inverse_fast_fourier_transform.cpp

/**
 * @file
 * @brief [An inverse fast Fourier transform
 * (IFFT)](https://www.geeksforgeeks.org/python-inverse-fast-fourier-transformation/)
 * is an algorithm that computes the inverse fourier transform.
 * @details
 * This algorithm has an application in use case scenario where a user wants
 * find coefficients of a function in a short time by just using points
 * generated by DFT. Time complexity this algorithm computes the IDFT in
 * O(nlogn) time in comparison to traditional O(n^2).
 * @author [Ameya Chawla](https://github.com/ameyachawlaggsipu)
 */

#include <cassert>   /// for assert
#include <cmath>     /// for mathematical-related functions
#include <complex>   /// for storing points and coefficents
#include <cstdint>
#include <iostream>  /// for IO operations
#include <vector>    /// for std::vector

/**
 * @namespace numerical_methods
 * @brief Numerical algorithms/methods
 */
namespace numerical_methods {
/**
 * @brief InverseFastFourierTransform is a recursive function which returns list
 * of complex numbers
 * @param p List of Coefficents in form of complex numbers
 * @param n Count of elements in list p
 * @returns p if n==1
 * @returns y if n!=1
 */
std::complex<double> *InverseFastFourierTransform(std::complex<double> *p,
                                                  uint8_t n) {
    if (n == 1) {
        return p;  /// Base Case To return
    }

    double pi = 2 * asin(1.0);  /// Declaring value of pi

    std::complex<double> om = std::complex<double>(
        cos(2 * pi / n), sin(2 * pi / n));  /// Calculating value of omega

    om.real(om.real() / n);  /// One change in comparison with DFT
    om.imag(om.imag() / n);  /// One change in comparison with DFT

    auto *pe = new std::complex<double>[n / 2];  /// Coefficients of even power

    auto *po = new std::complex<double>[n / 2];  /// Coefficients of odd power

    int k1 = 0, k2 = 0;
    for (int j = 0; j < n; j++) {
        if (j % 2 == 0) {
            pe[k1++] = p[j];  /// Assigning values of even Coefficients

        } else {
            po[k2++] = p[j];  /// Assigning value of odd Coefficients
        }
    }

    std::complex<double> *ye =
        InverseFastFourierTransform(pe, n / 2);  /// Recursive Call

    std::complex<double> *yo =
        InverseFastFourierTransform(po, n / 2);  /// Recursive Call

    auto *y = new std::complex<double>[n];  /// Final value representation list

    k1 = 0, k2 = 0;

    for (int i = 0; i < n / 2; i++) {
        y[i] =
            ye[k1] + pow(om, i) * yo[k2];  /// Updating the first n/2 elements
        y[i + n / 2] =
            ye[k1] - pow(om, i) * yo[k2];  /// Updating the last n/2 elements

        k1++;
        k2++;
    }

    if (n != 2) {
        delete[] pe;
        delete[] po;
    }

    delete[] ye;  /// Deleting dynamic array ye
    delete[] yo;  /// Deleting dynamic array yo
    return y;
}

}  // namespace numerical_methods

/**
 * @brief Self-test implementations
 * @details
 * Declaring two test cases and checking for the error
 * in predicted and true value is less than 0.000000000001.
 * @returns void
 */
static void test() {
    /* descriptions of the following test */

    auto *t1 = new std::complex<double>[2];  /// Test case 1
    auto *t2 = new std::complex<double>[4];  /// Test case 2

    t1[0] = {3, 0};
    t1[1] = {-1, 0};
    t2[0] = {10, 0};
    t2[1] = {-2, -2};
    t2[2] = {-2, 0};
    t2[3] = {-2, 2};

    uint8_t n1 = 2;
    uint8_t n2 = 4;
    std::vector<std::complex<double>> r1 = {
        {1, 0}, {2, 0}};  /// True Answer for test case 1

    std::vector<std::complex<double>> r2 = {
        {1, 0}, {2, 0}, {3, 0}, {4, 0}};  /// True Answer for test case 2

    std::complex<double> *o1 =
        numerical_methods::InverseFastFourierTransform(t1, n1);

    std::complex<double> *o2 =
        numerical_methods::InverseFastFourierTransform(t2, n2);

    for (uint8_t i = 0; i < n1; i++) {
        assert((r1[i].real() - o1[i].real() < 0.000000000001) &&
               (r1[i].imag() - o1[i].imag() <
                0.000000000001));  /// Comparing for both real and imaginary
                                   /// values for test case 1
    }

    for (uint8_t i = 0; i < n2; i++) {
        assert((r2[i].real() - o2[i].real() < 0.000000000001) &&
               (r2[i].imag() - o2[i].imag() <
                0.000000000001));  /// Comparing for both real and imaginary
                                   /// values for test case 2
    }

    delete[] t1;
    delete[] t2;
    delete[] o1;
    delete[] o2;
    std::cout << "All tests have successfully passed!\n";
}

/**
 * @brief Main function
 * @param argc commandline argument count (ignored)
 * @param argv commandline array of arguments (ignored)
 * calls automated test function to test the working of fast fourier transform.
 * @returns 0 on exit
 */

int main(int argc, char const *argv[]) {
    test();  //  run self-test implementations
             //  with 2 defined test cases
    return 0;
}