From acceleration data I made a Power Spectral Density (PSD) and calculated the RMS via:

$$a_{RMS} = \left[\displaystyle\int_{f_1}^{f_2} a_{PSD}(f) df\right] ^{\frac 12}\tag{1}$$

Now, I want to calculate the RMQ (Root Mean Quad) from the PSD. Considering the fourth-power relation for the RMS I think the following formula should translate the PSD to the RMS:

$$a_{RMQ} = \left[\int_{f_1}^{f_2} a_{PSD}(f)^2 df\right] ^{\frac 14}\tag{2}$$

But I did not manage to prove this.

Can someone provide me some extra information/a prove about this problem. I can not find anything about calculating the RMQ from the PSD. Thanks in advance!


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