f
^
t=∑
n
c
n
ei
w
o
nt
f
^
t
n
c
n
w
o
n
t
(1)-
In the late 1800's many machines, (mechanical) computers, were built for computing FS coefficients
c
n
c
n
and resynthesizing
f
^
t
f
^
t
.
-
Albert Michelson's 1898 machine could compute
c
o
c
o
up to
c
±
79
c
±
79
and resynthesize:
f
^
N
t=∑
n
=−NN
c
n
ei
w
o
nt
f
^
N
t
n
N
N
c
n
w
o
n
t
-
Clearly hoped that
f
N
t→ft
f
N
t
f
t
as
N→∞
N
.
-
It performed well in almost all test except those involving
discontinuous functions
f
N
t
f
N
t
for various
N
N:
- Even as
N→∞
N
the height of the ripple
never decreases below 9% of pulse height.
-
But energy in ripple
→0
0
since we know
limit
N
N→∞∥
f
^
N
−f∥=0
N
N
f
^
N
f
0
. ie: (zoom in)
∥
f
N
−f∥→0
f
N
f
0
→ area inside ripples
→0
0
as
N→∞
N
, but area
→0
0
by width
→0
0
. In fact, height
→
nonzero constant (9% of jump height).
-
This is called Gibb's phenomena after the
dude who first explained it in 1899. (read Lathi pp 205-206
for more history)
- More technically, Gibb's
phenomena is a case of nonuniform
convergence.
