2.1. INTRODUCTION 9
Euclidean norm. Given,
x=
x1
.
.
.
xn
,
the Euclidean (or 2-norm) of x, denoted by kxk, is defined by,
kxk= n
X
i=1
|xi|!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 valueds ignal,
x(t)=
x
1
(t)
.
.
.
x
n
(t)
.
As well as the issue of the spatial norm, we nowhave the issue of a time norm. In the
theory given here,we concentrate on the 2-norm in the time domain. In otherwords,
kxi(t)k=Z∞
−∞
|xi(t)|2dt1/2
.
This is simply the energy of the signal. This norm is sometimes denoted by a subscript
of two, i.e. kxi(t)k2. Parseval’s relationship means that we can also express this norm in
the Laplace domain as follows,
kxi(s)k=1
2πZ∞
−∞
|xi(ω)|2dω1/2
.