D2D-Aided Multi-Antenna Multicasting under Generalized CSIT
11 D2D-Aided Multi-Antenna Multicasting underGeneralized CSIT
Placido Mursia,
Student Member, IEEE,
Italo Atzeni,
Member, IEEE ,Mari Kobayashi,
Senior Member, IEEE, and David Gesbert,
Fellow, IEEE
Abstract
Multicasting, where a base station (BS) wishes to convey the same message to several user equip-ments (UEs), represents a common yet highly challenging wireless scenario. In fact, guaranteeingdecodability by the whole UE population proves to be a major performance bottleneck since the UEsin poor channel conditions ultimately determine the achievable rate. To overcome this issue, two-phasecooperative multicasting schemes, which use conventional multicasting in a first phase and leveragedevice-to-device (D2D) communications in a second phase to effectively spread the message, have beenextensively studied. However, most works are limited either to the simple case of single-antenna BS orto a specific channel state information at the transmitter (CSIT) setup. This paper proposes a generaltwo-phase framework that is applicable to the cases of perfect, statistical, and topological CSIT in thepresence of multiple antennas at the BS. The proposed method exploits the precoding capabilities at theBS, which enable targeting specific UEs that can effectively serve as D2D relays towards the remainingUEs, and maximize the multicast rate under some outage constraint. Numerical results show that ourschemes bring substantial gains over traditional single-phase multicasting and overcome the worst-UEbottleneck behavior in all the considered CSIT configurations.
Index terms —Cooperative communications, device-to-device communications, multicasting, multiple-input multiple-output, statistical precoding.
P. Mursia and D. Gesbert are with the Communication Systems Department, EURECOM, France (email: { placido.mursia,david.gesbert } @eurecom.fr). I. Atzeni is with the Centre for Wireless Communications, University of Oulu, Finland (email:italo.atzeni@oulu.fi). M. Kobayashi is with the Technical University of Munich, Germany (email: [email protected]).The work of P. Mursia was supported by Marie Skłodowska-Curie Actions (MSCA-ITN-ETN 722788 SPOTLIGHT). The workof I. Atzeni was supported by the Marie Skłodowska-Curie Actions (MSCA-IF 897938 DELIGHT). The work of M. Kobayashiand D. Gesbert was supported by the French-German Academy towards Industry 4.0 (SeCIF project) under Institut Mines-Telecom. Part of this work has been presented at IEEE ICC 2019 [1] and at ASILOMAR 2019 [2]. a r X i v : . [ c s . I T ] F e b I. I
NTRODUCTION
Multicast services, where a base station (BS) needs to convey a common valuable messageto a set of user equipments (UEs), arise naturally in many wireless scenarios [3]–[8]. Notableexamples are wireless edge caching, where popular media are cached during off-peak hoursand subsequently streamed via multicasting [9], [10], and the broadcasting of mission-criticalmessages in vehicular networks [11]. However, it is well known that multicasting over wirelesschannels is hindered by the worst-user-kills-all effect, whereby the multicast capacity vanishesas the number of UEs K increases for a fixed number of BS antennas [3], [4]. In fact, sincethe message transmitted by the BS must be decoded by all the UEs, the multicast capacity islimited by the UEs with the smallest fading gain and the latter tends to decrease with the systemdimension. In particular, for the case of i.i.d. Rayleigh fading channels, the multicast capacityvanishes quickly as it scales inversely proportional to K [3].To overcome this issue, different approaches have been considered in the literature (e.g.,[3], [5]–[8], [12]–[18]), which can be roughly classified into three groups. In the first group, asubset of UEs in good channel conditions is selected to be served, whereas the UEs in poorchannel conditions are neglected [12], [13]. However, not only does such an approach resultin limited network coverage, but it also implies solving a combinatorial problem to find thebest subset of UEs. The second group exploits multiple antennas at the transmitter and theresulting channel hardening to mitigate the variance of the individual received signal power asthe number of UEs increases [3], [14]. However, such an approach is based on the assumptionof i.i.d. Rayleigh fading channels and requires that the number of BS antennas grows at least as log( K ) . Lastly, the third group builds on the UE cooperation enabled by device-to-device (D2D)links. Indeed, D2D communications hold the potential to counteract the performance limitationsof several emerging applications in fifth-generation (5G) wireless systems such as multicasting,machine-to-machine communication, and cellular-offloading [19]–[23]. In the relevant case ofmulticasting, D2D communications between the UEs can be leveraged to overcome the vanishingbehavior of the multicast capacity by dividing the total transmission time in two phases. Here,conventional multicasting occurs only in the first phase, where the BS transmits at such a ratethat the common message is received by a subset of UEs in favorable channel conditions. Then,these UEs act as opportunistic relays and cooperatively retransmit the message in the secondphase. This approach has been extensively studied in the literature under specific channel state information at the transmitter (CSIT) assumptions and by focusing on the simple case of single-antenna transmitter [5]–[8], [15]–[18], as detailed next.Theoretical analysis of two-phase cooperative multicasting can be found in [5]–[7], [15],[18]. More specifically, [5] established the multicast capacity by using a two-phase cooperativescheme for a simple network with i.i.d. Rayleigh fading channels. The multicast scaling wasanalyzed in [6] for two different network models, where the multicast capacity was shownto grow as log(log( K )) in the case of dense network (i.e., a scenario in which the numberof receivers increases over a fixed network area) with spatially i.i.d. channels. Recently, [18]characterized the multicast scaling for a more general network topology (capturing the pathloss)and showed that, with statistical CSIT, the average multicast rate increases as log(log( K )) . Asimilar analysis can be found in [7] for IEEE 802.16-based wireless metropolitan area networks.Furthermore, [15] characterized the achievable multicast rate of an interactive scheme based onfull-duplex and non-orthogonal cooperation links. Another two-phase scheme was presented in[8], which focused on minimizing the total power consumption while guaranteeing a certaincoverage under perfect CSIT. On the other hand, [16] considered a two-layer multicast messagestructure with a high-priority, low-rate part and a low-priority, high-rate part, such that the UEswho are able to decode the entire message assist the others by acting as opportunistic relays.The time allocation between the two phases was investigated in [17], which showed that moretime should be dedicated to the second phase as the UEs move away from the BS. Finally, asimilar two-phase cooperative scheme with multiple antennas at the BS was proposed in [24] inthe context of broadcasting under perfect CSIT. By exploiting rate splitting, this scheme formsa virtual common message to be multicast in the first phase and retransmitted via opportunisticrelaying in the second phase.In summary, existing works have demonstrated the benefits of two-phase cooperative schemeseither for specific CSIT configurations or for the simple case of single-antenna BS. This motivatesus to study the two-phase cooperative multicasting by exploiting multiple antennas at the BSunder various CSIT configurations ranging from perfect CSIT to topological CSIT, where onlythe map of the network area and the UE distribution are available at the BS. A. Contribution
In this paper, we propose a general two-phase cooperative multicasting framework that lever-ages both multi-antenna transmission at the BS and D2D communications between the UEs.
