2370 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 6, JUNE 2016

Transcription

1 370 IEEE ASACIOS O COMMUICAIOS, VO. 64, O. 6, JUE 016 Channe Esimaion via Ohogona Maching Pusui fo ybid MIMO Sysems in Miimee Wave Communicaions Junho ee, ye-ae i, Membe, IEEE, and Yong. ee, Senio Membe, IEEE Absac We popose an efficien open-oop channe esimao fo a miimee-wave mm-wave hybid muipe-inpu muipeoupu MIMO sysem consising of adio-fequency beamfomes wih age anenna aays foowed by a baseband MIMO pocesso. A spase signa ecovey pobem expoiing he spase naue of mm-wave channes is fomuaed fo channe esimaion based on he paameic channe mode wih quanized anges of depaues/aivas AoDs/AoAs, caed he ange gids. he pobem is soved by he ohogona maching pusui OMP agoihm empoying a edundan dicionay consising of aay esponse vecos wih finey quanized ange gids. We sugges he use of non-unifomy quanized ange gids and show ha such gids educe he coheence of he edundan dicionay. he owe and uppe bounds of he sum-of-squaed eos of he poposed OMP-based esimao ae deived anayicay: he owe bound is deived by consideing he oace esimao ha assumes he knowedge of AoDs/AoAs, and he uppe bound is deived based on he esus of he OMP pefomance guaanees. he design of aining vecos o sensing maix is paicuay impoan in hybid MIMO sysems, because he beamfome pevens he use of independen and idenicay disibued andom aining vecos, which ae popua in compessed sensing. We design aining vecos so ha he oa coheence of he equivaen sensing maix is minimized fo a given beamfoming maix, which is assumed o be uniay. I is obseved ha he esimaion accuacy can be impoved significany by andomy pemuing he coumns of he beamfoming maix. he simuaion esus demonsae he advanage of he poposed OMP wih a edundan dicionay ove he exising mehods such as he eas squaes mehod and he OMP based on he viua channe mode. Index ems Channe esimaion, hybid /baseband pocessing, miimee wave communicaion, ohogona maching pusui, spasiy. I. IODUCIO ECEY, miimee-wave mm-wave communicaion sysems ae emeging as a pomising echnoogy fo nexgeneaion wieess communicaions [1], []. By empoying Manuscip eceived ovembe, 015; evised Mach 14, 016; acceped Api 13, 016. Dae of pubicaion Api 0, 016; dae of cuen vesion June 14, 016. his wok was suppoed by IC &D pogam of MSIP/IIP [B , Deveopmen of sma basesaion suppoing muipe seams based on E-A sysems]. his pape was pesened a he IEEE oba Communicaions Confeence, Dec he associae edio coodinaing he eview of his pape and appoving i fo pubicaion was X. ao. Coesponding auho: Yong. ee. J. ee and Y.. ee ae wih he Depamen of Eecica Engineeing, Koea Advanced Insiue of Science and echnoogy, Daejeon , Souh Koea e-mai: junho515@kais.ac.k; junho515@kais.ac.k..-. i is wih he Insiue fo Infomaion echnoogy Convegence, Koea Advanced Insiue of Science and echnoogy, Daejeon , Souh Koea e-mai: gaegi@kais.ac.k. Coo vesions of one o moe of he figues in his pape ae avaiabe onine a hp://ieeexpoe.ieee.og. Digia Objec Idenifie /COMM age anenna aays ha can be packed ino a vey sma aea hanks o he sho wave-engh, mm-wave sysems can achieve he beamfoming gain needed o ovecome he popagaion oss which is highe han he exising mico-wave communicaion sysems. uhemoe, mm-wave sysems can uiize he vas bandwidh avaiabe in he mm-wave specum. hese chaaceisics aow he design of mui-bps mm-wave sysems, as demonsaed in indoo wieess sysems such as wieess oca aea newok A [3] and pesona aea newok PA [4]. he channes of mm-wave communicaions ae spase in he sense ha impuse esponses ae dominaed by a sma numbe of cuses of significan pahs. his spasiy causes ankdeficien channe maices having a few dominan singua vaues, and he effecive ank of an mm-wave channe, which is he numbe of dominan singua vaues, ends o be consideaby smae han he numbe of ansmi/eceive anennas [5]. Based on his fac, use of a hybid MIMO pocesso consising of an anaog beamfome in adio fequency domain cascaded wih a digia MIMO pocesso in baseband has been poposed fo mm-wave communicaions [6] [8]. In a hybid MIMO pocesso, he numbe of chains is deemined by he effecive ank of he channe, and is impemenaion can be much simpe han conveniona MIMO sysems ha need one chain pe anenna. As in conveniona muipe-inpu muipe-oupu MIMO sysems, channe sae infomaion CSI is usefu fo designing efficien hybid MIMO pocessos of mm-wave sysems. o exampe, if CSI is given, he beamfome and he baseband MIMO pocesso can be joiny designed [6], [7], esuing in bee pefomance han he beam aining-based hybid beamfomes which fis see he beams o ange of depaue/aiva AoD/AoA diecions and hen opimize he baseband MIMO pocesso. oweve, esimaing CSI in mm-wave sysems is chaenging because he signa-o-noise aio S befoe beamfoming is ow and he numbe of anennas is age. o incease he S i woud be necessay o empoy pope beamfoming duing channe esimaion; and o eieve he effec of age anenna size he spasiy of mm-wave channes woud be expoied: insead of esimaing a he enies of he channe maix, one woud esimae ony he AoDs/AoAs of dominan pahs and he coesponding pah gains. In mm-wave communicaion, cosed-oop beam aining-based mehods ae poposed fo CSI esimaion [3], [4], [9] [11]. hese mehods fis esimae he AoDs/AoAs by cosed-oop beam aining and hen esimae IEEE. Pesona use is pemied, bu epubicaion/edisibuion equies IEEE pemission. See hp:// fo moe infomaion.

2 EE e a.: CAE ESIMAIO VIA OOOA MACI PUSUI O YBID MIMO SYSEMS 371 he pah gain associaed wih each pai of AoD and AOA. he cosed-oop beam aining is a muisage pocess ha can avoid an exhausive beam seach. A each sage he ansmie emis he pio beams, and he eceive seecs he bes beam and feeds back is decision. his pocess sas wih wide beams ha cove a of he anges of inees and makes he beams fine ony aound he anges whee AoDs/AoAs ae pesen. Whie cosed-oop beam aining-based mehods have been adoped in pacica sysems [3], [4], hei pefomance ends o be imied by he aining beam paens o codebook; fuhemoe, in muiuse MIMO sysems, hei aining ovehead ends o incease ineay wih he numbe of uses. An aenaive appoach o AoD/AoA esimaion of mmwave channes is o appy he compessed sensing CS echniques in [1] [14], which have been poposed fo spase muipah esimaion in angua domain of massive MIMO channes. hese CS-based mehods ae open-oop echniques ha pefom expici channe esimaion: he ansmie emis pio vecos fo channe esimaion, and he eceive esimaes he channe fom he eceived pio signas. 1 he pio ovehead of he CS-based mehods is given by O n, whee is he numbe of non-zeo spaia channe pahs o he spasiy eve and is he numbe of ansmi anennas [16], [17]. When, which is ofen ue in massive MIMO sysems, his pio ovehead of he CS-based mehods epesens subsania educion of he ovehead as compaed wih he conveniona eas squaes S mehod, whose ovehead is given by O. In addiion, he pio ovehead O n emains he same ieevan o he numbe of uses in muiuse MIMO sysems, due o he open-oop chaaceisic of he CS echniques. hese CS-based channe esimaos have been deived based on he viua angua domain epesenaion of MIMO channes [18], [19], caed he viua channe mode, which descibes he channe wih espec o fixed basis funcions coesponding o he anges whose esouion is deemined by he spaia esouion of aays. As a esu, hei esimaion accuacy is imied by he aay esouion. Anohe difficuy encouneed when appying he CS-based schemes o mm-wave channe esimaion is caused by hei pio vecos: hey assume conveniona micowave communicaion whee he S wihou beamfoming is easonaby high and sugges he use of i.i.d. ademache vecos as pios. In hybid MIMO sysems ove a mm-wave channe use of such i.i.d. andom pios is difficu because he beamfomes, which ae empoyed o incease he S, yied dieciona aining beams ha ae coeaed wih each ohe. In his pape, we deveop an aenaive CS-based openoop channe esimao fo mm-wave hybid MIMO sysems. his scheme is based on he paameic channe mode wih quanized AoDs/AoAs, caed he ange gids, and he pio 1 A cosed-oop echnique ha pefoms coase channe esimaion by beam aining and fine channe esimaion by CS is inoduced in [15]. his ovehead, which is he numbe of measuemens needed fo ecoveing he spase veco in he CS fomuaion, has been deived fo diffeen seings, which incude ecovey wih high pobabiiy using he ohogona maching pusui OMP in [16] and he basis-pusui BP in [17]. vecos of he poposed scheme is geneaed by he cascade of he beamfome and he baseband MIMO pocesso. o channe esimaion, we adop he gid-based ohogona maching pusui OMP agoihm which has been poposed fo esimaing channes of muicaie undewae acousic channes [0] and ime-vaying ohogona fequency division muipexing ODM sysems [1]. he conibuions of his pape ae descibed as foows: We fomuae a spase signa ecovey pobem and deveop a gid-based OMP agoihm fo esimaing channes of hybid MIMO sysems. In he CS fomuaion, we sugges he use of a edundan dicionay consising of aay esponse vecos wih finey quanized ange gids which ae non-unifomy disibued in [0,π]. Specificay, we deemine he ange gids, denoed as { g },so ha { cos g } ae unifomy disibued in [1, 1. he angua esouion of he poposed scheme can be much fine han ha of he viua channe mode, and he nonunifomy disibued ange gids ae shown o educe he coheence of he edundan dicionay. he owe- and uppe-bounds of he sum-of-squaed eos SSE of he poposed OMP-based esimao ae deived anayicay: he owe-bound is deived by consideing he oace esimao ha assumes he knowedge of AoDs/AoAs, and he uppe-bound is deived based on he esus of he OMP pefomance guaanees. he esus indicae ha he SSE inceases wih he numbe of channe pahs o he spasiy eve. Efficien pio vecos ae designed fo given beamfomes: we design he baseband MIMO pocesso so ha he oa coheence of he equivaen sensing maix is minimized fo a given beamfoming maix which is assumed o be uniay. ee, he equivaen sensing maix is he poduc of he sensing maix and he dicionay, and he oa coheence is he sum of he squaed inne poducs of a pais of coumns in he equivaen sensing maix []. I is obseved ha he esimaion accuacy can be impoved significany by andomy pemuing he coumns of he beamfoming maix. he oganizaion of he pape is as foows. Secion II pesens he sysem mode. In Secion III, we fomuae he CS based channe esimaion pobem wih a edundan dicionay and pesen he gid-based OMP agoihm. he SSE is anayzed in Secion IV. In Secion V, he pio beam paens ae designed unde he beam powe consains fo a given beamfoming maix. Simuaion esus showing he advanages of he poposed schemes ove he exising mehods in ems of boh he esimaion accuacy and he aining ovehead ae pesened in Secion VI. inay, he concusion is pesened in Secion VII. oaions: Bod uppecase A denoes a maix and bod owecase a denoes a veco. Supescips A, A, A, A 1, A denoe he conjugae, he anspose, he conjugae anspose, he invese, and he pseudo-invese of a maix A, especivey. diag A 1,...,A epesens a bock diagona maix whose diagona enies ae given by {A 1,...,A }. a 0 and a ae he 0 and noms, especivey, and a n denoes he n-h eny of a veco a. he suppo of a veco a is defined as

