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The Gibbs Phenomenon

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The Gibbs Phenomenongibbs_icon.png 

 

To describe a signal with a discontinuity in the time domain requires infinite frequency content. In practice, it is not possible to sample infinite frequency content. The truncation of frequency content causes a time domain ringing artifact on the signal, which is called the “Gibbs phenomenon”.

 

Any signal with a sudden discontinuity or step will show the Gibbs phenomenon. The examples in this article use a square wave.  Some real world signals that could exhibit the Gibbs phenomenon include forces from driving over a pothole, the sound from an explosion, or the vibration from a golf club when hitting a ball.

 

Background

 

To understand the Gibbs phenomenon, it is important to understand the Fourier Transform. According to the Fourier Transform, every signal is a summation of a unique set of sinusoidal waves. Some examples are shown in Figure 1.

 

FFTs_of_different_signals.pngFigure 1: Various time signals (left) and their equivalent frequency content (right).

Of the signals in Figure 1, notice that some signals, the square wave and impulse, have infinite frequency content. Both of these signals have discontinuities, or a sudden step, contained in them. 

 

This sudden transient change, and the associated infinite frequency content, is difficult to replicate with data acquisition systems with finite frequency capabilities. This can give rise to the Gibbs phenomenon in measurement systems.

 

The Gibbs phenomenon manifests itself in measured signals as a ringing artifact at the step/transition location in the time domain signal as shown in Figure 2.

 

gibbs_square.pngFigure 2: Gibbs phenomenon on a Square wave.

In digital data acquisition systems, this ringing can be inconsistent at each step function in the signal.  It can vary in amplitude or not occur at all.  This depends on timing of the transient relative to the data samples.

 

This ringing artifact is created when describing the signal with less than perfect frequency information. In practice, there are a few situations which create less than ideal frequency information:

  • Truncation of Frequency Content – In a measurement system, it is not possible to measure infinite bandwidth. For example, a square wave should contain an infinite number of harmonic frequencies.  When measuring a signal, an infinite number of harmonic frequencies cannot be measured, which truncates the frequency content present in the signal. 
  • Filter Shape - Filters (for example, anti-aliasing) are often used in measurement devices, which can introduce ringing effects related to the sharpness of the filter.

At each discontinuity or step in the time signal, there is overshoot and undershoot ringing around the original signal.  In the time domain, from an amplitude point of view, this ringing is not always desirable.  The measured/observed amplitude could be different than the actual signal.

 

Causes and Control of the Gibbs Phenomenon

 

The ringing artifact created by the Gibbs phenomenon can be described in two ways:

  • Amplitude – How much the signal under and over shoots the original signal
  • Duration – How long the ringing phenomenon lasts

These two quantities are illustrated in Figure 3.

 

gibbs_amp_duration.pngFigure 3: The Gibbs phenomenon ringing artifact has an amplitude and duration.The duration of the ringing is controlled by the frequency content used to describe the signal, while the amplitude is affect by the type of filter used. In the following examples, a square wave, which has infinite frequency content, will be used to illustrate the Gibbs phenomenon. 

 

Duration of Ringing: Truncation of Harmonics

 

A square wave consists of odd harmonics as shown in Figure 4. If some of the harmonics are removed (i.e., truncated), then the time domain representation of the signal is not exactly square.

 

square_wave.pngFigure 4: Square is infinite number of oddly spaced harmonics.

Removing these harmonics introduces the Gibbs phenomenon.  A ringing artifact is created at the square wave transition.  In Figure 5, the same square wave is shown with all harmonics, harmonics truncated at 2000 Hz, harmonics truncated at 750 Hz. This is done by applying a low pass filter to the signal.

 

frequency_truncation.pngFigure 5: Top – Square wave spectrum (blue), low pass filtered at 2000 Hz (red), low pass filtered at 750 Hz (green). Bottom – As more frequency harmonics are truncated, the ringing artifact has a longer duration and the transition is less sharp.

The graph in the lower half of Figure 5 shows that the ringing artifact lasts longer in duration as more harmonics are removed. In addition, the step, or discontinuity, in the square is more gradual without the higher frequency harmonics.  In Figure 5, the slope of the green line at the step is less steep than the blue or red signals.

 

The signal being measured has an important role in determining if the Gibbs effect occurs.  If the signal has no frequency content that is truncated, the Gibbs phenomenon does not occur. For example, applying low pass filters to a single frequency sine wave creates no ringing artifact as shown in Figure 6. 

 

gibbs_sine.pngFigure 6: The type of signal also affects if the Gibbs phenomenon occurs. For example, the same set of frequency filters will not affect a sine wave as no frequency content is truncated, and thus the Gibbs phenomenon does not occur.In this case, a sine wave (same frequency as square wave of Figure 5 ) is not affected by using the same set of lowpass filters that caused the Gibbs phenomenon to occur in the square wave.

 

When the Gibbs effect does occur, the amplitude of the ringing is also partially controlled by the shape of the anti-aliasing low pass filter used during acquisition.

 

Amplitude of Ringing: Filter Shape 

 

An anti-aliasing filter is often used when measuring a signal.  The shape of this low pass anti-aliasing filter is important in determining the amplitude of the ringing artifacts of the Gibbs phenomenon.  The sharper the filter, the greater the amplitude of the ringing.

 

In Figure 7, the shape of two different filters is overlaid: a Bessel and a Butterworth filter.  The Bessel filter has a more gradual rolloff than the Butterworth filter. 

 

bessel_vs_butterworth.pngFigure 7: Bessel and Butterworth have same 3 dB rolloff point (where they cross), but the Bessel rolloff is more gradual than the Butterworth.

