1348 IEEE TRANSACTIONS ON SIGNAL PROCESSING, …lwang1/pub/TSP2014.pdf · · 2015-10-06Asymptotic and Non-Asymptotic Analysis of Uplink SumRateforRelay-AssistedMIMOCellularSystems

1348 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 6, MARCH 15, 2014

Asymptotic and Non-Asymptotic Analysis of UplinkSum Rate for Relay-Assisted MIMOCellular Systems

Hao Wu, Liming Wang, Member, IEEE, Xiaodong Wang, Fellow, IEEE, and Xiaohu You, Fellow, IEEE

Abstract—We consider uplink relay-assistedMIMO cellular sys-tems with multiple relays deployed in each cell that perform theamplify-and-forward operations. We are interested in obtainingdeterministic expressions for the ergodic sum rate of such systems.We first consider the large MIMO dimension scenario, and obtaintwo asymptotic sum rate expressions, corresponding to the cases offixed number of users and large number of users, respectively. Wethen consider the case of arbitrary MIMO dimension and largenumber of users, and obtain an upper and lower bound that thesum rate lies in betweenwith high probability. The bounds are tightwhen the number of users is large. Both single-cell and multi-cellsystems are considered, where the latter assumes full base stationcooperation. Numerical experiments show that these deterministicsum rate expressions match well with the Monte Carlo simulationresults. Therefore they can be useful tools for the design and anal-ysis of relay-assisted MIMO cellular systems.

Index Terms—Uplink sum rate, MIMO, relay, multi-cell,random matrix, asymptotic analysis, non-asymptotic analysis.

I. INTRODUCTION

R ELAY-ASSISTED cooperative communications have re-ceived significant recent interests due to their potential of

enhancing the system throughput. Among the several existingrelay strategies, the amplify-and-forward (AF) scheme is easyto implement and amenable to theoretical analysis. In particular,the rate of wireless AF systems is analyzed in [1]. In [2] and [3],the asymptotic rate of a MIMO AF system with finite numberof relay hops is analyzed when the number of antennas is large.Optimal precoding at the source and relays is also discussed tomaximize the asymptotic sum rate in [2]. However, these worksconsider only the case of a single transmitter and receiver, andthe direct link from source to sink is ignored. On the other hand,the achievable rate of a MIMO relay system employing the AFstrategy is discussed in [4]–[7] and [8], where the direct link

Manuscript received March 04, 2013; revised June 03, 2013 and June 20,2013; accepted July 01, 2013. Date of publication July 24, 2013; date of currentversion February 21, 2014. The associate editor coordinating the review of thismanuscript and approving it for publication was Prof. Jian-Kang Zhang. Thiswork was supported in part by the Natural Science Foundation of China undergrants 61221002 and 61101086 and 61329101.H. Wu and X. You are with the National Communications Research Lab,

Southeast University, Nanjing 210096, China (e-mail: [email protected];[email protected]).L. Wang is with the Department of Electrical and Computer Engineering,

Duke University, Durham, NC 27708 USA (e-mail: [email protected]).X.Wang is with the Electrical EngineeringDepartment, ColumbiaUniversity,

New York, NY 10027 USA, and also with King Abdulaziz University, Jeddah,Saudi Arabia (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2013.2274642

from the source to the destination is considered. Furthermore,the ergodic capacity and the optimal input covariance matrix toachieve the capacity are analyzed in [9] with the presence ofmultiple relays and direct link.In this paper, we are interested in computing the sum rate of

an uplink MIMO AF system in a single-cell or multi-cell net-work with multiple relays. The multi-user capacities of uplinkand downlink MIMO systems are given in [10]. More recently,multi-cell systems have been considered [11]. In [12], a multi-cell processing model is considered where a full BS coopera-tion strategy is employed resulting in an augmented multi-usermodel. The asymptotic sum rate for such system is then ana-lyzed using the random matrix theory. Moreover, the sum rateof an uplink TDMAmulti-cell systemwith non-regenerative AFrelays is analyzed in [13], where it is assumed that the mobileusers serve as relays to help each other. In this paper we areinterested in systems with fixed relay stations that perform AFoperations.Specifically, we perform both asymptotic and non-asymptotic

