4.4 Fourier series approximation of signals  (Page 2/2)

More than just decaying slowly, Fourier series approximation shown in [link] exhibits interesting behavior.

Fourier series approximation of a square wave

Consider this mathematical question intuitively: Can a discontinuous function, like the square wave, be expressed as asum, even an infinite one, of continuous signals? One should at least be suspicious, and in fact, it can't be thusexpressed. This issue brought Fourier much criticism from the French Academy of Science (Laplace,Lagrange, Monge and LaCroix comprised the review committee) for several years after its presentation on 1807. It was notresolved for almost 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. 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. However, mathematicians later in the nineteenth century showed that the rms error ofthe Fourier series was always zero. $$\lim_{K\to }K\to$$∞ rms ε K 0 What this means is that the error between a signal and its Fourier series approximation may not be zero, but that its rmsvalue will be zero! It is through the eyes of the rms value that we redefine equality: The usual definition of equality iscalled pointwise equality : Two signals $s_1(t)$ , $s_2(t)$ are said to be equal pointwise if $s_1(t)=s_2(t)$ for all values of $t$ . A new definition of equality is mean-square equality : Two signals are said to be equal in the mean square if $\mathrm{rms}(s_1-s_2)=0$ . For Fourier series, Gibb's phenomenon peaks have finite height and zero width. The error differs from zero onlyat isolated points—whenever the periodic signal contains discontinuities—and equals about 9% of the size of thediscontinuity. The value of a function at a finite set of points does not affect its integral. This effect underlies the reasonwhy defining the value of a discontinuous function, like we refrained from doing in defining the step function , at its discontinuity is meaningless. Whatever you pick for a value hasno practical relevance for either the signal's spectrum or for how a system responds to the signal. The Fourier series value"at" the discontinuity is the average of the values on either side of the jump.