Max-Min Fair Hybrid Precoding for Multi-group Multicasting in Millimeter-Wave Channel
aa r X i v : . [ c s . I T ] F e b Max-Min Fair Hybrid Precoding for Multi-groupMulticasting in Millimeter-Wave Channel
Fawwaz Alsubaie
Abstract —The potential of using millimeter-wave (mmWave) toencounter the current bandwidth shortage has motivated packingmore antenna elements in the same physical size which permitsthe advent of massive multiple-input-multiple-output (MIMO)for mmWave communication. However, with increasing numberof antenna elements, the ability of allocating a single RF-chainper antenna becomes infeasible and unaffordable. As a cost-effective alternative, the design of hybrid precoding has beenconsidered where the limited-scattering signals are captured bya high-dimensional RF precoder realized by an analog phase-shifter network followed by a low-dimensional digital precoder atbaseband. In this paper, the max-min fair problem is consideredto design a low-complexity hybrid precoder for multi-groupmulticasting systems in mmWave channels. The problem isnon-trivial due to two main reasons: the original max-minproblem for multi-group multicasting for a fully-digital precoderis non-convex, and the analog precoder places constant modulesconstraint which restricts the feasible set of the precoders in thedesign problem. Therefore, we consider a low complexity hybridprecoder design to tackle and benefit from the mmWave channelstructure. Each analog beamformer was designed to maximize theminimum matching component for users within a given group.Once obtained, the digital precoder was attained by solving themax-min problem of the equivalent channel.
Index Terms —Multicasting, Limited RF chains, Hybrid pre-coding, Low complexity algorithm.
I. INTRODUCTIONThe event of common data targeting a mass of users or relaystations, in the case of relay networks, has become popular incurrent wireless systems. Services like video/audio streaming,news, common clips and messages are expected to furthergrow in next-generation wireless systems [1]. In a scenariowhere a group of users demand the same information, i.e.single-group multicasting (SGM) or broadcasting, the basestation (BS) can provide the service by a point-to-multipointconnection. In the case of multi-group multicasting (MGM),groups with disjoint sets of users are entitled to differentcommon messages. In the case of a single user per group, thescenario is known as multi-user (MU) downlink beamforming.The provision of multicasting services, both SGM and MGM,has been introduced by the Global System for Mobile com-munication (GSM) and Universal Mobile TelecommunicationsSystem (UMTS) as a form of multimedia broadcast/multicastservice [2], [3]. In addition, a form of multicasting has beenproposed by the Worldwide Interoperability for MicrowaveAccess (WiMAX) [4].When considering a serving-network where users can re-quest same common messages (multicasting), the support ofsuch services can be provided under a wired network. How-ever, from a cost and feasibility prospective, it is impossibleto extend cables to every user or a relay station. The solution is to utilize the wireless medium to perform multicasting.The apparent wireless nature as a broadcast medium causescross-talk or co-channel interference among users scheduledat the same time and frequency slots, not mentioning fadingand shadowing. Then, to mitigate interference, users usedto be scheduled among the orthogonal resources, e.g. timeand frequency. This resulted in increasing network traffic anddecreasing spectral efficiency.To address this issue, in MU-beamforming, smart antennaarray with adjustable antenna weights has been implemented atthe transmitter to beamform, or precode, to the served users.Hence, users can be simultaneously scheduled at the sametime-frequency slot while experiencing minimal interference.The later technique utilizes the third degree of freedom (DoF)in the system, i.e. space, and formally called spatial divisionmultiple access (SDMA). Techniques to perform SDMA todesign optimal and sub-optimal precoders and with variousscenarios have been extensively studied in the literature [5]–[10]. When considering multicasting, the concept of physical-layer multicasting (PHY-multicasting) was firstly introduced in[11] for SGM and in [12] for MGM. Moreover, the discusseddesign problems were far from trivial compared to MU-beamforming and are detailed later in this paper.In addition to the future need for multimedia/ multicastingservices, the demand for low-latency application and deliveryof high-quality multimedia content is a genuine challenge.With the rapid growth of smartphones, available carrier fre-quency slots have become ever more scarce. To overcomethe bandwidth shortage, the underutilized millimeter wave(mmWave) has been strongly suggested for the next-generationfrequency spectrum use [13]. The mmWave signal experi-ences orders-of-magnitude path loss due to a ten-fold carrierfrequency increase. With this difficulty comes the interestingfeature of mmWave that is packing more antennas at the samephysical