Compressive Sensing (CS)-Assisted Low-Complexity Beamspace Hybrid Precoding for Millimeter-Wave MIMO Systems

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1 Compessve ensng (C-Asssed ow-complexy Beamspace yd Pecodng fo Mllmee-Wave MIMO ysems Chang-en Chen, Cheng-Rung Tsa, uden Meme, IEEE, Yu-sn u, We-un ung, and An- Yeu (Andy Wu, ellow, IEEE Asac yd analog/dgal pecodng s a pomsng echnque o educe he hadwae cos of R componens compaed o he convenonal full-dgal pecodng appoach n mllmee-wave (mmwave mulple-npu mulple oupu (MIMO sysems. oweve, he lage anenna dmensons of he hyd pecode desgn makes dffcul o acque an opmal full-dgal pecode. Moeove, also eques max nveson, whch leads o hgh complexy n he hyd pecode desgn. In hs pape, we popose a low-complexy opmal full-dgal pecode acquson algohm, named eamspace-vd ha saves powe fo he ase saon (B and use equpmen (UE. We explo educed-dmenson eamspace channel sae nfomaon (CI gven y Compessve ensng (C-ased channel esmaos. Then, we popose a C-asssed eamspace hyd pecodng (C-BP algohm ha leveages C-ased CI. mulaon esuls show ha he poposed eamspace-vd educes complexy y 99.4% compaed o an opmal full-dgal pecode acquson usng full-dmenson VD. uhemoe, he poposed C-BP educes he complexy of he sae-of-he-a appoach y 99.6% and has less han 5% pefomance loss compaed o an opmal full-dgal pecode. Index Tems MIMO, mllmee-wave communcaon, hyd pecodng, eamfomng, low-complexy. I. ITRODUCTIO E mllmee-wave (mmwave mulple-npu mulple Toupu (MIMO sysem s a pomsng echnology o allow nex-geneaon mole communcaons acheve gga-pesecond daa aes fo fuue ndoo and oudoo communcaons []-[5]. Convenonal MIMO sysems ae ealzed enely a aseand (. oweve, he facaon cos and enegy consumpon of hgh-fequency mxed sgnal componens make full-dgal pecodes mpaccal due o he lage nume of ado-fequency (R chans n mmwave MIMO sysems wh ens-o-hundeds of anennas. To addess he R chan ssues, a moe effcen hyd analog/dgal sucue s poposed n [6]-[9]. g. shows a hyd sucue, whee he aay sgnal pocessng s Ths wok was fnancally suppoed y he Mnsy of cence and Technology of Tawan unde Gans MOT -6-E--4 and I was also sponsoed y MedaTek Inc., sn-chu, Tawan, and OVATEK ellowshp. The auhos ae wh he Gaduae Insue of Eleconcs Engneeng and Depamen of Eleccal Engneeng, aonal Tawan Unvesy, Tape, 67, Tawan (e-mal: {james, aulshephed, mke, syvane}@access.ee.nu.edu.w; andywu@nu.edu.w. g.. ysem lock dagam of a mmwave MIMO sysem usng a hyd analog/dgal sucue a oh he ansme and eceve. paoned no a R pecode cascaded wh a aseand pecode. The R pecode conols he phases of he sgnals ha avel no and ou of he anenna elemens va pue analog phase shfes o geneae seveal eams owad he domnan pahs n mmwave channel. Meanwhle, he pecode povdes an addonal level of flexly ove he consangan/phase-only opeaons a he R pecode. Due o he spase naue of mmwave channels [6], he nume of R chans equed fo he hyd sucue s much lowe han he nume of anennas whle achevng a smla pefomance wh a full-dgal pecode. oweve, hese soluons [6]-[9] have wo majo ssues n pacce: gh complexy fo acqung an opmal full-dgal pecode: Desgnng a hyd pecode eques an opmal full-dgal pecode, whch s usually oaned fom he domnan sngula vecos of an explc spaal doman channel max. In mmwave MIMO, he sngula value decomposon (VD compuaons of he explc channel max ae complcaed due o he lage nume of anennas. Also, hyd pecode desgns ae usually done a he eceve o educe feedack ovehead [6]. Theefoe, he mplemenaon of VD ecomes nfeasle. In addon, [6]- [9] also eque full channel sae nfomaon (CI o calculae he opmal full-dgal pecode. A hgh anng ovehead fo explc channel esmaon s expeced due o he hgh-dmensonal space n mmwave MIMO X (c 6 IEEE. Pesonal use s pemed, u epulcaon/edsuon eques IEEE pemsson. ee hp:// fo moe nfomaon.

2 gh complexy fo he hyd pecode desgn: Afe acqung he opmal full-dgal pecode, he hyd pecode desgn can e fomulaed as a spase opmzaon polem n [6]. The hyd pecode desgn can hen e solved y smulaneous ohogonal machng pusu (OMP [], a modfed mulple-measuemen-veco (MMV-solve of ohogonal machng pusu (OMP []. Ths appoach can acheve nea-opmal pefomance. oweve, OMP eques max nveson fo calculang he opmal aseand pecode, whch leads o hgh complexy fom a hadwae pespecve. ence, seveal appoaches [8]-[9] have een poposed o smplfy he max nveson compuaon and o educe he eaons n machng pusu y applyng he chu-banachewcz lockwse nveson n []. eveheless, [8]-[9] have a longe calculaon laency fo he eave max updang of lockwse nveson. In summay, [6]-[9] canno compleely avod he max nveson calculaons. In hs pape, we popose a novel hyd pecode ased on Compessve ensng (C-asssed eamspace CI fo mmwave MIMO sysems. eveal mehods ased on C echnques []-[4] have een poposed o esmae he paamees of he spase channel pahs n he mmwave channel wh low anng ovehead, ncludng Angle-of-Depaue (AoD, Angle-of-Aval (AoA, and a educed-dmensonal eamspace channel. In [5], s also menoned ha C s expeced o e a key echnque o desgn compuaonal effcen hyd pecodng algohms. Based on he concep of eamspace-mimo [7], [8], [9] whch was developed fo dscee lens aays, communcaons n mmwave channel happen n a lowe dmensonal suspace,.e., eamspace. Insped y aove woks, he poposed eamspace hyd pecodng algohm explos he spase naue of he mmwave channel o susanally educe complexy. The complexy of he poposed algohm s popoonal only o he nume of channel sgnal popagaon pahs, e.g., channel spasy, ahe han he nume of anennas n convenonal schemes. The man conuons of hs pape ae summazed as follows: Beamspace-VD fo opmal full-dgal pecode acquson: Based on he CI fom he C-ased channel esmao, we popose an algohm o acque he opmal full-dgal pecode y pefomng VD mplcly on he educed-dmensonal eamspace channel ahe han explcly on he lage-dmensonal spaal doman channel. The oveall complexy s on he ode of he ems of he channel spasy, whch s much lowe han he nume of anennas n mmwave MIMO. Ths can educe he compuaonal complexy of acqung he opmal fulldgal pecode y 99%. C-asssed eamspace hyd pecodng (C-BP: By comnng eamspace-vd and a