Research Article Multiple Beam Selection for Combing M2M Communication Networks and Cellular Networks with Limited Feedback

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1 International Jornal of Distribted Sensor Networks Volme 2013, Article ID , 11 pages Research Article Mltiple Beam Selection for Combing M2M Commnication Networks and Celllar Networks with Limited Feedback Xiaoning Zhang, 1 Lin Bai, 2 and Wenyang Gan 1 1 State Key Laboratory of Advanced Optical Commnication Systems and Networks, Peking University, Beijing , China 2 School of Electronic and Information Engineering, Beihang University, Beijing , China Correspondence shold be addressed to Lin Bai; l.bai@baa.ed.cn and Wenyang Gan; ganwenyang@pk.ed.cn Received 18 Agst 2013; Accepted 19 October 2013 Academic Editor: Jianha He Copyright 2013 Xiaoning Zhang et al. This is an open access article distribted nder the Creative Commons Attribtion License, which permits nrestricted se, distribtion, and reprodction in any medim, provided the original work is properly cited. We stdy the scenario in which a large nmber of machine-type commnication devices (MTCDs) commnicate with each other by tilizing the help of the base station (BS) throgh some MTCD gateways. We consider an overlay mode of an orthogonal freqency division mltiple access (OFDMA) based celllar system sing orthogonal beamforming to provide broadband wireless access for the MTCD gateways. In order to avoid the interference with mobile sers, the beamforming vectors to the MTCD gateways have to be orthogonal to the channel vectors of mobile sers, which become a beamforming constraint for the MTCD gateways. However, with limited feedback of channel state information (CSI) at a BS, the orthogonal beamforming constraints may not be achieved. In sch a practical case, conventional feedback schemes are feasible bt not efficient de to the orthogonality constraints. In this paper, we propose a novel mltiple beam selection (MBS) approach with limited feedback for MTCD gateways by taking into accont the previos orthogonality constraints. Simlation reslts show that the performance improvement of the proposed approach over the conventional ones is generally abot 10% when a BS is eqipped with an array of 6 elements. 1. Introdction Machine-to-machine (M2M) commnication networks have been proposed to connect machines withot the intervention from hman beings. In recent years, the research of combing M2M networks and mobile celllar networks draws mch attention [1, 2]. In an M2M network, there are lots of machine-type commnication devices (MTCDs). These MTCDs access a celllar network by some MTCD gateways [3],whichareassmedassecondarysersinthecelllar network. There are hndreds of MTCDs connecting to one MTCD gateway. In the combined M2M celllar networks, the MTCD gateway may need to commnicate with the BS at a high data rate. As the mobile sers in a celllar system have a higher priority than the MTCD gateways, the beamforming vectors for the mobile sers are designed to maximize the beamforming gains. In the overlay mode, since the signals transmitted to the MTCD gateways shold not interfere with themobilesers[4], the beamforming vectors have to be orthogonal to the channel vectors to the mobile sers, which become a constraint in finding the beamforming vectors for the MTCD gateways. The throghpt performance can be improved when the orthogonal beamforming is combined with mltiser schedling over mltiple sbcarriers, where the sers with good channel conditions are selected for each transmission. Channel state information (CSI) is reqired in both beamforming and mltiser schedling. However, nfortnately, in a practical system, the CSI available at the BS is imperfect de to specific system configrations. Therefore, it is worthwhile to design orthogonal beamforming sing imperfect CSI for this kind of hybrid network Related Work. There are mainly two ways for the BS to get information from each ser, that is, channel reciprocity and feedback channel. In a time division dplexing (TDD) system, it is possible to se the channel reciprocity at the BS, which refers to estimating the downlink channel from the plink signal. However, the channel reciprocity reqires calibration of the radio devices of both the receiver and the transmitter, for example, A/D converters, mixers, filters, and antennas. This reqires extra special hardware as well as special protocols to facilitate the channel measrement and calibration [5, 6]. Therefore, most of the research papers