size of the original microwave antenna. Large antennaarray offers beamforming gain to overcome path loss andallows spatial multiplexing which could improve the overallsystem spectral efficiency [14]–[16].So far, the adaptive antenna array permits flexible configura-tion in the RF domain of both magnitude and phase. However,in mmWave channels, the BS is equipped with a large antennaarray and it is rather infeasible to dedicate a single RF-chain per antenna element [17]. Therefore, precoding has beendivided into two stages: digital low-dimensional baseband pre-coding followed by high-dimensional RF-precoding realizedby an analog phase shifter network. Since the introductionof hybrid precoding in [18], it has been a strong candidatefor the next-generation wireless system [19]. In the hybridstructure, the RF-precoding has a fewer number of RF-chains fully or partially-connected with all antennas whereas thedigital-processing is done at the baseband. The Full analysisof the minimal number of RF chains to realize a fully-digitalprecoder is given in [20]–[22]. Works on designing a hybridprecoder in mmWave channels for various systems are givenin [23]–[28] while a low-complexity hybrid precoder designunder independent and identically distributed (i.i.d) Rayleighfading channel is considered in [29].II. R
ELATED P REVIOUS W ORK
In multicasting, SGM or MGM, the existing work have ad-dressed various frameworks to specify what-called the optimalprecoder. Herein, we mention the existing design problems formulticasting systems: • Quality of Service (QoS): the QoS problem considers aminimum service level for each individual user in themulticasting system. The goal is to minimize the totaltransmit power subject to the minimum service levels forall users in the multicasting system. The optimal precoderin the QoS sense provides the minimum service level(s)with the minimum power consumption. • Max-min fair (max-min): the max-min fair problem isconcerned with a fair performance in a multicastingsystem. The goal is to maximizes the minimum QoSlevel(s) for all users in the multicasting system subject toa total transmit power. Conceptually, this is achieved byreducing the power for the good-condition-channel userscompared to the worst channel(s). The optimal precoderin the fairness sense ensures a fair performance amongusers and satisfies the total transmit power constraint withequality. • Sum-Rate Maximization (SR): the SR problem considersthe optimal precoder for which the multicasting systemexperiences the ultimate throughput. Along this line, theSR problem doesn’t consider fairness among users orgroups. For example, in the case of MGM, low-channel-condition group can be set to service unavailability. Inother words, the power is not consumed to compensatefor channel conditions.One or more of the above problems could be solved underPer Antenna Constraint (PAC). The motivation of investigatingsuch a constraint comes from a practical system implementa-tion aspect. Power flexibility is not always feasible at the trans-mitter due to different antennas having individual amplifiersand hence the need to specify the power consumed by eachantenna element. Also, different antenna amplifiers can havedifferent power range, i.e. different saturation power levels,and thus PAC problem can help in controlling the power ofeach antenna element.
A. Single-group multicasting (SGM):
In SGM, the cell performance characterized by total trans-mission power or overall throughput is constrained by theworst-condition user in the group. In SGM, a single commonmessage addresses a group of co-channel users and thus cross-talk is not an issue. While SGM can be a special case of MGM and a work on MGM would generally apply on SGM,we mention specific works on SGM systems.Physical layer multicasting was firstly introduced by Lopezin [30], where the sum of signal-to-noise ratio (SNR) wasmaximized for all users. The optimization was equivalent tomaximizing an average SNR not considering individual userswhich boiled down to a simple eigenvalue problem. However,the work in [30] has inspired the development of various algo-rithms and design problems concerning multicasting systems.The QoS problem for SGM has been firslty introducedin [11], where the problem is formulated as a non-convexQuadratically Constrained Quadratic Program (QCQP) andshown to be an NP-hard problem. The problem is relaxed, newvariables are introduced, and reformulated as a semi-definiterelxation program (SDR) [31]. Following that, if the solutionis not rank-one, Gaussian randomization with scaling are usedto find the optimal precoder. The max-min problem has alsobeen addressed in [11], and the solution is found followingthe previous strategy.In [32], the max-min fair problem has been formulated tofind a sub-optimal hybrid precoder design under mmWavechannel. The RF-precoder and baseband precoder are decou-pled. The RF-precoder is designed based on a codebook tomaximize an upper bound assuming a fixed baseband precoder.Once obtained, the digital precoder is obtained following thestrategy in [11].
