Digital Signal Processing Reference
In-Depth Information
For this class of sequences it can also be proved that the convergence of
X
M
(
is continuous. While absolute
summability is a sufficient condition, it can be shown that the sum in (4.15)
is convergent also for all
square-summable
sequences, i.e. for sequences
whose energy is finite; this is very important to us with respect to the dis-
cussion in Section 3.4.3 where we defined the Hilbert space
e
j
ω
)
to
X
(
e
j
ω
)
is uniform and that
X
(
e
j
ω
)
l
g
r
,
y
i
d
.
,
©
,
L
s
.Inthe
case of square summability only, however, the convergence of (4.15) is no
longer uniform but takes place only in the mean-square sense, i.e.
(
)
2
π
X
M
2
e
j
ω
)
−
e
j
ω
)
(
(
ω
=
lim
M
X
d
0
(4.18)
→∞
−
π
Convergence in the mean square sense implies that, while the total energy
of the error signal becomes zero, the pointwise values of the partial summay
never approach the values of the limit. Onemanifestation of this odd behav-
ior is called the
Gibbs phenomenon
, which has important consequences in
our approach to filter design, as we will see later. Furthermore, in the case
of square-summable sequences,
X
e
j
ω
)
(
is no longer guaranteed to be con-
tinuous.
As an example, consider the sequence:
1 r
−
N
≤
n
≤
N
x
[
n
]=
(4.19)
0 rwie
Its DTFT can be computed as the sum
(4)
N
e
j
ω
)=
e
−j
ω
n
(
X
n
=
−
N
N
N
e
j
ω
n
e
−j
ω
n
=
+
n
=
1
n
=
0
1
−
e
−j
ω
(
N
+
1
)
1
−
e
j
ω
(
N
+
1
)
=
e
−j
ω
+
e
j
ω
−
1
1
−
1
−
e
−j
ω
(
N
+
1
)
e
j
ω/
2
e
j
ω
(
N
+
1
)
e
−j
ω/
2
e
j
ω/
2
1
−
e
−j
ω/
2
1
−
=
e
−j
ω/
2
+
e
j
ω/
2
−
1
−
−
1
2
)
1
2
)
e
j
ω
(
N
+
e
−j
ω
(
N
+
−
=
e
j
ω/
2
e
−j
ω/
2
−
sin
N
2
1
ω
+
=
sin
(
ω/
2
)
N
1
−
x
N
+
1
x
n
(4)
Remember that
=
.
1
−
x
n
=
0
