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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.

Spectrum  obtained from ccd-matrix

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    $\begingroup$ "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. $\endgroup$
    – Jdip
    Commented Nov 14, 2022 at 11:55
  • $\begingroup$ 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? $\endgroup$ Commented Nov 14, 2022 at 12:13

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I only have the real part of the input spectrum

Then you can't reconstruct the time domain signal. You need both the magnitude and the phase (or real and imaginary part) of the spectrum.

There are a few exceptions (like the time domain impulse response of a zero phase filter), but in almost all cases you need a complex spectrum.

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  • $\begingroup$ In fact we obtain the spectrum on the ccd-matrix as abs(F1(k) + F1(k))^2 (F1(k), F2(k) - spectra of two origin interfering broadband signals), because it is assumed that ccd works like square-law detector and can only procude the real-valued spectrum. $\endgroup$ Commented Nov 14, 2022 at 12:07
  • $\begingroup$ It would help if you describe your application in more detail. I'm assuming that your transform variable is space which you are transforming into spatial frequency. The units of your signal may very well be "spectral intensity" or power but it would be dependent on a location, not on frequency. $\endgroup$
    – Hilmar
    Commented Nov 14, 2022 at 12:57
  • $\begingroup$ You are absolutely right. This is the problem of optical low-coherence fourier-domain tomography. There is an michelson-type interferometer with two arms - reference and sample arms. Using broad band source of light we obtain a low-coherence fring patterns on the spectrometer ccd-matrix. Because of the non-zero distance dz between reference mirror and investigated sample, the spectrum has special cosine modulated waves of frequancy f. I try to translate this cosine waves from wavelength domain to spatial domain bursts with distance dz. $\endgroup$ Commented Nov 14, 2022 at 14:49

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