3 37 IEEE ASACIOS O COMMUICAIOS, VO. 64, O. 6, JUE 016 ig. 1. A mm-wave sysem empoying hybid MIMO pocessos. supp a = {n a n = 0}. A is he obenius nom; A is he ace; and A n and A m, n denoe he n-h coumn and he m, n-h eny of a maix A, especivey. A I is he sub-maix of a maix A ha ony conains he coumns whose indices ae incuded in a se of coumn indices I. Simiay, a I is he sub-veco of a veco a ha ony conains he enies whose indices ae incuded in a se of indices I. I denoes he ideniy maix and O M, denoes he M a-zeo maix. vec A is a veco obained hough he vecoizaion of a maix A, andvec 1 A epesens a maix obained by he invese of vecoizaion. In addiion, fo an squae maix A, vecd A is an -dimensiona veco consising of he diagona enies of A he n-h eny of vecd A is given by A n, n. λ max A and λ min A epesen he maximum and he minimum eigenvaues of A, especivey. o M maices A and B, A B denoes he M maix of Konecke poduc beween A and B; anda B denoes he M maix of Khai-ao poduc [3] defined as A B = [A 1 B 1,...,A B ]. E [ ] ishe expecaion opeao, and P {B} denoes he pobabiiy ha an even B occus. II. SYSEM MODE In his secion, we pesen he signa mode fo he openoop beam aining and he channe mode in mm-wave communicaions. A. Signa Mode fo Open-oop aining We conside he hybid MIMO sysem shown in ig. 1, whee he ansmie and he eceive ae equipped wih and anennas, especivey, and boh of hem have chains 3 whee min,. ee, we assume ha and ae muipes of, and denoe and by Bock and Bock, especivey. he beamfomes ae assumed o be phased aay beamfomes, eaized using anaog phase shifes. o channe esimaion, he ansmie { uses pio beam paens denoed as f p 1 : } fp = 1, p = 1,...,,and he { eceive uses beam paens denoed as w q 1 : } wq = 1, q = 1,...,. ee, we aso assume ha and ae muipes of. 3 o simpiciy, we assume he same numbe of chains a he ansmie and he eceive. he poposed famewok can be exended o he case whee diffeen numbes of chains ae empoyed. Duing he aining peiod, he ansmie successivey emis is aining beams { } f p, and a he eceive each aining beam is eceived hough is beam paens { } w q. Since he eceive has chains, i can geneae beams{ simuaneousy and eceives he veco y q,p C 1, q 1,..., }, fo he p-h ansmi beam. Specificay, he q-h eceived veco fo he p-h ansmi beam is given by y q,p = W q f px p + W q n q,p, 1 whee [ x p is he ansmied ] pio symbo, W q = w q w q C, C epesens he channe maix, and n q,p C 1 is a noise veco wih C { 0,σn I. Coecing y q,p fo q 1,..., },we ge y p C 1 given by y p = W f p x p + diag W 1,..., W / [ ] n1,p,...,n, /,p ] / [ whee W = W 1,..., W C.oepesen he eceived signas } fo a ansmi beams, we coec y p yieding fo p { 1,..., Y = W X +, 3 [ ] whee Y = y 1,...,y C, = [ ] f 1,...,f C,and C is he noise maix given by [ [ ],..., diag W 1,..., W n / 1,1,...,n /,1 [ ] ] n 1,,...,n /,. he maix X C is a diagona maix wih { } x p on is diagona. houghou he pape, we assume idenica pio symbos so ha X = PI whee P is he pio powe. In he hybid MIMO famewok, he ansmi and eceive pocessing maices ae decomposed as = BB and W = W W BB, and he eceived signa 3 is ewien as Y = PW BB W BB +, 4 whee C and W C denoe he ansmi and eceive beamfoming maices, especivey, and

4 EE e a.: CAE ESIMAIO VIA OOOA MACI PUSUI O YBID MIMO SYSEMS 373 BB C and W BB C denoe he ansmi and eceive baseband pocessing maices, especivey. hese maices ae designed unde he foowing assumpions: i and W ae uniay maices whose enies have uni magniude; ii -beam pio beams geneaed by he coumns of cove he fu ange of AoDs, [ π,π]; and iii -beam pio beams geneaed by he coumns of W cove he fu ange of AoAs, [ π,π]; iv since hee ae chains, and W ae [ paiioned ino Bock ] and Bock sub- beams: =,1,...,, Bock and W = [ ] W,1,...,W, Bock whee, p C and W, q C. Simiay, BB and W BB ae bock diagona maices given by BB = diag BB,1,..., BB, Bock and W BB = diag W BB,1,...,W BB, Bock,whee BB, p C / Bock and W BB, q C heefoe, = BB can be epesened as [ = 1,..., Bock / Bock. ], 5 whee p =, p BB, p and 1 p Bock. Simiay W = W W BB can be epesened as [ ] W = W 1,...,W Bock, 6 whee W q = W, q W BB, q and 1 q Bock. 4 B. Channe Mode We use he paameic channe mode [6], [4] given by = α a θ a θ, 7 =1 whee is he numbe of scaees ha is assumed o be ess han he anenna sizes, < min,, α is he compex gain, and θ and θ ae he AoA and AoD of he -h pah, especivey. ee, we assume he use of a unifom inea aay UA whose aay esponse vecos ae denoed as a θ C 1 fo he eceive and a θ C 1 fo he ansmie. Dopping he subscip/supescip,,, he aay esponse veco of an UA is epesened as a θ = 1 [ 1, e j π λ d cosθ,...,e j π λ 1d cosθ], 8 whee λ is he signa waveengh, and d is he anenna spacing whichisassumedobed = λ. oe ha he aay esponse veco is an even funcion of θ. o simpiciy, each scaee is assumed o conibue a singe popagaion pah. he channe gains {α } =1 ae modeed by i.i.d. andom vaiabes wih disibuion C 0,σα. he AoAs and AoDs ae modeed by he apacian disibuion whose mean is unifomy disibued ove [0, π, and angua 4 he paiion of W in 6 is diffeen fom he paiion in : he fome paiions W ino Bock sub-maices, whie he ae paiions W ino / sub-maices. sandad deviaion is σ AS [6], [4]. he channe mode in 7 can be wien in maix fom as = A a A, 9 whee a = diag α 1,...,α, A = [ a θ 1,..., ] a θ C, and A = [ ] a θ 1,...,a θ C. In veco fom, 9 can be ewien as vec = A A vec a 10 = A A vecd a, 11 whee he fis equaiy foows fom he ideniy, vec ABC = C A vec B; he second equaiy hods because a is a diagona maix and A A C. C. S and Oace Esimaos o fomuae he channe esimaion pobem, i is necessay o vecoize he eceived signa maix Y in 4. Denoing vec Y by ȳ C 1, 4 is ewien as ȳ = P BB W BB W vec + n = PQ vec + n, 1 whee he fis equaiy comes fom he ideniy, vec ABC = C A vec B, n = vec, and Q = BB W BB W C. iven 1, a naua appoach o esimaing vec is o use he S mehod which diecy esimaes he enies of vec : he S esimae, denoed as vec S,is given by vec S = 1 Q Q 1 Q ȳ. 13 P he S souion equies so ha Q Q has fu ank. Use of he S souion fo mm-wave communicaion is difficu because, ae usuay age ineges and he dimension of he esimaion pobem can be excessivey age. his difficuy can be ovecome in he CS appoach because he numbe of enies o be esimaed in he CS fomuaion is popoiona o he spasiy eve whichismuchesshan. he oace esimao of esimaes he diagona channe gain maix a in 9 unde he assumpion ha AoDs and AoAs o equivaeny, A and A ae exacy known [5]. o deive he oace esimao, ȳ in 1 is ewien as foows: using 11 in 1 ȳ = PQ A A vecd a + n = PQ o vecd a + n, 14 C. he maix Q o whee Q o = Q A A can be epesened as Q o = BB A W BB W A, 15 because Q = BB W BB W and A BC D = AC BD. he oace esimao esimaes vecd a in

