大规模阵列支持的多对全双工中继系统功率优化与效率分析
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本文档深入探讨了"Multipair Two-Way Full-Duplex Relaying with Massive Array and Power Allocation"这一主题，发表于2017年10月的IEEE Transactions on Vehicular Technology，卷66，第10期。研究关注的是一个多对多全双工中继网络，其中多个全双工用户通过共享一个大规模天线阵列的中继站进行信息交换。每个用户拥有一个共享天线，而中继站则配备了M个共享天线。
在这个系统中，采用的是放大转发（Amplify-and-Forward, AF）协议，同时利用最大比合并（Maximal Ratio Combining, MRC）或零对于准空间分集接收/发送（Zero Forcing Reception/Transmission, ZF-Rx/Tx）技术来进行信号处理。当中继站的天线数量趋向于无限大或者足够多时，作者针对这种情况进行了深入分析。
研究的核心内容包括在无限或大量天线阵列下，推导中继站的极限发射功率以及在不同策略下的谱效（spectral efficiency）。特别地，当考虑用户总发射功率约束时，论文探讨了如何最优地分配用户的发射功率，以最大化整个系统的谱效。通过大量的模拟实验，作者验证了这些理论结果在大规模天线阵列环境中的有效性，并展示了其逼近实际性能的优越性。
这项研究对于无线通信系统设计具有重要意义，尤其是在全双工技术日益受到重视的背景下，大规模天线阵列的应用能够显著提升多对多通信的效率和可靠性。它为未来无线网络优化设计提供了理论依据，有助于实现更高效的频谱利用率和能量管理。
8928 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 66, NO. 10, OCTOBER 2017
where G =[g
1
, g
2
,...,g
2K
] denotes the M × 2K chan-
nel matrix from users to the relay with g
k
∈ C
M ×1
; A =
diag(
√
p
1
,
√
p
2
,...,
√
p
2K
) is the square-root matrix of users’
transmit powers, here diag(·) represents a diagonal operation
with
√
p
k
on the main diagonal; G
R
is an M × M LI chan-
nel matrix (after SIC) at the relay; n
R
(n) ∼CN(0,σ
2
r
I
2K
) is
the additive white Gaussian noise (AWGN) vector at the relay;
s
R
(n + d) is the broadcast signal at time (n + d), and the posi-
tive integer d (d ≥ 1) denotes the processing delay [17], [21]; ρ
is the amplification factor, which is a constant and determined
by the RF hardware [19]; F is a transmit beamforming matrix
relying on the state of channel matrix G at time (n + d); P is a
2K × 2K permutation matrix given by P =[0
K
, I
K
; I
K
, 0
K
];
W is a signal detection matrix at the relay depending on the
state of channel matrix G at time n. Note that F and W
T
are
independent of each other, because they are derived from the
signal channel matrix belonging to different coherence times.
Meanwhile, the received signals y
D
at users can be written as
y
D
(n)=G
T
s
R
(n)+G
D
Ax(n)+n
D
(n), (3)
where G
D
is a 2K × 2K SI channel at users, which is given
by G
D
=[G
d
, 0
K
; 0
K
, G
d
], G
d
and G
d
are the K × K ma-
trices, and their elements are distributed as CN(0,σ
2
k,i
) and
CN(0,σ
2
k
,i
), respectively [21], with k, i ∈{1,...,K}, i
=
K + i, ∀k, k
,i,i
. Since the massive array can greatly prolong
the transmission distance from the source to the destination, the
interferences between users on different sides are very weak and
they can be omitted. n
D
(n) is AWGN vectors at users, whose
elements are i.i.d. CN (0,σ
2
d
) random variables.
A. Channel Model
The channel matrix G can be modeled as [12], [20], [21]
G = HD
1/2
, (4)
where H represents the small-scale fading matrix with i.i.d.
CN(0, 1) entries, D is a diagonal matrix representing the large-
scale fading and the kth element is denoted as β
k
.
