Continuous Power Spectral Density (PSD) from a Discrete PSD

Hi everyone,

it is possible that I already posted a similar question in the past , but this one is a slightly different problem.  The physical background is the fatigue calculation derived from random vibration with a known PSD. There are many literature sources for that, I will just mention a good practical summary by Tom Irvine (hope the link will work), or just google "Estimating Fatigue Damage from Stress Power Spectral Density Function".

Now my problem is: the input for these approaches is always a continuous integrable function like for example this example from that paper:

Obviously the units of these function are the square of the input quantitydivided by Hz (for stress for example ksi**2/Hz or MPa**2/Hz). In this way the square root of the integral of the function will provide the RMS of the input quantity (in this example 6.64 ksi).

Now let's say I have a measured time signal (it is actually a vibration velocity, but it would be no problem to convert it in stress, so that is not the point).  I use pspectrum (from the Signal Analysis Extension Pack) to compute its "discrete PSD".  Using /cfft**2/ gives similar results, so the computation should be ok.

I say discrete, because while the sum of the "spectral lines" divided by their number is constant (and equal to the sum of the square of the input values, also divided by their number, see illustration from the "spectral analysis chapter of the e-book "Signal Analysis Extension Pack"),

well, now depending on the number of intervals that I chose, I get different "heights".  Here an example:

This is obvious, because the sum must remain constant, and so the function, though shown with a track, like it was continuous, is actually made from many vertical "lines" separated one from the other.  In other words, the units of this psd output are (I believe) the square of the input quantity, and not, as in the analytical/numerical PSD example above, the quantity divided by Hz.

Now I would like to be able to fit some kind of envelope to the measured PSD, in order to do some computations, something like this:

Possibly the integral of the fitted function should be only slightly above the RMS of the input signal, to avoid overconservativity, but that is another subject.

Now finally my question: how do I get the "right" units [MPa**2/Hz] or [ksi**2/Hz] on the vertical axis? It may be so a dummy question that the answer is: just divide the peaks by their frequency and that's it, but somehow I am unsure if that's all what I need to do.

Thanks a lot for any hints

Best regards

Claudio