Millimeter Wave Hybrid Beamforming with DFT-MUB Aided Precoder Codebook Design

Transcription

1 Mllmeter Wave Hybrd Beamformng wth DFT-MUB Aded Precoder Codebook Desgn K Satyanarayana, Mohammed El-Hajjar, Png-Heng Kuo, Alan Mourad, Lajos Hanzo Department of Electroncs and Computer Scence, Unversty of Southampton, UK InterDgtal, Europe, London, UK Emal: {ksr5, meh, lh}@ecssotonacuk, {png-hengkuo, alanmourad}@interdgtalcom Abstract The mllmeter wave (mmwave) frequency band offers substantal htherto unused spectral resources for future wreless communcaton systems n order to meet the ncreasng capacty demand However, mmwave frequences suffer from hgh propagaton losses, whch may be mtgated by drectonal beamformng n addton to baseband precodng Ths s usually referred to as hybrd beamformng In ths paper, we nvestgate the so-called dscrete Fourer transform-mutually unbased bases (DFT-MUB) aded codebook desgonceved for lmtedfeedback mmwave systems, where the MUB aded codebook s appled n the baseband, whle the DFT codebook s nvoked for analog phase shfters We demonstrate that our DFT- MUB codebook desgn performs smlarly to the optmal dgtal precodng matrx, where the precoder s selected as the rght sngular matrx of the channel However, our soluton sgnfcantly reduces the search complexty n the baseband, whle performng wthn 25 db from the optmal dgtal precoder I INTRODUCTION Recent studes show a dearth of spectral resources to meet the ncreasng data rate demands of moble users n the mcrowave band [] Hence next generaton systems am for addtonally harnessng mmwave frequences [2] as a beneft of the abundant avalablty of bandwdth However, mmwave frequences suffer from a hgh pathloss because of the attenuaton due to water vapour, oxygen absorpton and folage As a result, the sgnal-to-nose (SNR) rato n mmwave communcatons would typcally be low at the recever [2] To compensate the propagaton losses, drectonal beamformng s consdered to be a promsng soluton [3] Furthermore, snce the λ/2 spacng between the antennas would be on the order of mllmeters at mmwave frequences, large number of antennas can be accommodated n a compact space for achevng hgh beamformng gans Employng dgtal beamformng usng a large number of antennas, where both the ampltude and phase are controlled dgtally usng ADCs/DACs, would result n a hgh cost and hardware complexty [4] On the other hand, employng analog beamformng, where the antenna array s connected to an chan through amplfers and controllable analog phase shfters, reduces the hardware complexty sgnfcantly compared to the dgtal beamformng However, n analog beamformng, the angular resoluton s typcally naccurate [2] Hence hybrd beamformng combnng analog and dgtal beamformng for mmwave systems was conceved n [5], where the ensung dgtal processng The fnancal support of InterDgtal, UK and of Beam-Me-Up project s gratefully acknowledged s capable of correctng the analog mperfectons In hybrd solutons, analog beamformng s carred out at the front end, where the dgtally precoded sgnals are phase-shfted and then fed to the transmt antennas Smlarly, at the recever front end the receved sgnals are phase-shfted and then fed to the dgtal combner operatng n the baseband Two dfferent hybrd beamformng desgns have been reported n [6], namely a so-called full-array-connected desgn and, subarray-connected desgn [6] In the full-array-connected desgn the dgtal precoded sgnals are phase-shfted and then fed to all the transmt antennas By contrast, n the sub-arrayconnected desgn, after phase shftng the dgtally precoded sgnals are fed only to a subset of the antennas Hence, the sub-array-connected mplementaton s more energy effcent than the full-array-connected desgn [6] In frequency-dvson duplex (FDD) systems, the recever has to feed back the channel estmaton to the transmtter for the transmt precoder (TPC) to compensate the channel-effects about to be encountered However, feedng back the channel coeffcents to the transmtter would result n large overhead An appealng practcal soluton for feedng back the channel nformaton to the transmtter s based on a carefully desgned TPC quantzed weght codebook, whch s known both to the transmtter and the recever Explctly, based upon the channel mpulse response (CIR) estmate the recever selects the best TPC matrx from ts codebook, and reports the selected ndex to the transmtter over