5 374 IEEE ASACIOS O COMMUICAIOS, VO. 64, O. 6, JUE 016 he S sense. om 14, he S esimae of vecd a, denoed as vec a o,isgivenby vecd a o 1 Q = o 1 Q o Q o ȳ. 16 P he oace esimao equies so ha Q o Q o has fu ank. hen he oace esimae of is given by o = A o a A, 17 whee a o is he diagona maix wih diagona enies given by vecd a o. he oace esimao is usefu fo examining he pefomance owe-bound of a CS-based channe esimao. III. POPOSED SPASE CAE ESIMAIO A. Pobem omuaion o appy he OMP agoihm o channe esimaion, we fis seec he se of quanized ange paamees, caed he gids, defined as = { g : g [0,π], g = 1,..., }, 18 whee 1 = 0 and = π. he gids { } g in 18 ae deemined so ha { cos } g ha appea in he definiion of he aay esponse veco in 8 ae unifomy disibued in [1, 1: specificay, g is deemined o saisfy cos g = g 1 1, 19 fo g {1,,..., }. hese gids ae disibued nonunifomy in he angua ineva [0,π]. oe ha he anges in 18 ae imied o [0,π] because he aay esponse veco in 8 is an even funcion; any g [ π,0] can be epaced wih g, giving he same aay esponse veco. I is assumed ha he numbe of gids is age han he anenna size, i.e., max {, }. Coecing a he aay esponse vecos wih anges { } g, we define he aay esponse maices Ā = [a 1,...,a ] and Ā = [a 1,...,a ], which consiue he dicionay of ou CS fomuaion. he ows of hese aay esponse maices ae ohogona, as shown beow. emma 1: When he ange gids { } g saisfy 19 and d = λ, he aay esponse maices Ā and Ā mee he foowing: Ā Ā = I and Ā Ā = I. 0 Poof: om 8 and 19, he n-h coumn of Ā C can be wien as Ā n = 1 [ e jπn 1, e jπ 1 n 1,..., e jπ g 1 1 n 1 jπ ] 1 1 n 1,...,e, 1 and he m, n-h eny of Ā Ā C is given by ] [Ā Ā m, n = 1 e jπm n + e jπ 1 m n + +e jπ 1 1 m n = g=1 jπ g 1 1 m n e. hus, he main m = n and off-diagona m = n enies of Ā Ā ae given by and ± e jπm n 1 = 0, especivey. his indicaes Ā Ā = e jπm n/ 1 I. In a simia manne, Ā Ā = I can be poved. his ype of ohogonaiy does no hod fo conveniona aay esponse maices wih unifomy disibued ange gids in [0,π] [10]. I wi be shown in Secions IV and V ha he ohogonaiy popey in emma 1 heps educe he coheence of he poposed CS fomuaion and design he pio beam paens. Using he aay esponse maices, he channe maix in 9 can be epesened as = Ā a Ā + E = + E, whee = Ā a Ā, which is caed he quanized-channe, is he appoximaed channe maix defined on he quanized ange space; a C is an -spase maix having non-zeo enies coesponding o he AoDs/AoAs; and E C is he quanizaion eo maix caused by quanizing he AoDs/AoAs by he gid poins in. oe ha a C is no a diagona maix, unike he diagona maix a C in 9. he fis em in he igh-handside S of educes o he viua channe mode when = = in [18], [19]. efeing o 10, can be ewien in veco fom as vec = Ā Ā vec a + vec E 3 Using 3 in 1, we have ȳ = PQ Ā Ā vec a +ē + n = P Q vec a +ē + n, 4 whee Q = Q Ā Ā C and ē = PQ vec E C 1. he maix Q can be ewien as Q = BB Ā W BB W Ā, 5 because Q = BB W BB W and A B C D =AC BD. Esimaing he -spase veco vec a C 1 in 4 is a CS pobem wih a edundan dicionay, because Ā Ā C and < max {, }. he maix Q is he sensing maix consising of he and baseband pocessos of he hybid MIMO sysem. he poduc of he sensing maix and he dicionay is efeed o as he equivaen sensing maix, denoed as Q. 5 When = =, he edundan 5 Compaing Q o C in 15 and Q C in 5, he fome is he sysem maix fo he S pobem and he ae is he equivaen sensing maix fo he CS pobem.

6 EE e a.: CAE ESIMAIO VIA OOOA MACI PUSUI O YBID MIMO SYSEMS 375 Agoihm 1 OMP Based mmwave Channe Esimao equie: sensing maix Q, measuemen veco ȳ, and a heshod δ 1: I 0 = empy se, esidua 1 = 0, 0 =ȳ, se he ieaion coune = 1 : whie 1 >δdo 3: j = ag max Qi 1 ind AoD/AoA pai i=1,..., 4: I = I 1 { j} Updae AoD/AoA pai se ȳ 5: h = ag min P Q I h Esimae channe h gains 6: =ȳ P Q I h Updae esidua 7: = + 1 8: end whie 9: ĥ a i = h 1 i fo i I 1 and ĥ a i = 0ohewise 10: eun a CS = vec 1 ĥ a dicionay Ā Ā educes o he ohonoma dicionay ha can be deived fom he viua channe mode. he opimizaion pobem fo CS based channe esimaion can be wien as vec a CS ȳ = ag min P Q vec a a subjec o vec 0 a =, 6 whee a CS denoes he CS esimae of a. om 6 he CS esimae, denoed as CS, of he desied quanized-channe is given by CS = Ā a CS Ā, 7 and he SSE is defined as he obenius nom of he diffeence beween and CS, i.e., CS. he opimizaion pobem in 6 is a non-convex opimizaion wih 0 nom and is difficu o sove. he OMP agoihm pesened beow is one of he mos common and simpes geedy CS agoihms ha can sove 6. B. Channe Esimaion via OMP he OMP agoihm soving 6 is summaized in Agoihm 1. A he -h ieaion, his agoihm chooses he coumn of Q ha is mos songy coeaed wih he esidua 1 sep 3, and updaes he coumn index se sep 4. Each coumn index obained in sep 3 coesponds o a pai of gids epesening he quanized AoD/AoA. hen, he channe gains associaed wih he chosen gid poins ae obained by soving he S pobem in sep 5. In sep 6, he conibuions of he chosen coumn vecos o ȳ ae subaced o updae he esidua 1. his pocedue is epeaed uni 1 fas beow he pedeemined heshod δ. ee we assume ha he spasiy eve is unknown. In he case whee is known, a sopping cieion ha depends on woud be pefeed. In sep 9, he veco ĥ a C 1 is consuced so ha is i-h eemen, 1 i, ĥ a i = h 1 i if i I 1 and ĥ a i = 0, ohewise. he desied esimae is given by a CS = vec 1 ĥ a. he compuaiona oad of he OMP agoihm is popoiona o and can be excessive in mm-wave channe esimaion whee > max {, } and he anenna sizes ae age. Specificay, in sep 3, he muipicaion of Qi and 1 needs O compuaions and he maximum seecion needs O compuaions; in sep 5, assuming he use of he Q facoizaion of Q I 1 and he modified am-schmid agoihm, soving he S pobems equies O I,whee I denoes he cadinaiy of I. he compuaiona compexiy of he OMP agoihm can be educed by empoying efficien OMP agoihms such as he mui-sage OMP [1]. IV. AAYSIS O SUM O SQUAED EOS In his secion, we anayze he SSE of he poposed OMP esimao, defined as CS. Specificay, a pefomance owe-bound is deived by consideing he mean SSE, E [ o ], of he oace esimao, and a pefomance uppe-bound is deived based on he esus of OMP pefomance guaanees [4]. he vaidiy of he deived bounds wi be examined hough compue simuaion in Secion VI. A. owe-bound Anaysis Conside he oace esimao which esimaes he channe gain a in 9 unde he assumpion ha A and A ae known. he mean SSE of he oace esimao can be wien as [ E o ] [ vec = E vec o ] [ = E A A vecd a vecd a o ], 8 whee he second equaiy foows fom 11 and 17. A owebound of he mean SSE is deived as foows. emma : Suppose ha E [ n n ] = σn I in 14. he mean SSE in 8 is owe-bounded as fo n [ E o ] κ σn P 9 whee κ = λ min B B / Q o Q o and B = A A. Poof: o simpify noaions, e B = A A and e = vecd a vecd a o. he mean SSE in 8 can be wien as [ ] [ ] E Be λ min B B E e, 30 whee he inequaiy hods due o he ayeigh-iz aio [7]. Using 14 in 16, vecd a o can be ewien as vecd a o = vecd a 1 C n, P whee C = Q o Q o 1 Q o. hus e = 1 C n, and P [ ] E e = 1 [ ] P E C n = σ n P CC. 31