2 2 International Jornal of Distribted Sensor Networks assme that a feedback channel exists from each ser to the BS for carrying feedback information as those in [7 9]. It is generally assmed that the feedback channel from each ser to the BS is not perfect. In [10 12], the delayed feedback channel is considered. The effects of otdated CSI at the BS are analyzed by means of Markov model of the channel temporal correlation in [11, 12]. The effects of the noisy feedback channel are considered in [13 15], where the feedback channel is assmed as a discrete symmetric channel. Plentyofthepiecesofliteratrefocsonthefeedbackchannel with finite rate(see, e.g.,[16] and the references therein), which is referred to as limited feedback channel. From the limited feedback channel, each ser is allowed to send a small nmber of information bits back to the BS. Most of the conventional feedback schemes focs on the qantization of the channel direction information (CDI) [9] sing some vector qantization algorithms [17]. In [8], a codebook of orthogonal beamforming matrices is sed for the qantization. Each ser qantizes the channel directions according to a beamforming codebook with a priori knowledge by both the BS and the sers. Possible ways of redcing the nmber of feedback bits have been proposed in [18, 19], where only the mean or covariance of CSI is sent back to the BS. Asfortheoverlaymodebasedontheorthogonalbeamforming that we consider in this paper, the conventional feedback strategies are no longer sitable; sing conventional feedback schemes, the BS acqires an approximation of each sers channel directions the reslting beamforming vectors designed according to this approximation will not be best aligned with the MTCD gateways channel direction. Besides, de to the orthogonality constraint, the conventional feedback strategies become inefficient. That is, the codebook sed and designed in [8] is niformly distribted in the whole space, while the actal beamforming vector lies in a sbspace de to the orthogonality constraint. Ths, the vectors in the codebook that lie ot of the sbspace provide redndant information and reslt in excessive feedback. Therefore, a new feedback strategy is needed in the overlay mode for MTCD gateways. Inthisstdy,weproposeanewmltiplebeamselection (MBS) strategy with limited feedback of CSI. A (downlink) broadcasting channel is sed to inform MTCD gateways with alltheavailablebeams.eachserchoosesthebestoneot of the available beams and sends the index of the chosen beam back to the BS. As the beams are dynamically generated at the BS according to the interference pattern of the sers, the orthogonality is garanteed and the CSI can be fed back throgh the feedback channel to the BS more efficiently. This paper addresses joint beamforming, schedling, and feedback for broadband wireless access to provide high throghptformtcdgateways.thedesignofbeamforming vectors and limited feedback from each ser are closely connecteddetotheorthogonalconstraintformtcdgateways. In conventional feedback schemes, the beamforming vectors are calclated after receiving the feedback information from each ser. However, in the proposed MBS scheme, the beamforming vectors are calclated in advance, awaiting for each ser s choice. A similar thoght has been applied to opportnistic space division mltiple access (OSDMA) in [20, 21] Organization and Notations. The remainder of this paper is organized as follows. The system model is given in Section 2. InSection 3, a detailed stdy on the limited feedbackformtcdgatewaysispresentedandthembsstrategy is proposed. The performance analysis is given in Section 4 followed by a comprehensive evalation of the proposed MBS strategy sing Monte Carlo simlation in Section 5.We conclde the paper with some remarks in Section 6. The sperscripts T and H standforthetransposeand Hermitian transpose, respectively. Upper and lower boldfaced letters are sed for matrices and colmn vectors, respectively. Denote by x the absolte vale of a scalar x or cardinality of x if x is a set, x the 2-norm of vector x, perpendiclarity, and I the identity matrix of a certain size implicitly given by the context. We denote by [w 1, w 2,...,w N ] the concatenation of N colmn vectors and by [W, c] the concatenation of a matrix W and a vector c of the same colmn size. Elements of a set are enmerated as {a 1,...,a N }. E[ ] denotes the statistical expectation. CN(a, R) represents the distribtion of circlarly symmetric complex Gassian (CSCG) random vectors with mean vector a and covariance matrix R, andc n denotes n-dimensional complex space. We denote the probability of an event X by Pr(X). logand log 2 represent the natral logarithm and binary logarithm, respectively. 2. System Model As in [22], in this paper, we consider downlink beamforming formtcdgateways.spposethatabsinacelllarnetworkis eqipped with an antenna array of L elements. For downlink channels, there are M orthogonal sbcarriers. The BS can spport two different grops of sers. One grop has ordinary mobile sers, which is called Grop I. The other grop consists of MTCD gateways, which is called Grop II. The sers in Grop II want to have high data rate services. It isassmedthatthesersingropihavehigherpriority than those in Grop II. In the overlay mode, the downlink transmissions to the sers in Grop II shold not affect the sers in Grop I. It is assmed that there are K sers in Grop I, and Q sers ot of K are chosen for data transmissions in each sbcarrier. ThesearecalledactivesersinGropI.Thenmberofsers in Grop II is denoted by U.Onlyoneofthesesersisselected for transmission throgh each sbcarrier. In general, Q is small (sally Q=1if no spatial mltiplexing is considered and U is also small as the nmber of active fixed sbscribers at a time wold be limited, especially in a rral area where there are not so many mobile sers). The transmitted signal vectors from the antenna array to sers in Grops I and II throgh sbcarrier m, denoted by s I,m and s II,m,aregivenby s I,m = Q q=1 w I,m,q a m,q ; s II,m = w II,m b m, (1)