analyses on the sum rate forMIMO cellular communication sys-tems employing fixed AF relay stations and multi-cell coopera-tion. We start from a single-cell uplink MIMO system with mul-tiple AF relays and multiple users in Gaussian fading channels.The relays just amplify and forward the signals, and we considerprecoding matrices at both users and relays. At the receivingend, the base station (BS) receives and combines signals fromall users and all relays. All channel links including users to BS,users to relays and relays to BS (Fig. 1) are taken into account.In addition, the Kronecker model [14] is employed to modelcorrelated MIMO channels. The asymptotic results that we ob-tain are useful in calculating the sum rate of a system with largenumber of antennas. On the other hand, the non-asymptotic re-sults are useful for systems with any number of antennas andlarge number of users. Extension to multi-cell systems is alsogiven.The remainder of the paper is organized as follows. In

Section II, we give the system model. In Section III, we providetwo asymptotic analyses on the sum rate. The non-asymptoticanalysis is given in Section IV. Numerical results are providedin Section V. Section VI contains the conclusion.

II. SYSTEM MODEL

We start by considering a single-cell MIMO system with re-lays. Specifically, the cell has one BS at the center, mo-bile users at random locations within the cell and relays atsome fixed positions. The amplify-and-forward (AF) strategy isadopted at each relay, where a precoding matrix is applied to

1053-587X © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

WU et al.: RELAY-ASSISTED MIMO CELLULAR SYSTEMS 1349

Fig. 1. An uplink single-cell MIMO system with relays.

Fig. 2. The two-stage amplify-and-forward (AF) relay strategy.

the received signal which is then forwarded to the BS. The BSreceives the signals from all users and all relays. Moreover, weassume perfect synchronization among the relays so that thereis no interference among them.The system model is depicted in Fig. 1, with various param-

eters annotated. Specifically, we assume that each user hasantennas, each relay has antennas and the BS has an-tennas. User transmits information signal through a pre-coding matrix , which is received by the BS and all relays.Weassume unit transmit power, i.e., . The channelmatrix between user and the BS is , and the channel matrixbetween user and relay is . Upon receiving the superpo-sition of all user signals, relay forwards it to the BS using aprecoding matrix , and the channel matrix between relayand the BS is . Although the BS also receives the noise for-warded by each relay, it was shown in [1] that the impact ofsuch forwarded noise on the system capacity is negligible. InSection V, simulation results demonstrate that the system ca-pacity is hardly affected by the relay noise. Therefore we onlyconsider the additive noise at the BS.We employ a two-stage AF relay strategy as in [4]–[8] shown

in Fig. 2. In the first slot, each user transmits its signal and allrelays and BS receive the signals, and the relays do not transmit.In the second slot, relays forward the signals received during thefirst slot and the BS receives the forward signals. The users donot transmit during the second slot.According to the description above, by stacking the received

signals by the BS in the two slots, we obtain the following signalmodel

(1)

(2)

where is the additive complex Gaussian noise with zeromean and covariance matrix and denotes

the signal-to-noise ratio (SNR). In practical systems, some ofthe users may not use relaying for uplink transmission becauseof the half-duplex penalty in rate. In that case we can simplyset the relay powers for the users who choose not to employrelaying to zero. Hence the analysis developed in this paper canbe applied to the general network scenario where some usersemploy relaying and others do not.We assume that the MIMO channels are in general spacially

correlated according to the Kronecker model [14] and can beexpressed as

(3)

(4)

(5)

In (3)–(5), the matrices and contain zero-meanindependent and identically distributed (i.i.d.) Gaussian entries;thematrices and are fixed correlationma-trices. We assume that the precoding matrices and are in-dependent of and . This is the case when covariancefeedback is employed, i.e., when the precoders are designedbased on the channel covariance matrices [15]. This assump-tion is also reasonable when codebook-based precoders are em-ployed [16]–[19], e.g., the DFT codebook, in which case thecorrelation between the precoder and the channel is sufficientlyweak. Furthermore, the path loss is taken into account in thechannel model, so that different matrices have different vari-ance profile functions [2]. In particular, we have

(6)

(7)

