几类信道不确定性的鲁棒MIMO预编码资源-CSDN文库

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标题中提到的“鲁棒MIMO预编码”涉及的是无线通信技术中的关键概念。MIMO（多输入多输出）技术是现代通信系统中的一项重要技术，它允许多个发送和接收天线之间的数据传输，大大提高了系统的数据传输速率和频谱效率。然而，为了有效利用MIMO技术，发送端需要准确的信道状态信息（CSIT），这在实际中往往很难做到，因为会有各种不确定性的存在，如信号的反射、散射、折射等因素，都会影响信号的传播路径，从而影响到信道的状态。 “信道不确定性”是指信道状态信息存在偏差或不完整的情况。当存在信道不确定性时，传统预编码技术可能会产生性能下降，因此需要设计能够抵抗这种不确定性的鲁棒预编码技术。鲁棒预编码的目标是使系统能够抵御一定的信道波动，以确保在不利信道条件下系统性能不会显著下降。在描述中提到的“鲁棒预编码设计”遵循的是“最坏情况鲁棒性”设计哲学，即最大化最坏情况下的信噪比，或最小化最坏情况下的误码率。文章提出了一种新的设计框架，它能够将不同的不完整CSIT情形纳入考虑，通过建立一个最大化最小化问题来求解鲁棒预编码器。文章还提出了一个单一的凸优化问题来有效计算鲁棒预编码器的解决方案。在内容部分，作者们提到了“凸不确定性集”、“单位不变凸不确定性集”、“Schatten范数”等关键概念。这些概念在数学和信号处理中非常重要，与优化问题的求解有着密切关系。例如，凸集具有很好的数学特性，凸优化问题在数学上可以通过各种成熟的算法来求解，保证了找到全局最优解的可能性。而单位不变性意味着在特定的数学变换下，系统的结构能够保持不变，这对于设计具有普遍适用性的鲁棒预编码算法至关重要。Schatten范数是对矩阵的一种度量，它可以用来定义矩阵不确定性的集合，这对于处理信道状态信息的不确定性提供了数学工具。研究论文中还提到，在单位不变凸不确定性集的情况下，通道对角化结构的最优性被证明了。这意味着通过将多维的复数矩阵问题简化为实数向量的功率分配问题，可以通过水填充算法来解析求解。水填充算法是一种在给定功率约束条件下寻找最优功率分配的方法，因其直观形象而得名。至于Schatten范数定义的不确定性集，作者们给出了一个完全封闭形式的解决方案，这使得波束形成和均匀功率传输的鲁棒性分析成为可能。文章中提到的“索引术语”，包括“凸不确定性集”、“不完整的CSIT”、“最大化最小化”、“MIMO”、“最小最大化”、“鞍点”、“Schatten范数”、“单位不变不确定性集”和“最坏情况鲁棒性”，这些术语在理解和研究无线通信、信号处理以及优化算法领域是不可或缺的。总结来看，这篇论文介绍了鲁棒MIMO预编码设计的最新研究成果，将多种信道不确定性情况纳入统一框架，并给出了有效的求解算法和理论分析。这对于提升通信系统的鲁棒性和可靠性具有重要的实践价值和理论意义。

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3056 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 12, JUNE 15, 2013

