import numpy as np

def fourier_analysis(borg_vol):
  L = (borg_vol.ranges[1]-borg_vol.ranges[0])
  N = borg_vol.density.shape[0]
    
  return np.fft.rfftn(borg_vol.density)*(L/N)**3, L, N

def half_pixel_shift(borg):

    dhat,L,N = fourier_analysis(borg)
    return dhat, L

    ik = np.fft.fftfreq(N,d=L/N)*2*np.pi
    phi = 0.5*L/N*(ik[:,None,None]+ik[None,:,None]+ik[None,None,:(N/2+1)])
    phase = np.cos(phi)+1j*np.sin(phi)
    
    return dhat*phase, L