In Figure 8, the amplitude of the ringing artifact of a Bessel filtered square wave is less than the amplitude ringing of the same square wave using a Butterworth filter. In fact, the Butterworth filter is designed to have fixed overshoot.

 

bessel_vs_butter_square.pngFigure 8: The more gradual Bessel filter does not introduce time domain ringing artifacts, unlike the sharper Butterworth filter.

The sharper the filter, the more likely the Gibbs phenomenon is seen in the time domain data. Why is this? Ultimately, it is related to the time domain shape of the filter.

 

When an inverse Fourier Transform is performed on the frequency domain filter shape, the result is called an ‘impulse’ or ‘time step function’. The broader the filter in the frequency domain, the shorter the duration impulse in the time domain. The ringing artifacts are more pronounced as impulses become shorter.  This is why the steeper Butterworth filter has ringing, while the more gradual Bessel does not.

 

Simcenter Testlab and the Gibbs Phenomenon

 

The Gibbs phenomenon can be greatly reduced in Simcenter Testlab (formerly called LMS Test.lab) by applying a low pass Bessel filter to the incoming signal. In the Channel Setup worksheet, choose “Tools -> Channel Setup Visibility…” (Figure 9) to activate the low pass filter field settings.

 channel_setup2.pngFigure 9: To add the low pass filter setting fields to channel setup, choose “Tools -> Add-ins -> Channel Setup Visibility

In the Channel Setup visibility menu, highlight the ‘LPCutoff’, ‘LPFilterCharacteristics’, ‘LPFilterOn’, and ‘LPFilterOrder’ fields and press “Add” to make them visible as shown in Figure 10.

 

channel_setup1.pngFigure 10: Channel Setup Visibility menu.

Four new columns are added to the channel information in the Channel Setup worksheet as shown in Figure 11.

 

lowpass_channel_settings.pngFigure 11: Lowpass (LP) filter selections in Channel Setup.

 The following parameters can be set:

  • LPCutOff – The Bessel has a gradual rolloff. A good “rule of thumb” is to set the lowpass filter cutoff ten times less than the sampling frequency of the measurement.
  • LPFilterCharacteristics – Either Bessel or Butterworth can be selected. Bessel is less steep.
  • LPFilterOn – Turns the filter ON or OFF.
  • LPFilterOrder – The lower the order, the more gradual the filter rolloff. Second order is the most gradual/lowest setting available.

With a second order Bessel applied to an incoming square wave, the Gibbs phenomena is reduced greatly as shown in Figure 12.

 

square_wave_final.pngFigure 12: Square wave with default anti-aliasing filter (red) and additional Bessel filter (green)The default anti-aliasing filter is very steep. By applying the additional Bessel filter to the incoming signal, the Gibbs phenomenon is mitigated.

 

These low pass filter settings are available in the Simcenter SCADAS VB8-E series and V8-E series of cards. Be sure to check the product information sheet or your local support if you have questions about your hardware.

 

Conclusions

 

The effects of the Gibbs phenomenon on measured time signals can be greatly reduced or eliminated. A few things to keep in mind:

  • The Gibbs phenomenon, which creates ringing artifacts with unpredictable overshoot, can be controlled through the proper usage of anti-aliasing low pass filters during data acquisition. Both the filter frequency truncation and filter shape affect the amplitude and duration of the Gibbs phenomenon.
  • The Gibbs phenomenon is observable with transient signals that contain a step or discontinuity. The phenomenon does not occur on sinusoidal signals. It only occurs when a part of the frequency content of a signal is removed during data acquisition.
  • The Gibbs phenomenon primary effects are in the time domain, not frequency domain.

 Questions?  Email peter.schaldenbrand@siemens.com or contact Siemens PLM GTAC support.

 

About Gibbsgibbs_photo.png 

 

Josiah Willard Gibbs (1839-1903) was an American scientist from Yale University.  In 1899, he published observations about the undershoot and the overshoot of a step function in the Fourier series, which later became known as the “Gibbs Phenomenon”.  Later it was discovered that this had already been described by an English mathematician, Henry Wilbraham, in 1848.  Despite this revelation, the phenomenon continued to be named after Gibbs.

 

Gibbs was awarded the first American doctorate in Engineering. He specialized in mathematical physics, and his work affected diverse fields from chemical thermodynamics to physical optics.

 

Addendum #1: Pre-Ringing

 

When viewing signals with the Gibbs phenomenon, sometimes a ringing artifact can be observed before the discontinuity or step in the signal, other times the ringing artifact might only be seen after the discontinuity as shown in Figure 13.

 

pre_ring.pngFigure 13: Square wave on left exhibits no pre-ringing, square wave on right has pre-ringing.

This can be caused by analog versus digital filters. When a digital filter is used in a “zero phase” mode, where the data is fed through the filter forward and backward, pre-ringing is created.  With an analog filter, pre-ringing is not possible.

 

Addendum #2: Analog to Digital Converters

 

Sometimes there is confusion about a Successive Approximation Register (SAR) versus Sigma-Delta analog to digital converters and Gibbs phenomenon. Many Sigma-Delta converters have sharp anti-aliasing filters which prevent alias errors.  But these sharp filters are not inherent to Sigma-Delta converters, any type of filter can be used. 

 

Using a filter with a gradual roll-off with any analog to digital converter reduces or eliminates the Gibbs phenomenon.  The effect of the filter should not be confused with the type of analog to digital converter. In fact, a lowpass filter can even be used after the acquisition on a digitized signal containing Gibbs to remove the phenomenon.

 

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