Dscee oue Tansfom (DT-ased R eamfomng codeook, we popose a hyd pecodng algohm decly ased on Casssed eamspace CI. The C-BP algohm elmnaes max nveson compuaons and machng pusu eaons. Moeove, he oveall algohm s pocessed n he low-dmensonal eamspace CI. Theefoe, he C- BP algohm can educe he compuaonal complexy y aou 99% compaed wh [8]-[9]. The es of hs pape s oganzed as follows. We noduce he sysem model and elaed hyd pecodng desgn algohms n econ II. In econ III, we pesen he eamspace-vd algohm. The C-BP algohm s poposed n econ IV. In econ V, smulaon esuls demonsang he pefomance of he poposed algohms ae gven. nally, we conclude he pape n econ VI. II. REVIEW O MMWAVE MIMO YBRID PRECODIG/COMBIIG YTEM A. oaons The noaons used n hs pape ae as follows: A s a max. a s a veco. a s a scala. s a se. A s he oenus nom of A. A s he deemnan of A. ace( A s he ace of A. (:, l A s he lh column of A. [ A] s he expecaon of A. dag( A s a veco composed of dagonal elemens of A. (, A o A(, s he sumax of max A wh ndex ses and. T - A, A, and A denoe he anspose, conjugae anspose and nvese of a max A, especvely. ( a, A s a complex Gaussan veco wh mean a and covaance max A. [ AB ] s he hozonal concaenaon. a ( s he h elemen of a veco a. s he cadnaly of. ( means he ode s. I denoes a deny max. B. ysem Model Consde he sngle-use mmwave MIMO sysem shown n g.. The B and UE ae equpped wh ansm and eceve anennas, especvely. The B ansms daa seams o he UE. and R R chans ae equpped a R he B and UE especvely o enale mul-seam ansmsson such ha δ R δ, δ {, }. The B s equpped wh a hyd pecode, whch s composed of a R aseand dgal pecode,, and a R R analog pecode, R. The R pecode R s ealzed y pue analog phase shfes wh un nom (, j consans such ha R = /, {...., }. The oal ansm powe consan sasfes R =.The pecoded dscee-me ansmed sgnals hough he channel can e wen as y = ρ s + n ( R. s s he npu sgnal, ha [ ] =, s he channel such y s he ansmed sgnal, 5-587X (c 6 IEEE. Pesonal use s pemed, u epulcaon/edsuon eques IEEE pemsson. ee hp:// fo moe nfomaon.

3 n~ (, σ n I s he nose eceved a he UE, and ρ s he aveage pah loss eween he B and UE. nally, he eceved sgnal pos-pocessed y he hyd comne W W a UE s expessed as ( R yˆ = ρ W W s+w W n. ( R R R C. Mllmee-Wave Channel Model [7], [], [6] Because of he spase popey of he mmwave channel, we adop a wdely used D naowand fequency-nonselecve mmwave channel model wh ndvdual scaees, whee each scaee s modeled y a sngle popagaon pah [7], [], [6]. The channel n hs model, whch can e expessed as α l ~ (,, s he complex pah gan of he lh scaee. ϕ l and ϕ l epesen he azmuh angles of depaue (AoD and he aval (AoA of he lh scaee, especvely. a ( ϕ l and ( a ϕ l ae he aay esponse vecos (eamfomng vecos ha coespond o he AoD and AoA of he lh scaee. We assume ha oh he B and UE adop an δ -elemens unfom lnea aay (UA, hence a ( ϕ l and a ( ϕ l can e expessed as whee λ and d epesen he sgnal wavelengh and he anenna spacng especvely. Exenson o a unfom plana aay s possle and sagh fowad [6]. oe ha he channel n ( can also e epesened as = A dag( α A, (5 whee α = / α [, α,..., α ] T s a veco ha conans he powe on each popagaon pah. The ansm canddae max A and he eceve canddae max A have he fom α a ϕ a ϕ ( = l= l ( l ( l. a ϕl = e e (4 j(π/λ dsn( δ l j( (π/λ dsn( δ δ ϕ δ ϕl T δ ( [,,..., ], δ A = [ a ( ϕ, a ( ϕ,..., a ( ϕ ]. (6 δ δ δ δ δ δ δ Typcally, he channel knowledge of he AoDs, he AoAs, and he pah gan can e acqued y he C-ased channel esmao a he UE []-[4], whee he channel esmaon pocess s fomulaed as a spase opmzaon polem, and s desced n deal n Appendx. o a C-ased channel esmao, each esmaed AoA ϕ ˆn and AoD ϕ ae aken ˆm fom unfom quanzed gds such ha ˆ ϕ = πm/ G, m m=,,..., G and ˆ ϕ = πn/ G, n n=,,..., G, whee G δ s he angula esoluon []-[4]. The quanzed G ansm canddae max A and he G eceve canddae max A have he fom ˆ δ δ δ A=[a δ δ( ˆ ϕ, aδ( ϕ,..., a δ( ϕ ]. (7 Theefoe, he esmaed channel can e wen as ˆ = Aˆ ˆ A ˆ, (8 whee Ĥ s a G G spase max. Due o he hgh pah loss of he mmwave channel, hee ae K GG nonzeo ( K - G δ spase esmaed pah gans ˆ α = [ ˆ α, ˆ α,..., ˆ α ] T K locaed a he quanzed AoA se and quanzed AoD se such ha ˆ ˆ (, j α m,f ( ( j =,, elsewhee whee m=,,..., K s he ndex of he esmaed pah gan. o acaly, we assume =,.e., Ĥ has only columns assocaed wh he esmaed AoDs ha conan nonzeo enes. Afe pefomng C-ased channel esmaon, he UE wll have nfomaon fo he esmaed he AoD se, he AoA se, and he eamspace channel Ĥ. D. Revew of yd Pecode Desgns [6]-[9] The ojecve of [6]-[9] s o desgn a hyd pecode wh low-complexy o maxmze he achevale ae ove all possle soluons of (,, W, W, whch s gven y ρ R = log I + R n WWRR R WRW, ( whee ρ / σ n s he eceved R, and Rn = σ nwwrwrw s he nose covaance max afe comnng. Based on he mahemacal devaons, he desgns of he pecode and he comne ae decoupled o avod an nacale soluon [6]. Ths means ha we can focus on he desgn of a hyd pecode R wh an opmal full-dgal comne. Thus, he desgn polem can e efomulaed as: (, = agmn op op R op R R,.. R(:, { Acan (:,, }, R R =. s k k R R ( op = [ v, v,..., v ] s he opmal full-dgal pecode wh pefec channel knowledge, whch consss of gh sngula vecos assocaed wh he lages egenvalues of. Tha s, = mn(, = m= U V σ u v m m m, (9 ( whee U= [ u, u,..., u ] and V = [ v, v,..., v ] ae he lef and gh sngula maces of especvely. s a dagonal max wh all egenvalues of n a non-nceasng ode such ha σ σ... σ. A mn(, can s a canddae max whee ansm aay esponse vecos ae chosen fom. In R [6], A can s equal o A n (6 such ha ecomes an AoD ased canddae max. On he ohe hand, ( s fomulaed as an opmzaon polem wh spasy consans n [6]-[9]. In [6], ( s solved y OMP. The hyd pecode desgned y OMP s summazed n Algohm. Alhough OMP exhs nea-opmal pefomance close o he opmal full-dgal pecode, eques a max nveson n ep 7 o calculae he leas squaes soluon wh (( R complexy, whch s no suale fo hadwae mplemenaon. To solve hs ssue, a paallel-ndex-selecon max-nveson-ypass 5-587X (c 6 IEEE. Pesonal use s pemed, u epulcaon/edsuon eques IEEE pemsson. ee hp:// fo moe nfomaon.

