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**Spectrum versus Autopower**

When performing a Fourier Transform, there are several types of spectral functions that can be computed. Two of these functions, *Autopower *and *Spectrum *can yield very different results.

When selecting between the two functions, as shown in *Figure 1*, great care should be taken to understand how the functions are calculated, and how the results can vary.

Results can be affected by amplitude, phase, and averaging. This article explains the background and differences between the two functions and when to use them.

**Background**

Both the *Spectrum* and *Autopower* functions produce results of amplitude versus frequency.

The main difference between the two functions is in the handling of phase:

- Spectrum function includes phase
- Autopower function eliminates the phase

*Figure 2* shows a Spectrum and an Autopower for the same signal:

In *Figure 2*, a single average was performed, and the amplitude portion of both functions is identical. The only difference is that the Spectrum has phase, while the Autopower does not contain phase information (phase is zero at all frequencies).

To understand how important the phase can be when processing spectral data, it is helpful to review the Fourier Transform and how it handles phase.

**Fourier Transform and Phase**

When performing a Fourier Transform of any time signal, it is broken down into a unique set of individual sine waves. These sine waves, when summed together, equal the original time signal (*Figure 3*).

Each individual sine wave not only has an amplitude, but has a phase value as well. Both the phase and amplitude values are uniquely determined for each sine wave by the Fourier Transform. The amplitude and phase values are determined so that sum of the sine waves is equal to the original signal.

Because every signal is a collection of single sine waves, much of this article shows examples using single sine waves.

Take the two sine waves in *Figure 4*. They have the same amplitude, but the phase is different by 45 degrees.

The phase is relative *to the value at the beginning* of the time block. In *Figure 4*, the red sine wave is considered to have zero degrees phase and the blue sine wave is considered to have 45 degrees phase.

This shift in the time domain is evident in the Fourier Transform of both signals as shown in *Figure 5*.

The fact that the Spectrum preserves differences in phase while the Autopower function does not, are key to understanding when to use one or the other.

The next section covers how a Fourier Transform is related to a Spectrum or an Autopower.

**Mathematical Difference between Spectrum and Autopower**

In many digital signal processing algorithms, the Fourier Transform results in a double sided spectrum which is mirrored about 0 Hertz. This mirroring around zero Hertz is eliminated, so the Spectrum (S_{x}). only contains frequencies from 0 Hertz and higher. The Spectrum is a complex function having both amplitude and phase, which can be expressed as real and imaginary (a+ib), versus frequency.

The Autopower (G_{xx}) goes one step further and performs a multiplication of the complex conjugate of the Spectrum as shown *Equation 1*. This conjugate is calculated separately at each frequency line in the Spectrum.

By performing the complex conjugate operation, the Spectrum becomes an Autopower function which only contains amplitude, but is without phase.

The complex conjugate leaves the units squared (Example: g^{2} or *power units*). It is common to take the square root so that linear units are restored (Example: g). This is called a ‘Autopower Linear’ function. ‘Linear’ indicates the square root was performed after the complex conjugate multiplication.

Selecting either *Autopower* or *Spectrum* is also an important consideration if averaging of multiple functions or samples will be performed, which is discussed in the next section.

**Averaging Considerations**

In *Figure 6*, the time domain and frequency domain of two different sine waves are shown. The two sine waves are 180 degrees out of phase.

If the two sine waves are averaged, the result would be zero. Whether performing the average in the time domain or in the frequency domain, the average would be zero. Using a Autopower, however, it can be made so the average will not be zero in the frequency domain.

In *Figure 7*, the Autopower of both sine waves is shown. Notice that the phase of the Autopower of both sine waves is zero, and identical. The phase of the Autopower in the frequency domain is the same, even though the two sine waves are clearly out of phase in the time domain.

In this case, while the average in the time domain would be zero, the average of the Autopower functions in the frequency domain would be correct. The average would *not be* zero in the frequency domain, thanks to the Autopower function!

**Correct Amplitude by Eliminating Phase**

How is the correct amplitude of the average made possible by the Autopower function? How does eliminating phase do this? *Figure 8 *contains an* illustrative* example. For clarity, the example is based on time data, but it holds true in the frequency domain:

- On the left side of
*Figure 8*, there are two sine waves shown that are out of phase. They clearly average to zero. Even in the frequency domain, they would average to zero because of their 180 degree inverse phase relationship. - On the right side of
*Figure 8*, the two sine waves are squared before averaging. This is equivalent to the multiplication by the complex conjugate in the frequency domain. By squaring the sine waves, they become all positive functions. They cannot average to zero.

In fact, in this example, the two squared time functions are completely identical for a very consistent average.

