不完美反馈下时空编码MIMO系统的功率控制策略
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"这篇研究论文探讨了时空编码的MIMO（多输入多输出）系统在联合发射-接收相关信道上使用不完美反馈进行功率控制的策略。文章提出了在平均功率约束下最小化误比特率（BER）的优化方法，并考虑了仅基于均值反馈和仅基于协方差反馈的情况。"
在无线通信领域，时空编码MIMO系统因其能显著提高频谱效率和抗干扰能力而备受关注。然而，在实际应用中，由于信道条件的不确定性，获取完全准确的信道状态信息（CSI）通常是困难的，这导致了所谓的“不完美反馈”。该论文针对这一问题，提出了一种基于近似误比特率界的空间唯一功率控制（PC）策略。
首先，文章阐述了在联合发射-接收相关信道的背景下，如何利用不完美的信道信息（包括均值反馈和协方差反馈）来适应性地调整发射功率。发射-接收相关信道是指发射端和接收端之间的信道具有一定的相关性，这在地理上接近的通信节点间是常见的现象，例如移动通信中的小区内干扰。
接着，作者提出了一个最优的空间唯一功率控制策略，这个策略是基于对误比特率界的近似计算。通过分析误比特率与发射功率的关系，他们构建了一个优化问题，目标是在满足平均功率约束的同时，尽可能减小误比特率。考虑到仅使用均值反馈或协方差反馈的情况，他们分别设计了相应的功率控制算法，这些算法能够在不完全了解信道状态的情况下，动态调整发射机的功率分配。
此外，论文可能还深入分析了这些策略的性能，可能包括理论上的误比特率表现、与理想情况下的性能差距以及实际环境中的鲁棒性。可能还进行了仿真验证，对比了不同功率控制策略在不同信道条件下的效果，以展示所提方法的有效性和实用性。
最后，论文可能讨论了未来的研究方向，如如何进一步改进功率控制算法以减少由于不完美反馈引起的性能损失，或者探索在更复杂信道模型下的应用。这项工作对于理解和改进MIMO系统在现实世界条件下的性能具有重要意义，有助于推动无线通信技术的发展。
0018-9545 (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See
http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/TVT.2014.2323295, IEEE Transactions on Vehicular Technology
3
P
1
P
N
Space-
time
coding
Beam
Forming
U
. . .
Rx1Tx1
TxN
Space-
time
decoding
. . .
Channel
estimation
. . .
Feedback path
RxK
Power
control
Output
data
Input
data
. . .
Fig. 1. System diagram.
to exploit the available CSI. A complex orthogonal STBC
(OSTBC), which is represented by an N × T transmission
matrix D, is used to encode L input symbols into an N-
dimensional vector sequence of T time slots. The matrix D
is a linear combination of L symbols satisfying the complex
orthogonality: DD
H
= ε(|d
1
|
2
+ ··· + |d
L
|
2
)I
N
, where
{d
l
}
l=1,...,L
are the L input symbols, and ε is a constant
which depends on the STBC transmission matrix [4]. Hence,
the transmission rate of the STBC is r = L/T .
In this paper, we assume that CSI is perfectly known at
the receiver, and the transmitter has the knowledge of R
t
and
receives R
r
and imperfect (time-delayed) mean feedback from
the receiver. The channel matrix H
w
in H can be related to
its τ time-delayed version
ˆ
H
f
at the transmitter as [34]
H
w
= ρ
ˆ
H
f
+ E
w
=
ˆ
H
w
+ E
w
, (1)
where E
w
is a K × N channel error matrix independent of
ˆ
H
f
. The entries of the K ×N matrix
ˆ
H
f
are i.i.d. zero mean
complex Gaussian r.v.s of unit variance, while the entries of
E
w
are i.i.d. complex Gaussian r.v.s with zero mean and
variance σ
2
e
= 1 − ρ
2
, where ρ is the correlation coefficient
given by ρ = J
0
(2πf
d
τ), J
0
(·) is the zero-order Bessel
function of the first kind [37], and f
d
is the maximum Doppler
frequency [38]. The correlation between the entries of
ˆ
H
f
and the entries of H
w
is given by the correlation coefficient
ρ. The entries of
ˆ
H
w
= ρ
ˆ
H
f
∼ CN(0, ρ
2
). Based on the
Kronecker relationship, the channel matrix H is expressed as
H = R
1/2
r
(
ˆ
H
w
+ E
w
)R
1/2
t
=
ˆ
H + E, where K × N matrix
ˆ
H = R
1/2
r
ˆ
H
w
R
1/2
t
is the channel mean feedback [7, 26], and
E = R
1/2
r
E
w
R
1/2
t
. The correlation matrix of H conditioned
on
ˆ
H is given by
e
R
h
= E{H
H
H|
ˆ
H} =
ˆ
H
H
ˆ
H + E{E
H
E} (2)
where
e
R
h
is a N × N matrix. Utilizing E = R
1/2
r
E
w
R
1/2
t
,
E{E
H
E} in (2) can be expressed as
E{E
H
E} = E{(R
1/2
t
)
H
E
H
w
(R
1/2
r
)
H
R
1/2
r
E
w
R
1/2
t
}
= R
1/2
t
E{E
H
w
R
r
E
w
}R
1/2
t
= σ
2
e
P
K
k=1
ζ
r,k
R
t
(3)
where {ζ
r,k
, k = 1, . . . , K} are the eigenvalues of R
r
. The
last equality in (3) can be obtained from the following Lemma.