4 4 Algohm : yd Pecode Reconsucon usng OMP [6]-[7] Reque : op, A can : R = Empy Max : es = op : fo R do 4: Ψ - = A can es (, l l 5: = agmax ( Ψ Ψ k l=,,..., (:, k 6: R = [ R A can ] 7: ( = RR R es op R 8: es = op R 9: end fo : = : Reun R, R smulaneous ohogonal machng pusu (PI-MIB-OMP ased hyd pecode s poposed o calculae he nvese max eavely n [8]-[9]. oweve, algohms n [8]-[9] wh m degees of paallelsm eque he m m max nvese o e calculaed dung each eaon. Theefoe s dffcul o exend [8]-[9] o lage anenna aay. III. PROPOED BEAMPACE-VD AGORITM In hs pape, we assume ha a C-ased channel esmao can successfully deec he se wh coespondng esmaed eamspace channel Ĥ. Afe he channel s esmaed usng he algohm n [], R s desgned ased on he opmal full dgal pecode ˆ op, whch consss of he gh sngula vecos assocaed wh he lages egenvalues of he esmaed channel Ĥ. In hs secon, we popose a low-complexy algohm fo acqung he appoxmae opmal gh sngula vecos of he spaal doman channel. A. ull-dgal Pecode Acquson usng Beamspace-VD Insead of pefomng full-dmensonal VD on he spaal doman channel Ĥ n g. (a, we seek o decly pefom VD on he eamspace equvalen channel wh educed dmenson, as shown n g. (. Ths can educe he compuaonal complexy and feedack ovehead. The eamspace equvalen channel can e expessed as: A A ( =,,DT,DT whee A,DT and A,DT ae vual ansmed and eceved eamfomng vecos conssng of DT maces [7]. DT maces coespond o (7 when G =, and G =. ence, we have Aˆ = A,DT, Aˆ = A,DT, ˆ, and ˆ. oe ha Ĥ s a spase max due o he chaacescs of he mmwave channel and ˆ ( j, s he popagaon gan measued wh he jh DT eamfomng veco a he UE when he B ansms symol sn = on he h DT eamfomng veco. nce enes locaed ousde of he quanzed AoA se and he quanzed AoD se ae equal o zeo, he nfomaon conans n Ĥ s he same as he lowe dmenson channel = ˆ (:,, (4 whee s he lowe dmenson eamspace suchannel esmaed y he UE when he B uses DT eamfomng vecos assocaed wh he se. In paccal mplemenaon schemes, he ovehead fo acqung ˆ op ases fom he fac ha: The complexy of pefomng VD on Ĥ s (, whch s exemely hgh snce he nume of anennas s much lage han he spasy of he mmwave channel. Ĥ conans enes, whch eques a huge amoun of feedack. To ovecome hese polems, we fs popose decly pefomng VD on he esmaed eamspace su-channel, and hen devng he elaon eween he gh sngula vecos of and ˆ op. The VD of s gven y: = U V (5, whee U, V, and ae he lef and gh sngula maces, and he egenvalue of he eamspace su-channel especvely. Then, he auocoelaon max of s gven y R = = V U U V = V. V mlaly, he auocoelaon max of Ĥ s gven y Rˆ = ˆ ˆ = V U U V = V V. (6 (7 On he ohe hand, snce Ĥ can e decomposed no (:, A A, ˆR can e also expessed as,dt,dt Because R ˆ = ˆ ˆ = A A A A. (:, (:,,DT,DT,DT,DT A,DT s DT eamfomng max, we have g.. Pocessng flow of (a convenonal full-dmenson VD, and ( eamspace-vd. In geneal, he nume of channel pahs s much lowe han he nume of ansm anennas unde mmwave MIMO sysems. ( X (c 6 IEEE. Pesonal use s pemed, u epulcaon/edsuon eques IEEE pemsson. ee hp:// fo moe nfomaon.

5 5 A A I (9,DT,DT =. usung (6 and (9 no (8, ˆR can e smplfed as Rˆ = A A A A = A A = A V V A. (:, (:,,DT,DT,DT,DT (:, (:,,DT,DT (:, (:,,DT,DT ( Compang (7 wh (, we oseve ha oh V and (:, A,DTV fom he column space of ˆR. Base on (, we popose an opmal full-dgal pecode acquson algohm,.e., eamspace-vd, whch s summazed n Algohm. ence, ˆ op can e vewed as he ansfom of he eamspace gh sngula max V. Moeove, f he quanzed ansm canddae max  s assumed o e known o oh he B and he UE, he feedack nfomaon conans only V and. Ths eques ( enes ahe han ( enes when feedack ˆ op s gven explcly. g. plos he eam paens of dffeen opmal full-dgal pecode acquson algohms. We can oseve ha he eam paens of g. (a and g. ( ae he same, whch sasfes he mahemacal devaon. B. Complexy Analyss of Beamspace-VD In hs secon, we analyze he compuaonal complexy of he poposed eamspace-vd. We compae eamspace-vd wh adonal full-dmenson VD n Tale I. In Tale I, he compuaonal complexy of VD s analyzed usng a supelnea convegence VD (-VD algohm [] as he pefomance mec. The compuaons, excep fo he eamfome mulplcaon, ae pocedues ha elong o he -VD algohm. In [], he asc compuaonal cos un Algohm : Opmal ull-dgal Pecode Acquson usng Beamspace-VD cheme (:, Reque :, A,DT : = U V (:, : V= A,DTV : ˆ op = V (:,: Reun ˆ, V 4: op s one complex mulple and adde (CMAC. The paamee s he nume of ansm anennas, s he nume of eceve anennas, s he nume of daa seams, and e s he maxmum eaon fo calculang each gh sngula veco,.e., V (:, and V (:,, usng -VD. g. 4 compaes he compuaonal complexy of fulldmenson VD and he poposed eamspace-vd vesus unde dffeen nume of pahs ( = 8,, 6. Ths analyss ses =, = 8, and e = 4, whee vaes fom 6 o 64. We use hs seng ecause he nume of pahs s much lowe han he nume of ansm anennas fo mmwave MIMO communcaon [6]. o eamspace-vd, snce he eamspace gh sngula vecos V ae calculaed y pefomng VD on he esmaed eamspace channel, we need a max mulplcaon wh CMACs o calculae ˆ op n ep of Algohm. On he ohe hand, fo fulldmenson VD, snce C-ased channel esmao deec nsead of Ĥ, ( + addonal CMACs ae equed fo he eamfome mulplcaon such ha ˆ (:, = A,DTA,DT. I can e seen ha he poposed eamspace-vd educes he compuaonal complexy y 99.4% (98.5%, 96.9% wh = 8 (, 6, = 64 when oh gh sngula max acquson algohms calculae he esmaed fulldgal pecode ˆ op. Ths ases fom he dffeence eween he npu dmensons of VD, whee < << sasfes he spase chaacescs of he mmwave MIMO scenao []. ex, we compae daa ansfe and soage equemens. om g., shows ha hee s a g gap n he npu dmensons of a full-dmenson VD and eamspace VD. Tha s, he eamspace channel now conans only complex enes, nsead of enes fo he ognal spaal doman channel Ĥ. The oveall soage of CI can e TABE I UMBER O CMAC OR DIERET VD AGORITM Compuaon ull-dmenson VD Beamspace-VD Inal age / / Max Auocoelaon e / ( e / ( Deflaon / ( / ( Gam-chmd Pocess Beamfome Mulplcaon + g.. Beam paens of (a opmal full-dgal pecode wh esmaed channel, and ( poposed eamspace-vd wh esmaed channel. g. 4. Compuaonal complexy of full-dmenson VD and eamspace- VD vesus unde dffeen nume of pahs ( = 8,, X (c 6 IEEE. Pesonal use s pemed, u epulcaon/edsuon eques IEEE pemsson. ee hp:// fo moe nfomaon.

