Digital Signal Processing Reference
In-Depth Information
The Space of Bandlimited Signals.
The class of
N
-bandlimited func-
tions of finite energy forms a Hilbert space, with the inner product defined
in (9.1). An orthogonal basis for the space of
Ω
N
-bandlimited functions can
easily be obtained from the prototypical bandlimited function, the sinc; in-
deed, consider the family:
Ω
l
g
r
,
y
i
d
.
,
©
,
L
s
sinc
t −nT
s
T
s
,
ϕ
(
n
)
(
t
)=
n
∈
(9.19)
ϕ
(
n
)
(
)=
ϕ
(
0
)
(
where, once again,
T
s
)
so that each basis function is simply a translated version of the prototype
basis function
=
π/
Ω
N
. Notethatwehave
t
t
−
nT
s
ϕ
(
0
)
. Orthogonality can easily be proved as follows: first of all,
because of the symmetry of the sinc function and the time-invariance of the
convolution, we can write
ϕ
(
n
)
(
)
=
ϕ
(
0
)
(
)
ϕ
(
m
)
(
ϕ
(
0
)
(
t
)
,
t
t
−
nT
s
)
,
t
−
mT
s
=
ϕ
(
0
)
(
)
ϕ
(
0
)
(
nT
s
−
t
)
,
mT
s
−
t
=(
ϕ
(
0
)
∗
ϕ
(
0
)
)
(
T
s
n
−
m
)
We can now apply the convolution theorem and (9.9) to obtain:
∞
π
Ω
rect
Ω
Ω
ϕ
(
n
)
(
)
=
2
1
2
ϕ
(
m
)
(
e
j
Ω(
n−m
)
T
s
d
t
)
,
t
Ω
π
N
N
−∞
Ω
N
=
π
2
e
j
Ω(
n−m
)
T
s
d
Ω
2
N
Ω
−
Ω
N
⎧
⎨
⎩
π
Ω
N
=
T
s
if
n
=
m
=
0
if
n
=
m
so that
ϕ
(
n
)
(
)
n∈
is orthogonal with normalization factor
t
Ω
/π
(or, equiv-
N
alently, 1
/
T
s
).
In order to show that the space of
Ω
N
-bandlimited functions is indeed
a Hilbert space, we should also prove that the space is complete. This is a
more delicate notion to show
(7)
and here it will simply be assumed.
(7)
Completeness of the sinc basis can be proven as a consequence of the completeness of
the Fourier series in the continuous-time domain.