3 International Jornal of Distribted Sensor Networks 3 where a m,q and b m are independent data symbols for sers in Grops I and II, respectively, and w I,m,q and w II,m are the corresponding beamforming weight vectors. Throghot the paper, we assme that the beamforming vectors are normalized; that is, w I,m,q = w II,m =1for all m. Denote P I = E[ a m,q 2 ] and = E[ b m 2 ] as the transmitted powers for Grops I and II, respectively. We denote by k m theindexofaseringropiwho receives the signal throgh sbcarrier m.the received signal at this ser is given by x m,km = h H m,k m (s I,m + s II,m )+V m,km, (2) where h H m,k m is the 1 Ldownlink channel vector of sbcarrier m from the transmit antenna array to the k m th ser in Grop I and V m,km CN(0, σ 2 ) is an independent backgrond noise. If a ser in Grop II also receives a signal throgh sbcarrier m whose index is denoted by m,thereceived signal at this ser is given by y m,m = g H m, m (s I,m + s II,m )+V m,m, (3) where g H m, m is the 1 Ldownlink channel vector of sbcarrier m from the transmit antenna array to the m th ser in Grop II and V m,m CN(0, σ 2 ) is also an independent backgrond noise. Note that the characteristics of the channel vectors to sers in Grops I and II are different. In general, g m,m is not varying rapidly, while h m,km is time varying as Grop I sers are mobile. The beamforming weight vectors w I,m,q and w II,m are decided by the BS according to the information sent back by each ser. If the feedback channel is perfect, the CSI of each ser, that is, h m,km and g m,m,isknownbythebs. Throghot the paper, we focs on the data transmissions of the MTCD gateways. Generally we assme that the h m,km s areavailableatthebsforbeamforminganddiscssthe impact of limited feedback with g m,m.notethattheimpact of imperfect h m,km is considered in [8, 9]. We do not consider this isse in detail. 3. Beamforming with Limited Feedback In this section, the beamforming algorithms for MTCD gateways are proposed. First, the algorithms with perfect CSI of the Grop II sers are briefly described in Section 3.1. The motivations for the new feedback scheme are given in Section 3.2. In Section 3.3, the beamforming algorithms with limited feedback CSI of the MTCD gateways are proposed Orthogonal Beamforming with Perfect CSI. As sers in Grop I have a higher priority than those in Grop II, the signals to sers in Grop II shold not interfere with the signals to sers in Grop I. For broadband access, it is reqired that the data rate to sers in Grop II from the BS is sfficiently high, while the nmber of sers in Grop II wold not be large. In order to deal with these isses, we consider orthogonal beamforming. Using orthogonal beamforming, space division mltiple access (SDMA) [23] canbe implemented for downlink transmission to sers in Grops I and II. The main difference of the orthogonal beamforming approachinthispaperfromthatfortheconventionalsdma is that the overlay mode based beamforming approach shold take into accont the priority between sers in Grops I and II and maximize the data rate to sers in Grop II. Note that a ser schedling problem exists in Grop I as Q<K. When Q>1, this selection problem cannot be solved by simply choosing the sers with the best signal-tointerference-pls-noise ratio (SINR) or signal-to-noise ratio (SNR). Fortnately, there is an existing approach in [24]. In [24], the semiorthogonal ser selection (SUS) algorithm is sed for schedling Q active sers ot of K sers throgh each sbcarrier. The SUS algorithm reslts in a grop of sers with channel vectors h m,k q m, q=1,...,q,and hh m,k h m i j m,k 1 m h m,km 2 i h 2 >1 ε, m,km j 2 i,j=1,...,q,i =j, where ε is a small positive constant. Note that ε is a parameter related to the nmber of sers K similar to α in [24], which is sed to find semiorthogonal sers. The SUS algorithm chooses the sers with semiorthogonal channel directions. Ths, hereafter, we assme that the chosen sers in Grop I have orthogonal channel directions; that is, h m,k i m h m,k j m, m=1,...,m, i,j=1,...,q, i =j. This becomes tre when the nmber of sers K is sfficiently large [24]. We denote by Q m thesetofthegropiactivesers in each sbcarrier m.from(5), the beamforming vectors for the Q selected sers are simply represented by h m,k i m (4) w I,m,i = h, i=1,...,q. (6) m,km i At a ser in Grop II, the SINR is given by SINR II,m,m = (5) gh 2 m, m w II,m PII. (7) Q i=1 gh 2 m, m w I,m,i PI +σ 2 Since w I,m,i is decided, this SINR can be maximized by maximizing the nmerator. That is, maximize gh m, m w II,m, (8) sbject to w II,m h m,k i m, i=1,...,q. (9) The optimal beamforming vector is given by where w II,m = w II,m =(I Q i=1 1 w II,m w II,m, (10) h m,k i m h H m,km i h 2 ) g m, m m,km i

