Simple cross correlation implementation

Complex.java

/******************************************************************************
 *  Compilation:  javac Complex.java
 *  Execution:    java Complex
 *    
 *  Data type for complex numbers.
 *
 *  The data type is "immutable" so once you create and initialize
 *  a Complex object, you cannot change it. The "final" keyword
 *  when declaring re and im enforces this rule, making it a
 *  compile-time error to change the .re or .im fields after
 *  they've been initialized.
 *
 *  % java Complex
 *  a            = 5.0 + 6.0i
 *  b            = -3.0 + 4.0i
 *  Re(a)        = 5.0
 *  Im(a)        = 6.0
 *  b + a        = 2.0 + 10.0i
 *  a - b        = 8.0 + 2.0i
 *  a * b        = -39.0 + 2.0i
 *  b * a        = -39.0 + 2.0i
 *  a / b        = 0.36 - 1.52i
 *  (a / b) * b  = 5.0 + 6.0i
 *  conj(a)      = 5.0 - 6.0i
 *  |a|          = 7.810249675906654
 *  tan(a)       = -6.685231390246571E-6 + 1.0000103108981198i
 *
 ******************************************************************************/

/**
 *  The {@code Complex} class represents a complex number.
 *  Complex numbers are immutable: their values cannot be changed after they
 *  are created.
 *  It includes methods for addition, subtraction, multiplication, division,
 *  conjugation, and other common functions on complex numbers.
 *  <p>
 *  For additional documentation, see <a href="http://algs4.cs.princeton.edu/99scientific">Section 9.9</a> of
 *  <i>Algorithms, 4th Edition</i> by Robert Sedgewick and Kevin Wayne.
 *
 *  @author Robert Sedgewick
 *  @author Kevin Wayne
 */
public class Complex {
    private final double re;   // the real part
    private final double im;   // the imaginary part

    /**
     * Initializes a complex number from the specified real and imaginary parts.
     *
     * @param real the real part
     * @param imag the imaginary part
     */
    public Complex(double real, double imag) {
        re = real;
        im = imag;
    }

    /**
     * Returns a string representation of this complex number.
     *
     * @return a string representation of this complex number,
     *         of the form 34 - 56i.
     */
    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";
    }

    /**
     * Returns the absolute value of this complex number.
     * This quantity is also known as the <em>modulus</em> or <em>magnitude</em>.
     *
     * @return the absolute value of this complex number
     */
    public double abs() {
        return Math.hypot(re, im);
    }

    /**
     * Returns the phase of this complex number.
     * This quantity is also known as the <em>angle</em> or <em>argument</em>.
     *
     * @return the phase of this complex number, a real number between -pi and pi
     */
    public double phase() {
        return Math.atan2(im, re);
    }

    /**
     * Returns the sum of this complex number and the specified complex number.
     *
     * @param  that the other complex number
     * @return the complex number whose value is {@code (this + that)}
     */
    public Complex plus(Complex that) {
        double real = this.re + that.re;
        double imag = this.im + that.im;
        return new Complex(real, imag);
    }

    /**
     * Returns the result of subtracting the specified complex number from
     * this complex number.
     *
     * @param  that the other complex number
     * @return the complex number whose value is {@code (this - that)}
     */
    public Complex minus(Complex that) {
        double real = this.re - that.re;
        double imag = this.im - that.im;
        return new Complex(real, imag);
    }

    /**
     * Returns the product of this complex number and the specified complex number.
     *
     * @param  that the other complex number
     * @return the complex number whose value is {@code (this * that)}
     */
    public Complex times(Complex that) {
        double real = this.re * that.re - this.im * that.im;
        double imag = this.re * that.im + this.im * that.re;
        return new Complex(real, imag);
    }

    /**
     * Returns the product of this complex number and the specified scalar.
     *
     * @param  alpha the scalar
     * @return the complex number whose value is {@code (alpha * this)}
     */
    public Complex scale(double alpha) {
        return new Complex(alpha * re, alpha * im);
    }

