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Figure 5.7
Tensor of third-order moments.
order. There exists a wide variety of higher-order noise models but the simplest model
is
fourth-order white (and circular) noise
, which is defined as follows.
Definition 5.1.
A random vector
n
is called
fourth-order white
(and circular) if the only
nonzero moments up to fourth order are
E
(
n
i
n
∗
j
)
=
δ
ij
(5.168)
E
(
n
i
n
j
n
k
n
l
)
=
δ
ik
δ
jl
+
δ
il
δ
jk
+
K
δ
ijkl
,
(5.169)
where
δ
ijkl
=
1
if i
=
j
=
k
=
l and zero otherwise, and K is the kurtosis of the noise.
This means that all moments must have the same number of conjugated and non-
conjugated terms. This type of noise is “Gaussian-like” because it shares some of the
properties of proper Gaussian random vectors (compare this with Result
2.10
). However,
Gaussian noise always has zero kurtosis,
K
=
0, whereas fourth-order white noise is
allowed to have
K
=
0.
Notes
1 The introduction of channel, analysis, and synthesis models for the representation of composite
covariance matrices of signal and measurement is fairly standard, but we have guided by Mullis
and Scharf (1996).
2 The solution of LMMSE problems requires the solution of normal equations
WR
yy
=
R
xy
.
There has been a flurry of interest in “multistage” approximations to these equations, beginning
with the work of
Goldstein
et al
. (1998
). The connection between multistage and conjugate
gradient algorithms has been clarified by Dietl
et al.
(2001), Weippert
et al.
(2002), and
Scharf
et al
. (2008
). The attraction of these algorithms is that they converge rapidly (in
N
steps) if the
matrix
R
yy
has only a small number
N
of distinct eigenvalues.
3 WLMMSE estimation was introduced by
Picinbono and Chevalier (1995
), but widely linear (or
linear-linear-conjugate) filtering had been considered already by
Brown and Crane (1969
). The
performance comparison between LMMSE and WLMMSE estimation is due to
Schreier
et al
.
(2005
).
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