4 4 International Jornal of Distribted Sensor Networks =(I Q i=1 w I,m,k i m w H I,m,k i m ) g m, m =(I W I,m W H I,m ) g m, m. Here W I,m =[w I,m,1, w I,m,2,...,w I,m,Q ]. (11) h m,km w I,m 3.2. Motivations for the New Feedback Scheme. With perfect CSI, the optimal beamforming vectors for sers in Grop II are designed sbject to the orthogonal constraints (9). The reslting beamforming vector (11) lies in the nll space formedbythecolmnvectorsofw I,m and is best aligned to the channel vector g m,m. However, with limited feedback, the exact vale of g m,m is not available at the BS. In the conventional feedback scheme in [8], each g m,m is decomposed into two components, that is, the gain and the direction, as follows: Nll space w II,m g m,m g m,m = g m, m g m, m, (12) where g m,m = g m,m / g m,m is the channel direction. The ser comptes the SINR and the channel direction, qantizes them according to a codebook and sends them back to the BS. In conventional feedback schemes, a codebook-based qantizer with a codebook comprised of mltiple sets of orthonormal vectors in C L is sed to qantize the channel direction g m,m / g m,m. Denote the codebook by F, F = B b=1 V (b), (13) where V (b) is the bth orthonormal set in the codebook F. There are L mtally orthogonal vectors in each V (b), b = 1,...,B. Ths, the codebook size can be denoted by F = BL. For each weight vector k in the codebook F, a distortion fnction d(k, g m,m ) is defined as d(k, g m,m )=1 kh g m,m 2. (14) For the qantization of the channel direction, the member of F that has the smallest distortion is chosen as the feedback channel direction; that is, ĝ m,m = arg min d(k, g m, k F m ). (15) For the qantization of the SINR, G bits are sed to qantize this scalar vale. For analysis convenience, the SINR isalwaysassmedtobeknownperfectlybythebs.this assmption will be jstified in Section 5 where the simlation resltsshowthatthethroghptlossdetolimitedsinr feedback is marginal compared with the case in which perfect SINR feedback is available. In the orthogonal beamforming for sers in Grop II, the optimal beamforming vectors are designed sbject to orthogonal constraints which is illstrated in Figre 1.Ifw I,m Unite ball Figre 1: An example of orthogonal beamforming for MTCD gateways. w II,m shold lie in the nll space of h m,km and be best aligned with g m,m. and g m,m are given, the beamforming vector for g m,m shold lie in the nll space of w I,m,thatis,thediskperpendiclarto w I,m.Thebestbeamfor m is w II,m in Figre 1. Withlimitedfeedback,ifwecanqantizeg m,m to the vector w II,m as depicted in Figre 1, thebsobtainssfficient information to determine the beamforming vector. It is eqivalent to say that if the qantization codebook F lies in thenllspaceformedbythecolmnvectorsofw I,m,weare more likely to get a qantized version of g m,m as close to w II,m as possible. In conventional feedback schemes, the codebook is predesigned and known at both the BS and each ser. The vectors in F that lie ot of nll space of W I,m provide redndant information and reslt in excessive feedback bits. Meanwhile it is not possible to change F flexibly according to agivenw I,m. A new feedback scheme is needed in orthogonal beamforming for MTCD gateways Orthogonal Beamforming with Limited Feedback. We propose a new MBS strategy where a grop of beamforming vectors for the MTCD gateways is generated by the BS and broadcasted throgh the downlink channel. Sppose that the Q activesersingropihavebeen chosen and the beamforming matrix is denoted by W I,m.The

5 International Jornal of Distribted Sensor Networks 5 size of W I,m is L Q. For an extra ser from Grop II, we generate N new candidate beamforming vectors as follows: C II,m = C II,m C II,m C II,m, =(I W I,m W H I,m ) U m, (16) where U m is a random matrix of size L N. Each colmn vector of C II,m isanormalizedweightvector.thesetofthe colmn vectors of C II,m is denoted by {c m,1,...,c m,n }. For the schedling of a MTCD gateway, the BS shold transmit N pilot signals to let them choose the best beam among N beams, {c m,1,...,c m,n }.Theresltingbeamselection problem can be given by where the SINR is given by SINR II,m,m,n = max SINR n II,m,m,n, (17) gh 2 m, m c m,n PII. (18) Q i=1 gh 2 m, m w I,m,i PI +σ 2 The SINR expressions in (17) and (18) are replaced by SINR II,m,m,n in order to emphasize their dependence on n or c m,n, the beamforming candidate. The detailed process is described as follows. (1) The BS broadcasts c m,1,...,c m,n when choosing a ser in Grop II by employing a series of the beamforming matrices, {[W I,m, c 1 ],...,[W I,m, c N ]} for N symbol drations. At