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Why are the units of a PSD in g2/Hz PDF Print E-mail
Monday, 18 June 2007 17:20

Imagine an acceleration response in units of g through time of a hard point on a structure subject to some loading, which might have been obtained from a test. The data is in the time domain as we see the variation of acceleration in time.


The signal does not appear to have any pattern to it and varies randomly with time; it is also a stationary process. As this is a randomly varying signal, we cannot predict precisely what is going to happen next. Therefore we must employ some statistical (averaging) method so that we can predict the chance or probabilitiy that the signal will be within certain limits.

If the process is ergodic, we sample the data at time t1, t2, t3, etc.., for many points, and determine the mean (average) value. In practice, the process will probably not be ergodic, and we take the average of an ensemble of signals which are taken from different parts of the acceleration response.

Having obtained the mean value of the signal, we subtract this value from the data sampled at each of the many points in order to normalize the data to this mean, and then square the result; the average of these is the mean-square value and the r.m.s value is obtained by taking the square root of the mean square value. From this we get a mean-square value in g².


Why do we normalize to the mean ?

In practice, the PSD curve is calculated using a Fourier tranform. If we normalize to the mean (subtract the mean value from the data sampled at each of the many points) the autocorrelation function will tend to zero as dt (time between samples) tends to infinity. This makes the autocorrelation function absolutely integrable, allowing classical Fourier analysis to be performed.

If the process we want to analyse is a random stationary process, we cannot perform Fourier analysis on this directly because, by definition the process never ends and cannot tend to zero as dt tends to infinity.


Why do we square the sampled data ?

This is in order to obtain a positive quantity. Imagine the signal was a truely random one, containing all frequencies at all amplitudes between zero and the peak. This is known as white noise, the analogy being white light which contains all visible frequencies in equal amounts and noise describing no specific pattern of amplitude. If this were the case, then the mean value would be zero, because a sampled point would be equally as likely to have a positive value as a negative value and the net result after sampling many, many points would be zero.

To give us an idea about the variation of the mean-square value of g² across the frequency range, we progressively filter the signal from 0Hz to some upper bound that we decide. For instance, first we could stop all freqencies above 10Hz and just look at the mean-square value of g² below 10Hz. If we then divide the mean-square value of g² up to 10Hz by what is called the bandwidth of the filter (10Hz in this case) we get the spectral density of the signal in g²/Hz, up to 10Hz. Now we can increase the filter bandwidth to 20Hz and look at the mean-square value now. Again we divide this mean-square value by the filter bandwidth to get the spectral density up to 20 Hz. This is repeated this for 30Hz, 40Hz etc.. If we plot the quantity of mean-square divided by bandwidth (g²/Hz) against frequency (Hz) we get a familiar looking PSD curve.

Actually PSD (Power Spectral Density) is a bit of a mis-nomer. It should be correclty termed ASD for Acceleration Spectral Density. It has become known as PSD where the "power" originates from the output from the accelerometers during structural test.


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