Robust MIMO Precoding for Several Classes of

Channel Uncertainty

Jiaheng Wang, Member, IEEE, Mats Bengtsson, Senior Member, IEEE,BjörnOtt

ersten, Fellow, IEEE,and

Daniel P. Palomar, Fellow, IEEE

Abstract—The full potential of multi-input multi-output

(MIMO) communication systems relies on exploiting channel state

information at the tra

nsmitter (CSIT), which is, however, often

subject to some uncertainty. In this paper, following the worst-case

robust philosophy, we consider a robust MIMO precoding design

with deterministic

imperfect CSIT, formulated as a maximin

problem, to maximize the worst-case received signal-to-noise ratio

or minimize the worst-case error probability. Given different types

of imperfect C SIT

in practice, a uniﬁed framework is lacking in

the literature to tackle various channel uncertainty. In this paper,

we address this open problem by considering several classes of

uncertainty

sets that include most deterministic imperfect CSIT

as special cases. We show that, for general convex uncertainty

sets, the robust precoder, as the solution to the maximin problem,

can be efﬁc

iently computed by solving a single convex optimiza-

tion problem. Furthermore, when it comes to unitarily-invariant

convex uncertainty sets, we prove the optimality of a channel-di-

agonali

zing structure and simplify the complex-matrix problem

to a real-vector power allocation problem, which is then analyt-

ically solved in a waterﬁlling manner. Finally, for uncertainty

sets d

eﬁned by a generic matrix norm, called the Schatten norm,

we provide a fully closed-form solution to the robust precoding

design, based on which the robustness of beamforming and uni-

for

m-power transmission is investigated.

Inde

xTerms—Convex uncertainty set, imperfect CSIT, max-

imin, MIMO, minimax, saddle point, Schatten norm, unitarily-in-

variant uncertainty set, worst-case robustness.

Manuscript received July 17, 2012; revised January 5, 2013; accepted March

10, 2013. Date of publication April 12, 2013; date of current version May

20, 2013. The associate editor coordinating the review of this manuscript

and approving it for publication was Prof. Y.-W.Peter Hong. This work was

supported in part by the National Basic Research Program of China (973) under

2013CB336600 and 2013CB329204, National Natural Science Foundation of

China under 61201174, Natural Science Foundation of Jiangsu, China under

BK2012325, and by the European Research Council under the European

Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant

agreement no. 228044, and by the Hong Kong RGC 617911 research grant,

and by the National Key Special Program of China no. 2012ZX03003005-003.

J. Wang was with the Signal Processing Laboratory, ACCESS Linnaeus

Center, KTH Royal Institute of Technology, SE-100 44 S toc kh o lm, Sweden.

He is now with the National Mobile Com munications Research La boratory,

Southeast University, Nanjing, China (e-mail: jhwang@seu.edu.cn).

M. Bengtsson is with A CCESS L inn aeu s Center, Signal Processing Lab-

oratory, KTH Royal Institute of Technology, S to ckholm SE- 10 0 44, Sweden

(e-mail: mats.bengtsson@ee.kth.se).

B. Ottersten is with ACCESS Linnae us Center, Signal Processing Laboratory,

KTH Ro y a l Institute of Te chnology, Stockholm SE-1 00 44, Sweden , and a lso

with th e Interdisciplinary Centre for Secur ity, Reliability and Trust (SnT) , Uni-

versity of Luxembourg, Luxembourg-Kirchberg L-1359, Luxembourg (email:

bjorn.ottersten@ee.kth.se; bjorn.ottersten@uni.lu).

D. P. Palomar is with the Hong Kong University of Science and Technology,

Hong Kong, and also with the Centre Te cnológic de Te lecom municacions de

Catalunya-Hong Kon g (CTTC- HK), Hong Kong (e-mail: paloma r@ust.hk).

Color v ersions of one or more of the ﬁgures in this paper are available online

at http://ieeexplore.ieee.org.

Dig

ital Object Identiﬁer 10.1109/TSP.2013.2258016

I. INTRODUCTION

T is well known that the performance of multi-in

put multi-

output (MIMO) comm unication system s depends

,toasub-

stantial extent, on the quality of the channe

l stat e information

(CSI) [1]. The full beneﬁts of MIMO channels

are achieved by

exploiting CSI at the transm itt er (CSIT) an

d adopting proper

precoding techniques [2], [3]. With p

erfect CSIT, the optimal

MIMO precoding has been well studied

under various criteria

[4]–[6] for either single-user or mu

lti-user com munications. In

practice, however, CSIT is seldom p

erfect but subject to some

uncertainty due to many practical

issues, such as i naccurate

channel estimation, quantizati

on o f CSI, erroneous or outdated

feedback, and time delays or fre

quency offsets between the re-

ciprocal channels. Therefo

re, the imperfection of CSIT has to

be considered in MIMO preco

ding designs so that the commu-

nication system, on one han

d, can fully utilize CSIT, and on the

other hand, is robust to va

rious i m perfect CSIT.

