6 ms·

Gibbs Phenomenon

40 points by luu 5y ago

myWindoonn 5y ago
See also https://en.wikipedia.org/wiki/Countersteering https://en.wikipedia.org/wiki/Countersteering
- bananabiscuit 5y ago
  I think this is only superficially similar but actually very distinct. In countersteering, no oscillation is required, even for the most sudden direction changes, only a single movement in the opposite direction is all it takes.
opwieurposiu 5y ago
Those sharp corners are dangerous. At my old work we got a batch of control interface cards for some industrial equipment that suddenly stopped working. A chip receiving ~100khz serial data through a short cable would randomly stop working and get hot. Power cycle would fix it. Classic signs of latchup. Turns out the chip sending the data had gone through a process shrink that caused it to switch much faster and produce and put much sharper corners. These sharper corners would ring so hard at the other end they would latch up the receiver chip. We added some termination to mitigate the ring and the problem was fixed.
jphoward 5y ago
This is a beautiful example of how bad Wikipedia is within mathematics and a lot of the physical sciences. An introduction such as this would be unacceptable on a page about medical sciences: "...is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. The nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself" I have a PhD in machine learning and also an interest in magnetic resonance imaging. Sometimes Gibbs phenomenon leads to an artefact (called Gibbs artefact) that we see in our images of the heart when we are doing a certain sequence called stress perfusion cardiac MRI. I am not an idiot, and yet I actually cannot understand the first paragraph of this page. Mathematics articles on Wikipedia so frequently read in a way that is only interpretable by mathematicians or physicists. This is not the point of Wikipedia! You don't even have to pick an advanced topic - the Taylor Series is taught in high school maths in the UK, and look at the introduction to its Wikpedia article, it's ridiculous! Articles about relatively esoteric topics in the biological sciences are almost always much better, and at least try to first introduce the topic in a way the average reader can understand, e.g. https://en.wikipedia.org/wiki/Supraventricular_tachycardia https://en.wikipedia.org/wiki/Supraventricular_tachycardia
- jakeva 5y ago
  I'm just a lowly software engineer with a M.Sc. in Computing, but I didn't find that first paragraph particularly difficult to understand. The images further support it. Naturally, to understand it you need to some understanding of the terms used but that's why they each link to their own wikipedia pages.
  - dstr 5y ago
    OC’s background seems to be cardiology. I recall having read the proof of gibbs in second year undergrad. Generally I think undergraduate education in engineering or physics goes into a lot more details than one may admit. I suspect this is also true for undergraduate education in other fields for a person from engineering when they go to an interdisciplinary field.
- mhh__ 5y ago
  To be fair to Wikipedia, Taylor Series are taught to students in the UK at A-level but only for about (thanks to the horrors of A-level mathematics) an A4 page worth of material. As for the section you quote - I saw what it was trying to say in my head (graphically-ish) before I actually parsed the sentence properly, that might be why it's remained poor. It's not a bad explanation it just isn't long enough.
  - GreenWatermelon 5y ago
    it's less of an explanation and more of a summary in my opinion.
- dwohnitmok 5y ago
  Am I suffering from knowledge blindness? The Taylor series article on Wikipedia doesn't seem to have a particularly impenetrable introduction. There's a couple phrases here and there that read a little bit awkwardly, but I think all the concepts here are covered in most high school mathematics curricula, except for the very end that briefly alludes to some introductory analysis concepts (the "analytic" portion). (Note that some of the concepts such as convergence here have links to additional articles) > In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after Brook Taylor, who introduced them in 1715. > If zero is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. > The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk). https://en.wikipedia.org/wiki/Taylor_series https://en.wikipedia.org/wiki/Taylor_series
korethr 5y ago
This is a phenomenon that happens in real-world analog signals, too. In this Xiph.org video in which Monty explains various bit about analog and digital signals[1], in a section starting at 17:20, he demonstrates how the Gibbs phenomenon happens in band-limited signals. 1. https://www.xiph.org/video/vid2.shtml https://www.xiph.org/video/vid2.shtml
- LeoPanthera 5y ago
  This video taught me more about digital audio in 20 minutes than days of reading Wikipedia would do. I recommend it to anyone even vaguely interested in the subject.
phreeza 5y ago
Went down the rabbit hole to the initial paper describing this phenomenon [1] from 1848. Turns out the paper immediately preceding it in the journal is "The calculus of logic" by George Boole! I wonder what the density of papers that will be of any practical relevance in 170 years is in today's journals... [1] https://www.google.com/books?id=JrQ4AAAAMAAJ&pg=PA198#v=onepage&q&f=false https://www.google.com/books?id=JrQ4AAAAMAAJ&pg=PA198#v=onep...
kevinventullo 5y ago
I wonder if there are analogues of the Gibbs Phenomenon when you use a basis for L^2 other than the Fourier basis.
lxe 5y ago
Wish there was an explorable explanation for this concept that's not as verbose for the layman like me. It's still unclear to me about how the mathematical explanation (approximating a continuous function with finite series) explain the physically observable "ringing artifacts" and square wave jitters.

