I am trying to retrieve the origin signal in time domain (or path-length domain) from a spectrum obtained by CCD-matrix of interferometer.
I understand how to transform a real signal in the time domain into a complex spectrum in frequency domain. But it's difficult to do the opposite. I only have the real part of the input spectrum with F_min, F_max, df, Nf_points. I need to obtain t_0, t_max, dt, Nt_points.
I suppose if we have a real spectrum, then the signal in the time domain is an even and real function. So would a simple inverse cosine transform work? How would I recover the limits and dt in the time domain?
EDIT: What is the real problem? I have a spectrum of two interfering signals on ccd-matrix in Michelson interferometer. This spectrum is real in wavelength-domain because spectrometer ccd-matrix works like square-law detector. Output of the ccd-matrix (in nanometres) is shown here. I try to translate this cosine wave of wavelength f into burst in spacial domain with distance dz between sample and reference arms.