7 376 IEEE ASACIOS O COMMUICAIOS, VO. 64, O. 6, JUE 016 Since CC = Q o Q o 1 = Q o Q o 1 C,hen CC = Q λ o 1 Q o =1 1 Q = λ o 1 Q o =1 =1 Q 1/λ o Q o 1 = =1 λ Q o Q o = Q o Q o whee he inequaiy comes fom he inequaiy beween he aihmeic and hamonic means [8], and he hid equaiy hods because he -h eigenvaue of Q o Q o 1 can be wien as λ Q o Q o 1 = 1/λ Q o Q o. hus E [ e ] in 31 saisfies [ ] E e σ n P Q o 3 Qo he owe-bound in 9 can be obained by using 3 in 30. B. Uppe-Bound Anaysis We deive an uppe-bound of he SSE unde he foowing assumpions: i = and = ; ii he ansmi and eceive maices, = BB and W = W W BB in 4, ae uniay; iii boh a and E in 3 ae unknown deeminisic maices; and iv he spasiy eve of a is known. he SSE of he spase esimaion pobem in 6 can be examined based on he coheence of he equivaen sensing maix Q defined as μ Q = max Qm Q n m,n,m =n his coheence can be decomposed in ems of he coheences of he ansmi and eceive aay esponse maices dicionaies, denoed as μ Ā and μ Ā, especivey, as shown beow. emma 3: When he ange gids { } g saisfy 19 and d = λ, he coheence of he equivaen dicionay Q = BB Ā W BB Ā W in 5 can be expessed as μ Q { } = max μ BB Ā,μ WBB W Ā = max { μ } Ā,μ Ā { } π = max D, π D, 34 and D φ is he Diiche kene defined as 1 D φ = 1 e jnφ = 1 jφ 1/ sin φ/ e sin φ/. 35 n=0 Poof: he fis equaiy in 34 foows fom he decomposiion of he coheence of Konecke poduc [9], and he ig.. he coheence μ Q agains he numbe of gids. second equaiy is due o he assumpion ha BB and W W BB ae uniay. he coheence of each aay esponse maix is μ Ā = max a i a j 1 i, j,i = j a = max 1 i, j,i = j D π cos j cos i b = D π cos i+1 cos i π = D, 36 whee he equaiy a foows fom 8 when d = λ ; he equaiy b is due o he foowing facs: i cos i cos j depends ony on i j, because of 19; ii D π cos i cos j is maximized when j = i + 1. In a simia manne, μ Ā = D π can be deived. Since D φ monoonicay inceases and conveges o one as φ deceases fom π/ o zeo, he coheences μ Ā, μ Ā and μ Q monoonicay incease and convege o one as inceases fom max {, } o infiniy. his is iusaed in ig. fo μ Q. If he gids wee chosen such ha { } g ae unifomy disibued in [0,π], he coheence μ Ā woud be D π cos 0 cos π which is in 34, because 1 cos π < π. age han π D his obsevaion indicaes ha he poposed non-unifomy disibued ange gids educe he coheence μ Q. Ou deivaion fo he SSE uppe-bound sas wih he foowing inequaiy showing ha he SSE is uppe-bounded by he sum of he quanized-channe esimaion eo and he ange quanizaion eo: CS = + E CS = vec + E vec CS vec vec CS + vece, }{{}}{{} Quanizaion eo Quanized channe esimaion eo 37

8 EE e a.: CAE ESIMAIO VIA OOOA MACI PUSUI O YBID MIMO SYSEMS 377 whee is he quanized-channe in, and CS is he CS esimae of in 7. Since vec = Ā Ā vec a and vec CS =Ā Ā vec a CS, he quanized-channe esimaion eo in he S of 37 can be epesened as vec vec CS = Ā Ā vec a vec a CS. 38 We deive an uppe bound of he quanized-channe esimaion eo when supp vec a = supp vec a CS he condiion unde which his equaiy hods wi be pesened in emma 5 beow. emma 4: Suppose ha he suppos of vec a and vec a CS ae he same. e I denoe hei suppo: I = supp vec a = supp vec a CS and I =. hen he quanized-channe esimaion eo in 38 saisfies vec vec CS μ Ā Ā vec a vec a CS. 39 Poof: he S of 38 saisfies he foowing: Ā Ā vec a vec a CS = Ā Ā I vec a I vec a CS I Ā Ā I vec a I vec a CS I = Ā Ā I vec a vec a CS, whee he inequaiy comes fom he Cauchy-schwaz inequaiy. o show ha Ā Ā I 1+ 1 μ Ā Ā, Ā Ā we define I = Ā I Ā I. hen he squaed speca nom saisfies Ā Ā I = λ max I.Since a he diagona enies of I ae equa o one, I can be epesened as I = I + Ɣ, whee Ɣ C is a maix having zeo diagona enies. om he definiion of he coheence in 33, we obseve ha he off-diagona enies of Ɣ o equivaeny I ae uppe-bounded by μ Ā Ā = μ Ā Ā and define Ɣ C by epacing a he off-diagona enies of Ɣ by he uppe-bound μ Ā Ā. ow we appy he eshgoin cice heoem [30] o I + Ɣ o find an uppe-bound of λ max I + Ɣ. o be specific, e A = I + Ɣ. hen he eshgoin cice heoem saes ha A D, whee { A is he se of eigenvaues of A and D = z C : z 1 Ɣ m=1, m }. hus λ max I + Ɣ μ Ā Ā. he eshgoin cice heoem aso indicaes ha he uppe-bound of λ max I is ess han o equa o ha of λ max I + Ɣ. hus, he squaed speca nom saisfies Ā Ā I μ Ā Ā which is he desied esu. =1 In he S of 39, vec a vec a CS epesens he spase esimaion eo of he CS pobem in 6. In wha foows, an uppe-bound of his esimaion eo is deived by exending he OMP pefomance guaanees in [6] o he case of 4 whee boh he quanizaion eo ē and noise n exis. emma 5 OMP Pefomance uaanees: Conside he spase esimaion pobem in 6 wih he spasiy eve. Suppose ha P min a 1 1 μ Q σ n 1 + β og + P vec E, 40 whee min a is he smaes magniude of he non-zeo enies of a in 3, σ n is he sandad deviaion of n, and β is a posiive consan. hen, wih pobabiiy a eas 1 β π 1 he OMP coecy idenifies he supp vec a, and vec a CS saisfies vec a vec a CS 1 1 μ Q 1 + β σ n P og + vec E. 41 { Poof: We define an even B = max Q m n +ē } <τ fo τ = 1 m σ n 1 + β og + P vec E and fis show ha P {B} 1 β π 1. oe ha max Q m n +ē 1 m max Q m n + max Q m ē 1 m 1 m max Q m n + max Q m 1 m 1 m ē = max Q m n + P vec E, 1 m whee he equaiy hods because Q m = 1foam and ē = P vec E. ence, { } P {B} = P max Q m n +ē <τ 1 m { P max Q m n + P vec E 1 m <σ n 1 + β og + } P vec E { } = P max Q m n <σ n 1 + β og 1 m 1, 1 π β whee he as inequaiy foows fom [31, Poposiion 5]. ex, we deive a condiion ha he OMP coecy idenifies he suppo I = supp vec a. Suppose ha he even B occus and he condiion 40 is saisfied. hen, by modifying he poof of [6, emma 3] o ake accoun of compex

