Simcenter Testing Solutions The Gibbs Phenomenon

Direct YouTube link: https://youtu.be/s16Yiaf34JY
 

The Gibbs Phenomenon

To describe a signal with a sharp transient in the time domain requires infinite frequency content. In practice, it is not possible to sample infinite frequency content. The truncation of higher frequency content causes a time domain ringing artifact which is often referred to as the “Gibbs phenomenon” or "overshoot".

In this article, this phenomenon will be explained more in-depth.  Terms used in the explanation are defined here:

• Signal: The term signal will refer to an acquired measurement, where a voltage is captured as function of time. The voltage relates to some engineering quantity like acceleration or strain.
• Function: The term function will refer to a mathematical relationship between a "x" and "y" quantity.  In this article, the "x" quantity will be time while "y" will refer to an engineering unit/voltage.  
• Gibbs and Overshoot: From a technical point of view, these are two separate items, However, one helps understand the other.  Gibbs can be seen in the finite reconstruction of a function via the Fourier series. Overshoot is caused by filters on measured signals.  In both cases, a portion of the higher frequency content is attenuated or lost, which can cause a ringing effect in the time domain.

The examples in this article often use a square wave, but real world filtered signals can exhibit overshoot.  These 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 causes of ringing in the time domain, it is important to understand the Fourier Transform and Fourier Series. According to the Fourier Transform, every periodic function is a summation of a unique set of sinusoidal waves. Some examples are shown in Figure 1.

Figure 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 their time domain signals.

This sudden transient change, and the associated infinite frequency content, is difficult to replicate with data acquisition systems which always have finite frequency measurement capabilities. Depending on their design, the filters involved in the acquisition process can cause overshoot, giving a similar ringing artifact in measured signals at the sharp transient in the time domain (Figure 2). This overshoot is similar to the Gibbs phenomenon observed when creating a function from a finite set of its Fourier Series.
 

Figure 2: Certain acquisition filters, depending on how they attenuate high frequency content, overshoot when measuring a square wave signal.

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 and on the filter design.

This ringing artifact is created when describing a signal with less than perfect frequency information. In practice, this can be caused by:

• Truncation of Frequency Content – Certain functions (sharp transients or signals with discontinuities) are composed of an infinite number of harmonic frequencies. If the Fourier Series used to describe the original function is not infinite, the reconstruction shows a ringing artifact called the Gibbs phenomenon.
• Filter Shape - Filters (for example, anti-aliasing) are often used in measurement devices, which can overshoot and introduce ringing effects.  These overshoots are related to the sharpness of the filter and the high frequency content of the measured signals.

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 because of overshoot introduced by the ringing artifact.

Causes and Control of Overshoot

The ringing artifact 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.

Figure 3: Overshoot 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 effected by the type of filter used.

In the following examples, a square wave, which has infinite frequency content, will be used to illustrate overshoot.

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.

Figure 4: A square wave function 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, a measured square wave is shown with all harmonics, harmonics lowpass filtered at 2000 Hz, harmonics lowpass filtered at 750 Hz. 

Figure 5: Using a sharp filter: 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 filtered, 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 attenuated. 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 a ringing artifact with overshoot occurs. If the signal has no frequency content that is filtered, the ringing (i.e., apparent 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.

Figure 6: The type of signal also affects if ringing occurs. For example, the same filter will not create a ringing artifact when used to measure a single sine wave, but will create a ringing artifact when measuring a square wave which has higher frequency content.

For a high frequency content signal, the shape of the anti-aliasing low pass filter used during acquisition will then determine if ringing artifacts occur.

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 if any ringing artifacts (i.e., apparent Gibbs phenomenon) will appear in the measured signal. 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 is is less sharp than the Butterworth filter.

Figure 7: Bessel and Butterworth have same 3 dB rolloff point (where they cross), but the Butterworth is more sharp than the Bessel filter

The Butterworth is considered sharper than the Bessel because it is flatter over a larger frequency range of the pass band than the Bessel.  In the stop band of the filter, the rolloff (or slope) of both filters is the same.

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.

Figure 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 overshoot (or apparent Gibbs phenomenon) is seen in the time domain data. Why is this? Ultimately, it is related to the time domain shape of the filter.

Simcenter Testlab and the Gibbs Phenomenon

Ringing artifacts 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.
 

Figure 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.

Figure 10: Channel Setup Visibility menu.

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

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

The following parameters can be set:

• LPCutOff – The LPCutOff is the frequency where the filter has attenuated the signal by 3 dB in amplitude. 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 sharp than the Butterworth.
• 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 overshoot (apparent Gibbs phenomenon) is reduced greatly as shown in Figure 12.

Figure 12: Square wave with default anti-aliasing filter (red) and additional Bessel filter (green)
 

The default anti-aliasing filter is very sharp and steep. By applying the additional Bessel filter to the incoming signal, the apparent 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 Gibbs phenomenon helps illustrate why sharp filters tend to overshoot in the presence of a signal with fast transients. Overshoot effects on measured time signals can be greatly reduced or eliminated. A few things to keep in mind:

• Ringing artifacts amplitude can be controlled through the proper usage of anti-aliasing low pass filters during data acquisition. Both the filter frequency cut-off and filter shape affect the amplitude and duration of the ringing artifact phenomenon.
• Overshoot is observable with transient signals that contain a step or discontinuity. It does not occur on sinusoidal signals. It only occurs when a part of the frequency content of a signal is attenuated by sharp filters during data acquisition.
• Overshoot primary effects are in the time domain, not frequency domain.

Questions? Email peter.schaldenbrand@siemens.com or contact Siemens Support Center.

 

About Gibbs

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 demonstrating the so-called Gibbs phenomenon, sometimes a ringing artifact can also be observed before the transient or step in the signal as shown on the right side of Figure 13.  Other times the ringing artifact might only be seen after the transient as shown on the left side of Figure 13.

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

If overshoots occur in a signal, pre-ringing can occur as follows:

• Analog filters never exhibit pre-ringing.
• Some Digital linear FIR filters exhibit pre-ringing.  Digital IIR filters do not.
• Pre-ringing is possible for any digital filter being used to post process acquired digital data.  It can be created when using a "zero phase" filter where the data is passed forward and backward (offline, not realtime) through the filter to remove phase shift.

More about digital filters in this knowledge article: "Introduction to Filters: FIR versus IIR".

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 smooth gradual filter with any analog to digital converter reduces or eliminates the "observed" 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 ringing artifacts to remove the overshoots.

Related Digital Signal Processing links:

• On-Demand Webinar: Digital Signal Processing Fundamentals
• Index of Testing Knowledge Articles
• Simcenter SCADAS Mobile and Recorder Hardware
• Gain, Range, Quantization
• Anti-Aliasing Filters
• Acquisition Filters and Overshoot
• Overloads
• Averaging Types: What's the difference?
• What is Fourier Transform?
• Time-Frequency Analysis: Wavelets
• Spectrum versus Autopower
• Autopower Function...Demystified!
• Power Spectral Density
• Shock Response Spectrum (SRS)
• Windows and Leakage
• Window Types
• Window correction factors
• Exponential Window Correction Factors
• RMS Calculations
• The Gibbs Phenomenon
• Introduction to Filters: FIR and IIR