Lemma: If E
w
is a complex Gaussian random matrix
whose entries are i.i.d. with zero-mean and variance σ
2
e
,
then E{E
H
w
R
r
E
w
} = σ
2
e
P
K
k=1
ζ
r,k
I
N
, where {ζ
r,k
} are the
eigenvalues of R
r
.
Proof: Let R
r
= V
r
Λ
r
V
H
r
be the eigenvalue decompo-
sition (EVD) of R
r
, where Λ
r
= diag{ζ
r,1
, ..., ζ
r,K
}, then
we have:
E
H
w
R
r
E
w
= E
H
w
V
r
Λ
r
V
H
r
E
w
=
e
E
H
w
Λ
r
e
E
w
=
K
X
k=1
ζ
r,k
e
w,k
e
H
w,k
(4)
where the K × N matrix
e
E
w
= V
H
r
E
w
, and {e
w,k
} are the
N ×1 column vectors of
e
E
H
w
. Since V
r
is a unitary matrix,
˜
E
w
and E
w
have the same statistical distribution. Thus the entries
of
˜
E
w
are also i.i.d. complex Gaussian r.v.s with zero mean
and variance σ
2
e
= 1 − ρ
2
. Hence,
E{E
H
w
R
r
E
w
} =
X
K
k=1
ζ
r,k
E{e
w,k
e
H
w,k
} = σ
2
e
X
K
k=1
ζ
r,k
I
N
.
(5)
Substituting (3) into (2) gives
˜
R
h
=
ˆ
H
H
ˆ
H + σ
2
e
X
K
k=1
ζ
r,k
R
t
(6)
Using EVD,
˜
R
h
can be expressed as
˜
R
h
= V
h
Λ
h
V
H
h
(7)
where the N × N unitary matrix V
h
contains the N eigen-
vectors of
˜
R
h
, Λ
h
= diag{a
1
, ..., a
N
} is the diagonal matrix
containing the nonnegative eigenvalues {a
n
} sorted in a de-
creasing order.
With beamforming U and power control P, the transmitted
signal matrix can be expressed as
X = UPD (8)
where the N × N unitary matrix U (whose elements are
u
ij
, i, j = 1, . . . , N) is set equal to the orthogonal eigenvector
matrix V
h
of
˜
R
h
[26], P = diag(
√
P
1
,
√
P
2
, ...,
√
P
N
) is
a diagonal N × N matrix, where {P
n
, n = 1, . . . , N} is the
power control to the N eigen-beams with the following power
constraint.
X
N
n=1
P
n
= 1, P
n
≥ 0. (9)
Thus, the received signal matrix can be written as
Y = HX + Z = HUPD + Z (10)
where Y is a K × T received signal matrix, Z is a K × T
noise matrix with i.i.d. entries modeled as complex Gaussian
r.v.s with zero-mean and variance σ
2
n
. The equivalent received
signal vector after the space-time decoder of the receiver can
be expressed as [4, 5]
y = ε||HUP||
2
F
d + ˜z (11)
where y is an L ×1 received signal vector, d = [d
1
, . . . , d
L
]
T
is the L × 1 input data symbol vector. Each symbol in the
transmission matrix has an average power T
¯
S/(εL), where
¯
S is the average transmitted power radiated from N transmit
antennas, ||HUP||
2
F
=
P
N
n=1
P
n
P
K
k=1
|
P
N
i=1
h
ki
u
in
|
2
, and
˜z is an L ×1 noise vector whose elements are i.i.d. complex
Gaussian r.v.s with zero-mean and variance ε||HUP||
2
F
σ
2
n
.
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