Heterogeneous networks (HetNets) employing massive multiple-input multiple-output (MIMO) and millimeter-wave (mmWave) technologies have emerged as a promising solution to enhance the network capacity and coverage of next-generation 5G cellular networks. However, the use of traditional fully-digital MIMO beamforming methods, which require one radio frequency (RF) chain per antenna element, is not practical for large-scale antenna arrays, due to the high cost and high power consumption. To reduce the number of RF chains, hybrid analog and digital beamforming has been proposed as an alternative structure. In this paper, therefore, we consider a HetNet formed with one macro-cell base station (MBS) and multiple small-cell base stations (SBSs) equipped with large-scale antenna arrays that employ hybrid analog and digital beamforming. The analog beamforming weight vectors of the MBS and the SBSs correspond to the the best-fixed multi-beams obtained by eigendecomposition schemes.
electronics Article Hybrid Beamforming for Millimeter-Wave Heterogeneous Networks Mostafa Hefnawi Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, ON K7K 7B4, Canada; hefnawi@rmc.ca Received: 23 December 2018; Accepted: 23 January 2019; Published: 28 January 2019 Abstract: Heterogeneous networks (HetNets) employing massive multiple-input multiple-output (MIMO) and millimeter-wave (mmWave) technologies have emerged as a promising solution to enhance the network capacity and coverage of next-generation 5G cellular networks However, the use of traditional fully-digital MIMO beamforming methods, which require one radio frequency (RF) chain per antenna element, is not practical for large-scale antenna arrays, due to the high cost and high power consumption To reduce the number of RF chains, hybrid analog and digital beamforming has been proposed as an alternative structure In this paper, therefore, we consider a HetNet formed with one macro-cell base station (MBS) and multiple small-cell base stations (SBSs) equipped with large-scale antenna arrays that employ hybrid analog and digital beamforming The analog beamforming weight vectors of the MBS and the SBSs correspond to the the best-fixed multi-beams obtained by eigendecomposition schemes On the other hand, digital beamforming weights are optimized to maximize the receive signal-to-interference-plus-noise ratio (SINR) of the effective channels consisting of the cascade of the analog beamforming weights and the actual channel The performance is evaluated in terms of the beampatterns and the ergodic channel capacity and shows that the proposed hybrid beamforming scheme achieves near-optimal performance with only four RF chains while requiring considerably less computational complexity Keywords: hybrid beamforming; massive MIMO; HetNets; mmWaves Introduction Recently, heterogeneous networks (HetNets) that use massive multiple-input multiple-output (MIMO) and millimeter-wave (mmWave) technologies has emerged as a promising solution to enhance the network capacity and coverage of next-generation 5G cellular networks [1–6] Small cell deployment in HetNets can achieve high signal to interference plus noise ratio (SINR) and dense spectrum reuse, mmWave can address the current challenge of bandwidth shortage, and the large number of antenna arrays [7–10] are essential for mmWaves to compensate for channel attenuation In Reference [11] we applied the concept of massive multiuser (MU)-MIMO to enhance both the access and the backhaul links in HetNets, and it was shown that such a concept could significantly improve the system performance in terms of link reliability, spectral efficiency, and energy efficiency Traditional MIMO-beamforming systems require a dedicated radio frequency (RF) chain for each antenna element, which becomes impractical with massive MIMO systems due to either cost or power consumption To reduce the number of RF chains, hybrid beamforming (HBF), which combines RF analog and baseband digital beamformers, has been proposed as a promising solution [12–17] Figure shows a general hybrid configuration that connects