In particular, we highlight how endowing the BS with multiple antennas radically transformsthe problem of cooperative multicasting. Indeed, the precoding capabilities at the BS introduceadditional degrees of freedom for spatial selectivity that, exploited together with the D2D links,modify the nature and the performance of the two-phase schemes described in the previoussection. However, this implies the joint optimization of the precoding strategy at the BS and themulticast rate, which is, at first glance, highly complex to tackle: to the best of our knowledge,this is the first work that addresses such a scenario.We consider a general system model (in terms of both channel model and network topology)and explicitly optimize the precoding strategy at the BS and the multicast rate over the two phases.More specifically, we propose several schemes to tackle different CSIT configurations, namely: i) perfect CSIT, where the instantaneous channels are perfectly known; ii) statistical CSIT, whereonly the long-term channel statistics are available; and iii) topological CSIT, where only the mapof the network area and the UE distribution are accessible. Note that statistical CSIT applies toscenarios with a large number of UEs or limited feedback in frequency-division duplex mode,while topological CSIT applies to scenarios where neither instantaneous nor statistical CSIT isavailable and only the UE distribution across the network can be considered for the optimization(see, e.g., [25]). In addition, following [18], we use the notion of target outage in the optimizationof the multicast service, by which the multicast rate is maximized while guaranteeing decodabilityby most UEs up to the desired success level. In this way, we strategically avoid wasting resourceson a small amount of UEs with particularly unfavorable channel conditions [26]. Numericalresults show that the proposed schemes significantly outperform conventional single-phase multi-antenna multicasting in all the considered CSIT configurations. Remarkably, they allow toeffectively overcome the vanishing behavior of the multicast rate and achieve an increasingperformance as the UE population grows large.The contributions of this paper are summarized as follows: • Assuming a general channel model and network topology, we propose a two-phase coop-erative multicasting framework with multi-antenna transmission at the BS. We tackle thejoint optimization of the precoding strategy at the BS and the multicast rate subject to someoutage constraint. This framework is particularized to three different CSIT configurations,i.e., perfect, statistical, and topological CSIT. An interesting feature of our algorithms isto provide, as by-product, a selection of the UEs that are best positioned to serve as D2Drelays to the remaining UEs without the need for any explicit relay selection scheme. • For the case of perfect CSIT, we propose a low-complexity iterative algorithm that jointlyselects a subset of UEs to be served by the BS in the first phase and optimizes the multicastrate while guaranteeing the desired success level. This algorithm, referred to as
D2D-MAM ,is shown to converge to a locally optimal solution. • For the case of statistical CSIT, we propose a low-complexity algorithm that relies onlong-term channel statistics without requiring costly instantaneous CSIT, which is a majoradvantage in scenarios with a large number of UEs or limited feedback. For this algorithm,referred to as
D2D-SMAM , we study the scaling of the resulting multicast rate as a functionof the number of UEs and BS antennas and show that this is non-vanishing in the case ofdense network. • For the case of topological CSIT, we propose an algorithm based on Monte Carlo samplingthat relies uniquely on the map of the network area and the probability density function (pdf)of the UE locations. This approach is desirable in scenarios where neither instantaneousnor statistical CSIT is available and only the UE distribution across the network can beconsidered for the optimization. The proposed algorithm, referred to as
D2D-TMAM , runsthe D2D-MAM algorithm on several sets of UE locations and channels generated accordingto the UE distribution, and the outputs are averaged to obtain the actual precoding strategyat the BS and multicast rate. • We present a comprehensive numerical evaluation of the proposed schemes showing sub-stantial gains compared to the reference single-phase multi-antenna multicasting in the threedifferent CSIT configurations.
B. Outline and Notation
The rest of the paper is organized as follows. Section II describes the system model. Section IIIdeals with the case of perfect CSIT and introduces the D2D-MAM algorithm. Section IV tacklesthe case of statistical CSIT and presents the D2D-SMAM algorithm. Section V considers the caseof topological CSIT and proposes the D2D-TMAM algorithm. Then, Section VI provides numeri-cal results assessing the performance of the proposed schemes in the various CSIT configurations.Finally, Section VII summarizes our contributions and draws some concluding remarks.Throughout the paper, scalars are denoted by italic letters, while (column) vectors and matricesare denoted by boldface lowercase and uppercase letters, respectively. C represents the set ofcomplex numbers, whereas C N × M denotes the set of ( N × M ) -dimensional complex matrices. BS with M antennasFirst phaseSecond phase K single-antenna UEsRelay set U = { , , } h , γ θ h , γ h , γ Fig. 1: A BS equipped with M antennas multicasts a common message to a subset of UEs with a properly designedprecoding strategy in the first phase (solid lines). The UEs who successfully decode the message in the first phaseretransmit it in the second phase to the remaining UEs via D2D links (dashed lines). ( · ) T , ( · ) H , and ( · ) ∗ are the transpose, Hermitian transpose, and conjugate operators, respectively. and represent the all-one vector and the all-zero matrix, respectively, of proper dimensions.The N -dimensional identity matrix is denoted by I N , whereas e n indicates its n th column. (cid:107) · (cid:107) represents the Euclidean norm for vectors, whereas E [ · ] and and [ · ] are the expectationoperator and the indicator function, respectively. Furthermore, [ a , . . . , a N ] denotes horizontalconcatenation, whereas { a , . . . , a N } or { a n } n ∈N denote the set of elements in the argument.Lastly, X P → ¯ X denotes convergence in probability of the random variable X , whereas f ( (cid:15) ) ∼ (cid:15) → g ( (cid:15) ) means that lim (cid:15) → f ( (cid:15) ) g ( (cid:15) ) = 1 . II. S YSTEM M ODEL
A. Two-Phase Cooperative Multicasting
We consider a wireless network where a BS equipped with M antennas aims at transmittinga common valuable message to a set of single-antenna UEs K (cid:44) { , . . . , K } , where h k ∈ C M × denotes the downlink channel between the BS and UE k . The UEs are also connected to eachother via D2D links in half-duplex mode, where h jk ∈ C denotes the D2D channel betweenUEs j and k . We adopt a dense network scenario, i.e., where the number of receivers increasesover a fixed network area, and we assume that K (cid:29) M . For the sake of simplicity, we follow[5], [6] and focus on a cooperative scheme divided into two phases of equal length. Such ascheme is depicted in Fig. 1 and the two phases are described next. First phase.
The BS transmits the message x ∈ C M × at rate r , referred to as multicast rate , and with transmit covariance Γ (cid:44) E [ xx H ] , with tr( Γ ) ≤ . The receive signal at UE k in the first phase is given by y k, (cid:44) (cid:112) ξ h H k x + n k ∈ C (1)where ξ is the transmit power at the BS and, since we assume the additive white Gaussiannoise (AWGN) noise term n k to be distributed as CN (0 , , it can be interpreted as thetransmit signal-to-noise-ratio (SNR) at the BS. The message is decoded by UE k if itsachievable rate in the first phase is greater than or equal to the multicast rate r , i.e., if log (1 + ξ h H k Γh k ) ≥ r . We define the subset of UEs whose achievable rate in the firstphase is at least r for a given transmit covariance as U (cid:44) (cid:8) k ∈ K : log (1 + ξ h H k Γh k ) ≥ r (cid:9) . (2) Second phase.
The UEs who were able to decode the message in the first phase jointlyretransmit the message in an isotropic fashion, thus acting as opportunistic relays. Hence,the receive signal at UE k in the second phase is a non-coherent sum of the D2D transmitsignals and is given by y k, = (cid:88) j ∈U (cid:112) ξ j h jk x j + n k ∈ C , ∀ k ∈ K \ U (3)where ξ j is the transmit power at UE j and can be interpreted as the transmit SNR at UE j (cf. (1)); moreover, x j is the message transmitted by UE j , with E [ | x j | ] = 1 . The messageis successfully decoded by UE k if its achievable rate in the second phase is greater thanor equal to r , i.e., if log (cid:0) | (cid:80) j ∈U (cid:112) ξ j h jk | (cid:1) ≥ r . B. Single-Phase Multicasting
As a special case of the above, we describe a single-phase multicasting scheme, which werefer to as baseline scheme . This will serve as a means to assess the benefits brought by adding asecond phase of D2D communications to traditional multi-antenna multicasting. In this scheme,the BS simply transmits the common message aiming at reaching all the UEs. The receive signalat UE k is the same as (1) and the multicast capacity is given by (see [3]) C ( H ) (cid:44) max Γ (cid:23) : tr( Γ ) ≤ min k ∈K log (1 + ξ h H k Γh k ) (4) = log (cid:16) ξ max Γ (cid:23) : tr( Γ ) ≤ min k ∈K h H k Γh k (cid:17) (5) We assume that the UEs retransmit the message with fixed power and do not perform any power control in the second phase. where H = [ h , . . . , h K ] ∈ C M × K . Although a closed-form expression of the multicast capacityis not available, C ( H ) is convex in Γ and, therefore, it can be computed via semidefiniteprogramming. The main drawback of this single-phase scheme is that the multicast capacity islimited by the UE with the worst channel conditions. In particular, for the case of i.i.d. Rayleighfading channels and when the number of BS antennas M is fixed, the multicast capacity scalesas K − /M [3]. C. Channel Model
Following the millimeter wave (mmWave) one-ring channel model (see, e.g., [27] and refer-ences therein), let us express the direct channel to UE k as h k (cid:44) η k √ γ k a k ∈ C M × (6)where η k ∼ CN (0 , is the small-scale fading coefficient, γ k is the average channel powergain, and a k ∈ C M × is the array response vector at the BS for the steering angle θ k , with || a k || = M . Here, we have γ k = d − αk in case of line-of-sight (LoS) conditions and γ k = d − βk in case of non-line-of-sight (NLoS) conditions, where d k denotes the distance between the BSand UE k and α (resp. β ) is the LoS (resp. NLoS) pathloss exponent. For simplicity, we assumethat the BS is equipped with a uniform linear array (ULA), such that a k = [1 , e − j πδ cos( θ k ) , . . . , e − j πδ ( M −
1) cos( θ k ) ] T ∈ C M × (7)where δ = 0 . is the ratio between the antenna spacing and the signal wavelength. On the otherhand, the D2D channel between UEs k and j is represented as h jk (cid:44) η jk √ γ jk ∈ C (8)where η kj ∼ CN (0 , is the small-scale fading coefficient and γ jk is the average channel powergain. Here, we have γ jk = d − αkj in case of LoS conditions and γ jk = d − βkj in case of NLoSconditions, where d jk denotes the distance between UEs k and j (cf. (6)). D. CSIT Configurations
In this paper, we consider several configurations of CSIT that may be available at the BSunder different application scenarios. i) Perfect CSIT [Section III].
The knowledge of both the direct channels, i.e., { h k } k ∈K , andthe D2D channels, i.e., { h jk } k,j ∈K , is assumed. ii) Statistical CSIT [Section IV].
The knowledge of the UE locations is assumed. From this information, the BS can extract long-term statistics such as the average channel power gainsof both the direct channels, i.e., { γ k } k ∈K , and the D2D channels, i.e., { γ jk } k,j ∈K , togetherwith the steering angles { θ k } k ∈K . iii) Topological CSIT [Section V].
The knowledge of the map of the network area, i.e., thelocation and size of the obstacles (such as buildings) within its coverage area, and of thepdf of the UE locations is assumed.The above configurations correspond to settings with decreasing requirements on the informationavailable at the BS. While configuration i) is relevant for the case of moderate (or finite) numberof UEs and low mobility, configuration iii) is relevant for the case of large number of UE and highmobility: for instance, these features arise in vehicular networks, where the BS multi-antennabeam pattern ought to be designed on the basis of a city map and road traffic distribution. Lastly,configuration ii) can be considered as an intermediate case between i) and iii).
E. Performance Metrics
We propose two different performance metrics in terms of service reliability. In order to reflectthe inherent difficulty to guarantee a given data rate in a wireless setting with uncertainties onthe channel conditions across the UEs, we introduce the target outage (cid:15) ∈ [0 , , which describesthe trade-off between the multicast rate and the reliability level at which we can maintain sucha rate. Furthermore, let P k, ( r, Γ ) and P k, ( r, Γ ) denote the probabilities that UE k successfullydecodes in the first and in the second phase, respectively. a) Average multicast rate.
We define the average success probability as the probability thata randomly chosen UE successfully decodes over the two phases, which is given by P A ( r, Γ ) (cid:44) K (cid:88) k ∈K (cid:2) P k, ( r, Γ ) + (cid:0) − P k, ( r, Γ ) (cid:1) P k, ( r, Γ ) (cid:3) . (9)Hence, the average multicast rate is defined as the maximum transmission rate at whicha randomly chosen UE successfully decodes with probability at least − (cid:15) over the twophases, which can be expressed as R A ( r, Γ ) (cid:44) r with r solution to P A ( r, Γ ) ≥ − (cid:15). (10) b) Outage multicast rate.
Let us introduce the binary variables z k, ( r, Γ ) and z k, ( r, Γ ) , whichare equal to if UE k successfully decodes in the first and in the second phase, respectively,and to otherwise. Furthermore, let z ( r, Γ ) (cid:44) [ z , ( r, Γ ) . . . z K, ( r, Γ )] . We define the jointsuccess probability as the probability that all the UEs successfully decode over the two phases, which is given by P J ( r, Γ ) (cid:44) E (cid:20) (cid:89) k ∈K P (cid:20) log (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j (cid:54) = k ξ j h kj (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) ≥ r (cid:0) − z k, ( r, Γ ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) z ( r, Γ ) (cid:21)(cid:21) . (11)Hence, the outage multicast rate is defined as the maximum transmission rate at which allthe UEs successfully decode with probability at least − (cid:15) over the two phases, which canbe expressed as R O ( r, Γ ) (cid:44) r with r solution to P J ( r, Γ ) ≥ − (cid:15). (12) F. Problem Formulation
Our objective is to jointly optimize the multicast rate r and the transmit covariance Γ underone of the above outage constraints over the two phases. Such a problem can be formalized as max r> , Γ (cid:23) r s . t . tr( Γ ) ≤ , P T ( r, Γ ) ≥ − (cid:15) (13)where T ∈ { A , J } . Hence, when T = A , we recover the average multicast rate R A ( r, Γ ) definedin (10) and, when T = J , we recover the outage multicast rate R O ( r, Γ ) defined in (12). Notethat problem (13) is non-convex in both optimization variables due to the non-convex outageconstraint and is thus highly complex to solve. In the following, we detail our proposed methodsto tackle problem (13) in the three CSIT configurations described in Section II-D.III. D2D-A IDED M ULTI -A NTENNA M ULTICASTING WITH P ERFECT
CSITIn this section, we consider the case where all the direct channels, i.e., { h k } k ∈K , and all theD2D channels, i.e., { h jk } k,j ∈K , are perfectly known at the BS. For each UE k , let us define thebinary variables z k, ( r, Γ ) (cid:44) (cid:2) log (1 + ξ h H k Γh k ) ≥ r (cid:3) , (14) z k, ( r, Γ ) (cid:44) (cid:20) log (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ∈K\{ k } z j, ( r, Γ ) (cid:112) ξ j h jk (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) ≥ r (cid:21) (15)which are equal to if the UE successfully decodes in the first and in the second phase,respectively, and to otherwise. Hence, the probabilities that UE k successfully decodes in the The factor in the objective describes the equal time division between the two phases and is irrelevant for the optimization. first and in the second phase are given by P k, ( r, Γ ) = z k, ( r, Γ ) , (16) P k, ( r, Γ ) = z k, ( r, Γ ) (17)respectively: these stem from the fact that, with perfect CSIT, the decodability of each UE ineach phase is deterministic. In this context, the average success probability in (9) can be writtenas P A ( r, Γ ) = 1 K (cid:88) k ∈K (cid:0) z k, ( r, Γ ) + (cid:0) − z k, ( r, Γ ) (cid:1) z k, ( r, Γ ) (cid:1) . (18)On the other hand, the joint success probability in (11) becomes a product of binary variables,which is equal to if even a single UE does not decode the message over the two phases: hence,it is not suited to accommodate any target outage in the case of perfect CSIT. For this reason,in the rest of the section, we focus on maximizing the average multicast rate in (10). A. Multi-Antenna Multicasting (MAM) Algorithm
Considering the single-phase baseline scheme described in Section II-B, problem (13) with
T = A and perfect CSIT can be solved by selecting the best subset of K with size (1 − (cid:15) ) K tobe served by the BS and computing the transmit covariance that maximizes the multicast rateover such a subset of UEs. Note that, in this case, the outage constraint in (13) can be simplyexpressed as (cid:80) k ∈K z k, ( r, Γ ) ≥ (1 − (cid:15) ) K . While this problem formulation is also novel, it mainlyserves as a benchmark to demonstrate the gains obtained by the adding a second phase of UEcooperation enabled by D2D links in Section VI. However, the problem of deriving the optimalUE selection strategy is NP-hard since it requires to evaluate all possible subsets of K with size (1 − (cid:15) ) K . To reduce the complexity, we build on the intuition described in the following lemmato derive a suboptimal UE selection scheme. Lemma 1.
For a class of channels satisfying E [ h k h H k ] = γ k I M , ∀ k ∈ K , which includes (6) ,the optimal UE selection strategy with statistical channel knowledge is the one choosing the (1 − (cid:15) ) K UEs with the highest average channel power gains among { γ k } k ∈K . Without loss of generality, one can assume that (cid:15) is chosen such that (1 − (cid:15) ) K is an integer number. Proof: If { γ k > } k ∈K are known at the BS, we have max U⊂K : |U| =(1 − (cid:15) ) K E (cid:104) max Γ (cid:23) : tr( Γ ) ≤ min k ∈U h H k Γh k (cid:105) ≤ max U⊂K : |U| =(1 − (cid:15) ) K max Γ (cid:23) : tr( Γ ) ≤ min k ∈U E [ h H k Γh k ] (19) = max U⊂K : |U| =(1 − (cid:15) ) K max Γ (cid:23) : tr( Γ ) ≤ min k ∈U tr (cid:0) Γ E [ h k h H k ] (cid:1) (20) = max U⊂K : |U| =(1 − (cid:15) ) K min k ∈U γ k (21)where (19) follows from the concavity of min k ∈U h H k Γh k and (21) is due to the fact that theoptimal Γ satisfies tr( Γ ) = 1 . Finally, the solution presented in the lemma readily follows from(21).Lemma 1 states that, if the channels can be ordered statistically based on the average channelpower gains { γ k } k ∈K , the exhaustive search over all possible subsets of K with size (1 − (cid:15) ) K reduces to choosing the (1 − (cid:15) ) K UEs with the highest γ k . Motivated by this observation, wethus propose to apply such a UE selection strategy to the case of perfect CSIT and obtain the multi-antenna multicasting (MAM) algorithm . More specifically, we build U ⊂ K by selecting the (1 − (cid:15) ) K UEs with the highest channel power gain (cid:107) h k (cid:107) and compute the transmit covariancethat achieves the multicast capacity over U , i.e., Γ = argmax Γ (cid:23) : tr( Γ ) ≤ min k ∈U h H k Γh k . (22)Since the whole time resource is dedicated to the first phase, the resulting average multicast rateis given by r = log (cid:16) ξ min k ∈U h H k Γ h k (cid:17) . (23) B. D2D-Aided Multi-Antenna Multicasting (D2D-MAM) Algorithm
To solve problem (13) with
T = A and perfect CSIT, we resort to the alternating optimizationof the multicast rate r and the transmit covariance Γ . In this respect, we propose an efficientiterative algorithm whose goal is to serve a subset of UEs (which are suitably selected bymeans of precoding at the BS) in the first phase such that the multicast rate is maximized.At each iteration n , the transmit covariance Γ ( n ) that achieves the multicast capacity over apredetermined subset U ( n − ⊂ K is computed (see (4)–(5)). Then, the multicast rate r ( n ) isobtained as the maximum rate that guarantees the outage constraint over the two phases giventhe transmit covariance computed in the previous step, i.e., such that P A ( r ( n ) , Γ ( n ) ) ≥ − (cid:15) . The Algorithm 1 (D2D-MAM)
Data:
Direct channels { h k } k ∈K and D2D channels { h jk } k,j ∈K . Fix U (0) = K and n = 1 . (S.1) Optimize the transmit covariance as Γ ( n ) = argmax Γ (cid:23) : tr( Γ ) ≤ min k ∈U ( n − h H k Γh k . (S.2) Maximize the multicast rate as r ( n ) = max (cid:8) r : P A ( r, Γ ( n ) ) = 1 − (cid:15) (cid:9) . (S.3) Update the subset of UEs successfully decoding in the first phase as U ( n ) = (cid:8) k : log (1 + ξ h H k Γ ( n ) h k ) ≥ r ( n ) (cid:9) . (S.4) If r ( n ) = r ( n − : fix Γ = Γ ( n ) and r = r ( n ) ; Stop . Else : n ← n + 1 ; Go to (S.1) .new r ( n ) yields an updated U ( n ) of UEs that are able to decode in the first phase and, therefore,an improved transmit covariance can be obtained by optimizing over U ( n ) . This procedure isiterated until the multicast rate converges. The proposed algorithm is referred to as D2D-aidedmulti-antenna multicasting (D2D-MAM) algorithm and is formally described in Algorithm 1. TheD2D-MAM algorithm has the key advantage of not requiring any tuning parameter selection.Furthermore, it converges to a local optimum of problem (13) with
T = A , as formalized in thefollowing theorem.
Theorem 1.
The D2D-MAM algorithm converges to a local optimum of problem (13) with
T = A .Proof:
Since step (S.1) of Algorithm 1 optimizes Γ ( n ) over U ( n − , we have min k ∈U ( n − h H k Γ ( n ) h k ≥ min k ∈U ( n − h H k Γ ( n − h k (24)i.e., the minimum rate achievable by the UEs in U ( n − increases with the new transmit covariance Γ ( n ) . Furthermore, at each iteration n of the D2D-MAM algorithm, the following holds: r ( n ) ≥ log (cid:16) ρ min k ∈U ( n − h H k Γ ( n ) h k (cid:17) (25) ≥ log (cid:16) ρ min k ∈U ( n − h H k Γ ( n − h k (cid:17) (26) ≥ r ( n − (27)where (25) follows from step (S.2) of Algorithm 1 (by which it is possible to increase the multicast rate as long as the outage constraint is guaranteed), (26) is a direct consequence of(24), and (27) stems from the fact that U ( n − contains the UEs whose achievable rate in thefirst phase is at least r ( n − . Hence, the multicast rate cannot decrease between consecutiveiterations. Finally, if U ( n ) = U ( n − , then it is not possible to further increase the multicast rate,i.e., r ( n ) = r ( n − , which implies that convergence is reached.Regarding the optimization of the multicast rate in step (S.2) of Algorithm 1, we have r ( n ) ∈ (cid:104) r ( n − , log (cid:16) ρ max k ∈U ( n − h H k Γ ( n ) h k (cid:17)(cid:105) (28)where the lower bound follows from Theorem 1 and the upper bound is necessary to guaranteethat at least one UE is served in the first phase: thus, r ( n ) can be efficiently computed by meansof bisection over the above interval. Accordingly, every iteration of the D2D-MAM algorithmrequires the solution of a convex problem in step (S.1) and a linear search in step (S.2); inaddition, for the settings considered for our simulations in Section VI, convergence is reachedafter a small number of iterations. Hence, the D2D-MAM algorithm provides a locally optimalsolution of problem (13) with T = A with very low complexity.IV. D2D-A
IDED M ULTI -A NTENNA M ULTICASTING WITH S TATISTICAL
CSITIn this section, we consider the case where only the UE locations are known at the BS. Fromthis information, the BS can extract long-term statistics such as the average channel power gainsof both the direct channels, i.e., { γ k } k ∈K , and the D2D channels, i.e., { γ jk } k,j ∈K , together with thesteering angles { θ k } k ∈K . On the other hand, the BS has no knowledge of the small-scale fadingcoefficients, i.e., { η k } k ∈K and { η jk } k,j ∈K . Under statistical CSIT, we characterize the servicereliability in terms of the joint success probability in (11) and, accordingly, we maximize theoutage multicast rate in (12). To alleviate the task of dealing with the involved expression of thejoint success probability, we derive its deterministic equivalent in the following proposition. Proposition 1.
Assuming that all (direct and D2D) channels are independent, we have P J ( r, Γ ) P → K →∞ ¯P J ( r, Γ ) (29) where ¯P J ( r, Γ ) (cid:44) exp (cid:18) − (cid:88) k ∈K (2 r − (cid:0) − P k, ( r, Γ ) (cid:1)(cid:80) j ∈K\{ k } P j, ( r, Γ ) γ jk ξ j (cid:19) (30) is the deterministic equivalent of P J ( r, Γ ) in (11) .Proof: The proof follows similar steps as the proof of [18, Thm. 4] and is thus omitted. Note that, with statistical CSIT, the probability that UE k successfully decodes in the first phaseis given by P k, ( r, Γ ) = P (cid:2) z k, ( r, Γ ) = 1 (cid:3) (31) = P (cid:2) log (1 + ξ γ k | η k | a H k Γa k ) ≥ r (cid:3) (32) = exp (cid:18) − r − ξ γ k a H k Γa k (cid:19) . (33)with z k, ( r, Γ ) defined in (14) and where (33) follows from the exponential distribution of | η k | .In the rest of the section, we replace P J ( r, Γ ) with its deterministic equivalent ¯P J ( r, Γ ) in (30). A. Statistical Multi-Antenna Multicasting (SMAM) Algorithm
Considering the single-phase baseline scheme described in Section II-B, problem (13) with
T = J and statistical CSIT can be solved by computing the transmit covariance that maximizesthe outage multicast rate. Note that, in this case, the outage constraint in (13) can be simplyexpressed as (cid:81) k ∈K P k, ( r, Γ ) ≥ − (cid:15) . Since this problem is convex in Γ for a fixed r and viceversa, we decouple the optimization over the two variables in the following way. For a giventransmit covariance Γ , the outage multicast rate, denoted in this context by R O , ( r , Γ ) , ismaximized when the outage constraint is satisfied with equality, leading to R O , ( r , Γ ) = log (cid:18) ξ log (cid:18) − (cid:15) (cid:19)(cid:18) (cid:88) k ∈K γ k a H k Γ a k (cid:19) − (cid:19) . (34)Then, the optimal transmit covariance is obtained by solving min Γ (cid:23) (cid:88) k ∈K γ k a H k Γ a k s . t . tr( Γ ) ≤ (35)by means of semidefinite programming. As in Section III-A, this problem formulation mainlyserves for the comparative purposes in Section VI. The resulting algorithm is referred to as statistical multi-antenna multicasting (SMAM) algorithm .The following proposition derives a tractable expression of Γ and will be useful in the nextsection. Proposition 2.
Assume that K consists of M UEs exhibiting mutually orthogonal array re-sponses, i.e., (cid:88) k ∈K a k a H k = M I M . (36) Then, the optimal transmit covariance for problem (35) can be written in closed form as Γ = 1 M ν K (cid:88) k ∈K √ γ k a k a H k (37) with ν K (cid:44) (cid:80) k ∈K √ γ k .Proof: See Appendix A.A set of array response vectors satisfying (36) can be obtained as the columns of the M -dimensional discrete Fourier transform (DFT) matrix or, alternatively, it can be constructedalong specific virtual angles as described in [28]. B. D2D-Aided Statistical Multi-Antenna Multicasting (D2D-SMAM) Algorithm
To solve problem (13) with
T = J and statistical CSIT, we use the deterministic equivalentderived in Proposition 1 and, to further reduce the complexity, we decouple the optimizationacross the two phases in the following way. First, we carefully select a subset
U ⊂ K of UEs withfavorable statistical properties to be served in the first phase by the BS. In particular, assuminglarge K and uniform UE distribution in the angular domain, we build on Proposition 2 andconstruct U by selecting M UEs satisfying the condition in (36): by doing so, the BS spreadsits transmit power along a set of orthogonal directions spanning the whole angular domain. Inthis setting, the transmit covariance that maximizes the multicast rate over U is given by Γ in(37). Next, we fix the joint success probability in the first phase over U to a given value − (cid:15) and obtain the corresponding multicast rate r ( (cid:15) ) from (34). Finally, we optimize (cid:15) in order toobtain the desired joint success probability − (cid:15) over the two phases.Let us first focus on maximizing the outage multicast rate over U in the first phase, i.e., max r ( (cid:15) ) > , Γ (cid:23) r ( (cid:15) )s . t . tr( Γ ) ≤ , exp (cid:18) − (cid:88) k ∈U r ( (cid:15) ) − ξ γ k a H k Γa k (cid:19) ≥ − (cid:15) . (38)Since the outage constraint is convex in Γ , we can solve problem (38) by decoupling theoptimization of r ( (cid:15) ) and Γ . Letting the outage constraint be satisfied with equality, we havethat the multicast rate becomes r ( (cid:15) ) = log (cid:18) ξ log (cid:18) − (cid:15) (cid:19)(cid:18) (cid:88) k ∈U γ k a H k Γa k (cid:19) − (cid:19) (39) Since K is large, we assume that it is always possible to select M UEs whose steering angles satisfy (41). Algorithm 2 (D2D-SMAM)
Data:
Build U by selecting M UEs such that (41) holds. (S.1)
Compute the transmit covariance as in (42) with weights given in (43). (S.2)
Find (cid:15) by solving (45). (S.3) Compute the multicast rate as in (44).and problem (38) reduces to finding the transmit covariance Γ by solving min Γ (cid:23) (cid:88) k ∈U γ k a H k Γa k s . t . tr( Γ ) ≤ . (40)From Proposition 2, the transmit covariance resulting from the above problem is known to havea simple closed-form expression when |U | = M and the UEs in U exhibit orthogonal arrayresponse vectors, i.e., (cid:88) k ∈U a k a H k = M I M . (41)Since K is large, we assume that it is always possible to build U by selecting M UEs satisfyingthe condition in (41). In this case, the optimal transmit covariance is given by Γ = (cid:88) j ∈U w j a j a H j (42)with weights given by w j (cid:44) M ¯ γ U √ γ j ∀ j ∈ U (43)and where we have defined ¯ γ U (cid:44) (cid:80) k ∈U √ γ k . Finally, plugging (42) and (43) into (39), we obtain r ( (cid:15) ) = log (cid:18) ξ log (cid:18) − (cid:15) (cid:19) M ¯ γ U (cid:19) . (44)Let us now focus on deriving (cid:15) that achieves the desired joint success probability − (cid:15) overthe two phases. This can be done by solving the following expression for (cid:15) ∈ [0 , (e.g., bymeans of bisection): (2 r ( (cid:15) ) − (cid:88) k ∈K − exp (cid:0) − r ( (cid:15) − ξ γ k a H k Γa k (cid:1)(cid:80) j ∈K\{ k } exp (cid:0) − r ( (cid:15) − ξ γ k a H k Γa k (cid:1) γ jk ξ j ≤ log (cid:18) − (cid:15) (cid:19) . (45)The proposed algorithm is referred to as D2D-aided statistical multi-antenna multicasting (D2D-SMAM) algorithm and is formally described in Algorithm 2. In the next section, we illustrate apossible way to derive an approximation of the optimal (cid:15) . C. Asymptotic Behavior of the D2D-SMAM Algorithm
Let us assume that (cid:15) → and, consequently, that (cid:15) → . By applying the Taylor approxima-tion exp (cid:16) − r ( (cid:15) − ξ γ k a H k Γa k (cid:17) ≈ − r ( (cid:15) − ξ γ k a H k Γa k to (45), we have (cid:15) ∼ (cid:15) → − exp (cid:32) − ¯ γ U M ξ (cid:118)(cid:117)(cid:117)(cid:116) log (cid:0) − (cid:15) (cid:1)(cid:80) k ∈K ξ γ k a H k Γa k (cid:0) (cid:80) j ∈K\{ k } γ jk ξ j (cid:1) − (cid:33) (46)and, hence r ∼ (cid:15) →
12 log (cid:32) (cid:118)(cid:117)(cid:117)(cid:116) ξ log (cid:0) − (cid:15) (cid:1)(cid:80) k ∈K γ k a H k Γa k (cid:0) (cid:80) j ∈K\{ k } γ jk ξ j (cid:1) − (cid:33) (47) (cid:44) ˜ r. (48)Now, assume that d k ∈ [ R min , R max ] , ∀ k ∈ K , where R min and R max denote the minimumand maximum distance, respectively, between each UE and the BS. It follows that the averagechannel power gains can be bounded as γ k ∈ [ R − β max , R − α min ] , ∀ k ∈ K , (49) γ jk ∈ (cid:2) (2 R max ) − β , (2 R min ) − α ] , ∀ k, j ∈ K . (50)In this setting, we have a H k Γa k = 1 M ¯ γ U (cid:88) j ∈U √ γ j | a H k a j | (51) ≥ M ¯ γ U R α/ (52)where (52) follows from assuming that all the UEs in U are at distance R min from the BS, i.e., { γ j = R − α min } j ∈U . Hence, we have that ˜ r defined in (47)–(48) can be lower bounded as ˜ r ≥ log (cid:32) (cid:118)(cid:117)(cid:117)(cid:116) ξ ξ UE ( K − M log (cid:0) − (cid:15) (cid:1) R α/ ¯ γ U (2 R max ) β R α/ + 2 β ¯ γ U R β max ( K − M ) (cid:33) (53)where, for simplicity, we have assumed that { ξ k = ξ UE } k ∈K (i.e., all the UEs have the sametransmit SNR in the second phase). Finally, we consider the asymptotic behavior of ˜ r in thecase where both K and M increase with fixed ratio c (cid:44) KM > as well as in the case where K increases for a fixed M . Hence, (53) behaves as ˜ r → K →∞ log (cid:18) (cid:114) ξ ξ UE M log (cid:0) − (cid:15) (cid:1) R α/ β ¯ γ U R β max (cid:19) for fixed M log (cid:18) (cid:114) ξ ξ UE log (cid:0) − (cid:15) (cid:1) R α/ β ¯ γ U R β max ( c − K (cid:19) for M = Kc , with c > (54)which is non-vanishing in the first case as in [18] and increasing as log (cid:0) √ K (cid:1) in the secondcase. V. D2D-A IDED M ULTI -A NTENNA M ULTICASTING WITH T OPOLOGICAL
CSITIn this section, we consider the case where only the map of the network area, i.e., the locationand size of the obstacles (such as buildings) within its coverage area, and the pdf of the UElocations are known at the BS. Such pdf can be obtained on the basis of the city map and long-term information on the traffic distribution. This setting describes a scenario with a high densityof UEs (e.g, cars or terminals) where it may not be feasible to design a precoding solution thatadapts instantaneously to the channels or, in the longer term, to the channel statistics. In thiscase, it is meaningful to derive the precoding strategy at the BS based solely on the networktopology and on the UE distribution.First, we slightly adapt the channel model described in Section II-C to express all the parame-ters as functions of the possible UE locations within the map. Let
A ⊂ R denote the continuousset of points representing the network area and let p = ( θ, ρ ) be a random variable denoting apossible position within A in which a UE can be located, where θ and ρ represent the steeringangle and the distance from the BS, respectively. In this setting, we use f ( p ) to denote the pdfof the UE locations, which describes the probability of finding a UE in the position identifiedby p . Focusing on the first phase, let us write the direct channel to position p as (cf. (6)) h ( p ) = η (cid:112) γ ( p ) a ( θ ) (55)where η ∼ CN (0 , is the small-scale fading coefficient, γ ( p ) is the average channel powergain at position p , and a ( θ ) is the array response vector at the BS for the steering angle θ . Here,we have γ ( p ) = ρ − α in case of LoS conditions and γ ( p ) = ρ − β in case of NLoS conditions.The receive SNR at position p in the first phase can be expressed as SNR ( p , Γ ) (cid:44) | η | γ ( p ) ξ a H ( θ ) Γa ( θ ) . (56)Note that, if position p falls within the area occupied by an obstacle (e.g., a building), thecorresponding receive SNR is zero. Hence, the probability that a UE located at position p successfully decodes in the first phase is given by P ( p , r, Γ ) (cid:44) P (cid:2) log (cid:0) ( p , Γ ) (cid:1) ≥ r (cid:3) (57) = exp (cid:18) − (2 r − γ ( p ) ξ a H ( θ ) Γa ( θ ) (cid:19) . (58)Focusing on the second phase, let us write the D2D channel between positions p and p (cid:48) as(cf. (8)) h ( p , p (cid:48) ) = η (cid:112) γ ( p , p (cid:48) ) (59)where γ ( p , p (cid:48) ) is the average channel power gain. Here, we have γ ( p , p (cid:48) ) = d ( p , p (cid:48) ) − α in caseof LoS conditions and γ ( p , p (cid:48) ) = d ( p , p (cid:48) ) − β in case of NLoS conditions, where d ( p , p (cid:48) ) denotesthe distance between positions p and p (cid:48) . For simplicity, let us assume that all the UEs in anyposition within A have the same transmit SNR ξ UE in the second phase. Furthermore, let U ⊂ A be the subset of positions where a potential UE could successfully decode in the first phase.Hence, the probability that a UE located at position p successfully decodes the message in thesecond phase is given by P ( p , r, Γ ) (cid:44) P (cid:34) log (cid:32) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) U (cid:112) ξ UE f ( p (cid:48) ) h ( p , p (cid:48) )d p (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) ≥ r (cid:21) (60) = E (cid:20) exp (cid:18) − r − ξ UE (cid:82) U f ( p (cid:48) ) γ ( p , p (cid:48) )d p (cid:48) (cid:19)(cid:21) (61)where the expectation is over all the possible combinations of U .In the context of topological CSIT, the average success probability in (9) can be written as P A ( r, Γ ) = (cid:90) A f ( p ) (cid:0) P ( p , r, Γ ) + (1 − P ( p , r, Γ ))P ( p , r, Γ ) (cid:1) d p . (62)On the other hand, the joint success probability in (11) turns out to be impractical when A isconnected, i.e., when the network area contains infinite points. For this reason, in the rest of thesection, we focus on maximizing the average multicast rate in (10). Since (62) is quite difficultto handle even for simple UE distribution models (e.g., uniform), in the next section, we detaila heuristic approach to maximize the average multicast rate based on the Monte Carlo samplingof f ( p ) . A. D2D-Aided Topological Multi-Antenna Multicasting (D2D-TMAM) Algorithm
To solve problem (13) with
T = A and topological CSIT, we resort to the Monte Carlosampling of the pdf of the UE locations to generate a set of test points within the map and the Algorithm 3 (D2D-TMAM)
Data:
Map of the network area and pdf of the UE locations f ( p ) . Fix l = 1 . For l = 1 , . . . , L : (S.1) Generate T test points together with the corresponding direct and D2D channelsaccording to (55) and (59), respectively. (S.2) Execute Algorithm 1 with the channels generated in step (S.1) as input data andobtain the multicast rate r ( (cid:96) ) and the transmit covariance Γ ( (cid:96) ) as output data. End(S.3)
Fix r = L (cid:80) L(cid:96) =1 r ( (cid:96) ) and Γ = L (cid:80) L(cid:96) =1 Γ ( (cid:96) ) .corresponding artificial channels, which are subsequently used to run the D2D-MAM algorithmdescribed in Algorithm 1 (see Section III-B). More specifically, we produce L batches of T testpoints each, where T is a random variable that describes the number of UEs and whose distribu-tion depends on f ( p ) . For each batch (cid:96) , we artificially generate the direct channels for each testpoint according to (55) as well as the D2D channels for each pair of test points according to (59).Then, such channels are used as input data to Algorithm 1, which produces the multicast rate r ( (cid:96) ) and the transmit covariance Γ ( (cid:96) ) as output data. Finally, the final multicast rate and transmitcovariance are obtained by averaging the output data of the L batches, i.e., r = L (cid:80) L(cid:96) =1 r ( (cid:96) ) and Γ = L (cid:80) L(cid:96) =1 Γ ( (cid:96) ) , which provides an approximate solutions to problem (13) with T = A . Theproposed algorithm is referred to as
D2D-aided topological multi-antenna multicasting (D2D-TMAM) algorithm and is formally described in Algorithm 3. Evidently, evaluating more batchesof test points allows to achieve a more precise representation of the long-term network statistics,which produces a more accurate result in terms of average success probability. Since the D2D-TMAM algorithm involves L instances of Algorithm 1, its computational complexity may bequite high. However, it is worth observing that this procedure is based on slowly varying networkstatistics and needs to be updated only when the UE distribution changes significantly. Therefore,it can be conveniently executed offline using a large value of L .VI. N UMERICAL R ESULTS
In this section, we present numerical results to validate the proposed algorithms in the threedifferent CSIT configurations, i.e., perfect CSIT (described in Section III), imperfect CSIT(described in Section IV), and topological CSIT (described in Section V). Unless otherwisestated, the considered network topology consists of a semicircular area with radius R max = 100 m Fig. 2: Evaluation scenario: the white area and the dotted areas are in LoS and NLoS conditions, respectively,whereas the UEs are not admitted in the regions occupied by the buildings. where four rectangular buildings are positioned in a Manhattan-like grid, as shown in Fig. 2. Weassume that the UEs are distributed uniformly within the network area with the exception of theregions occupied by the buildings and with a minimum distance from the BS of R min = 5 m. Thedirect and D2D links whose line of sight is obstructed by one or more buildings are consideredto be in NLoS conditions both in the first and in the second phase. The LoS and NLoS pathlossexponents are fixed to α = 2 and β = 4 , respectively. For simplicity, we assume that all the UEshave the same transmit SNR in the second phase, i.e., { ξ k = ξ UE } k ∈K , and we set ξ = 30 dB and ξ UE = 20 dB. Moreover, unless otherwise stated, the BS is equipped with M = 32 antennas andthe target outage is fixed to (cid:15) = 0 . . Lastly, all the numerical results are averaged over × independent UE drops. A. Perfect CSIT
In the case of perfect CSIT, we evaluate the performance of the proposed D2D-MAM algorithmin Algorithm 1 versus the single-phase MAM algorithm described in Section III-A. Interestingly,the D2D-MAM algorithm converges in very few iterations (typically between and ) evenfor large values of K . Fig. 3(a) plots the average multicast rate against the number of UEs fordifferent values of (cid:15) . Indeed, the second phase of D2D communications brings substantial gainswith respect to traditional multi-antenna multicasting. In particular, the average multicast rateobtained with the D2D-MAM algorithm increases with K , whereas that resulting from the MAMalgorithm quickly vanishes. Hence, the D2D-MAM algorithm effectively overcomes the worst-UE bottleneck behavior of conventional single-phase multicasting and remarkably achieves anincreasing trend of the multicast rate. In the same setting of Fig. 3(a), Fig. 3(b) shows that theaverage number of UEs who are able to decode in the first phase varies between and of the total UEs depending on the target outage. Lastly, Fig. 3(c) illustrates the average multicast
10 20 30 40 50 60 70 80 90 10001234567891011 10 -2 D2D-MAMMAM (a) Average multicast rate against the number of UEs with M = 32 and for different values of (cid:15) .
