Introduction
Hopefully you are familiar with the notion of the eigenvectors
of a "matrix system," if not they do a quick review of
eigen-stuff. We can
develop the same ideas for LTI systems acting on signals. A
linear time invariant
(LTI) system
ℋ
ℋ
operating on a continuous input
ft
f
t
to produce continuous time output
yt
y
t
ℋft=yt
ℋ
f
t
y
t
(1)
is mathematically analogous to an
NNx
NN
matrix
A
A
operating on a vector
x∈
ℂ
N
x
ℂ
N
to produce another vector
b∈
ℂ
N
b
ℂ
N
(see
Matrices and LTI
Systems
for an overview).
Ax=b
A
x
b
(2)
Just as an
eigenvector
of
A
A
is a
v∈
ℂ
N
v
ℂ
N
such that
Av=λv
A
v
λ
v
,
λ∈ℂ
λ
,
we can define an
eigenfunction (or
eigensignal) of an LTI system
ℋ
ℋ
to be a signal
ft
f
t
such that
∀λ,λ∈ℂ:ℋft=λft
λ
λ
ℋ
f
t
λ
f
t
(3)
Eigenfunctions are the simplest possible
signals for
ℋ
ℋ
to operate on: to calculate the output, we simply multiply the
input by a complex number
λ
λ.
Eigenfunctions of any LTI System
The class of LTI systems has a set of eigenfunctions in common:
the
complex exponentials
ⅇst
s
t
,
s∈ℂ
s
are eigenfunctions for
all LTI systems.
ℋⅇst=
λ
s
ⅇst
ℋ
s
t
λ
s
s
t
(4)
Note:
While
∀s,s∈ℂ:ⅇst
s
s
s
t
are always eigenfunctions of an LTI system, they are not
necessarily the only eigenfunctions.
We can prove
Equation 4
by expressing the output as a
convolution
of the input
ⅇst
s
t
and the
impulse response
ht
h
t
of
ℋ
ℋ:
ℋⅇst=∫-∞∞hτⅇst-τdτ=∫-∞∞hτⅇstⅇ-sτdτ=ⅇst∫-∞∞hτⅇ-sτdτ
ℋ
s
t
τ
h
τ
s
t
τ
τ
h
τ
s
t
s
τ
s
t
τ
h
τ
s
τ
(5)
Since the expression on the right hand side does not depend on
t
t,
it is a constant,
λ
s
λ
s
. Therefore
ℋⅇst=
λ
s
ⅇst
ℋ
s
t
λ
s
s
t
(6)
The eigenvalue
λ
s
λ
s
is a complex number that depends on the exponent
s
s
and, of course, the system
ℋ
ℋ.
To make these dependencies explicit, we will use the notation
Hs≡
λ
s
H
s
λ
s
.
Since the action of an LTI operator on its eigenfunctions
ⅇst
s
t
is easy to calculate and interpret, it is convenient to
represent an arbitrary signal
ft
f
t
as a linear combination of complex exponentials. The
Fourier series
gives us this representation for periodic continuous time
signals, while the (slightly more complicated)
Fourier transform
lets us expand arbitrary continuous time signals.
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