(8)

where denotes the distance between user and the BS, de-notes the distance between user and relay , and denotes thedistance between relay and the BS, is the path loss exponent.The system given by (1)–(2) is essentially a MIMO MAC

model and its ergodic sum rate is given by

(9)

where is the transmit covariance matrix of user. In general, we need to resort to Monte Carlo methods to com-pute the sum rate in (9). Our goal is to find some deterministicexpressions that can well approximate in (9), especially for largeMIMO systems and/or large number of users.Before we move to the asymptotic and non-asymptotic anal-

ysis, we introduce the Shannon transform of the random matrix, given by

(10)

which is simply the sum rate per receive antenna at the BS.


III. ASYMPTOTIC ANALYSIS

In this section, we perform the asymptotic analysis and derivethe deterministic equivalents for the per-receive-antenna sumrate given in (10). We consider two asymptotic cases. The firstcase corresponds to large MIMO dimensions but fixed numberof users; whereas the second case corresponds to large MIMOdimensions and/or large number of users.A deterministic equivalent is a deterministic approximation

to some transform of a random matrix (e.g., the Shannon trans-form in (10)); and the approximation error goes to zero withprobability one when the dimension of the random matrix goesto infinity. Specifically, suppose that is a se-quence of random Hermitian matrices, and is a sequenceof functionals of matrices. A deterministic equivalent of

for the functional is a sequence of deterministic matrices, such that

with probability one. Note that does not need to have alimit as .

A. Large MIMO Dimensions, Fixed Number of Users

In this subsection, we derive the deterministic equivalent ofthe per-receive-antenna sum rate when the MIMO dimension islarge but the number of users is fixed. In [3], the deterministicequivalent of the rate of a multi-hop relay system is analyzed.However, the direct link between the source and destination isignored, whereas in this work we consider multiple two-hoprelay systems with direct links.We will make use of the results in [20] and [21] on the deter-

ministic equivalent of the Shannon transform of a random ma-trix of the following form:

(11)

where

(12)

(13)

has i.i.d. entries and has zero mean

and unit variance; and are non-negative definite Hermitian matrices. Under certain conditions,the deterministic equivalent of the Shannon transform of :

for exists when andall grow large, and it depends on and theunique solution of a fixed point equation. In [20], the determin-istic equivalent is obtained under the condition that and

’s are Gaussian. This result is extended to random matriceswith and arbitrary distribution in [21], where it is shownthat if the entries of ’s have finite 6th-order moments, thenthe difference between the Shannon transform of in (11)

and its deterministic equivalent has the order . If ’s

are Gaussian, the difference becomes . We have the fol-lowing result.Theorem 1: Consider the single-cell MIMO system with re-

lays defined in Section II with one BS, users and relays,where the BS has antennas, each user has antennas andeach relay has antennas. Assume that the channel correla-tion matrices satisfy for and . Assumefurther that the eigenvalues of and are bounded.Define

(14)

(15)

Then for fixed and fixed ratio , as, the per-receive-antenna uplink sum rate defined in (10) sat-

isfies

(16)

where the deterministic equivalent is given by

(17)

where

(18)

(19)

form the unique solution to thefollowing set of fixed-point equations:

(20)

(21)

(22)

Proof: We will transform the random matrixinto the form of (11) and then directly apply


the result of [21] to obtain its deterministic equivalent. Byusing (2)–(5), we can write

(23)

(24)

where (24) follows from (14)–(15) and the assumption thatfor and . Note that is an

random matrix with the -th entry given by

(25)

Since the entries of and are zero-mean i.i.d.Gaussian random variables, it follows that the entries ofand are i.i.d. with zero mean and variance and .Hence we have

(26)

where the entries of and are i.i.d. with zero mean and

variance , which is exactly the same form as (11). Theorem1 then follows directly from the result in [21].Given , the channel correlation matrices

and the variances , bysolving the fixed-point (20)–(22), we can then obtain the

asymptotic per-receive-antenna sum rate using (17). This pro-vides an efficient way to evaluate the system capacity since noMonte Carlo simulations are needed.

