Evaluating Fourier transform of a Gaussian with FFTW3 and C++

Question

I have tried to calculate the Fourier Transform of a Gaussian function by using FFTW3 in C++. Here is the main part of my code

main(int argc, char** argv)
{
   fftw_plan p;
   complex<double> *in,*out;
   long N=8;

   //allocation of in and the fftw plan called 
   in=(complex<double>*) calloc(N,sizeof(complex<double>));
   p=fftw_plan_dft_1d(N,(fftw_complex*)in,(fftw_complex*)in,FFTW_BACKWARD,FFTW_ESTIMATE);

   //initialize the array in with the values of a Gaussian function
   gaussianf(in,N);
   //Fourier transform in
   fftw_execute(p);  
   //write the result into a file
   writeft(in,N);
   fftw_destroy_plan(p);
}

Since the array has been initialized with the values of a Gaussian, I would expect that the output is also a Gaussian but actually only the envelope has a Gaussian shape. As I show in the data below, where it is possible to see that some negative values have appeared.

#input values
#x       real part     imag part

-10     3.72008e-44     0
-7.5    3.72336e-25     0
-5      1.38879e-11     0
-2.5    0.00193045      0
0       1       0
2.5     0.00193045      0
5       1.38879e-11     0
7.5     3.72336e-25     0

#output
#x       real part     imag part
-10     1.00386 0
-7.5    -1.00273        0
-5      1       0
-2.5    -0.99727        0
0       0.996139        0
2.5     -0.99727        0
5       1       0
7.5     -1.00273        0

Could anyone tell me what I am doing wrong? I would really appreciate your help. Thanks a lot.

Answer 1

You're not doing anything wrong in the sense of the C programming or FFTW calls: those are the correct values. The real part of the FFT of a Gaussian curve does oscillate around zero. If you plot the absolute values, that might look a little more like you expect.

Answer 2

These oscillations are expected. In practice, one needs to use a window function to reduce these oscillations. http://en.wikipedia.org/wiki/Window_function