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The Autopower Function… Demystified!

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Autopowers Power versus Autopower Linear? Power Spectral Density versus Energy Spectral Density? What’s the difference? Which one to choose?

Ever notice the different options for autopowers when you get to the Testlab Online Processing worksheet? It can be tricky to choose what to use if you aren’t familiar with the differences between AutoPowers Linear, AutoPowers PSD, AutoPowers Power, and Autopowers ESD.

Figure 1: There are four distinct autopower options in Online Processing.

Autopowers Power (Autopower Spectrum)

The autopower power (also called autopower spectrum) is the squared magnitude of the frequency spectrum.

Equation 1: GXX is the autopower spectrum. SX is the frequency spectrum. Sx* is the complex conjugate of the frequency spectrum. The autopower spectrum is created by multiplying a frequency spectrum by its complex conjugate.

This results in the autopower spectrum equaling the magnitude of the frequency spectrum squared. There is no phase information (as the autopower spectrum is all real, no imaginary component).

Figure 2: Example autopower power: The y-axis is the amplitude unit squared. The x-axis is in frequency. There is no phase information.

Consider an average of these two functions: a sine wave and an inverse sine wave. If they are averaged, the result will be zero.

Figure 3: A sine wave and inverse sine wave sum to zero amplitude.

Before averaging, suppose the functions were squared (Figure 4). Would the functions still average to zero?  No, they would not!

Figure 4: The squared sine wave and inverse sine wave do not average to zero.

By squaring before averaging, the functions can be averaged without the resulting amplitude being wrong (in this case, it would be zero).  This is the benefit of “power” in the autopower – squaring removes the phase between averages, so an averaged amplitude can be calculated.

Both sine waves had the same amplitude, but their phase relationship was different.  Averaging to zero might be unexpected.  If the two measurements had 5 g's of vibration each, one would not expect the average to be zero vibration!  To make vibration "go away", you would just have to average measurements to remove it!

When acquiring data, the phase at any given frequency can vary from average to average.  Even these slight changes in phase will reduce the averaged amplitude, but not as drastically as the example with the two averages being 180 degrees out of phase.  For this reason, an autopower is useful when performing averaging to insure a meaningful averaged amplitude.

Calculating a complex conjugate on the spectrum is equivalent to "squaring" in the frequency domain.

An autopower function is used ensure a meaningful amplitude when averaging multiple measurements.  This is done by removing the phase via "squaring" the individual measurements before averaging.

When viewing the averaged results, many people prefer not to have the amplitude unit squared ("5 g's" makes more sense than "25 g's squared"). That is where the autopower linear comes in….

Autopower Linear

The autopower linear is obtained by taking the square root of the amplitude unit of the autopower power. Therefore, the amplitude units will not be squared as in the autopower power.

Equation 2: Autopower linear is obtained by taking the square root of the autopower power. The resultant amplitude units will be linear as opposed to squared.

Figure 5: Example linear autopower: The y-axis is the amplitude unit. The x-axis is in frequency.

A linear autopower function is the square root of a autopower "power" function. This makes the units (ie, 25 g2) easier to understand (5 g).

Another issue with the autopower linear and autopower power is the frequency resolution can have a great impact on the amplitude of the resulting spectrum.

Autopower Power Spectral Density (PSD)

Look at the figure below (Figure 6). The three broadband random signals are identical in the time domain, but because of the difference in frequency resolution, they have greatly different amplitudes in the frequency domain. See this article for details on why this happens.

Figure 6: The amplitude of the spectra can be greatly affected by the frequency resolution.

To normalize the amplitude despite the frequency resolution, an autopower power spectral density (PSD) is used….

The autopower PSD normalizes the amplitude level with respect to the frequency resolution. This helps minimize amplitude differences that arise from using a specific frequency resolution with broadband random signals.

Equation 3: Δf is the frequency resolution. The autopower PSD takes the autopower amplitude values and divides them by the frequency resolution.

