When comparing the square wave to its Fourier series
representation it is not clear that the two are equal. The fact
that the square wave's Fourier series requires more terms for a
given representation accuracy is not important. However, close
inspection of Figure 1 does reveal a
potential issue: Does the Fourier series really equal the square
wave at all values of
t
t
? In particular, at each step-change in the square wave, the
Fourier series exhibits a peak followed by rapid
oscillations. As more terms are added to the series, the
oscillations seem to become more rapid and smaller, but the
peaks are not decreasing. Consider this mathematical question
intuitively: Can a discontinuous function, like the square wave,
be expressed as a sum, even an infinite one, of continuous 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 the review
committee) for several years after its presentation on 1807. It
was not resolved for also 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 is
discontinuous, 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
t
--equality is not guaranteed. What
mathematicians later in the nineteenth century showed was that
the rms error of the Fourier series was always zero.
limK→∞rms
ε
K
=0
K
rms
ε
K
0
(1)
What this means is that the difference between an actual signal
and 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
1
t
,
s
2
t
s
2
t
are said to be equal in the
mean square if
rms
s
1
-
s
2
=0
rms
s
1
s
2
0
. These signals are said to be equal
pointwise if
s
1
t=
s
2
t
s
1
t
s
2
t
for all values of
t
t
. For Fourier series, Gibb's phenomenon peaks have finite height
and zero width: The error differs from zero only at isolated
points--whenever the periodic signal contains
discontinuities--and equals about 9% of the size of the
discontinuity. The value of a function at a finite set of points
does not affect its integral. This effect underlies the reason
why 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 has
no 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.
