#### Inverse Fast Fourier Transform

L
2
A
```/**
* @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 <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;
}
```