Heriot-Watt University

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1 Heriot-Watt Univerity Heriot-Watt Univerity Reearch Gateway Hybrid Beamforming with Reduced umber of Phae Shifter for aive IO Sytem Payami, Sohail; Ghoraihi, ir; Dianati, ehrdad; Sellathurai, athini Publihed in: IEEE Tranaction on Vehicular Technology DOI: /TVT Publication date: 018 Document Verion Peer reviewed verion ink to publication in Heriot-Watt Univerity Reearch Portal Citation for publihed verion (APA): Payami, S., Ghoraihi,., Dianati,., & Sellathurai,. (018). Hybrid Beamforming with Reduced umber of Phae Shifter for aive IO Sytem. IEEE Tranaction on Vehicular Technology. DOI: /TVT General right Copyright and moral right for the publication made acceible in the public portal are retained by the author and/or other copyright owner and it i a condition of acceing publication that uer recognie and abide by the legal requirement aociated with thee right. If you believe that thi document breache copyright pleae contact u providing detail, and we will remove acce to the work immediately and invetigate your claim.

2 1 Hybrid Beamforming with Reduced umber of Phae Shifter for aive IO Sytem Sohail Payami, ir Ghoraihi, ehrdad Dianati, athini Sellathurai Heriot-Watt Univerity, Univerity of Surrey, Univerity of Warwick {.payami, Abtract- In thi paper, two novel hybrid beamforming method are propoed to reduce the cot and power conumption of hybrid beamformer with ubconnected phae hifter network tructure in maive multiple-input multiple-output (IO) ytem. Thi i achieved by replacing ome of the phae hifter with witche which, in general, are cheaper and have lower power conumption compared to phae hifter. The propoed method and the cloed-form expreion of their performance are derived according to the propertie of the element of the ingular vector of the channel matrix. In the firt approach, it i hown that by combining the ubconnected phae hifter network with a fully-connected witch architecture, the number of the phae hifter can be reduced up to 50% while the pectral efficiency i preerved. Then, in order to implify the tructure of the witch network, the fully-connected witche i replaced by ubconnected witch network, e.g. binary witche. The analytical and imulation reult indicate that jut by uing 5% of phae hifter 90% pectral efficiency can be achieved. Finally, imulation reult indicate that imilar behavior i oberved when the wirele channel i conidered to be pare or correlated. Index Term Antenna election, hybrid beamforming, maive IO. I. ITRODUCTIO aive multiple-input multiple-output (IO) technology with digital beamforming can increae the pectral efficiency in wirele communication ytem. However, a dedicated RF chain per antenna increae the cot of thi technology. In order to reduce the number of RF chain, hard and oft antenna election technique have been propoed [1]. In the hard election, the RF chain are connected to the antenna by a network of witche. The drawback of thi approach i that large beamforming gain cannot be achieved a only a mall fraction of the antenna are ued [1], []. In the oft antenna election, alo known a hybrid beamforming, the RF chain and the antenna are connected through a network of phae hifter [1], [3] [5]. Such architecture have lower cot and power conumption compared to digital beamformer and they achieve a higher pectral efficiency compared to hard election. There are two type of phae hifter network known a fully-connected and ubconnected [5]. In the fullyconnected tructure, each RF chain i connected to all the antenna a in [1], [3], [4]. It can exploit the full array gain, however, it power conumption can be very high due to the maive number of the phae hifter it require [3], [5]. In the ubconnected configuration, each RF chain i connected to a ubet of antenna which reult in a impler circuit but with lower pectral efficiency compared to that of the fully-connected configuration [5]. In general, the deign of the optimal oft antenna election cheme i a challenging tak due to the nonconvex contant modulu contraint impoed by the phae hifter [1], [3] [5]. In thi context, [3], [5], [6] ued witche that are cheaper and power efficient alternative to phae hifter. Similar to the phae hifter network, witch network have fully-connected and ubconnected tructure. However, due to the requirement of large number of witche, a fully-connected configuration with witche ha high hardware complexity and experience inertion loe and cro talk ditortion [7]. Hence, ubconnected witch network,for example binary witche, i alo preferred in practice depite the le degree of freedom in antenna election [7]. It i noted that the work in [1], [], [4] [7], and reference therein, only focu on antenna election with either phae hifter or witche, and do not conider the combination of the two method. Recently, the author in [8] ued for the firt time a combination of witche and non-tunable phae hifter