Digital Signal Processing Reference
In-Depth Information
N
the Euclidean distance is defined by the expression:
In particular, for
N−
1
d
(
x
,
y
)=
0
|x
n
−y
n
|
2
(3.33)
l
g
r
,
y
i
d
.
,
©
,
L
s
n
=
whereas for
L
2
[
−
π
π
]
we have
,
π
x
2
d
(
x
,
y
)=
(
t
)
−
y
(
t
)
dt
(3.34)
−
π
In the practice of signal processing, the Euclidean distance is referred to as
the
root mean square error
;
(4)
this is a global, quantitative goodness-of-fit
measure when trying to approximate signal
y
with
x
.
Incidentally, there are other types of distance measures which do not
rely on a notion of inner product; for example in
N
we could define
(
)=
d
x
,
y
max
N
|
x
n
−
y
n
|
(3.35)
0
≤
n
<
This distance is based on the supremum norm and is usually indicated by
x
−
y
∞
; however, it can be shown that there is no inner product fromwhich
this norm can be derived and therefore no Hilbert space can be constructed
where
·
∞
is the natural norm. Nonetheless, this normwill reappear later,
in the context of optimal filter design.
3.3
Subspaces, Bases, Projections
Now that we have defined the properties of Hilbert space, it is only nat-
ural to start looking at the consequent inner
structure
of such a space. The
best way to do so is by introducing the concept of
basis
. You can think of
a basis as the “skeleton” of a vector space, i.e. a structure which holds ev-
erything together; yet, this skeleton is flexible and we can twist it, stretch
it and rotate it in order to highlight some particular structure of the space
and facilitate access to particular information that we may be seeking. All
this is accomplished by a linear transformation called a
change of basis
;to
anticipate the topic of the next Chapter, we will see shortly that the Fourier
transform is an instance of basis change.
Sometimes, we are interested in exploring in more detail a specific sub-
set of a given vector space; this is accomplished via the concept of
subspace
.
A subspace is, as the name implies, a restricted region of the global space,
(4)
Almost always, the square distance is considered instead; its name is then the
mean
square error
,orMSE.
