最近做音频信号处理的时候，需要对数据做fft变换。关于fft原理：
请参考：FFT算法讲解——麻麻我终于会FFT了！
matlab实现fft函数很简单，直接调用fft即可。但java实现起来就有点难了。参考了比较好两篇java实现的博客：
A.Java实现算法导论中快速傅里叶变换FFT递归算法
B.快速傅里叶变换及java实现
通过比对。A博客只实现了数组长度是2的幂次的函数，其他就没有实现，B博客是实现了所有的，但其中有几处错误。最后实现方法如下：
1）FFT函数
这个函数A和B基本上差不多，但B博客的n为1时，返回要改一下。
public static Complex[] fft(Complex[] x) {
int n = x.length;
// 因为exp(-2i*n*PI)=1，n=1时递归原点
if (n == 1){
// 这里和B博客中有一点变化
return new Complex[]{x[0]};
}
// 如果信号数为奇数，使用dft计算
if (n % 2 != 0) {
return dft(x);
}
// 提取下标为偶数的原始信号值进行递归fft计算
Complex[] even = new Complex[n / 2];
for (int k = 0; k < n / 2; k++) {
even[k] = x[2 * k];
}
Complex[] evenValue = fft(even);
// 提取下标为奇数的原始信号值进行fft计算
// 节约内存
Complex[] odd = even;
for (int k = 0; k < n / 2; k++) {
odd[k] = x[2 * k + 1];
}
Complex[] oddValue = fft(odd);
// 偶数+奇数
Complex[] result = new Complex[n];
for (int k = 0; k < n / 2; k++) {
// 使用欧拉公式e^(-i*2pi*k/N) = cos(-2pi*k/N) + i*sin(-2pi*k/N)
double p = -2 * k * Math.PI / n;
Complex m = new Complex(Math.cos(p), Math.sin(p));
result[k] = evenValue[k].plus(m.multiple(oddValue[k]));
// exp(-2*(k+n/2)*PI/n) 相当于 -exp(-2*k*PI/n)，其中exp(-n*PI)=-1(欧拉公式);
result[k + n / 2] = evenValue[k].minus(m.multiple(oddValue[k]));
}
return result;
}
2）DFT函数
这个函数主要参考B博客的，修改的地方有：
a.当长度为1时，返回值
b.算cos，sin函数参数的时候，把-2 * k* Math.PI / n改为-2 * i * k* Math.PI / n;;
public static Complex[] dft(Complex[] x) {
int n = x.length;
// exp(-2i*n*PI)=cos(-2*n*PI)+i*sin(-2*n*PI)=1
if (n == 1)
return new Complex[]{x[0]};
Complex[] result = new Complex[n];
for (int i = 0; i < n; i++) {
result[i] = new Complex(0, 0);
for (int k = 0; k < n; k++) {
//使用欧拉公式e^(-i*2pi*k/N) = cos(-2pi*k/N) + i*sin(-2pi*k/N)
double p = -2 * i * k* Math.PI / n;
Complex m = new Complex(Math.cos(p), Math.sin(p));
result[i] = result[i].plus(x[k].multiple(m));
}
}
return result;
}
3）Complex代码
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 multiple(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);
}
// scalar multiplication
// return a new object whose value is (this * alpha)
public Complex multiple(double alpha) {
return new Complex(alpha * re, alpha * im);
}
// 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.multiple(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);
}
}
最后运行情况
matlab
matlab运行结果
上边是输入的数组，下边是fft输出数组。
java
java运行结果
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