9 378 IEEE ASACIOS O COMMUICAIOS, VO. 64, O. 6, JUE 016 vaiabes, i can be shown ha Q m ȳ > max Q m ȳ. 4 max m I m / I he inequaiy 4 is he condiion ha he OMP coecy idenifies I = supp vec a [6]. When he suppo I is successfuy idenified, he CS esimao becomes he oace esimao epesened as vec a o I = P 1/ Q Iȳ, and is squaed-eo is wien as vec a I vec a o I = vec a I Q I Q I vec a I P 1/ Q I n +ē = P 1/ Q I n +ē = P 1/ Q I Q 1 I Q I n +ē Q I Q 1 I P 1/ Q m n +ē m I 1 1 μ Q 1 + β σ n P og + vec E, whee he as inequaiy is deived by using he definiion of B and he fac ha Q I Q 1 = I 1/λ min Q I Q I 1/ 1 1 μ Q. ee, he inequaiy can be poved by appying he eshgoin cice heoem o λ min Q I Q I as in he poof of emma 4. he condiion in 40 equies min a o be age han he sum of a consan muipe of he noise sandad deviaion and he quanizaion noise. his is needed o ensue ha he suppo of vec a wi be idenified wih high pobabiiy. uhemoe, we can make some ineesing obsevaions fom his condiion. Because he S of 40 is posiive, he condiion can be educed o he we-known bound [3] given by < μ Q 43 o μ Q 1 < he bounds in 43 and 44 ead o he foowing inequaiies: and 1 μ Q + μ Q > μ Q > 0, 45 μ Q < 1 + μ Q Due o 45, he denominao in he S of 41 is aways nonzeo. In addiion, he inequaiy in 44 imis he gid size because he S of 44 is much ess han one even fo asma and μ Q is a monoonicay inceasing funcion of. emma 5 is vaid ony fo hose vaues ha make μ Q mee 45. he foowing exampe iusaes emma 5. Exampe 1: e = = 3, a min = 3, = 3, β = 0.5, vec E = 1, and 10og 10 P/σ n = 0dB. hen μ Q , and he condiion 40 is saisfied fo 38. When = 38, he pobabiiy fo successfuy idenifying he supp vec a is given by 1 β π 1 = 0.99 and he uppe-bound in 41 is ex, we deive an uppe-bound of he quanizaion eo vec E in 37. om 7 and 9, vec = vec a, a θ =1 a θ = vec a, a θ =1 a θ = a, a =1 θ a θ, whee he hid equaiy hods due o he ideniy, vec ABC = C A vec B we se A = a θ, B = a,, C = a θ. o simpify noaions, we define g = a, a θ a θ. hen, vec = g. 47 =1 Simiay, vec can be epesened as vec = ḡ, 48 whee ḡ = a, a a and and ae he quanized anges of θ and θ, especivey. Suppose ha ḡ is he pojecion of g ono he ine hough he veco a a and ha and ae seeced fom he se in 18 accoding o he foowing: {, } = ag max a, θ a θ a a. 49 hen, since a a = 1, we ge a, = a a g. 50 Using 47-50, an uppe-bound of he quanizaion eo is deived as foows. emma 6: Suppose ha ḡ is obained via he pojecion associaed wih 50. When he ange gids { } g saisfy 19 and d = λ, he quanizaion eo in 37 is bounded by =1 vec E a, =1 π π 1 D D = EUB, 51 fo a in 9.

10 EE e a.: CAE ESIMAIO VIA OOOA MACI PUSUI O YBID MIMO SYSEMS 379 Poof: vec E = vec vec = =1 g =1 ḡ = =1 a, a θ a θ =1 a, a a. om 50, a, = a a g = a, a a a θ a θ 5 = a, a a θ a a θ a = a, D π cos cos θ D π cos cos θ = a, D π D π 53 whee he equaiy a foows fom 8 when d = λ ; = cos cos θ and = cos cos θ. ee, he hid equaiy hods due o he ideniy A BC D = AC BD. he quanizaion eo is epesened as vec E = a, a =1 =1 θ a θ a, a a a a, a =1 θ a θ D π D π a a b = a, 1 D =1 π D π c π π a, 1 D =1 D, 54 whee a foows fom he Cauchy-Schwaz inequaiy. he equaiy b hods because a θ a θ = 1, a a = 1, and a θ a θ a a = D π D π whee he hid equaiy foows fom 5 and 53; he inequaiy c hods because boh and ae ess han o equa o 1/, due o 19. he Diiche kene D π/ in 51 monoonicay inceases and conveges o one as inceases fom / o infiniy. hus, as expeced, he quanizaion eo uppe bound monoonicay deceases o zeo as inceases. Combining emmas 3-6, we have he desied uppe-bound fo CS in 37. heoem 1: Suppose ha he condiion in 40 is saisfied fo given a and E, and ha he anges ae quanized accoding o 19. he SSE in 37 is uppe-bounded as CS μ Ā Ā 1 1 μ Q 1 + β σ n P og + E UB + E UB μ Q μ Q 1 + β σ n P og + E UB + E UB, 55 wih pobabiiy a eas 1 β π 1, whee μ Q = μ Ā Ā = max { D π/, D π/ } fom emma 3; E UB = vec E =1 a, 1 D π π D fom emma 6; and he second inequaiy comes fom 45. Poof: his heoem is a consequence of emmas 3-6. Cooay 1: In he high S egime whee σ n P og is appoximaey zeo, he uppe-bound in 55 educes o 1 + E UB in he imi as μ Q appoaches one. Poof: In he high S egime, he imi of he bound in 55 as μ Q 1 can be wien as 1 + μ Q μ Q E UB = 1 + μ 1 Q E UB im μ Q 1 im μ Q 1/ = 1 + E UB. Since 0 < μ Q < 1 fom 45 and 46, 1 is he supemum of μ Q and 1 + E UB is an SSE uppebound when μ Q appoaches is maximum vaue. his bound indicaes he foowing: i he SSE ends o incease wih he numbe of scaees and ii he SSE ends o deceases as he numbe of gids inceases. hese obsevaions ae inuiivey easonabe and wi be confimed hough compue simuaion in Secion VI. V. PIO BEAM PAE DESI In [], and [33] [35], i has been obseved ha a caefuy designed sensing maix, which minimizes he oa coheence of he equivaen dicionay, can impove he CS pefomance, as compaed wih andomized sensing maices which ae popua in CS and have been used fo he CS-based channe esimaos in [1] and [13]. In his secion, we conside he design of he sensing maix Q in 4, fo a given dicionay Ā Ā, by minimizing he oa coheence unde he beam consains fo f p and w q in 1. his design pobem invoves join opimizaion of he and baseband pocessos, which is a difficu ask. o avoid he join design, we sha focus on he design of he baseband pocessos, BB and W BB, unde he assumpion ha he pocessos, and W,aegiven in 4 hey ae assumed o be uniay maices. he oa coheence, denoed as μ Q,isdefinedby μ Q, = Qm Q n 56 m n,m =n whee Q is he equivaen sensing maix in 5. he oa coheence is he sum of he squaed inne poducs of a

