The PROB module in DATS for Windows provides, amongst other options, a probability density analysis. Also, the signal generation suite has a module, GENPRB, which generates a classical gaussian probability density curve (and others). How then may these be used to compare the probability density of our measured signal with that of a true Gaussian one. The method is quite straight forward and is a matter of scaling.
A probability density curve is normally drawn with the probability density as the y axis and the signal amplitude as the x axis. For a Gaussian signal the position and scaling of the x axis is determined entirely by the mean and standard deviation of the equivalent time history signal. For reference purposes some people refer to the mean as the dc component. Also, the standard deviation is the same as the rms level for a signal with a zero dc component. If we wish to compare a measured probability density with the theoretical Gaussian probability density then we have to match the means and standard deviations. An example of a theoretical Gaussian probability density curve is shown in Figure 1.
In order to compare the measured data and the theoretical values one’s approach is to first use the Statistics module (STAT) to find and record the mean and standard deviation of the measured signal. In the following example the mean and standard deviation of the measured signal were calculated by STAT as 0.2510 and 0.3984 respectively.
When carrying out probability analyses on measured data, one generally needs quite long signals. Typically, I use around 100,000 data values if I can. Experience also shows that using 101 bins (the “Dalmatian” approach!!) gives a reasonable estimate of the probability density. The region which will be most erratic will be the ‘tails’ of the distribution as there are less extreme values. Because of this it is sometimes useful to limit the analysis range to the equivalent of (mean – 4 * standard deviations) to (mean + 4 * standard deviations).
PROB is then used to get the measured probability density curve. Having found the probability density function of the measured data using PROB then we use the GENPRB module to create the corresponding theoretical distribution. To do this we specify the same mean and standard deviation as found in the measured signal using STAT. Again, it is useful to limit to about ±4 standard deviations which is the usual default.
We now have 2 curves which may be overlayed for a graphical comparison. Alternatively, we may subtract one from the other and then use STAT to find the rms value of the difference curve. This gives a single value which quantifies the difference. In this example it was 0.2037. Figure 2 shows the probability density function of a typical measured Gaussian signal overlaid with the theoretical probability density.
The other method of doing the comparison is essentially the same. Instead of using STAT to find the mean and standard deviation, module NORM is used to reduce the measured signal to zero mean and unit standard deviation. This reduced signal is analysed by PROB to get the probability density using a range of ±4. We now only need one theoretical Gaussian curve for any comparison by computing the theoretical one with zero mean and unity standard deviation. Again, the curves may be compared graphically and by using the rms of the difference. If a report were done it would be useful to include the mean and standard deviation of the original signal as well as the normalised rms difference. This is shown in Figure 3 below.
Table 1: Comparison of Theoretical and Actual
As shown in Table 1 above the rms difference between the normalised results is very small.
Chief Signal Processing Analyst (Retired) at Prosig
Dr Colin Mercer was formerly at the Institute of Sound and Vibration Research (ISVR), University of Southampton where he founded the Data Analysis Centre. He then went on to found Prosig in 1977. Colin retired as Chief Signal Processing Analyst at Prosig in December 2016. He is a Chartered Engineer and a Fellow of the British Computer Society.