Gibbs Phenomenon

40 points by luu 5y ago

myWindoonn 5y ago

See also https://en.wikipedia.org/wiki/Countersteering https://en.wikipedia.org/wiki/Countersteering

bananabiscuit 5y ago
I think this is only superficially similar but actually very distinct. In countersteering, no oscillation is required, even for the most sudden direction changes, only a single movement in the opposite direction is all it takes.

opwieurposiu 5y ago

Those sharp corners are dangerous. At my old work we got a batch of control interface cards for some industrial equipment that suddenly stopped working. A chip receiving ~100khz serial data through a short cable would randomly stop working and get hot. Power cycle would fix it. Classic signs of latchup. Turns out the chip sending the data had gone through a process shrink that caused it to switch much faster and produce and put much sharper corners. These sharper corners would ring so hard at the other end they would latch up the receiver chip. We added some termination to mitigate the ring and the problem was fixed.

jphoward 5y ago

This is a beautiful example of how bad Wikipedia is within mathematics and a lot of the physical sciences. An introduction such as this would be unacceptable on a page about medical sciences: "...is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. The nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself" I have a PhD in machine learning and also an interest in magnetic resonance imaging. Sometimes Gibbs phenomenon leads to an artefact (called Gibbs artefact) that we see in our images of the heart when we are doing a certain sequence called stress perfusion cardiac MRI. I am not an idiot, and yet I actually cannot understand the first paragraph of this page. Mathematics articles on Wikipedia so frequently read in a way that is only interpretable by mathematicians or physicists. This is not the point of Wikipedia! You don't even have to pick an advanced topic - the Taylor Series is taught in high school maths in the UK, and look at the introduction to its Wikpedia article, it's ridiculous! Articles about relatively esoteric topics in the biological sciences are almost always much better, and at least try to first introduce the topic in a way the average reader can understand, e.g. https://en.wikipedia.org/wiki/Supraventricular_tachycardia https://en.wikipedia.org/wiki/Supraventricular_tachycardia

jakeva 5y ago
I'm just a lowly software engineer with a M.Sc. in Computing, but I didn't find that first paragraph particularly difficult to understand. The images further support it. Naturally, to understand it you need to some understanding of the terms used but that's why they each link to their own wikipedia pages.
- dstr 5y ago
  OC’s background seems to be cardiology. I recall having read the proof of gibbs in second year undergrad. Generally I think undergraduate education in engineering or physics goes into a lot more details than one may admit. I suspect this is also true for undergraduate education in other fields for a person from engineering when they go to an interdisciplinary field.
mhh__ 5y ago
To be fair to Wikipedia, Taylor Series are taught to students in the UK at A-level but only for about (thanks to the horrors of A-level mathematics) an A4 page worth of material. As for the section you quote - I saw what it was trying to say in my head (graphically-ish) before I actually parsed the sentence properly, that might be why it's remained poor. It's not a bad explanation it just isn't long enough.
- GreenWatermelon 5y ago
  it's less of an explanation and more of a summary in my opinion.
dwohnitmok 5y ago
Am I suffering from knowledge blindness? The Taylor series article on Wikipedia doesn't seem to have a particularly impenetrable introduction. There's a couple phrases here and there that read a little bit awkwardly, but I think all the concepts here are covered in most high school mathematics curricula, except for the very end that briefly alludes to some introductory analysis concepts (the "analytic" portion). (Note that some of the concepts such as convergence here have links to additional articles) > In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after Brook Taylor, who introduced them in 1715. > If zero is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. > The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk). https://en.wikipedia.org/wiki/Taylor_series https://en.wikipedia.org/wiki/Taylor_series

korethr 5y ago

This is a phenomenon that happens in real-world analog signals, too. In this Xiph.org video in which Monty explains various bit about analog and digital signals[1], in a section starting at 17:20, he demonstrates how the Gibbs phenomenon happens in band-limited signals. 1. https://www.xiph.org/video/vid2.shtml https://www.xiph.org/video/vid2.shtml

LeoPanthera 5y ago
This video taught me more about digital audio in 20 minutes than days of reading Wikipedia would do. I recommend it to anyone even vaguely interested in the subject.

phreeza 5y ago

Went down the rabbit hole to the initial paper describing this phenomenon [1] from 1848. Turns out the paper immediately preceding it in the journal is "The calculus of logic" by George Boole! I wonder what the density of papers that will be of any practical relevance in 170 years is in today's journals... [1] https://www.google.com/books?id=JrQ4AAAAMAAJ&pg=PA198#v=onepage&q&f=false https://www.google.com/books?id=JrQ4AAAAMAAJ&pg=PA198#v=onep...

kevinventullo 5y ago

I wonder if there are analogues of the Gibbs Phenomenon when you use a basis for L^2 other than the Fourier basis.

lxe 5y ago

Wish there was an explorable explanation for this concept that's not as verbose for the layman like me. It's still unclear to me about how the mathematical explanation (approximating a continuous function with finite series) explain the physically observable "ringing artifacts" and square wave jitters.