Digital Signal Processing Reference
In-Depth Information
will encounter later. It is generally better to envisage a calculation of the LS and
TLS solutions via the singular value decomposition (SVD) of the matrix
ˆ
R
,
particularly when the system is not well conditioned, because the SVD proves a
greater numeric stability. The LS and TLS solutions calculated by the SVD are:
ˆ
ˆ
ˆ
ˆ
H
ˆ
(
)
RU V
Σ
Σ
σσ
σσ
ˆ
≥
ˆ
ˆ
ˆ
=
=
diag
,,
x
ppp
p
k
k
+
1
p
(
)
( )
−
1
( )
−
1
ˆ
ˆ
ˆ
H
ˆ
ˆ
ˆ
1
1
LSYW
aV
=−
Σ
Ur
Σ
=
diag
,
…
p
,
pp
px
p
ˆ
ˆ
σ
σ
1
[6.16]
ˆ
v
a
⎡⎤
=
p
+
1
TLSYW
⎢⎥
(
)
1
v
p
+
1
⎣⎦
p
+
1
where
is the eigenvector associated to the smallest eigenvalue of the matrix
V
p
+
H
ˆ
ˆ
ˆ
p
+
⎡ ⎤ ⎡ ⎤
⎣ ⎦ ⎣ ⎦
Rr
, i.e. the (
p
+ 1)
th
column vector of the matrix
ˆ
ˆ
V
so that
Rr
x
x
x
x
1
ˆ
ˆ
ˆ
ˆ
H
⎡ ⎤
=
Rr
ˆ
U
V
.
Σ
⎣ ⎦
x
x
p
+++
1
p
1
p
1
There exists an extension of the Levinson-Durbin algorithm for solving LSYW,
it is what we call the least-squares
lattice
[MAK 77, PRO 92]. The interest of this
algorithm is that it is time and order recursive, which makes it possible to implement
parameter estimation adaptive procedures.
We can very easily notice that these methods are biased because the correlation
at 0
th
lag makes the power of the white noise
2
σ
intervene. The approached value of
this bias for
(6.14) is:
(
)
1
−
ˆ
2
a
ˆ
=−
R
σ
I
r
ˆ
+
x
−
u
u
x
−
u
22
ˆ
−
21
ˆ
−
aa
ˆ
≈+
σ
ˆ
uxuxu
R r
=−
a
σ
R a
[6.17]
−−
uxu
−
where
x - u
is the noiseless signal. Certain methods [KAY 80, SAK 79] exploit this
relation in order to try to eliminate this bias, but these require an estimation of
2
σ
.
The simplest solution for obtaining a non-biased estimation of the
AR
parameters
consists of not making the zero lag correlation intervene in the estimation:
()
()
(
)
⎡
γ
ˆ
p
γ
ˆ
1
⎤⎡
γ
ˆ
p
+
1
⎤
xx
xx
xx
⎢
⎥⎢
⎥
a
ˆ
=−
−
1
[6.18]
⎢
⎥⎢
⎥
(
)
( )
()
⎢
⎥⎢
⎥
γ
ˆ
21
p
−
γ
ˆ
p
γ
ˆ
2
p
⎣
⎦⎣
⎦
xx
xx
xx
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