Transmit Precoding in a Noncoherent Relay Channel With Channel Mean Feedback

382 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 1, JANUARY 2012

Transmit Precoding in a Noncoherent Relay ChannelWith Channel Mean Feedback

Minhua Ding, Member, IEEE, and Q. T. Zhang, Fellow, IEEE

Abstract—This paper investigates the optimal source transmitprecoding to maximize the ergodic mutual information between theinput and the output of a noncoherent amplify-and-forward relaychannel, with multiple antennas at the source and with a singleantenna at both the relay and the destination. It is assumed thatonly the source-relay channel mean information is available at thesource. The challenge here is that relaying introduces a nonconvexstructure in the objective function. Therefore, previous methodsdealing with channel mean feedback, which generally require theconcavity of the objective function, cannot be applied. To circum-vent the difficulty at hand, a different approach based on stochasticordering is employed. The stochastic optimization problem hereis ultimately transformed into the comparison of two nonnegativerandom variables in the Laplace transform order. It is shown thatthe optimal source transmit strategy is to transmit along the knownchannel mean and its orthogonal eigen-channels. Our result sub-sumes as an asymptotic case the optimal precoding for multiple-input single-output (MISO) channels without relaying under meanfeedback. Furthermore, the analysis can be partially extended tothe case with multiple antennas at the relay. Numerical examplesare provided to complement and corroborate the analysis.

Index Terms—Channel mean feedback, Laplace transformorder, multiantenna systems, mutual information, relay channels,stochastic optimization, stochastic orders.

I. INTRODUCTION

W IRELESS cooperative relay systems have been subjectof recent intensive research, due to their potential of

providing distributed diversity, extended coverage or great flex-ibility in the tradeoff between system performance and com-plexity [1], [2]. Efficient relaying protocols have been devel-oped, among which the most popular are the amplify-and-for-ward (AF) and decode-and-forward protocols [2].

To further enhance system performance, multiple antennasare deployed on one or more nodes of cooperative networks [3].However, in multiantenna systems, the optimal system struc-ture and the resultant performance (e.g., data rate, error rate,mean-square error, etc.) depend heavily on the nature of channelfading and the channel state information (CSI) available at thetransceiver [4]–[6]. In particular, by properly incorporating theavailable partial CSI at the transmitter into the transmitter de-sign, the corresponding system performance can be significantly

Manuscript received June 15, 2010; revised March 14, 2011; accepted August02, 2011. Date of current version January 06, 2012. This work was supportedby a grant from the Research Grants Council of the Hong Kong Special Admin-istrative Region, P. R. China (Project No. CityU 124408). The material in thispaper was presented in part at the IEEE International Conference on Commu-nications, Kyoto, Japan, June 5–9, 2011.

The authors are with the Department of Electronic Engineering, CityUniversity of Hong Kong, Hong Kong (e-mail: [email protected],[email protected]).

Communicated by M. Skoglund, Associate Editor for Communications.Digital Object Identifier 10.1109/TIT.2011.2169301

improved [4]. In fast fading channels, the transmitter may ac-quire the long-term channel covariance information (CCI), andrelated precoder designs can be found in [7]–[11]. In slowlyfading channels, channel mean information (CMI) can be trans-ferred to the transmitter, and relevant studies can be found in[10]–[15]. The CMI and CCI available at the transmitter arealso referred to as channel mean and covariance feedback, re-spectively. Parallel to that on traditional multiantenna commu-nications, the research on multiantenna relay communicationshas progressed from early works assuming perfect CSI [3], [16],[17] to recent investigations assuming more realistic partial CSI[18], [19].

Consider a two-hop half-duplex AF relay link with an-tennas at the source, a single antenna at the relay anda single antenna at the destination without a direct link. This sce-nario typically occurs when a traditional multiple-input single-output (MISO) source-destination link is obstructed, which is byno means uncommon in wireless transmissions, and thus a relayis used to maintain the connectivity. It is assumed that the desti-nation has full knowledge of the source-relay and relay-destina-tion channels, whereas the relay and the source are only awareof the long-term statistics of the source-relay channel. Relayingwithout instantaneous CSI (as in this scenario) is commonly re-ferred to as noncoherent relaying. No complicated processing isrequired at the relay.