the feedback channel Ths technque s referred to as lmted-feedback desgn Song et al [7] proposed a codebook desgn for mmwave systems usng the algorthm referred to as orthogonal matchng pursut (OMP) As a futher soluton, Rajashekar and Hanzo [8] desgned a codebook desgn usng vector quantzaton conceved for fnte modulaton alphabets In [9], Alkhateeb and Heath proposed codebook desgns for lmted feedback mmwave systems By explotng the channels recprocty, a heurstc codebook desgn was advocated by Eltayeb et al [0] The effcency of a practcal communcaton system crtcally hnges on ts performance vs complexty tradeoff Moreover, the aforementoned desgns mmwave system desgns mpose a hgh complexty The performance of a practcal codebook desgn depends on the choce of the codebook entres, as detaled by Clerckx et al n [] Aganst ths background, n ths contrbuton, we conceve a lowcomplexty hybrd beamformng codebook desgn

2 More explctly, we desgn a codebook for dgtal beamformng based on mutually unbased bases (MUBs) combned wth dscrete Fourer transform (DFT) based analog beamformng for a lmted-feedback aded mmwave MIMO system The MUB asssted codebook mposes lower search complexty than the exstng dgtal beamformng desgns wthout compromsng the attanable performance A MUB asssted codebook was desgned n [2], [3] for a sub-6 GHz scenaro By contrast, we desgn a codebook based on MUBs combned wth DFT aded analog beamformng for a lmted-feedback mmwave MIMO system The ratonale behnd employng the DFT for analog beamformng s ts hgh effcency n spatally correlated channels [4] Our smulaton results demonstrate that despte of ts low complexty, the proposed MUB-DFT codebook desgn performs close to the equvalent benchmarker applyng dgtal precodng and relyng on the smplfyng assumpton of havng perfect channel state nformaton (CSI) The rest of the paper s organzed as follows In Sec II, we dscuss our system model as well as the desgn of the transmtter and recever n addton to the mmwave channel model In Sec III, we frst dscuss the DFT aded codebook desgonceved for analog beamformng Later we focus our attenton on MUB asssted codebook desgn for the baseband Our smulaton results and conclusons are presented n Sec IV and Sec V, respectvely Notatons: We use upper case boldface, A, for matrces and lower case boldface, a, for vectors We use < >, T, H, for the nner product, transpose, Hermtan transpose and elementwse multplcaton, respectvely Fnally, we use CN, U, and d to denote complex-valued normal dstrbuton, unform dstrbuton, and ndependent and dentcal dstrbuton, respectvely II SYSTEM MODEL Consder a sngle-user mmwave MIMO system shown n Fg, where the transmtter s equpped wth N t antennas and the recever wth N r antennas Then the receved sgnal vector after and baseband combnng s gven by: y = P W H BBW H HF F BB s W H BBW H n, () where H s the statstcal channel model expressed as [5]: N N r N c N t ray H = αn nray N c N c a r (φ nray )a T t (φ nray ), (2) ray = n ray= and the transmtter s equpped wth chans and the recever wth chans, where F s the beamformer matrx of sze N t, F BB s the baseband TPC matrx of sze N s, W H s the combner matrx of sze N r, W H BB s the baseband combner of sze N s, y s the N s receved vector, s s the symbol vector of sze N s, where N s < and n s the nose vector of d CN (0, σ 2 ) Furthermore, H s the statstcal channel matrx of sze N r N t so that E[ H 2 F ] = N tn r, whle αn nray c CN (0, ) s a complex-valued Gaussan random varable, whose ampltude and phase are Raylegh and unform dstrbuted, respectvely For a unform lnear array (ULA) wth N r and N t antenna elements the response vectors a r and a t are expressed as: a r (φ r ) = [ e j 2π λ d cos(φr) e j 2π λ ( )d cos(φr) ] T, (3) a t (φ t ) = [ e j 2π λ d cos(φt) e j 2π λ ( )d cos(φt) ] T (4) Fnally, N c and N ray are the number of CIR clusters and rays, respectvely Ns FBB Baseband Precodng F Analog Precodng mmwave Channel Nc, ay H W Analog Combnng WBB Ns Baseband Combnng Fg : Hybrd beamformng transcever archtecture In mmwave communcaton, F and W are used for analog beamformng, where the beams are steered n the drecton of the desred user by phase shfters, whle F BB and W BB are the baseband precoder and combner weghts used for mtgatng the nter-stream nterference Wth the ad of ths system model, the achevable rate n bts per second per channel use (bps/cu) s gven by: det C = log 2 (INs (5) ) P, R n W H N BBW H HF F BB F H BBF H H H W W BB s where R n = σ 2 W H BBW H W W BB, and P s the sgnal power III CODEBOOK DESIGN Fg shows the transcever desgn relyng both on a TPC and on analog beamformng In ths desgn, before the streams are fed to the transmt antennas, they are precoded n the baseband usng a lnear TPC and then the precoded streams are fed to the beamformer, where the streams are phase shfted accordngly and fnally fed to the transmt antennas In ths secton, we dscuss the desgn of beamformng based on the DFT, where the antenna element weghts are selected for steerng the beam n the desred drecton Then, we descrbe the constructon of the baseband TPC codebook relyng on MUBs A Analog Beamformer Desgn Usng DFT To derve the beamformng gans so as to compensate for the propagaton losses due to water vapour, oxygen absorpton and folage, the antennas are placed at a separaton of λ/2 We consder the DFT based codebook for beamformng, because the codewords n the DFT based beamformng codebook match wth the statstcal dstrbuton of the optmal beamformng weght vectors, whch makes the DFT based codebook more effcent n spatally correlated channels [4] Furthermore, t satsfes the constant modulus constrant mposed by the phase shfters to avod any power mbalance for

3 the system Therefore, t s reasonable to employ a DFT based codebook for analog beamformng usng phase shfters On the other hand, for the baseband TPC matrx, we nvoke a codebook desgonstructed from MUBs as a beneft of ts effcency and low complexty, whch wll be detaled later n ths secton The columns of the analog beamformer matrx F are selected from the N t N t DFT matrx whch exhbts maxmum correlaton wth the columns of the rght sngular matrx of the channel H=UΣV H, e, F (:, ) = max <DFT (:, ), V >, N t, (6) where V s the rght sngular matrx of the channel and DFT (:, ) s the th column of the N t N t DFT matrx Smlarly, the columns of W are selected from the N r N r DFT matrx whch exhbts maxmum correlaton wth the columns of the left sngular matrx of the channel H, e, W (:, ) = max <DFT (:, ), U>, N r, (7) where U s the left sngular matrx of the channel and DFT (:, ) s the th column of the N r N r DFT matrx After applyng the analog beamformer weghts, the channel seen by the baseband TPC after beamformng s the effectve channel gven by: H eff = W H HF We now desgn the baseband codebook usng MUBs descrbed n the next secton B Baseband Codebook Desgn Usng MUBs The theory of MUB and the assocated codebook desgn are descrbed n ths secton Let B and B be a par of orthonormal bases of an N-dmensonal Hlbert space Then, B and B are sad to be mutually unbased f and only f [6]: < b, b > 2 = N, b B and b B, (8) where b and b are vectors n B and B, respectvely The codebook constructed from MUBs ensures havng low dstances between the codewords [2] However, the number of MUBs for a gven dmenson s lmted, whch affects the sze of the codebook Furthermore, the number of antenna elements should be a power of a prme [6], then the number of MUBs avalable for dmenson s [6] Moreover, f the number of chans s a power of 2, then the entres of the MUB matrx s observed to be composed of fnte alphabets e,{,,, } Addtonally, f {F,, F N } are the codewords, then the mnmum dstance d mn between the codewords s gven by: d mn = mn F kf H k F l F H l, (9) k,l;k l whch s close to the theoretcal bound [2] Codebook constructon: For a dmenson of = 2 n, where n s a natural number, we commence our dscourse from a Hadamard matrx A of dmenson 2 n, whch satsfes the property of a MUB Usng the Hadamard matrx A, we caonstruct another bass B by fndng a column vector v, so that all the elements n A H v have a magntude equal to Then we obtan the bass B by element-wse multplcaton of v wth A, e, f A = [a a 2 a 2 n] then B = v A 2 n = [va va 2 va 2 n] We contnue ths procedure untl all the 2 n bases are obtaned [2] For example, we consder the scenaro where the transmtter s equpped wth = 4 chans For = 4, the Hadamard matrx A s gven by: A = 2 (0) The factor /2 s to ensure that (0) satsfes (8) Now the vector v s chosen such that the magntude of all elements of A H v s Solvng for the vectors results n three possble vectors, whch are: v =, v 2 =, v 3 = Then, the set of MUBs s gven by: B = v A, C = v 2 A, and D = v 3 A Gven that Hadamard matrx A n (0) and by applyng element-wse operaton on A wth vectors v, v 2, and v 3, we arrve at B = 2, C = 2, D = 2 Note that the dentty matrx s also an MUB, snce t satsfes (8) Thus, we have obtaned = 5 MUBs However, snce the dentty matrx does not explctly perform any lnear processng of the nput streams, t can be dscarded n order to reduce the feedback overhead Nonetheless, to leverage any benefts offered by the dentty matrx, the transmtter can always keep a check on t by comparng ts performance to that of non-trval MUBs performance The left crcular shft of the columns of each matrx A, B, C and D by p modulo 4 places, where p =, 2, 3, 4, forms the codebook entres, wth cardnalty C = 6, relyng on a 4-bt feedback Thus, the set of TPC matrces of the codebook F after employng the left crcular shft by p modulo 4 places to each matrx, the codebook can be expressed as F = {A 0, A, A 2, A 3, B 0, B, B 2, B 3, C 0, C, C 2, C 3, D 0, D, D 2, D 3 } An alternate method for constructon of MUBs s presented n [6]

4 Havng obtaned the codebook F, we select the frst N s columns of the specfc TPC matrx from the codebook whch maxmzes the mnmum SNR 2 of the effectve channel The SNR of the k th stream usng a ZF recever s gven by [7]: P SNR k = N s N o (F BBH effh eff F BB ), () kk where P and N o are the sgnal and the nose powers, respectvely Thus, the TPC F BB whch maxmzes the mnmum SNR s selected from the codebook F as follows: F BB = arg max F BB F Λ mn{h eff F BB }, (2) where Λ mn s the mnmum sngular value of H eff F BB Snce the mmwave channel of the adjacent antenna elements s correlated due to ther λ/2 spacng, employng MUB baseband TPC amalgamated wth DFT aded beamformng sgnfcantly reduces the complexty, whle havng an mproved performance n spatally correlated channels The pseudo code for the proposed desgn s shown below Algorthm Proposed and Baseband Codebook Desgn Acqure codebook F and store n memory : Obtan vectors v from Hadamard matrx A N such that t magntude of all elements of A H v = ; 2: Obtan MUBs by element-wse multplcaton of v wth A, ; 3: Obtan the codebook F by left crcular shft of each MUB to p places, where p =,, ; Selecton of TPC from the codebook : We have H = UΣV H ; 2: Obtaolumns of F and W from DFT such that F = max < DFT (:, ), V >,, and W = max < DFT (:, ), U >, N r ; 3: Set the effectve channel to H eff = W H HF ; 4: Select the TPC from the codebook whch satsfes: F BB = arg max FBB F Λ mn {H eff F BB }; 5: W BB s chosen as ZF combner C Complexty and Propertes of the MUB Codebook The codebook search complexty s quantfed n terms of the number of computatons requred for fndng the best TPC matrx To fnd the best TPC from the codebook we have to perform ) complex-valued addtons [3], where C s the cardnalty of the codebook, but no complex multplcatons are requred On the other hand, codebooks such as the Grassmannan and Fourer desgns would requre CN s ( ) 2 complex multplcatons plus C( ) 2 (N s ) addtons [7], CN r ( whle a typcal dgtal TPC operatng n the baseband relyng on perfect CSI, such as [8] usng the same number of chans would need N s ( ) complex-valued addtons and N s multplcatons Thus, the MUB based codebook sgnfcantly reduces the overall search complexty Furthermore, the entres of the codebook are constructed from a fnte alphabet X = {,,, }, whch meets the 2 Mnmum SNR of all lnks between each Tx and Rx antenna constant modulus constrant, and hence avods any potental power mbalance for the system Addtonally, the codebook exhbts a nested property, whch means that the computatons that are performed for hgher-rank transmssoan be reused for lower-rank transmsson In other words, the rank TPC s a subset of the rank 2 TPC, the rank 2 TPC s a subset of the rank 3 and the rank 3 TPC s a subset of the rank 4 TPC [2] as llustrated below Rank Rank 2 Rank 3 Rank 4 (3) IV SIMULATION RESULTS In ths secton, we present our smulaton results for characterzng both the performance and the achevable rate when the DFT-MUB based and baseband precodng s employed We performed Monte Carlo smulatons for performance comparson between the baseband TPC based on SVD of the channel and the hybrd precodng relyng on lmted-feedback The system confguraton and parameters, such as the number of chans, antennas, and modulaton schemes are shown n Table I TABLE I: Smulaton parameters Parameters Values Modulaton 4 QAM N c 4 N ray 6 N t 32, 8 N r 6, 8 N s φ n ray U[0, 2π) Fg 2 plots the