In the literatur e , imperfe

ct CSI is modeled by either stochastic

or de terministic approac

hes. The stochastic model assumes that

the channel is a random q

uantity and its instantaneous value

is unknown but its stat

istics, such as the mean and/or the co-

variance, is known by

the transmitter. In this case, the robust

design usually aims a

t optimizing either the long-term average

performance [7]–[1

0] or the outage performance [11]–[14]. On

the o ther hand, the

deterministic model, which is mo re suit-

able to characteri

ze instantaneous CSI with errors, assumes that

the actual chann

el lies in the neighborhood, often called the

uncertainty se

t or region, of a nominal channel known by the

transmitter. T

he size of this set represents the amount of un-

certainty on t

he channel, i.e., the larger the set is the more un-

certainty th

ere is. In this case, a precoding design is said to

be ro bust if i

t can achieve the best performance in the worst

channel wi

thin the uncertai nty set, or equ ival entl y can guar-

antee a per

formance level for any channel in the uncertainty set.

Such robu

st precoding desig ns can be achieved by optimizing

the worst

-case performance [15], often leading to a maximin or

minimax

problem [16]–[32]. Note that, worst-case robustness is

to statistical robustness in some situat ion . For example,

many ou

tage based robust designs are often transformed into d e-

termi

nistic formulations by deﬁning an uncertainty set that has

a cert

ain probability [11], [12], [14].

The fo

cus of this paper is on worst-case robust precoding de-

signs

based on deterministic imperfect CSIT. As an important

bran

ch of robust designs [15], the philosophy of worst-case

rob

ustness has been widely used in signal processing [16]–[19]

WA NG et al.: ROBUST MIMO PRECODING FOR SEVERAL CLA SSE S OF C HANNEL UNCERTAINTY 3057

and communications [20]– [3 2]. In terms of MIMO com -

munications, [20] studied the compound capacity [3 3] of a

MIMO channel for an isotropically unconstrained uncertainty

set, which was recently generalized to a class of norm-based

uncertainty sets [21]. The worst-case robust minim um mean

square error (MMSE ) preco der was proposed in [22 ]. In [23]

and [24], the authors tried to maximize the worst-case received

signal-to-noise ratio (SNR) but only focused on a simpliﬁed

power allocation problem by ﬁxing the transmit directions.

Interestingly, it was recently f ound in [25] and [26] that the

transmit directions imposed in [23], [24] are optim al in some

situations, which leads to fully analytical robust precoders

along with some interesting insights. The worst-case robust

precoding was also studied for MIMO m ultiaccess channels

[27], broadcasting channels [28], multi-cell systems [30], [31],

and cognitive radio systems [29], [32].

As a c onsequence of many practical factors that may cause

imperfect CSIT, there are various channel uncertainty models

commonly used in t he literature. For example, the ch annel error

induced by quantizing CSI is generally regarded to fall into a

polyhedron [23]. As the most frequently used imperfect CSI

model [16]–[29], the channel e rro r is often covered by a sph ere

or ellipsoi d uncertainty set that is usually deﬁnedbyamatrixor

vector norm, such as the Frobenius n orm [22]–[25], [ 27 ]–[29]

or the spectral norm [17], [21], [26], where the shape of the un-

certainty set or the coverage of possible imperfection is deter-

mined by which norm is used. An uncertainty set can also be

deﬁned by other means, for example the Kullback-Leibler diver-

gence [19]. Despite different types of imp erfect CSIT that may

be encountered in practice, most existin g works on worst-case

robust designs focused o nly on one or a few particular uncer-

tainty sets, mainly based on the Frobenius and spectral norms

due to their amenability. So far there lacks a uniﬁed framework

on worst-case robust MIMO precoding that is applicable to var-

ious channel uncertainty. The major goal of this paper is to ad-

dress this open problem.