B. Multi-group multicasting (MGM):
A fundamental difference in MGM compared to SGM: is theexisting of disjoint sets of users demanding different commonmessages and hence interference becomes an issue. However,the problem is very general where it includes many scenarios,SGM, for example. In addition, the SGM design problemsare always feasible, whereas for MGM systems, the designproblems can be infeasible. On the other hand, some of theMGM design problems are more flexible in which the servicelevels for different groups can be adjusted to fulfill someoptimization goals.The QoS and MMF problems for MGM have been firstlyintroduced in [12]. Due to interference and possible infeasibil-ity, the QoS problem for MGM is different from the SGM onepresented in [11]. The SDR method is proposed to solve theQoS problem similar to [11]. However, the randomization ismore involved since the scaling can enhance the interference.Therefore, a power control program is solved to provide theright scaling factors. For the max-min problem, the SDRprogram is non-linear and is solved, if feasible, by a bisectionmethod followed by a multi-group power control program. Inthe case of Vandermonde channels, which is the case in line-of-sight (LoS) scenarios, the channel matrices are shown to berank-one and the relaxation is tight. Hence, if the problem isfeasible, the optimal solution is always obtainable [33].To account for practical system limitation, the MGM sum-rate problem with PAC has been presented in [34] while themax-min fair problem with PAC has been discussed in [35],[36].In order to mitigate SDR complexity, specially as number ofusers increases, Successive Convex Approximation (SCA) has been proposed to solve the QoS problem in [37] and max-minfair problem in [38]. Also, the max-min sum rate and max-minwith PAC have been investigated in [39] and [40], respectively.Similar to SGM, the solution might not be optimal but it is alow-complexity design compared to SDR.Recent works have considered hybrid precoding for MGMsystems under mmWave channels [41]–[44].In this paper, we consider the problem for MGM systemswith hybrid precoding introduced in [18] under mmWavechannels. We give a low complexity max-min fair hybridprecoding design for MGM systems under mmWave channels.System performance is simulated and discussed.
Notations:
We use the following notation throughout thispaper: A is a matrix; a is vector; a is a scalar; I N is the N × N identity matrix; k . k F is the Frobenius norm; ( . ) T is thetranspose operator; ( . ) H is the conjugate transpose operator; tr( . ) is the trace operator; E [ . ] is the expectation and arg( . ) is the phase of the entries for matrix or vectors.III. S YSTEM M ODEL
Consider a single-cell system where the base station (BS)is equipped with N antenna elements and N RF RF-chainscommunicating with M single-antenna users. Further, let therebe a total of ≤ G ≤ N RF multicasting groups where themulticast group index set writes as I = {G , . . . , G G } and G k is the set of users belonging to the k th multicast group, k = { , . . . , G } . Each user can belong to one group only andhence, G i T G j = 0 , ∀ i, j ∈ { , . . . , G } and S Gi =1 G i = M .We assume a single RF chain is dedicated to each group andthere are as many groups as number of chains at a given timeinstant and thus N RF = G .On the downlink, the base station applies the N RF × G digitalbaseband precoder W = [ w , w , . . . , w G ] to the sampledtransmitted data s ∈ C G × and up-converts the processedsignal to the carrier frequency by applying the N × N RF RF precoder F = [ f , f , . . . , f N RF ] . The precoded transmittedsignal x ∈ C N × writes as: x = FWs , (1)where E [ ss ∗ ] = I G . The RF precoder F is implemented usingan analog phase shifter network and its entries has the constantmodulus constraint such that F ( i, j ) = e θ i,j and | F ( i, j ) | =1 . The total transmission power P is allocated such that k FW k F = P .For simplicity, we focus on the narrow-band block fadingchannel model in which the m th user, belonging to the k thgroup, observes the following received signal: y m = h Hm Fw k s k | {z } Intended-signal + G X i = k,i =1 h Hm Fw i s i | {z } Interference-term + n m |{z} noise-term , (2)where h m ∈ C N × is the mmWave channel response vectorbetween the BS antenna elements and the m th single-antennauser. We adopt a geometric finite-scattering channel modelwith L propagation paths between the BS and a mobileterminal. The mmWave channels are expected to have finite scattering for a single propagation path and hence we assumethat each scatter