B. Large MIMO Dimensions, Large Number of Users

In the previous subsection, the number of users is fixed andwe considered the large MIMO dimension regime, i.e.,and . In this subsection, we consider another

asymptotic scenario where the MIMO dimension is large andthe number of users can also be large. Specifically, we let

, and keep the ratio fixed.Our derivation of the deterministic equivalent of the asymp-

totic per-receive-antenna sum rate in this case is based on thefollowing result [22], [23]. Suppose that an random ma-trix whose entries are independent zero-mean random vari-ables satisfies the Lindeberg condition

(27)

for any when . Then the deterministic equiv-alent of and its Shannon transform for large and can beobtained, which is a function of the variances of the entries of .We focus on the randommatrix in (23) and its Shannon trans-

form . Our basic idea here is to transform the matrix sum-mation in (23) tomatrix product and then explore the asymptoticcharacteristic of the transformed random matrix. Define

(28)

(29)

(30)

Then we can rewrite (23) as

(31)

where is a matrix, given by

(32)

Define a matrix whose entries are given by

(33)

where the variance of the -th entry of can be calculatedas (34), shown at the bottom of the page, withand being the ceiling operator.

(34)


We have the following result.Theorem 2: Consider the single-cell system with relays de-

fined in Section II, where the BS has antennas, each userhas antennas, and the cell has users and relays. As-sume that the random matrix in (32) satisfies the Lindebergcondition in (27). Then by fixing the ratio , and when

, the per-receive-antenna sum ratein (10) satisfies

(35)

where the deterministic equivalent is given by

(36)

where denotes an -dimensional all-1 column vector, anddenotes a column vector whose entries are the logarithms

of the corresponding entries in . The -dimensional columnvector and the -dimensional column vector form theunique solution to the following fixed-point equations

(37)

(38)

Proof: See Appendix A.Note that unlike Theorem 1, Theorem 2 makes no assump-

tions on the MIMO correlation matrices. On the other hand, theasymptotic regime of , sub-sumes a special case of fixed and , which is thesame asymptotic regime considered in Theorem 1. Numericalresults in Section V indicates that with fixed , the determin-istic equivalent given by Theorem 1 is more accurate than thatgiven by Theorem 2 when and are large.

IV. NON-ASYMPTOTIC ANALYSIS

The asymptotic analysis in the previous section considers theregime of large MIMO dimensions, with the number of userseither fixed or large. In this section, we consider another regimewith the fixed MIMO dimensions, and large number of users.We will treat both single-cell and multi-cell cases. The tool em-ployed for this case is the non-asymptotic analysis [24], and wewill obtain deterministic upper and lower bounds which the sumrate lies in between with high probability.Asymptotic methods explore the limiting characteristics of

the eigenvalues or singular values of the random matrices. Onthe other hand, non-asymptotic methods try to find upper andlower bounds to constrain the singular values of the randommatrices by cutting the tail of the probability distribution func-tion (pdf) of the entries. The singular values can be constrainedwithin the bounds with high probability because the tail of thepdf decays fast.

A. Single-Cell

In this subsection, we still focus on the random matrix de-fined in (32). Each entry of is the sum of Gaussian randomvariables and products of Gaussian random variables. It will beshown that the pdf of the entries of has sub-exponential de-caying feature. Therefore we can obtain the bounds for the sin-gular values of through cutting off the tail of the pdf of itsentries.In order to obtain the bounds on the sum rate through the

bounds on the singular values of , we rewrite the sum rate interm of the singular values:

(39)

where denotes the -th eigenvalue of , and de-notes the -th singular value of .Note that from (32), we can write

(40)

where

(41)

(42)

Then similar to (34), we have (43) and (44),

(43)

(44)

whereWe have the following result on the sum rate bounds for fixed

MIMO dimensions and large number of users.Theorem 3: Consider the single-cell system with relays de-

fined in Section II, where the BS has antennas, each userhas antennas, and the cell has users and relays. Forany and large , with probability at least

(45)

we have

(46)


If we choose to ensure the lower bound in (46) is positive, wethen have with probability in (45)

(47)

where and are constants satisfying (48)–(50).Proof: See Appendix B.

