To characterize a subtle oscillation embedded in a time varying voltage signal measured in microvolts, I took the second derivative of the PSD (which I computed as the fft of the autocorrelation) which nicely amplified the peak so it could be easily detected. I then integrated the PSD across the frequency band of interest to arrive at a single number that characterized the sharpness and strength of the peak in this band. I have two questions: (i) What is the unit of this measure (microvolt^2/Hz^3?)? (ii) What would be the best way to describe this measure in terms of energy?
2026-03-31 17:58:43.1774979923
Unit of the second derivative of the power spectral density
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1
Say you have volts in the time domain. Consider what the DFT does to get its values. It weights and sums the time samples. If you sum volts, you still have volts. Now, however, rather than having volts at an instance in time, you have volts at instance of frequency. If you differentiate volts w.r.t. frequency, you will have units of volts per Hertz, just like if you differentiated volts w.r.t. time you have volts per second.