Na antenna elements to Nd RF chains, where Nd < Na , using an analog RF beamforming matrix built from only phase-shifters Two widely-used analog beamformer architectures for hybrid beamforming are shown in Figure The fully-connected hybrid beamforming structure of Figure 2a provides a full beamforming gain per transceiver—but with Electronics 2019, 8, 133; doi:10.3390/electronics8020133 www.mdpi.com/journal/electronics Electronics 2019, 8, 133 of 10 Electronics 2018, 7, x FOR PEER REVIEW of 10 Electronics 2018, 7, x FOR PEER REVIEW each RF chain to all antennas through a network of N × N phase of 10 high complexity—by connecting a d shifters [12–15] on the other hand, shows partially-connected where each RF network of 𝑁Figure × 𝑁 2b, phase shifters [12–15] Figure a2b, on the other hand, structure, shows a partially-connected network of 𝑁 × 𝑁 phase shifters [12–15] Figure 2b, on the other hand, shows a partially-connected chain is connected toeach Na /N sub-arrays Such has a lower hardware complexity d number structure, where RF chain is of connected to 𝑁 /𝑁 a structure number of sub-arrays Such a structure has a structure, where each RF chain is connected to 𝑁 /𝑁 number of sub-arrays Such a structure has a at the price of reduced beamforming gain lower hardware complexity at the price of reduced beamforming gain lower hardware complexity at the price of reduced beamforming gain Figure Hybrid beamforming Figure Hybrid beamforming Figure Hybrid beamforming (a) (a) (b) (b) Figure The architecture of analog beamformers: Fully-connected structure; partiallyFigure The architecture of analog beamformers: (a) (a) Fully-connected structure; (b) (b) partiallyFigure The architecture of analog beamformers: (a) Fully-connected structure; (b) partiallyconnected structure connected structure connected structure Previous studies on massive hybrid MIMO mainly focused on single-user systems [12–14].[12–14] On the On Previous studies on massive hybrid MIMO mainly focused on single-user systems Previous studies on massive hybrid MIMO mainly focused on single-user systems [12–14] On other cases were studied References [15–17] In Reference [15] a scheme [15] called thehand, otherMU-MIMO hand, MU-MIMO cases wereinstudied in References [15–17] In Reference a Joint scheme the other hand, MU-MIMO cases were studied in References [15–17] In Reference [15] a scheme Spatial Division Multiplexing (JSDM) was proposed to create multiple virtual sectors whichvirtual reducesectors the called Joint Spatial Division Multiplexing (JSDM) was proposed to create multiple called Joint Spatial Division Multiplexing (JSDM) was proposed to create multiple virtual sectors overhead and computational complexity of downlink training and uplink feedback In References [16,17] which reduce the overhead and computational complexity of downlink training and uplink feedback which reduce the overhead and computational complexity of downlink training and uplink feedback it was shown that the required number RFrequired chains only needs to chains be twice theneeds number data the In References [16,17] it was shown thatofthe number of RF only to beoftwice In References [16,17] it was shown that the required number of RF chains only needs to be twice the streams to achieve the sametoperformance of anyperformance fully-digitalofbeamforming scheme These studies, number of data streams achieve the same any fully-digital beamforming scheme number of data streams to achieve the same performance of any fully-digital beamforming scheme however, did not consider in the contextHBF of HetNets and focused primarily on macro-cellular These studies, however,HBF did not consider in the context of HetNets and focused primarily on These studies, however, did not consider HBF in the context of HetNets and focused primarily on systems In this paper, we In consider a HetNet where aboth the macro-cell stations (MBSs) and macro-cellular systems this paper, we consider HetNet where bothbase the macro-cell base stations macro-cellular systems In this paper, we consider a HetNet where both the macro-cell base stations