Digital Signal Processing Reference
In-Depth Information
The DOAs are estimated using the
H
k
(
t
) values during an estimation period P, which
is defined to be a set of
L
mini-slots. Each estimation period is divided into
Q
subestima-
tion periods P
i
,
i
= 1,
…
,
Q
, such that P = P
1
∪P
2
∪
…
∪
P
Q
and P
i
∩P
j
= 0,
i
≠
j
. Each P
q
is composed of
L
q
mini-slots with
∑
1
Q
L
=
L
q
.
q
=
In subestimation period P
q
, the artificial phase shift ϕ(
t
) of adaptive opportunistic
beamforming is set to ϕ
q
, i.e., ϕ(
t
) = ϕ
q
∈[0,2π), for
t
∈P
q
,
q
= 1, 2,
…
,
Q
.* For example,
we can set ϕ
q
= (
q
/
Q
)2π, for
q
= 1, 2,
…
,
Q
. We define A
k,q
, Ω
k,q
, and K
k,q
as follows:
A
k,q
=
A
k
(
t
)|
t
∈P
q
, Ω
k,q
= Ω
k
(
t
)|
t
∈P
q
=
A
k,q
+ σ
2
, and K
k,q
= K
k
(
t
)|
t
∈P
q
.
Given σ
k
and
A
k,q
at the base station, the ML estimation of θ
k
is given by
Q
L
q
2
2
HtA
()
+
2
HtA
()
∏
2
∏
P
ˆ
k
k q
,
k
k q
,
θ
ML
=
argmax
H
(
t
)exp
−
I
(7. 2 6)
k
k
0
σ
2
σ
2
σ
k
2
θ
∈
[,
02
π
)
k
k
k
q
=
1
t
∈
q
for
k
= 1,
…
,
M
. Taking logarithm, a simpler estimator is given by
Q
2
HtA
()
∑
ˆ
ML
k
k q
,
2
θ
=
argmax
ln
I
−
A
kq
(7. 27)
k
0
,
2
σ
θ
∈ 02
[,
π
)
k
k
q
=
1
tP
∈
q
for
k
= 1,
…
,
M
. In order to reduce the computational complexity further at the loss of
the ML optimality, suboptimum estimators may be considered. In particular, noting
that
G
k
(
t
) is maximized when ϕ
q
= -θ
k
, an efficient and very simple suboptimum estima-
tor can be given by
ˆ
=−
( )
θ
SUB
qQ
∗
2
π
,
(7. 2 8)
k
1
∑
∗
2
q
=
argmax
Ht
().
(7. 29)
k
L
=
…
q
1
,,
Q
q
t
q
∈
7.4.3 Estimation of
K
-Factors of the Physical Channels
In the proposed DOA estimation algorithms, it was assumed that the exact {
K
k
}
k
=1
values were known at the base station. In practical systems, however, each user needs
* Recall that in conventional opportunistic beamforming, ϕ(
t
) is randomly chosen from [0,2π).
Also, note that the pilot overhead is the same for the adaptive and conventional schemes, because
only one mini-slot of fixed length is used for pilot signaling in each time slot for both schemes.
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