to how that the pectral efficiency of hybrid beamforming with fullyconnected phae hifter network can be achieved uing uch combination. While the approach in [8] require a relatively low computational complexity and low power conumption, it demand a maive number of RF route which could reult in a complex hardware and high level of crotalk. Within the ame context of deigning the joint witche and phae-hifter baed hybrid beamforming, [3] conider a fully-connected phae hifter network where each phae hifter wa equipped with a witch. It wa hown that it i poible to reduce the power conumption of the RF beamformer by turning off almot 50% of the phae hifter while achieving the ame pectral efficiency. With the motivation to reduce the power conumption of maive IO ytem while achieving high pectral efficiency, there i till a need for invetigating new technique to jointly deign the beamforming weight at the baeband a well a the phae hifter and witch network. In thi paper, two uch novel combination of phae hifter and witche are propoed and their performance in term of the achievable um-rate are evaluated. To thi end, firt we derive the cloed-form expreion of the beamformer and it achievable um-rate over uncorrelated independent and

3 identically ditributed (i.i.d.) channel when the RF beamformer ha ubconnected tructure. Thi approach offer lower computational complexity and imilar performance compared to the ucceive interference cancellation baed method in [5] when a mall number of RF chain are connected to a large number of antenna. Second, baed on the preented approach for the ubconnected phae hifter network and uing phae hifter election technique, in [3], for fully-connected phae hifter network, it i hown that a combination of ubconnected phae hifter and fully-connected witch network can reduce the number of the phae hifter by 50% without any performance lo over uncorrelated i.i.d. Rayleigh fading channel. The imulation reult for pare cattering and correlated Rayleigh fading channel indicate that the achievable umrate almot remain at the ame level when the number of the phae hifter are reduced to half. However, a the propoed tructure require a fully-connected witch network, which may not be uitable for practical application, it i deirable to ubtitute the complex witch network with impler tructure, for example binary witche. Hence, we preent another novel beamforming method that provide a lightly lower performance but with a much impler hardware tructure. In thi approach, the fully-connected witche are replaced with imple ubconnected witche, e.g. binary witche. Finally, the imulation reult indicate that the aymptotic cloedform expreion of the pectral efficiency provide a good approximation of the performance for moderate number of antenna and phae hifter. Thi paper i organized a following, the ytem model and hybrid beamforming with ubconnected phae hifter network are decribed in ection II and III. In ection IV, the propoed method for hybrid beamforming with ubconnected phae hifter and fully-connected witche are preented. The analyi for the ubconnected phae hifter that are connected to ubconnected witch network are dicued in V. Finally, the imulation reult and concluion are preented in ection VI and VII. otation: Bold capital and mall letter A and a repreent a matrix and a vector, repectively. A mn denote the (m, n)- th element of A, a m i the m-th column of A and A 1:m i a matrix containing the firt m column of A. det(a), A H and trace(a) denote determinant, Hermitian and trace of A, repectively. oreover, A and A denote the magnitude and angle of complex number A. I m i an m m identity matrix. Finally, f A (a), F A (a) and E[A] denote the probability denity function (pdf), cumulative ditribution function (cdf) and expected value of A, repectively. II. SYSTE ODE In thi work, a narrowband ingle-cell multiuer cenario in downlink where the bae tation with omni-directional antenna erve K ingle-antenna uer i conidered. The wirele channel matrix H C K follow an uncorrelated Rayleigh fading model with i.i.d. element a H kn C (0, 1), k {1,..., K} and n {1,..., }. In thi cae, the relationhip between the channel input vector x C 1 and output vector y C K 1 i expreed a y Hx + z where z C K 1 i i.i.d. additive white Gauian noie vector with z k C (0, σz ) and noie variance of σz. It i aumed that the tranmitter ha perfect channel tate information. A vector of K ymbol u C K 1 with E[uu H ] I K are precoded uing the precoding matrix F. Then, the ignal at the tranmitter antenna i x P Γ Fu where P i the total tranmit power per tream and Γ E [ trace(ff H ) ] /K i a power normalization factor. Auming that equal power i allocated to the uer, the ergodic um-capacity of downlink channel i [9] [ C(H, P) E ( log det I K + P σz HH H)]. (1) In maive IO ytem, it ha been hown that linear precoder uch a zero-forcing (ZF) can achieve a cloe to optimal performance [10]. Applying ZF precoding matrix H H (HH H ) 1, the um-rate become [3] ( P ) R ZF K log 1 +, () Γ ZF σz where Γ ZF E [ trace ( (HH H ) 1)] /K i the power normalization factor for ZF precoder. When H follow i.i.d. Rayleigh fading model, then Γ ZF 1/( K) a for central complex Wihart matrice it wa hown that E[trace ( (HH H ) 1) K/( K) [11]. To maximize multiplexing gain in the high ignal-to-noie ratio (SR) regime, it i aumed that K where i the number of the RF chain. Figure 1(a) preent the diagram of a fully-connected antenna election tructure where each RF chain i connected to all antenna through a network of witche or phae hifter. Depending on the performance metric, e.g. maximizing the pectral efficiency, the hard antenna election chooe the bet out of antenna uing it witching network. In thi paper, a matrix that include the tate of the witche, i.e. on or off which are repreented by zero and one, i referred a the elect matrix. The diadvantage of hard antenna election i that large array gain cannot be achieved when. In general, oft antenna election technique provide a better performance compared to hard election [1], [3], [4]. However, the fully-connected tructure in Fig. 1(a) require witche or phae hifter which become very large in maive IO cenario [5]. Thi introduce high inertion loe and hardware complexity. Hence, the ubconnected configuration, a hown in Fig. 1(b), i preferred in practice. The precoder matrix F F ub F B for the tructure of Fig. 1(b) conit of a block diagonal RF beamforming matrix F ub C and a baeband precoder F B C K. The RF beamformer ha to be deigned uch that the pectral efficiency R ub i maximized ubject to F ub,nm e jθnm, θ nm [0, π) and n I m where I m { (m 1) + 1,..., m}, otherwie F ub,nm 0. In thi cae, Γ ub [ trace(f ub F H ub) ] / /. In general, oft election (hybird beamforming) i a challenging tak a the maximization of the pectral efficiency i a nonconvex problem due to the contant modulu contraint impoed by the phae hifter [1], [3] [5]. In thi paper, firtly the cloed-form expreion for an aymptotically optimal beamformer will be preented. In order to reduce the power conumption of the tructure in Fig.

4 3 Fig K F B Switch /Phae Shifter etwork (a) Fully-connected network of witche/phae hifter, 1 K 1 K F B F B (b) Subconnected tructure, (c) Subconnected tructure with fully-connected witch network. 1 K F B (d) Subconnected phae hifter and witche. Block diagram of antenna election technique. 1(b), it will be hown that the configuration of Fig. 1(c) can replace 50% of the phae hifter with witche and without a performance lo. Finally, in Fig. 1(d), we propoe a impler tructure that the complicated fully-connected witch network i replaced with low-cot 1-out-of-S witche where S i the ratio of the number output-to-input port. III. SUBCOECTED STRUCTURE WITH PHASE SHIFTERS The ingular value decompoition (SVD) of the channel matrix i denoted a H UΣV H, where V C and U C K K contain the right and left ingular vector. The diagonal matrix Σ R K include the ingular value of H. It i noted that H UΣ 1: V H 1: a H ha only nonzero S ingular value. The tatitical propertie of the element of V when wa analyzed in [3] and it wa hown that 1) V nn C (0, 1), n, n {1,..., } are i.i.d., ) V nn i a Rayleigh variable with parameter 1, 3) E[ V nn ] π/. When the impact of the beamforming matrix of the ubconnected phae hifter network i conidered, the achievable um-rate in (1) i expreed a ( R ub log det I K + P Γ ub σz HF ub F H ubh H) (3) ( log det I K + P Γ ub σz UΣV H F ub F H ubvσ H U H) ( (a) log det I K + P ΣH 1: Σ 1: V H 1: F ub F H ) ubv 1: Γ ub σz (b) ( ) log (1 + ρσmmq mm ), m1 where (a) reult from log det(i+ab) log det(i+ba), matrix Q i defined a Q VH 1: FubFH ub V 1:, and the inequality (b) come from linear algebra a for any poitive emidefinite matrix A C, det(a) m A mm. If Q i a diagonal matrix, then (b) in (3) turn into equality. Hence, F ub that upper bound R ub hould i) diagonalize Q GG H where G /V H 1: F ub. Thi require / v H m f ub,m 0 when m m and m, m {1,..., }, ii) maximize the diagonal element of GG H. In Appendix A, it i hown that thee condition are held and (3) i maximized when { e j Vnm if n I m, F ub,nm (4) 0 if n / I m. oreover, according to Appendix A, uing (4) reult in n I lim m V nm E[ V nm ] π, (5) due to the law of large number. Similarly, lim v H nf ub,m 0, m n a E[ V mn ] 0. Remark 1: The propoed RF beamformer in (4) for Fig. 1(b), i derived under the aumption that i fixed and. Hence, it provide a uboptimal olution for relatively mall value of compared to the RF beamformer of [5]. It i noted that the computational complexity of [5] grow with O( ) wherea the complexity of calculating V 1: i related to O( 3 ) [3], [1]. In our cenario of interet where i fixed and, the aociated computational complexitie of (4) and [5] are proportional to O() and O( ), repectively. In thi following, the pectral efficiency for the configuration of Fig. 1(b) will be calculated when (4) i ued. From (5), it could be eaily hown that /HF ub π/( )UΣ1:. Applying ZF to /HF ub to cancel the interference between the uer, the precoding matrix become F F ub F B where F B (HF ub ) 1. Then, the power normalization factor i [ ( )] Γ E trace F ub (HF ub ) 1 (F H ubh H ) 1 F H ub / (6)