For the above relay-assisted communication system, the op-timal transmit precoding at the source to maximize the ergodicmutual information between the source-relay channel input andrelay-destination channel output has been determined in [18],[19], where the source and relay are provided with the long-term covariance information of the correlated Rayleigh fadingsource-relay channel. However, when the mean information ofa slowly varying Rician fading source-relay channel is availableat the source and relay, the optimal precoding for maximum er-godic mutual information remains an open problem, which willbe solved in this paper.

With statistical CSI such as CMI and CCI, transmitter de-signs in multiple-input multiple-output (MIMO) or MISO sys-tems are often formulated as stochastic optimization problems[35]. As mean feedback and covariance feedback impose dif-ferent constraints on the optimization, techniques to deal withthem are also different. To achieve the ergodic capacity of tra-ditional MIMO or MISO Rician channels with channel meanfeedback (without relaying), various methods have been em-ployed in the literature. The optimization method in [10], [11] isbased on calculus of variations, whereas in [12], an expressionfor the ergodic mutual information is first obtained and then or-dinary calculus is applied in the optimization. The tool in [13]is the Löwner partial order for positive semidefinite matrices.

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DING AND ZHANG: TRANSMIT PRECODING IN A NONCOHERENT RELAY CHANNEL 383

In all these works, the concavity of the objective function in theoptimization variable is utilized to obtain the optimal designs.With source-relay channel mean feedback at the source in theaforementioned MISO relay channel, the resultant optimizationproblem becomes more involved, as the noncoherent relayinginduces a nonconvex structure in the objective function. Hence,previous methods mentioned above cannot be applied.

In this paper, a completely different approach based onstochastic ordering is proposed to obtain the optimal sourcetransmit precoding for the MISO relay channel with meanfeedback. The key idea here is to transform the stochasticoptimization problem into the comparison of two nonnegativerandom variables in the Laplace transform order [20]. Usingthis approach, it is shown that the optimal transmission strategyis to transmit along the known channel mean and its orthogonaleigen-channels. The obtained result gracefully matches andsubsumes as a special case that in [10, Theorem 3.1] [12].The impact of CSI at the source transmitter on the maximumergodic mutual information is further investigated. The analysisis also partially extended to the case with multiple antennas atthe relay.

The remainder of this paper is organized as follows.Section II describes the system model and the problem formu-lation. Section III presents the solution to the proposed problemas well as the proof. Further insight, discussions and numericalexamples based on the optimal solution are also provided there.Conclusions are drawn in Section IV.

Notation: Upper (lower) case boldface letters are for ma-trices (vectors); denotes the statistical expectation andthe trace of a matrix; stands for the Euclidean norm ofvector ; and denote the transpose and complex con-jugate transpose (Hermitian), respectively; is reserved forthe identity matrix; represents the circularlysymmetric complex Gaussian distribution; means that

is majorized by , whereas means that is posi-tive semidefinite; denotes a diagonal matrixwith diagonal entries given by ;and stand for the -dimensional complex matrixspace, the -dimensional real vector space and the -dimen-sional nonnegative (real) vector space, respectively; denotesthe natural logarithm.

II. SYSTEM MODEL AND PROBLEM FORMULATION

A. System Model

We focus on a two-hop half-duplex AF link with antennasat the source and with a single antenna at both the relay and thedestination. The source-relay (backward) link and the relay-des-tination (forward) link are MISO and single-input single-outputflat-fading channels, respectively. We assume that there is no di-rect link between the source and the destination. The backwardchannel is modeled as [10]:

(1)

where represents the source-relay channel mean,is a positive scaling constant , and denotesthe scattering component . Only the knowledge of

and is provided to the source, which we refer to as source-

relay CMI (or, simply, channel mean feedback) throughout thispaper.

In the first time slot, the received signal at the relay is

where is the source transmit signal vector distributedas , and is the noise at the relay.The precoding (shaping) matrix is employed at thesource, and a transmit power constraint is imposed on the sourcesuch that , where is the total sourcetransmit power. Define the source transmit covariance matrixas , and then the power constraint can bewritten as

(2)

which yields the constraint on . Note that some-times it is convenient to consider rather than as the opti-mization variable. Hereafter we will refer to both the optimiza-tion over and that over as precoding. At the relay, only thebackward CMI ( and ) is assumed to be available, which leadsto noncoherent relaying. Denote the relay amplifying factor as

. Due to the noncoherent relaying, a long-term powerconstraint is imposed on the relay, i.e.

