Hello Sherman Shan Chen ,

I am not sure what article you read, but the term "Gibbs phenomenon" only refers to what happens when you try to express a periodic function with finite discontinuities in terms of its Fourier's series terms. The American mathematical physicist Josiah Willard Gibbs (1839-1903) showed that the Fourier series did not converge uniformily to the periodic function f(t) in the neighborhood of the finite discontinuities no matter how many terms you included.

Consider, for example, a periodic square wave or sawtooth wave. If you write the Fourier series for either of these two waveforms, you will note that at the finite jump, the Fourier series representation will produce an overshoot before and after the jump. And this overshoot does not vanish as you add more terms from the Fourier series; the overshoots just move closer to the discontinuity.

In the case of the RLC circuit excited by a step function (an aperiodic function), the ringing at the output represents the low pass filtering of the input signal, which removes the higher fequencies that woud normally be present in the Fourier transform spectrum of the output signal. When you remove the higher frequency components, the sharp, finite jump of the input step function is smeared out ( the jump takes a finite amount of time instead being instantaneous) and ringing appears in what would have been the flat potions of the step at the output. In addition, the output waveform is delayed with respect to the input waveform.

But perhaps the article you read applied a periodic square wave to the input of the RLC circuit. Is that how it decided that the output waveform was related to the Gibbs phenomenon? I think that the point of confusion is that even if the circuit connecting the input terminals to the output terminals were simply a piece of wire, the Gibbs phenomenon would still occur for the reasons mentioned in the first paragraph and it is not related to the bandwidth of the circuit.

I am not sure this answers your question, but perhaps it will do until someone else can provide a better answer.

Regards,

Tom Cuff

Hello Sherman Shan Chen ,

I forgot to mention that the attribution "Gibbs phenomenon" is, like so much in science and technology, a misattribution. This phenomenon was originally discovered by Henry Wilbraham in 1848 and then independently rediscovered by Josiah Williard Gibbs in 1898-1899. This oversight was announced to the English speaking world in October 1925 in two consecutive articles by C. N. Moore and H. S. Carslaw, respectively, in the Bulletin of the American Mathematical Society, Vol. 31, No. 8, although, it had already been known in Europe much earlier.

Regards,

Tom Cuff

Hello Sherman Shan Chen ,

I have added a complete derivation of the Gibbs phenomenon to my RG page, Method Gibbs Phenomenon

Regards,

Tom Cuff

As much as I know the term "Gibbs phenomenon" means a behavioral feature of the Fourier series transformation at a jump discontinuity. Please see https://mathworld.wolfram.com/GibbsPhenomenon.html.

Hello Sherman Shan Chen and Tahir Midhat Lazimov ,

I am surprised at how few responses this question received, even though it is a very good question. Until I read this question, I don't believe I ever thought about how easy it was to confuse the Gibbs phenomenon with bandwidth limitations (Gibbs-like phenomenon). But after consulting six textbooks in my library, I believe I can see how the confusion arises.

Most textbooks that mention the Gibbs phenomenon - and there are many that do not - present an incomplete (or sometimes erroneous) description, and only one textbook [1] in my sample even favored the reader with a derivation. None of the textbooks mention that the "Gibbs phenomenon" is a misnomer since it was actually first discovered by Wilbraham in 1848, a fact first mentioned to English speakers in 1925. Most of the books in this sample that mention the Gibbs phenomenon are electrical engineering textbooks [1, 2, 4, & 5], although, it is very easy to find EE textbooks that make no mention of this phenomenon, for example, the graduate level textbook on signal processing by Couch, [7]. One EE textbook [2] refers to a "Sir Willard Gibbs" even though there is no evidence that Josiah Willard Gibbs was knighted. Some of the books [3 & 6] leave the reader, especially this reader, confused about what they are actually saying, i.e., they seem to be implying that Gibbs phenomenon has something to do with bandwidth limiting.

