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Below we will use the square wave, along with its Fourier Series representation, and show several figures that revealthis phenomenon more mathematically.

Square wave

The Fourier series representation of a square signal below says that the left and right sides are "equal." In order tounderstand Gibbs Phenomenon we will need to redefine the way we look at equality.

s t a 0 k 1 a k 2 k t T k 1 b k 2 k t T

Figure 1 shows several Fourier series approximations of the square wave using a varied number of terms, denoted by K :

Fourier series approximations of a square wave

Fourier series approximation to sq t . The number of terms in the Fourier sum is indicated in each plot, and the square wave is shown as adashed line over two periods.
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When comparing the square wave to its Fourier series representation in [link] , it is not clear that the two are equal. The fact that thesquare wave's Fourier series requires more terms for a given representation accuracy is not important. However, closeinspection of [link] does reveal a potential issue: Does the Fourier series reallyequal the square wave at all values of t ? In particular, at each step-change in the square wave, theFourier series exhibits a peak followed by rapid oscillations. As more terms are added to the series, theoscillations seem to become more rapid and smaller, but the peaks are not decreasing. Consider this mathematicalquestion intuitively: Can a discontinuous function, like the square wave, be expressed as a sum, even an infinite one, ofcontinuous 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 thereview committee) for several years after its presentation on 1807. It was not resolved for also a century, and itsresolution 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. They occur whenever the signal isdiscontinuous, and will always be present whenever the signal has jumps.

Redefine equality

Let's return to the question of equality; how can the equal sign in the definition of the Fourier series be justified? The partial answer is that pointwise--each and every value of t --equality is not guaranteed. What mathematicians later in the nineteenth century showed was that the rmserror of the Fourier series was always zero.

K rms ε K 0
What this means is that the difference between an actual signaland its Fourier series representation may not be zero, but the square of this quantity has zero integral! It is through the eyes of the rms value that we define equality:Two signals s 1 t , s 2 t are said to be equal in the mean square if rms s 1 s 2 0 . These signals are said to be equal pointwise if s 1 t s 2 t for all values of t . For Fourier series, Gibb's phenomenon peaks have finite height and zero width: Theerror differs from zero only at isolated points--whenever the periodic signal contains discontinuities--and equalsabout 9% of the size of the discontinuity. The value of a function at a finite set of points does not affect itsintegral. This effect underlies the reason why defining the value of a discontinuous function at its discontinuity ismeaningless. Whatever you pick for a value has no practical relevance for either the signal's spectrum or for how asystem responds to the signal. The Fourier series value "at" the discontinuity is the average of the values oneither side of the jump.

Visualizing gibb's phenomena

The following VI demonstrates the occurrence of Gibb's Phenomena. Note how the wiggles near the square pulse to the left remain even if you drastically increase the order of the approximation, even though they do become narrower. Also notice how the approximation of the smooth region in the middle is much better than that of the discontinuous region, especially at lower orders.

Interact (when online) with a Mathematica CDF demonstrating Gibbs Phenomena. To download, right click and save as .cdf.


We can approximate a function by re-synthesizing using only some of the Fourier coefficients(truncating the F.S.)

f N t n n N c n ω 0 n t
This approximation works well where f t is continuous, but not so well where f t is discontinuous. In the regions of discontinuity, we will always find Gibb's Phenomena, which never decrease below 9% of the height of the discontinuity, but become narrower and narrower as we add more terms.

Questions & Answers

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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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