线性接收器MIMO Nakagami-m衰落信道总速率分析

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"这篇论文研究了在MIMO(多输入多输出)系统中，采用线性接收器如均方误差(MSE)接收器和迫零(ZF)接收器时，Nakagami-m衰落信道的遍历总速率。在双发射天线配置下，文章提供了新式的、闭合形式的总速率上限。此外，文中还精确计算了在低信噪比(SNR)条件下的两个关键性能指标：每信息比特传输所需的最小能量和宽带斜率，以揭示线性接收器相对于最优接收器的性能差距，并分析模型参数对总速率的影响。" 在无线通信领域，MIMO技术因其能显著提高信道容量和数据传输速率而备受关注。Nakagami-m衰落模型是一种通用的信道模型，它可以描述多种衰落环境，包括Rayleigh、Rician等特殊情况。线性接收器，如MSE和ZF接收器，因其实现简单且计算复杂度相对较低，在实际应用中广泛采用。然而，它们的性能通常逊色于最优的接收策略，如最大似然(ML)解码。论文首先探讨了在Nakagami-m衰落环境下，MIMO系统的总速率性能。对于双天线设置，作者给出了两种线性接收器的总速率上界，这是通过数学分析和优化得到的闭合形式表达式。这样的结果对于理解和预测系统在不同信道条件下的性能非常有用。进一步，论文关注低SNR条件下的系统行为，这是无线通信中常见的工作区间。文中定义并计算了两个关键参数：一是最小能量每信息比特，它表示在保证可靠传输的前提下，发送每个信息比特所需的最低平均能量；二是宽带斜率，这描述了SNR与总速率之间的关系，特别是在SNR较低时的增益。这两个参数对于理解系统在资源有限情况下的性能极限至关重要。通过对比线性接收器与最优接收器的性能，作者展示了线性接收器的次优性。他们指出，尽管线性接收器在实现上具有优势，但在某些情况下，其性能损失可能不容忽视。此外，通过分析模型参数，比如天线的数量和Nakagami-m分布的形状参数m，可以为系统设计提供指导，以优化总速率和能效。这篇论文为MIMO系统在Nakagami-m衰落信道中的性能评估提供了一种新的视角，特别是对于线性接收器的设计和优化。其结果对于无线通信系统的设计者和研究人员来说，是理解和改进MIMO系统性能的重要参考。

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 10, OCTOBER 2012 3651

On the Sum Rate of MIMO Nakagami-m

Fading Channels with Linear Receivers

Caijun Zhong, Member, IEEE, Michail Matthaiou, Member, IEEE, Aiping Huang, Senior Member, IEEE,

and Zhaoyang Zhang, Member, IEEE

Abstract—We investigate the ergodic sum rate of multiple-

input multiple-output Nakagami-m fading channels with linear

receivers. In particular, both mean square error and zero-

forcing receivers are considered. For dual transmit antenna

conﬁgurations, we present new, closed-form upper bounds on

the ergodic sum rate of both receivers. Moreover, we derive

exact expressions for the two key parameters dictating the sum

rate behavior in the low signal to noise ratio regime, namely

the minimum energy per information bit to reliably convey any

positive rate and the wideband slope. By doing so, we are able

to explicitly demonstrate the sub-optimality of linear receivers

compared to optimal receivers, and draw useful insights into the

impact of model parameters (e.g., number of antennas, fading

parameters).

Index Terms—Linear receivers, MIMO systems, Nakagami-m

fading, sum rate.

I. INTRODUCTION

T has been demonstrated that the capacity of multiple-

input multiple-output (MIMO) systems increases linearly

with the minimum number of the antennas in a rich scattering

environment [1]. This seminal discovery has triggered enor-

mous interest in understanding the capacity limits of MIMO

systems in various practical propagation environments. Thus

far, the effects of antenna correlation, line-of-sight, frequency

selectivity and keyhole phenomenon on the capacity of MIMO

systems have been well studied (see for example [2–5] and

references therein). The common characteristic of the above

mentioned works is that they all assume the use of optimal

nonlinear maximum likelihood (ML) receivers.

