Digital Signal Processing Reference
In-Depth Information
the mean-square weight estimator is given by
w ¼ C
1
p
or by
w
¼ P
1
q
where the covariance and pseudo covariance matrices
C
and
P
as well as the
cross covariance vectors
p
and
q
are defined in Section 1.4. Consequently,
show that the mean-square error difference between a linear and a widely
linear MSE filter (
J
diff
¼ J
L
,min
-
J
WL
,min
) for this case is exactly zero, that is,
using a widely linear filter does not provide any additional advantage even
when the signal is noncircular.
1.7
The conclusion in Problem 1.6 can be extended to prediction of an autoregres-
sive process given by
X
(
n
)
þ
X
N
a
k
X
(
nk
1)
¼ V
(
n
)
k¼
1
where
V
(
n
) is the white Gaussian noise. For simplicity, assume one-step ahead
predictor and show that
J
diff
¼
0 as long as
V
(
n
) is a doubly white random pro-
cess, that is, the covariance and the pseudo covariance functions of
V
(
n
) satisfy
c
(
k
)
¼ c
(0)
d
(
k
) and
p
(
k
)
¼ p
(0)
d
(
k
) respectively.
1.8
For the widely linear weight vector error difference
1
(
n
)
¼ v
(
n
)-
v
opt
, show that
we can write the expression for the modes of the widely linear LMS algorithm
given in (1.41) as
E
{
1
0
k
(
n
)}
¼ 1
0
k
(0)(1
ml
k
)
n
and
mJ
WL
,min
mJ
WL
, min
2
ml
k
2
ml
k
þ
(1
ml
k
)
2
n
E
{
j1
0
k
(
n
)
j
2
}
¼
j1
0
k
(0)
j
2
as shown in [16, 43] for the linear LMS algorithm. Make sure you clearly ident-
ify all assumptions that lead to the expressions given above.
1.9
Explain the importance of the correlation matrix eigenvalues on the performance
of the linear and widely linear LMS filter (
l
and
¯
). Let input
x
(
n
) be a first order
autoregressive process (
N ¼
1 for the AR process given in Problem 1.7) but let
the white Gaussian noise
v
(
n
) be noncircular such that the pseudo-covariance
Ef
v
2
(
n
)
g
=
0. Show that when the pseudo-covariance matrix is nonzero, the
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