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u
1
y
=(
σ
H
(1) +
u
1
Ev
1
)
+
u
1
n
(4)
In the general case the doppler spread of the signal is greater than the pulse
bandwidth. In a typical environment the MIMO channel is fast-fading. Assuming
a MIMO flat fast-fading transmission each sub-channel can be formulated as
follows [11]:
x
h
mn
(
t
)=
h
mn
(
k
)+
jh
mn
(
k
)
(5)
where in-phase component is represent as
h
I
mn
(
k
)=
2
M
M
cos(2
πf
d
k
sin(
α
n
)+
Φ
n
)
(6)
n
=1
and the quadrature component can be written as
h
mn
(
k
)=
2
M
M
cos(2
πf
d
k
sin(
α
n
)+
Ψ
n
)
(7)
n
=1
2
πn−π
+
Θ
4
M
where
α
n
=
and
Φ
n
,Ψ
n
,
Θ
are
U
[
−π, π
].
3 The Multidimensional Recurrent LS-SVM
TherecurrentLS-SVMbasedonthesum squared error (SSE) to deal with
the function approximation and prediction has been proposed by Suykens and
Vandewalle [8]. However, the so defined recurrent LS-SVM will not be adequate
for the channel prediction in the MIMO system. It is caused by the lack of the
high-dimensional reconstructed embedding phase space.
In order to extend the recurrent least squares vector machine to a multidi-
mensional recurrent LS-SVM we introduce scalar time series
{s
1
,s
2
,...,s
T
}
in
the form
k
=
m
,m
+1
,...,N
+
m
−
1 (8)
where
m
,
N
are referred to the embedding dimension and the number of train-
ing data, respectively.
The function approximation is given by
s
k
=
f
(
s
k−
1
)
,
T
φ
i
(
s
k−
1
)+
b,
k
=
m
,m
+1
,...,N
+
m
−
i
=1
,
2
,...,m
s
k
=
w
1
,
(9)
where
=[
w
1
,w
2
,...,w
m
] is the output weight vector,
b ∈ R
is the bias term
φ
(
.
) is the nonlinear mapping function estimated by means of using training
data.
The recurrent LS-SVM can be formulated as the quadratic optimization
problem:
w
m
N
+
m
−
m
1
(
w
i
,b
i
,e
k,i
)=
1
2
w
i
w
i
+
γ
2
e
k,i
w
i
,b
i
,e
k,i
J
min
(10)
i
=1
i
=1
k
=
m
+1
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