JavaAlgorithms/Maths/FFT.java

package com.maths;

import java.util.ArrayList;
import java.util.Collections;

/**
 * Class for calculating the Fast Fourier Transform (FFT) of a discrete signal using the
 * Cooley-Tukey algorithm.
 *
 * @author Ioannis Karavitsis
 * @version 1.0
 */
public class FFT {
  /**
   * This class represents a complex number and has methods for basic operations.
   *
   * <p>More info: https://introcs.cs.princeton.edu/java/32class/Complex.java.html
   */
  static class Complex {
    private double real, img;

    /** Default Constructor. Creates the complex number 0. */
    public Complex() {
      real = 0;
      img = 0;
    }

    /**
     * Constructor. Creates a complex number.
     *
     * @param r The real part of the number.
     * @param i The imaginary part of the number.
     */
    public Complex(double r, double i) {
      real = r;
      img = i;
    }

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

    /**
     * Returns the imaginary part of the complex number.
     *
     * @return The imaginary part of the complex number.
     */
    public double getImaginary() {
      return img;
    }

    /**
     * Adds this complex number to another.
     *
     * @param z The number to be added.
     * @return The sum.
     */
    public Complex add(Complex z) {
      Complex temp = new Complex();
      temp.real = this.real + z.real;
      temp.img = this.img + z.img;
      return temp;
    }

    /**
     * Subtracts a number from this complex number.
     *
     * @param z The number to be subtracted.
     * @return The difference.
     */
    public Complex subtract(Complex z) {
      Complex temp = new Complex();
      temp.real = this.real - z.real;
      temp.img = this.img - z.img;
      return temp;
    }

    /**
     * Multiplies this complex number by another.
     *
     * @param z The number to be multiplied.
     * @return The product.
     */
    public Complex multiply(Complex z) {
      Complex temp = new Complex();
      temp.real = this.real * z.real - this.img * z.img;
      temp.img = this.real * z.img + this.img * z.real;
      return temp;
    }

    /**
     * Multiplies this complex number by a scalar.
     *
     * @param n The real number to be multiplied.
     * @return The product.
     */
    public Complex multiply(double n) {
      Complex temp = new Complex();
      temp.real = this.real * n;
      temp.img = this.img * n;
      return temp;
    }

    /**
     * Finds the conjugate of this complex number.
     *
     * @return The conjugate.
     */
    public Complex conjugate() {
      Complex temp = new Complex();
      temp.real = this.real;
      temp.img = -this.img;
      return temp;
    }

    /**
     * Finds the magnitude of the complex number.
     *
     * @return The magnitude.
     */
    public double abs() {
      return Math.hypot(this.real, this.img);
    }

    /**
     * Divides this complex number by another.
     *
     * @param z The divisor.
     * @return The quotient.
     */
    public Complex divide(Complex z) {
      Complex temp = new Complex();
      temp.real = (this.real * z.real + this.img * z.img) / (z.abs() * z.abs());
      temp.img = (this.img * z.real - this.real * z.img) / (z.abs() * z.abs());
      return temp;
    }

    /**
     * Divides this complex number by a scalar.
     *
     * @param n The divisor which is a real number.
     * @return The quotient.
     */
    public Complex divide(double n) {
      Complex temp = new Complex();
      temp.real = this.real / n;
      temp.img = this.img / n;
      return temp;
    }
  }

  /**
   * Iterative In-Place Radix-2 Cooley-Tukey Fast Fourier Transform Algorithm with Bit-Reversal. The
   * size of the input signal must be a power of 2. If it isn't then it is padded with zeros and the
   * output FFT will be bigger than the input signal.
   *
   * <p>More info: https://www.algorithm-archive.org/contents/cooley_tukey/cooley_tukey.html
   * https://www.geeksforgeeks.org/iterative-fast-fourier-transformation-polynomial-multiplication/
   * https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm
   * https://cp-algorithms.com/algebra/fft.html
   *
   * @param x The discrete signal which is then converted to the FFT or the IFFT of signal x.
   * @param inverse True if you want to find the inverse FFT.
   */
  public static void fft(ArrayList<Complex> x, boolean inverse) {
    /* Pad the signal with zeros if necessary */
    paddingPowerOfTwo(x);
    int N = x.size();

    /* Find the log2(N) */
    int log2N = 0;
    while ((1 << log2N) < N) log2N++;

    /* Swap the values of the signal with bit-reversal method */
    int reverse;
    for (int i = 0; i < N; i++) {
      reverse = reverseBits(i, log2N);
      if (i < reverse) Collections.swap(x, i, reverse);
    }

    int direction = inverse ? -1 : 1;

    /* Main loop of the algorithm */
    for (int len = 2; len <= N; len *= 2) {
      double angle = -2 * Math.PI / len * direction;
      Complex wlen = new Complex(Math.cos(angle), Math.sin(angle));
      for (int i = 0; i < N; i += len) {
        Complex w = new Complex(1, 0);
        for (int j = 0; j < len / 2; j++) {
          Complex u = x.get(i + j);
          Complex v = w.multiply(x.get(i + j + len / 2));
          x.set(i + j, u.add(v));
          x.set(i + j + len / 2, u.subtract(v));
          w = w.multiply(wlen);
        }
      }
    }

    /* Divide by N if we want the inverse FFT */
    if (inverse) {
      for (int i = 0; i < x.size(); i++) {
        Complex z = x.get(i);
        x.set(i, z.divide(N));
      }
    }
  }

  /**
   * This function reverses the bits of a number. It is used in Cooley-Tukey FFT algorithm.
   *
   * <p>E.g. num = 13 = 00001101 in binary log2N = 8 Then reversed = 176 = 10110000 in binary
   *
   * <p>More info: https://cp-algorithms.com/algebra/fft.html
   * https://www.geeksforgeeks.org/write-an-efficient-c-program-to-reverse-bits-of-a-number/
   *
   * @param num The integer you want to reverse its bits.
   * @param log2N The number of bits you want to reverse.
   * @return The reversed number
   */
  private static int reverseBits(int num, int log2N) {
    int reversed = 0;
    for (int i = 0; i < log2N; i++) {
      if ((num & (1 << i)) != 0) reversed |= 1 << (log2N - 1 - i);
    }
    return reversed;
  }

  /**
   * This method pads an ArrayList with zeros in order to have a size equal to the next power of two
   * of the previous size.
   *
   * @param x The ArrayList to be padded.
   */
  private static void paddingPowerOfTwo(ArrayList<Complex> x) {
    int n = 1;
    int oldSize = x.size();
    while (n < oldSize) n *= 2;
    for (int i = 0; i < n - oldSize; i++) x.add(new Complex());
  }
}