6 6 sgnfcanly saved. o example, f he wod lengh of each eny s denoed y W s, and we se = 64, = 6, = 6, and W = 6. Then, he oal nume of equed s pe channel can e educed fom,768 s ( Ĥ o,7 s (, whch leads o gea educon n oh daa soage and ansmsson of he CI daa and paamees n execung ou algohm. In concluson, he poposed eamspace-vd has followng mes: Complexy s educed y 99.4% snce he VD s pefomed on ahe han on Ĥ. eedack enes ae educed fom ( o (. Daa ansfe and soage equemens of he max daa can e educed fom ( W o ( W. IV. COMPREIVE EIG-AITED OW-COMPEXITY YBRID PRECODIG AGORITM In hs secon, we pesen an ohogonal eamfomng codeook and a low-complexy eamspace hyd pecodng algohm fo he sysems shown n g.. s, a hadwaefendly ohogonal eamfomng codeook s desgned ased on DT. econd, we avod he max nveson n spaally spase pecodng y usng ohogonal eamfomng codeook. Thd, ased on he eamspace gh sngula veco and he DT eamfomng codeook, a low-complexy eamspace hyd pecodng algohm s poposed o appoxmae he achevale ae of he opmal full-dgal pecode. A. Ohogonal Beamfomng Codeook Desgn ased on DT In hs susecon, we popose a codeook ha avods he max nveson n OMP y choosng a pope canddae max A can. The polems of canddae maces n [6]-[9] ae: Max nveson s equed o elmnae he nefeence fom he coelaed canddae max A, Accuae nfomaon fo he AoDs,.e., A, s unavalale n eal wod applcaons, and nfne esoluon s equed o epesen. To avod hese ssues, a suale choce of A can mus sasfy followng popees: The column vecos of A can need o e ohogonal o each ohe o avod max nveson, and The column vecos of eamfomng codeook consss of pedefned enes. Due o he spaally spase chaacescs of he mmwave channel, he enegy of he opmal gh sngula vecos wll spead on few AoDs. In [6], s suggesed ha A can mus e ale o span he gh sngula max of channel Ĥ. nce he gh sngula max s spanned y he ansm aay esponse veco A, a lnea comnaon of columns of an elgle canddae max mus e ale o synhesze aay A. Movaed y hese popees and he afoemenoned eamspace-mimo, we popose usng DT codeook as he canddae max,.e., Acan = A,DT, whch has he fom j π ( k j π ( k ( T (:, k A,DT = [, e,..., e ], ( whee k =,,...,. Based on [], s well known ha he DT codeook s he ass fo he space ha s spanned y aay esponse vecos wh aay decons on he D suface. By employng he DT codeook, he B s ale o see sgnals n ndependen decons. In g. 5, a 6- (:, elemens UA s used o see he DT eam paen of A,DT (:, 9 and A,DT especvely. The ohogonaly of he DT codeook means ha hee s no coelaon eween s column vecos. Because he DT codeook has a consan column sze and pedefned enes nsead of nfomaon fom he AoDs, s suale fo hadwae mplemenaon. B. paally pase Pecodng wh Ohogonal Beamfomng Codeook [] In hs susecon, we noduce he me of spaally spase pecodng wh an ohogonal eamfomng codeook. Ths coesponds o adopng he DT codeook as he canddae max A can fo OMP. The famewok s llusaed n g. 6. o OMP (Algohm, o choose aay esponse R vecos fom a canddae max and econsuc he assocaed aseand pecode, he coelaon max eween he opmal full-dgal pecode op and he canddae max A s calculaed fs. Then, a he h eaon n eps 4 o 6, he canddae aay esponse veco ha has he hghes coelaon powe wh he opmal pecode s seleced as he h column of R. Afe selecng he hghes coelaed column, we can econsuc he coespondng aseand pecode y oanng he leas squaes soluon n ep 7. In ep 8, snce he columns of he AoD ased canddae max A ae coelaed, s necessay o pefom nefeence cancellaon. Ths means ha he esdual max es s ecalculaed y (a ( g. 5. R eam paens wh UA phase shfes seeed y (a fs and ( nnh DT eamfomng code. g. 6. Illusaon of he (a coelaon sep, and ( paallel ndex selecon sep of OMP wh he DT eamfomng codeook X (c 6 IEEE. Pesonal use s pemed, u epulcaon/edsuon eques IEEE pemsson. ee hp:// fo moe nfomaon.