small-cell base stations (SBSs) are equipped with fully-connected massive hybrid antenna arrays, (MBSs) and small-cell base stations (SBSs) are equipped with fully-connected massive hybrid antenna (MBSs) and small-cell base stations (SBSs) are equipped with fully-connected massive hybrid antenna while all mobile have users a single antenna Weantenna proposeWe a low-complexity HBF that is fully-based arrays, whileusers all mobile have a single propose a low-complexity HBF that is on fullyarrays, while all mobile users have a single antenna We propose a low-complexity HBF that is fullyeigenbeamforming The MBSs and the select the the best-fixed by eigendecomposition based on eigenbeamforming TheSBSs MBSs and SBSs multi-beams select the best-fixed multi-beams by based on eigenbeamforming The MBSs and the SBSs select the best-fixed multi-beams by of the access and backhaul channels The selected are then used by beams the digital beamformers, eigendecomposition of the access and backhaulbeams channels The selected are then used by the eigendecomposition of the access and backhaul channels The selected beams are then used by the which are based on the maximization of theon receive SINR of the effective channels consisting the digital beamformers, which are based the maximization of the receive SINR of the of effective digital beamformers, which are based on the maximization of the receive SINR of the effective cascade of the analog beamforming weights the actual channel weights [18,19] and the actual channel [18– channels consisting of the cascade of theand analog beamforming channels consisting of the cascade of the analog beamforming weights and the actual channel [18– 19] 19] System Model 2.We System Model consider System Model the access and backhaul uplinks in the HetNet of Figure 3, where K cognitive small cells andWe their Ls small-cell users are concurrently sharing with one consider the access and(SUs) backhaul uplinks in the HetNetthe of same Figurefrequency 3, where band 𝐾 cognitive small We consider the access and backhaul uplinks in the HetNet of Figure 3, where 𝐾 cognitive small MBS and their L p 𝐿macro-cell users (PUs) It is assumed MBS and SBSs areone cells and their small-cellprimary users (SUs) are concurrently sharingthat the both samethe frequency band with cells and their 𝐿massive small-cell users (SUs) are concurrently sharing thePUs same band with one equipped with hybrid antenna SUs and arefrequency equipped with a single MBS and their 𝐿 macro-cell primaryarrays users while (PUs).the It is assumed that both the MBS and SBSs are MBS and their 𝐿 macro-cell primary users (PUs) It is assumed that both the MBS and SBSs are equipped with massive hybrid antenna arrays while the SUs and PUs are equipped with a single equipped with massive hybrid antenna arrays while the SUs and PUs are equipped with a single antenna The SBSs are acting as smart relays between the SUs and the MBS with 𝑁 − element antenna The SBSs are acting as smart relays between the SUs and the MBS with 𝑁 − element transmitting/receiving antenna arrays and 𝑁 RF chains The MBS is equipped with 𝑀 − element transmitting/receiving antenna arrays and 𝑁 RF chains The MBS is equipped with 𝑀 − element Electronics 2019, 8, 133 of 10 antenna The SBSs are acting as smart relays between the SUs and the MBS with Na − element Electronics 2018, 7, x FOR PEER REVIEW of 10 transmitting/receiving antenna arrays and Nd RF chains The MBS is equipped with Ma − element antenna arrays and Md RF chains It is also assumed that both the SBS and the MBS perform arrays transmission and 𝑀 RF chains is also assumed that both the SBS and thesubcarriers MBS perform anantenna OFDM-based and thatItthe analog beamformers are identical for all whilean OFDM-based transmission and that the analog beamformers are identical for all subcarriers while adapting digital beamformers in each subcarrier adapting digital beamformers in each subcarrier p p p Let xs [ f n ] = x1s , x2s , · · · , x sLs and x p [ f n ] = x1 , x2 , · · · , x L p denote, respectively, the set of Ls Let 𝒙 𝑓 = 𝑥 , 𝑥 , ⋯ , 𝑥 and 𝒙 𝑓 = 𝑥 , 