Anyway, the student looking for a clear and complete discussion of the Gibbs phenomenon should be very wary and very persistant; the truth is out there, somewhere.

[1] George E. Owen, P. W. Keaton; Fundamentals of Electronics, Volume 1; Harper & Row; 1966; pp. 39-41.

[2] M. E. Van Valkenburg; Network Analysis 3rd Edition; Prentice Hall; 1974; pp. 468-470.

[3] F. Paul Carlson; Introduction to Applied Optics for Engineers; Academic Press; 1977; pp. 74-75.

[4] Ferrel G. Stremler; Introduction to Communications Systems; Addison-Wesley Publishing Company; 1977; pp. 62-64.

[5] Constantine A. Balanis; Antenna Theory, Analysis and Design; Harper & Row; 1982; p. 673.

[6] Barbara Burke Hubbard; The World Accoring to Wavelets; A. K. Peters; 1996; pp. 104-108.

[7] Leon W. Couch II; Digital and Analog Communication Systems, Second Edition; Macmillan Publishing Company; 1987.

Regards,

Tom Cuff

Thomas Cuff I think you are right.

First time I read on this phenomenon in the widely-known book of Andre Angot (Complements de Mathematiques, Paris, 1957). This is interesting that now I cannot remember any purely mathematical book describing the Gibbs phenomenon itself, with no concern to applications.

Regards

Hello Tahir Midhat Lazimov ,

You bring up a very good point: mathematics books seemingly do not discuss the subject of Gibbs phenomenon. I am inclined to agree with you because, when I checked, after reading your answer, I was only able to find this subject in two mathematics reference books [1 & 2], not mathematics textbooks:

[1] Glenn James, Robert C. James (editors); Mathematics Dictionary, Students Edition; D. Van Nostrand Company, Inc.; 1959; p. 176.

[2] Kiyosi Itô; Encyclopedic Dictionary of Mathematics, Volume 1, Second Edition; The MIT Press; 1987; p. 159 D.

Note, the entries in both these reference books are terse to the point of incomprehensibility. Looking through my library, I happened on a book [3] on Fourier integrals, which both discussed the Gibbs phenomenon and proivded a proof of the Gibbs phenomenon for an aperiodic function, the unit step function, which I mentioned in my first answer. It is the only book of which I aware that does this particular proof.

[3] Athanasios Papoulis; The Fourier Integral and its Applications; McGraw-Hill Book Company, Inc.; 1962; pp. 30-31.

The funny thing is that this book is not a mathematics textbook. It was part of the McGraw-Hill Electronic [Engineering] Sciences Series and it was written by a Professor, Papoulis, of Electrical Engineering at the Polytechnic Institute of Brooklyn. All of this seems to support your contention that the Gibbs phenomenon is not to be found in pure - as opposed to applied - mathematics textbooks.

It turns out, though, that most of the discussion of Gibbs phenomenon by pure mathematicians took place in journal articles not textbooks. For example, Maxime Bôcher, who coined the term 'Gibbs phenomenon' did so in 1906 in the following journal,

[4] Maxime Bôcher; Introduction to the theory of Fourier Series; Annals of Mathematics; Vol. 7 (2nd Series); No. 3; 1906; pp. 81-152, see in particular p. 129.

I have more to say about all of this, but it will have to wait for another day. Just remember, the more you look, the more you find.

Regards,

Tom Cuff

I always thought the Gibbs phenomenon was the spurious sidelobes at the base of the spectral line

Hello Mark Sitkowski ,

I would like to explore why you think the sidelobes of the magnitude plot of the FFT (Fast Fourier Transform) of the rectangular pulse (?) indicates the Gibbs phenomenon. So far, we have found examples of the mention of the Gibbs phenomenon mainly in EE textbooks, usually analog signal processing textbooks. But you have brought up an interesting point, namely, what about DSP (Digital Signal Processing) textbooks? I have chosen three DSP textbooks.

[1] William D. Stanley; Digital Signal Processing; Reston Publishing Company, Inc.; 1975.