In practice, the average of a Spectrum is not always zero after two acquisitions. For example, when measuring a broadband random signal, the amplitude of the signal is reduced *gradually *with averaging. The more averages, the lower the amplitude as shown in *Figure 9*.

Some observations:

- With 10, 25, and 100 averages, the variation in the Autopower amplitude is reduced, and the overall amplitude is the same
- With 10, 25, and 100 averages, the Spectrum has a lot of amplitude variation and progressively lower amplitude

Using a Spectrum function, the amplitude gets progressively lower by averaging! Since getting the correct amplitude is of importance, the use of averaging with a Spectrum should be avoided, unless special precautions have been taken to assure the phase is consistent (i.e., a trigger or something similar).

**Phase Referenced Spectrum**

What if correct amplitude, phase, and averaging are desired? For example, an Operational Deflection Shape (ODS) analysis requires correct amplitude and phase with averaging:

- If phase is required, an Autopower function will not work, so a Spectrum must be used
- If averaging is required a Spectrum will not produce correct amplitude, unless the phasing between averages can be controlled

The best alternative is an option called *Phase Referenced Spectrum*.

With a Phase Referenced Spectrum, instead of relying on the *phase relative to start of each acqusition*, a different type of phase will be used. The *phase between different data channels* on a vibrating structure will be used instead to make averaging possible.

Consider the vibrating structure animation in *Figure 10*:

- Measurement location
is in phase with*plate:1:Z**plate13:Z* - Measurement location
is out of phase with*plate 15:Z**plate 13:Z*

By taking advantage of the fixed phase relationship between different locations on a vibrating structure, it is possible to do averaging, even using a Spectrum function.

With a *Phase Referenced Spectrum* one measurement location of a multiple channel acquisition is designated as the phase reference. The reference channel is used to keep the phase consistent between all channels to allow averaging.

Data was acquired on the vibrating structure and is shown in *Figure 11*:

- The start time (A, B, and C) of each acquisition produced a different phase between acquisition 1, 2, and 3.
- The relative phase
*between data channels*is the same. For example, Plate:15:Z is always 180 degrees out of phase with plate:13:Z.

Even though all the sine waves shown have the same amplitude, any resulting average *will not* have the correct amplitude.

The *phase referenced spectrum* will make this data consistent so it can be averaged by (*Figure 12*):

- For each frequency, subtract enough phase from the phase reference channel to set the phase to zero. This is done independently for each frequency line in the spectrum, and done after each average/acquisition.
- The same amount of phase subtracted from the reference channel is subtracted from every other data channel. This is also done separately for each frequency/spectral line, and after each average/acquisition.

Because there is a fixed relationship in phase between the data channels and the reference channel, the inconsistent data will be made consistent, as shown in *Figure 12.*

This allows the averaging to be done properly and results in the correct amplitude value. Of course, at frequencies where there is no relationship between channels, this will not be the case.

For example, phase referencing will not work with data channels being acquired on a vehicle if the phase reference channel is on the laboratory floor. The phase reference channel instead should be a data channel on the vehicle that is very active.

Will the phase referenced spectrum and an autopower have exactly the same amplitude Probably not. Any random vibration in the structure that has no fixed phase relationship to other locations will be averaged out. So there can be differences depending on how much the vibration of different locations on the structure are related to each other.

Note that this example is shown for one frequency in the time domain for *illustrative* purposes. The ‘Phase Referenced Spectrum’ will work on all frequencies independently in the spectrum. The amount of phase subtracted is not always 0, 90, or 180 degrees. It can be any phase amount (33, 48, 56 degrees) that is present in the vibrating structure between different locations.

**Phase Referenced Spectrum in LMS Test.Lab**

To calculate a *Phase Referenced Spectrum *in LMS Test.Lab, change the function to ‘Spectrum’ and click on the checkbox next to ‘Phase Referenced Spectrum’ as shown in *Figure 13*.Under ‘References’ click the ‘Define…’ button and select a single channel to use as a reference.

**Conclusions**

A* Spectrum* and *Autopower* are both functions of amplitude versus frequency. In the case of Spectrum, the phase is also preserved, while with an Autopower it is not.

- Be careful using Spectrum when averaging data. It is easy to get the wrong amplitude.
- Use Autopower for applications where only amplitude is required.
- Use a Phase Referenced Spectrum when phasing and averaging are both required.

Enjoy Spectrums and Autopowers!

Questions? Contact us!

**Digital Signal Processing**

- Digital Signal Processing: Sampling Rates, Bandwidth, Spectral Lines, and more...
- Gain, Range, Quantization
- Aliasing
- Overloads
- Averaging Types: What's the difference?
- Autopower Function...Demystified!
- Power Spectral Density
- What is an Operational Deflection Shape?
- LMS Test.Lab Throughput Processing Tips

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