11 380 IEEE ASACIOS O COMMUICAIOS, VO. 64, O. 6, JUE 016 pais of coumns in Q o equivaeny he sum of a squaed off-diagona enies of Q Q. As in he case of he coheence in emma 3, he oa coheence can be epesened in ems of he ansmi and eceive oa coheences, which ae denoed as μ BB and μ Ā WBB Ā W, especivey, as shown beow. emma 7: he oa coheence of equivaen sensing maix Q = BB Ā W BB Ā W in 5 is uppe-bounded as μ Q μ BB Ā μ WBB W Ā 57 Poof: Using he noaions = BB Ā and W = WBB W Ā in he S of 57, we ge m n,m =n = m n,m =n a = m n,m =n b m n,m =n m n,m =n m n,m =n m n,m =n Qm Q n m Wm n W n m n Wm W n m n 4 Wm W n 4 m n, Wm W n 58 whee he equaiy a foows fom A BC D = AC BD and he inequaiy b hods due o he Cauchy-Schwaz inequaiy. Based on his emma, we decompose he design pobem fo minimizing μ Q ino wo sepaae designs: he design of BB minimizing μ BB and ha of Ā WBB fo minimizing μ WBB W Ā. ex we descibe he pocess fo designing BB, saing wih he foowing emma. emma 8: When he ange gids { } g saisfy 19 and d = λ, minimizing he oa coheence μ BB is Ā equivaen o minimizing BB BB I. Poof: Minimizing μ BB Ā wih espec o BB is equivaen o minimizing Ā BB BB Ā I because he off-diagona ems in Ā BB BB Ā ae he inne poducs ha appea in 56. ow, Ā BB BB Ā I = + I = + I + = BB Ā Ā BB I + = BB BB I +, 59 whee he fis hee equaiies foow fom he eaion beween he obenious nom and he ace, and he as equaiy hods because Ā Ā = I, as shown in emma 1, and = I. his compees he poof. he opimizaion pobem unde he beam nomaizaion consain in 1 is wien as min BB BB BB I subjec o BB m = 1, m = 1,...,. 60 o he bock diagona BB = diag BB,1,..., BB, Bock assumed in ou sysem mode, he objecive funcion can be decomposed ino he foowing bock opimizaion pobems: min BB, p BB, p BB, p I subjec o BB, p m = 1, / Bock m = 1,...,. 61 Bock his opimizaion pobem 61 is non-convex due o he muipe spheica consains, and may no be efficieny sovabe [36]. ounaey, we can eax he individua powe consains in 61 ino a singe consain, as shown beow. heoem : he souion of 61 can be obained by soving he foowing opimizaion pobem: min BB, p BB, p BB, p I / Bock subjec o / Bock m=1 BB, p m =. 6 Is souion is given by MC BB, p = Ū I Bock O Bock, Bock Bock V, 63 whee 1 p bock =, Ū C, and V C / Bock / Bock ae abiay uniay maices. he supescip MC in he ef-hand-side S of 63 sands fo minima oa coheence MC. he coumns of MC BB, p saisfy he individua consains of 61. Poof: Conside he singua vaue decomposiion SVD of BB, p = U V whee U C and V C / Bock / Bock ae uniay maices and = diag λ 1,...,λ Bock O Bock, Bock

12 EE e a.: CAE ESIMAIO VIA OOOA MACI PUSUI O YBID MIMO SYSEMS 381 wih singua vaues λ j. he objecive funcion in 6 can be ewien as min BB, p BB, p BB, p I / Bock = min V U U V I / Bock = min V V I / Bock = min V I / Bock V = min I / Bock. In a simia manne, he consain of 6 can be wien as / Bock m=1 BB, p m = / Bock j=1 λ j. ence, he pobem 6 educes o / Bock min λ λ j=1 j j subjec o / Bock j=1 λ j = Bock. 64 his pobem can be soved by he mehod of agange muipies. he agangian is given by λ j,ν = / Bock λ j=1 j + ν / Bock j=1 λ j Bock, 65 whee ν is he agange muipie. Since λ i,ν is a convex quadaic funcion of λ j, we can find he opima λ j by soving λ λ j j,ν = λ j + ν = 0. he opima λ j is given by λ j = 1foa j = 1,...,. heefoe, he maix Bock of opima singua vaues is, I Bock O Bock, Bock and he opima MC BB, p he opima MC BB, p MC BB, p MC BB, p = VI / V Bock = I / Bock. In a simia manne, we can deive he opima W BB minimizing μ WBB Ā W.he q-h bock of he opima W BB wih MC is given by W MC BB, q = Ū can be wien as 63. o each m, m saisfies he consain in 61 because I Bock whee 1 q bock O Bock =, Bock V, 66, and boh Ū C and V C / Bock / Bock ae abiay uniay maices. ow, we compae he oa coheences of he poposed MC beam paens and he conveniona beams geneaed by empoying andomized BB and W BB consising of i.i.d. ig. 3. isogams of he magniudes of off-diagona enies of Q Q when = = 3, = = 4, = 8, = 100, and 3 3 D maix is used fo pocessing. a andomized BB and W BB consising of i.i.d. ademache andom vaiabes. b Poposed opima BB and W BB wih MC when Ū and V ae D maices. ademache andom vaiabes [1], [13]. Boh he beam paens use he D maix fo pocessing and W. ig. 3 shows hisogams of he magniudes of off-diagona enies of Q Q when = = 3, = = 4, = 8, and = 100. he poposed opimizaion ends o educe he magniudes of off-diagona enies and eshapes he hisogam such ha hee is a maked shif owads he oigin. he educion of he oa coheence is significan: he nomaized oa coheences, μ Q /, ae fo he andom BB /W BB, and fo he opima BB /W BB wih MC. Some addiiona emaks of inees iusaing he chaaceisics of he opima aining and eceiving beams ae pesened beow. 1 Ahough any uniay Ū and V in 63 and 66 minimizes he oa coheence, diffeen ypes of uniay Ū, V maices esu in diffeen coheence vaues, defined in 33. o exampe, if we empoy he D maix as Ū, V, he coheence is fo he paamees in ig. 3. he coheence inceases o 0.8 and fo he discee cosine ansfom DC and he ideniy maix, especivey. Since he sysem pefomance can be impoved fuhe by educing he coheence, we sugges he use of he D maix as Ū, V in 63 and 66. Due o he bocking naue of hybid MIMO pocesso, he magniudes of he enies of Q Q exhibi a bocking paen, as shown in ig. 4a, fo he foowing paamees: = = 3, = = 4, = 8, and = 3. In his figue, age vaues of he inne poducs enies of Q Q ae cuseed, esuing

13 38 IEEE ASACIOS O COMMUICAIOS, VO. 64, O. 6, JUE 016 ig. 4. Magniudes of he enies of Q Q when = = 3, = = 4, = 8, and = D maix is used fo pocessing. a o coumn pemuaion. b andomy pemuing he coumns of and W. in a bocking paen. his paen ends o cause pefomance degadaion because he channe consiss of cuses of significan pahs and he pefomance fo esimaing he AoDs/AoAs of each cuse of pahs ends o be imied by one o wo cuses of age inne poduc vaues. heefoe, i is desiabe o emove he paen by speading ou he age vaues. ounaey, his can be achieved by andomy pemuing he coumns of and W. ig. 4b shows he magniudes of he enies of Q Q afe he andom pemuaion. oe ha age vaues of he inne poducs ae spead ou, and hee is no bocking paen. he oa coheence of Q afe he pemuaion is equivaen o ha of Q wihou pemuaion. oweve, as demonsaed in he foowing secion hough simuaion, consideabe pefomance impovemen can be achieved by andomy pemuing he coumns of and W. 3 he opima baseband pocesso WBB, MC q in conjuncion wih he coesponding pocesso W, q exhibis a desiabe popey ha he join and baseband pocessing peseves he whieness of noise: specificay, [ ] E WBB, MC q W, q nn W, q WBB, MC q = σn WBB, MC q W, q W, q WBB, MC q = σ n I / Bock, 67 whee n C 1 is he whie noise veco wih C 0,σn I. 67 indicaes ha he noise veco afe he and baseband pocessing emains i.i.d. aussian wih C 0,σn I / Bock so ha no pewhiening is needed. In conas, his popey does no hod when andomized W BB, q is empoyed. 4 he ansmiing beams and eceiving beams ae used fo obaining he measuemens which ae he enies of ȳ in 1 and 4. In he case of S esimaion in 13, he numbe of measuemens needed fo a non-singua Q Q is a eas =. On he ohe hand, he OMP-based esimao needs O n measuemens ha can be much ess han [37], [38]. o exampe, if = = 3, = 5, and = 100, hen n = =. eeafe, he cases wih = and < ae efeed o as fu-aining and paia-aining, especivey. 5 he opima paens in 63 and 66 ae designed by minimizing he oa coheence and shoud be usefu fo any ype of CS agoihms [5] ha can be appied o sove 6. uhemoe, i can be seen fom [39] ha he opima beam paens ae aso opima fo =. he S esimao in 13 when In ou simuaion, he deived beam paens ae used fo boh he OMP and S esimaos. VI. SIMUAIO ESUS he pefomance of he poposed channe esimao is examined hough compue simuaion wih he foowing paamees. he ansmie and he eceive ae equipped wih UAs wih = {3, 64} and = 8 Bock = Bock = 4. o pocessing and W, he D maices ae empoyed. he pio beam paens MC BB, p and WMC BB, q ae designed using 63 and 66, whie seing Ū and V a D maices. he esus in his simuaion ae obained hough 500 channe eaizaions wih σα = 1and σ AS = 15. We se δ = 0.1σn in sep of Agoihm 1. he poposed OMP esimao, emed he OMP-edundan, is compaed wih he S esimao in 13, he oace esimao in 16, and he OMP esimao based on he viua channe mode [13], emed he OMP-uniay. he meics fo pefomance compaison ae he nomaized mean squae eo MSE and he achievabe[ speca efficiency ASE, ] which ae defined as 10og 10 E Ĥ / and I og + P n 1 W W, especivey, fo he channe esimae Ĥ. ee and W ae he opima pecode and combine, especivey, which ae designed via he SVD