contribute a single path between the BS and amobile user. Additionally, we consider an Uniformally-Linear-Array (ULA) structure at the BS. Then, the channel for the m th user is written as: h m = r NL L X l =1 β m,l a t ( φ m,l ) , (3)where a t ( φ m,l ) ∈ C N × is the array response vector at thebase station due to the transmitted (beamformed) signal at the φ m,l angle of departure (AoD) and can be expressed as: a t ( φ m,l ) = 1 √ N h , e j π dλ sin( φ m,l ) , . . . , e j ( N − π dλ sin( φ m,l ) i T , and β m,l is the channel gain of the m th user in the direction ofthe steered signal. The term n m is an Additive White GaussianNoise (AWGN) at the m th receiver such that E [ n m n ∗ m ] = σ m ,and we assume without the lose of generality σ = σ = · · · = σ M = σ n . The Signal to Interference plus Noise Ratio (SINR)experienced by the m th is given by: SINR m = (cid:12)(cid:12) h Hm Fw k (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G P i = k,i =1 h Hm Fw i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + σ n (4)Assuming i.i.d Gaussian input-streams, the instantaneous rateexperienced by the m th user writes as: R m = log (1 + SINR m ) (5)IV. PROBLEM FORMULATIONThe main objective is to design a low-complexity hybridprecoder for multi-group multicasting systems in mmWavechannels. In this work, we consider the max-min fairnessproblem to design the hybrid precoder to ensure minimumservice level for all users regardless to the group they belongto. The max-min fair problem is formally defined as: max F , W min k ∈{ ,...,G } min m ∈G k (cid:12)(cid:12) h Hm Fw k (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P i = k h Hm Fw i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + σ n (6) s . t . : F ∈ F , k FW k F ≤ P, where F is the set of N × G matrices with constant modulesentries and W is the N RF × G digital precoder. The totaltransmission power constraint is explicitly specified in (6).In the fully digital system, i.e, when FW = V D where V D ∈ C N × G , the problem in (6) has been shown to be non-convex (NP-hard) and thus cannot be solved efficiently usingconvex optimization solvers, e.g, interior-point method [12].Moreover, restricting the search space by adding the specialstructure of the RF precoder results in an even more difficultproblem. In addition, the baseband precoder W needs to bejointly designed with the RF precoder F and the optimizationis often found intractable [23]. Therefore, it is often advisedto decouple the problem into two parts. First, we design the RF precoder F by assuming a fixed baseband precoder W and then F is fixed and W is designed [28]. Assuming F ⋆ isattained, and for clarity we denote it as F , the program in (6)can be re-written as: W ⋆ = arg max W min k ∈{ ,...,G } min m ∈G k (cid:12)(cid:12)(cid:12) h H eff ,m w k (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P i = k h H eff ,m w i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + σ n s . t . : W ∈ C N RF × G , (7) k FW k F ≤ P, where h H eff ,m = h Hm F . The program in (7) can be equiva-lently written as a Semi-Definite Program (SDP). Defining n Q eff ,m = h eff ,m h H eff ,m o m = Mm =1 and (cid:8) X k = w k w Hk (cid:9) k = Gk =1 .Thus, (cid:12)(cid:12)(cid:12) h H eff ,m w k (cid:12)(cid:12)(cid:12) = tr( Q eff ,m X k ) , X k ≥ and noting k FW k F = P Gk =1 tr( FX k F H ) , (7) is then reformulated as: { X ⋆k } Gk =1 = arg max t { X k } Gk =1 (8) s . t . : tr( Q eff ,m X k ) P i = k tr( Q eff ,m X i ) + σ n ≥ t, G X k =1 tr( FX k F H ) ≤ P, X k ≥ , t ≥ , rank ( X k ) = 1 , ∀ k ∈ { , . . . , G } , ∀ m ∈ { , . . . , M } . where (8) is a reminiscent of the max-min fairness problemdiscussed in [12] where the non-convex rank-one constraintsare dropped and the problem is solved, if feasible, by bisectionto yield an upper bound for the maximum SINR experiencedby all users. The relaxed problem is re-casted as: P main : max t { X k } Gk =1 s . t . : tr( Q eff ,m X k ) − t X i = k tr( Q eff,m X i ) + σ n ≥ , G X k =1 tr( FX k F H ) ≤ P, X k ≥ , t ≥ , ∀ k ∈ { , . . . , G } , ∀ m ∈ { , . . . , M } . Therefore, in this paper, we aim to design a low-complexityhybrid precoder design for multi-group multicasting systemsto approach the upper bound achieved by the fully-digitalprecoder bearing in mind the special structure of the channel,namely, sparse multipath channels or mmWave channels.V. PROPOSED SOLUTIONSProblem P main can be solved to attain a local optimaldigital baseband precoder W ⋆ . However, prior solving for W ⋆ , we need to design the RF precoder and constructthe effective channel(s) Q eff . Fortunately, the