Digital Signal Processing Reference
In-Depth Information
If we associate it with the first eigenvector, we obtain:
ρ
(0)
2
ρ
(0)
tan(
θ
)=
1
/c
=
1
/c
+
√
Δ
λ
−
c
−
If we associate it with the second eigenvector, we obtain:
tan(
θ
)=
ρ
(0)
c
2
ρ
(0)
−
√
Δ
In the two cases, if we make use of the relation:
λ
=
−
c
−
1
/c
2tan(
θ
)
tan(2
θ
)=
tan
2
(
θ
)
1
−
we obtain:
θ
=
1
2
arctan
2
ρ
(0)
c
−
1
/c
the angle between
−
π/
4 and +
π/
4.When
θ
is positive, that is, when
ρ
(0) and
c
−
1
/c
have the same sign:
s
(
n
)=cos(
θ
)
x
1
(
n
)+sin(
θ
)
x
2
(
n
)
When
θ
is negative, we must choose:
s
(
n
)=
−
sin(
θ
)
x
1
(
n
)+cos(
θ
)
x
2
(
n
)
.
It is always more straightforward to calculate the dominant signal using the first
relation by choosing:
θ
opt
=m d
1
2
arctan
2
ρ
(0)
,
π
2
[9.6]
c −
1
/c
When
θ
is chosen as
θ
=
θ
opt
, relation [9.5] is simply the Karhunen Loeve
transform applied to the vector
X
(
n
):
S
(
n
)
S
(
n
)
=
V
t
X
1
(
n
)
X
2
(
n
)
We h ave :
S
(
n
)
S
(
n
)
[
S
(
n
)
S
(
n
)]
=
V
t
Γ
V
=
σ
X
1
σ
X
2
Λ
E
The two signals
S
(
n
) and
S
(
n
), therefore, have powers
σ
S
=
σ
X
1
σ
X
2
λ
1
and
σ
S
=
σ
X
1
σ
X
2
λ
2
respectively and these are not correlated. We can note that the
relationship between the two eigenvalues, that is between the two powers, is given by:
−
√
Δ
−
√
μ
1+
√
μ
with
μ
=1+
4
ρ
2
(0)
λ
λ
1
=
c
+1
/c
c
+1
/c
+
√
Δ
=
1
4
(
c
+1
/c
)
2
−
[9.7]
