示例

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

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());  }}