10 20 30 40 50 60 70 80 90 10005101520253035404550 (b) Average number of UEs who are able to decode in thefirst phase against the number of UEs with M = 32 and fordifferent values of (cid:15) . -2 D2D-MAMMAM (c) Average multicast rate against the number of BS antennaswith (cid:15) = 0 . and for different values of K . Fig. 3: Perfect CSIT: D2D-MAM algorithm versus MAM algorithm. rate against the number of BS antennas for different values of K . Evidently, the BS can betterfocus its transmit power as M increases, which results in an overall improved performance.Here, the lowest value corresponds to M = 1 , i.e., when the BS has no beamforming capabilityand can only transmit in an isotropic fashion in the first phase. B. Statistical CSIT
In the case of statistical CSIT, we evaluate the performance of the proposed D2D-SMAMalgorithm in Algorithm 2 versus the single-phase SMAM described in Section IV-A. In addition,we compare the asymptotic expressions obtained in Section IV-C with numerical simulations.For the D2D-SMAM algorithm, we build the set U by identifying M UEs whose steering anglessatisfy the condition in (36), while their distance from the BS is uniformly distributed. We
10 30 50 70 90 110 130 15000.20.40.60.811.2 10 -3 D2D-SMAMAsymptotic behaviorSMAM (a) Outage multicast rate against the number of UEs fordifferent values of c = KM .
10 30 50 70 90 110 130 15000.20.40.60.811.2 10 -3 D2D-SMAMAsymptotic behaviorSMAM (b) Outage multicast rate against number of UEs for differentvalues M .
10 30 50 70 90 110 130 15055.566.577.588.599.5 10 -1 SimulationDeterministic equivalent (c) Joint success probability in (11) and its deterministicequivalent in (30) for different values of c = KM . Fig. 4: Statistical CSIT: D2D-SMAM algorithm versus SMAM algorithm. consider two cases of interest, i.e., where both the number of UEs K and the number of BSantennas M increase with a fixed ratio c = KM > and where K increases for a fixed M . Thefirst case is depicted in Fig. 4(a), which shows that the outage multicast rate always grows aslong as M grows together with K . The second case is illustrated in Fig. 4(b), which shows howincreasing M is always beneficial for any given number of UEs K . Here, the outage multicastrate obtained with the D2D-SMAM algorithm grows with K and reaches a constant value forlarge K : this is confirmed by its asymptotic behavior, which is constant with K . On the contrary,the SMAM algorithm produces a vanishing outage multicast rate and even increasing M doesnot fundamentally solve this issue. Lastly, Fig. 4(c) compares the joint success probability in(11) with its deterministic equivalent in (30) for different values of c . Here, the approximationis tight for sufficiently large values of K . (a) Evaluation scenario. (b) Antenna diagram of the transmit covariance with M = 32 . Fig. 5: Toy example with topological CSIT: the UEs are admitted only in the two white sectors.
C. Topological CSIT
In the case of topological CSIT, we evaluate the performance of the proposed D2D-TMAMalgorithm in Algorithm 3 versus the D2D-MAM algorithm in Algorithm 1, where the latteris based on the assumption of perfect CSIT. Although unfair to the D2D-TMAM algorithm,this comparison demonstrates how the proposed approach with topological CSIT can accuratelysample the long-term network statistics. In turn, this enables to effectively design the precodingstrategy at the BS with minimal CSIT requirements and no training overhead without excessivelycompromising the performance. Let A denote the area of the network excluding the regionsoccupied by the buildings (expressed in m ) and let us consider a uniform UE distribution withdensity λ (expressed in UEs/m ). In this setting, we assume that each UE drop consists of K UEs,where K is a Poisson random variable with mean ¯ K = λA . Recall that, for the D2D-TMAMalgorithm, the transmit covariance and the multicast rate are computed offline by averaging theoutput of the D2D-MAM algorithm over L batches of T test points, where we fix L = 10 ; onthe other hand, the D2D-MAM algorithm is executed for each UE drop. • Toy example.
As a first experiment to verify the effectiveness of the proposed method,we consider the simplified network topology depicted in Fig. 5(a), with R max = 20 m andwhere only two sectors admit the presence of UEs. In this setting, we have A = 100 m and, fixing λ = 0 . UEs/m , the average number of UEs in the network is ¯ K = 50 ;moreover, we assume that all the links are in LoS conditions. Fig. 5(b) shows the antennadiagram of the transmit covariance obtained with the D2D-TMAM algorithm with T = ¯ K test points for each batch: as expected, the multi-antenna beam pattern uniformly coversthe two sectors in which the UEs are concentrated. Now, we evaluate the average multicastrate and the average success probability as T varies in order to verify which value gives
20 30 40 50 60 70 802.533.544.555.5 (a) Average multicast rate against number of test points with M = 32 .
20 30 40 50 60 70 8077.588.599.510 10 -1 (b) Average success probability against number of test pointswith M = 32 . Fig. 6: Topological CSIT applied to the toy example in Fig. 5(a): D2D-TMAM algorithm versus D2D-MAMalgorithm, where the latter relies on perfect CSIT. the best performance. Fig. 6 shows that, when T is too small, the algorithm is overcautiousand selects a low multicast rate corresponding to an average success probability above thetarget; on the other hand, when T is too large, the algorithm is overaggressive and selectsa high multicast rate corresponding to an average success probability below the target. Asexpected, the target outage is reached for T = ¯ K and the corresponding mean value of theaverage multicast rate is very close to that obtained with the D2D-MAM algorithm (whichrelies on perfect CSIT).Now, let us go back to the original evaluation scenario depicted in Fig. 2 and compare theproposed D2D-TMAM algorithm with the D2D-MAM algorithm. Fig. 7(a) illustrates the averagemulticast rate against the UE density for different values of (cid:15) . First of all, we observe that bothschemes benefit from increasing the number of UEs, thus effectively overcoming the worst-UEbottleneck behavior of conventional single-phase multicasting. Furthermore, the performance gapbetween the D2D-TMAM algorithm and the D2D-MAM algorithm is remarkably small despitethe huge difference in the CSIT requirements of the two schemes. Lastly, Fig. 7(b) plots theaverage success probability against the UE density for different values of (cid:15) , showing that thetarget success probability is achieved more accurately by the D2D-TMAM algorithm as the UEdensity increases. VII. C ONCLUSION
This paper proposes a general two-phase cooperative multicasting framework that leveragesboth multi-antenna transmission at the BS and D2D communications between the UEs. We -3 -1 (cid:15) = 0 . (cid:15) = 0 . (cid:15) = 0 (a) Average multicast rate against UE density with M = 32 and for different values of (cid:15) . -3 -1 (cid:15) = 0 (cid:15) = 0 . (cid:15) = 0 . (b) Average success probability against UE density with M =32 and for different values of (cid:15) . Fig. 7: Topological CSIT applied to the evaluation scenario in Fig. 2: D2D-TMAM algorithm versus D2D-MAMalgorithm, where the latter relies on perfect CSIT. explicitly optimize the precoding strategy at the BS and the multicast rate over the two phasessubject to some outage constraint. In particular, we devise efficient algorithms to tackle threedifferent CSIT configurations, i.e., perfect CSIT, statistical CSIT, and topological CSIT. Numer-ical results show that the proposed schemes significantly outperform conventional single-phasemulti-antenna multicasting in all the considered CSIT configurations. Remarkably, they allowto effectively overcome the vanishing behavior of the multicast rate and achieve an increasingperformance as the UE population grows large.A
