Digital Signal Processing Reference
In-Depth Information
()
(
m
)
−
m
However, if
() ()
∑
f
N f
+
N
≡
0mod 1 ,
N
∆
f
=
1
thus:
2
k
1
k
2
k
q
=
1
q
(
)
(
)
(
)
()
()
( )
cum
df
N df
,
N
→
S f
k
1
k
2
x
k
1
m >
2: using a similar reasoning, we can verify that the cumulants of order
strictly higher than 2 always tend towards 0.
The cumulants of the Fourier transform tend towards those of a complex process
which is jointly Gaussian and independent at the same time. Once the asymptotic
distribution of the Fourier transform is obtained, that of the periodogram is obtained
directly by noting that:
2
2
{
}
{
}
()
()
()
I
f
=
Re
d
f
+
Im
d
f
RESULT 3.5.
Under the same hypotheses as used previously, the I
(
f
k
(
N
))
are
asymptotically independent and distributed according to
()
2
2
Sf
χ
/2
w e
χ
xk
2
2
designates a chi squared distribution with two degrees of freedom.
{
}
(
)
2
,
()
χ
E
distribution
tends towards
S
x
(
f
k
)
but
var
(
I
(
f
k
(
N
)))
also tends towards
S
x
(
f
k
)
2
which shows that
the periodogram
is not a consistent estimator of the power spectral density.
However, the
independence property of the periodogram bias is the basis of numerous methods
generating a consistent estimator of the power spectral density by local averaging of
the periodogram [BRI 81].
According to properties of the
If N
k
3.4. Analysis of estimators based on
()
ˆ
xx
cm
The theoretical results of section 3.2 are the basis from which we can analyze the
statistical performance of a large number of estimators used in spectral analysis. In
fact, the majority of the spectral analysis techniques are based on the use of a set of
estimated covariances
M
{
}
()
ˆ
cm
=
or, in other words the estimator may be written
xx
m
0
as:
ˆ
()
ˆ
θ
=
g c
We note that the vector
c
considered here consists of estimates of the
covariance function. Nevertheless, the approach recommended below can easily be
generalized in the case where, for example,
c
contains estimates of moments of
higher order, as soon as we know the asymptotic properties of the vector
c
.
Generally, there are two cases:
1) either the function
g
is sufficiently simple (linear for example) and we can
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