(3)

from which we obtain

(4)

In the derivation of (3), we have used the backward CMI1 and

the identity: , since .In the second time slot, the received signal at the destination isgiven by

where is the complex forward channel coefficient, andis the noise at the destination, independent of . Since

the main analysis in this paper does not depend on the spe-cific distribution of , we assume that the distribution of isknown but otherwise arbitrary.

Moreover, the channel coefficients and in each blocktransmission are assumed to be independent and identically dis-tributed (i.i.d.) [5], [10], [11], [12]. Perfect knowledge ofand is assumed at the destination. No knowledge of the re-alization of is available at the source or relay. Since thenoise variances are all normalized to unity, denotes the sourcetransmit signal-to-noise ratio (SNR).

B. Problem Formulation

For the above relay channel, the maximization of the ergodicmutual information (in nats per channel use) between the

1Note that the relay utilizes the backward CMI only to obtain the amplifyingfactor �. Alternatively, to potentially reduce the overhead, the factor � can beevaluated at the source or at the destination and then transferred to the relay.This does not affect the subsequent problem formulation.


source-relay channel input and the relay-destination channeloutput over the source input covariance matrix is formulated asthe following stochastic optimization problem2 [18]

(5)

where accounts for the half-duplex assumption. By substi-tuting (4) into (5) and applying similar techniques to those in[22, pp. 131–132, Sec. 5.8.6], we can show that the inequalityconstraint in (5) can be equivalently replaced by the equalityconstraint [23]. Thus, we equivalently reformulate(5) as the following optimization problem:

(6)

where

(7)Our goal is to find the optimal for the problem defined by(6)–(7).

The existing methods to determine the optimal all benefitfrom the concavity of the objective function [10]–[13], and thuscannot be applied to deal with our problem. This stems fromthe fact that the function inside the expectation operatorof (7) is nonconvex in [23]. To circumvent this difficulty,below we employ a method based on one of the stochastic or-ders—the Laplace transform order [20]. The mathematical pre-liminaries pertaining to the Laplace transform order are givenin Appendix I-1. Some auxiliary steps of the following anal-ysis involve the definitions of majorization and Schur-convex/Schur-concave functions, which are now more commonly usedin wireless communications (see, e.g., [24]–[26]). Their basicsare introduced in Appendix I-2, and details can be found in [27].

III. THE OPTIMAL TRANSMIT COVARIANCE MATRIX

A. The Optimal Solution to the Problem in (6)–(7)

We first present the main result of this section.Theorem 1: The optimal for the problem formulated in

(6)–(7), denoted as , has the following structure:

where

(8)

(9)

vectors are arbitrary orthonormal vectors orthog-onal to in the -dimensional complex space, and denotesthe optimal power allocated to the eigenmode . The max-imum ergodic mutual information is given by .

2The precoding procedure here is similar to that in [10], [11], which can be(broadly) considered as a special case of Proposition 1 of [21].

Proof: We first sketch the basic idea. According to theWeierstrass’ Theorem [28, p. 654, Prop. A.8], a globally op-timal solution for (6) must exist.

Unlike previous methods, in our approach we first divide thewhole feasible set (according tothe eigen-structures) into two disjoint subsets, represented by

and , respectively, and then exploit the difference in theeigenvectors. Specifically, denote as a feasible but other-wise arbitrary covariance matrix, which has none of its eigen-vectors aligned with . On the other hand, let have the sameeigenvectors as [see (8)], and thus has an eigenvectoraligned with . Our goal is to show that given any , by anappropriate power allocation in , we can always find asuch that . In particular, we will show that, tomaximize , the same amount of power must be allocatedto the transmissions along [see (8)–(9)]. We willfurther show that with its optimal power allocation achievesat least the same ergodic mutual information as does. Theoptimality of can then be established.