    /**
     * Returns the product of this complex number and the specified scalar.
     *
     * @param  alpha the scalar
     * @return the complex number whose value is {@code (alpha * this)}
     * @deprecated Replaced by {@link #scale(double)}.
     */
    @Deprecated
    public Complex times(double alpha) {
        return new Complex(alpha * re, alpha * im);
    }

    /**
     * Returns the complex conjugate of this complex number.
     *
     * @return the complex conjugate of this complex number
     */
    public Complex conjugate() {
        return new Complex(re, -im);
    }

    /**
     * Returns the reciprocal of this complex number.
     *
     * @return the complex number whose value is {@code (1 / this)}
     */
    public Complex reciprocal() {
        double scale = re*re + im*im;
        return new Complex(re / scale, -im / scale);
    }

    /**
     * Returns the real part of this complex number.
     *
     * @return the real part of this complex number
     */
    public double re() {
        return re;
    }

    /**
     * Returns the imaginary part of this complex number.
     *
     * @return the imaginary part of this complex number
     */
    public double im() {
        return im;
    }

    /**
     * Returns the result of dividing the specified complex number into
     * this complex number.
     *
     * @param  that the other complex number
     * @return the complex number whose value is {@code (this / that)}
     */
    public Complex divides(Complex that) {
        return this.times(that.reciprocal());
    }

    /**
     * Returns the complex exponential of this complex number.
     *
     * @return the complex exponential of this complex number
     */
    public Complex exp() {
        return new Complex(Math.exp(re) * Math.cos(im), Math.exp(re) * Math.sin(im));
    }

    /**
     * Returns the complex sine of this complex number.
     *
     * @return the complex sine of this complex number
     */
    public Complex sin() {
        return new Complex(Math.sin(re) * Math.cosh(im), Math.cos(re) * Math.sinh(im));
    }

    /**
     * Returns the complex cosine of this complex number.
     *
     * @return the complex cosine of this complex number
     */
    public Complex cos() {
        return new Complex(Math.cos(re) * Math.cosh(im), -Math.sin(re) * Math.sinh(im));
    }

    /**
     * Returns the complex tangent of this complex number.
     *
     * @return the complex tangent of this complex number
     */
    public Complex tan() {
        return sin().divides(cos());
    }


    /**
     * Unit tests the {@code Complex} data type.
     *
     * @param args the command-line arguments
     */
    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());
    }

}

/******************************************************************************
 *  Copyright 2002-2016, Robert Sedgewick and Kevin Wayne.
 *
 *  This file is part of algs4.jar, which accompanies the textbook
 *
 *      Algorithms, 4th edition by Robert Sedgewick and Kevin Wayne,
 *      Addison-Wesley Professional, 2011, ISBN 0-321-57351-X.
 *      http://algs4.cs.princeton.edu
 *
 *
 *  algs4.jar is free software: you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation, either version 3 of the License, or
 *  (at your option) any later version.
 *
 *  algs4.jar is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with algs4.jar.  If not, see http://www.gnu.org/licenses.
 ******************************************************************************/

CrossCorrelation.java

public class CrossCorrelation {
    public static void main(String[] args) {
        double[] source = {1, 2, 3, 4, 5, 6, 7, 8};
        double[] target = {1, 2, 3, 0, 0, 0, 0, 0};
        int n = source.length;
        Complex[] sourceComplex = new Complex[n];
        Complex[] targetComplex = new Complex[n];
        for (int i = 0; i < n; i++) {
            sourceComplex[i] = new Complex(source[i], 0);
            targetComplex[i] = new Complex(target[i], 0);
        }
        Complex[] fftS = FFT.fft(sourceComplex);
        Complex[] fftT = FFT.fft(targetComplex);
        for (int i = 0; i < fftS.length; i++) {
            fftS[i] = fftS[i].conjugate();
        }
        Complex[] timeProduct = new Complex[fftS.length];
        for (int i = 0; i < fftS.length; i++) {
            timeProduct[i] = fftS[i].times(fftT[i]);
        }
        Complex[] y = FFT.ifft(timeProduct);

