大规模多用户MIMO系统能源与频谱效率研究
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本文档深入探讨了"Very Large Multiuser MIMO Systems"（大规模多用户多输入多输出系统）的能源效率和频谱效率。大规模MIMO是一种先进的无线通信技术，其中众多的自主终端同时向一个紧凑型基站天线阵列发送数据流。关键创新在于，尽管基站利用从传输的训练信号中获取的不完美的信道状态信息（CSI），依然能够高效地分离出各个终端的数据流，无需显著牺牲性能。
论文的核心发现是，终端发射功率可以通过增加基站天线的数量来降低，具体来说，与单个天线系统相比，功率消耗可以按天线数量平方根的比例减少。然而，如果拥有完美的信道状态信息，功率消耗将更进一步，按天线数量的比例减小。文中还分析了不同接收技术，如最大比合并（MRC）、零 forcing（ZF）和最小均方误差（MMSE）的容量下限。通常情况下，MRC的性能不如ZF和MMSE，但随着功率水平的下降，MRC的交叉干扰最终会低于噪声水平，使得这种简单的接收方式变得可行。
文章的重点在于量化能源效率（以比特每焦耳衡量）和频谱效率（以比特每信道使用次数每个终端计）之间的权衡。在考虑了小型尺度衰落但忽略大型尺度衰落的通道模型下，研究发现适度规模的天线阵列能极大地提高这两个指标，相较于单天线系统，可能提升几个数量级的性能。这意味着在实际通信系统设计中，通过采用大规模MIMO技术，不仅能够提升数据传输速率，还能有效地管理能源消耗，从而实现更高的网络效率和可持续性。这对于5G、6G等未来通信技术的发展具有重要的理论指导意义。
1438 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 4, APRIL 2013
are satisfied with p
p
p and q
q
q being any two distinct colum ns of
G
G
G.Inthiscasewehave
G
G
G
H
G
G
G
M
= D
D
D
1/2
H
H
H
H
H
H
H
M
D
D
D
1/2
≈ D
D
D, M K
and we say that we have favorable propagation. Clearly, if all
fading coefficients are i.i.d. and zero mean, we have favorable
propagation. Recent channel measurements campaigns have
shown that multiuser MIMO systems with large antenna
arrays have characteristics that approximate the favorable-
propagation assumption fairly well [10], and therefore p rovide
experimental justification for this assumption.
To understand why favorable propagation is desirable, con-
sider an M × K uplink (multiple-access) MIMO channel H
H
H,
where M ≥ K, neglecting for now path loss and shadowing
factors in D
D
D. This channel can offer a sum-rate of
R =
K
k=1
log
2
1+p
u
λ
2
k
(6)
where p
u
is the power spent per terminal and {λ
k
}
K
k=1
are
the singular values of H
H
H, see [13]. If the channel matrix is
normalized such that |H
ij
|∼1 (where ∼ means equality of
the order of magnitude), then
K
k=1
λ
2
k
= H
H
H
2
≈ MK.
Under this constraint the rate R is bounded as
log
2
(1 + MKp
u
) ≤ R ≤ K log
2
(1 + Mp
u
) . (7)
The lower bound (left inequality) is satisfied with equality
if λ
2
1
= MK and λ
2
2
= ··· = λ
2
K
=0and corresponds
to a rank-one (line-of-sight) channel. The upper bound (right
inequality ) is achieved if λ
2
1
= ···= λ
2
K
= M. This occurs if
the columns of H
H
H are mutually orthogonal and have the same
norm, which is the case when we have favorable propagation.
III. A
CHIEVABLE RATE AND ASYMPTOTIC (M →∞)
P
OW E R EFFICIE NCY
By using a large antenna array, we can reduce the transmit-
ted power of the u sers as M grows large, while maintaining a
given, desired quality-of-service. In this section, we quantify
this potential for power decrease, and derive achievable rates
of the uplink. Theoretically, the BS can use the maximum-
likelihood detector to obtain optimal performance. However,
the complexity of this detector grows exponentially with K.
The interesting operating regime is when both M and K are
large, but M is still (much) larger than K, i.e., 1 K M.