Note that G
R
can be modeled as a Rayleigh fading channel
if the SIC is performed preliminarily [7], [17], [21]. Therefore,
the LI channel G
R
can be written as
G
R
= H
R
D
1/2
R
, (5)
where H
R
is the small-scale fading matrix with i.i.d. CN(0, 1)
elements, and D
R
is a diagonal matrix representing the large-
scale fading. Unlike the signal matrix, the large-scale fading in
the SI channel is assumed to be identical, namely, [D
R
]
mm
= β,
m ∈{1,...,M}, as the antenna array size is much smaller than
the distance to the scatters. This assumption is reasonable even
if the large-scale fading coefficients are different, because all
the elements of D
R
can be set as the maximum value β
max
,
which represents the worst case.
B. Channel Estimation
Channel estimation is a prerequisite for the signal detection
and beamforming. Commonly, a series of orthogonal training
sequences are sent from users to the relay in a short period
τ during the channel coherence time T . Thanks to the chan-
nel reciprocity of the shared-antenna system, both the uplink
and downlink channel state information (CSI) can be estimated
in the uplink training phase like the time division duplexing
(TDD) system. Therefore, the M × τ received pilot matrix for
the uplink channel can be written as
Y
R
=
P
P
GΦ + N
R
, (6)
where P
P
denotes the transmit power of pilot. Φ is a 2K ×τ
matrix whose rows are the pilot sequences for the CSI estimation
(τ ≥ 2K). The pilot matrices satisfy ΦΦ
†
= I
2K
. N
R
is an
M × τ matrix of AWGN with its elements following CN(0,σ
2
).
The uplink channel matrix G can be estimated by the min-
imum mean-square-error (MMSE) method, so the estimated
channel is given by [17]
ˆ
G =
1
√
P
P
Y
R
Φ
†
ˇ
D =
G +
1
√
P
P
N
ˇ
D, (7)
where N N
R
Φ
†
and
ˇ
D (
1
P
P
D
−1
+ I
K
)
−1
.LetΔ denote
the estimation error of G,wehave
G =
ˆ
G + Δ. (8)
Owning to the properties of MMSE estimation, the estimation
error Δ is independent of
ˆ
G. The variances of [
ˆ
G]
mk
and [Δ]
mk
are
ˆ
β
k
=
P
P
β
2
k
P
P
β
k
+σ
2
and
˜
β
k
=
β
k
σ
2
P
P
β
k
+σ
2
, respectively [17], [42].
C. Receive and Transmit Beamforming
Once the CSI is estimated, the relay can use it for the uplink
signal detection and downlink transmit beamforming. The relay
receiver is assumed to adopt either ZFR or MRC for the data
reception, and the relay transmitter uses either ZFT or MRT
scheme for the data transmission. The receive beamforming
matrix W
T
in Eq. (2) can be given as [12], [13]
W
T
=
⎧
⎪
⎨
⎪
⎩
1
M
ˆ
G
†
, for MRC,
(
ˆ
G
†
ˆ
G)
−1
ˆ
G
†
, for ZFR.
(9)
After the detection, matrix F is used for the transmit beamform-
ing [12], [13], i.e.,
F =
⎧
⎪
⎨
⎪
⎩
1
√
M
ˆ
G
∗
, for MRT,
√
M
ˆ
G
∗
(
ˆ
G
T
ˆ
G
∗
)
−1
, for ZFT,
(10)
where
ˆ
G
represents the estimated channel of
ˆ
G at time (n + d).
As mentioned previously,
ˆ
G and
ˆ
G
are independent of each
other.
Substituting (1) into (2), let FPW
T
= T and d = 1 [17],
[21], we can obtain
s
R
(n + 1)=ρT [GAx(n)+G
R
s
R
(n)+n
R
(n)] . (11)
Because the transmit power of signals is constrained by
E[Tr(s
R
(n)s
†
R
(n))] = P
R
,wehave
M
m =1
E[s
2
R,m
(n)] = P
R
,
where s
R,m
is the mth element of s
R
(n).
Proposition 1: The elements of LI
˜
s
R
(n)=G
R
s
R
(n) are
i.i.d. CN(0,βP
R
) random variables when M →∞.
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