nature of the mmWave suggests a way in selecting the RF precoder andhere we could indicate the following remarks: • In the case of a mmWave channel between the BS and aserved user and as number of transmit antenna increases,the dependence on other paths becomes less importantcompared to the strongest path [25]. Thus, we considersingle path mmWave channels. • In single path mmWave channels and as number of trans-mit antenna increases, different users with distinct AoD’sfrom the BS exhibit orthogonal channel vectors, thanksto the asymptotic orthogonality property of mmWavechannels [24]. • In order to realize a fully-digital precoder under any chan-nel structure, [18], [26] have shown that two RF-chainsare needed per a digital precoder, i.e, G = 2 N RF . Inaddition, in order to modulate G streams, it is necessaryto have at least G RF-chains and hence N RF , min = G .Therefore, we consider a single RF-chain per group, N RF = G . • In single path mmWave channels of users with distinctangles from the BS and as N → ∞ , the conjugate analogbeamformer per user, N RF = M , is optimal [45]. • In connection of the last mentioned remarks, we proposeto design the RF precoder as in the following program: max f k min m ∈G k (cid:12)(cid:12) h Hm f k (cid:12)(cid:12) (9) s . t . : | f k ( i ) | = 1 , ∀ k ∈ { , . . . , G } , ∀ i ∈ { , . . . , M } . However, the program in (9) is NP-hard due to the non-convex constant modules constraint. Therefore, we opt to relaxthe constraint as shown in (10). max u k min m ∈G k (cid:12)(cid:12) h Hm u k (cid:12)(cid:12) (10) s . t . : | u k | = N, ∀ k ∈ { , . . . , G } , ∀ m ∈ { , . . . , M } . where u k is now entirely digital and 10 is equivalently writtenas: F main : max Y k t s . t . : tr( Q m Y k ) ≥ t tr( Y k ) = N, ∀ k ∈ { , . . . , G } , ∀ m ∈ { , . . . , M } . where Y k = tr( u k u Hk ) and Q m = tr( h i h Hi ) . Problem F main represents a set of single group max-min fairnessmulticasting problems where the problem is always feasible[11]. In addition, if the solution is optimal, F main outputs arank-one solution. In fact, Problem F main always outputs arank-one solution since the channels are all rank-one. Hence,the principle component vectors of { Y k } Gk =1 gives the optimaldigital precoders { u ⋆k } Gk =1 . Once { u ⋆k } Gk =1 is obtained anddenoted U = [ u ⋆ , . . . , u ⋆N RF ] , the RF-precoder has a closed-form solution arg( F ) = arg( U ) (11) SNR (dB) -10 0 10 20 30 40 50 R a t e ( bp s / H z ) Digital N=4Hybrid N=4 , N RF =1Digital N=8Hybrid N=8 , N RF =1 Fig. 1: Max-min rate of a Single group multicasting systemwith G = 1 , M = 4 users.After finding the RF precoder based on thecriterion described above, the equivalent channels n Q eff ,m = h eff ,m h H eff ,m o m = Mm =1 are formed. Then, Problem P main is solved, if feasible, to obtain an upper bound for themax-min SINR value.VI. SIMULATION RESULTSIn this section, simulation results are presented to showthe performance of the hybrid proposed algorithm versus afully-digital system algorithm with ( N RF = N ) for variousmulticasting systems. A mmWave channel with L = 1 is con-sidered between the m th user and the BS and an Uniformally-Linear-Array (ULA) with N elements is assumed at the BS.The angle of departure (AoD) of the m th user, φ m , is drawnfrom a uniform random variable defined as φ m ∼ U (0 , π ) .In addition the channel gain of the m th user is drawn froma circularly symmetric Gaussian random variable defined as β m ∼ CN (0 , . The noise variance of the receivers isassumed as σ = σ = · · · = σ M = σ n = 1 . The transmitpower at the BS is varied from − to with increment.Fig. 1 shows the performance of the hybrid proposedalgorithm versus the fully-digital system algorithm for a singlegroup multicasting system (broadcasting). In this scenario,number of users in the group is fixed as M = 4 while numberof antennas N is varied to evaluate the performance. As canbe seen in Fig. 1, both systems with different N values havethe same slope of at high SNR values which indicates aninterference-free stream experienced by all users in the system.The proposed algorithm performance is close to the fully-digital system at low SNR values and with constant gap athigh SNR values. As N increases, the rate increases for bothsystems which is a result of the array gain offered by theMISO channel.Fig. 2 shows the performance of the hybrid proposedalgorithm versus the fully-digital system for a multi-userbeamforming system with N = 64 , G = 3 and M = 3 . As can SNR (dB) -10 0 10 20 30 40 50 R a t e ( bp s / H z ) N-64, N RF =3N=64 Fig. 2: Max-min rate of a