Based on the above sketch, given any , we need to con-struct a specific and compare the ergodic mutual informa-tion achieved by them. To facilitate subsequent comparisons,our first step is to exploit the eigen-structures of the transmit co-variance matrices. Denote the eigenvalue decomposition ofas , where

(10)

Here constitutes an arbitrary orthonormalbasis of the -dimensional complex space, among which noneis aligned with . In addition, , , so thatthe power constraint in (6) is satisfied. Let have the sameeigenvectors as given in (8), i.e., , where

(11)

is customized for the proof with and .Also define

(12)

which, in turn, gives . Based on (1), (10) and (12), itcan be readily shown that and

where and

(13)

Clearly, has the same distribution as [see (1)]. There-fore, , , are all noncentral chi-squarerandom variables with two degrees of freedom, with their non-centrality parameters given by , , respec-tively, [31, p. 43], and is a convex combination of them.Similarly, from (8) and (11), it can be shown that


and

where and

(14)

Here also has the same distribution as , and thus ,, share the same exponential distribution.

After analyzing the eigen-structures of and , we con-tinue the construction of a specific superior to a given interms of the ergodic mutual information (7). To this end, con-sider the subset of feasible covariance matrices represented by

, and inspect all possible power allocations for these ma-trices. It is shown in Appendix II that, given any ,among all matrices [see (11)]

(15)

maximizes ; i.e., to maximize , the same amountof power must be allocated to . Hence, here-after we use (15) as an important step in the construction of thespecific achieving our goal. From (14) and (15), we obtain

(16)

Up to now, we have obtained from (7) and the above deriva-tions that

(17)

(18)

where the optimal equal power allocation amonghas been used in , and

(19)

(20)

Note that (17) and (18) differ not only in and , butalso in and , making further comparison ofthem cumbersome. To overcome this difficulty, we deliberatelychoose the only free (unspecified) parameter in , i.e., , asfollows:

(21)

where we have used the fact [see (12)]. As a sanitycheck, note that

and thus (21) is a valid choice. Using (21), we obtain

Indeed, the choice of in (21) is a crucial step in the construc-tion of a specific to fulfill our goal.

To prove with constructed using (8),(15) and (21), our next step is to show that

(22)

A straightforward method is to calculate the expectations onboth sides and compare them. The difficulty with this methodis that the calculation involves the probability density function(pdf) of a convex combination of noncentral chi-squarerandom variables, which is too complicated to serve our pur-pose [29]. Furthermore, the constituent random variables of

and [see (13) and (16)] are noncentral chi-square withdifferent noncentrality parameters and thus nonexchangeable.This, in turn, prevents us from using the classical results instochastic majorization [27, Ch. 11]. On the other hand, theLaplace transforms of the pdfs of and do possessmore elegant structures [31]. If we can avoid the pdfs and usetheir Laplace transforms instead, we will be able to circumventthe difficulty at hand. It turns out that Lemma 1 in AppendixI-1 is the precise tool we need here.

Following Definition 2 (see Appendix I-1), we can verify thatis a differentiable function in with

a completely monotone derivative for . Based onLemma 1, to show (22), it suffices to show that

[recall (13), (16), and (21)]. According to Definition 1 (seeAppendix I-1), equivalently, we need to show that

(23)

with and being the Laplace transforms of the

pdfs of and , respectively. According to [31, p. 43],

(24)

(25)


Fig. 1. Illustration of the power allocation in (9) for� � �,� � ��,� � �� ,�� ,and � �

��. Beamforming is optimal when � � �� .

Since and are positive, , (23) furtheramounts to

After some manipulations, we obtain

where

(26)

(27)

In Appendix III and Appendix IV, we show, respectively, thatand . Thus, (23) is established.

Consequently, (22) is also shown.At this moment, the only parameter in available for fur-

ther optimization is . Though chosen as per (21) is suffi-cient to guarantee for a specific , it can bepotentially further optimized to obtain as in . Therefore,for (6)–(7), [see (8)–(9)] is the optimum with being nu-merically optimized according to the fading statistics of and

. This completes our proof of Theorem 1.

Remark 1: From the proof, it is clear that the fading distri-bution of the relay-destination channel has no effect on theeigenvectors of .

Remark 2: It is important to point out that, after obtaining, we can also determine the optimal precoding ma-

trix , denoted as , where is set to be the rank of .

In fact, according to (9), there are only two possible values ofand (corresponding to and ,

respectively). When , only one data stream is transmittedfrom the source, and , a simple weighting vectorcommonly known as a transmit beamformer [10], [11], [14].When , , where can be any

unitary matrix.

