大规模MIMO信道的稀疏时域正交特性分析

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"时域稀疏大规模MIMO信道的渐近正交分析" 在无线通信领域，大规模多输入多输出（MIMO）技术是提升系统容量和频谱效率的关键技术之一。然而，对于大规模MIMO系统的理论分析通常基于理想化的高斯信道矩阵假设，其中通道向量具有渐近正交性。这种假设简化了分析，但可能并不完全符合实际的通信环境。这篇研究论文，"时域稀疏大规模MIMO信道的渐近正交分析"，由Zhen Gao, Linglong Dai, Chau Yuen和Zhaocheng Wang撰写，发表在2015年10月的IEEE通信快报上，旨在探讨真实世界中的稀疏大规模MIMO信道是否也能提供有利的传播条件，并揭示在实际稀疏信道下与理想高斯信道矩阵假设下的大规模MIMO系统之间的容量差距。论文的核心内容是分析了在时域中表现出空间共同稀疏（SCS）特性的大规模MIMO信道。这种特性意味着在时间域和角度域中，用户与基站天线之间存在一定的空间相关性。作者们通过理论证明，在稀疏信道环境下，不同用户的通道向量依然满足渐近正交性这一关键性质。这表明，尽管实际信道环境复杂，但大规模MIMO系统仍然可以受益于某种形式的正交性，从而提高系统性能。此外，论文还通过仿真结果验证了理论分析的正确性。这些仿真结果不仅确认了通道向量的渐近正交性，还揭示了实际信道条件下，大规模MIMO系统的性能表现与理想化模型之间的差异。这对理解大规模MIMO在实际部署中的行为至关重要，也为优化系统设计提供了有价值的信息。关键词包括：大规模MIMO，有利传播条件，稀疏信道，渐近正交性，无线通信。该研究对理解和改进实际部署中的大规模MIMO系统性能有深远的影响，特别是在考虑信道建模和信号处理策略时，这些发现将有助于开发更适应现实环境的通信解决方案。

1826 IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 10, OCTOBER 2015

Asymptotic Orthogonality Analysis of Time-Domain

Sparse Massive MIMO Channels

Zhen Gao, Linglong Dai, Chau Yuen, and Zhaocheng Wang

Abstract—The theoretical analysis of downlink massive MIMO

usually assumes the ideal Gaussian channel matrix with asymp-

totic orthogonality of channel vectors. Meanwhile, recent experi-

ments have shown that massive MIMO channels between a certain

user and massive base station antennas appear the spatial common

sparsity (SCS) in both the time domain and angle domain. This

motivates us to investigate whether realistic sparse massive MIMO

channels could provide the favorable propagation condition, and

reveal the capacity gap between massive MIMO systems over re-

alistic sparse channels and that under the ideal Gaussian channel

matrix assumption. This letter theoretically proves that channel

vectors associated with different users in massive MIMO over

sparse channels satisfy the asymptotic orthogonality. Moreover,

the simulation results conﬁrm the theoretical analysis.

Index Terms—Massive MIMO, favorable propagation condi-

tion, spatial common sparsity (SCS), asymptotic orthogonality.

I. INTRODUCTION

HE theoretical performance of massive MIMO relies on

the favorable propagation condition, which is equivalent

to the asymptotic orthogonality of channel vectors associated

with different users [1]. For theoretical analysis, the channel

is usually modeled as the ideal Gaussian channel matrix, so

that channel vectors associated with different users are asymp-

totically orthogonal and can provide the favorable propagation

condition fo r massive MIMO [1], [2].