The valid ranges of the constants and are given asfollows based on Appendix B:

(48)

(49)

(50)

where

(51)

(52)

Since we would like to make the bounds tight and maximizethe probability that the sum rate lies within the bounds, wechoose the parameters as follows:

(53)

(54)

(55)

Moreover from (45) and (47), when is large, the probabilitythat is between the bounds is large, but the gap between thebounds is also large.If we use and to denote the upper and lower bounds

in Theorem 3, respectively, then the gap between the upper andlower bounds satisfies the following

(56)

The bounds are asymptotically tight from this point of view.Furthermore, if we choose the parameters according to(53)–(55), then as since the variances donot change with . Therefore the probability in (45) tends to 1as .Compared with the asymptotic results in the previous sec-

tion, the non-asymptotic bounds can be used for regular MIMO

Fig. 3. A multi-cell MIMO system with relays.

systems with small numbers of transmit and receive antennas.Moreover, the non-asymptotic bounds are even simpler tocompute than the asymptotic results since they do not involvesolving fixed-point equations. Note that although the bounds in(47) are expressed in terms of and , the constants

and are functions of as well as the correlationmatrices.

B. Multi-Cell

We next extend the result obtained from the previous subsec-tion for single cell to the multi-cell case, where the base sta-tions jointly process the received signals to mitigate the effectof inter-cell interference.The multi-cell system is depicted in Fig. 3, where there are

totally BSs, relays, and users. As in the single-cellcase, the relays receive signals from users and forward them tothe BSs. Each BS receives signals from all users and relays.We denote as the transmitted signal from user and

as the signal received by BS . The numbers of antennas ateach BS, relay and user are and , respectively. Thechannel matrices between user and BS , between user andrelay and between relay and BS are denotes asand , respectively. Moreover the precoding matrices of userand relay are denoted as and , respectively. As before,unit transmit power is assumed, i.e., .By stacking the received signals at all BSs in two time slots,

we have the following multi-cell uplink signal model

.... . .

...

.... . .

...

... (57)

where

(58)

(59)


is the additivecomplexGaussiannoisevectorwithcovariance matrix , where denotes the SNR.Under the Kronecker model, the channels are expressed as

(60)

(61)

(62)

where and are receive antenna correlation matrices,

and are transmit antenna correlation matrices, and

and are zero-mean i.i.d. Gaussian matrices. Fur-thermore, considering path loss, the variances of the entries ofGaussian matrices are

(63)

(64)

(65)

where is the path loss exponent, is the distance betweenuser and BS is the distance between user and relayand is the distance between relay and BS .

Denote . Theper-cell sum rate is then

(66)

where is the transmit covariance matrix of user

, and .Denote

(67)

(68)

.... . .

... (69)

.... . .

... (70)

Then we have

(71)

Furthermore, if we define

(72)

(73)

(74)

then we have (75) and (76),

(75)

(76)

where and .We now extend the non-asymptotic single-cell sum rate

bounds in Theorem 3 to the bounds on the multi-cell per-cellsum rate.Corollary 1: Consider a multi-cell system with relays de-

scribed above, where there are BSs, relays and users.Each BS has antennas and each user has antennas. Forany and large , with the probability at least

(77)

the singular values of the random matrix satisfy

(78)

If we choose to ensure the lower bound in (78) is positive, thenfor the per-cell sum rate we have with probability in (77)

(79)

where and are constants satisfying (80)–(82).Similar to (53)–(55), we set the constants as:

(80)

(81)

(82)

where

(83)

(84)

Similar to (56), the bounds become tight with probability one as.


Fig. 4. The impact of relay noise on the per-receive-antenna sum rate.

V. NUMERICAL RESULTS

In this section, we illustrate the accuracy of the asymptoticand non-asymptotic rate expressions derived in the previoussections, by comparing them with simulation results. In allsimulations, the radius of each cell is 10 meters. Moreover, forthe variance profiles in (7)–(8) and (64)–(65), we set .The correlation matrices have the form of , where isa random unitary matrix [25], and is a diagonal matrix withpositive entries. The DFT precoders [26], [27] are employedby the users and relays, and the codebook contains 4 DFTmatrices.