𝑥 , ⋯ , 𝑥 denote, respectively, the set of 𝐿 SUs SUs signals and L p PUs signals transmitted on each subcarrier, and f n , n = 1, · · · , Nc , where Nc signalsthe andnumber 𝐿 PUsofsignals transmitted on each subcarrier, and 𝑓 ,The 𝑛 =analysis 1, ⋯ , 𝑁 is , where 𝑁 denotes denotes subcarriers per OFDM symbol in the system done separately the number of subcarriers per OFDM symbol in the system The analysis is done separately on each on each subcarrier For brevity therefore, we drop the frequency index f n subcarrier For brevity therefore, we drop the frequency index 𝑓 Figure System model: hybrid beamforming-based HetNet with one macro-cell and K small-cells Figure System model: hybrid beamforming-based HetNet with one macro-cell and K small-cells 2.1 Access Link 2.1 Access Link The Na × received signal vector yk,SBS at the kth SBS is given by The 𝑁 × received signal vector 𝒚 , at the 𝑘 SBS is given by yk,SBS +G nk,SBS 𝒚 ,=G =k,SU 𝑮 ,xs 𝒙 +k,PU 𝑮 , x p𝒙+ + 𝒏 ,, (1)(1) , th SBS where 𝑮 , ∈∈CℂNa ××Ls isisthe thechannel channelmatrix matrixbetween betweenthe thek𝑘 SBS anditsitsLs𝐿users, users, 𝑮 ,∈∈ where Gk,SU and Gk,PU C Nℂa × L×p isis × th SBS the channel matrix between k and SBSthe and ∈ the ℂ transmitted is the transmitted signal the channel matrix between the kthe L p the PUs,𝐿xs PUs ∈ C L,s ×𝒙1 is signal vector of × L × th 𝐿 users in the 𝑘 x p small-cell, ∈ℂ is thesignal transmitted vectorinofthe 𝐿 HetNet, users in Lsvector users of in the k small-cell, ∈C p is𝒙 the transmitted vector signal of L p users Na ×1 thenHetNet, ℂ × iscomplex the received complex and additive whiteadditive Gaussianwhite noiseGaussian (AWGN)noise vector(AWGN) at the k,SBS ∈ Cand 𝒏is, the∈received th k vector SBS at the 𝑘 SBS shouldbe benoted noted that that in Equation between small-cells waswas neglected This It Itshould Equation(1), (1),the theinterference interference between small-cells neglected waswas justified by the fact fact that that small-cell basebase stations are using a large number of antennas, which This justified by the small-cell stations are using a large number of antennas, enables sharpsharp beamforming towards theirtheir usersusers without harming neighboring small-cells which enables beamforming towards without harming neighboring small-cells Thekth𝑘 SBS SBS received signal, 𝒚 , , is ,first is first applied to an ×d𝑁receive receive analog beamforming The received signal, yk,SBS applied to an Na 𝑁 ×N analog beamforming SBS weightmatrix, matrix,AR,k,l 𝑨 , s ,, whose , whoseoutput outputis isdirected directedtotoananNd𝑁× × receive digital beamformingweight weight weight N𝑁 digital beamforming d receive SBS th vector 𝑫 If we denote the combined digital-analog receive beamformer for the 𝑙 user vector DR,k,l,s , If we denote the combined digital-analog receive beamformer for the ls user asas th usersignal = 𝑨SBS 𝑫SBS signalby byits itskth𝑘 SBS SBS expressed w𝐰 cancan bebe expressed as as , , D , , , then the detection of the l𝑙s user R,k,l R,k,l, , = A R,k,l s s s = 𝐰 H, , 𝑮 , 𝒙 + 𝐰 H, , 𝑮 , 𝒙 + 𝐰 ,H, 𝒏 , 𝒓 , =𝐰 , , 𝒚 , rk,ls = wH R,k,ls yk,SBS = w R,k,ls Gk,SU xs + w R,k,ls Gk,PU x p + w R,k,ls nk,SBS Ls H = H𝑥 + 𝐰 H 𝑥 + H+ 𝐰 𝒈R,k,l 𝐰k,PU 𝒙 R,k,l = wR,k,l g 𝐰 x s +𝒈w, R,k,l n , , 𝒏 ∑ , , gk,ixs + w , s +G , w , , x𝑮p + s k,ls , ls, s s k,SBS i =1,i =ls where 𝒈 user , is the 𝑙 column of 𝑮 , i , , (2)(2) , that represents the channel between the 𝑘 SBS and its 𝑙 Electronics 2019, 8, 133 of 10 where gk,ls is the lsth column of Gk,SU that represents the channel between the kth SBS and its lsth user If we denote H AL,k,ls = ASBS R,k,ls H gk,ls as the effective access channel between the kth SBS and its H lsth user, then for a set of selected beams, i.e known ASBS R,k,ls the digital beamformer, DSBS R,k,ls , γSBS k,ls , the SINR can be expressed in terms of as DSBS R,k,ls = H H AL,k,ls xlss xlss DSBS R,k,ls H H SBS HH AL,k,ls D R,k,ls , (3) B AL DSBS R,k,ls where B AL is the covariance matrix of the interference-plus-noise given by Ls ∑ B AL = H (ASBS R,k,i ) i =1,i =ls H H SBS H H SBS SBS SBS gk,i xis ( xis ) H gk,i AR,k,i + (ASBS R,k,ls ) Gk,PU x p x p Gk,PU A R,k,ls + σn (A R,k,ls )A R,k,ls , (4) Interference from L p PUs Interference from Ls −1 SUs 2.2 Backhaul Link The hybrid beamforming weights at the backhaul link are obtained based on orthogonal pilot p signals transmitted from each SBS to the MBS The kth SBS applies its pilot signal sk ∈ C Nd ×1 to an Nd × Nd transmit digital beamforming weight vector DSBS T,k followed by an Na × Nd transmit analog SBS beamforming matrix AT,k If we denote the combined digital-analog transmit beamformer for the kth SBS SBS as wT,k = ASBS T,k DT,k , then the array output of the MBS can be written as K p yMBS = ∑ Hk,MBS wT,k sk + HPU,MBS x p + n MBS , p (5) k =1 p where yMBS is the Ma × vector containing the outputs of the Ma − element antenna array of the MBS, Hk,MBS is the Na × Ma channel matrix representing the transfer functions from the Na − element antenna array of the kth SBS to the Ma − element antenna array of the MBS, HPU,MBS is the Ma × Lp channel matrix from the L p PUs to the MBS’s Ma − element antenna array, and n MBS is the received Ma × complex AWGN vector at the MBS p The MBS detects the kth SBS signal by applying the output of the array yMBS to the Ma × Md receive MBS followed by the M × M receive digital beamforming weight vector D MBS analog weight matrix AR,k d d k,R MBS D MBS , If we denote the combined digital-analog receive beamformer for the kth SBS as ck = Ak,R k,R then the detection of the kth SBS signal by the MBS can be expressed as p p xˆ k = ckH yMBS = Sk + S Ik + S I p + ckH n MBS , p (6) p p where Sk = ckH Hk,MBS wT,k sk is the kth SBS signal, S Ik = ckH ∑iK=1,i =k Hk,MBS wT,k sk is the interference from K − other SBSs, and SIp = ckH HPU,MBS x p is the interference from L p PUs p Assuming that sk are complex-valued random variables with unit power, i.e., E MBS and denoting HBL,k = AR,k H Hk,MBS ASBS T,k H p sk = 1, as the effective channel between the kth SBS and the MBS, the SINR at the MBS for the kth SBS can be expressed as γkMBS = MBS DR,k H HBL,k DSBS T,k H ckH BBL ck MBS H DSBS T,k H BL,k D R,k , (7) Electronics 2019, 8, 133 of 10 H HH H H where BBL = ∑iK=1,i =k Hi,MBS wT,i wT,i i,MBS + H PU,MBS x p x p H PU,MBS + σn I Ma is the covariance matrix of the interference-plus-noise at the backhaul link 2.3 End-to-End SINR and Channel Capacity Once the hybrid beamforming weights of the backhaul link are obtained, they can be used to H forward the SUs signals to the MBS The kth SBS applies the received lsth user signal, wR,k,l g x s , to the s k,ls ls hybrid beamformer, wT,k The NT,a × transmitted signal sk,ls at the output of the antenna array can then be expressed as H sk,ls = wT,k wR,k,l g xs , (8) s k,ls ls and the expression for the array output of the MBS can be written as K y MBS = HPU,MBS x p + yk,MBS + ∑ yi,MBS + n MBS , (9) i =1, i =k H where yk,MBS = Hk,MBS wT,k wR,k,l y is the array output of the MBS from the kth SBS s k Using Equation (2), yk,MBS can be expressed in terms of the lsth user signal, xlss , as follows: H yk,MBS = Hk,MBS wT,k wR,k,l g xs s k,ls ls H + Hk,MBS wT,k wR,k,ls ∑iL=s 1,i =ls gk,i xis H + Hk,MBS wT,k wR,k,l G xp s k,PU H + Hk,MBS wT,k wR,k,l n , s k,SBS (10) When the MBS applies the output of the array, y MBS , to the hybrid weight, ckH , the detection of the user signal of the kth SBS by the MBS can be expressed as lsth PU xˆ k,ls = ckH yMBS = ckH Sk,ls + SSBSs + SSU I k,I + S I + N MBS , (11) where H Sk,ls = Hk,MBS wT,k wR,k,l g x s is the lsth user signal of the kth SBS, s k,ls ls Ls H s th SSU k,I = Hk,MBS wT,k w R,k,ls ∑i =1,i =ls gk,i xi is the interference from the Ls − other SUs of k SBS, H y SSBS = ∑iK=1, i =k Hi,MBS wT,i wR,i,l is the interference from the K − other SBSs I s i,SBS H H SPU I = Hk,MBS wT,k w R,k,ls Gk,PU x p + ck H PU,MBS x p H N MBS = Hk,MBS wT,k wR,k,l n + n MBS s k,SBS The end-to-end SINR at the MBS for the lsth user of the kth SBS can be expressed as MBS γk,l s = H ckH Hk,MBS wT,k wR,k,l g x s xlss s k,ls ls SU where B AL− BL = SSU k,I Sk,I H H H w H gk,l R,k,ls wT,k H k,MBS ck s ckH B AL− BL ck H + SSBS SSBS I I H + SPU SPU I I H , (12) + N MBS (N MBS ) H is the covariance matrix of the interference-plus-noise for the lsth user end-to-end link The ergodic channel capacity for each user, ls , is given by [19] C = E log2 + H H H w H ckH Hk,MBS wT,k wR,k,l g xs x Hs gk,l R,k,ls wT,k H k,MBS ck s k,ls s where E(.) denotes the expectation operator ckH B AL− BL ck , (13) Electronics 2019, 8, 133 of 10 2.4 Channel Model For the access and backhaul links, we consider mmWave propagation channels with limited scattering which can be modelled at each subcarrier by the clustered channel representation [13] We assume a scattering environment with Ncl scattering clusters randomly distributed in space and within each cluster, there are Nray closely located scatterers For the backhaul link, the channel matrix at subcarrier f n between the kth SBS and the MBS can be expressed as Hk,MBS, f n = Na Ma Ncl Nray N N ∑i cl ∑l=ray1 αil, fn a MBS, fn t t r , θi,l , ∅ri,l , θi,l a∗k,SBS, f n ∅i,l (14) where αil, f n = αil e− j2πi f n /Nc are the complex gains of the jth ray in the ith scattering cluster and αil are ) With σ2 representing the average power of the i th cluster, ∅r and ∅t are assumed i.i.d CN(0, σα,i i,l i,l α,i r and θ t are the elevation angles of arrival the azimuth angles of arrival and departure, respectively, θi,j i,j r and departure, respectively, and a MBS, f n ∅ri,l , θi,l t , θt and ak,SBS, f n ∅i,l i,l represent, respectively, the normalized receive and transmit array response vectors of the MBS and the kth SBS For the access link, the channel matrix at subcarrier f n between the kth SBS and its Ls users can be written as Gk,SU, f n = Ls Ma Ncl Nray N N ∑i cl ∑l=ray1 αil, fn aSBS, fn r t t ∅ri,l , θi,l a∗k,SU, f n ∅i,l , θi,l , (15) t , θt where ak,SU, f n ∅i,l i,l represents the spatial signature vector of the Ls single antenna users r,t The Nray azimuth and elevation angles, ∅r,t i,l and θi,l , within the cluster i are assumed to be r,t randomly distributed with a uniformly-random mean cluster angle of ∅r,t i and θi , respectively, and a constant angular spread of σ∅r,t and σθ r,t , respectively For simplicity, the access links between the MBS and its L p users (HPU,MBS ) and between PUs and th the k SBS (Gk,PU ) are modeled by convetional i.i.d MIMO channels Note that in this per-subcarrrier representation, it is assumed that for each subcarrier f n , the carrier r t , θt frequency f c is much larger than f c ± f n and that a MBS, f n ∅ri,l , θi,l and ak,SBS, f n ∅i,l can i,l approximately be considered equal for all subcarriers Consequently, the channel covariance matrices are approximately similar with the same set of eigenvectors for all subcarriers and can be Nc H replaced by the average of the covariance matrices, i.e., H H AL,k,ls H AL,k,ls = Nc ∑n=1 H AL,k,ls , f n H AL,k,ls , f n , H H HBL,k BL,k = Nc N H H ∑n=c HBL,k, f n H BL,k, f n , and H PU,MBS H PU,MBS = N Nc H ∑n=c HPU,MBS, f n H PU,MBS, f n Proposed Hybrid Beamforming 3.1 Access Link The kth SBS communicates with each SU through a set of selected beams that corresponds to a set of weight vectors These weight vectors are obtained using the eigenbeamforming scheme and can be expressed as SBS SBS SBS ASBS R,k,ls = a R,k,ls ,1 , a R,k,ls ,2 , · · · , a R,k,ls ,N subject to ASBS R,k,ls (i, j ) d , (16) =1 th th where aSBS R,k,ls ,i denote the i selected Ma × eigenvector corresponding to the i maximum eigen value of g Hk,ls gk,ls carrier frequency 𝑓 is much larger than 𝑓 𝑓 and that 𝒂 , ∅ , , 𝜃 , and 𝒂 , , ∅ , , 𝜃 , can approximately be considered equal for all subcarriers Consequently, the channel covariance matrices are approximately similar with the same set of eigenvectors for all subcarriers and can be replaced 𝐇 , , , 𝐇 , , , , by the average of the covariance matrices, i.e., 𝐇 , , 𝐇 , , = ∑ Electronics 2019, 8, 133 ∑ 𝐇 , 𝐇 , = 𝐇 , , 𝐇 , , , and 𝐇 , 𝐇 , = ∑ 𝐇 , , 𝐇 , , of 10 Proposed Hybrid Beamforming Since the analog beamforming matrix ASBS R,k,ls is realized using phase shifters only, its elements, ASBS satisfy ASBS (i, j),Link R,k,l R,k,ls (i, j ) = It should be noted that each SBS is using a different analog matrix s 3.1 Access for each user and that the system model shown in Figure focuses on the detection of the lsth user of the The 𝑘 SBS communicates with each SU through a set of selected beams that corresponds to a kth SBS and shows the analog beamformer and the RF chains for one user only The analog beamformer set of weight vectors These weight vectors are obtained using the eigenbeamforming scheme and can be implemented using the Butler matrix as shown in Figure 4, where four users ( Ls = 4) and can be expressed as four RF chains per user ( Nd = 4) are assumed Depending upon which ports are activated, beams are produced in specified directions channels, we should expect 𝑨 , , Since = 𝒂 we , 𝒂 assuming , , , are , , , , ⋯ , 𝒂 4, different , , , (16) different ports for each user 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑨 , , (𝑖, 𝑗) = Figure4.4.Hybrid Hybridbeamforming beamformingbased basedon onButler Butlermatrix matrixfor forthe theaccess accesslink link Figure SBS , are obtained analog beams are selected,𝑀 the×received optimalcorresponding digital weights, whereOnce 𝒂 , the selected eigenvector to Dthe maximum R,k,ls𝑖 , , denote the 𝑖 SBS based on the maximization of the access link receive SINR, γ , given by Equation (3): eigen value of 𝒈 , 𝒈 , k,ls −1 DSBS R,k,ls = B AL,k,ls V AL,k,ls , (17) where V AL,k,ls denote the eigen vector corresponding to the maximum eigenvalue of the effective access channel, H H AL,k,ls H AL,k,ls 3.2 Backhaul Link The transmit analog weights of the kth SBS are based on the eigenbeamforming scheme and are given by SBS SBS SBS ASBS T,k = aT,k,1 , aT,k,2 , · · · , aT,k,N subject to ASBS T,k (i, j ) d , (18) =1 th th where aSBS T,k,i denote the i selected Na × eigenvector corresponding to the i maximum eigenvalue H of H k,MBS Hk,MBS Assuming channel reciprocity with Na = Ma , the receive analog weight vectors of the MBS are MBS = ASBS It should also be noted that the MBS is using a different analog matrix for each given by AR,k T,k SBS, which can be implemented using the Butler matrix of Figure 4, where mobile users are replaced by SBSs Electronics 2019, 8, 133 of 10 MBS For fixed analog beamforming weights, ASBS T,k and A R,k , the transmit optimal digital weight vector MBS of the kth SBS, DSBS T,k , and the receive optimal digital weight vector of the MBS, D R,k , are obtained base on the maximization of the backhaul link receive SINR and are given by MBS −1 DSBS T,k = D R,k = B BL,k HBL,k V BL , (19) H H where VBL is the eigenvector corresponding to the maximum eigenvalue of HBL,k BL,k , with H BL,k representing the effective channel given by HBL,k = AMBS R H k Hk,MBS ASBS T H k Simulation Results In our simulation setups, we considered a HetNet organized into four SBSs (K = 4) and one macro-cell The SBSs and the MBS used the same number of antennas, Na = Ma = 64, and the same number of RF chains, Nd = Md = or Each SBS is serving Ls = users and the macro-cell is serving users, each transmitting with a single antenna We assumed QPSK modulation For the OFDM configurations, we assumed the 256-OFDM system (Nc = 256), which is widely deployed in broadband wireless access services Figure shows the beampattern of the proposed HBF with four RF chains and the optimal fully-digital one for the access link It is noted that the optimal beamformer has about five dominant beams, three of which are similar to the selected beams of the proposed HBF This beampattern means that the data streams can be successfully transmitted through those three beams using the proposed HBF and that near optimal performance could be achieved if we were to bring the number of RF chains close to the number of dominant beams of the optimal beamformer For the backhaul link, Figure Electronics 2018, 7, x FOR PEER REVIEW of 10 Electronics 2018, 7, x FOR PEER REVIEW with more dominant beams of 10 shows very similar beampatterns (a) (a) (b) (b) Figure Beampattern of the access link: (a) Proposed HBF, RF chains; (b) fully-digital Figure Beampattern ofof the the access access link: link: (a)(a)Proposed ProposedHBF, HBF,4 4RFRFchains; chains;(b)(b)fully-digital