The power spectral density (PSD) is one of the primary ways we characterize random or broadband signals. In many cases, a PSD is read from a signal analyzer and used qualitatively to describe the frequency content of a signal. But to do anything quantitative with a PSD, we need to understand its units.
I recently came across a very thorough discussion of PSDs and their units by Randall Peters. I highly recommend it to anyone working with PSDs. What follows is a brief explanation of some of the key points I’ve encountered while working with PSDs across a few disciplines.
Units Squared per Hertz and Parseval’s Theorem
PSDs should have units of signal squared per frequency, as in V
/Hz for a signal measured in Volts. (The term “power” originates from the measurement of voltage across a 1 Ohm load so one would properly have V/(
Hz), or power per unit frequency.) These units seem strange at first, but actually tie in nicely with a physical interpretation: The integral of the PSD over a frequency interval gives the mean-square value of the content of the signal within the frequency interval. (This is known as Parseval’s Theorem.) So to get the RMS value of a signal from its PSD, you just compute the area under the PSD and take the square root.
One-Sided vs Two-Sided PSDs
In fields like vibration and acoustics, it is customary to use one-sided PSDs, meaning all of the signal is obtained by integrating over positive frequencies. But you might occasionally come across two-sided PSDs, which give the spectral density across positive and negative frequencies. For real-valued signals, the two-sided PSD is an even function of frequency, so it is customary to work with the one-sided PSD, whose value is twice that of the two-sided PSD but is defined over only positive frequencies. When working across disciplines or using Fourier transform methods, it is worth double checking for that factor of two.
Root-Mean-Square Units
Sometimes the units of PSDs are expressed as Vrms or equivalent. This sort of unit is more properly “Vrms per frequency bin.” To illustrate, suppose a PSD is measured with a frequency resolution of 10 Hz. Then the Vrms value at 5 Hz is obtained by integrating the PSD (in V/Hz) from 0 to 10 Hz and taking the square root, the Vrms value at 15 Hz is obtained by integrating from 10 to 20 Hz and taking the square root, and so on. Note that the value of a PSD expressed in Vrms changes if we change the frequency spacing.
1/3rd Octave and Log-Frequency Bins
A very common way to express PSDs in acoustics and vibration is on 1/3rd-octave bands. These bands are obtained by dividing the audible range (20 Hz to 20 kHz) into 31 bands, each being 1/3rd of an octave. The specifics are laid out in an ANSI standard, and the numbers are tabulated in lots of places. To compute the spectrum on the 1/3rd octave (or any other non-uniformly spaced) set of bins, we simply integrate the PSD over each bin and take the square root. Note that the shape of the PSD plot on 1/3rd octave bands will be different from the shape on uniformly spaced bands.