Digital Signal Processing Reference
In-Depth Information
i
n
u
k
2
d
k
−
λ
+
u
k
2
d
k
−
λ
f
(λ)
=
1
+
k
=
1
k
=
i
+
1
=
1
+
Φ(λ)
+
Ψ(λ)
(13.15)
where coefficients
c
1
,
.In[
18
]the
above mentioned secular function has been represented similarly using the system's
matrix as:
c
2
and
c
3
are computed by interpolating
f
(λ)
1
ρ
uu
T
D
+
(13.16)
s
d
k
−
λ
+
S
d
k
+
1
−
λ
h
(λ)
=
ρ
+
r
+
R
+
(13.17)
s
d
k
−
λ
S
d
k
+
1
where
r
+
is approximated to
Ψ(λ)
and
R
+
is approximated to
Φ(λ)
.
−
λ
The initial guess
(λ
0
)
plays a very significant role in finding solution for secular
d
k
+
d
k
+
1
2
equation. For
k
<
n
the initial guess is calculated as
and for
k
=
n
the initial
uu
T
. So with
guess is
d
n
+
ρ
λ
0
, we calculate the value of
h
(λ)
, and if the rational
function is greater than zero, the root is closer to
d
i
and if it is less than zero, the
root is closer to
d
i
+
1
. So according to the placement of the root, if it is close to
d
i
then we shift each of
λ
0
,
d
i
and
d
i
+
1
by subtracting
d
i
from them. With the new
value of the initial guess
(λ
1
)
, we solve the quadratic equation formed by the two
rational functions, and
η
(iterative correction) is found. The desired root is found by
computing
. We know that the weights over the poles play a major role in the
solution of secular equation. One of the circumstances is associated with weights.
In order to overcome this, fixed weight method was implemented with the structure
same as middle way method. By combining
middle way method
and
fixed weight
method
, a hybrid scheme was designed to make iteration faster. In this scheme, the
function interpolation was carried out with
d
k
,
d
k
+
1
and
d
k
−
1
. The rational function
for hybrid scheme is given below.
(λ
1
+
η)
u
k
2
d
k
−
λ
+
s
d
k
−
1
−
λ
+
S
d
k
+
1
−
λ
h
(λ)
=
c
+
(13.18)
where
c
R
. Once the eigenvalues are obtained, eigenvectors can be
computed using the following equation
=
ρ
+
r
+
)
−
1
u
(λ
I
−
D
x
=
(13.19)
)
−
1
u
(λ
I
−
D
Unfortunately, the above equation does not produce accurate eigenvectors
[
13
,
22
] and the following equation is used for modifying
u
vectors where
u
vectors
are updated and used in Eq. (
13.19
)
