The most commonly encountered notion of the energy of a signal defined on Z[a,b]Z[a,b] is the l2l2 norm defined by the square root of the sumation of the square of the signal, for which the notation


xn


2
=
∑
n
=
a
b

xn

2
1
/
2
.


x
n


2
=
∑
n
=
a
b

x
n

2
1
/
2
.
(22)However, this idea can be generalized through definition of the lplp norm, which is given by


xn


p
=
∑
n
=
a
b

xn

p
1
/
p


x
n


p
=
∑
n
=
a
b

x
n

p
1
/
p
(23)for all 1≤p<∞1≤p<∞. Because of the behavior of this expression as pp approaches ∞∞, we furthermore define


xn


∞
=
sup
n
∈
Z
[
a
,
b
]

xn

,


x
n


∞
=
sup
n
∈
Z
[
a
,
b
]

x
n

,
(24)which is the least upper bound of xn

x
n
. In this case, this least upper bound is simply the maximum value of xn

x
n
. A signal xn
x
n
is said to belong to the vector space lp(Z[a,b])lp(Z[a,b]) if xn
p<∞
x
n
p<∞. The periodic extension of such a signal would have infinite energy but finite power.
For example, consider the function defined on Z[5,3]Z[5,3] by
xn
=
n

5
<
n
<
3
0
otherwise
.
x
n
=
n

5
<
n
<
3
0
otherwise
.
(25)The l1l1 norm is


xn


1
=
∑
n
=

5
3

xn

=
∑

5
3

n

=
21
.


x
n


1
=
∑
n
=

5
3

x
n

=
∑

5
3

n

=
21
.
(26)The l2l2 norm is


xn


2
=
∑

5
3

xn

2
1
/
2
=
∑

5
3

n

2
d
t
1
/
2
≈
8
.
31


x
n


2
=
∑

5
3

x
n

2
1
/
2
=
∑

5
3

n

2
d
t
1
/
2
≈
8
.
31
(27)The l∞l∞ norm is


xn


∞
=
sup
n
∈
Z
[

5
,
3
]

xn

=
5
.


x
n


∞
=
sup
n
∈
Z
[

5
,
3
]

x
n

=
5
.
(28)
"My introduction to signal processing course at Rice University."