[2] John G. Proakis, Dimitris G. Manolakis; Digital Signal Processing, Principles, Algorithms, and Applications, Third Edition; Prentice Hall; 1996; pp. 259 & 629.

[3] Sanjit K. Mitra; Digital Signal Processing, A Computer-Based Approach, Second Edition; McGraw-Hill/Irwin; 2001; pp. 122, 449-452, 461, & 471.

The book by Stanley, [1], has no entry for the term 'Gibbs phenomenon' in its Index section, which puts it in good company since not many EE textbooks mention this subject. The next two textbooks, [2-3], are unusual in two ways: 1) their respective Index sections both have an entry for 'Gibbs phenomenon', and 2) they both mention this term in two or more, widely separated parts of their respective textbooks. All the textbooks we have previously examined only mentioned the term 'Gibbs phenomenon' in one part of their textbook.

Both of these books, [2-3], have extensive Bibliography sections: pp. R1-R15 and pp. 837-854, respectively. But for all their citing of primary sources, neither book cites the two papers by Josiah Willard Gibbs nor the paper by Henry Wilbraham. For the purpose of full disclosure, I now present the citations to these three papers:

[4] J. Willard Gibbs; Fourier's series; Nature; Vol. 59; No. 1522; December 29, 1898; p. 200.

[5] Idem; Fourier's series; Nature; Vol. 59; No. 1539; April 27, 1899; p. 606.

[6] Henry Wilbraham; On a certain periodic function; The Cambridge and Dublin Mathematics Journal; Vol. 3; 1848; pp. 198-201. [Note, there are 4 pages of text and 1 plate. Warning, the Hathi Trust catalog is very confusing as it contains the following entries: V3 1848, V3-4 1841-1845, V3 1841-1843, & V3; choose the first entry: V3 1848.}

It should also be noted that textbooks [2-3] do not cite any derivative articles such as the two articles by Bôcher - one of which I cited in a previous answer - or the article by Hewitt and Hewitt:

[7] Maxime Bôcher; On Gibbs's [sic] phenomenon; Journal für reine und angewandt Mathematik [Journal for Pure and Applied Mathematics, a.k.a, Crelle's Journal]; Vol. 144; 1914; pp. 41-47.

[8] Edwin Hewitt, Robert E. Hewitt; The Gibbs-Wilbraham phenomenon; Archive for History of Exact Sciences; Vol. 21; No. 2; 1979; pp. 129-160.

In addition, [2-3] do not provide any mathematical proofs of Gibbs phenomenon. So exactly what is it they do provide to buttress their assertions? One example should suffice. According to p. 259 of [2], the oscillatory behavior of the partial series sum at the finite jump is given the moniker 'Gibbs phenomenon', and - wait for it - a "similar effect" appears when truncating the Fourier series. Notice, they say "similar effect" not same effect, i.e., it is Gibbs-like but not the Gibbs phenomenon. You might think that [3] says the same thing, but you would be wrong. [3] says in effect on p. 122 that the mean square convergence at the point of discontinuity is called the Gibbs phenomenon. In other words, these two books have different definitions. The definition in [3] actually follows from work by Dirichlet in 1829 and is not the same thing as the Gibbs phenomenon.

Now for my sermon. Both religion and science are "similar" in that they require faith. In science, the faith is manifest in our belief that the writers of textbooks have read the original source material and presented it faithfully. The difference between religion and science is that in science we can examine the original source materials for ourselves and correct our belief on that basis. In other words, we can ascertain the facts, which is not the same thing as truth. Or, we can simply sit in our assigned place in the pews and rely solely on faith. The choice is yours.

I had a lot more to say on the subject of the Gibbs phenomenon, but I am getting so little discussion from the followers of this question that I have decided to move on. I have simply gotten tired of doing all the heavy lifting by myself and I am, in any case, not an expert. Va con Dios.

Regard,

Tom Cuff