14 EE e a.: CAE ESIMAIO VIA OOOA MACI PUSUI O YBID MIMO SYSEMS 383 ig. 5. MSE pefomance agains P fo he S, OMP-uniay, and OMP-edundan esimaos fo he hee ypes of aining beams when = = 3 and = = 4. of Ĥ; n = σ n W W see he Appendix fo deais of ASE deivaion. wo ypes of signa-o-noise aios Ss ae consideed: one is he pio-o-noise aio P, defined as 10og 10 P/σ n ; and he ohe is he daa-o-noise aio D, 10og 10 P/σ n, whee P and P denoe he pio powe and he daa powe, especivey. We whi show he MSE cuves agains P and he ASE cuves agains D. his secion consiss of wo pas. In he fis pa, he pefomance of he poposed scheme is compaed wih hose of he ohe esimaos, whie empoying vaious aining beam paens. In he second pa, he poposed scheme wih he opima aining beam is anayzed, and he MSEs of he poposed scheme ae compaed wih he SSE owe- and uppe-bounds deived in Secion IV. A. Pefomance Compaison We conside hee ypes of aining beam paens. i BB /W BB ae obained by geneaing i.i.d. ademache andom vaiabes and /W ae he D maices wihou any coumn pemuaion. his beam is caed he andom beam. ii BB /W BB ae he MC beams in 63 and 66, and /W ae he D maices wihou any coumn pemuaion. his is caed he MC beam wihou pemuaion. iii BB /W BB ae he MC beams, and /W ae he pemued D maix whose coumns ae andomy pemued. his is caed he MC beam wih pemuaion. I is assumed ha = =. he numbe of scaees in he channe spasiy eve is se a = 5, and he numbe of ange gids fo he OMP-edundan is se a = 180. ig. 5 shows he MSEs of he S, oace, OMP-uniay, and OMP-edundan agains P when = = 3 and ig. 6. MSE pefomance agains he numbe of aining beams when = =,P= 0dB, and he MC aining beams wih pemuaion ae empoyed. a = = 3, b = = 64. = = 4 = 4. his is a case of paiaaining and, compaed o he fu-aining = 3,he numbe of measuemens is educed by 9/16. As expeced, he MSEs of he oace esimao, which assumes he knowedge of AoDs/AoAs, ae smae han hose of he ohes. Among he pacica esimaos, he poposed OMP-edundan aways oupefoms he ohes fo each aining beam paen. he S mehod pefoms he wos because of paia-aining. Compaing he beam paens, he MC beams wih pemuaion exhibi he bes pefomance, whie he andom beams exhibi he wos. he pefomance gain achieved by pemuaion is significan: in he case of he OMP-edundan, he MSE is educed by abou 10dB when P = 0dB. ig. 6 shows he MSEs agains he numbe of aining beams when P = 0dB, = {3, 64}, andhe

15 384 IEEE ASACIOS O COMMUICAIOS, VO. 64, O. 6, JUE 016 ig. 7. ASE agains D when Speca efficiency agains D when = = 3, = = 4, =, P = 0dB and he MC aining beams wih pemuaion ae empoyed. ig. 8. ASE agains he numbe of aining beams when = = 3, = =, =, P = D = 0dB, and he MC aining beams wih pemuaion ae empoyed. MC beams wih pemuaion ae empoyed. As he numbe of aining beams inceases, he MSEs of a hee esimaos decease monoonicay. In he case of he S esimaion, he MSE dops apidy when is inceased fom 56 o 64. his happens because Q Q in 13 becomes nonsingua when = 64. On he ohe hand, he OMP and he oace cuves end o decease sowy. Again, among he pacica esimaos, he OMP-edundan oupefoms he ohes; is MSE is consideaby ess han ha of he S mehod, which is ue even fo he cases of fu-aining. As a esu, he OMP-edundan wih paia-aining can pefom ike he S wih fu-aining. o exampe, in ig 6b, he pefomance of he poposed wih = 3 paia aining is compaabe o ha of he S wih = 64 fu-aining. his indicaes ha he aining ovehead can be educed significany by he OMP edundan. ig. 7 shows he ASE cuves agains D when = = 3, = = 4, =, P = 0dB, and he MC beams wih pemuaion ae empoyed. I is ineesing o noe ha he ASE cuve of he poposed OMP-edundan is cose o ha of he oace esimao fo a D vaues, whie he ae amos oveaps wih he idea ASE cuve assuming he pefec CSI. Compaing he ASEs of he pacica esimaos, he ASE gain achieved by he poposed scheme can be significan. o exampe, he gain ove he S mehod is abou 5bis/sec/z when D = 0dB, and he gain ends o incease wih he D. ig. 8 shows he ASE cuves agains he numbe of beams when = = 3, =, P = D = 0dB, and he MC beams wih pemuaion ae empoyed. he ASEs of he OMP-based esimaos end o emain consan fo 0, whie ha of he S esimao monoonicay inceases wih.when 0, he pefomance gap beween he poposed and he oace esimaos is ess han 0.4bis/sec/z. ig. 9. MSE pefomance agains he numbe of gids when = 5, P = 0dB, and {4, 3}. B. Anaysis of Poposed OMP he MSEs of he OMP-edundan ae obained fo vaious vaues of and, and he esus ae compaed wih he SSE owe-bound κ σn /P in 9 and he uppe bound, 1 + E UB, deived in Cooay 1. he P is se a P = 0dB. Boh fu-aining = 3 and paia-aining wih = 4 ae consideed. ig. 9 shows he MSE cuves of he OMP-edundan agains he numbe of gids. o compaison, he MSEs of he oace esimao and he fuy ained S esimao, which ae consan fo diffeen vaues of, and he SSE owe-/ uppe-bounds ae aso shown. Boh he fuy- and paiayained OMPs oupefom he S when 6, and he

16 EE e a.: CAE ESIMAIO VIA OOOA MACI PUSUI O YBID MIMO SYSEMS 385 uhe wok in his aea incudes he appicaion of moe sophisicaed CS agoihms han he OMP. Vaious geedy agoihms and 1 minimizaion agoihms [5] can be used fo soving he channe esimaion pobem in 6. uhemoe, he off he gid echniques ha can sove CS pobems wih inepoaion-based agoihms [40] o wihou defining he gids [41] woud be usefu fo educing he ange quanizaion eo, and hei appicaion o hybid MIMO channe esimaion needs invesigaion. ig. 10. MSE pefomance agains he spasiy eve when = 180, P = 0dB, and {4, 3}. MSEs of he OMPs monoonicay decease wih, as expeced by he SSE uppe-bound. ig. 10 shows he MSEs when vaies fom o 1. Again he OMPs oupefom he S, and hei MSEs monoonicay incease wih, as expeced by he SSE oweand uppe-bounds. inay, i is noed ha hee ae big gaps beween he owe-/uppe-bounds and simuaion esus in igs. 9 and 10. In he case of he owe-bound, he knowedge of AoDs/AoAs, which is assumed in oace esimaion, has a dominan effec on he gap. o he uppe-bound, he gap is caused by he inequaiies in equaions 37, 39, 41, and 51. In his case i is difficu o find ou which inequaiy has a dominan effec on he gap. VII. COCUSIO An open-oop channe esimao fo hybid MIMO sysems in mm-wave communicaions was poposed. Mm-wave channes ae spase in he sense ha hey ae dominaed by a few cuses of significan pahs. Expoiing his spasiy, we fomuaed a CS pobem ha esimaes he AoD, AoA, and he coesponding gain of each significan pah. his pobem is based on he paameic channe mode wih quanized ange gids and soved by he OMP mehod ha empoys a edundan dicionay consising of aay esponse vecos wih finey quanized ange gids. iven he beamfoming maix, which is assumed o be uniay, he baseband pocesso fo aining o he sensing maix was designed by minimizing he oa coheence of he equivaen sensing maix. I has been obseved ha he esimaion accuacy can be impoved consideaby by andomy pemuing he beamfoming maix. he compue simuaion esus demonsaed ha he poposed OMP mehod can oupefom exising mehods such as he S mehod and he OMP based on he viua channe mode. APPEDIX We conside he singe-use mm-wave sysem whee he ansmie sends seams o he eceive. he ansmied signa can be wien as x = s, whee C is he pecode maix, s C 1 is he symbo veco wih E [ ss ] = P I,and P is he aveage oa ansmi powe. he eceived signa afe combining can be wien as ỹ = W s + W ñ, whee W C is he combine maix, is he channe maix in 7, and ñ C 1 is he noise veco wih C 0,σn I. he ASE can be wien as = og I + P n 1 W W, whee n = σn W { W; and, W} ae designed via he SVD of Ĥ, denoed as Ĥ = Û ˆ ˆV.ee {Û, ˆV} ae he singua veco maices and ˆ is he singua vaue maix whose diagona enies ae aanged in deceasing ode. he pecode and he combine W consis of he fis coumns of ˆV and Û, especivey. EEECES [1] S. angan,. S. appapo, and E. Ekip, Miimee-wave ceua wieess newoks: Poenias and chaenges, Poc. IEEE, vo. 10, no. 3, pp , Ma []. Bai, A. Akhaeeb, and. W. eah, J., Coveage and capaciy of miimee-wave ceua newoks, IEEE Commun. Mag., vo. 5, no. 9, pp , Sep [3] Pa 11: Wieess A Medium Access Cono MAC and Physica aye PY Specificaions Amendmen 3: Enhancemens fo Vey igh houghpu in he 60 z Band, IEEE Sandad P80.11.ad, Jan [4] Pa 15.3: Wieess Medium Access Cono MAC and Physica aye PY Specificaions fo igh ae Wieess Pesona Aea ewoks WPAs Amendmen : Miimee-Wave Based Aenaive Physica aye Exension, IEEE Sandad c, Oc [5] M.. Akdeniz e a., Miimee wave channe modeing and ceua capaciy evauaion, IEEE J. Se. Aeas Commun., vo. 3, no. 6, pp , Jun [6] O. E Ayach, S. ajagopa, S. Abu-Sua, Z. Pi, and. W. eah, J., Spaiay spase pecoding in miimee wave MIMO sysems, IEEE ans. Wieess Commun., vo. 13, no. 3, pp , Ma [7] J. ee and Y.. ee, A eaying fo miimee wave communicaion sysems wih hybid /baseband MIMO pocessing, in Poc. IEEE In. Conf. Commun., Jun. 014, pp [8] S. an, I. Chih-in, Z. Xu, and C. owe, age-scae anenna sysems wih hybid anaog and digia beamfoming fo miimee wave 5, IEEE Commun. Mag., vo. 53, no. 1, pp , Jan [9] J. Wang e a., codebook based beamfoming pooco fo mui- bps miimee-wave WPA sysems, IEEE J. Se. Aeas Commun., vo. 7, no. 8, pp , Oc. 009.