A. Asymptotic Results

For the single-cell case, the BS is located at the center ofthe cell. Relays are placed uniformly on a circle centered at theBS with a radius of 5 meters, and users are located randomlyin the cell. The distances between users and the BS and thedistances between users and relays are computed based on theirpositions.In order to show how performance is affected by the relay

noise, we conduct a simulation on per-receive-antenna sum ratewith and without relay noise. We set the number of relays, the number of antennas at each relay and the numberof users . The numbers of antennas at each user and atthe BS are and . Signal power is set as 0 dB.The relay noise power we evaluate is dB, 2 dB and 6 dB.Fig. 4 verifies that the impact of relay noise on the sum rate issufficiently small to be ignored.In Fig. 5, we compare the the sum rates of two network sce-

narios: one is that all users employ relaying, and the other is thatonly half of the users employ relaying with the correspondingrelay powers added to the transmission powers to make a faircomparison. In this simulation, we set

and . It is seen that because of the re-lays’ half-duplex penalty, the sum rate for the case that all userschoose to use relaying is slightly lower than that for the othercase.

Fig. 5. Sum rates for two scenarios of relaying.

Fig. 6. Per-receive-antenna sum rate as a function of SNR.

Fig. 6 shows the per-receive-antenna sum rate obtained bysimulations and the deterministic equivalents given by Theo-rems 1 and 2. We set and . We consider thescenario that and . It is seen fromFig. 6 that both theorems provide fairly accurate deterministicapproximations to the per-receive-antenna sum rate.In Fig. 7, we show the convergence of the two theorems with

the increasing MIMO dimensions. We set and. Moreover we set and dB.

The per-receive-antenna sum rate is shown as a function ofin Fig. 7. It is seen that as the MIMO dimension grows, thedeterministic equivalents given by Theorems 1 and 2 convergeto the true rate.Fig. 8 illustrates the per-receive-antenna sum rate as a func-

tion of the number of relays . We set, and . It is seen that both determin-

istic equivalents provide good approximations to the true rate.Moreover, the rate increases as the number of relays increases,although the rate of the increase is quite slow.


Fig. 7. Per-receive-antenna sum rate as a function of .

Fig. 8. Per-receive-antenna sum rate as a function of the number of relays .

B. Non-Asymptotic Results

For non-asymptotic analysis, we consider both single-cell andmulti-cell cases. The single-cell setup is the same as before. Themulti-cell setup is shown in Fig. 3, where there arehexagonal cells with a BS at the center of each cell. The cellradius is 10 meters and in each cell the relays are located ona circle that is 5 meters from the BS. The users are randomlylocated in different cells.In Fig. 9 we plot the ranges of the constants and

according to (48)–(50) for a single-cell system. The number ofusers in the cell is and the number of relays in a cellis . Moreover, and . The spaceabove the curvy plane and below the flat plane is the region forthe constants. If we choose and according to (53)–(55),in Fig. 10, we plot the bound gap and the probabilityin (45) as a function of . It is seen that as increases, the prob-ability also increases, but the gap between the upper and lowerbounds becomes larger.

Fig. 9. The range of and specified by (48)–(50).

Fig. 10. The bound gap and the probability in (45) as a function of .

In Fig. 11 we plot the upper and lower bounds on the sum rateof a single-cell system with the corresponding probabilities. Weset , and . According to Fig. 9and Section IV, we set and choose and accordingto (53)–(55). The bounds corresponding to andare plotted in Fig. 11. In Fig. 12, with the same setup as thatin Fig. 11 and dB, we plot the convergence of thegap between the upper and lower bounds as a function of . Itis seen that as increases, the gap becomes smaller, i.e., thebounds become tighter.Finally, the per-cell sum rate performance for a multi-cell

system is shown in Fig. 13. The system parameters for each cellare the same as those for Fig. 11. Comparing Figs. 11 and 13 itis observed that the upper bound of the single-cell sum rate issmaller than the lower bound of the multi-cell per cell sum ratewith the same number of users per cell. Hence by employingjoint processing in multi-cell systems, the per-cell sum rate canbe made larger than that in a single-cell system.


Fig. 11. Upper and lower bounds on sum rate in a single-cell system as a func-tion of SNR.

Fig. 12. Upper and lower bounds on sum rate in a single-cell system as a func-tion of the number of users .