7 7 emovng he componen conued y he seleced canddae eamfomng vecos efoe selecng he nex one. eveheless, when he canddae max sasfes he equemen fo ohogonaly, e.g., A,DT, he eave pocess fo selecng he es eamfomng vecos can e R pefomed n paallel snce hee s no coelaon eween s columns vecos. Theefoe, he max nveson fo calculang he leas squae soluon can e avoded. Based on he ohogonaly of he DT codeook n g. 6(a, he powe of he gh sngula veco ha s dsued acoss eamfomng decons s calculaed n paallel y pefomng coelaon eween op and A. ence,,dt Ψ = A (,DT op. Ψ s he coelaon max. Then, he powe dsued n each eamfomng decon can e calculaed y β= dag( Ψ Ψ, ( whee β s he veco wh he h eny equal o he enegy dsuon of he gh sngula max op n he h eamfomng decon of DT codeook (:, A,DT. nce he DT eamfomng vecos ae uncoelaed and ohogonal o each ohe, he aay esponse vecos can e found n R paallel y choosng he columns fom DT codeook R = A (:,. (4 R,DT s he se coespondng o he lages enes of β R such ha = R, as shown n g. 6(. oe ha hs sep no longe eques an eave nefeence cancellaon pocess. As all he eamfomng vecos ae chosen, he R coespondng lowe dmensonal aseand pecode s calculaed va he leas squaes soluon: = (. (5 R R R op nce = (:, R A,DT consss of unay DT eamfomng vecos, he nvese of ( R R s equal o he deny max. Then we have = ( A A A (:, (:, (:,,DT,DT,DT op (:, = A,DT op = Ψ (,:. (6 om (6, can e oseved he leas squaes soluon s smplfed o meely selecng ows of he coelaon max Ψ assocaed wh he se. In ohe wods, he max Algohm : Ohogonaly Based Machng Pusu (OBMP [] Reque: op, A can,oh. : Ψ = A can,oh. op : β=dag( ΨΨ : = Empy e 4: fo R do 5: k = agmax β ( l l=,,..., = k 6: [ ] 7: β ( l = 8: end fo 9: = A (:, R can,oh. : = Ψ (,: : = : Reun R, nveson calculaon can e compleely omed y eplacng a coelaed canddae max wh an ohogonal one. Based on he aove devaons, we summaze a spaally spase hyd pecode desgn wh an ohogonal eamfomng codeook n Algohm, whch s ou pevous wok, ohogonaly ased machng pusu (OBMP [], whee A epesens he ohogonal canddae max. can,oh. R C. Poposed Compessve ensng Asssed Beamspace yd Pecodng (C-BP Alhough Algohm can avod max nveson, sll eques an opmal full-dgal pecode acqued y fulldmenson VD wh hgh compuaonal complexy. Based on he C-ased channel esmao, we develop a lowcomplexy C-asssed eamspace hyd pecodng algohm,.e., C-BP, whch comnes he mes of he poposed eamspace-vd and he DT codeook. Ths coesponds o g. 7(. The famewok of he C-BP algohm s llusaed n g. 8. g. 7. Pocessng flow of (a spaally spase hyd pecodng, and ( eamspace hyd pecodng. g. 8. Illusaon of he (a oue poduc sep, and ( paallel-ndexselecon sep of C-BP X (c 6 IEEE. Pesonal use s pemed, u epulcaon/edsuon eques IEEE pemsson. ee hp:// fo moe nfomaon.

8 8 om he ˆ op gven y he poposed eamspace-vd algohm n econ III, he desgn of he hyd pecode R can e followed y he spase pecodng lock n g. 7(a y adopng Algohm wh he educed compuaonal ovehead of full-dmenson VD and whou max nveson. oweve, we noe ha ˆ op s fs gven y he max (:, mulplcaon of A (:,:,DT and V, and hen decomposed no a wo sage hyd pecode. Ths mples some edundan calculaons. On he ohe hand, fo a DT canddae eamfomng max n Algohm, he coelaon max Ψ eques ( complex mulplcaons o calculae he enegy speads on decons. Moeove, snce he coelaon max fo OMP needs o e calculaed fo R eaons, he oveall complexy ecomes ( R, whch s much lage han n Algohm. Movaed y he eamspace-vd algohm, ˆ op can e vewed as he ansfom of he eamspace gh sngula vecos as deved n econ III, and we oseve ha he coelaon max Ψ n ( can e fuhe expessed as: Ψ = A ˆ,DT op (:, (:,: = A,DTA,DTV. (7 Because he DT max s ohogonal, only he enes whn he esmaed AoD se ae lef, whch can e expessed as (:, (:,, f A,D T A,D T =., elsewhee (8 By susung (8 no he coelaon max n (7, we have Ψ (,:= A A V (:, (:, (:,:,DT,DT V = ( k,:,f = ( k., elsewhee (9 om (9, can e seen ha he powe spead o he h (,: esmaed AoD Ψ s exacly he same as he enes of he ( k,: eamspace gh sngula vecos V f he ndex elongs o he esmaed AoD se, and s ohewse equals o zeo. Ths mples ha he max mulplcaon sep fo acqung ˆ op n ep of he eamspace-vd and he (:,: coelaon sep n ( ae avoded snce V s nally acqued fom he esmaed eamspace channel. nce he nfomaon fo Ψ s conaned n V, we oseve ha ( can e smplfed as (,: l (:,: l β( l = Ψ Ψ ( k,: ( k,: V V,f l = ( k ( =., elsewhee Ths means ha β ( l can e acqued y pefomng oue (:,: poduc on V, as shown n g. 8(a. Then, he R eamfomng veco selecon n eps 4-8 of Algohm educes o choosng he ndex assocaed wh he lages values fom dagonal enes of he oue R (:,: (:,: poduc max V V. Ths pocedue deemnes he se y selecng he es eamfomng decons fom R he decons assocaed wh he esmaed AoD se n paallel. Ths coesponds o g. 8(. Afe he se has een chosen, we have he R eamfomng vecos gven n ep 9 of Algohm. Then, n Algohm he aseand pecode s calculaed va he ows n he coelaon max assocaed wh he se. oweve, snce we have shown ha he nfomaon of Ψ s conaned n eamspace gh sngula vecos V, he aseand pecode can e fuhe deved as = Ψ (,: (:, = A ˆ,DT op = A A V (:, (:, (:,:,DT,DT = V (,:, ( (:,: whee s he se of ow vecos of V ha coesponds o he R eamfomng vecos whn he se such ha = R. Ths sep means ha s desgned y selecng (:,: ows of V wh he lages ow powes. As wh R Algohm, he max nvese s also avoded, and also need o e nomalzed as n ep of he OMP. Based on he aove devaons, he poposed C-BP algohm s summazed n Algohm 4. Wh he ad of he eamspace CI fom he C-ased channel esmao, he poposed C-BP desgned hyd (:, pecode mplcly ased on he esmaed AoDs A,DT and on (:,: he eamspace gh sngula vecos V acqued fom he eamspace-vd. Ths can educe edundan max mulplcaon when compaed o explc full-dmenson VD. Moeove, C-BP elmnaes he ovehead fo compung coelaon max Ψ n ( snce we only need o fnd R esmaed quanzed AoDs fom decons whn he se Algohm 4: yd Pecode Reconsucon usng C-BP (:,: (:, Reque: V, A,DT (:,: (:,: : γ = dag( V V : = Empy e : fo R do 4: k = agmax γ ( l l=,,..., = k 5: [ ] 6: γ ( l = 7: end fo 8: = A (:, R,DT 9: = V (,: : = : Reun R, R 5-587X (c 6 IEEE. Pesonal use s pemed, u epulcaon/edsuon eques IEEE pemsson. ee hp:// fo moe nfomaon.

9 9 TABE II UMBER O COMPEX MUTIPICATIO I TE T ITERATIO Compuaon PI-MIB-OMP [8] WI-MIB-OMP [9] Poposed C-BP Ψ ( ll, [ ( ] / R + Updae Ψ ( ( β=dag( Ψ Ψ ( Ψ Ψ ( / Updae A m m ( m ( Updae V m ( m ( m + / Updae M m m ( A V m ( m m ( AVA m [ m ( + ] m ( / ( AVM m ( m VM Updae G Updae X m m TABE III UMBER O COMPEX ADDITIO I TE T ITERATIO (a Compuaon PI-MIB-OMP [8] WI-MIB-OMP [9] Ψ ( β=dag( Ψ Ψ ( Ψ Ψ Updae Ψ ( ll, [ ( ] ( / ( ( + Poposed C-BP ahe han quanzed AoDs on he whole D suface. uhemoe, he max nvese fo calculang can also e avoded. D. Complexy Analyss of yd Pecodng In hs susecon, we analyze he complexy of he poposed C-BP. We compae wh PI-MIB-OMP [8] and wh sldng wndow-ndex-selecon max-nvesonypass smulaneous ohogonal machng pusu (WI-MIB- OMP [9] ecause only [8] has a hadwae mplemenaon, and s mpoved n [9] wh educed ndex-selecon complexy. We also compae wh ou pevous wok, ohogonaly ased machng pusu (OBMP []. We compae he complexy n ems of he nume of complex mulplcaons and addons n Tales II and III especvely [8], [9]. Ths analyss also consdes he degee of paallelsm. The paamee m n Tales II and III epesens he degee of paallelsm of he PI-MIB-OMP and WI-MIB-OMP R ( / ( Updae A m [ m ( ] m ( Updae V m ( m ( m + / Updae M m m ( A V m ( ( m m ( AVA ( m [ m ( + ] m ( / ( AVM m ( ( m VM m ( m Updae G m ( [ m ( + ] / Updae X m ( ( fo updang he leas squaes soluon. The compuaonal complexy of he hyd pecodng algohms s analyzed y followng he same seng n [8] such ha m =, 8, 6, = R = =, and = floo( /. On he ohe hand, he nume of complex mulplcaons and addons of OBMP ae gven y + and ( + ( especvely. ( g. 9. Compuaonal complexy of he poposed C-BP and elaed hyd pecodng algohms n ems of he nume of complex (a mulplcaons and ( addons unde dffeen degees of paallelsm. TABE IV COMPUTATIOA AVIG O COMPEX MUTIPICATIO WIT DIERET DEGREE O PARAEIM Paallelsm m = m = 8 m = 6 PI-MIB-OMP [8] 99.6% 99.6% 99.6% WI-MIB-OMP [9] 99.6% 99.6% 99.5% OBMP [] 98.5% TABE V COMPUTATIOA AVIG O COMPEX ADDITIO WIT DIERET DEGREE O PARAEIM Paallelsm m = m= 8 m = 6 PI-MIB-OMP [8] 99.6% 99.6% 99.6% WI-MIB-OMP [9] 99.6% 99.6% 99.6% OBMP [] 98.5% 5-587X (c 6 IEEE. Pesonal use s pemed, u epulcaon/edsuon eques IEEE pemsson. ee hp:// fo moe nfomaon.

10 TABE VI COMPARIO O EXITIG WORK We can oseve ha he max nveson paamees A, V, and G n [8], [9] ae elmnaed fo he case of C- BP, hs s ecause hee s no max nvese fo C-BP. In g. 9, we compae he compuaonal complexy of [8], [9], [], and C-BP vesus. In Tales IV and V, he nume of complex mulplcaons and he addons of C-BP ae educed y 99.6%, 99.6%, 98.5% wh = 64 fo m = n compason o PI-MIB-OMP, WI-MIB-OMP, and OBMP, especvely. Ths s ecause ha fo C-BP, he coelaon max Ψ s educed smply o a ow powe (:,: selecon on he eamspace gh sngula vecos V, and hee s no max nveson fo calculang. In summay, he poposed C-BP comnes he mes of eamspace-vd and DT codeook, and acheves: 99.6% complexy educon. o calculaon fo he coelaon max Ψ. o max nveson fo calculang. We summaze he compason among dffeen hyd pecodng algohms n Tale VI. V. PERORMACE AAYI In hs secon, we pesen smulaon esuls fom he poposed eamspace-vd algohm, he ohogonal eamfomng codeook, and he C-asssed eamspace hyd pecodng algohm menoned n econs III and IV. A. ysem Confguaon and Channel Model We consde a sngle-use MIMO sysem. The B s equpped wh = 64 anennas and he UE has = 6 anennas. Boh adop UA wh anenna spacng of d = λ /. We assume ha dung he esmaon phase, he UE adops pue analog eamfomes o esmae he channel, and dung he downlnk ansmsson phase, he UE adops a full-dgal MME comne [6]. The mmwave channel s modeled y egh popagaon pahs ( = 8 wh AoDs and AoAs assumed o e unfom dsued eween [, π ]. Ths sysem s assumed o opeae a 8 Gz cae fequency wh an Mz andwdh, pah-loss exponen s equal o []. We calculae he specal effcency of he poposed hyd pecodng algohm y ( wh full-dgal MME comne MME ( / { W = ρ + ( σ n / ρ I } adoped a he UE sde. can e ehe a full dgal pecode ( op / ˆ o a hyd pecode R. All smulaon esuls fo op cheme OMP [6][7] PI-MIB- OMP [8][9] OBMP [] Poposed C-BP Channel Knowledge paal Doman CI paal Doman CI paal Doman CI Beamspace CI Opmal ull-dgal Pecode ulldmenson VD ulldmenson VD ulldmenson VD Beamspace- VD Baseand Pecode Max Invese [6][7] Ieave Invese [8][9] Coelaon (Eq. 6 Oue Poduc (Eq. Complexy gh Medum ow owes g.. Achevale ae of poposed eamspace-vd and full-dmenson VD fo 64 6 mmwave MIMO sysems vs. R. g.. Achevale ae of he poposed C-BP and pevalng hyd pecodng algohms fo 64 6 mmwave MIMO sysems vs. R. specal effcency ae aveaged ove andom channel ealzaons. We assume ha he esmaed AoDs and AoAs ae quanzed wh gds such ha = G = 64 and = G = 6. The esmaed 6 8 eamspace channel and esmaed AoDs se ae acqued y he C-Channel esmaon algohm n j φ ( mn, [] wh dealed sengs as follows: P( m,n= e, whee φ ( mn, s unfomly and andomly chosen fom {, π / Q,..., ( Q π / Q}, whch s he quanzed phase wh Q = 6 paons. The nume of anng eamfome s se o M =. o smplcy, a he UE sde we se he measuemen eamfomng max Q=A,DT such ha he UE adops M = 6 DT eamfomng vecos fo each p n. The channel s ecoveed va OMP afe MM = 5 successve me slos. The hyd pecodes ae lae desgned ased on he esmaed eamspace channel and he quanzed AoD eamfomng vecos A. (:,,DT B. pecal Effcency vesus R In g., we compae he pefomance of full-dmenson VD wh (a pefec channel knowledge, ( esmaed 5-587X (c 6 IEEE. Pesonal use s pemed, u epulcaon/edsuon eques IEEE pemsson. ee hp:// fo moe nfomaon.

