We saw Parseval/Plancherel in the context of orthonormal basis expansions.
This begs the question, do FF and F-1F-1 just take signals and compute
their representation in another basis?
Let's look at F-1:L2[-π,π]→ℓ2(Z)F-1:L2[-π,π]→ℓ2(Z) first:
F
-
1
(
X
(
e
j
w
)
)
=
1
2
π
∫
-
π
π
X
(
e
j
ω
)
e
j
ω
n
d
ω
.
F
-
1
(
X
(
e
j
w
)
)
=
1
2
π
∫
-
π
π
X
(
e
j
ω
)
e
j
ω
n
d
ω
.
(1)Recall that X(ejω)X(ejω) is really just a function of ωω, so if we replace ωω
with tt, we get
F
-
1
(
X
(
t
)
)
=
1
2
π
∫
-
π
π
X
(
t
)
e
j
t
n
d
t
.
F
-
1
(
X
(
t
)
)
=
1
2
π
∫
-
π
π
X
(
t
)
e
j
t
n
d
t
.
(2)Does this seem familiar? If X(t)X(t) is a periodic function defined on [-π,π][-π,π],
then F-1(X(t))F-1(X(t)) is just computing (up to a reversal of the indicies) the continuous-time Fourier series of X(t)X(t)!
We said before that the Fourier series is a representation in an orthobasis,
the sequence of coefficients that we get are just the weights of the
different basis elements. Thus we have →x[n]=FF-1(X(t))→x[n]=FF-1(X(t)) and
X
(
t
)
=
∑
n
=
-
∞
∞
x
[
n
]
e
-
j
t
n
2
π
.
X
(
t
)
=
∑
n
=
-
∞
∞
x
[
n
]
e
-
j
t
n
2
π
.
(3)What about FF? In this case we are taking an x∈ℓ2(Z)x∈ℓ2(Z) and mapping it to an X∈L2[-π,π]X∈L2[-π,π]. XX represents an infinite
set of numbers, and when we weight the functions ejωnejωn by X(ω)X(ω) and sum
them all up, we get back the original signal
x
[
n
]
=
∫
-
π
π
X
(
ω
)
e
j
ω
n
2
π
d
ω
.
x
[
n
]
=
∫
-
π
π
X
(
ω
)
e
j
ω
n
2
π
d
ω
.
(4)Unfortunately, ejωn2π=∞ejωn2π=∞ (≠1≠1) so technically, we can't really
think of this as a change of basis.
However, as a unitary transformation, FF has everything we would ever want
in a basis and more: We can represent any x∈ℓ2(Z)x∈ℓ2(Z) using {ejωn}ω∈[-π,π]{ejωn}ω∈[-π,π], and since it is unitary,
we have Parseval and Plancherel Theorems as well. On top of that, we
already showed that the set of vectors {ejωn}ω∈[-π,π]{ejωn}ω∈[-π,π] are eigenvectors of LSI systems – if this really were a basis, it would
be called an eigenbasis.
Eigenbases are useful because once we represent a signal using an
eigenbasis, to compute the output of a system we just need to know what it
does to its eigenvectors (i.e., its eigenvalues). For an LSI system, H(ejω)H(ejω)
represents a set of eigenvalues that provide a complete
characterization of the system.
