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Delete inverse_fast_fourier_transform.cpp
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/**
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* @file
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* @brief [An inverse fast Fourier transform
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* (IFFT)](https://www.geeksforgeeks.org/python-inverse-fast-fourier-transformation/)
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* is an algorithm that computes the inverse fourier transform.
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* @details
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* This algorithm has an application in use case scenario where a user wants
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* find coefficients of a function in a short time by just using points
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* generated by DFT. Time complexity this algorithm computes the IDFT in
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* O(nlogn) time in comparison to traditional O(n^2).
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* @author [Ameya Chawla](https://github.com/ameyachawlaggsipu)
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*/
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#include <cassert> /// for assert
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#include <cmath> /// for mathematical-related functions
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#include <complex> /// for storing points and coefficents
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#include <iostream> /// for IO operations
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#include <vector> /// for std::vector
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/**
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* @namespace numerical_methods
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* @brief Numerical algorithms/methods
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*/
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namespace numerical_methods {
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/**
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* @brief InverseFastFourierTransform is a recursive function which returns list
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* of complex numbers
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* @param p List of Coefficents in form of complex numbers
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* @param n Count of elements in list p
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* @returns p if n==1
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* @returns y if n!=1
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*/
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std::complex<double> *InverseFastFourierTransform(std::complex<double> *p,
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uint8_t n) {
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if (n == 1) {
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return p; /// Base Case To return
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}
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double pi = 2 * asin(1.0); /// Declaring value of pi
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std::complex<double> om = std::complex<double>(
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cos(2 * pi / n), sin(2 * pi / n)); /// Calculating value of omega
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om.real(om.real() / n); /// One change in comparison with DFT
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om.imag(om.imag() / n); /// One change in comparison with DFT
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auto *pe = new std::complex<double>[n / 2]; /// Coefficients of even power
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auto *po = new std::complex<double>[n / 2]; /// Coefficients of odd power
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int k1 = 0, k2 = 0;
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for (int j = 0; j < n; j++) {
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if (j % 2 == 0) {
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pe[k1++] = p[j]; /// Assigning values of even Coefficients
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} else {
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po[k2++] = p[j]; /// Assigning value of odd Coefficients
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}
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}
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std::complex<double> *ye =
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InverseFastFourierTransform(pe, n / 2); /// Recursive Call
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std::complex<double> *yo =
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InverseFastFourierTransform(po, n / 2); /// Recursive Call
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auto *y = new std::complex<double>[n]; /// Final value representation list
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k1 = 0, k2 = 0;
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for (int i = 0; i < n / 2; i++) {
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y[i] =
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ye[k1] + pow(om, i) * yo[k2]; /// Updating the first n/2 elements
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y[i + n / 2] =
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ye[k1] - pow(om, i) * yo[k2]; /// Updating the last n/2 elements
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k1++;
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k2++;
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}
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if (n != 2) {
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delete[] pe;
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delete[] po;
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}
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delete[] ye; /// Deleting dynamic array ye
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delete[] yo; /// Deleting dynamic array yo
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return y;
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}
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} // namespace numerical_methods
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/**
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* @brief Self-test implementations
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* @details
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* Declaring two test cases and checking for the error
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* in predicted and true value is less than 0.000000000001.
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* @returns void
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*/
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static void test() {
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/* descriptions of the following test */
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auto *t1 = new std::complex<double>[2]; /// Test case 1
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auto *t2 = new std::complex<double>[4]; /// Test case 2
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t1[0] = {3, 0};
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t1[1] = {-1, 0};
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t2[0] = {10, 0};
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t2[1] = {-2, -2};
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t2[2] = {-2, 0};
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t2[3] = {-2, 2};
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uint8_t n1 = 2;
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uint8_t n2 = 4;
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std::vector<std::complex<double>> r1 = {
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{1, 0}, {2, 0}}; /// True Answer for test case 1
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std::vector<std::complex<double>> r2 = {
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{1, 0}, {2, 0}, {3, 0}, {4, 0}}; /// True Answer for test case 2
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std::complex<double> *o1 =
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numerical_methods::InverseFastFourierTransform(t1, n1);
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std::complex<double> *o2 =
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numerical_methods::InverseFastFourierTransform(t2, n2);
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for (uint8_t i = 0; i < n1; i++) {
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assert((r1[i].real() - o1[i].real() < 0.000000000001) &&
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(r1[i].imag() - o1[i].imag() <
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0.000000000001)); /// Comparing for both real and imaginary
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/// values for test case 1
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}
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for (uint8_t i = 0; i < n2; i++) {
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assert((r2[i].real() - o2[i].real() < 0.000000000001) &&
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(r2[i].imag() - o2[i].imag() <
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0.000000000001)); /// Comparing for both real and imaginary
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/// values for test case 2
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}
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delete[] t1;
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delete[] t2;
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delete[] o1;
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delete[] o2;
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std::cout << "All tests have successfully passed!\n";
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}
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/**
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* @brief Main function
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* @param argc commandline argument count (ignored)
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* @param argv commandline array of arguments (ignored)
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* calls automated test function to test the working of fast fourier transform.
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* @returns 0 on exit
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*/
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int main(int argc, char const *argv[]) {
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test(); // run self-test implementations
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// with 2 defined test cases
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return 0;
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}
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