5 4 [ ( E trace (F H ubh H ) 1 F H ubf ub (HF ub ) 1)] / [ ( E trace ( FH ubh H ) 1 ( HF ub) 1)] / 4 ( [trace π E (HH H ) 1)] 4 π Γ ZF, a lim /F H ubf ub I. Hence, the achievable umrate by the propoed hybrid beamformer i ( πp ) R ub log 1 +, (7) 4Γ ZF σz when. It i oberved that in the high SR regime, the fully-digital cheme reult in log ( π 4 ) bit/hz/ higher pectral efficiency compared to the hybrid beamforming with ubarray tructure. Remark : The preented approach to derive (7) will be ued in the ret of thi paper. Thee tep can be ummarized a 1) Diagonalization of Q. ) Ue the i.i.d. and zero-mean propertie of V nm to conclude 1/ Γ ub v H nf ub,m 0, m n. 3) Calculate lim 1/ Γ ub v H mf ub,m E[ V nn ]. 4) Calculate the power normalization factor of the hybrid beamforming, a in (6), when the RF beamformer and baeband ZF precoder are combined. 5) Replace Γ ZF in () with the power normalization factor from tep 4. IV. SUBCOECTED PHASE SHIFTER ETWORK - FUY-COECTED SWITCH ETWORKS The performance of the propoed oft election for Fig. 1(b) depend on V nm. It i noted that the phae hifter that are multiplied with maller V nm have a relatively maller contribution to the pectral efficiency. oreover, turning off uch hifter in Fig. 1(b) i equivalent to witching the correponding antenna off. Thu, the tructure of Fig. 1(c) i propoed to reduce the number of the phae hifter by employing witch network. By thi mean, the power conumption of the RF beamformer i reduced a witche require ignificantly maller power to operate compared to phae hifter [3], [6], [8]. et denote the number of the phae hifter connected to each RF chain, and α be a predefined threhold. Then, by employing a fully-connected witch network and phae hifter, the RF beamformer in (4) i modified uch that the phae hifter which are correponding to V nm α are turned off, i.e. F ub,nm 0, where α i a predefined threhold. Defining V a an i.i.d. random variable with the ame Rayleigh ditribution a V nm, then f V (α v) exp( α ) / i a meaure of the reduction in the number of the phae hifter. In the ret of the analyi, it i noted that α hould be choen carefully to make ure that,, are integer number. For the practical implementation, however, once the hardware parameter are et, the correponding α will be fixed. When the number of antenna goe large and the determinitic behavior of maive IO are oberved, f V (α v) exp( α ) / will hold. et F SF C denote the RF beamforming matrix for the ubconnected phae hifter with fully-connected witch network. For n I m, the element of F SF are expreed a { 0 if V nm α, F SF,nm exp(j V nm ) if α < (8) V nm. The received ignal power i related to 1 ΓSF v H mf SF,m where Γ SF f V (α v)/. Thi term can be obtained a a function of α v H mf SF,m ΓSF n I lim m VmnF SF,nm fv (α v)) E[Ṽ ] (a) fv (α v)) π (9) + αe α π erf(α), fv (α v)) where Ṽ i defined a { 0 if V α, Ṽ (10) V if α < V, and (a) in (9) i a conequence of emma 6 in [3]. In thi cae, (9) reult in 1/ HF SF E[Ṽ ]UΣ 1: / f V (α v)). When the impact of ZF at the baeband i conidered, the achievable rate R SF i R SF log (1 + ( π + αe α π f V (α v))γ ZF σz erf(α)) ) P. (11) It i noted that (11) i a generalization of (7) a for / (equivalently α 0), then R SF R ub. Remark 3: The propoed F SF include the effect of both ubconnected phae hifter and fully-connected witch network. In order to et the beamforming weight of the phae hifter and the elect matrix of the witche in Fig. 1(c) according to F SF, pleae refer to Appendix B. V. SUBCOECTED PHASE SHIFTER ETWORK - SUBCOECTED SWITCH ETWORKS A it will be dicued in the next ection, the performance of hybrid election with a fully-connected witch network and /() phae hifter i almot equal to the ubarray tructure. Thi i equivalent to 50% reduction in the number of phae hifter and ignificantly maller power conumption. However, employing a fully-connected witch network require a complex hardware with high inertion loe and crotalk ditortion. Hence, we evaluate the performance of the propoed hybrid beamformer when ubconnected witch network are employed. In thi tructure, a hown in Fig. 1(d), each phae hifter i connected to only one of the S adjacent antenna. In other word, the lth, l I m, phae hifter connected the mth RF chain i able to chooe one of the antenna which it index i in J q {(q 1)S + 1,..., qs}, J q I m where q {1,..., /S}. Following a imilar argument a for the phae hifter election technique, the lth phae hifter will be connected to the correponding antenna element ˆn according to ˆn arg max n J q V nm. et m J l I m be a et that contain ˆn, where it cardinality i, and ˆV be a random variable that ha the ame ditribution a max n J l V nm.

