Digital Signal Processing Reference
In-Depth Information
signal pdf and the corresponding parametric model) using the
Kullback-
Leibler divergence
(KLD). The KLD is given by the following expression [78]:
ln
p
Y
dy
,
∞
(
y
)
D
p
Y
(y)
)
=
p
Y
(
y
)
·
(5.107)
(
y
(
y
)
−∞
(
y
)
where the function
is the parametric model that fits the statistical
behavior of an ideally recovered signal [64].
Since we deal with discrete symbols in the presence of Gaussian noise, it
is suitable to pose
exp
y
s
i
2
S
−
1
2πσ
r
(
y
)
=
−
·
P
(
a
i
)
,
(5.108)
2σ
r
i
=
1
where
S
is the cardinality of the transmitted alphabet
P
is the probability of occurrence of a symbol
s
i
σ
r
is the variance of the Gaussian kernels we assume in the model
(
s
i
)
In this sense, the algorithm can be understood as an attempt to “equalize”
the pdfs of the transmitted signal and the filter output.
The proposed cost function, to be minimized, is
∞
ln
(
)
dy
J
FPC
(
w
(
n
))
=−
p
Y
(
y
)
y
−∞
E
ln
(
)
=−
y
(5.109)
where FPC stand for fitting pdf criterion.
A stochastic version for filter adaptation is given by
i
=
1
exp
2σ
r
P
−
y
s
i
a
i
)
y
a
i
x
2
(
n
)
−
/
(
(
n
)
−
(
n
)
σ
r
i
=
1
exp
2σ
r
P
∇
J
FPC
(
w
(
n
))
=
(5.110)
y
s
i
2
−
(
n
)
−
/
(
a
i
)
w
(
n
+
1
)
=
w
(
n
)
−
μ
∇
J
FPC
(
w
(
n
))
,
where μ is the algorithm step-size. This adaptive algorithm is referred to as
fitting pdf algorithm (FPA).
Based on the approach proposed in [228], we can use the criterion of
explicit decorrelation of beamformer outputs to establish the cost function
for the multiuser fitting pdf criterion (MU-FPC). Thus, we obtain the
following criterion for the MIMO scenario:
