GRAM SCHMIDT BASED GREEDY HYBRID PRECODING FOR FREQUENCY SELECTIVE MILLIMETER WAVE MIMO SYSTEMS. Ahmed Alkhateeb and Robert W. Heath Jr.

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1 GRAM SCHMIDT BASED GREEDY HYBRID RECODING OR REQUENCY SELECTIVE MILLIMETER WAVE MIMO SYSTEMS Ahmed Alkhateeb ad Robert W Heath Jr The Uiversity of Texas at Austi, TX, USA, {aalkhateeb, rheath}@utexasedu ABSTRACT Hybrid aalog/digital precodig allows millimeter wave MIMO systems to leverage large atea array gais while permittig low cost ad power cosumptio hardware Most prior work has focused o hybrid precodig for arrow-bad mmwave systems MmWave systems, however, will likely operate o widebad chaels with frequecy selectivity Therefore, this paper cosiders frequecy selective hybrid precodig with beamformig vectors take from a quatized codebook or this system, a low-complexity yet earoptimal greedy algorithm is developed for the desig of the hybrid aalog/digital precoders The proposed algorithm greedily selects the beamformig vectors usig Gram-Schmidt orthogoalizatio Simulatio results show that the developed precodig desig algorithm achieves very good performace compared with the ucostraied solutios while requirig less complexity Idex Terms Millimeter wave commuicatios, frequecy selective, Gram-Schmidt, hybrid precodig INTRODUCTION Millimeter wave mmwave commuicatio ca leverage the large badwidth potetially available at the high frequecy bads to provide high data rates 5 To guaratee sufficiet received sigal power at these high frequecies, though, large atea arrays eed to be deployed at both the trasmitter ad receiver, 3, 6 Desigig precodig ad combiig matrices for these mmwave widebad large MIMO systems differs from lower-frequecy solutios This is maily due to the differet hardware costraits o the mixed sigal compoets because of their high cost ad power cosumptio 7 Therefore, developig precodig schemes for widebad mmwave systems is importat for buildig these systems or the sake of low power cosumptio, hybrid aalog/digital precodig solutios, that divide the precodig betwee aalog ad digital domais, ad hece requirig smaller umber of chais, were proposed i 8 6 or geeral MIMO systems, hybrid precodig desig with diversity ad spatial multiplexig objectives were ivestigated i 8, 9 I 0, the sparse ature of mmwave chaels was exploited, ad low-complexity iterative algorithms based o matchig pursuit were devised, assumig perfect chael kowledge at the trasmitter Extesios to the case whe oly partial chael kowledge is required was cosidered i, Other heuristic algorithms that do ot rely o orthogoal matchig pursuit were also proposed i 3 5 for the hybrid precodig desig with perfect chael kowledge at the trasmitter The work i 0 4 assumed a arrow-bad mmwave chael, with perfect or partial This work is supported i part by the Natioal Sciece oudatio uder Grat No 39556, ad by a gift from Nokia chael kowledge at the trasmitter I 6, hybrid beamformig with oly a sigle-stream trasmissio over MIMO-ODM system was cosidered The solutio i 6 though relied o the joit exhaustive search over both ad basebad codebooks which results i high-complexity As mmwave commuicatio is expected to employ broadbad chaels, developig spatial multiplexig hybrid precodig algorithms for widebad mmwave systems is importat I this paper, we ivestigate the frequecy selective hybrid precodig desig to imize the achievable mutual iformatio give that the precoders are take from a quatized codebook We first derive the optimal basebad precoders as fuctios of the precoders The, we desig a greedy hybrid precodig algorithm based o Gram-Schmidt orthogoalizatio Despite its low-complexity, the proposed algorithm is illustrated to achieve a similar performace compared with the optimal hybrid precodig desig that requires a exhaustive search over the codebooks SYSTEM MODEL Cosider the ODM based system model i ig where a basestatio BS with N BS ateas ad N chais is assumed to commuicate with a sigle mobile statio MS with N MS ateas ad N chais The BS ad MS commuicate via legth- data symbol blocks, such that N N BS ad N N MS At the trasmitter, the data symbols s k at each subcarrier k,, are first precoded usig a N digital precodig matrix k, ad the symbol blocks are trasformed to the timedomai usig N -poit IT s Note that our model assumes that all subcarriers are used ad, therefore, the data block legth is equal to the umber of subcarriers A cyclic prefix of legth D is the added to the symbol blocks before