6 5 Algorithm Calculate the RF beamformer for Fig. 1(d) 1: F SS 0, : for m 1 : do 3: m, 4: I m { (m 1) + 1,..., m}, 5: for q 1 : /S do 6: J q {(l 1)S + 1,..., ls}, 7: if J q I m then 8: ˆn arg max V nm, n J q 9: F SS,ˆnm exp j Vˆnm, 10: m m {ˆn}, 11: end if 1: end for 13: end for 14: Return F SS. The RF beamforming matrix F SS C for thi cenario can be derived according to Algorithm 1. Since S and Γ SS, the received power at uer ide i related to v H mf SS 1 lim ΓSS ΓSS lim, 1 S n I m n m V mn F SS,nm (1) Vmn 1 E[ ˆV ]. S In order to calculate E[ ˆV ] + f ˆV (ˆv)ˆvdˆv, firt we calculate F ˆV (ˆv). Since ˆV i the maximum of S i.i.d. Rayleigh ditributed element when, then F ˆV (ˆv) F V (v) S (1 e ˆv ) S (13) where F V (v) 1 e V a V follow Rayleigh ditribution. Then, the pdf of ˆV i calculated a f ˆV (ˆv) d dˆv (1 e ˆv ) S Sˆv(1 e ˆv ) e ˆv (14) (b) ( ) S 1 Sˆve ˆv ( 1) e ˆv 0 ( ) S 1 Sˆv ( 1) e (+1)ˆv, 0 where (b) i the binomial expanion of (1 e ˆv ). The expected value of ˆV i expreed a E[ ˆV ] + 0 f ˆV (ˆv)ˆvdˆv (15) ( S 1 + S )( 1) ˆv e (+1)ˆv dˆv 0 0 (c) ( ) S 1 ( 1) S π ( + 1), 3/ where (c) reult from + 0 ˆv e (+1)ˆv dˆv π/4(+1) 3/ [13]. A a reult of (1) and (15), 1 HF SS UΣ 1: V H 1: F SS (16) ( ) S 1 ( 1) Sπ ( + 1) UΣ 3/ 1:. 0 The performance of the propoed ytem with ZF at the baeband can be derived following the tep in (6) and (7). In thi cae the pectral efficiency by the propoed cheme i R SS log (1 + ( 0 ( ) ( 1) (+1) 3/ ) P Sπ 4Γ ZF σ z ). (17) It i noted that (17) i a generalization of (7) a for S 1, then R SS R ub. Remark 4: The propoed F SS include the effect of the phae hifter and the ubconnected witche. In order to et the weight of the phae hifter and witche in Fig. 1(d) according to F SS, pleae refer to Appendix C. VI. SIUATIO RESUTS In thi ection, computer imulation are ued to evaluate the performance of the propoed antenna election technique for the tructure in Fig. 1(b) to Fig. 1(d). In addition, the cloed-form expreion in (7), (11) and (17) will be examined when doe not hold. onte-carlo imulation over 1000 realization for K 4 and P/σz 10 db are ued to ae the performance. At the end of thi ection, the performance of the propoed method over pare cattering channel a well a correlated Rayleigh fading channel will be examined. Figure how the tradeoff between the pectral efficiency and the total number of the phae hifter when i fixed. In order to guarantee that the propertie of maive IO are oberved and the hybrid beamformer of (4) i cloe to optimal, i et to a large number a 51. It i noted that the fully-connected witch network provide more flexibility between the number of the input and output port which i not poible with 1-out-of-S witche. When / 0.75, Fig. indicate that the fully-connected witch network with phae hifter election provide lightly higher pectral efficiency compared to the tructure of Fig. 1(b). In addition, compared to the cenario that each antenna ha a phae hifter, the number of the phae hifter can be reduced to 50% without a performance lo when a fullyconnected witch network with / 0.5 i ued. Figure alo how that when a imple binary witch i ued, i.e. S, the lo of the achievable rate i le than 1 bit/hz/ compared to oft election with ubconnected tructure. It i oberved that the propoed method with S 4, or equivalently 18 phae hifter, achieve around 93% of the pectral efficiency compared to the cenario that 51. Figure alo how that there i good match between the imulation reult and the cloed-form expreion of (7), (11) and (17) for variou ratio of the number of input to output. Figure 3 how the impact of the number of the antenna on the accuracy of the cloed-form expreion of pectral efficiency. It i aumed that the ratio of input to output i / 0.5. At 3, 1% error between the imulation reult and (11) and (17) i oberved. Thi i due to the fact that / 4 i mall and, hence, the law of