VI. CONCLUSION

We have considered MIMO cellular networks with mul-tiple relay stations deployed in each cell that help relay usersignals to the base stations by using the amplify-and-forwardstrategy. We have obtained several deterministic expressionsfor the uplink ergodic sum rate of such systems. In particular,using the random matrix analysis tool, we have obtained twodeterministic sum rate expressions for the case of large MIMOdimensions, with fixed or large number of users, respectively.And by using the non-asymptotic analysis tool, we have ob-tained tight upper and lower bounds on the sum rate for thecase of arbitrary MIMO dimensions and large number of users.These deterministic sum rate expressions can be evaluatedgiven the system parameters such as the numbers of relays,users, base stations, the numbers of antennas at the user, relay,

Fig. 13. Upper and lower bounds on per-cell sum rate in a multi-cell system asa function of SNR.

and base station, the locations of the users and relays, thetransmit precoding schemes employed, as well as the MIMOchannel correlation statistics. With these analytical expressions,we can evaluate the sum rate of a single-cell or multi-cell net-work without resorting to Monte Carlo simulations. Thereforethey can be useful tools for the design and analysis of relay-as-sisted MIMO cellular networks.

APPENDIX APROOF OF THEOREM 2

In order to use the results in [22] and [23], we first transformthe entries of the matrix in (33) into a two-dimensional function defined as

(85)

if and . Then, accordingto [22], [23], if satisfies the Lindeberg condition in (27), wehave with

(86)

where and form the unique solution to the followingfixed-point equations

(87)

(88)

We next express in terms of .


Using the definition of , we have

(89)

Similarly we write (87)–(88) in terms of :

(90)

(91)

If we define

(92)

(93)

and substitute them into (89), we obtain (36). Furthermore, ifwe integrate over from to on both sides of (90) andintegrate over from to on both sides of (91), anduse (92)–(93), then (90) and (91) can be transformed into (37)and (38). Therefore Theorem 2 is proved.

APPENDIX BPROOF OF THEOREM 3

The proof starts from the following lemma [24]Lemma 1: If a matrix satisfies

(94)

for some , where denotes the operator norm of amatrix, then

(95)

Setting , our goal is to find some and calculatethe probability

(96)

Then we can find the bounds on the singular values of aswell as the probability that the singular values are constrainedwithin the bounds. In order to obtain this probability, we needto explore the distributions of the entries of and in (40).We define , where and

. It is obvious from (42) that the entries of areindependent Gaussian random variables. The following lemmacharacterizes the distribution of the entries of in (42)

Lemma 2: The product of two independent zero-meanGaussian randomvariables is a sub-exponential randomvariable.

Proof: Assume , where and are inde-pendent zero-mean Gaussian random variables with variancesand respectively, so . Thus we have for

any

(97)

According to the definition of sub-exponential random variablesin [24], we obtain Lemma 2.By Lemma 2, the entries of in (42) are linear combina-

tions of sub-exponential random variables. Then we can boundthe probability in (96) by exploring the decaying feature of thepdfs of and .We have

(98)

where denotes the -norm of a vector, and denotes the-net of the unit sphere , which is a subset ofsuch that for every point , we can find some point

so that . Note that (98) follows fromthe fact that for the -net of and the operator norm ofany square matrix [24], where

denotes the inner product of two vectors. Furthermore, thecardinality of satisfies . Moreover, we have

(99)

(100)

where

(101)

(102)


with and denoting the complex conjugateoperation. Because the entries of are zero-mean Gaussianvariables and the entries of are are the linear combina-tion of zero-mean sub-exponential variables, it then follows that

is the linear combination of Gaussian variables and isthe linear combination of sub-exponential variables. The Ho-effding-type inequality and the Bernstein-type inequality char-acterize the distribution of the linear combination of differenttypes of zero-mean random variables with decaying pdf [24]. Inparticular, we have

(103)

(104)

where is the sub-Gaussian norm of the vector, given by

(105)

whereas is the sub-exponential norm of the vector, given by

(106)

According to Lemma 2 and the definition of sub-Gaussian andsub-exponential random variables [24], and can be com-puted as follows:

(107)

(108)

where and are given by(75)–(76). Thus is a sub-Gaussian random variable and thedecaying rate of the pdf of is at least sub-exponential.