applyig the N BS N precodig It is importat to emphasize here that the precodig matrix is the same for all subcarriers This meas that the precoder is assumed to be frequecy flat while the basebad precoders ca be differet for each subcarrier The discretetime trasmitted complex basebad sigal at subcarrier k ca therefore be writte as yk ksk, where sk is the trasmitted vector at subcarrier k, such that E sksk I NS, ad is the average total trasmit power Sice is implemeted usig aalog phase shifters, its etries are of costat modulus To reflect that, we ormalize the etries m, urther, we assume that the agles of the aalog phase shifters are quatized ad have a fiite set of possible values With these assumptios, m, e jφm,, where φ m is a quatized agle The hybrid precoders are assumed to have a uitary power costrait,ie, /6/$ IEEE 3396 ICASS 06

2 -poit IT Add Chai Chai Delete -poit T of 4 I the followig propositio, we show that the basebad precoders ca be writte optimally as a fuctio of the precoders Basebad Digital recodig recoder { k } N -poit IT Add N Chai recoder N BS Digital Combiig N MS Combier N N W { W k } ig A block diagram of the ODM based BS-MS trasceiver that employs hybrid aalog/digital precodig they meet k U NBS, with the set of semi-uitary matrices U NBS { U C N BS U U I } At the MS, assumig perfect carrier ad frequecy offset sychroizatio, the received sigal is first combied i the domai usig the N MSN combiig matrix W The, the cyclic prefix is removed, ad the symbols are retured back to the frequecy domai where the symbols at each subcarrier k are combied usig the N digital combiig matrix Wk Deotig the N MS N BS chael matrix at subcarrier k as Hk, the received sigal at subcarrier k after processig ca be the expressed as recoder Basebad Chai Delete -poit T yk Wk W Hk ksk Wk W k, where k N 0, σ NI is a Gaussia oise vector 3 ROBLEM ORMULATION The paper objective is to develop a low-complexity hybrid precodig desig to imize the achievable system spectral efficiecy Give the system model i Sectio or simplicity of expositio, we will assume that the receiver ca perform optimal earest eighbor decodig based o the N MS-dimesioal received sigal with fully digital hardware This allows decouplig the trasceiver desig problem, ad focusig o the hybrid precoders desig to imize the mutual iformatio of the system 0, defied as I, {k} k log I NMS ρ Hk kk Hk, 3 k where ρ is the SNR As combiig with fully digital hardware is ot a practical mmwave solutio, the hybrid combiig de- σ sig problem eeds also to be cosidered The desig ideas that will be give i this paper for the hybrid precoders, however, provide direct tools for costructig the hybrid combiig matrices, W, {Wk} k, ad is therefore omitted due to space limitatios If the beamformig vectors are take from a codebook that captures the hardware costraits, the the imum mutual iformatio uder the give hybrid precodig model is I I, {k} k,{k} k st :,r, r,, N k U NBS N, k,,, Oe challege of the hybrid precodig desig to solve the optimizatio problem i 4 is the couplig betwee basebad ad precoders that arises i the power costrait the secod costrait 4 ropositio Defie the SVD decompositios of the matrices Hk UkΣkVk ad ΣkVk UkΣkVk, the the basebad precoders {k} k that solve 4 are give by k Vk, k,,, 5 :,: roof: The proof follows usig chage of variables It is omitted due to space limitatio, but available i the joural versio 7 Give propositio, the optimal hybrid precodig based mutual iformatio ca be give by makig a exhaustive search over oly the precodig codebook To avoid this search we propose efficiet greedy hybrid precodig algorithms i the followig sectios 4 GREEDY HYBRID RECODING A atural greedy approach to costruct the hybrid precoder is to iteratively select the N beamformig vectors from the codebook to imize the mutual iformatio I this paper, we call this the direct greedy hybrid precodig DG- algorithm Let the N BS i matrix i deote the precodig matrix at the ed of the i th iteratio The by leveragig the optimal basebad precoder structure i 5, the objective of the ith iteratio is to select f that solves with I i f λ l H k ˆ i, ˆ i, i k l ˆi,, f log ρ N ˆ i, ˆi, H k, 6 The best vector f will be the i, f added to the precodig matrix to form i The achievable mutual iformatio with this algorithm is the I DG I N The mai limitatio of this algorithm is that it still requires a exhaustive search over ad eigevalues calculatio i each iteratio I the ext sectio, we will make a first step towards developig