7 6 Spectral Efficiency (bit/hz/s) / Fully-connected, Sim. Fully-connected, (11) 1-out-of S, Sim. 1-out-of S, (17) Fig.. Spectral efficiency by the propoed technique veru the number of the phae hifter, 51, 4 and P/σz 10 db. Spectral Efficiency (bit/hz/s) / Fully-connected, Sim. Fully-connected, (11) 1-out-of S, Sim. 1-out-of S, (17) Fig. 3. Spectral efficiency by the propoed technique veru the number of the antenna, / 0.5, 4 and P/σz 10 db. 60 large number doe not hold. When increae to 8, the error between the imulation and analytical reult reache to around 3%. Figure 3 indicate that equation (7), (11) and (17) can provide a good approximation of the performance when 16. Figure 4 preent the achievable rate by the propoed beamformer with binary witche when P/σz varie. Compared to the tructure of Fig. 1(b) with / phae hifter, it i oberved that the performance lo due to the ue of binary witche i almot negligible at the high SR regime. In addition, Fig. 4 provide a comparion between our RF beamformer in (4) for the tructure in Fig. 1(b) and it aymptotic performance expreion (7), which wa derived under aymptotically large number of antenna, and the RF beamformer of [5]. In both cae, ZF i applied to the effective channel matrix H e 1/ Γ ub HF ub at the baeband. It i oberved that [5] achieve a contant 1.5 bit/hz/ higher pectral efficiency, which i at the cot of higher complexity, compared to the beamformer in (4). When a imple binary witch i ued and the number of the phae hifter i reduced from 18 to 64, the achievable um-rate i around.7 bit/hz/ le than [5] with 18 phae hifter. Although i.i.d. Rayleigh fading channel model i commonly ued in the literature on maive IO to preent theoretical tudie, a in [14] and reference therein, thi channel model may not be uitable for many practical application. In order to further invetigate the performance of our propoed method under more realitic channel, in the following we preent the achievable um-rate by thee technique over correlated Rayleigh fading and pare geometry-baed channel model. In order to model correlation, we ue an exponentially correlated IO channel model. We aume that the correlation effect are oberved at the bae tation according to H H w R 1/ where H w C K i zero-mean i.i.d. Rayleigh fading Spectral Efficiency (bit/hz/s) /, Sim. /, (7) /, [5] 1-out-of S, Sim. 1-out-of S, (17) P/ z Fig. 4. Spectral efficiency by the propoed technique veru P/σ z for / and /, 4 and 18. channel matrix, and R C i the correlation matrix a 1 ρ ρ 1 ρ 1 ρ R......, (18) ρ 1 ρ 1 where 0 ρ 1 i the correlation coefficient. To evaluate the performance of the propoed method over pare channel, we ue the geometry-baed model with uniform and linearly paced antenna at the bae tation. Auming there are C multipath component (PC) in the channel between the bae tation and uer k, the channel vector h T k C 1 for uer k i C h T k β ck a (φ ck ), (19) C c1 where β ck C (0, 1) i the multipath coefficient, φ ck i the angle-of-departure of the cth multipath. The teering vector

8 7 Spectral Efficiency (bit/hz/s) / Fully-connected witch 1-out-of S ref [5] Spare Channel Correlated Rayleigh Uncorrelated Rayleigh Fig. 5. Spectral efficiency by the propoed technique veru the number of the phae hifter, 51, 4. The parameter for the pare channel are P/σz 0 db and C, for the correlated Rayleigh fading r 0.7 and P/σz 10 db, for the uncorrelated Rayleigh fading P/σz 0 db. a(φ ck ) for linear array i expreed a a(φ ck ) 1 (1, e jπd λ co(φck)..., e jπd λ ( 1) co(φck) ) T (0) where φ ck [0, π], λ i the wavelength and d i the antenna pacing. In our imulation, it i aumed that d λ/. Similar to Fig., the tradeoff between the pectral efficiency and the total number of the phae hifter i tudied in Fig. 5 when different channel model are conidered. In order to make a comprehenive comparion in a ingle plot, P/σz i et to 0 db, 10 db, and 0 db to perform the computer imulation to evaluate the performance over uncorrelated Rayleigh fading, correlated Rayleigh fading and pare channel, repectively. For the pare channel model, we aume that there are only PC, i.e. C in (19), from the bae tation to each uer. For the correlated Rayleigh fading channel, it