Multi-user beamforming system with N = 64 antennas, G = 3 groups, M = 3 users.be seen in Fig. 2, the performance of the proposed algorithm isclose to the fully-digital system algorithm at low SNR values.However, the gap increases as the transmit power increases.At high SNR values, the interference between different usersbecomes more dominant compared to the noise level and thusdegrades the overall performance of the proposed algorithm.The interference causes the slope of the rate curve to graduallydecrease as SNR value increases until it saturates. Conversely,at low SNR values, the interference is neglected compared tothe noise level and hence the performance of both algorithmsare relatively similar.Fig. 3 shows the performance of both systems for a multi-group multicasting system with G = 3 , N RF = 3 , and M = 6 . Users are equally distributed among groups, i.e. users per group. Number of antennas N is varied to evaluatethe performance of both systems. As seen in Fig. 3, at lowSNR values the performance of both algorithms are extremelyclose. However, at high SNR, the rate curves for the proposedalgorithm with different N values saturates while the fully-digital system experiences no saturation effect. The saturationis due to inter-group interference experienced by users belong-ing to different groups. The fully-digital system doesn’t sufferfrom the interference due to the minimal adequate amount ofantennas at the transmitter to fully null-out the interference.In specific, for equal user-group distribution, the minimumnumber of antennas at the transmitter to null-out the inter-group interference is M − |G| . Figure 4 illustrates theperformance for an asymmetric distribution specified in thefigure caption compared to the symmetric distribution (equaluser-group distribution). As can be seen in Figure 4, withan asymmetric distribution, the rate curve slope is largercompared to the symmetric case. Saturation occurs in bothcases but it would happen later in the asymmetric scenario andthe reason could attribute to the less interfering users in thegroups with smaller number of users compared to the group SNR (dB) -10 0 10 20 30 40 50 R a t e [ bp s / H z ] N=8, N RF = 3N=128,N RF = 3N=8N=128 Fig. 3: Max-min rate of a MGM system with G = 3 groups, M = 6 users (equal group distribution). SNR (dB) -10 0 10 20 30 40 50 R a t e [ bp s / H z ] N=8, AsymmetricN=8, N RF =3, AsymmetricN=8, symmetricN=8, N RF =3, symmetric Fig. 4: Max-min rate of a MGM system with N = 8 antennas, G = 3 groups, M = 6 users ( |G| = 3 , |G| = 2 , |G| = 1 ).with largest number of users.In Fig. 5, the performance of the two systems are shownfor a multi-group multicasting system with N = 8 , G = 3 , M = 6 and different number of paths per user’s channel. Thegoal is to evaluate the performance at an extreme mmWavecase when L = 1 and a moderate case when L = 15 . Forthe fully-digital system, the extra paths in the channel allowsfor an increase in the rate for all users. In contrast, for theproposed system design and at high SNR values, the extrapaths result in an early saturation effect compared to the singlepath channel performance. Because the design only accountsfor the dominant path of each user, increasing number ofpaths per user creates a greater inter-group interference andeventually results in a saturation effect. At low SNR values,the interference is minimal and a larger rate is observed when L = 15 compared to the performance when L = 1 . SNR (dB) -10 0 10 20 30 40 50 R a t e [ bp s / H z ] N=8, L=1,N RF =3N=8, L=15N=8, L=1N=8, L=15,N RF =3 Fig. 5: Max-min rate of a MGM system with N = 8 antennas, G = 3 groups, M = 6 users (equal group distribution).VII. CONCLUSIONIn this paper, a low complexity max-min fair hybrid pre-coder design was proposed for multi-group multicasting sys-tems. The design extended to multi-user systems as wellas broadcasting systems. The design was motivated by thespecial structure of the channel, namely, single-path mmWavechannel for each user. The RF precoder consisted of G analogbeamformer(s). Each analog beamformer was designed tomaximize the minimum matching component for users withina given group. Once obtained, the digital precoder was attainedby solving the max-min problem of the equivalent channel.The performance of the fully-digital system and the proposedsystem design were shown by simulations for multi-groupmulticasting, broadcasting and multi-user systems.R EFERENCES[1] Y. Silva and A. Klein, “Linear Transmit Beamforming Techniques forthe Multigroup Multicast Scenario,”
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