Digital Signal Processing Reference
In-Depth Information
so that the norm of
f
(
t
)
is
π
f
2
=
(
)
f
t
dt
(3.31)
l
g
r
,
y
i
d
.
,
©
,
L
s
−
π
to belong to
L
2
[
−
π
π
]
it must be
For
f
(
t
)
,
f
<
∞
.
3.2.3
Inner Products and Distances
The inner product is a fundamental tool in a vector space since it allows us
to introduce a notion of
distance
between vectors. The key intuition about
this is a typical instance in which a geometric construct helps us to gener-
alize a basic idea to much more abstract scenarios. Indeed, take the simple
Euclidean space
N
and a given vector
x
;foranyvector
y
N
the inner
∈
product
is the measure of the
orthogonal projection
of
y
over
x
.We
know that the orthogonal projection defines the point on
x
which is closest
to
y
and therefore this indicates how well we can approximate
y
by a simple
scaling of
x
. To illustrate this, it should be noted that
〈
x
,
y
〉
〈
x
,
y
〉
=
x
y
cos
θ
where
θ
is the angle between the two vectors (you can work out the expres-
2
to easily convince yourself of this; the result generalizes to any
other dimension). Clearly, if the vectors are orthogonal, the cosine is zero
and no approximation is possible. Since the inner product is dependent on
the angular separation between the vectors, it represents a first rough mea-
sure of similarity between
x
and
y
; in broad terms, it provides a measure of
the difference in
shape
between vectors.
In the context of signal processing, this is particularly relevant sincemost
of the times, we are interested in the difference in shape” between signals.
As we have said before, discrete-time signals
are
vectors; the computation of
their inner product will assume different names according to the processing
context in which we find ourselves: it will be called
filtering
,whenweare
trying to approximate or modify a signal or it will be called
correlation
when
we are trying to detect one particular signal amongst many. Yet, in all cases,
it will still be an inner product, i.e. a
qualitative
measure of similarity be-
tween vectors. In particular, the concept of orthogonality between signals
implies that the signals are perfectly distinguishable or, in other words, that
their shape is completely different.
The need for a
quantitative
measure of similarity in some applications
calls for the introduction of the Euclidean distance, which is derived from
the inner product as
sion in
1
/
2
d
(
x
,
y
)=
〈
x
−
y
,
x
−
y
〉
=
x
−
y
(3.32)