        System.out.println("id:\txcorr abs:\t\t\txcorr complexes:");
        for (int i = 0; i < y.length; i++) {
            Complex c = y[i];
            System.out.println(i + "\t" +c.abs() + "\t\t\t" + c.toString());
        }
        System.out.println("Max arg: " + argmax(y)); //I suppose it should equals 0
    }

    public static int argmax(Complex[] a)
        {
            double y = Double.MIN_VALUE;
            int idx = -1;

            for(int x = 0; x < a.length; x++)
            {
                if(a[x].abs() > y)
                {
                    y = a[x].abs();
                    idx = x;
                }
            }

            return idx;
        }
}

FFT.java

/******************************************************************************
 *  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)
 *  
 *
 *  % 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
 *
 ******************************************************************************/

/**
 *  The {@code FFT} class provides methods for computing the 
 *  FFT (Fast-Fourier Transform), inverse FFT, linear convolution,
 *  and circular convolution of a complex array.
 *  <p>
 *  It is a bare-bones implementation that runs in <em>n</em> log <em>n</em> time,
 *  where <em>n</em> is the length of the complex array. For simplicity,
 *  <em>n</em> must be a power of 2.
 *  Our goal is to optimize the clarity of the code, rather than performance.
 *  It is not the most memory efficient implementation because it uses
 *  objects to represents complex numbers and it it re-allocates memory
 *  for the subarray, instead of doing in-place or reusing a single temporary array.
 *  
 *  <p>
 *  For additional documentation, see <a href="http://algs4.cs.princeton.edu/99scientific">Section 9.9</a> of
 *  <i>Algorithms, 4th Edition</i> by Robert Sedgewick and Kevin Wayne.
 *
 *  @author Robert Sedgewick
 *  @author Kevin Wayne
 */
public class FFT {

    private static final Complex ZERO = new Complex(0, 0);

    // Do not instantiate.
    private FFT() { }

    /**
     * Returns the FFT of the specified complex array.
     *
     * @param  x the complex array
     * @return the FFT of the complex array {@code x}
     * @throws IllegalArgumentException if the length of {@code x} is not 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), 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;
    }


    /**
     * Returns the inverse FFT of the specified complex array.
     *
     * @param  x the complex array
     * @return the inverse FFT of the complex array {@code x}
     * @throws IllegalArgumentException if the length of {@code x} is not 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].scale(1.0 / n);
        }

        return y;

    }

    /**
     * Returns the circular convolution of the two specified complex arrays.
     *
     * @param  x one complex array
     * @param  y the other complex array
     * @return the circular convolution of {@code x} and {@code y}
     * @throws IllegalArgumentException if the length of {@code x} does not equal
     *         the length of {@code y} or if the length is not a power of 2
     */
    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 IllegalArgumentException("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);
    }

    /**
     * Returns the linear convolution of the two specified complex arrays.
     *
     * @param  x one complex array
     * @param  y the other complex array
     * @return the linear convolution of {@code x} and {@code y}
     * @throws IllegalArgumentException if the length of {@code x} does not equal
     *         the length of {@code y} or if the length is not a power of 2
     */
    public static Complex[] convolve(Complex[] x, Complex[] y) {
        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
    private static void show(Complex[] x, String title) {
        StdOut.println(title);
        StdOut.println("-------------------");
        for (int i = 0; i < x.length; i++) {
            StdOut.println(x[i]);
        }
        StdOut.println();
    }


   /***************************************************************************
    *  Test client.
    ***************************************************************************/

    /**
     * Unit tests the {@code FFT} class.
     *
     * @param args the command-line arguments
     */
    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(StdRandom.uniform(-1.0, 1.0), 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)");
    }

}

/******************************************************************************
 *  Copyright 2002-2016, Robert Sedgewick and Kevin Wayne.
 *
 *  This file is part of algs4.jar, which accompanies the textbook
 *
 *      Algorithms, 4th edition by Robert Sedgewick and Kevin Wayne,
 *      Addison-Wesley Professional, 2011, ISBN 0-321-57351-X.
 *      http://algs4.cs.princeton.edu
 *
 *
 *  algs4.jar is free software: you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation, either version 3 of the License, or
 *  (at your option) any later version.
 *
 *  algs4.jar is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with algs4.jar.  If not, see http://www.gnu.org/licenses.
 ******************************************************************************/