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Based on this answer. Once you have your DFT in the unit sound pressure RMS, what is the correct method for getting the sound pressure RMS sum over multiple bins (if for example you're trying to get the total RMS over a frequency band)?

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The power spectrum of the DFT is the square of the magnitudes and is related to the power spectrum of the discrete time signal by Parsevel's theorem:

$$E_x=\sum_{k=0}^{N-1}\lvert x[k]\rvert ^2=\frac{1}{N}\sum_{k}^{N-1}\lvert X[k]\rvert^2.$$

The RMS is just the square root of the power. (I assume you've already taken care of the 1/N since you mentioned your DFT is already in the correct units).

$$\textrm{RMS}_{x}=\sqrt{\sum_{k=0}^{N-1}\lvert X[k]\rvert^2}.$$

You can compute the RMS of a range of bins $[i,j]$ by constraining $k$.

$$\textrm{RMS}_{x_{ij}}=\sqrt{\sum_{k=i}^{j}\lvert X[k]\rvert^2}.$$

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