the ser m in Grop II, the received signal when the nth pilot signal is transmitted is given by Q y m,m = g H m, m ( w I,m,q a m,q,n + c m,n b m,n )+V m,m, (19) q=1 where a m,q,n and b m,n denote the nth pilot signals to the qth ser in Grop I and the ser in Grop II, respectively. (2) After receiving all the N pilot signals, each ser in Grop II estimates the local SINR, namely, SINR II,m,m,n, with different beams, n = 1,...,N.The SINR II,m,m,ns areassmedtobeknownperfectlyat each ser m. (3) The sers choose the best beam among all the pilot beams according to (17) and sends the index of the best beam n m to the BS. n m = arg max n {1,...,N} SINR II,m, m,n. (20) If there are more than one beam that are eqally the best, the ser randomly chooses one of them and feeds the index back. (4) At each ser, the SINRs of the best beam throgh each sbcarrier are qantized sing G bits as follows: SINR II,m,m,nm ŜINR II,m,m,nm. (21) Throgh sbcarrier m, only the qantized SINR s of thebestbeamateachser,namely,ŝinr II,m,m,nm,are sent back throgh the feedback channel. In order to minimize the impact of the interference from the signals to sers in Grop I or maximize the throghpt for sers in Grop II, the sbcarrier allocation for Grop II sers can be carried ot as follows: m = arg max log 2 (1 + ŜINR II,m,,n {1,...,U} ) = arg max ŜINR II,m,,n {1,...,U}, (22) and the corresponding beamforming vectors are selected as the final beam n m. (23) That is, for each sbcarrier, the ser in Grop II who maximizes the achievable rate, namely, log 2 (1 + ŜINR II,m,,n ), or the ŜINR II,m,,n is to be selected. The set of sbcarriers allocated to ser is given by M ={m m =,m=1,...,m}. (24) Ths, the estimated throghpt for ser is given by R = log 2 (1 + ŜINR II,m,,n ). m M 4. Asymptotic Throghpt Performance (25) In this section, we analyze the performance of the proposed MBS scheme asymptotically when N.When N, there is always a beam that is perfectly aligned to the channel direction of each ser. From (11) and(16), if one colmn of U m is aligned with g m,m / g m,m,thembsscheme achieves the ideal performance with perfect CSI at the BS. In this section the asymptotic throghpt performance is first derived with perfect CSI feedback. In order to obtain a sefl expression that characterizes the system performance with limited feedback, a lower bond on the asymptotic throghpt performance is then derived. Assmption 1. The elements of h m,km and g m,m are independent and identically distribted (i.i.d.) and CSCG random variables, CN(0, 1/L). Hereafter, we consider performance analysis for sers in Grop II nder the CSCG Assmption 1. Forconvenience, define the SNR at ser m in Grop II as γ II,m,m = g m, m 2 PII σ 2. (26)

6 6 International Jornal of Distribted Sensor Networks Frthermore, define Then, we have SINR II,m,m,Q m = ξ m (Q m, m ) =1 gh 2 m, m W I,m. (27) g 2 m, m ξ m (Q m, m ) (1 ξ m (Q m, m )) (P I / )+(1/γ II,m,m ). (28) In (28), SINR II,m,m, which is the SINR at ser m in Grop II, is replaced with SINR II,m,m,Q m in order to emphasize its dependence on Q m or W I,m. The proof of (28) isgivenin Appendix A. Note that the SINR for a ser in Grop II is pper-bonded as SINR II,m,m,Q m min { ξ m (Q m, m ),γ 1 ξ m (Q m, m ),m,m }. (29) I Assmption 2. The distribtion of 1 ξ m (Q m, m ) can be approximated by a beta distribtion with the parameters Q and L 1. Note that the approximation becomes more accrate when L. or That is, we have 1 ξ m (Q m, m ) B (Q, L 1) (30) 1 ξ m (Q m, m ) B (a, b) (1 y)a 1 y b 1, 0 y 1, (31) where B(a, b) is the B fnction and a=q, b=l 1. Now, we present nmerical reslts to verify Assmption 2. InFigre 2, the nmber of sers in Grop IissettoQ = 4. Both the approximate and empirical distribtions of 1 ξ m (Q m, m ) are illstrated for different nmber of transmitter antennas (L =10, 20, and40). When L = 40,approximation(30) fits well with the empirical distribtion. Figre 3 shows the reslts when the nmber of sersingropiissettoq=2. The same transmitter antenna arrays are considered as those in Figre 2. Itisshownthat the approximate distribtion is reasonable especially when there are less sers in Grop I. The probability density fnction (PDF) of z = ξ m (Q m, m )/(1 ξ m (Q m, m )) is given by ξ m (Q m, m ) 1 ξ m (Q m, m ) f(z) = 1 B (a, b) z b 1 (1+z) a+b, and the cmlative distribtion fnction (CDF) by 0 z< F (z) = Pr ( ξ m (Q m, m ) 1 ξ m (Q m, m ) z)=1 I 1/(1+z) (a, b), (32) (33) PDF f(x) Q=4 L = 10, 20, x Beta approximation Simlations Figre 2: Distribtion of 1 ξ m (Q m, m ), Q=4sers in Grop I. PDF f(x) Q=2 L = 10, 20, x Beta approximation Simlations Figre 3: Distribtion of 1 ξ m (Q m, m ), Q=2sers in Grop I. where I x (a, b) istheincompletebetafnction.thecdfof the maximm SINR among U independent sers in Grop II is as F max (z) = Pr (max ξ m (Q m,) 1 ξ m (Q m,) z) =F U (z) =[1 I 1/(1+z) (a, b)] U. (34) The otage probability of the maximm SINR is defined P ot (Γ) = Pr (max SINR II,m,,Qm Γ), (35) where Γ is the target threshold SINR.

