Digital Signal Processing Reference
In-Depth Information
Now consider the periodized version:
∞
f
(
t
)=
f
(
t −
2
nS
)
l
g
r
,
y
i
d
.
,
©
,
L
s
n
=
−∞
(
)=
|
>
>
since
f
T
0
the copies in the sum do not
overlap, as shown in Figure 9.15. If we compute the Fourier series expan-
sion (9.43) for the 2
S
-periodic
t
0for
|
t
T
0
, if we choose
S
f
(
t
)
we have
S
1
2
S
f
(
t
)
e
−j
(
π/
S
)
nt
dt
A
n
=
−
S
T
0
1
2
S
e
−j
(
π/
S
)
nt
dt
=
f
(
t
)
−
T
0
F
j
S
n
Sinceweassumedthat
f
=
(
t
)
is bandlimited, it is
Ω
0
S
π
A
n
=
0,
for
|
n
|
>
=
N
0
and therefore we can write the reconstruction formula (9.42):
N
0
f
A
n
e
j
(
π/
S
)
nt
(
t
)=
n
=
−
N
0
Now consider the complex-valued polynomial of degree 2
N
0
+
1
N
0
A
n
z
n
(
)=
P
z
n
=
−
N
0
obviously
P
e
j
(
π/
S
)
t
=
f
but we also know that
f
(
t
)
(
t
)
is identically zero
over the
[
T
0
,2
S
−
T
0
]
interval (Fig. 9.15). Now, a finite-degree polynomial
P
has only a finite number of roots and therefore it cannot be identi-
cally zero over an interval unless it is zero everywhere (see also Example 6.2).
Hence, either
f
(
z
)
(
t
)=
0everywhereor
f
(
t
)
cannot be both bandlimited and
time-limited.
f
(
)=
t
0
−
2
S
T
0
2
S
Figure 9.15
Finite support function
f
(
t
)
(black) and non-overlapping periodiza-
tion (gray).