Define and , whereis a constant that will be set later. If we set in Lemma 1

in this way, we can find that is small when is large, whichmeans the bounds become tighter as increase. Furthermore,from the later analysis, the probability in (96) becomes smalleras increases. We have (109)

(109)

In addition, when , for the tall matrices and wehave and [24], which means

and tend to zero-mean random variables.

Since are sub-Gaussian, so are sub-ex-ponential random variables. Therefore according to the Bern-stein-type inequality,

(110)

where is the sub-Gaussian norm of the vector, and

(111)

Similarly by (104) and the Bernstein-type inequality, we have

(112)

where is the sub-exponential norm of the vector, and

(113)

We define and

. According to the feature of , we have

. Therefore when we choose sufficiently largesatisfying

(114)

we have

(115)

(116)

If the lower bound of (116) is positive, we can obtain (47). Thiscompletes the proof.


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Hao Wu received the B.S. degree in mathematicsand applied mathematics from Beijing University ofPosts and Telecommunications, Beijing, China, in2010.He is currently working toward the Ph.D. degree

in communication and signal processing with theSchool of Information Science and Engineering,Southeast University, Nanjing, China. He was avisiting student at Columbia University, New York,during 2011–2012. His research interests includeMIMO wireless communications, large scale an-

tenna array systems, and iterative signal processing.

LimingWang (S’08–M’11) received the B.S. degreein electronic engineering from the Huazhong Univer-sity of Science and Technology, China, in 2006, theM.S. degree in mathematics, and the Ph.D. degree inelectrical and computer engineering from the Univer-sity of Illinois at Chicago, both in 2011.From January 2011 to April 2012, he was a

Postdoctoral Research Scientist in the Department ofElectrical Engineering, Columbia University, NewYork. Since May 2012, he has been a PostdoctoralAssociate in the Department of Electrical and

Computer Engineering, Duke University. His research interests are in high-di-mensional data processing; machine learning; statistical signal processing;compressive sensing; bioinformatics and genomic signal processing.

Xiaodong Wang (S’98–M’98–SM’04–F’08) re-ceived the Ph.D. degree in electrical engineeringfrom Princeton University.He is a Professor of Electrical Engineering at Co-

lumbia University, New York. His research interestsfall in the general areas of computing, signal pro-cessing and communications, and has published ex-tensively in these areas. Among his publications is abook entitled Wireless Communication Systems: Ad-vanced Techniques for Signal Reception (EnglewoodCliffs, NJ: Prentice-Hall, 2003). His current research

interests include wireless communications, statistical signal processing, and ge-nomic signal processing.Dr. Wang received the 1999 NSF CAREER Award, the 2001 IEEE Com-

munications Society and Information Theory Society Joint Paper Award, andthe 2011 IEEE Communication Society Award for Outstanding Paper on NewCommunication Topics. He has served as an Associate Editor for the IEEETRANSACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS ON WIRELESSCOMMUNICATIONS, the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and theIEEE TRANSACTIONS ON INFORMATION THEORY. He is listed as an ISI Highly-cited Author.

Xiaohu You (SM’11–F’12) received the B.S., M.S.,and Ph.D. degrees in electrical engineering fromNanjing Institute of Technology, Nanjing, China, in1982, 1985, and 1989, respectively.From 1987 to 1989, he was with the Nanjing Insti-

tute of Technology as a Lecturer. From 1990 to thepresent, he has been with Southeast University, firstas an Associate Professor and later as a Professor.His research interests include mobile communica-tions, adaptive signal processing, and artificial neuralnetworks with applications to communications and

biomedical engineering. He is the Chief of the Technical Group of China3G/B3G Mobile Communication R&D Project.Dr. You received the excellent paper prize from the China Institute of Com-

munications in 1987, the Elite Outstanding Young Teacher Awards from South-east University in 1990, 1991, and 1993, and the National Technological Inven-tion Award of China in 2011. He was also a recipient of the 1989 Young TeacherAward of Fok Ying Tung Education Foundation, State Education Commissionof China.

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