7 International Jornal of Distribted Sensor Networks 7 Using the pper bond in (29), a lower bond on the otageprobabilityofthesinrcanbefond.since max SINR II,m,,Qm max min {max min { ξ m (Q m,) 1 ξ m (Q m,) ξ m (Q m,) 1 ξ m (Q m,) P I,γ II,m, }, max P I γ II,m, }, (36) Throghpt per ser we have P ot (Γ) 1 (1 F max ( P IΓ )) (1 F U χ (2L) (2Lσ2 Γ )), (37) where F χ (n)(z) denotes the chi-sqare CDF with n degrees of freedom. The proof of (37) in details is given in Appendix B. The average achievable rate per sbcarrier for Grop II is r=e [log 2 (1 + max SINR II,m,,Qm )] (bps/hz), (38) and the average throghpt per ser becomes E [R ]= M r. (39) U We consider a lower-bond on the average achievable rate sing the otage probability in (37). For a given target threshold SINR Γ,wehave r=e [log 2 (1 + max SINR II,m,,Qm )] Pr (max SINR II,m,,Qm Γ) 0 + Pr (max SINR II,m,,Qm >Γ)log 2 (1+Γ) =(1 P ot (Γ)) log 2 (1+Γ). (40) Atightlower-bondonr canbeachievedifthelower-bond is maximized with respect to Γ,whichresltsin E [R ] M U max (1 P ot (Γ)) log Γ 2 (1+Γ). (41) Note that the asymptotic throghpt performance is first characterized by (39) sing(28).thisresltisthenlower- bonded in (41). We will se (41) as an indication of the system performance with limited feedback. 5. Simlation Reslts In this section we present varios nmerical reslts to frther investigate the performance of the proposed MBS scheme. The simlation settings are described as follows: the channel elements are independent and identically distribted (i.i.d.) CSCG random variables, that is, CN(0, 1/L); the celllar radis is normalized as one; the base band channel is considered in the simlation withot a specified radio freqency Nmber of sers in grop II Beam selection N=1 Beam selection N=10 Beam selection N=30 Beam selection N=60 Beam selection N=90 Beam selection N = 1000 Lower bond of asymptotic performance Figre 4: Throghpt per MTCD gateway verss U(Q = 1, L=10, M = 128,and /σ 2 =20dB). In the following two sbsections, we will first assme that the BS knows each sers SINR: in Section 5.1, theeffect of increasing feedback bits, that is, increasing the nmber of candidate beams N, is investigated; in Section 5.2, the throghpt performance of the proposed MBS is compared with that of the conventional feedback scheme. Then, in the last sbsection, the effect of limited feedback of SINR is shown with different nmber of sers in Grop II. The simlation reslts are presented nder Assmption 1 with M = 128, P I =,and /σ 2 =20dB Effect of Increasing Feedback Bits. Simlation reslts for different nmber of active sers in Grop I are presented to show the effect of increasing feedback bits. As conceived, the throghpt per MTCD gateway increases as the increase of the nmber of beams generated by the BS, nlike the reslts in [8] where increasing channel shape feedback does not necessarilyleadtobetterperformance. In Figre 4, the nmber of transmitter antennas is set to L=10.OnlysingleactiveserisconsideredinGropI.From Figre 4, when the nmber of beams increases from 1 to 10 beams with U=2,wecanseethatthethroghptincreases abot 80%. The lower bond of the asymptotic performance is also plotted in Figre 4.Theoreticalreslt(41)givesagood prediction of the system performance with limited feedback. In Figres 5 and 6, the nmber of active sers in Grop I are 2 and 4, respectively. As expected, theoretical reslt (41) canbesedasagoodindicationofthesystemperformance with limited feedback Comparison with Conventional Feedback Schemes. From the analysis in Section 3, the proposed MBS scheme achieves better performance than the conventional feedback schemes. In this sbsection, nmerical reslts are given to verify or analysis.

8 8 International Jornal of Distribted Sensor Networks Throghpt per ser Throghpt per ser Nmber of active sers in grop I increases Q = 2, 3, 4, Nmber of sers in grop II Beam selection N=1 Beam selection N=10 Beam selection N=30 Beam selection N=60 Beam selection N=90 Beam selection N = 1000 Lower bond of asymptotic performance Figre 5: Throghpt per MTCD gateway verss U(Q = 2, L=10, M = 128,and /σ 2 =20dB) Nmber of sers in grop II Proposed feedback Conventional feedback Figre 7: Throghpt per MTCD gateway verss U(N = 256, L=6, M = 128,and /σ 2 =20dB); the nmber of active sers in Grop I varies from p to bottom (Q =2, 3, 4,and5, for each pair of crves resp.). Throghpt per ser Nmber of sers in grop II Beam selection N=1 Beam selection N=10 Beam selection N=30 Beam selection N=60 Beam selection N=90 Beam selection N = 1000 Lower bond of asymptotic performance Figre 6: Throghpt per MTCD gateway verss U(Q = 4, L=10, M = 128,and /σ 2 =20dB). In Figre 7, the performances of the proposed feedback scheme and the conventional one are compared with an increasing nmber of sers in Grop II. The nmber of beams generated by the BS in or proposed scheme and the nmber of the vectors in the codebook of the conventional feedback scheme are the same and set to N = 256. The orthogonal beamforming matrices of the conventional feedback are generated sing a method in [8]. As expected, the proposed feedback scheme otperforms the conventional one,especiallywhenthenmberofactivesersinthegrop I is small. In general, a throghpt increase of abot 10% over the conventional feedback scheme is achieved by the proposed MBS Effect of SINR Qantization. The analyses and simlations above are based on the assmption that the BS has perfect knowledge of each ser s SINR. The effect of qantization of SINR on the performance is shown sing nmerical simlations in this sbsection. Figre 8 compares the cases of perfect and qantized SINR feedback with G-bit qantization, G = 2,3,and4. A simple scalar qantizer with eqally spaced codebook is sed to qantize each ser s SINR on the best beam. The nmber of beams generated by the BS N=10,andnmber of transmitter antennas L = 10.From Figre 8, we can see that the performance degradation is small. A magnified observation in Figre 8 shows that 4-bit feedback of SINR is sfficient for a performance as good as perfect feedback. From Figre 8, the throghpt loss de to limited feedback of SINR is shown to be considerably low, even by employingfewnmberoffeedbackbits.thesameobservation is made in the conventional feedback of per ser nitary and rate control (PU2RC). In [8],3bitsforSINRfeedback is fond to be sfficient to make the capacity loss de to negligible SINR qantization. 6. Conclsions In this paper, we propose a scheme to combine celllar networks and M2M networks. The devices in an M2M network access a celllar network throgh some gateways, which areassmedassecondarysersinthecelllarnetwork. With limited feedback of CSI at the BS, the new MBS method is proposed for the MTCD gateways with an overlaid

9 International Jornal of Distribted Sensor Networks 9 Throghpt per ser Throghpt per ser Nmber of sers in grop II Perfect SINR SINR qantization G=4 SINR qantization G=3 SINR qantization G=2 Figre 8: Throghpt per MTCD gateway verss U(N = 10, L=10, M = 128,and /σ 2 =20dB). transmission mode. The asymptotic performance of the proposed MBS method is characterized. Using a lower bondonthethroghptwithperfectcsi,weobtaina sefl expression that shows the system performance with limited feedback. Simlation reslts show that, when the BS is eqipped with 6 antenna elements, the throghpt was improved by abot 10% when the proposed MBS method is employed compared with the conventional ones. Appendices A. Proof of (28) In order to make the proving process convenient, we omit the sbscripts in the following derivations. Denote SINR II,m,m,Q m, g m,m, w II,m,andW I,m by SINR, g, w,andw I, respectively. From (7), the SINR is SINR = gh w 2 PII gh W I 2 P I +σ 2. (A.1) From (10), the beamforming vector for the MTCD gateway is where w = 1 w w, (A.2) Eqation (a) is de to the following featres of the matrix A= (I W I W H I ): (1) A H =A (2) A n =A, n = 1, 2,.... The nmerator of SINR is gh 2 w PII = = The denominator of SINR is Ths, the SINR is SINR = gh (I W I W H I ) g 2 w 2 gh (I W I W H I ) g 2 g 2 gh W I 2 = g 2 gh W I 2 g 2 gh W I 2 2 =( g 2 gh W I 2 )PII. (A.4) gh 2 W I PI +σ 2. (A.5) (1 ( gh W I 2 / g 2 )) ( gh W I 2 / g 2 )(P I / )+(σ 2 / g 2 ). (A.6) Define ξ=1 ( g H W I 2 / g 2 ),andγ II =( g 2 )/σ 2.The SINR expression becomes Ths, we have SINR = SINR II,m,m,Q m = B. Proof of (37) Consider P ot (Γ) ξ (1 ξ) (P I / )+(1/γ II ). (A.7) ξ m (Q m, m ) (1 ξ m (Q m, m )) (P I / )+(1/γ II,m,m ). = Pr (max SINR II,m,,Qm Γ) Pr (min {max =1 Pr (min{max ξ m (Q m,) 1 ξ m (Q m,) ξ m (Q m,) 1 ξ m (Q m,), max P I, max P I γ II,m, } Γ) (A.8) γ II,m, }>Γ) w =(I W I W H I ) g, w = w H w = g H (I W I W H I )H (I W I W H I ) g (a) = g H (I W I W H I ) g = g 2 gh W I 2. (A.3) =1 Pr (max ξ m (Q m,) 1 ξ m (Q m,) Pr (max γ II,m, >Γ). P I >Γ) (B.1)

