Specification on power spectral density for population data

What is the best way to put a specification on the single-side auto-power spectral density (PSD)?

We have a product for which we have a time signal. For this signal we calculate the PSD (or respectively the cumulative PSD or 3 sigma cumulative PSD) to determine the frequency behavior. However, each product shows slightly different behavior. Currently analysis to determine if behavior is OK is done on visual inspection of the PSD.

PSD

Cumulative PSD

Now we would like to put a specification on this for the frequency range. Such that we can more easily determine if something is OK or NOK.

Does someone know how specifications are set on a PSD? How is this is done in industry for frequency data?

For continuous time domain signal you can simply set a lower and upper spec limit. Furthermore you can apply statistical process control for discrete data.

What I currently thought of; get a population of good products. Use the PSD results to calculate an mean PSD and the variance of the PSD (for each frequency point). Then all frequency data can be normalized by the standard score $$(X - \mu)/\sigma$$. As such giving flat lines for data which is OK, and for data which is NOK you would see deviations.

Then the spec, as an example, could be mean plus four times the standard deviation, for which we could get triggered.

• I can think of several ways to create a test, even with a possible grading system. But I'm not sure if there is a "best way"; maybe there are theoretical descriptions on which you base such an estimator (or rather: estimator distribution). In practice I'd go for something practical. – cherub Oct 29 '19 at 10:33
• Do you have an example? – WG- Oct 30 '19 at 10:26
• Every choice depends on what kind of information you want to extract. As one example you can get a somewhat "frequency resolved" deviation by comparing your reference cumulative PSD with the current cumulative PSD. If you normalize for total area/summed amplitude values, you can interpret the deviation as relative. Since the total number at the end of the spectrum needs to be the same, you can directly identify the frequency (ranges) where they deviate. – cherub Nov 1 '19 at 14:10