实现傅里叶变换，请用java写出实现代码

好的，以下是 Java 中实现傅里叶变换的代码示例：

import java.util.ArrayList;
import java.util.List;

public class FFT {
    public static List<Complex> fft(List<Complex> x) {
        int n = x.size();

        // 基数为 2 的情况
        if (n == 2) {
            List<Complex> y = new ArrayList<>(2);
            y.add(x.get(0).add(x.get(1)));
            y.add(x.get(0).subtract(x.get(1)));
            return y;
        }

        // 否则递归计算
        List<Complex> even = new ArrayList<>(n / 2);
        List<Complex> odd = new ArrayList<>(n / 2);
        for (int i = 0; i < n; i++) {
            if (i % 2 == 0) {
                even.add(x.get(i));
            } else {
                odd.add(x.get(i));
            }
        }

        List<Complex> q = fft(even);
        List<Complex> r = fft(odd);

        List<Complex> y = new ArrayList<>(n);
        for (int i = 0; i < n / 2; i++) {
            double k = -2 * i * Math.PI / n;
            Complex z = new Complex(Math.cos(k), Math.sin(k));
            y.add(q.get(i).add(z.multiply(r.get(i))));
            y.add(q.get(i).subtract(z.multiply(r.get(i))));
        }
        return y;
    }

    private static class Complex {
        double real;
        double imag;

        Complex(double real, double imag) {
            this.real = real;
            this.imag = imag;
        }

        Complex add(Complex b) {
            return new Complex(real + b.real, imag + b.imag);
        }

        Complex subtract(Complex b) {
            return new Complex(real - b.real, imag - b.imag);
        }

        Complex multiply(Complex b) {
            return new Complex(real * b.real - imag * b.imag, real * b.imag + imag * b.real);
        }
    }
}

在这个代码中，FFT 类实现了基于递归的 Cooley-Tukey 分治算法。它首先判断输入序列的长度是否为 2，如果是，则直接进行傅里叶变换；否则