Is there a generally accepted way of calculating an impulse response of an LTI system from its frequency and phase spectra? Of course, the obvious thing to do is an inverse Fourier transform, but since the impulse response shows a discontinuity at zero time I expect Gibbs phenomenon -- under-/overshooting and ringing -- to superpose the real impulse response. Indeed, I find a precursor (or wraparound) in the time domain, but since the expected impulse response is that of an underdamped harmonic oscillator plus some spurious events I cannot distinguish between true and "numerical" ringing.
(1): Do a cosine transform of the real part of the spectrum, get an even function and chop off the mirror impulse response. The tail of the impulse response is small enough to not contaminate the mirror impulse response.
(2): I know, I basically apply a boxcar function to the spectrum, which amounts to a convolution with a sinc function in the time domain. Shall I apply another convolution with an inverse sinc function to extract the true impulse response, and could I get the inverse sinc function by Wiener deconvolution of a sinc function with an impulse?
Basically, I am wondering how to proceed correctly.