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.
*
* 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.
*
*
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 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