Averaging

March 29, 2018

With a random vibration signal, the digitized time sample used to calculate the FFT gives just one estimate of the PSD. A single time sample does not provide an accurate determination of the magnitude of the PSD. In fact, the statistical error in the magnitude of the PSD from one sample of white noise is 100%, in that the standard deviation is equal to the mean value.

This can be seen in Figure 2.24, where the PSD is computed using one sample of a white noise signal, low pass filtered at 1000Hz. In this case, the PSD is plotted using linear axes. The RMS value of the signal is 1.0, so the PSD should have a magnitude of 0.001 from 0 to 1000Hz. In this example, the mean value of the PSD equals 0.00093 and the standard deviation equals 0.0010.

PSD computed from one sample of a white noise signal

Figure 2.24. PSD computed from one sample of a white noise signal.

The accuracy of the PSD can be improved by averaging the PSD computed from successive samples of the signal. The standard deviation of M averages of a random variable decreases as  M.  This is illustrated in Figure 2.25 where the PSD is computed from 25 and 100 averages of the signal used in Figure 2.24 and the standard deviation of the PSD is reduced approximately by factors of 5 and 10, respectively.

PSD computed from the averages of a white noise signal

Figure 2.25. PSD computed from the averages of a white noise signal. 25 Averages: Mean Value of PSD = 0.00097, Standard Deviation = 0.00020. 100 Averages: Mean Value of PSD = 0.00098, Standard Deviation = 0.00011.

When the signal is windowed with something like the Hanning window, part of the information from the beginning and end of the digital sample is lost. So it is common to compute averages using overlapping samples. This is illustrated in Figure 2.26 for a “50% overlap.”

Successive digital samples of a signal using a Hanning window with 50% overlap

Figure 2.26. Successive digital samples of a signal using a Hanning window with 50% overlap.