achevable rates of dgtal precodng usng SVD, of OMP [8], of DFT-MUB, and DFT-Identty, where the dentty matrx s chosen as the baseband TPC, usng (5) for a 8 8 and 32 6 MIMO sytem In ths fgure, the DFT-MUB based codebook desgn usng 4-bt feedback s nvestgated In ths confguraton, two spatal streams were transmtted usng 4 chans both at the transmtter and at the recever It s seen from Fg 2 that the performance gap between the SVD based TPC and the DFT-MUB aded codebook s around 25 db for the 32 6 and as low as 08 db for the 8 8 MIMO confguratons consdered Furthermore, there s a gap of around db aganst the OMP aded precodng desgn for a 32 6 MIMO, whereas for an 8 8 MIMO the performance of the DFT-MUB overlaps wth the OMP We also compared our results aganst the DFT-Identty based system, where the TPC s constructed by selectng frst N s columns of the dentty matrx Aganst the DFT-Identty aded system, there s an approxmately 05 db gan when DFT-MUB s performed Addtonally, t s mportant to quantfy the complexty of dfferent desgns consdered For = 4 and = 4, the MUB based TPC would need only 92 complex addtons On the other hand, the Grassmannan and Fourer codebooks based TPC would need 52 complex addtons plus 256 complex multplcatons, as descrbed n Sec III-C Although the TPC relyng on perfect CSI such as OMP [8] would requre 24

5 Achevable Rate (bps/hz) Unconstraned Precodng (SVD) Orthogonal Matchng Pursut DFT-MUB Codebook DFT-Identty Sold lne: 32 6 MIMO Dashed lne: 8 8 MIMO 32 6 MIMO 8 8 MIMO SNR [db] Fg 2: Comparson of achevable rate usng DFT-MUB based codebook desgn wth 4-bt feedback and dfferent other methods relyng on perfect CSI for 32 6 and 8 8 MIMO confguratons In ths confguraton, we have N s = 2 and = = 4 complex addtons plus 32 complex multplcatons, t mposes a hgh overhead as t depends on perfect CSI Furthermore, the SVD based TPC reles completely on dgtal processng elements, where DACs and automatc gaontrols (AGCs) are needed for each antenna element, whch mposes a hgh power consumpton and hgh hardware complexty Bt Error Rate Unconstraned Precodng (SVD) DFT-MUB Codebook DFT-Identty Orthogonal Matchng Pursut SNR [db] Fg 3: Comparson of uncoded bt error rate (BER) of a 8 8 MIMO between on DFT-MUB based codebook desgn wth 4-bt feedback and dfferent other methods relyng on perfect CSI In ths confguraton, we have N s = 2 and = = 4 We also performed Monte Carlo smulatons to nvestgate the attanable uncoded bt error rate (BER) performance for an 8 8 MIMO system usng 4 QAM In ths confguraton, two spatal streams usng 4 QAM were transmtted over = 4 and = 4 chans Fg 3 shows the comparson between the BERs of the DFT-MUB aded codebook desgn usng 4-bt feedback, of the SVD based TPC, of the DFT-Identty and of OMP It s seen that at low SNRs the DFT-MUB codebook desgn outperforms the SVD based TPC It s nstructve to note that the SVD based TPC s sub-optmal n terms of BER Furthermore, at the BER of 0 3 the performance gap between the DFT-MUB based desgn and the SVD desgn s around 25 db, despte the DFT-MUB scheme s constant modulus nature and sgnfcantly lower complexty DFT-MUB also outperforms the DFT-Identty and OMP desgns V CONCLUSIONS In ths paper, we nvestgated a hybrd codebook desgn based on the combnaton of the DFT n the and MUB n the baseband for mmwave MIMO systems We demonstrated that the proposed desgn performs close to the optmal unconstraned precoder (SVD) n terms of achevable rate Furthermore, our desgn has lower BER than that of OMP Addtonally, our system performs wthn 25 db from the SVD based precodng wth sgnfcantly lower complexty than orthogonal matchng pursut and SVD REFERENCES [] A Ghosh, R Ratasuk, B Mondal, N Mangalvedhe, and T Thomas, Lte-advanced: next-generaton wreless broadband technology, IEEE Wreless Communcatons, vol 7, no 3, pp 0 22, June 200 [2] Y Nu et al, A survey of mllmeter wave (mmwave) communcatons for 5G: Opportuntes and challenges, CoRR, vol abs/ , 205 [3] G Maccartney, M Samm, and T Rappaport, Explotng drectonalty for mllmeter-wave wreless system mprovement, n IEEE Internatonal Conference on Communcatons (ICC), June 205, pp [4] B Le et al, Analog-to-dgtal converters, IEEE Sgnal Processng Magazne, vol 22, no 6, pp 69 77, 2005 [5] X Zhang, A F Molsch, and S-Y Kung, Varable-phase-shft-based -baseband codesgn for MIMO antenna selecton, 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