Digital Signal Processing Reference
In-Depth Information
rapidly (depending on
N
)
k
ln (
N
). This criterion is named MDL
(Minimum
Description Length):
()
() ()
MDL
kN
=
1n
ˆ
+
kN
ln
[6.35]
ρ
k
This criterion is consistent and gives better results than AIC [WAX 85].
It would be tedious to present all the criteria that were developed; for more
information on this, see the following references: [BRO 85, BUR 85, FUC 88, PUK
88, YIN 87, WAX 88].
[WAX 85] expressed the AIC and MDL criteria depending on the eigenvalues of
the autocorrelation matrix
ˆ
R
:
⎛
1
⎞
⎛
p
⎞
pk
−
⎜
⎟
ˆ
∏
⎜
λ
⎟
⎜
⎟
⎜
t
⎟
⎝
⎠
tk
=+
1
() ( )
⎜
⎟
(
)
AIC k
=
N
p
−
k
ln
+
k
2
p
−
k
k
=
1,
,
p
−
1
[6.36]
⎜
p
⎟
ˆ
∑
1
λ
⎜
⎟
t
pk
tk
−
⎜
⎟
⎜
⎟
=+
1
⎝
⎠
⎛
1
⎞
⎛
p
⎞
pk
−
⎜
⎟
ˆ
∏
⎜
λ
⎟
⎜
⎟
⎜
t
⎟
1
⎝
⎠
tk
=+
1
() ( )
⎜
⎟
(
) ( )
MDL k
=
N
p
−
k
ln
+
k
2
p
−
k
ln
N
[6.37]
⎜
p
⎟
2
ˆ
∑
1
λ
⎜
⎟
t
pk
tk
−
⎜
⎟
⎜
⎟
=+
1
⎝
⎠
ˆ
λ are the ordered eigenvalues
(
)
ˆ
ˆ
ˆ
y
Rp
×
.
It is
possible to define these criteria according to the singular values ˆσ
of the matrix
(
of the matrix
(
)
where
λλ
+
≥
t
t
1
ˆ
ˆ
)
2
y
RMpM p
×
,
>
by replacing
λ σ
in the matrices [6.36] and [6.37]
by
ˆ
l
t
[HAY 89].
The increase in the number of lines of the matrix
ˆ
R
evidently makes an
improvement of the order estimation performances possible. We have compared the
two expressions [6.35] and [6.37] of the MDL
criterion on the following example.
We consider
N
= 100 samples of a signal made up of two sinusoids of identical
amplitudes of frequencies 0.1 and 0.2 and of one white noise. The dimension of the
matrix
ˆ
R
is (30 x 30). The simulations were carried out for signal-to-noise ratios of
10 dB and 0 dB and are presented in Figures 6.3 and 6.4.
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