a low-complexity algorithm that has a similar or very close performace to this DG- algorithm 5 GRAM-SCHMIDT GREEDY HYBRID RECODING I hybrid aalog/digital precodig architectures, the effective chael see at the basebad is through the precoders les This gives the ituitio that it is better for the beamformig vectors to be orthogoal or close to orthogoal, as this physically meas that the effective chael will have a better coverage over the domiat subspaces belogig to the actual chael matrix This ituitio is also cofirmed by the structure of the optimal basebad precoder discussed i ropositio, as the overall matrix has a semi-uitary structure This ote meas that i each iteratio i of the greedy hybrid precodig algorithm i 6 with a selected codeword f, the additioal mutual iformatio gai over the previous iteratios is due to the cotributio of the compoet of f that is orthogoal o the existig precodig matrix i This is similar to the greedy user schedulig i MIMO broadcast chaels based o the orthogoal 3397

3 chael compoets 8, but i a differet cotext Based o that, we modify the DG- algorithm by addig a Gram-Schmidt orthogoalizatio step i each iteratio i to project the cadidate beamformig codewords o the orthogoal complemet of the subspace spaed by the selected codewords i i This ca be simply doe by multiplyig the cadidate vectors by the projectio matrix i I i i i i i Give the optimal precoder desig i 5, the mutual iformatio at the ith iteratio of the modified Gram-Schmidt hybrid precodig GS- algorithm ca be writte as I i a f f k l i, k l log ρ λ l H k i, N i, Hk i f i, H k, 7 log ρ N λ l T i f i H k, 8 with T i H k i i i i H k, ad i, i, i f Note that T i is a costat matrix at iteratio i, ad a follows from the Gram-Schmidt orthogoalizatio which allows the matrix i, i, i, to i be writte as, i f Hece, i i the eigevalues calculatio i 8 ca be calculated as a rak- update of the previous iteratio eigevalues, which reduces the overall complexity 9 The best vector f will be the added to the precodig matrix to form i i, f At the ed of the N iteratios, we get I GS I N I the followig propositio, we prove that this Gram-Schmidt hybrid precodig algorithm is exactly equivalet to the DG- algorithm ropositio The achieved mutual iformatio of the direct greedy hybrid precodig algorithm i 6 ad the Gram-Schmidt hybrid precodig algorithm i 7 are exactly equal, ie, I DG I GS roof: See Sectio 9 6 AROXIMATE GRAM-SCHMIDT BASED GREEDY HYBRID RECODING The mai advatage of the Gram-Schmidt hybrid precodig desig i Sectio 5 is that it leads to a ear-optimal low-complexity desig of the frequecy selective hybrid precodig as will be discussed i this sectio Give the optimal basebad precodig solutio i 5, the mutual iformatio at the ith iteratio i 7 ca be writte as I i a f f k k l l i, log ρ λ l H k i, N i, i, log ρ λ l Σk Ṽ k H k, 9 Algorithm Approximate Gram-Schmidt Greedy Hybrid recodig Iitializatio Costruct Π Σ H Ṽ H, with Σ H Σ,, Σ ad Ṽ H Ṽ,, Ṽ Set Empty Matrix Set A CB f,, fn CB v, where f,,, NCB v are the codewords i recoder Desig or i, i,, N a Ψ Π A CB b Ψ:, arg,,n v CB c i i f d Π Π I i i i i i Digital recoder Desig 3 k N N N Vk, k :,:,,, with Vk defied i 5 i, b i, i, i, log I ρ Σk tr Σk k f ΣkṼk i, k Ṽk Σ k, 0 i, i, where a is by cosiderig oly the first domiat sigular values of Hk, Σk Σk:,:NS, Ṽk Vk :,:NS, ad b follows from usig the large mmwave MIMO approximatios used i 0 The objective of the ith iteratio is the to solve f arg f ΣkṼk i, k arg Σ H ṼH i f f i i, i, i, 3 a arg Σ H Ṽ i f, 4 where Σ H Σ,, Σ, ṼH Ṽ,, Ṽ, ad a is a result of the Gram-Schmidt processig as described i Sectio 5 The problem i 4 is simple to solve with just a imum projectio step We call this algorithm the approximate Gram-Schmidt hybrid precodig Approximate GS- algorithm As show i Algorithm, the developed algorithm sequetially build the ad basebad precodig matrices i two separate stages irst, the beamformig vectors are iteratively selected to solve 4 The, the basebad precoder is optimally desiged accordig to 5 Despite its sequetial desig of the ad basebad precoders, which reduces the complexity whe compared with prior solutios that mostly deped o the joit desig of the basebad ad precodig matrices 0,, Algorithm achieves a sigificat gai over prior solutios, ad gives a very close performace to the optimal solutio give by exhaustive search, as will be show i Sectio