To be m ore exact, in this paper we consider a worst-case

robust MIMO precoding design, form ulated as a maximin

problem, to maximize the worst-case received SNR or to

minimize the worst-case pairwise e rr or prob ability (PEP) if a

space-time block code (STBC) [34], [35] is used. In contrast

with the existing works [16]–[22], [24]–[29] that are based on

particular channel uncertainty sets, we try to take into account

various channel u ncertainty within a uniﬁed framework by con-

sidering several general classes of uncertainty sets that differ

in generality and tractability. We provide robust precoding de -

signs for arbitrary convex uncertainty sets, unitarily-invariant

convex uncertainty sets, and the uncertainty sets deﬁnedbya

generic matrix norm termed the Schatten norm. These general

uncertainty sets contain all aforementioned d eterministic im-

perfect CSIT models as special cases. Therefore, the previous

works, e.g., [23]–[26], are included as special cases in this

uniﬁed framework. We show that the formulated worst-case

robust MIMO precoding design with the general uncertainty

sets can be elegantly handled through convex o ptimizatio n

[36]. The main contributions of this paper are as follows.

We start from the m ost general case where the uncertainty set

is a nonempty compact convex set, which covers all existing un-

certainty sets, e.g., [16]–[22], [24]–[29], and may also be used

to m odel more complicated deterministic imperfect CSIT. Note

that [23] also considered a convex uncertainty set bu t only fo -

cused on a simpliﬁed power allocation p ro blem by imposin g

some ﬁxed transmit directions without knowing whether they

are optimal or not. So far the optimal worst-case rob ust M IMO

precoder for a general convex uncertainty set h as still been un-

known. In this paper, we solve this open problem by relating

the formulated maximin problem to a minimax problem from

the dual perspective, and showing that the robust precoder can

be efﬁciently foun d by solving a single co nvex optimization

problem. As a byproduct, the worst channel for the robust pre-

coder can also be obtained simultaneously. Furthermore, we

provide a practical reformulatio n of the convex problem that

can be efﬁciently handled by general optimization methods as

well as software packages.

Then, we consider the case where the convex uncertainty

set is unitar ily -invariant, which contains many uncertainty sets

based on matrix norms, e.g., [16]–[18], [20]–[29]. It is shown

that the robust precoder results in a favorable channel-diagonal-

izing structure, and thus the precoding design can be simpliﬁed

to a power allocation problem without any loss of optimality.

Note that such a desirable structure was only found to be optimal

for the uncertainty sets based on s om e speci

ﬁcmatrixnorms

in [25] and [26]. We fu rth e r sh ow that, given a particular form

of the unitarily-invariant convex set, both the robust precoder

and the worst channel can be simultaneously diagonalized, thus

fully reducing the com plex-matrix robust precoder design to a

real-vector convex problem. Moreover, we show that the solu-

tion of such a real-vector problem can be analytically obtained

via a convenient waterﬁlling fashion.

Finally, we consider uncertainty sets deﬁned by the Schatten

norm [37], which is a special case of unitarily -invariant convex

sets, but still general enough to contain most frequently used

uncertainty sets based on, e.g., the Frobenius norm [22]–[25],

[27]–[29] or the spectral norm [17], [21], [26]. As a generic

norm, the Schatten norm includes not only the Frobenius and

spectral norms, but also other common matrix norms such as

the nuclear norm [38]. In this case, we provide fully closed-form

solutions to the robust precoder, based on which we also investi-

gate the robustness of beamform ing and equal power transmis-

sions.

The paper is organized as follows. Section II i ntroduces the

system model, problem formulation, and various channel uncer-

tainty models. Section III fo cuses on how to achi eve the optimal

robust precoder for general convex uncertainty sets in an efﬁ-

cient way. Section IV considers un itarily-invarian t convex un -

certainty sets and shows the optimality of the eigenmode trans-

mission, whi le Section V focuses on uncertain ty sets deﬁned by

the Schatten norm. Numerical results are provided in Section VI

and Section VII concludes t he paper.

Notation: Uppercase and lowercase boldface denote m atrices

and vectors, respectively.

, ,and denote the sets of real

numbers, complex numbers, and positive semide ﬁnite matrices,

respectively.

and represent the identity m atrix and the vector

of ones, respectively.

denotes the ( th, th) element of a

matrix

,and denotes the th column of the identity m atrix.

or , we mean that is a positive semideﬁ-

3058 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 12, JUNE 15, 2013

nite or deﬁnite matrix, respectively. The operators and are

deﬁned componentwise for vectors and matrices. The operators

, ,and denote the Herm itian, inverse, and trace

operations, respectively.

and represent the vectors

of singular values and eigenvalues of a matrix

, respectively.