i aumed that ρ 0.7 in (18). In addition, we ue the RF beamformer of [5] a reference to evaluate the performance of our method. Figure 5 how that the RF beamformer in (4) and [5] have almot the ame performance for the pare channel, and the difference in um-rate i le than 1. bit/hz/ for the other two channel. When the tructure in Fig. 1(c) i ued and the number of the phae hifter i reduced by 50% and in the wort cae cenario, the performance of our method i 1.3 bit/hz/ lower than [5] which require 51 phae hifter. For 75% reduction, the beamformer of [5], with 51 phae hifter, ha 4.65 bit/hz/, 3.03 bit/hz/ and.55 bit/hz/ higher achievable rate compared to the pare, correlated and uncorrelated channel, repectively. Replacing the fully-connected with binary witche, the performance difference between our method and [5] become.43 bit/hz/,.86 bit/hz/ and 1.73 bit/hz/ for the pare, correlated and uncorrelated channel, repectively. VII. COCUSIO In thi paper, we invetigated the performance of hybrid beamformer when the RF beamformer conit of a combination of ubconnected phae hifter network with fullyconnected/ubconnected witch network. The propoed beamforming method and the cloed-form expreion of their pectral efficiencie were derived baed on the propertie of the ingular vector of the channel matrix when the propagation environment i modeled by Rayleigh fading. Our imulation reult indicated that the propoed method can perform well when channel parity and correlation effect are conidered. Such tructure reduce the power conumption of hybrid beamformer with phae hifter only a witche require ignificantly lower power to operate compared to the phae hifter. Specially, in maive IO ytem where the number of the required phae hifter i large. It wa hown that the fullyconnected witch network provide lightly better performance compared to the ubbonnected tructure. However, due to the implicity of the ubconnected approach and lower inertion loe and crotalk, it i preferred in practice. The power conumption of witche i negligible compared to phae hifter, a a reult, it i expected that the power conumption of the propoed method will be roughly reduced according to the reduction in the number of the phae hifter. On the other hand, uch tructure can complicate the channel etimation procedure. Hence, in future we are aiming to analyze the joint optimization of pectral and energy efficiencie to chooe the ytem parameter, i.e. number of the witche, phae hifter and antenna. In addition, further reearch i required to invetigate the impact of channel etimation when the propoed tructure are ued. APPEDIX A DIAGOAIZATIO OF Q I (3) In order to analyze Q, we invetigate the behavior of the element of G C, defined a G /V H 1: F ub. It i noted that our choice of f ub,m will reult in either cae 1: vh mf ub,m 0, or cae : vh mf ub,m 0, m m. In the firt cae that vh mf RF,m 0, m m, it could be eaily hown that all of the element of G except the G mm become zero, and (b) in (3) turn into equality. Then, G mm v H m f ub,m n I m V nm, where the equality hold when { e j Vnm if n I m, F ub,nm 0 if n / I m. n I m V nme jθnm (1) () Uing thi beamforming matrix and in the limit of large number, when, (1) become n I lim m V nm E[ V nm ] π. (3)

9 8 In the following, we analyze the impact of etting F ub, according to (), on the off-diagonal element of G. For the uncorrelated i.i.d. Rayleigh channel, the element of ingular vector of the channel matrix are zero-mean i.i.d. random variable and their phae are uniformly ditributed over [0, π] [3]. A a conequence of law of large number lim vh mf ub,m lim t n I m E[ V nm ] 0. V nm e j V nm (4) A a reult, it could be concluded that all of the element of G except the diagonal element become zero. Thi mean that chooing F ub according to () will fulfill the condition of cae 1. oreover, thi matrix can diagonalize Q and maximize it diagonal element. Hence, the achievable um-rate in (3) i maximized. APPEDIX B THE WEIGHTS FOR THE SWITCHES AD PHASE SHIFTERS OF FIGURE 1(C) In order to et the phae of the phae hifter and the elect network of the witche in Fig. 1(c) according to F SF, the following procedure can be applied: 1) et f SF C 1 a f SF,i F SF,nm, i {1,..., /}, and n I m. ) Create the vector C 1 which include all the f SF f SF f SF nonzero element of. The vector contain the beamforming weight of the phae hifter that are connected the mth RF chain. 