B. Further Discussions

1) The Power Allocation: Using (8) and (15), with slightabuse of the notation , the ergodic mutual information asa function of can be written as

(28)

where

(29)

Then can be determined as

and the maximum ergodic mutual information is given by .This optimization can be solved using one-dimensional searchmethods [10], [11], [32, Ch. 7]. In our simulations, we simplyuse exhaustive searches.

An example of the power allocation versus the sourcetransmit SNR is shown in Fig. 1, where the numerical inte-gration has been performed using the software Mathematica.

2) The Effect of System Parameters on the Maximum ErgodicMutual Information: It is interesting to see from (28)–(29) thatthe maximum ergodic mutual information depends on onlythrough its Euclidean norm .


Fig. 2. Illustration of the effect of �� and channel state information on maximum ergodic mutual information for � � �, � � ��, and� � �� . In Case I: �� , �� .In Case II, �� , �� . In Case III,�� , �� . � �

��. Here “Perfect CSI” denotes the casewhen perfect knowledge of � and � is available at the source, whereas “CMI” denotes the case when the mean information of � is available at the source.

In Fig. 2, we plot the maximum ergodic mutual informationwith mean feedback versus the source transmit SNR for aset of parameters (see the dashed curves there). When andother parameters are fixed, the quality of CSI (reflected by theratio [10], [11]) increases with . Correspondingly,the maximum ergodic mutual information increases with ,as demonstrated in Fig. 2. In addition, the maximum ergodicmutual information with mean feedback converges at high SNRto the asymptotic limit with in [see(28)–(29)].

Remark 3: When (i.e., when the relay gain tendsto be very large), our main problem as given in (6)–(7) becomes

(30)

which is the same mathematical problem as in [10, Theorem3.1]. Thus, our result subsumes as an asymptotic case the op-timal (traditional) MISO precoding with channel mean feed-back, and it is not surprising to see the result in Theorem 1and that in [10, Theorem 3.1] share the same structure. Similarobservations have also been reported in [18] with channel co-variance feedback. In particular, our proof here can also serveto prove [10, Theorem 3.1]. However, the method used in [10]cannot be applied to obtain Theorem 1 in this paper.3

In Fig. 3, the variation of the maximum ergodic mutual infor-mation with both and is demonstrated, where the curve for

is obtained using (30). As can be seen, the curves forand almost overlap. Clearly, the simulation

results shown in Fig. 3 agree with Remark 3.

3It is probably worth mentioning that, similar to [10], we have taken a “guess-and-proof” approach to the problem given by (6)–(7). Our guess on the structureof the optimal� relies on the insight obtained from an asymptotic analysis.

3) The Effect of Channel Knowledge: When the knowledgeof and at the source becomes perfect and the channelknowledge at the relay remains the same [see (3)], the problemin (6) becomes the maximization of the instantaneous mutualinformation for each realization of and [21], i.e.

(31)

Since , (31) is simply a special case of whathas been solved in [30], and the closed-form optimal sourcetransmit covariance matrix is given by

(32)

Clearly, is rank-one, which implies that beamforming inthe direction of is optimal for (31). The maximum er-godic mutual information here can then be calculated as

(33)

In Fig. 2, we have also plotted versus the transmit SNR. While the rate performance improves with perfect CSI, in

practice, it is hardly possible to make perfect CSI available atthe source. However, the results in Fig. 2 suggest that higherCSI quality than mean feedback is desired at high transmit


Fig. 3. The impact of the relay gain � on maximum ergodic mutual information for � � �, � � ��,�� , �� , and� �

��.

SNR. We notice that at high SNR, also saturates to aconstant, which can be easily obtained from (32) and (33) byletting .

4) The Case With Multiple Antennas at the Relay: The resultin Theorem 1 can be extended to the case with antennas areat the source, antennas at the relay and a singleantenna at the destination, where the MIMO backward channelmean is of rank one.

Specifically, the MIMO backward channel is modeled as [11,Eq. (10)]

and the specular component (mean) is modeled

as: , with and being the array responses at thesource transmitter and at the relay receiver, respectively.4 As-sume that perfect knowledge of and is available at thedestination, where denotes the MISO forward channel. Fur-ther assume that the source and relay are both provided withthe knowledge of . In addition to the above, we assumethat the source is provided with full knowledge of the forwardchannel . Similar constraints to (2) and (3) are imposed onthe source and the relay, respectively. The noise vector at therelay is distributed as , and the noise at the destina-tion is . The source signals are also Gaussian.

