[Java]高能-绝对专业-快速傅立叶变换(FFT/DFT)算法实现

2021-02-10 66点热度 0人点赞 0条评论
高能⚠️

(如果您没有听过或不是要寻找快速傅立叶变换Fast Fourier transform(FTT)/离散傅里叶变换Discrete Fourier Transform(DFT)相关的问题,请直接无视这篇文章)

遇到一个通讯专业-波形频谱信号📶分析的项目,需要把一组数据做快速傅立叶变换(FTT)后,画出图形(如下图)

我也不是通信专业的,不懂什么FFT变换,在专业老师的指导下,网上找了相关的算法的代码(如果没有方向,而且专业性特强,很难找),故分享到这里,供有需要的朋友参考。

我一共找到了2个FFT算法,第一个客户说不太对,后来又找到一个,客户说可以了,这边都分享下,供参考选择。

  1. 第一套
    Java文件 Complex.java

    /**
    * @description 代表一个复数
    */
    public class Complex {
        private final double re;   // the real part 实部
        private final double im;   // the imaginary part 虚部
    
        // create a new object with the given real and imaginary parts 传入实部和虚部,组成一个复数的构造函数
        public Complex(double real, double imag) {
            re = real;
            im = imag;
        }
    
        // return a string representation of the invoking Complex object
        @Override
        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 and angle/phase/argument
        public double abs()   { return Math.hypot(re, im); }  // Math.sqrt(re*re + im*im)
        public double phase() { return Math.atan2(im, re); }  // between -pi and pi
    
        // 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);
        }
    
        // scalar multiplication
        // return a new object whose value is (this * alpha)
        public Complex times(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;
        }
    
    
    
        // 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());
        }
    
    }
    Java文件:FFT.java, FTT变换的测试在main方法里
    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)");
        }
    }
  2. 上面那套,客户看了生成的图后,感觉说不对,所以又找了一套
    新建DFT.java

    /*
     * Free FFT and convolution (Java)
     *
     * Copyright (c) 2017 Project Nayuki. (MIT License)
     * https://www.nayuki.io/page/free-small-fft-in-multiple-languages
     *
     * Permission is hereby granted, free of charge, to any person obtaining a copy of
     * this software and associated documentation files (the "Software"), to deal in
     * the Software without restriction, including without limitation the rights to
     * use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of
     * the Software, and to permit persons to whom the Software is furnished to do so,
     * subject to the following conditions:
     * - The above copyright notice and this permission notice shall be included in
     *   all copies or substantial portions of the Software.
     * - The Software is provided "as is", without warranty of any kind, express or
     *   implied, including but not limited to the warranties of merchantability,
     *   fitness for a particular purpose and noninfringement. In no event shall the
     *   authors or copyright holders be liable for any claim, damages or other
     *   liability, whether in an action of contract, tort or otherwise, arising from,
     *   out of or in connection with the Software or the use or other dealings in the
     *   Software.
     */
    
    /**
     * Downloaded from: https://www.nayuki.io/res/free-small-fft-in-multiple-languages/Fft.java
     */
    
    public class DFT {
        /*
         * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
         * The vector can have any length. This is a wrapper function.
         *
         */
        public static void transform(double[] real, double[] imag) {
            int n = real.length;
            if (n != imag.length)
                throw new IllegalArgumentException("Mismatched lengths");
            if (n == 0)
                return;
            else if ((n & (n - 1)) == 0)  // Is power of 2
                transformRadix2(real, imag);
            else  // More complicated algorithm for arbitrary sizes
                transformBluestein(real, imag);
        }
    
    
        /*
         * Computes the inverse discrete Fourier transform (IDFT) of the given complex vector, storing the result back into the vector.
         * The vector can have any length. This is a wrapper function. This transform does not perform scaling, so the inverse is not a true inverse.
         */
        public static void inverseTransform(double[] real, double[] imag) {
            transform(imag, real);
        }
    
    
        /*
         * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
         * The vector's length must be a power of 2. Uses the Cooley-Tukey decimation-in-time radix-2 algorithm.
         */
        public static void transformRadix2(double[] real, double[] imag) {
            // Length variables
            int n = real.length;
            if (n != imag.length)
                throw new IllegalArgumentException("Mismatched lengths");
            int levels = 31 - Integer.numberOfLeadingZeros(n);  // Equal to floor(log2(n))
            if (1 << levels != n)
                throw new IllegalArgumentException("Length is not a power of 2");
    
            // Trigonometric tables
            double[] cosTable = new double[n / 2];
            double[] sinTable = new double[n / 2];
            for (int i = 0; i < n / 2; i++) {
                cosTable[i] = Math.cos(2 * Math.PI * i / n);
                sinTable[i] = Math.sin(2 * Math.PI * i / n);
            }
    
            // Bit-reversed addressing permutation
            for (int i = 0; i < n; i++) {
                int j = Integer.reverse(i) >>> (32 - levels);
                if (j > i) {
                    double temp = real[i];
                    real[i] = real[j];
                    real[j] = temp;
                    temp = imag[i];
                    imag[i] = imag[j];
                    imag[j] = temp;
                }
            }
    
