Changing the center of a gaussian using FFTW in C++ - c++

Question

For a project I need to propagate a gaussian in real space using the Fourier transform of a gaussian centered at the origin using
What I want to calculate
Here is the latex code, since I can't include images yet
N(x | \mu, \sigma) = F^{-1}{F{ N(x |0, \sigma)} e^{-i\ 2\pi \mu\omega} \right},
where \omega is the frequency in Fourier space.
Now the problem I am having is I don't know how to calculate the frequency for some bin after doing the fft with fftw. Here is the code of what I am trying to do.
int main(){
int N = 128; //Number of "pixels" in real space
int N_fft = N/2 + 1;
double *in, *result, *x;
fftw_complex *out;
fftw_plan fw, bw;
in = (double*) fftw_malloc(sizeof(double) * N);
x = (double*) malloc(sizeof(double) * N);
out = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N_fft);
result = (double*) fftw_malloc(sizeof(double) * N);
fw = fftw_plan_dft_r2c_1d(N, in, out, FFTW_ESTIMATE);
bw = fftw_plan_dft_c2r_1d(N, out, result, FFTW_ESTIMATE);
double min_x = -9.0, max_x = 9.0; //Limits in real space
for (int i=0; i<N; i++){
x[i] = min_x + 2*max_x*i / (N - 1);
in[i] = std::exp(-(x[i]) * (x[i]));
}
for (int i=0; i<N_fft; i++){
out[i][0] = 0.0;
out[i][1] = 0.0;
}
fftw_execute(fw);
double w;
fftw_complex z;
double w_norm;
for (int i=0; i<N_fft; i++){
w = -2*M_PI*i / (max_x - min_x); //WHAT I DON'T KNOW
// Calculating the product with the exponential for translating the gaussian
z[0] = out[i][0]*std::cos(w) - out[i][1]*std::sin(w);
z[1] = out[i][0]*std::sin(w) + out[i][0]*std::cos(w);
out[i][0] = z[0];
out[i][1] = z[1];
}
fftw_execute(bw);
for (int i=0; i<N; i++){
std::cout << x[i] << " " << result[i]/N << " " << std::exp(-x[i] * x[i]) << std::endl;
}
fftw_destroy_plan(fw);
fftw_free(in);
fftw_free(out);
return 0;
}
For the moment I've tried using w-nth = -2*np.pi * 1/(max_x - min_x) * n, which worked in python, but for some reason it doesn't work in c++
Here is the result I am obtaining with c++
result
Here ref is the gaussian centered at 0, the one I obtaing should be centered at 1.0, but that's clearly is not happening.
Here is the latex code, since I can't include images yet
(Here is the latex code, since I can't include images yet)

Answer 1

Generally, the more obvious is the mistake, the more time it takes to find it.
This was verified here.
The mistake is simply here:
z[1] = out[i][0]*std::sin(w) + out[i][0]*std::cos(w);
It should be:
z[1] = out[i][0]*std::sin(w) + out[i][1]*std::cos(w);
Besides, I don't know why you didn't use N__ft = N, but I guess it is related the way fftw works.