Representation of complex numbers in C++ for Discrete Fourier Transformation

Question

I am currently writing a small tool which should help me check whether my manually calculated fourier vectors are correct. Now i need the n-th Root of Unity specified by omega = exp(2*pi*i / n) . Can somebody explain me how to represent this omega as a complex in C++?

Answer 1

Use Euler's formula :

exp(2πi/n) = cos(2π/n) + i sin(2π/n)

Then it's easy:

complex<double> rootOfUnity(cos(TWOPI/n), sin(TWOPI/n));

(replace TWOPI with either a macro available on your system or just the value of 2π however you see fit).

Answer 2

Well, the real and imaginary parts of the twiddle factor omega is just:

double angle = 2*pi/n;

double real = cos(angle);
double imaj = sin(angle);

complex<double> omega(real, imaj);

Answer 3

There is a function that returns a complex number using polar coordinates:

#include<complex>
complex polar(const T& rho)
complex polar(const T& rho, const T& theta)

where rho is the magnitude, and theta is the angle in radians.

In this case, rho is always 1.0.

const double pi = 3.141592653589793238462643383279;
double omega = polar(1.0, 2*pi*i/n);