The maximum eigenvalue of a Hermitian matrix is denoted by

. denotes a general matrix norm as well as the Eu -

clidean norm of a vector.

, ,and denote the nu-

clear, Frobenius, and spectral norms of a matrix, respectively.

and denote the real and i mage parts of a complex

value, respectively. We deﬁne

II. P

ROBLEM STATEMENT

A. System Model

Consider a narrowband point-to-point M IMO com m unica-

tion system equipped with

transmit and receive antennas.

Mathematically, the baseband, symbol-sampled system can be

represented by a linear model

(1)

where

and are the transmitted and re-

ceived signals, respectively,

is the channel ma-

trix, and

is a circularly symmetric comp lex Gaussian

noise vector with zero mean and covariance m atrix

,i.e.,

. The transmit strategy or precoding is de-

termined by the tra nsmit covariance matrix

Indeed, via decomposing

, the transmitted symbol

vector

,with , can be linearly precoded by ,re-

sulting in

. In practice, the transm itter should satisfy the

power constraint

where

(2)

and

is the bud get on the total transmit power.

Under the assumption of perfect CSIT, i.e., the channel

is perfectly known at th e transmitter, th e optimal MIMO

precoding h as been well studied fo r various criteria [4], [5].

However, due to many practical issues, CSIT is seldom perfect,

which thus calls fo r rob ust p recoding designs that can utilize

CSIT and at the same tim e combat against its imperfection. To

model imperfect CSIT, we consider a compound channel model

[33] assuming that

belongs to a known set , often called

an uncer tain ty set, of possible va lues but otherwise unknown.

In the literature, this im per fect channel model has been w idely

used in robust designs, an d the philosophy behind these robust

designs is the so-called worst-case r obu stness [15], which is

achieved by optimizing the system performance for the worst

channel in

[16]–[29].

Speciﬁcally, w e denote the system perform ance measure by

a utility or payoff function

. Then, the worst-case ro-

bust transmit strategy i s given by the soluti on to the following

maximin problem:

(3)

which, namely, offers the best performan c e for the worst

channel within

. As a counterpart of the maximin problem,

we also introduce the following minimax problem:

(4)

which is, name ly, to ﬁnd the worst channel for the best one of all

possible transmit strategies. We will show later that the maximin

problem (3) and the minimax problem (4) are closely related.

In this paper, we assume perfect CSI at the receiver (CSIR)

and adop t the following payoff or utility function:

(5)

which is proportional to the received SNR. It can be veriﬁed (see

Section II in [25]) that maxim izing

corresponds to: 1)

maximizing the received SNR; 2) minimizing the pairwise error

probability (PEP) i f a space-time block code (STBC) [34], [35]

is used at the transmitter; 3) maximizing a low-SNR approx-

imation of the mutual inform ation; 4) minimizing a low-SNR

approximation of the MSE if a linear MMSE equalizer is used

at the receiver.

B. Channel U ncertain ty M odels

In the literature [16]–[29], the uncertainty set

is often mod-

eled as a neighborhood of a nomin al channel

known by the

transmitter, whe re the nominal channel

couldbeanestimate

or feedback of the actual channel

.Bydeﬁning the channel

error

as the difference between the nominal channel and the

actual channel as

, th e uncertainty can be

equally described by

for some set . Co rrespon din gly,

we can rewrite the utility function in (5) based on

(6)

and thus the m axim in and minimax problems (3) and (4) based

can be expressed as

(7)

and

(8)

based on

, respectively.

The channel uncertainty set

provides a convenient way to

characterize different types of imperfect CSIT that are caused

by different reasons in practice. However, most existing works

on wo rst-case robust MIMO precoding d esigns, e.g., [16]– [22],

[24]–[29], considered only one or several particular choices

of the u ncert a int y set

, mostly focusing on the uncertainty

sets based on the Frobenius and sp ectral norms du e to their

tractability. So far a uniﬁed framework that is applicable to

various channel uncertainty sets is still absent. The major goal

of this paper is to address this open problem. To be more exact,

we consider various kin ds of channel uncertainty sets that

differ in terms o f generality and tractability. S peciﬁcally, the

uncertainty set

could be given in any one o f the followin g

forms:

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