Figure 7: The autopower power spectral density normalizes the amplitude by the frequency resolution so no matter what frequency resolution you use, you will get similar amplitude data in the frequency domain. This is the same data as in Figure 6, but now the amplitude levels for the three signals are similar despite their different frequency resolutions.

Autpower PSDs are commonly used to quantify random signals.  For example, PSDs are used in shaker testing to quantify fatigue created from random vibration on components.

An autopower PSD normalizes by the spectral resolution to make broadband random data acquired at different spectral resolutions appear to have similar amplitudes.  The units are divided by the spectral resolution (ie, 25 g2/Hz).

Dividing by frequency resolution is not the only way to normalize an autopower...

Autopower Energy Spectral Density (ESD)

The autopower ESD not only normalizes the amplitude by the frequency resolution but also by acquisition time (T).

The ESD is obtained by multiplying the Autopower PSD by the sample interval, T (also known as the acquisition block time). This is especially useful when measuring transient signals: the ESD allows for the analysis of the total energy in a transient signal rather than the power or average energy per unit time.

Equation 4: T is the sample interval (time it takes to acquire one block / sample). The autopower ESD is obtained by multiplying the autopower PSD by the sample interval.

Figure 8: Example ESD: The amplitude units are the autopower PSD units (Amplitude2/Δf) multiplied by the acquisition time (s).

An autopower ESD normalizes by the spectral resolution and measurement time.

Think that using different types of autopowers is the only way to change the frequency domain amplitude?  Unfortunately, the amplitude mode format also affects the levels.

Amplitude Mode Formats

Now it is clear what the different autopower types are. What are the different amplitude format types? How do they effect the signal?

According to the Fourier theory, any signal can be represented by a specific set of sine waves. The amplitude format describes how the amplitude (A) of the sine wave will be expressed in the resulting spectrum.  It can be either RMS, Peak, or Peak-to-Peak as shown below.

Figure 9: A sine wave with an amplitude, A, of 5. The peak amplitude is 5. The RMS amplitude is 3.5. The peak-to-peak amplitude is 10.

Peak:

Peak format is simply the amplitude value A.

RMS:

RMS format transforms the peak amplitude (A) to the peak amplitude divided by the square root of two.

Peak to Peak:

Peak to peak format transforms the peak amplitude (A), to two times the peak amplitude, 2A.

Depending on the application, or by convention in a particular industry, different amplitude modes might be used.  For example, RMS is commonly used in acoustic applications.  For many worst case durability assessments, Peak-to-Peak vibration is used.

Summary

Below is a table of all autopower types and amplitude formats:

Table 1: Amplitude output depending on autopower type and amplitude mode.When comparing measurements taken at different times or from different acquistion systems, it is important that both the autopower type and amplitude mode were set the same to make valid comparisons. For example, if one measurement was performed in peak-to-peak, while another is performed in peak, there will be a factor of 2 difference in the results even if an identical signal was measured.

Simcenter Testlab

In Simcenter Testlab, it is possible to change the Amplitude mode of a plot.

Right click on the y-axis and select processing. Under “Spectrum & Section Scaling”, you can change the amplitude mode.

Figure 10: Change the amplitude mode right from the plot.

Conclusion:

The amplitude of an autopower is a function of several variables: Autopower type, Amplitude mode, the spectral resolution, and any window (like Hanning, Flattop, etc.) applied during the Fourier transformation. Some things to keep in mind:

• Autopower power versus Autopower linear - While squaring allows spectrums to be averaged for meaningful amplitude, the results squared units ("25 g2") can be confusing.  The autopower linear takes the square root ("5 g").
• Autopower PSD - An autopower PSD is commonly used to quantify broadband random signals.  The PSD normalizes the amplitudes for differing spectral resolutions to make comparison easier
• Amplitude Mode - An autopower spectrum can be scaled whether it is RMS, Peak or Peak-to-Peak units.  This is in addition to any changes in amplitude caused by selecting a particular autopower type.

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