Digital Signal Processing Reference
In-Depth Information
H
=σ
2
E{
nn
()
t
()}
t
I
,
(8.2)
where
I
is the identity matrix, E{·} denotes the statistical expectation, and (·)
H
stands for
the Hermitian transpose.
The baseband complex signal at the output of the receive beamformer can be written as
L
∑
H
H
H
yt
()
=
wx
()
t
=
s t
()
wa wn
+
()
t
.
(8.3)
l
l
ll
=
8.2.2
The Downlink Case
Now, let us consider the transmit beamforming mode with single BS and the same signal
sent to all users. The baseband signal received by the
l
th
user can be expressed as
H
zt st
()
=
()
hw
+
nt
()
,
(8.4)
l
l
l
where
s
(
t
) is the transmit baseband signal,
w
is the BS weight vector, and
n
l
(
t
) is additive
noise at the
l
th
user.
The latter model can be further extended to the case of
K
different BSs and
L
mobile
users. Let
w
l
be the weight vector used at the BS assigned to the
l
th
user to transmit the
baseband signal
s
l
(
t
) to this user. Let us also define the BS cell site index
c
(
l
) as the index
of the particular BS that is assigned to the
l
th
user. Note that
c
(
l
) =
c
(
m
) if both the
l
th
and
m
th
users are assigned to the same BS, and
c
(
l
) ≠
c
(
m
) if these users are assigned to
different BSs. Using these notations, the vector of signals transmitted from the
k
th
BS
can be expressed as [14]
∑
G
x
()
t
=
s t
()
w
,
(8.5)
k
i
i
i
∈
()
k
where
G
() {()}
ki ci
: =
k
(8.6)
is the set of indices of all weight vectors that are used at the
k
th
BS (or equivalently, the
set of indices of all users that are assigned to this BS). Equation (8.5) implies that the
k
th
BS transmits only to the users that are assigned to it rather than to all the users in the
cellular network.
Using (8.5), the baseband signal received by the
l
th
user can be modeled as [14]
K
∑
1
H
zt
()
=
hx
()
tnt
+
()
,
(8.7)
l
lk
,
k
l
k
=
where
h
l,k
is the downlink channel vector between the
k
th
BS and the
l
th
user.
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