            // Cooley-Tukey decimation-in-time radix-2 FFT
            for (int size = 2; size <= n; size *= 2) {
                int halfsize = size / 2;
                int tablestep = n / size;
                for (int i = 0; i < n; i += size) {
                    for (int j = i, k = 0; j < i + halfsize; j++, k += tablestep) {
                        int l = j + halfsize;
                        double tpre = real[l] * cosTable[k] + imag[l] * sinTable[k];
                        double tpim = -real[l] * sinTable[k] + imag[l] * cosTable[k];
                        real[l] = real[j] - tpre;
                        imag[l] = imag[j] - tpim;
                        real[j] += tpre;
                        imag[j] += tpim;
                    }
                }
                if (size == n)  // Prevent overflow in 'size *= 2'
                    break;
            }
        }
    
    
        /*
         * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
         * The vector can have any length. This requires the convolution function, which in turn requires the radix-2 FFT function.
         * Uses Bluestein's chirp z-transform algorithm.
         */
        public static void transformBluestein(double[] real, double[] imag) {
            // Find a power-of-2 convolution length m such that m >= n * 2 + 1
            int n = real.length;
            if (n != imag.length)
                throw new IllegalArgumentException("Mismatched lengths");
            if (n >= 0x20000000)
                throw new IllegalArgumentException("Array too large");
            int m = Integer.highestOneBit(n) * 4;
    
            // Trignometric tables
            double[] cosTable = new double[n];
            double[] sinTable = new double[n];
            for (int i = 0; i < n; i++) {
                int j = (int) ((long) i * i % (n * 2));  // This is more accurate than j = i * i
                cosTable[i] = Math.cos(Math.PI * j / n);
                sinTable[i] = Math.sin(Math.PI * j / n);
            }
    
            // Temporary vectors and preprocessing
            double[] areal = new double[m];
            double[] aimag = new double[m];
            for (int i = 0; i < n; i++) {
                areal[i] = real[i] * cosTable[i] + imag[i] * sinTable[i];
                aimag[i] = -real[i] * sinTable[i] + imag[i] * cosTable[i];
            }
            double[] breal = new double[m];
            double[] bimag = new double[m];
            breal[0] = cosTable[0];
            bimag[0] = sinTable[0];
            for (int i = 1; i < n; i++) {
                breal[i] = breal[m - i] = cosTable[i];
                bimag[i] = bimag[m - i] = sinTable[i];
            }
    
            // Convolution
            double[] creal = new double[m];
            double[] cimag = new double[m];
            convolve(areal, aimag, breal, bimag, creal, cimag);
    
            // Postprocessing
            for (int i = 0; i < n; i++) {
                real[i] = creal[i] * cosTable[i] + cimag[i] * sinTable[i];
                imag[i] = -creal[i] * sinTable[i] + cimag[i] * cosTable[i];
            }
        }
    
    
        /*
         * Computes the circular convolution of the given real vectors. Each vector's length must be the same.
         */
        public static void convolve(double[] x, double[] y, double[] out) {
            int n = x.length;
            if (n != y.length || n != out.length)
                throw new IllegalArgumentException("Mismatched lengths");
            convolve(x, new double[n], y, new double[n], out, new double[n]);
        }
    
    
        /*
         * Computes the circular convolution of the given complex vectors. Each vector's length must be the same.
         */
        public static void convolve(double[] xreal, double[] ximag,
                                    double[] yreal, double[] yimag, double[] outreal, double[] outimag) {
    
            int n = xreal.length;
            if (n != ximag.length || n != yreal.length || n != yimag.length
                    || n != outreal.length || n != outimag.length)
                throw new IllegalArgumentException("Mismatched lengths");
    
            xreal = xreal.clone();
            ximag = ximag.clone();
            yreal = yreal.clone();
            yimag = yimag.clone();
            transform(xreal, ximag);
            transform(yreal, yimag);
    
            for (int i = 0; i < n; i++) {
                double temp = xreal[i] * yreal[i] - ximag[i] * yimag[i];
                ximag[i] = ximag[i] * yreal[i] + xreal[i] * yimag[i];
                xreal[i] = temp;
            }
            inverseTransform(xreal, ximag);
    
            for (int i = 0; i < n; i++) {  // Scaling (because this FFT implementation omits it)
                outreal[i] = xreal[i] / n;
                outimag[i] = ximag[i] / n;
            }
        }
    }

    调用举例

    private void drawFFTChart() {
        //用FFT.java做好像有问题,现在用DFT.java做
    //        int n = findN();
    //        Complex source[] = new Complex[n];
    //        log.info("fft sample num: " + n + ", source sample num: " + ifData.length);
    
        int ifDataLength = ifData.length;
        double[] real = new double[ifDataLength]; // 实部
        double[] imag = new double[ifDataLength]; // 虚部
        for (int i = 0; i < ifDataLength; i++) {
            real[i] = ifData[i];
        }
        DFT.transform(real, imag);
        int fftLength = ifDataLength / 2;
        double[] fftData = new double[fftLength];
        for (int i = 0; i < fftLength; i++) {
            double abs = Math.hypot(real[i], imag[i]);
            fftData[i] = 20D * Math.log10(abs);
        }
        ((IFFFTChartPane) fftChartPanel).updateChartData(fftData);
    }

     

 

 

 

Terry

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