It is known that in this case, linear detector s (MRC, ZF
and MMSE) per form fairly well [8] and therefore we will
restrict consideration to those detectors in this paper. We treat
the cases of perfect CSI (Section III-A) and estimated CSI
(Section III-B) separately.
A. Perfect Channel State Information
We first consider the case when the BS has perfect CSI,
i.e. it knows G
G
G.LetA
A
A be an M × K linear detector matrix
which depends on the channel G
G
G. By usin g the linear detector,
the received signal is separated into streams by multiplying it
with A
A
A
H
as follows
r
r
r = A
A
A
H
y
y
y. (8)
We consider three conventional linear detectors MRC, ZF, and
MMSE, i.e.,
A
A
A =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
G
G
G for MRC
G
G
G
G
G
G
H
G
G
G
−1
for ZF
G
G
G
G
G
G
H
G
G
G +
1
p
u
I
I
I
K
−1
for MMSE
(9)
From (1) and (8), the received vector after using the linear
detector is given by
r
r
r =
√
p
u
A
A
A
H
G
G
Gx
x
x + A
A
A
H
n
n
n. (10)
Let r
k
and x
k
be the kth elements of the K ×1 vectors r
r
r and
x
x
x, respectively. Then,
r
k
=
√
p
u
a
a
a
H
k
g
g
g
k
x
k
+
√
p
u
K
i=1,i=k
a
a
a
H
k
g
g
g
i
x
i
+ a
a
a
H
k
n
n
n (11)
where a
a
a
k
and g
g
g
k
are the kth columns of the matrices A
A
A and
G
G
G, respectively. For a fixed channel realization G
G
G, the noise-
plus-interference term is a random variable with zero mean
and variance p
u
K
i=1,i=k
|a
a
a
H
k
g
g
g
i
|
2
+ a
a
a
k
2
. By modeling this
term as additive Gaussian noise independent of x
k
we can
obtain a lower bound on the achievable rate. Assuming further
that the channel is ergodic so that each codeword spans over a
large ( infinite) number of realizations of the fast-fading factor
of G
G
G, the ergodic achievable uplink rate of the kth user is
R
P,k
=E
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
log
2
⎛
⎜
⎜
⎜
⎝
1+
p
u
|a
a
a
H
k
g
g
g
k
|
2
p
u
K
i=1,i=k
|a
a
a
H
k
g
g
g
i
|
2
+ a
a
a
k
2
⎞
⎟
⎟
⎟
⎠
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
. (12)
To approach this capacity lower bound, the message has to be
encoded over many realizations of all sources of randomness
that enter the model (noise and channel). In practice, assuming
wideband operation, this can be achieved by coding over the
frequency domain, using, for example coded OFDM.
Proposition 1: Assume that the BS h as perfect CSI and that
the transmit power of each user is scaled with M according
to p
u
=
E
u
M
,whereE
u
is fixed. Then,
3
R
P,k
→ log
2
(1 + β
k
E
u
) ,M →∞. (13)
Proof: We give the proof for the case of an MRC receiver.
With MRC, A
A
A = G
G
G so a
a
a
k
= g
g
g
k
. From (12), the achievable
uplink rate of the kth user is
R
mrc
P,k
= E
log
2
1+
p
u
g
g
g
k
4
p
u
K
i=1,i=k
|g
g
g
H
k
g
g
g
i
|
2
+ g
g
g
k
2
.
(14)
Substituting p
u
=
E
u
M
into (14), and using (4), we obtain (13).
By using the law of large numbers, we can arrive at the same
result for the ZF and MMSE receivers. Note from (3) and (4)
that when M grows large,
1
M
G
G
G
H
G
G
G tends to D
D
D, and hence the
ZF and MMSE filters tend to that of the MRC.
Proposition 1 shows that with perfect CSI at the BS and a
large M, the performance of a MU-MIMO system with M
3
As mentioned after (1), p
u
has the interpretation of normalized transmit
SNR, and it is dimensionless. Therefore E
u
is dimensionless too.
Authorized licensed use limited to: JILIN UNIVERSITY. Downloaded on June 02,2020 at 10:41:29 UTC from IEEE Xplore. Restrictions apply.
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