# What is inverse Fourier transform of real spectrum?

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.

• "I suppose if we have a real spectrum, then the signal in the time domain is an even and real function." That's only true If your spectrum is real and even. But if it's real only, then your time-domain signal could be complex, more specifically, real even and imaginary odd.
– Jdip
Commented Nov 14, 2022 at 11:55
• Does it mean that the recovery signal is redundant and I can truncate the latter half of the complex signal in time and use only former half? Commented Nov 14, 2022 at 12:13