Digital Signal Processing Reference
In-Depth Information
8.3. Determination criteria of the number of complex sine waves
The theory tells us that the number of complex sine waves, given the order of the
model, is equal to the number of eigenvalues strictly higher than the smallest
eigenvalue of the covariance matrix of the observed signals. However, because we
have only a finite number of samples of the signal to analyze and thus only an
estimate of the covariance and also because the computer programs of eigenelement
research introduce numeric errors, the order of the model is difficult to determine. It
seems delicate to choose a threshold without risking not to detecting a frequency
contained in the signal. Actually, the performance of the MUSIC method is
conditioned by the determination of the number of estimated complex sine waves.
In order to remedy this limitation, criteria resulted from the information theory
were proposed in order to determine the order of a model [RIS 78, SCH 78]. We
simply give here the two most well known criteria (Akaike, MDL). They are based
on the likelihood function of the observations
1
.
AIC criterion (Akaike Information Criterion).
Akaike [AKA 73] proposed this
criterion to determine the order of an AR model. It consists in minimizing the
following quantity in relation to the supposed number
p
of complex sine waves:
M
ˆ
∏
λ
i
ip
=+
1
()
(
)
p
N
p
Mp
[8.27]
AIC
=−
log
+
2
−
(
)
Mp
−
M
∑
ˆ
1
λ
i
Mp
−
ip
=+
1
where
p
is an estimation of
P
,
N
the number of observations and
ˆ
λ
the eigenvalues
of
ˆ
x
Γ
arranged in descending order. For this criterion, the estimation of the noise
variance is given by the average of the
M - p
smallest eigenvalues of the covariance
matrix, given:
−
M
1
2
()
ˆ
ˆσ
p
=
λ
i
Mp
ip
=+
1
MDL criterion (Minimum Description Length).
This criterion is inspired from
[RIS 78, SCH 78] to determine the order of a model. The MDL criterion differs
from the AIC criterion by the second term, which is introduced:
M
ˆ
∏
λ
1
i
ip
=+
1
()
(
)
MDL
p
N
log
p
2
Mp
log
N
[8.28]
=−
+
−
)
(
Mp
−
2
M
∑
ˆ
1
λ
i
Mp
−
ip
=+
1
1
For more details see Chapter 6.
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