Digital Signal Processing Reference
In-Depth Information
u
i
T
~
i
-
1
2
θ
i
-
1
θ
i
~
i
2
FIGURE 1.1
An interpretation of the energy-conservation relation (Equation 1.10) by means of an analogy
with Snell's law of optics.
w
i
,
u
i
Let
θ
i
denote the acute angle between the column vectors
{
}
. Likewise,
w
i
−
1
,
u
i
let
θ
1
denote the acute angle between
{
}
. Then
i
−
e
a
(
2
2
cos
2
e
p
(
2
2
cos
2
i
)
=
u
i
·
w
i
−
1
·
(θ
i
−
1
)
,
and
i
)
=
u
i
·
w
i
·
(θ
i
).
Substituting into Equation 1.86 and collecting terms we find that it reduces to
2
sin
2
2
sin
2
(θ
)
=
(θ
).
w
i
−
1
w
i
(1.87)
i
−
1
i
Equality 1.87 resembles a famous result in optics, known as Snell's law,
which relates the refraction indices of two media with the sines of the incident
and refracted rays of light, i.e.,
η
θ
=
η
θ
1
sin
2
sin
2
,
1
where
θ
2
are the angles of incidence and refraction, respectively; both
angles are measured relative to the direction that is orthogonal to the surface
separating both media. This analogy suggests that we can relate the operation
of an adaptive filter, at each iteration, to that of a fictitious ray traveling from
one medium to another. The magnitudes
θ
1
and
w
i
−
1
and
w
i
play the role of
refraction indices of the media, while
play the role of the incidence
and refraction angles of the ray. Alternatively, we can interpret the result
(Equation 1.87) as shown in Figure 1.1. An incident vector of norm
{
θ
1
,
θ
}
i
−
i
w
i
−
1
1
with respect to
u
i
, while
impinges on the separation layer at an angle
θ
i
−
a refracted vector of norm
w
i
leaves the layer at an angle
θ
i
, also with
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