Before looking at this module, hopefully you have become fully
convinced of the fact that any periodic function,
ft
f
t
, can be represented as a sum of complex sinusoids. If you
are not, then try looking back at eigen-stuff in a nutshell or eigenfunctions of LTI
systems. We have shown that we can represent a signal
as the sum of exponentials through the Fourier Series equations below:
ft=∑n
c
n
ⅇⅈ
ω
0
nt
f
t
n
c
n
ω
0
n
t
(1)
c
n
=1T∫0Tftⅇ-ⅈ
ω
0
ntdt
c
n
1
T
t
T
0
f
t
ω
0
n
t
(2)
Joseph
Fourier insisted that these equations were true,
but could not prove it. Lagrange publicly ridiculed
Fourier, and said that only continuous functions can be
represented by
Equation 1 (indeed he
proved that
Equation 1 holds for
continuous-time functions). However, we know now that
the real truth lies in between Fourier and Lagrange's
positions.
Formulating our question mathematically, let
f
N
′t=∑n=-NN
c
n
ⅇⅈ
ω
0
nt
f
N
t
n
N
N
c
n
ω
0
n
t
where
c
n
c
n
equals the Fourier coefficients of
ft
f
t
(see Equation 2).
f
N
′t
f
N
t
is a "partial reconstruction" of
ft
f
t
using the first
2N+1
2
N
1
Fourier coefficients.
f
N
′t
f
N
t
approximates
ft
f
t
, with the approximation getting better and better as
NN gets large. Therefore, we can
think of the set
∀N,N=
01…
:
f
N
′t
N
N
0
1
…
f
N
t
as a sequence of functions, each one
approximating
ft
f
t
better than the one before.
The question is, does this sequence converge to
ft
f
t
? Does
f
N
′t→ft
f
N
t
f
t
as
N→∞
N
? We will try to answer this question by thinking
about convergence in two different ways:
-
Looking at the energy of the error signal:
e
N
t=ft−
f
N
′t
e
N
t
f
t
f
N
t
-
Looking at
limN→∞
f
N
′t
N
f
N
t
at each point and comparing to
ft
f
t
.
Let
e
N
t
e
N
t
be the difference (i.e. error) between the signal
ft
f
t
and its partial reconstruction
f
N
′t
f
N
t
e
N
t=ft−
f
N
′t
e
N
t
f
t
f
N
t
(3)
If
ft∈
L
2
0T
f
t
L
2
0
T
(finite energy), then the energy of
e
N
t→0
e
N
t
0
as
N→∞
N
is
∫0T|
e
N
t|2dt=∫0Tft−
f
N
′t2dt→0
t
T
0
e
N
t
2
t
T
0
f
t
f
N
t
2
0
(4)
We can prove this equation using Parseval's relation:
limN→∞∫0T|ft−
f
N
′t|2dt=limN→∞∑N=-∞∞|
ℱ
n
ft
−
ℱ
n
f
N
′t
|2=limN→∞∑|n|>N|
c
n
|2=0
N
t
T
0
f
t
f
N
t
2
N
N
ℱ
n
f
t
ℱ
n
f
N
t
2
N
n
n
N
c
n
2
0
where the last equation before zero is the tail sum of the
Fourier Series, which approaches zero because
ft∈
L
2
0T
f
t
L
2
0
T
.
Since physical systems respond to energy, the
Fourier Series provides an adequate representation for all
ft∈
L
2
0T
f
t
L
2
0
T
equaling finite energy over one period.
The fact that
e
N
→0
e
N
0
says nothing about
ft
f
t
and
limN→∞
f
N
′t
N
f
N
t
being equal at a given point. Take the
two functions graphed below for example:
Given these two functions,
ft
f
t
and
gt
g
t
, then we can see that for all t
t,
ft≠gt
f
t
g
t
, but
∫0T|ft−gt|2dt=0
t
T
0
f
t
g
t
2
0
From this we can see the following relationships:
energy convergence≠pointwise convergence
energy convergence
pointwise convergence
pointwise convergence⇒
convergence in L
2
0T
pointwise convergence
convergence in L
2
0
T
However, the reverse of the above statement does not hold true.
It turns out that if
ft
f
t
has a discontinuity (as can be seen in figure
of
gt
g
t
above) at
t
0
t
0
, then
f
t
0
≠limN→∞
f
N
′
t
0
f
t
0
N
f
N
t
0
But as long as
ft
f
t
meets some other fairly mild conditions, then
f
t
′
=limN→∞
f
N
′
t
′
f
t
′
N
f
N
t
′
if
ft
f
t
is continuous at
t=
t
′
t
t
′
.