3) et the matrix S C repreent the elect network of the fully-connected witch on the mth RF f SF chain where S f SF. Initially, et all the element of S to zero. 4) If f SF,i 0, then et S il 1, i {1,..., /} and l {1,..., }. APPEDIX C THE WEIGHTS FOR THE SWITCHES AD PHASE SHIFTERS OF FIGURE 1(D) In order to et the phae of the phae hifter and the elect network of the witche in Fig. 1(d) according to F SS, the following procedure can be applied: f (ml) SS f (ml) 1) Create C S 1 a SS, F SF,nm where n {(m 1)S + (l 1)S + 1,..., (m 1)S + ls}. et ŝ {1,..., S} denote the index of the only nonzero (ml) element of f SS. ) Set the lth phae hifter on the mth RF chain according (ml) to the nonzero element of f SS. 3) et vector (ml) C S 1, repreent the elect network of the ubconnected witch which connect the lth phae hifter on the mth RF chain to one of the nth antenna where n {(m 1)S+(l 1)S+1,..., (m 1)S+lS. 4) Except (ml) ŝ 1, et all the element of (ml) to zero. ACKOWEDGET The reearch leading to thee reult ha received funding from the European Union Seventh Framework Programme (FP7/ ) under grant agreement n (iwaves). We would alo like to acknowledge the upport of the Univerity of Surrey 5GIC member for thi work. REFERECES [1] X. Zhang, A. olich, and S.-Y. Kung, Variable-phae-hift-baed RFbaeband codeign for IO antenna election, IEEE Tranaction on Signal Proceing, vol. 53, no. 11, pp , ovember 005. [] X. Gao, O. Edfor, F. Tufveon, and E. G. aron, aive IO in real propagation environment: Do all antenna contribute equally? IEEE Tranaction on Communication, vol. 63, no. 11, pp , ovember 015. [3] S. Payami,. Ghoraihi, and. Dianati, Hybrid beamforming for large antenna array with phae hifter election, IEEE Tranaction on Wirele Communication, vol. PP, no. 99, pp. 1 1, 016. [4] O. El Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. Heath, Spatially pare precoding in millimeter wave IO ytem, IEEE Tranaction on Wirele Communication, vol. 13, no. 3, pp , arch 014. [5] X. Gao,. Dai, S. Han, C.. I, and R. W. Heath, Energy-efficient hybrid analog and digital precoding for mwave IO ytem with large antenna array, IEEE Journal on Selected Area in Communication, vol. 34, no. 4, pp , April 016. [6] R. endez-rial, C. Ruu, A. Alkhateeb,. Gonzalez-Prelcic, and R. W. Heath, Channel etimation and hybrid combining for mmwave: Phae hifter or witche? Information Theory and Application Workhop (ITA), 015, pp , February 015. [7] X. Gao, O. Edfor, F. Tufveon, and E. G. aron, ulti-witch for antenna election in maive mimo, 015 IEEE Global Communication Conference (GOBECO), pp. 1 6, Dec 015. [8] A. Alkhateeb, Y. H. am, J. Zhang, and R. W. Heath, aive IO combining with witche, IEEE Wirele Communication etter, vol. 5, no. 3, pp. 3 35, June 016. [9] C. B. Peel, B.. Hochwald, and A.. Swindlehurt, A vector-perturbation technique for near-capacity multiantenna multiuer communication-part I: channel inverion and regularization, IEEE Tranaction on Communication, vol. 53, no. 1, pp , Jan 005. [10] F. Ruek, D. Peron, B. K. au, E. G. aron, T.. arzetta, O. Edfor, and F. Tufveon, Scaling up mimo: Opportunitie and challenge with very large array, IEEE Signal Proceing agazine, vol. 30, no. 1, pp , Jan 013. [11] A.. Tulino and S. Verdú, Random matrix theory and wirele communication, ow Publiher Inc., 004. [1]. Brand, Fat low-rank modification of the thin ingular value decompoition, inear Algebra and it Application, vol. 415, no. 1, pp. 0 30, 006. [13] I. S. Gradhteyn and I.. Ryzhik, Table of integral, erie, and product, Elevier Academic Pre, Amterdam, 007. [14] T.. arzetta, E. G. aron, H. Yang, and H. Q. go, Fundamental of aive IO. Cambridge Univerity Pre, 016.