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What is the PSD?

March 29, 2018

In vibration analysis the PSD stands for the Power Spectral Density of a signal. Each word is chosen to represent an essential component of the PSD.

Power refers to the fact that the magnitude of the PSD is the mean-square value of the signal being analyzed. It does not refer to the physical quantity  power (as in watts or horsepower). But since power is proportional to the mean-square value of some quantity (such as the square of current or voltage in an electrical circuit), the mean-square value of any quantity has become known as the power of that quantity.

Spectral refers to the fact that the PSD is a function of frequency. The PSD represents the distribution of a signal over a spectrum of frequencies just like a rainbow represents the distribution of light over a spectrum of wavelengths (or colors).

Density refers to the fact that the magnitude of the PSD is normalized to a single hertz bandwidth. For example, with a signal measuring acceleration in units of G’s, the PSD has units of G2Hz.

Since the name PSD does not include the quantity being measured, the word power is sometimes replaced by the name of the quantity being measured. For example, the PSD of an acceleration signal is sometimes referred to as the Acceleration Spectral Density.

Why Power?

The mean-square value (power) is a convenient measure of the strength of a signal. This is illustrated in Fig. 1 which shows the vibration time history for a car’s floor panel, as measured by an accelerometer. The average amplitude of the signal cannot be specified by the mean value since this is near zero. Instead the signal is squared (resulting in a positive quantity) and then the mean value is computed. To obtain a linear value (in G’s for this case) the square root is taken to obtain the RMS (root-mean-square) value.

why-power-figure1-randomPSD-rev-125pct Figure 1. Vibration of Car Floor Panel, Mean-Square Value = 0.0053 G2, RMS value = 0.073 G

The mean-square value must be used when combining signals of different frequencies. This is illustrated in Fig. 2 where two sine waves of different frequencies are added together. The mean-square value of a unit sine wave is 0.5 and the RMS value is 0.707. When the two are added the mean-square value is 1.0 and the RMS value is 1.0.

why-power-figure2-randomPSD-rev-125pct

Mathematically this is based on a general result for two independent variables, A and B. The square of the sum is given by (A+B)2 = A2 + 2AB + B2.  If the two variables are independent and have a mean value of zero, then the mean value of 2AB is zero, as illustrated in Fig. 3.

why-power-figure3-randomPSD-rev-125pct

It should be noted that there are cases where the PSD is represented by the square root of its computed value, so an acceleration PSD might have units, G/(Hz)1/2. Care must be taken to determine the units used for the PSD.

Why Spectral?

The frequency distribution of a signal is very useful information when dealing with systems having resonances. This is illustrated in Fig. 4 where a cantilever beam is being driven at the base by a broad band signal (having a wide distribution of frequencies) and an accelerometer is measuring the tip vibration. It is difficult to determine from the time history of the signal the values of the resonance frequencies of the beam. However, the peaks in the frequency spectrum of the tip vibration clearly show the  resonance frequencies.

why-spectral-figure1-rev-125pct

Why Density?

The magnitude of the frequency distribution of a signal depends on the number of frequency bands in the distribution. This is illustrated in Fig. 5 where the frequency spectrum of the car vibration signal is computed with three different frequency bandwidths.  The squared magnitudes of the spectra are proportional to the frequency bandwidth. To overcome this variation, the PSD divides the squared magnitude by the frequency bandwidth to give a consistent value independent of the band width used.

Figure 5. Dependence of the Frequency Spectrum on the Frequency Bandwidth

It should be noted that there are cases where the bandwidth dependent frequency spectrum and the PSD are confused. Care must be taken to determine whether or not the bandwidth has been used to normalize the magnitude, which is part of a PSD calculation.