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2.1. INTRODUCTION 9
Euclidean norm. Given,
x =
x
1
.
.
.
x
n
,
the Euclidean (or 2-norm) of x, denoted by x, is defined by,
x =
n
i=1
|x
i
|
1/2
.
Many other norms are also options; more detail on the easily calculated norms can be
found in the on-line help for the norm function. The term spatial-norm is often applied
when we are looking at norms over the components of a vector.
Now consider a vector valued signal,
x(t)=
x
1
(t)
.
.
.
x
n
(t)
.
As well as the issue of the spatial norm, we now have the issue of a time norm. In the
theory given here, we concentrate on the 2-norm in the time domain. In otherwords,
x
i
(t) =
∞
−∞
|x
i
(t)|
2
dt
1/2
.
This is simply the energy of the signal. This norm is sometimes denoted by a subscript
of two, i.e. x
i
(t)
2
. Parseval’s relationship means that we can also express this norm in
the Laplace domain as follows,
x
i
(s) =
1
2π
∞
−∞
|x
i
(ω)|
2
dω
1/2
.