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subject to the following constraints:
⎧
⎨
⎩
s
(1)
k
=
w
1
φ
1
(
s
k−
1
− e
k−
1
)+
b
1
− e
k,
1
s
(2)
k
=
w
2
φ
2
(
s
k−
1
− e
k−
1
)+
b
2
.
− e
k,
2
(11)
s
m
k
− e
k,m
=
w
m
φ
(
s
k−
1
− e
k−
1
)+
b
m
where
k
=
m
+1
,m
=2
,...,N
+
m
−
1,
e
k
=
s
k
−
s
k
. Generally, the error
term here is defined as
e
k
=
x
k
− f
(
x
k−
1
)
(12)
4
Experimental Results
In order to test the performance of the MIMO channel prediction, we used the
received signal-to-noise ratio (SNR) in the general form
σ
x
σ
e
+
σ
n
ρ
=
(13)
where
σ
x
is the average received signal power,
σ
e
is the predictive error,
σ
n
is the average noise variance. Thus, after some algebraic manipulations for the
un-coded system we can obtain
ˆ
2
E
Hx
ρ
uc
=
(14)
E
Ex
2
+
E
n
2
and after several manipulations
i
=1
E
[
σ
2
(
i
)]
H
ρ
uc
=
(15)
i
=1
E
[
σ
2
(
i
)] +
N
r
N
0
E
s
E
ˆ
where
N
=
rank
(
E
),
σ
H
(
i
)and
σ
E
(
i
)arethe
i
-th non-zero singular values of
H
and
, respectively.
The MIMO beam-forming can be formulated as follows
E
=(
σ
1
+
u
1
)
x
+
n
(16)
Then, after a similar technique as the one used before, we can state the received
SNR for the MIMO beam-forming system
u
1
y
ˆ
E
[
σ
1
]
ρ
bf
=
(17)
u
1
UDV
H
v
1
−
2
+
N
r
N
0
E
s
E |
ˆ
σ
max
|
Thus, comparing the above equation with the Eq. (19) we get the value of the
prediction error
σ
e
for the beam-forming prediction, namely
ˆ
ˆ
σ
e
=
E
Ex
2
2
=
E
U
H
E
2
2
Vx
(18)
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