实时光域稀疏MIMO信道的渐近正交特性研究

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本文是一篇发表在2015年10月《IEEE通信快报》上的研究论文，标题为"时域稀疏大规模MIMO信道的渐近正交分析"，作者是Zhen Gao、Linglong Dai、Chau Yuen和Zhaocheng Wang。该论文主要探讨了大规模多输入多输出（Massive MIMO）通信系统中的一个重要理论问题，即在现实场景下，当信道呈现出时域和角度空间共同稀疏性（Spatial Common Sparsity, SCS）特征时，其性能与理想情况下的高斯通道矩阵假设相比有何不同。传统的理论分析通常假设大规模MIMO系统的下行信道矩阵接近于高斯分布，其中各个用户通道向量具有理想的渐近正交性。然而，随着技术的发展，实验研究表明，在实际的无线环境中，特别是大规模基站（Base Station, BS）与特定用户之间的信道，可能存在显著的空间稀疏性。这表明，真实世界中的信道特性可能并不完全符合传统的理想模型。本文的主要贡献在于，它首先理论证明了在时域稀疏的MIMO信道环境下，不同用户的通道向量仍然能够满足渐近正交性的特性。这意味着尽管信道存在稀疏性，大规模MIMO系统仍然能在一定程度上保持信号独立，这对于系统性能的分析和设计具有重要的指导意义。此外，通过仿真结果的验证，论文进一步证实了理论分析的正确性和有效性。研究者们不仅揭示了在现实稀疏信道条件下，大规模MIMO系统与理想高斯信道矩阵假设下的容量差距，还为优化无线通信系统的资源分配和信号处理策略提供了新的依据。这篇论文深入研究了大规模MIMO系统在时域稀疏信道环境中的性能，对于理解并充分利用这一新兴技术的潜力具有重要的学术价值和工程应用价值。未来的研究和实践可能会沿着这个方向进一步探索如何最大化地利用信道的稀疏性，提升通信系统的效率和可靠性。

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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