11 g.. Achevale ae of he poposed C-BP fo MIMO sysems vs. R mmwave g. 4. B eo ae of he poposed C-BP fo sysems vs. R mmwave MIMO g.. Achevale ae of he poposed C-BP wh dffeen numes of R chans fo 64 6 mmwave MIMO sysems vs. R. channel knowledge, and (c eamspace-vd wh esmaed channel knowledge unde 64 6 mmwave MIMO sysems. ee he B and he UE communcae va = and = daa seams. om hs fgue we can oseve ha ( and (c exh same pefomance, whch acheves 95% of (a a R = db. Ths esul sasfes he mahemacal devaon n econ III. I also means ha we can employ eamspace-vd o acque he esmaed full-dgal pecode ˆ op,.e., he lue lne n g., wh educed complexy. The pefomance loss manly ases fom he angle quanzaon eo and he esmaon eo, whch do no gealy affec he pefomance. In g., all hyd pecodng algohms excep fo he AoD ased OMP hyd pecode apply he DT codeook as he canddae eamfomng max. ee he nume of R chans s se o R = 4 fo all hyd pecodng algohms. The poposed C-BP acheves same pefomance as OMP, PI-MIB-OMP, WI-MIB-OMP, and OBMP wh he DT canddae max. Ths esul s conssen wh he mahemacal devaon epesened n econ IV. nce each AoD n AoD ased canddae eamfomng max s epesened wh nfne esoluon, he hyd pecode desgn ased on can fom eam paens ha have ee pefomance han he DT codeook. eveheless, adopng DT codeook can gealy educe he compuaonal complexy wh a mee 5% pefomance loss. Theefoe, s a hadwae-fendly canddae max. C. Pefomance Gap eween C-BP and Opmal ulldgal Pecode In g., we compae he pefomance of he poposed C BP when he nume of R chans vaes fom R = o R = 8. The R s se o db. ee, he C-BP adops he DT canddae max. We oseve ha as he nume of R chans nceases, he specal effcency of he hyd pecode wll appoxmae ˆ op,.e., he opmal full-dgal pecode wh esmaed channel knowledge. The pefomance gap eween C-BP and he opmal fulldgal pecode op ases fom he fac ha he hyd pecode s calculaed ased on he esmaed channel, whch s easonale nuvely. D. pecal Effcency of C-BP vesus R and R In g., we compae he pefomance of he poposed C BP wh he nume of R chans vayng fom R = o R = 4 vesus R. ee we adop he DT codeook as he canddae max. We can oseve ha as he nume of R chans nceases, he pefomance gaps eween he poposed C-BP and ecome smalle fo dffeen Rs. ˆ op E. Be Eo Rae of vesus R In g. 4, we compae he poposed hyd pecodng algohm wh exsng woks n ems of BER. ee he nume of R chans s se o R = 4 fo all hyd pecodng algohms, whch s he same as n g.. The modulaon scheme hee s QPK. We oseve ha he poposed C-BP exhs same BER pefomance as OMP, PI-MIB-OMP, WI-MIB-OMP, and OBMP wh DT canddae max gven dffeen numes of daa seams. mlaly, he hyd pecode algohms ha adops he DT canddae max has a slghly hghe pefomance loss han he AoD ased OMP hyd pecode. Ths s conssen wh g X (c 6 IEEE. Pesonal use s pemed, u epulcaon/edsuon eques IEEE pemsson. ee hp:// fo moe nfomaon.

12 VI. COCUIO In hs pape, we pesen a C-asssed low-complexy opmal full-dgal pecode acquson algohm and a eamspace hyd pecodng algohm fo a sngle-use mmwave MIMO sysem. We popose usng eamspace-vd o acque he esmaed opmal full-dgal pecode wh he asssance of a C-ased channel esmao. We noduce a hyd pecodng algohm ased on he ohogonal eamfomng codeook ha avods he max nveson n he aseand pecode desgn. We consde he paccal hadwae mplemenaon consans as well. We popose a C-BP ha comnes he mes of he poposed eamspace-vd and he DT-ased ohogonal eamfomng codeook, and desgns he hyd pecode mplcly wh eamspace CI ahe han explcly n he spaal doman. The poposed algohms exh same pefomance as elaed woks va smulaon esuls wh gealy educed compuaonal complexy. Ths wok focuses on he analyss of poposed algohms wh sofwae smulaon. The analyss of he daa ansfe ae unde cean mplemenaon may e consdeed as fuue wok. REERECE [] T. Rappapo, R. W. eah J., T. Danels, and J. Mudock, Mllmee wave weless communcaons. Pence all, 4. [] T.. Rappapo,. un, R. Mayzus,. Zhao, Y. Aza, K. Wang, G. Wong, J. chulz, M. amm, and. Gueez, Mllmee wave mole communcaons fo 5G cellula: I wll wok! IEEE Access, vol., pp. 5 49,. [] W. Roh, J.-Y. eol, J. Pak, B. ee, J. ee, Y. Km, J. Cho, K. Cheun, and. Ayanfa, Mllmee-wave eamfomng as an enalng echnology fo 5G cellula communcaons: Theoecal feasly and pooype esuls, IEEE Commun. Mag., vol. 5, no., pp. 6, e. 4. [4] Z. P and. Khan, An noducon o mllmee-wave mole oadand sysems, IEEE Commun. Mag., vol. 49, no. 6, pp. 7, Jun.. [5]. Rangan, T.. Rappapo, and E. Ekp, Mllmee-wave cellula weless newoks: Poenals and challenges, Poc. IEEE, vol., no., pp , Ma. 4. [6] O. E. Ayach,. Rajagopal,. Au-ua, Z. P, and R. W. eah, J, paally spase pecodng n mllmee wave MIMO sysems, IEEE Tans. Weless Commun., vol., no., pp , Ma. 4. [7] O. E. Ayach, R. W. eah, J.,. Au-ua,. Rajagopal, and Z. P, "ow-complexy pecodng fo lage mllmee wave MIMO sysems," n Poc. IEEE In. Conf. Communcaons (ICC, Jun., pp [8] Y.-Y. ee, C.-. Wang, and Y.-. uang, A hyd R/aseand pecodng pocesso ased on paallel-ndex-selecon max-nvesonypass smulaneous ohogonal machng pusu fo mllmee wave MIMO sysems," IEEE Tans. gnal Pocess., vol.6, no., pp.5-7, Jan, 5. [9] K.-. su, C.-. Wang, Y.-Y. ee, and Y.-. uang, ow-complexy hyd eamfomng and pecodng fo D plana anenna aay mmwave sysems, n Poc. IEEE Wokshop on gnal Pocess. ys. (P, ep. 5. pp. -6. [] W.-. ung, C.-. Chen, C.-C. ao, C.-R. Tsa, and A.