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I have a signal recorded by an accelerometer in m/s^2. It is basically accelerations over a period of time. I have calculated the power spectrum of the signal over in a certain frequency band and then transferred into a 20log(x) scale for plotting. I am wondering what should be unit for it?

Since it has been transferred into a log scale, I could use dB, but the standard formula for dB calculation is something like 20log(x/x0), where x0 is the reference and cancels out the unit leaving us with just dB. I am not sure if I can the same thing in the above case since I don't have any reference power? Can I just assume something like x0=1 and report the numbers in dB?

If not, then I am not sure what should accompany dB. In one paper, I have seen for accelerations reported in gravities 1g = 9.8m/s^2,dB g^2/2 was used as the unit for the power spectrum? This makes me guess maybe I should use dB m/2s^2 or weirdly dB m^2/2s^4? Any ideas are greatly appreciated.

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The typical practice with decibels is to append the unit. Unfortunately, it's informal, so you'll see things like "dBm" in RF circuits which refers to "dB with a reference of 1mW", not "dB with a reference of 1 meter".

you could:

  • Put a note in your graph that $0\mathrm{dB} = \mathrm{1 m/s^2}$
  • Use "dBg" (and use 1g as your reference)
  • Just keep the y-axis as the log of acceleration squared (and yes, $\mathrm{m^2/s^4}$ is awkward, but it's the correct unit).
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