fully-digital Figure5.5 Beampattern beamforming (optimal) beamforming(optimal) (optimal) beamforming (a) (a) (b) (b) Figure Beampattern of of the the backhaul backhaul link: (a) Figure 6 Beampattern (a) Proposed ProposedHBF, HBF, 44 RF RF chains; chains;(b) (b)fully-digital fully-digital Figure Beampattern of the backhaul link: (a) Proposed HBF, RF chains; (b) fully-digital beamforming (optimal) beamforming (optimal) beamforming (optimal) Figure 7, on the other hand, compares the ergodic channel capacity of the proposed HBF and the Figure 7, on the other hand, compares the ergodic channel capacity of the proposed HBF and the optimal fully-digital one It is observed that for both cases the optimal beamformer is outperforming optimal fully-digital one It is observed that for both cases the optimal beamformer is outperforming the proposed HBF However, as we increase the number of RF chains, the performance gap between the proposed HBF However, as we increase the number of RF chains, the performance gap between the two schemes was reduced, and a near-optimal solution was achieved by the proposed HBF using the two schemes was reduced, and a near-optimal solution was achieved by the proposed HBF using Electronics 2019, 8, 133 of 10 Figure 7, on the other hand, compares the ergodic channel capacity of the proposed HBF and the optimal fully-digital one It is observed that for both cases the optimal beamformer is outperforming the proposed HBF However, as we increase the number of RF chains, the performance gap between the two schemes was reduced, and a near-optimal solution was achieved by the proposed HBF using four RF chains On the other hand, for the single cell MU-MIMO case presented in References [12–14], near optimal performance was achieved with only five RF chains, and for the MU-MIMO case in [16,17], it was shown that the required number RF chains could be reduced to two to achieve fully digital beamforming performance However, unlike our case, where we have assumed a HetNet with a macro cell and multiple small cognitive cells, these studies focused primarily on macro-cellular systems and Electronics 2018, 7, x FOR PEER REVIEW of 10 did not consider HBF in the context of HetNets 50 45 Proposed HBF: Number of RF chains = Optimal Proposed HBF: Number of RF chains = 40 35 30 25 20 15 10 -20 -10 10 20 30 40 SNR (dB) Figure 7 Ergodic Ergodic channel channel capacity capacity of of the the proposed proposed HBF Figure HBF for for different different number number of of RF RF chains chains Conclusions Conclusion In this paper, we employed hybrid beamforming at the access and backhaul links of a mmWave In this paper, we employed hybrid beamforming at the access and backhaul links of a mmWave HetNet system We proposed a low-complexity HBF that was fully-based on MRT/MRC EigenHetNet system We proposed a low-complexity HBF that was fully-based on MRT/MRC Eigenbeamforming schemes The performance evaluation in terms of the beam patterns and the ergodic beamforming schemes The performance evaluation in terms of the beam patterns and the ergodic channel capacity showed that the proposed HBF scheme achieved near-optimal performance with channel capacity showed that the proposed HBF scheme achieved near-optimal performance with only four RF chains and required considerably less computational complexity only four RF chains and required considerably less computational complexity Funding: This research received no external funding Funding: This research received no external funding Acknowledgments: The author would like to thank the Canadian Microelectronics Corporation (CMC) for Acknowledgments: The author would like totothank thecomputationally-intensive Canadian Microelectronics Corporation (CMC) for providing the Heterogeneous Parallel Platform run the Monte-Carlo Simulations providing the Heterogeneous Parallel Platform to run the computationally-intensive Monte-Carlo Simulations Conflicts of Interest: The authors declare no conflict of interest Conflicts of Interest: The authors declare no 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