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What is a Power Spectral Density (PSD)? How is it different than an autopower?
A Power Spectral Density (PSD) is the measure of signal's power content versus frequency. A PSD is typically used to characterize broadband random signals. The amplitude of the PSD is normalized by the spectral resolution employed to digitize the signal.
For vibration data, a PSD has amplitude units of g2/Hz. While this unit may not seem intuitive at first, it helps ensure that random data can be overlaid and compared independently of the spectral resolution used to measure the data. This article explains how this is done.
To understand a Power Spectral Density (PSD), it is helpful to understand some limitations of an autopower function when analyzing data with differing spectral resolutions:
Scenario:
Identical broad band data was measured three different times. For each measurement, only the frequency resolution was changed. It was acquired with a 1 Hz frequency resolution, then a 4 Hz frequency resolution, and finally an 8 Hz frequency resolution.
The resulting autopower data has very different amplitudes (Picture 1). Why? Which one is correct?
Answer:
All of the autopowers are correct! They appear visually to have different amplitudes however, which is confusing.
As the frequency resolution (sometimes referred to as spectral resolution) gets finer (starting with 8 Hz, then 4 Hz, then 1 Hz), more data points are being used to measure the signal. The same signal is being divided up into smaller parts, but the total remains the same.
While the amplitude at individual frequencies appears to be different, the total summation of data across the frequency range is identical.
The “total” of the signal is reflected in the RMS summation of the entire spectrum as shown in Picture 2. The RMS summation is identical for all three autopower spectrums.
Even a partial RMS summation of the data (Picture 3), based on a smaller frequency range, is identical for all three measurements.
Spectral Lines
Spectral lines are the key to understanding why the plotted amplitudes for these identical signals looks to different, but sum to the same amount.
An autopower spectrum, measured with a 1 Hz resolution and a bandwidth of 6000 Hz, would have 6000 data points or spectral lines as shown in Equation 1.
Spectral lines are discrete points in the frequency domain used to digitize the spectrum (remember that a computer cannot store a continuous analog function, it must break any data into discrete points). The entire signal is divided up among these 6000 data points (i.e., spectral lines).
For the three separate measurements (each with a max frequency of 6000 Hz), the following is observed:
Note that the amplitude is really a function of the number of spectral lines. The more spectral lines, the lower the amplitude of each spectral line.
The results are correct for a given frequency resolution, although this is not readily apparent when viewing the spectrum. By zooming into a narrower frequency range (Picture 4), the different frequency resolutions between the three measurements start to become more obvious. One can see that the blue line (8 Hz resolution) has less data points than the red line (1 Hz resolution).
The manner in which the data is presented disguises the differences between the data curves. By default, most FFT analyzers display data with lines connecting the data points. Instead of connecting with the data with lines as in Picture 4, the same data can instead be viewed as block outlines (Picture 5). Block outlines allow the individual spectral lines to be seen.
Using the block outlines, the differences in the three measurements is more obvious (Picture 5). One can see that in the blue curve, measured with 8 Hz frequency resolution, that the levels of each spectral line are higher, but there are fewer data points over the frequency range. In the red curve, there are more data points, but each point/line is lower in amplitude.The green curve is in the middle.
Party Analogy
A party where beverages are being served can be used as an analogy (Picture 6) to explain this relationship between frequency resolution and amplitude in the autopower.
Imagine that the signal being measured is a fixed quantity of beverage to be served. The number of glasses held by attendees is analogous to the number of spectral lines.
To simulate a broadband signal, the beverage is distributed evenly among the glasses. The more glasses (i.e., spectral lines) for distribution, the lower the amount in each glass. The total amount of beverage (i.e., RMS) served remains the same.
The Power Spectral Density function will now be used to remove/reduce the apparent difference in the three autopower spectrums. Remember, the Autopower and Power Spectral Density are both correct, only the representation of the data is being changed by switching functions.
Power Spectral Density
Despite the fact that the total amount of signal (as shown by the RMS) is identical, it is often desired that the amplitudes shown in the autopower graph also look similar.
Power Spectral Density (PSD) normalizes the amplitudes by the frequency resolution to give the amplitudes a similar appearance (Picture 7).
The term “normalizing” by the frequency resolution means dividing the amplitude of each spectral line by the frequency resolution.
The amplitude is always shown divided by Hertz as result, as in 25 g^{2}/Hz.
Because a PSD gives similar amplitudes for random data, it is often used as the control function in random vibration control testing.
By convention, the amplitude of the data in a Power Spectral Density is squared. For example, if one were measuring a 5 g amplitude (rms) sine wave, the amplitude shown in a PSD would be 25 g^{2}/Hz.
Sinusoidal Data
Everything is the opposite for sinusoidal data!
Going back to our party analogy, in the case of a sinusoid, all the signal in the pitcher is put into a single glass (spectral line). For example, a 200 Hz sine wave acquired with a 1 Hz, 4 Hz, or 8 Hz frequency resolution puts all the signal in a single spectral line (200 is evenly divisible by 1, 4, and 8).
For a sinusoid, the amplitude will not vary greatly with changes in the number of spectral lines. Because the signal is always placed in one “glass” (ie, spectral line), the amplitude in the glass does not change as more glasses are added.
In fact, a Power Spectral Density (PSD) of a sinusoidal signal would actually change the apparent amplitude of a sine wave drastically as in Picture 10.
This is because the amplitudes of the sine waves are being divided by their respective frequency resolutions (delta f). Taking the same amplitude and dividing it by different frequency resolution (delta f) values makes the amplitudes different.
The PSD in which we divided the amplitude value by largest frequency resolution (in this case 8 Hz), results in the lowest amplitude.
The RMS summation of the PSD of the sine waves remains the same. This is because the RMS summation functionality adjusts for the frequency resolution division automatically to get the correct value.
Conclusion
Because computers digitize signals into discrete points, some interesting digital signal processing phenomenon are created.
Broadband (random) signals:
Sinusoidal (periodic) signals:
Table 1 summarizes these observations about Autopowers versus Power Spectral Densities.
In all cases, the RMS summation of data over the frequency range is the same in all cases. When evaluating a spectral function, it is best practice to uses a RMS for comparison purposes, since the RMS amplitudes take into account frequency resolution and other adjustments to produce consistent and useable values.
In practice, typically one finds in industry:
Simcenter Testlab Format Conversion
In Simcenter Testlab (formerly LMS Test.Lab), an autopower can be converted to a PSD and vice versa.
In the conditioning toolbar (Picture 11), the format of a spectral function can be converted by selecting the "FFT Format Conversion" button in Simcenter Testlab.
In the menu, the following can be changed:
While the mode and format can be changed, the spectral resolution cannot. To change the spectral resolution, the data must be reprocessed from the original time history.
Questions? Contact peter.schaldenbrand@siemens.com
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