10 10 International Jornal of Distribted Sensor Networks Using the CDF of the maximm SINR among U independent sers (34), the first mltiplier in (B.1) is Pr (max ξ m (Q m,) 1 ξ m (Q m,) P I >Γ)=1 F max ( P I Γ). (B.2) Nowwecalclatethesecondmltiplierin(B.1). From (26) and Assmption 1, γ II,m, is χ 2 distribted with 2L degrees of freedom and 1/2L variance. Denote F χ (n)(z) as the chi-sqare CDF with n degrees of freedom and nit variance. From [25], z 1 F χ (n) (z) = 0 2 n/2 Γ ((1/2) n) t(n 2)/2 e t/2 dt, (B.3) where Γ(x) is the gamma fnction. The CDF of γ II,m, is F χ (2L)(2Lz).Ths, Pr (max γ II,m, >Γ)=1 F U Γ χ (2L) (2Lσ2 ). (B.4) From (B.1), (B.2), and (B.4), we have P ot (Γ) 1 (1 F max ( P IΓ )) (1 F U Γ χ P (2L) (2Lσ2 )). II Acknowledgments (B.5) The athors wold like to thank the anonymos reviewers for their valable comments that have helped in improving the overall qality of the paper. This work has been spported by the Specialized Research Fnd for the Doctoral Program of the Ministry of Edcation of China (Grand no ) and the National Natral Science Fondation of China (Grants no , no , and no ). References [1] J. Zhang, L. Shan, H. H, and Y. Yang, Mobile celllar networks and wireless sensor networks: toward convergence, IEEE Commnications Magazine,vol.50,no.3,pp ,2012. [2] G. Wang, X. Zhong, S. Mei, and J. Wang, An adaptive medim access control mechanism for celllar based machine to machine (M2M) commnication, in Proceedings of the IEEE International Conference on Wireless Information Technology and Systems (ICWITS 10), pp , September [3] K.Zheng,F.H,W.Wang,W.Xiang,andM.Dohler, Radio resorce allocation in LTE-advanced celllar networks with M2M commnications, IEEE Commnications Magazine, vol. 50,no.7,pp ,2012. [4] 3GPP TR v0.5.1, System Improvements for Machine- Type Commnications (Release 10), Jly [5]N.Tyler,B.Allen,andH.Aghvami, Adaptiveantennas:the calibration problem, IEEE Commnications Magazine,vol.42, no.12,pp ,2004. [6] S. Nanda, R. Walton, J. Ketchm, M. Wallace, and S. Howard, A high-performance MIMO OFDM wireless LAN, IEEE Commnications Magazine,vol.43,no.2,pp ,2005. [7] D. J. Love, R. W. Heath, W. Santipach, and M. L. Honig, What is the vale of limited feedback for MIMO channels? IEEE Commnications Magazine,vol.42,no.10,pp.54 59,2004. [8] K.Hang,J.G.Andrews,andR.W.Heath, Performanceoforthogonal beamforming for SDMA with limited feedback, IEEE Transactions on Vehiclar Technology,vol.58,no.1,pp , [9] T. Yoo, N. Jindal, and A. Goldsmith, Mlti-antenna downlink channels with limited feedback and ser selection, IEEE Jornal on Selected Areas in Commnications,vol.25,no.7,pp , [10] Y. Ko and C. Tepedelenliogl, Orthogonal space-time block coded rate-adaptive modlation with otdated feedback, IEEE Transactions on Wireless Commnications,vol.5,no.2,pp , [11] Y. Ma, D. Zhang, A. Leith, and Z. Wang, Error performance of transmit beamforming with delayed and limited feedback, IEEE Transactions on Wireless Commnications,vol.8,no.3,pp , [12] R. Bhagavatla and R. W. Heath, Adaptive bit partitioning for mlticell intercell interference nlling with delayed limited feedback, IEEE Transactions on Signal Processing,vol.59,no.8, pp , [13] C.R.Mrthy,J.Zheng,andB.D.Rao, Performanceofqantized eqal gain transmission with noisy feedback channels, IEEE Transactions on Signal Processing,vol.56,no.6,pp , [14]T.WandV.K.N.La, Robstrate,powerandprecoder adaptation for slow fading MIMO channels with noisy limited feedback, IEEE Transactions on Wireless Commnications,vol. 7, no. 6, pp , [15] S. Ekbatani, F. Etemadi, and H. Jafarkhani, Otage behavior of qasi-static fading channels with partial power control and noisy feedback, in Proceedings of the 50th Annal IEEE Global Telecommnications Conference (GLOBECOM 07), pp , November [16]D.J.Love,R.W.Heath,V.K.N.La,D.Gesbert,B.D.Rao, and M. Andrews, An overview of limited feedback in wireless commnication systems, IEEE Jornal on Selected Areas in Commnications,vol.26,no.8,pp ,2008. [17] C. K. A-Yeng and D. J. Love, On the performance of random vector qantization limited feedback beamforming in a MISO system, IEEE Transactions on Wireless Commnications, vol. 6, no. 2, pp , [18] S. H. Simon and A. L. Mostakas, Optimizing MIMO antenna systems with channel covariance feedback, IEEE Jornal on Selected Areas in Commnications, vol.21,no.3,pp , [19] E. A. Jorswieck and H. Boche, Channel capacity and capacityrange of beamforming in MIMO wireless systems nder correlated fading with covariance feedback, IEEE Transactions on Wireless Commnications,vol.3,no.5,pp ,2004. [20] N. Sharma and L. H. Ozarow, A stdy of opportnism for mltiple-antenna systems, IEEE Transactions on Information Theory,vol.51,no.5,pp ,2005. [21] W.Choi,A.Forenza,J.G.Andrews,andR.W.Heath, Opportnistic space-division mltiple access with beam selection, IEEE Transactions on Commnications,vol.55,no.12,pp , 2007.

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