Digital Signal Processing Reference
In-Depth Information
THEOREM 3.5 (BARTLETT).
Using the absolute summability hypotheses of the
covariances and cumulants, we have:
∞
∑
(
)
() ()
() (
)
ˆ
ˆ
lim
N cmc
,
=
cncnm
+ −
cov
xx
xx
xx
xx
N
→∞
n
=−∞
(
) (
)
+
cnmcn
nmn
+
−
[3.17]
xx
xx
(
)
+
+
,,
cum
The previous theorem is known as Bartlett's formula. We note that if the process
is Gaussian, the 4
th
order cumulants are zero and only the first two terms are
retained. Let
T
() ()
( )
c
ˆ
=
⎡
c
ˆ
0
c
ˆ
1
…
c
ˆ
M
be the estimated vector of
⎤
⎣
⎦
xx
xx
xx
T
() ()
( )
c
=
⎡
c
0
c
1
…
c
M
constructed from [3.13].
⎤
⎣
⎦
xx
xx
xx
ˆ
cc
as
(
)
∼
( )
RESULT 3.2.
If the process is linear,
0
Σ
where the covariance
is given by Bartlett's formula; see [BRO 91, p. 228]. This result can also be applied
if we suppose that the process is Gaussian [POR 94, p. 105].
N
The results given until now concern only random wide sense stationary processes
(possibly with linear process hypothesis). Similar results can be demonstrated for
other classes of signals, particularly sine wave signals, which are commonly used in
spectral analysis. Let us consider the following signal:
p
∑
()
(
) () () ()
x n
=
A
sin 2
π
nf
+
φ
+
b n
=
s n
+
b n
[3.18]
k
k
k
k
=
1
where the {φ
k
} are assumed to be independent and uniformly distributed over [0,2π[
and
b
(
n
) is a sequence of random variables independent and identically distributed,
with zero mean and variance σ
2
. The covariance function is then:
p
1
∑
()
2
(
)
2
() ()
2
()
cm
=
A
cos 2
π
mf
+
σδ
mcm
=
+
σδ
m
[3.19]
xx
k
k
ss
2
k
=
1
We can thus show that (see for example [STO 89b]):
ˆ
cc
as
(
)
∼
( )
RESULT 3.3.
If x
(
n
)
verifies [3.18],
N
0
Σ
where the element
( )
m
of
Σ
may be written as:
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