Recent researches have shown that, due to the ﬁnite number

of signiﬁcant scatterers and the compact antenna array at the

base station (BS), realistic massive MIMO chan nels between a

certain user and massive base station (BS) antennas exhibit the

intrinsically spatial common sparsity (SCS) [3]–[8]. However,

in [4]–[6], the proposed channel estimation schemes only con-

sider the massive MIMO channels with SCS in the time domain,

while [7] and [8] only consider the SCS of massive MIMO

channels in the angle domain. For massive MIMO channels,

both the path delay in the time domain and angle of departure

(AoD)/angle of arrival (AoA) in the angle domain are essential

parameters to represent the channels. Therefore, in this letter,

we will jointly consider the SCS of massive MIMO channels

in both the time domain and angle domain. Clearly, the actual

Manuscript received January 31, 2015; revised June 24, 2015; accepted

July 16, 2015. Date of publication July 24, 2015; date of current version

October 8, 2015. This work was supported in part by the National Key Basic

Research Program of China (Grant No. 2013CB329203), Singapore A

∗

STAR

SERC Project (Grant No. 142 02 00043), National Natural Science Foundation

of China (Grant Nos. 61201185 and 61271266), and Beijing Natural Science

Foundation (Grant No. 4142027). The associate editor coordinating the revie w

of this paper and approving it for publication was M. Matthaiou.

Z. Gao, L. Dai, and Z. Wang are with the Tsinghua National Laboratory

for Information Science and Technology (TNList), Department of Electronic

Engineering, Tsinghua University, Beijing 100084, China (e-mail: gao-z11@

mails.tsinghua.edu.cn; daill@tsinghua.edu.cn; zcwang@tsinghua.edu.cn).

C. Yuen is with Singapore University of Technology and Design, Singapore

138682 (e-mail: yuenchau@sutd.edu.sg).

Digital Object Identiﬁer 10.1109/LCOMM.2015.2460243

massive MIMO channel experienced by practical systems is

different from the ideal Gaussian channel m atrix for theoretical

analysis. Such difference motivates us to investigate the propa-

gation cond ition of realistic massive MIMO chan nels with SCS,

and reveal the capacity gap between massive MIMO systems

over actual sparse channels and that under the ideal Gaussian

channel matrix assumption. Speciﬁcally, we ﬁrst introduce the

SCS of massive MIMO channels in both the time domain and

angle domain. Then, we p rove that channel vectors over sparse

channels are asymptotically orthogonal, where our analysis is

based on the widely used isotropic uniform linear array (ULA)

without loss of generality [3], [7], [8]. The asymptotic orthogo-

nality of different c hannel vectors indicates that massive MIMO

channels with SCS could provide the favorable propagation

condition, which is also conﬁrmed by simulations.

II. S

YSTEM MODEL

In typical massive MIMO systems, the BS employs M anten-

nas to simultaneously serve K single-antenna users. Usually, M

is large, and much larger than K,e.g.,M = 128 and K = 16 [1].

In the downlink, the BS transmits data x ∈ C

M×1

to K single-

antenna users, and the received signal y ∈ C

K×1

for K users can

be expressed as

y =

√

Gx+n =

√

1/2

Hx +n, (1)

where ρ

is the transmit power, G ∈ C

K×M

is the downlink

channel matrix, n ∈ C

K×1

is the additive white Gaussian noise

(AWGN) vector, the diagonal matrix D

1/2

∈ C

K×K

with the kth

diagonal element

√

represents the large-scale channel fading

due to shadow fading and path loss, while H ∈ C

K×M

denotes

the small-scale channel fading matrix due to channel variation.

Note that when the widely used OFDM is considered, (1) is

valid for every frequency-domain subcarrier.

To fully exploit advantages of massive MIMO, the BS re-

quires the downlink channel information for precoding, user

scheduling, etc. The propagation con dition of downlink channel

matrix determines the achievable capacity of downlink massive

MIMO, which can be expressed as [1]

C =log

det(I

+ρ

∗

), (2)

where I

∈ C

K×K

is the unit matrix, and (·)

∗

is the Hermitian

transpose operation. It can be observed from (2) that the

achievable capacity heavily depends on the property of channel

matrix G.

For the ideal Gaussian channel matrix, G provides the favor-

able propagation condition, where elements of the small-scale

channel fading matrix H follow the independent and identically

distributed (i.i.d.) circularly symmetric complex Gaussian dis-

tribution CN (0, 1). It has been shown that row vectors of G are

asymptotically orthogonal [1], i.e.,

lim

M→∞

∗

/M = lim

M→∞

1/2

∗

1/2

/M =D. (3)

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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