JAVA实现FFT算法

关于快速傅里叶变换（FFT）和傅里叶变换的理论知识这里我就不提了，本文主要讲解FFT实现：

之前想找一个FFT代码，在网上找了很多都是有问题的，下面我完善了一个供大家学习交流；

首先粘贴FFT的代码如下：
/******************************************************************************

• Compilation: javac FFT.java

• Execution: java FFT n

• Dependencies: Complex.java

• Compute the FFT and inverse FFT of a length n complex sequence

• using the radix 2 Cooley-Tukey algorithm.

• 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)
  

• For an in-place radix 2 Cooley-Tukey FFT, see

• https://introcs.cs.princeton.edu/java/97data/InplaceFFT.java.html

******************************************************************************/
import edu.princeton.cs.algs4.StdDraw;
import edu.princeton.cs.algs4.StdOut;
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 IllegalArgumentException("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),