Denote the source transmit covariance matrix as . By con-

sidering as a single random variable, applyingthe same method as that for Theorem 1, we can show that themaximum ergodic mutual information between the backward

4In general, � should be a matrix with an arbitrary rank. However, de-pending on the multipath cluster angle spread, � can also be rank-one [33],[34, Ch. 7]. Analysis with rank-one mean MIMO Rician channels can be foundin the literature (see, e.g., [11] and references therein).

channel input and the forward channel output is achieved by, where

and

. Here are arbitrary orthonormalvectors orthogonal to in the -dimensional complex space.The value of can be determined using one-dimensional searchmethods. It is interesting to note that only the array responseat the source affects the source transmit directions, which isreminiscent of [11, Theorem 3].

IV. CONCLUSION

In this paper, we have employed a method based on theLaplace transform order to derive the optimal source inputcovariance matrix (or, equivalently, the precoding matrix)which achieves the maximum ergodic mutual information ofa noncoherent two-hop half-duplex AF MISO relay channel,assuming the source-relay CMI at the source. It turns outthat the optimal transmission strategy is to transmit along theknown channel mean and its orthogonal eigen-channels. Ourresult agrees with and subsumes as an asymptotic case theoptimal precoding for a traditional MISO link. The analysisis also partially extended to the case with multiple antennasat the relay. The effect of channel state information has beeninvestigated as well. Numerical results show that the maximumergodic mutual information increases with the Euclidean normof channel mean with other parameters being fixed,i.e., it increases with the quality of channel state information.


Furthermore, we expect that the powerful stochastic orderingapproach [20] (e.g., the Laplace transform ordering approachused in this paper) will find many applications pertaining to sto-chastic optimization problems in wireless communications andsignal processing.

APPENDIX IMATHEMATICAL PRELIMINARIES

1) The Laplace Transform Order:Definition 1 [20, p. 233]: Let and be two nonnegative

random variables such that

Then is said to be smaller than in the Laplace transformorder, denoted as .

The Laplace transform order5 is implied by the usual sto-chastic order [20], [27]. Therefore, it is potentially suitable for awider range of applications. The usual stochastic order has beenused in [37].

Definition 2 [20, p. 234]: A functionis said to be completely monotone if for all and

, its derivatives exist and .Note that .

Lemma 1 [20, p. 235, Theorem 5.A.4]: Let and betwo nonnegative random variables. Then if and onlyif for all differentiable functions on

with a completely monotone derivative, provided that theexpectations exist.

2) Majorization and Schur-Convexity:

Definition 3 [27, p. 8, Ch. 1, A.1. Definition]: Let ,. Let denote the

components of in decreasing order. Similarly, letdenote the components of

in increasing order. Then is majorized by , denoted as, if either of the following holds:

• Majorization definition 1)

(34)

• Majorization definition 2)

(35)

Definition 4 [27, p. 80, Ch. 3, A.1. Definition]: A real-valued function defined on a set is said to beSchur-convex on if

Furthermore, is Schur-concave if and only if is Schur-convex.

5Note that in the standard definition of the Laplace transform, � is in generalcomplex. However, to apply the Laplace transform order, � is considered as apositive (real) number [20, p. 233]. Throughout this paper, we consider � � �

only.

APPENDIX II

Our goal here is to prove the following result.Let , , , and be defined as in (11), (15), (14),

and (20), respectively. Then given a fixed

(i.e., ) is the solution to the following problem:

(36)

Proof: Recall that has been defined in (14). Clearly,from (14), are exchangeable random vari-ables in the sense that their joint density remains the same underany permutation of its arguments [27, p. 392]. Furthermore,given and given any specific realizations of and , thefollowing function:

is continuous and concave in . According to [27, p. 395,Ch. 11, B.2.c]

is symmetric and concave. Thus, is Schur-concave in. Since , from [27, p. 9, Eq. (8)],

Using the Schur-concavity of

Since this holds for any realizations of and , we concludethat, given

is also maximized by .