-Y.Wu, "owcomplexy hyd pecodng algohm ased on ohogonal eamfomng codeook," n Poc. IEEE Wokshop on gnal Pocess. ys. (P, ep. 5. pp. -5. [] J. A. Topp, A. C. Gle, and M. J. auss, Algohms fo smulaneous spase appoxmaon: pa I: geedy pusu, gnal Pocess. vol. 86, pp , Ma. 6. [] A. Alkhaee, O. E. Ayach, G. eus, and R. W. eah, J., Channel esmaon and hyd pecodng fo mllmee wave cellula sysems, IEEE J. el. Topcs gnal Pocess., vol.8, no.5, pp.8-846, Oc. 4. [] A. Alkhaee, G. eus, and R. W. eah, J., "Compessed sensng ased mul-use mllmee wave sysems: ow many measuemens ae needed?," n Poc. IEEE Inenaonal Conf. on Acouscs, peech and gnal Pocessng (ICAP, Ap. 5, pp [4] J. ee, G.-T. Gl, and Y.. ee, "Explong spaal spasy fo esmang channels of hyd MIMO sysems n mllmee wave communcaons," n Poc. IEEE Gloal Telecommuncaons Confeence (GOBECOM, Dec 4, pp. 6-. [5]. Kuy and D. en, Beamfomng fo mllmee wave communcaons: An nclusve suvey, IEEE Commun. uveys Tus., vol.8, no., pp , May. 6. [6] V. Raghavan and A. ayeed, ulnea capacy scalng laws fo spase MIMO channels, IEEE Tans. Inf. Theoy, vol. 57, no., pp , Jan. [7] A. M. ayeed and. Behdad, Connuous apeue phased MIMO: asc heoy and applcaons, n Poc. Alleon Conf. Comm., Conol, and Compu., ep., pp [8] J.Bady,. Behdad, A.M. ayeed, "Beamspace MIMO fo mllmeewave communcaons: sysem achecue, modelng, analyss, and measuemens," IEEE Tans. Anennas Popag., vol.6, no.7, pp.84-87, Jul.. [9] A. M. ayeed and. Behdad, Connuous apeue phased MIMO: A new achecue fo opmum lne-of-sgh lnks, n Poc. IEEE In.ymp. An. Popag. (AP, Jul., pp [] D. Tse and P. Vswanah, undamenals of Weless Communcaon. Camdge Unvesy Pess, 5. [] T. T. Ca and. Wang, Ohogonal machng pusu fo spase sgnal ecovey wh nose, IEEE Tans. Inf. Theoy, vol. 57, no. 7, pp , Jul.. [] C.-Z. Zhan, Y.-. Chen, and A.-Y. Wu, Ieave supelneaconvegence VD eamfomng algohm and VI achecue fo MIMO-ODM sysems, IEEE Tans. gnal Pocess., vol. 6, no. 6, pp.64-77, Jun.. [] A. Bjöck, umecal mehods fo leas squaes polems, IAM, 996. APPEDIX Revew of C-ased Channel Esmaon []-[4] The man ojecve of he C-ased channel esmao n []-[4] s o esmae he channel s AoD, AoA, and pah gan wh low anng ovehead. Dung he esmaon phase he B connuously ansms M anng eamfomng vecos p n such ha P= [ p, p,..., p M ] ecomes a anng eamfomng max, whle he UE adops M measuemen eamfomng vecos Q= [ q, q,..., q M ] o eceve each p n. Boh he B and UE adop a sngle R chan o ansm and eceve sgnals fo each anng me slo []. Afe MM successve me slos, he eceved sgnal a he UE ecomes Y = P Q P + E, ( M M whee Y s a max ha consss of eceved sgnals, M M P s he oal ansm powe, and E= Q s he nose max especvely. Based on he deny vec( ABC T = ( C Avec( B, he eceved max Y hen s vecozed o y = ΦΨα + e, ( T whee y=vec( Y, e=vec( E, Φ = P( P Q, and Ψ s a max whose column vecos conss of * a( ϕl a ( ϕl, l =,,...,,.e., he Konecke poduc of ansm and eceve aay esponse vecos. Wh he assumpon ha AoAs and AoA ae aken fom unfom quanzed gds []-[4], y can e appoxmaed as 5-587X (c 6 IEEE. Pesonal use s pemed, u epulcaon/edsuon eques IEEE pemsson. ee hp:// fo moe nfomaon.

13 ˆ vec( x + e, y = ΦΨ (4 whee Ψˆ G G s a canddae max whee each column veco consss of a * (ϕˆ m a (ϕˆ n, and x G G s a spase veco whch conans he nfomaon of popagaon loss fo each quanzed AoA and AoD pa. Then he esmaed AoAs and AoDs ae esmaed y solvng ˆ Φ y, supp( x = agm ax( Ψ (5 whee s u p p ( x s he suppo,.e., ndex se of non-zeo enes of x esmaed y C algohms such as Ohogonal Machng Pusu (OMP [], hen he eamspace channel s ˆ = vec ( x. ecoveed y Chang-en Chen eceved hs B.. degee n eleccal engneeng fom aonal Tsng ua Unvesy, snchu, Tawan, n 4. e s cuenly pusung a M.. degee n he Gaduae Insue of Eleconcs Engneeng, aonal Tawan Unvesy, Tape, Tawan. s eseach neess ae n VI mplemenaon of DP algohms, MIMO sysems, mmwave communcaon, and sgnal pocessng fo weless communcaons. An-Yeu (Andy Wu (M 96-M - 5 eceved he B.. degee fom aonal Tawan Unvesy n 987, and he M.. and Ph.D. degees fom he Unvesy of Mayland, College Pak n 99 and 995, especvely, all n Eleccal Engneeng. In Augus, he joned he faculy of he Depamen of Eleccal Engneeng and he Gaduae Insue of Eleconcs Engneeng, aonal Tawan Unvesy (TU, whee he s cuenly a Pofesso. s eseach neess nclude low-powe/hgh-pefomance VI achecues fo DP and communcaon applcaons, adapve/mulae sgnal pocessng, econfguale oadand access sysems and achecues, o-medcal sgnal pocessng, and ysemon-chp (oc/ewok-on-chp (oc plafom fo sofwae/hadwae codesgn. om Augus 7 o Dec. 9, he was on leave fom TU and seved as he Depuy Geneal Deco of oc Technology Cene (TC, Indusal Technology Reseach Insue (ITRI, snchu, TAIWA, supevsng Paallel Coe Achecue (PAC VIW DP Pocesso, and Andod/Mulcoe oc plafom pojecs. In, D. Wu eceved Ousandng EE Pofesso Awad fom The Chnese Insue of Eleccal Engneeng (CIEE, Tawan. In 5, Pof. Wu s elevaed o IEEE ellow fo hs conuons o DP algohms and VI desgns fo communcaon IC/oC. Cheng-Rung Tsa eceved hs B.. degee n eleccal engneeng fom Chang Gung Unvesy, Taoyuan, Tawan, n. e s cuenly pusung a Ph.D. degee n he Gaduae Insue of Eleconcs Engneeng, aonal Tawan Unvesy, Tape, Tawan. s cuen eseach neess ae MIMO sysems, mllmee-wave communcaon, and sgnal pocessng fo weless communcaons. Yu-sn u eceved hs B.. degee n eleccal engneeng fom aonal Tsng ua Unvesy, snchu, Tawan, n 5. e s cuenly pusung a M.. degee n he Gaduae Insue of Eleconcs Engneeng, aonal Tawan Unvesy, Tape, s eseach neess nclude VI Tawan. mplemenaon of DP algohms and mmwave communcaon. We-un ung eceved hs B.. degee n Eleccal Engneeng and Compue cence Undegaduae onos Pogam fom aonal Chao Tung Unvesy, snchu, Tawan, n, and hs M.. degee fom he Gaduae Insue of Eleconcs Engneeng, aonal Tawan Unvesy, Tape, Tawan, n 5. e cuenly woks a Medaek. s eseach neess nclude dgal sgnal pocessng, VI and mmwave communcaon X (c 6 IEEE. Pesonal use s pemed, u epulcaon/edsuon eques IEEE pemsson. ee hp:// fo moe nfomaon.