PPENDIX AP ROOF OF P ROPOSITION Γ is optimal if and only if it satisfies the Karush–Kuhn–Tucker (KKT) conditions. Let us define the Lagrangian and its gradient as L ( Γ , µ, Ψ ) (cid:44) (cid:88) k ∈K γ k a H k Γ a k + µ (cid:0) tr( Γ ) − (cid:1) − tr( ΨΓ ) , (63) ∇L ( Γ , µ, Ψ ) (cid:44) − (cid:88) k ∈K γ k ( a H k Γ a k ) a k a H k + µ I M − Ψ (64) respectively, where we have introduced the dual variables µ ∈ R and Ψ ∈ C M × M . The KKTconditions of problem (35) can be written as (cid:88) k ∈K γ k ( a H k Γ a k ) a k a H k = µ I M − Ψ , (65a) tr( Γ ) ≤ , Γ (cid:23) , (65b) µ ≥ , Ψ (cid:23) , (65c) µ (cid:0) tr( Γ ) − (cid:1) = 0 , ΨΓ = . (65d)The condition in (65a) suggests that the transmit covariance has the structure Γ = (cid:88) k ∈K w k a k a H k (66)where (cid:80) k ∈K w k = 1 /M implies tr( Γ ) = 1 and { w k ≥ } k ∈K implies Γ (cid:23) . From (66), wecan write a H k Γ a k = (cid:88) j ∈K w j φ kj (67)where we have defined φ kj (cid:44) | a H k a j | , with Φ (cid:44) [ φ kj ] k,j ∈K ∈ C K × K being a symmetric matrixwith diagonal elements equal to M . Plugging (66) into (65), the KKT conditions become (cid:88) k ∈K γ k (cid:0) (cid:80) j ∈K w j φ kj (cid:1) a k a H k = µ I M − Ψ , (68a) (cid:88) k ∈K w k = 1 M , { w k ≥ } k ∈K , (68b) µ ≥ , Ψ (cid:23) , (68c) µ (cid:18) (cid:88) k ∈K w k − M (cid:19) = 0 , Ψ (cid:88) k w k a k a H k = . (68d)Let us define w (cid:44) [ w , . . . , w K ] T ∈ R K × . Choosing the weights that satisfy (68b) allows us toset Ψ = and, from (68a), we can show that w = 1 √ µM Φ − b (69)where we have defined b (cid:44) (cid:20) (cid:112) γ T Φ − e , . . . , (cid:112) γ K T Φ − e K (cid:21) T . (70)On the other hand, µ can be obtained by plugging (69) into the first condition in (68d), i.e., µ = M ( T Φ − b ) (71) and, by plugging (71) into (69), we obtain w k = e T k Φ − b M T Φ − b , ∀ k ∈ K . (72)Finally, choosing { w k } k ∈K as in (72), µ as in (71), and Ψ = readily satisfies (68b)–(68d),whereas (68a) yields (cid:88) k ∈K ( T Φ − e k ) a k a H k = 1 M I M . (73)The latter is satisfied when Φ = M I K , i.e., when K = M and the steering angles of the UEsare such that a H k a j = 0 , ∀ k (cid:54) = j (see, e.g., [28] for more details). In this setting, it follows from(72) that w k = 1 / ( M √ γ k ν K ) , from which we obtain the expression of the optimal transmitcovariance in (37). R EFERENCES [1] P. Mursia, I. Atzeni, D. Gesbert, and M. Kobayashi, “D2D-aided multi-antenna multicasting,” in
Proc. IEEE Int. Conf.Commun. (ICC) , Shanghai, China, May 2019.[2] P. Mursia, I. Atzeni, M. Kobayashi, and D. Gesbert, “D2D-aided multi-antenna multicasting in a dense network,” in
Proc.Asilomar Conf. Signals, Syst., and Comput. (ASILOMAR) , Pacific Grove, USA, Nov. 2019.[3] N. Jindal and Z.-Q. Luo, “Capacity limits of multiple antenna multicast,” in
Proc. IEEE Int. Symp. Inf. Theory (ISIT) ,Barcelona, Spain, July 2006.[4] N. D. Sidiropoulos, T. N. Davidson, and Z.-Q. Luo, “Transmit beamforming for physical-layer multicasting,”
IEEE Trans.Signal Process. , vol. 54, no. 6, pp. 2239–2251, June 2006.[5] A. Khisti, U. Erez, and G. Wornell, “Fundamental limits and scaling behavior of cooperative multicasting in wirelessnetworks,”
IEEE Trans. Inf. Theory , vol. 52, no. 6, pp. 2762–2770, June 2006.[6] B. Sirkeci-Mergen and M. C. Gastpar, “On the broadcast capacity of wireless networks with cooperative relays,”
IEEETrans. Inf. Theory , vol. 56, no. 8, pp. 3847–3861, July 2010.[7] F. Hou, L. X. Cai, P. Ho, X. Shen, and J. Zhang, “A cooperative multicast scheduling scheme for multimedia services inIEEE 802.16 networks,”
IEEE Trans. Wireless Commun. , vol. 8, no. 3, pp. 1508–1519, Mar. 2009.[8] Y. Zhou, H. Liu, Z. Pan, L. Tian, J. Shi, and G. Yang, “Two-stage cooperative multicast transmission with optimized powerconsumption and guaranteed coverage,”
IEEE J. Sel. Areas Commun. , vol. 32, no. 2, pp. 274–284, Feb. 2014.[9] M. A. Maddah-Ali and U. Niesen, “Fundamental limits of caching,”
IEEE Trans. Inf. Theory , vol. 60, no. 5, pp. 2856–2867,May 2014.[10] G. Paschos, E. Bas¸tu˘g, I. Land, G. Caire, and M. Debbah, “Wireless caching: Technical misconceptions and businessbarriers,”
IEEE Commun. Mag. , vol. 54, no. 8, pp. 16–22, Aug. 2016.[11] G. Araniti, C. Campolo, M. Condoluci, A. Iera, and A. Molinaro, “LTE for vehicular networking: A survey,”
IEEE Commun.Mag. , vol. 51, no. 5, pp. 148–157, May 2013.[12] V. Ntranos, N. D. Sidiropoulos, and L. Tassiulas, “On multicast beamforming for minimum outage,”
IEEE Trans. WirelessCommun. , vol. 8, no. 6, pp. 3172–3181, June 2009.[13] O. Mehanna, N. D. Sidiropoulos, and G. B. Giannakis, “Joint multicast beamforming and antenna selection,”
IEEE Trans.Signal Process. , vol. 61, no. 10, pp. 2660–2674, May 2013. [14] K.-H. Ngo, S. Yang, and M. Kobayashi, “Scalable content delivery with coded caching in multi-antenna fading channels,” IEEE Trans. Wireless Commun. , vol. 17, no. 1, pp. 548–562, Jan. 2018.[15] V. Exposito, S. Yang, and N. Gresset, “An information-theoretic analysis of the Gaussian multicast channel with interactiveuser cooperation,”
IEEE Trans. Wireless Commun. , vol. 17, no. 2, pp. 899–913, Feb. 2018.[16] L. Yang, Q. Ni, L. Lv, J. Chen, X. Xue, H. Zhang, and H. Jiang, “Cooperative non-orthogonal layered multicast multipleaccess for heterogeneous networks,”
IEEE Trans. Commun. , vol. 67, no. 2, pp. 1148–1165, Feb. 2019.[17] C. Yin, Y. Wang, W. Lin, and J. Xu, “Device-to-device assisted two-stage cooperative multicast with optimal resourceutilization,” in
Proc. IEEE Global Commun. Conf. (GLOBECOM) , Austin, TX, USA, Dec. 2014.[18] T. V. Santana, R. Combes, and M. Kobayashi, “Performance analysis of device-to-device aided multicasting in generalnetwork topologies,”
IEEE Trans. Commun. , vol. 68, no. 1, pp. 137–149, Oct. 2020.[19] A. Asadi, Q. Wang, and V. Mancuso, “A survey on device-to-device communication in cellular networks,”
IEEE Commun.Surveys Tutorials , vol. 16, no. 4, pp. 1801–1819, 4th quarter 2014.[20] M. N. Tehrani, M. Uysal, and H. Yanikomeroglu, “Device-to-device communication in 5G cellular networks: Challenges,solutions, and future directions,”
IEEE Commun. Mag. , vol. 52, no. 5, pp. 86–92, May 2014.[21] F. Boccardi, R. W. Heath, A. Lozano, T. L. Marzetta, and P. Popovski, “Five disruptive technology directions for 5G,”
IEEE Commun. Mag. , vol. 52, no. 2, pp. 74–80, Feb. 2014.[22] E. Bas¸tu˘g, M. Bennis, and M. Debbah, “Living on the edge: The role of proactive caching in 5G wireless networks,”
IEEECommun. Mag. , vol. 52, no. 8, pp. 82–89, Aug. 2014.[23] A. Gupta and R. K. Jha, “A survey of 5G network: Architecture and emerging technologies,”
IEEE Access , vol. 3, pp.1206–1232, July 2015.[24] Y. Mao, B. Clerckx, J. Zhang, V. O. Li, and M. Arafah, “Max-min fairness of K -user cooperative rate-splitting in MISObroadcast channel with user relaying,” IEEE Trans. Wireless Commun. , vol. 19, no. 10, pp. 6362–6376, Oct. 2020.[25] J. Harri, F. Filali, and C. Bonnet, “Mobility models for vehicular ad hoc networks: A survey and taxonomy,”
IEEE Commun.Surveys Tuts. , vol. 11, no. 4, pp. 19–41, Dec. 2009.[26] A. Destounis, G. S. Paschos, and D. Gesbert, “Selective fair scheduling over fading channels,” in
Proc. Int. Symp. Model.and Optim. in Mobile, Ad Hoc and Wireless Netw. (WiOpt) , May 2018.[27] A. N. Uwaechia and N. M. Mahyuddin, “A comprehensive survey on millimeter wave communications for fifth-generationwireless networks: Feasibility and challenges,”
IEEE Access , vol. 8, pp. 62 367–62 414, Mar. 2020.[28] A. M. Sayeed, “Deconstructing multiantenna fading channels,”