While ML receivers minimize the error probability when

all data vectors are equally likely by performing an exhaustive

search, they also entail the highest implementation complex-

ity, which may be prohibitive for practical cost-constrained

Manuscript received November 18, 2011; revised March 19 and May 19,

2012; accepted July 2, 2012. The associate editor coordinating the review of

this paper and approving it for publication was M.-O. Pun.

This work was supported in part by the National Key Basic Research

Program of China (No. 2012CB316104), the Nature Science Foundation of

China (61201229, 60972058 and 60972057), the Zhejiang Provincial Natural

Science Foundation of China (No. LQ12F01006 and No. LR12F01002), the

Fundamental Research Funds for Central Universities (2012QNA5011), and

the Supporting Program for New Century Excellent Talents in University

(NCET-09-0701). The work of M. Matthaiou was supported in part by the

Swedish Governmental Agency for Innovation Systems (VINNOVA) within

the VINN Excellence Center Chase. Part of the paper has been presented

at the IEEE Sensor and Multichannel Signal Processing Workshop (SAM),

Hoboken, NJ, June 2012.

C. Zhong, A. Huang, and Z. Zhang are with the Institute of Information and

Communication Engineering, Zhejiang University, Hangzhou, 310027, China

(e-mail: {caijunzhong, aiping.huang, ning

ming}@zju.edu.cn).

M. Matthaiou is with the Department of Signals and Systems, Chalmers

University of Technology, SE-412 96, Gothenburg, Sweden (e-mail:

michail.matthaiou@chalmers.se).

Digital Object Identiﬁer 10.1109/TWC.2012.090312.112054

communications systems. In light of this fact, low complexity

linear receivers, such as minimum mean square error (MMSE)

receiver and zero-forcing (ZF) receivers, have also received

a considerable amount of interest. For Rayleigh and Rician

fading channels, [6] proposed a generic framework for an-

alyzing the ergodic sum rate of MIMO systems employing

MMSE receivers, while a similar framework for ZF receivers

appeared recently in [7]. Yet, these studies were limited to

the case where the entries of the channel matrix are Gaussian

distributed.

It is well known that the Nakagami-m distribution rep-

resents a more general fading model, which encompasses

Rayleigh fading as a special case and can accurately ap-

proximate the Rician fading model [8]. Despite its practical

importance, very few works have studied the ergodic capacity

of MIMO Nakagami-m fading channels. This can be mainly

attributed to the difﬁculty in statistically characterizing the

eigenstatistics of the instantaneous MIMO correlation matrix

†

when the elements of H are not Gaussian distributed.

We recall that for H being Gaussian distributed, we can draw

upon the vast body of existing results on Wishart matrices. For

a contemporary review of Wishart theory and its application

to wireless communications, see [9].

In the context of Nakagami-m MIMO fading channels, [10]

derived the ergodic capacity of 2 ×2 and 2 ×3 MIMO conﬁg-

urations, while [11] proposed generic ergodic capacity upper

and lower bounds for arbitrary size conﬁgurations. Most lately,

[12] addressed the case with non-uniform phase distribution.

However, all these works have assumed the use of optimal

ML receivers. To the best of the authors’ knowledge, there

has been no prior work investigating the sum rate performance

of MIMO Nakagami-m fading channels with linear receivers.

Motivated by this, we hereafter study the ergodic sum rate

of MIMO Nakagami-m fading channels with MMSE and ZF

receivers. In particular, we are interested in the scenario where

the receiver has perfect channel state information, while the

transmitter does not have channel state information.

In this paper, we mainly focus on dual antenna conﬁgura-

tions, which are of high practical importance. For this reason,

these conﬁgurations have been well studied in the open liter-

ature [13–15]. More importantly, they have been employed in

next generation wireless standards, such as WiMAX and LTE-

advanced [16]. In the following, we derive new, closed-form

upper bounds on the ergodic sum rate of MIMO Nakagami-

m fading channels with 2 transmit antennas and N

receive

antennas for both MMSE and ZF receivers. In the low signal to

We note that H represents the MIMO fading matrix while H

†

denotes the

Hermitian transpose of H.

1536-1276/12$31.00

 2012 IEEE

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