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. 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. 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. Figure Channel Setup Visibility menu. Four new columns are added to the channel information in the Channel Setup worksheet as shown in Figure With a second order Bessel applied to an incoming square wave, the overshoot apparent Gibbs phenomenon is reduced greatly as shown in Figure Figure 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. Be sure to check the product information sheet or your local support if you have questions about your hardware. 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:. Email peter. Later it was discovered that this had already been described by an English mathematician, Henry Wilbraham, in 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.
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 Other times the ringing artifact might only be seen after the transient as shown on the left side of Figure Figure Square wave on left exhibits no pre-ringing, square wave on right has pre-ringing.
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. Log in to post to this feed. Oct 5, 1. Topics Show Topics. Siemens Digital Industries Software. Search the Community. Sign in to ask the community. But what does the Fourier series do near a discontinuity?
If a function is piecewise continuous, then the Fourier series at a jump discontinuity converges to the average of the limits from the left and from the right at that point. What the Fourier series does on either side of the discontinuity is more interesting. You can see high-frequency oscillations on either side. The series will overshoot on the high side of the jump and undershoot on the low side of the jump.
The exact proportion, in the limit, is given by the Wilbraham-Gibbs constant. Gibbs phenomenon is usually demonstrated with examples that have a single discontinuity at the end of their period, such as a square wave or a saw tooth wave. But Gibbs phenomenon occurs at every discontinuity, wherever located, no matter how many there are. As we consider even more coefficients, we notice that the ripples narrow, but do not shorten.
As we approach an infinite number of coefficients, this rippling still does not go away. This is when we apply the idea of almost everywhere. This means that their width is approaching zero and we can assert that the reconstruction is exactly the original except at the points of discontinuity. Since the Dirichlet conditions assert that there may only be a finite number of discontinuities, we can conclude that the principle of almost everywhere is met.
This phenomenon is a specific case of nonuniform convergence. Below we will use the square wave, along with its Fourier Series representation, and show several figures that reveal this phenomenon more mathematically.
The Fourier series representation of a square signal below says that the left and right sides are "equal. The fact that the square wave's Fourier series requires more terms for a given representation accuracy is not important. In particular, at each step-change in the square wave, the Fourier series exhibits a peak followed by rapid oscillations.
As more terms are added to the series, the oscillations seem to become more rapid and smaller, but the peaks are not decreasing. Consider this mathematical question intuitively: Can a discontinuous function, like the square wave, be expressed as a sum, even an infinite one, of continuous ones?
One should at least be suspicious, and in fact, it can't be thus expressed. This issue brought Fourier much criticism from the French Academy of Science Laplace, Legendre, and Lagrange comprised the review committee for several years after its presentation on It was not resolved for also a century, and its resolution is interesting and important to understand from a practical viewpoint.
The extraneous peaks in the square wave's Fourier series never disappear; they are termed Gibb's phenomenon after the American physicist Josiah Willard Gibbs.
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