Generate timehistory from PSD

I'm trying to generate a time-history from a synthetic PSD.

My idea (happy to accept other solutions) was to turn the PSD into an FFT and then use ifft to turn it into a time-history. Note: I know that in the PSD the phase information is missing. In my case, however, a randomly generated phase could do the trick.

I'm having however some difficuties in the normalization of the amplitude in the PSD->FFT passage.

At the moment my code looks like this:

[S, f] = pwelch(x,[],[],[],fsamp);
mag = sqrt(S);

fft_mag = abs(fft(x))/length(x);

However, if I compare these two, I see that they are still off by roughtly a factor of 10.

Where is the error? Is there a better way to generate a time-history from an arbitrary PSD?