It is worth pointing out that there exists an alternative proofbased on the Laplace transform order. Let be defined in(16). To show the desired result, it is equivalently, we need toshow that, given

which can be proved following similar lines to those in the prooffor (22).6

6The method used in the first proof in Appendix II was pointed out by oneof the anonymous reviewers. This method has been used in the literature (see,e.g., [26]). The alternative proof was used in our original submission of thispaper. The method in the alternative proof represents another application of theLaplace transform order (see also [36]).


APPENDIX III

Here we want to prove that , subject to (21).Toward this end, we first establish the following result.

Let and be thevectors corresponding to the diagonal power allocation matricesin and , respectively [see (10) and (15)], where

is the power allocated to the eigenvector in .Then under the condition (21), i.e., given

, the following holds:

(37)

Proof: Without loss of generality, assume that. For a thorough treatment, consider the following two

cases.Case 1) , i.e.,

Since the entries of and both sum to unit, according toDefinition 3-1) [see (34)], to show (37), we need to furtherestablish that

.(38)

Due to (21)

and thus . On the other hand, since ,

we have [27, p. 9,Eq. (8)], which implies that for

(39)

Thus, for

(40)

(41)

where (40) follows since and is pos-itive for , and (41) follows from (39).Therefore, (38) is established, and (37) holds in Case 1).Case 2) , i.e.,

Note that here is the smallest among all the entries of .From Definition 3-2) [see (35)], to show (37), equivalently,we need to show that

.(42)

Due to (21)

and thus . Apply [27, p. 9, Eq. (8)] again to obtain

, which impliesthat for ,

(43)

Thus, for

(44)

(45)

where (44) follows since , and (45) follows from(43). Thus, (42) is established, and (37) holds in Case 2).This also concludes the proof of the first result in thisappendix.

To further prove that , we consider thefunction

(46)

where is an arbitrary positive constant . Clearly,remains the same for any permutation of the elements of , i.e.,it is symmetric. Since is convex in , according to

[27, p. 92, Ch. 3, C.1. Proposition], is Schur-convex in .Based on (37) and the Schur-convexity of (46)

This concludes the proof.

APPENDIX IV

Here we prove that under (21), .From (21),

, and thus we obtain

(47)

On the other hand,


However,

(48)

(49)

where (48) is obtained by exchanging the summation order overand , and (49) is valid since , are dummy variables here.

Therefore,

(50)

Since (50) is always nonnegative, (47) is also nonnegative. Thus, the proof is concluded.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers fortheir helpful comments on the paper. The authors also wish tothank R. K. Mallik and M. R. McKay, for helpful discussions,and P. Dharmawansa, for helpful suggestions on the manuscript.

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Minhua Ding (S’00–M’08) received the B.S. and M.S. degrees from BeijingUniversity of Posts and Telecommunications (BUPT), Beijing, China, in 1999and 2002, respectively, and the Ph.D. degree from Queen’s University, Kingston,ON, Canada, in 2008, all in electrical engineering.

Her research interests include communication theory, information theory, andstatistical signal processing.

Q. T. Zhang (S’84–M’85–SM’95–F’09) received the B.Eng. degree from Ts-inghua University, Beijing, and the M.Eng. degree from South China Univer-sity of Technology, Guangzhou, China, both in wireless communications, andthe Ph.D. degree in electrical engineering from McMaster University, Hamilton,Ontario, Canada.

After graduation from McMaster University in 1986, he held a research po-sition and Adjunct Assistant Professorship at the same institution. In January1992, he joined the Spar Aerospace Ltd., Satellite and Communication SystemsDivision, Montreal, as a Senior Member of Technical Staff. At Spar Aerospace,he participated in the development and manufacturing of the Radar Satellite(Radarsat). He joined Ryerson University, Toronto, in 1993, and became a Pro-fessor in 1999. In 1999, he took one-year sabbatical leave at the National Uni-versity of Singapore. He is now a Chair Professor of Information Engineeringat the City University of Hong Kong. His research interest is on wireless com-munications with current focus on wireless MIMO, cooperative systems, andcognitive radio.

Dr. Zhang was an Associate Editor of the IEEE COMMUNICATIONS LETTERS

from 2000 to 2007 .

Transmit Precoding in a Noncoherent Relay Channel With Channel Mean Feedback

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