Recover High Frequency Loss in a Simulated Power Spectrum

I'd like to ask a theoretical question about power spectra. In MATLAB, by means of a particular approach, I'm trying to generate a Gaussian variable which embodies a particular power spectrum. The theoretical power spectrum has been calculated over an illimited range ([-Inf,Inf]), whereas the simulated one is obtained only in a short wave number vector range.

Generally, theoretical power spectrum and simulated one look like in the figure below

Red and green plots represent the simulated spectrum for two different configuration; the blue line stays for the theoretical spectrum.

I'd like to know if there is any easy way to recover the spectrum loss in the high frequency region, in order to force the simulated spectrum perfectly match the theoretical. I was thinking of a fairly easy filter working this way

filter = sqrt(S_th.^2./S_sim.^2);

S_rec = filter.*S_sim;


Do you believe such a procedure is allowed and physically reasonable?

there

• Gaussian random variables do not have power spectra; Gaussian processes (sequences of Gaussian random variables) do. If your simulated Gaussian random process has power spectral density $S_{\scriptstyle{\text{sim}}}(f)$ and you want to convert the process to one that has power spectral density $S_{\scriptstyle{\text{desired}}}(f)$, you have to pass the process through a linear time-invariant filter whose transfer function is $$\sqrt{\frac{S_{\scriptstyle{\text{desired}}}(f)}{S_{\scriptstyle{\text{sim}}}(f)}}$$ not just multiply the power spectral density by filter the way you have it. – Dilip Sarwate Jan 27 '13 at 23:19
• @DilipSarwate: you are right when talking of Gaussian processes compared to Gaussian variable. I'm sorry for the mistake. Do you believe what you proposed is feasable? Or does it have any negative impact to keep into account? – fpe Jan 28 '13 at 7:56