Digital Signal Processing Reference
In-Depth Information
The filter frequency f
c
is close to the frequency of the exponential f
exp
Given the fact that the term (
f
c
-
f
exp
) is weak, the smoothing factor
D
(
f
c
-
f
exp
),
contrary to the preceding case, plays a predominant role. The impulse response of
equation [7.12] generally has two predominant lobes. This response has a maximum
which is greater than 1 and which is no longer at the filter frequency but at a
frequency ranged between
f
exp
and
f
c
[DUR 00]. The filter is no longer a narrow band
filter.
The preceding approximations were validated with simulations [DUR 00]. They
lead to a key conclusion. The filter generated by the minimum variance method does
not lead to narrow band filters, which was often claimed in the literature [KAY 88,
LAG 86]. Actually, the constraints imposed by the minimum variance are not those
of a narrow band filter. This characteristic has important consequences when an
equivalent bandwidth should be calculated (see section 7.5.1).
Figure 7.5 illustrates the MV filters calculated when analyzing a mixed signal,
sum of a narrow band filter and of a wide band filter. Figure 7.5 represented the
signal power estimated by the MV method at order 12. It was necessary to calculate
a frequency response at all frequencies of this spectrum. The other five figures are
some examples of frequency responses at specific frequencies in relation to the
studied signal. It is clear that only the frequency response calculated at the sinusoid
frequency (Figure 7.5f) is similar to that of a narrow band filter. All the others
(Figures 7.5b to 7.5e) have lobes whose maximal amplitude can reach 25, which is
greatly superior to 1. Figures 7.5c and 7.5e are examples where, at the filter
frequency, the lobe is not at its maximum.
7.2.2.
Probability density of the MV estimator
Knowing the probability density of the MV estimator is the best way
for
accessing all the estimator moments. We will consider a more general signal than in
the preceding paragraph. Let a complex signal
x
(
k
) be the
sum of an ordinary
determinist signal
d
(
k
)
and of a Gaussian centered random noise
b
(
k
). The vector
(
(
)
T
)
(
)
(
)
B
=
bk
−
1,
…
,
bk M
−
is
normally
distributed
where
N
0,
R
b
(
)
H
RB B
represents the covariance matrix
M
×
M
of the noise
b
(
k
)
.
The
vector
X
defined by
X
T
= (
x
(
k
- 1),...,
x
(
k
-
M
)) is therefore normally distributed
(
=
E
⋅
b
(
)
)
T
Dd
(
)
(
)
DR
,
where
=
k
−
1,
…
,
d
k
−
M
.
N
b
Search WWH ::
Custom Search