4 Spectral Efficiecy bps/ Hz Ucostraied recodig erfect CSI Optimal Hybrid recodig Direct Greedy Hybrid recodig Gram Schmidt Hybrid recodig Approximate GS Algorithm Hybrid recodig Desig i SNR db ig The performace of the approximate Gram-Schmidt hybrid precodig desig i Algorithm compared with the optimal hybrid precodig solutio, the ucostraied SVD solutio, ad the prior work i 0 The system has N BS 3 ateas, N MS 6 ateas, ad N 3 7 SIMULATION RESULTS I this sectio, we evaluate the performace of the proposed algorithm usig umerical simulatios We adopt a widebad mmwave chael model that cosists of L 6 clusters The ceter AoAs/AoDs of the L clusters θ l, φ l are assumed to be uiformly distributed i 0, π Each cluster has R l 5 rays with Laplacia distributed AoAs/AoDs 0,0, ad agle spread of 0 o The umber of system subcarriers equals 5, ad the cyclic prefix legth is D 8, which is similar to 80ad The paths delay is uiformly distributed i 0, DT s Both the BS ad MS have ULAs with N 3 I ig, we validate the result i ropositio, i additio to evaluatig the approximate Gram-Schmidt based hybrid precodig algorithm The spectral efficiecies achieved by these greedy algorithms are compared with the optimal hybrid precodig desig give by the exhaustive search over the codebooks The rates are also compared with the prior solutio i 0 or a fair compariso, we assume that each beamformig vector is selected from a beamsteerig codebook with a size N CB 64 irst, ig shows that the direct greedy ad Gram-Schmidt based hybrid precodig algorithms achieve exactly the same performace which verifies ropositio Despite its low-complexity, the developed approximate Gram-Schmidt hybrid precodig desig i Algorithm achieves very close performace to the exhaustive-search based optimal solutio We emphasize here that ay hybrid precodig desig ca ot perform better that the show optimal hybrid precodig solutio with the cosidered codebook, which cofirms the earoptimal result of the proposed algorithm This is also clear from the cosiderable gai obtaied by the proposed algorithm compared with the prior solutio i 0 Also, it is worth metioig that the developed hybrid precodig algorithms i this paper ca be applied to ay large MIMO system ot specifically mmwave systems 8 CONCLUSION I this paper, we ivestigated hybrid precodig desig for widebad mmwave systems irst, we derived the optimal hybrid precodig desig that imizes the achievable mutual iformatio for ay give codebook, ad showed that the optimal basebad structure ca be decomposed ito a precoder depedet matrix ad a uitary matrix Secod, we developed a ovel greedy hybrid precodig algorithm based o Gram-Schmidt orthogoalizatio Thaks to this Gram-Schmidt orthogoalizatio, we showed that oly sequetial desig of the ad basebad precoders is required to achieve the same performace of more sophisticated algorithms that requires a joit desig of the ad basebad precoders i each step Simulatio results illustrated that the proposed precodig algorithms improve over prior work ad stay withi a small gap from the ucostraied perfect chael kowledge solutios 9 ROO O ROOSITION roof: To prove that I GS I DG, it is sufficiet to prove that N of the GS- ad DG- algorithms are equal To do that, we will show that both the algorithms choose the same beamformig vector i each iteratio, ie, i is equal for i,, N This ca be proved usig mathematical iductio as follows At the first iteratio, the two algorithms do the exhaustive search over the same codebook, ad cosequetly select the same beamformig vectors Now, suppose that the two algorithms reach the same precodig matrix i at iteratio i, we eed to prove that they both select the same beamformig vector at iteratio i, ie, we eed to prove that both 6 ad 7 choose beamformig vectors with the same idex To prove that, it is eough to show that the cotributios of the th beamformig vector f from v i 6 ad 7 are equalgive the optimal basebad precoder i 5, ad deotig the SVD of as ˆ i, l Ûi, ˆΣ i, i, ˆV log ρ λ l Hkˆ i, l, equatio 6 ca be writte as ˆi, ˆ i, ˆ i, ˆ i, Hk, 5 log ρ λ l N HkÛi, Ûi, Hk 6 S Equatio 7 ca be similarly writte, but with Ûi, U i, where i, prove that Ûi, Ûi, U i, U i, Σ i, V i, U i, deote the last colum of i, Gram-Schmidt operatios, we ca write f replaced by Hece, we eed to Let f i f As f is a result of successive f i α, where α is a vector results from the Gram-Schmidt process Cosequetly, ˆi, i, ca be writte as ˆ i, E C, I α where E C 0 T is a elemetary colum operatio matrix i, E CE C matrix Now, we ote that Û i, Ûi, i, U i, U i, ˆ i, i, ˆ, as E C is a i i full-rak 3399

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