Java版本的FFT和Inverse FFT

最新推荐文章于 2025-06-14 00:15:00 发布

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#fft #java #dependencies #compilation #numbers #algorithm

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本文提供了一个Java版本的快速傅立叶变换(FFT)及其逆变换(Inverse FFT)的实现示例。该实现假设输入序列长度为2的幂，并以清晰的代码为目标，而非极致性能。

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最近有些朋友在一些项目中需要用到Java版本的FFT和Inverse FFT，玄机逸士在网上找到了一个版本，供大家参考，现抄录如下：

（原文地址：http://www.cs.princeton.edu/introcs/97data/FFT.java.html）

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

* 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].times(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

Complex[] a = fft(x);

Complex[] b = fft(y);

// point-wise multiply

Complex[] c = new Complex[N];

for (int i = 0; i < N; i++) {

c[i] = a[i].times(b[i]);

}

// compute inverse FFT

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];

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();

}

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

* Test client and sample execution

* % java FFT 4

* x

* -------------------

* -0.03480425839330703

* 0.07910192950176387

* 0.7233322451735928

* 0.1659819820667019

* y = fft(x)

* -------------------

* 0.9336118983487516

* -0.7581365035668999 + 0.08688005256493803i

* 0.44344407521182005

* -0.7581365035668999 - 0.08688005256493803i

* z = ifft(y)

* -------------------

* -0.03480425839330703

* 0.07910192950176387 + 2.6599344570851287E-18i

* 0.7233322451735928

* 0.1659819820667019 - 2.6599344570851287E-18i

* c = cconvolve(x, x)

* -------------------

* 0.5506798633981853

* 0.23461407150576394 - 4.033186818023279E-18i

* -0.016542951108772352

* 0.10288019294318276 + 4.033186818023279E-18i

* d = convolve(x, x)

* -------------------

* 0.001211336402308083 - 3.122502256758253E-17i

* -0.005506167987577068 - 5.058885073636224E-17i

* -0.044092969479563274 + 2.1934338938072244E-18i

* 0.10288019294318276 - 3.6147323062478115E-17i

* 0.5494685269958772 + 3.122502256758253E-17i

* 0.240120239493341 + 4.655566391833896E-17i

* 0.02755001837079092 - 2.1934338938072244E-18i

* 4.01805098805014E-17i

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

public static void main(String[] args) {

int N = Integer.parseInt(args[0]);

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)");

}