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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.
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.
Figure 1 shows several Fourier series approximations of the square wave using a varied number of terms, denoted by $K$ :
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.
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.
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.
We can approximate a function by re-synthesizing using only some of the Fourier coefficients(truncating the F.S.)
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