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)

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=∫−∞∞hτes(t−τ)d
τ
=∫−∞∞hτeste−(sτ)d
τ
=est∫−∞∞hτe−(sτ)d
τ
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.