Meaning of dB units in periodogram (FFT)

Question

When I calculate spectrum of a voltage signal using FFT and then I calculate periodogram using $$S_k=10\cdot log_{10}(\frac{1}{N}|X_k|^2)$$ where \$X_k\$ if the result of FFT and \$N\$ is number of samples, the resulting graph usually has "dB" on vertical axis. But what is the meaning of dB in this context?

Decibels usually refer to either power (dBm) or amplification/attenuation or multiple of some reference value. But in this case, it seems to me that it is neither of these.

It linearly depends on the number of samples (Because the result of Fourier transform is linearly dependent on \$N\$, then I square it, but only divide by \$N\$). So is it somehow proportional to the energy of the signal? If so, how is voltage converted to energy/power? It would require some impedance value to get power from voltage. Or should it be somehow normalized (is there a definition of what 0 dB should mean)?

Answer 1

Originally the Bel was defined as the logarithm of the ratio of two power levels. Thus 1 Bel (B) represents a power ratio of 10. That unit proved to be too large so the decibel (dB) was introduced and defined as 10 times the logarithm of the ratio of two power levels. Thus 10 dB represents a power ratio of 10. If the two power levels are measured across the same impedance, then the power ratio is equal to the square of the voltage ratio. Thus, we can say that the number of dB of a voltage ratio is equal to 20 times the logarithm of the ratio. Engineers being engineers, however, and over time for convenience, this definition of the decibel has been corrupted so that it is normal practice to define 20 times the logarithm of any two voltages as decibels. The same formula is used for converting amplifier gain (or any gain for that matter) into dB. If you want an absolute value for your dB, then you have to define a reference level for 0 dB. This is completely arbitrary but some reasonable standards exist such as for audio levels (0.0002 ubar). In your case you are using dB to represent a number representing the relative level in a spectrum. If you want an absolute level, then you have to specify the level corresponding to 0 dB. There are no rules for this; it depends on the application and your needs. As I said, the original definition of dB has all but been lost to further convenience in presenting data and easing calculations especially with widely varying data.