要结合算法导论理解，参考：http://blog.csdn.net/fjssharpsword/article/details/53281889
代码中算法思路：输入n位（2的幂）向量，分别求值FFT和插值逆FFT，并计算卷积。
package sk.mlib;
/*************************************************************************
* 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("-------------------");
for (int i = 0; i < x.length; i++) {
System.out.println(x[i]);
}
System.out.println();
}
public static void main(String[] args) {
//int N = Integer.parseInt(args[0]);
int N=8;
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);
}
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
*********************************************************************/
依赖Complex.java 复数操作类的实现：
package sk.mlib;
/******************************************************************************
* 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(5.0, 6.0);
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());
}
}