17 386 IEEE ASACIOS O COMMUICAIOS, VO. 64, O. 6, JUE 016 [10] A. Akhaeeb, O. E Ayach,. eus, and. W. eah, J., Channe esimaion and hybid pecoding fo miimee wave ceua sysems, IEEE J. Se. opics Signa Pocess., vo. 8, no. 5, pp , Oc [11]. W. eah, J.,. onzáez-pecic, S. angan, W. oh, and A. M. Sayeed, An oveview of signa pocessing echniques fo miimee wave MIMO sysems, IEEE J. Se. opics Signa Pocess., vo. 10, no. 3, pp , Ap [1] W. U. Bajwa, J. aup, A. M. Sayeed, and. owak, Compessed channe sensing: A new appoach o esimaing spase muipah channes, Poc. IEEE, vo. 98, no. 6, pp , Jun [13] X. ao and V. K.. au, Disibued compessive CSI esimaion and feedback fo DD mui-use massive MIMO sysems, IEEE ans. Signa Pocess., vo. 6, no. 1, pp , Jun [14] Z. ao,. Dai, Z. Wang, and S. Chen, Spaiay common spasiy based adapive channe esimaion and feedback fo DD massive MIMO, IEEE ans. Signa Pocess., vo. 63, no. 3, pp , Dec [15] Z. ao,. Dai, D. Mi, Z. Wang, M. A. Iman, and M. Z. Shaki, MmWave massive-mimo-based wieess backhau fo he 5 ua-dense newok, IEEE Wieess Commun., vo., no. 5, pp. 13 1, Oc [16] J. A. opp and A. C. ibe, Signa ecovey fom andom measuemens via ohogona maching pusui, IEEE ans. Inf. heoy, vo. 53, no. 1, pp , Dec [17] D.. Donoho, Compessed sensing, IEEE ans. Inf. heoy, vo. 5, no. 4, pp , Ap [18] A. M. Sayeed, Deconsucing muianenna fading channes, IEEE ans. Signa Pocess., vo. 50, no. 10, pp , Oc. 00. [19] D. se and P. Viswanah, undamenas of Wieess Communicaion. Cambidge, U.K.: Cambidge Univ. Pess, 005. [0] C.. Bege, S. Zhou, J. C. Peisig, and P. Wie, Spase channe esimaion fo muicaie undewae acousic communicaion: om subspace mehods o compessed sensing, IEEE ans. Signa Pocess., vo. 58, no. 3, pp , Ma [1] D. u, X. Wang, and. e, A new spase channe esimaion and acking mehod fo ime-vaying ODM sysems, IEEE ans. Veh. echno., vo. 6, no. 9, pp , ov []. Zenik-Mano, K. osenbum, and Y. C. Eda, Sensing maix opimizaion fo bock-spase decoding, IEEE ans. Signa Pocess., vo. 59, no. 9, pp , Sep [3] S. i, Maix esus on he Khai-ao and acy-singh poducs, inea Ageba App., vo. 89, pp , Ma [4] A. oenza, D. J. ove, and. W. eah, J., Simpified spaia coeaion modes fo cuseed MIMO channes wih diffeen aay configuaions, IEEE ans. Veh. echno., vo. 56, no. 4, pp , Ju [5] Y. C. Eda and. Kuyniok, Eds., Compessed Sensing: heoy and Appicaions. ew Yok, Y, USA: Cambidge Univ. Pess, 01. [6] Z. Ben-aim, Y. C. Eda, and M. Ead, Coheence-based pefomance guaanees fo esimaing a spase veco unde andom noise, IEEE ans. Signa Pocess., vo. 58, no. 10, pp , Oc [7]. A. on and C.. Johnson, Maix Anaysis. ew Yok, Y, USA: Cambidge Univ. Pess, [8] P. S. Buen, andbook of Means and hei Inequaiies. Boson, MA, USA: Kuwe, 003. [9] S. Joka and V. Mehmann, Spase souions o undedeemined Konecke poduc sysems, inea Ageba App., vo. 431, no. 1, pp , Dec [30].. oub and C.. Van oan, Maix Compuaions, 3d ed. Baimoe, MD, USA: he Johns opkins Univ. Pess, [31] Y. Chi and. Cadebank, Coheence-based pefomance guaanees of ohogona maching pusui, in Poc. 50h Annu. Aeon Conf. Commun., Cono, Compu., Sep. 01, pp [3] J. A. opp, eed is good: Agoihmic esus fo spase appoximaion, IEEE ans. Inf. heoy, vo. 50, no. 10, pp. 31 4, Oc [33] M. Ead, Opimized pojecions fo compessed sensing, IEEE ans. Signa Pocess., vo. 55, no. 1, pp , Dec [34] J. M. Duae-Cavajaino and. Sapio, eaning o sense spase signas: Simuaneous sensing maix and spasifying dicionay opimizaion, IEEE ans. Image Pocess., vo. 18, no. 7, pp , Ju [35]. i, Z. Zhu, D. Yang,. Chang, and. Bai, On pojecion maix opimizaion fo compessive sensing sysems, IEEE ans. Signa Pocess., vo. 61, no. 11, pp , Jun [36]. ai and S. Oshe, A spiing mehod fo ohogonaiy consained pobems, J. Sci. Compu., vo. 58, no., pp , eb [37]. auhu, K. Schnass, and P. Vandegheyns, Compessed sensing and edundan dicionaies, IEEE ans. Inf. heoy, vo. 54, no. 5, pp , May 008. [38] M. A. Davenpo, D. eede, and M. B. Wakin, Signa space CoSaMP fo spase ecovey wih edundan dicionaies, IEEE ans. Inf. heoy, vo. 59, no. 10, pp , Oc [39] B. assibi and B. M. ochwad, ow much aining is needed in muipe-anenna wieess inks? IEEE ans. Inf. heoy, vo. 49, no. 4, pp , Ap [40] K. yhn, M.. Duae, and S.. Jensen, Compessive paamee esimaion fo spase ansaion-invaian signas using poa inepoaion, IEEE ans. Signa Pocess., vo. 63, no. 4, pp , eb [41]. ang, B.. Bhaska, P. Shah, and B. ech, Compessed sensing off he gid, IEEE ans. Inf. heoy, vo. 59, no. 11, pp , ov Junho ee eceived he B.S. degee in eecica engineeing fom Kwangwoon Univesiy, Seou, Souh Koea, in 010, and he M.S. degee fom he Univesiy of Science and echnoogy Eeconics and eecommunicaions eseach Insiue EI, Daejeon, Souh Koea, in 013. e is cueny pusuing he Ph.D. degee wih he Koea Advanced Insiue of Science and echnoogy, Daejeon. om 010 o 011, he was an Inen wih he Sma adio eseach eam, EI. is eseach ineess incude spasiy-awae saisica signa pocessing and is appicaions, especiay in wieess communicaions. ye-ae i M 04 was bon in Daejeon, Souh Koea, in e eceived he B.S. degee in eeconic communicaion engineeing fom anyang Univesiy, Seou, Souh Koea, in 1989, and he M.S. and Ph.D. degees in eecica engineeing fom he Koea Advanced Insiue of Science and echnoogy KAIS, Souh Koea, in 199 and 004, especivey. Since 1991, he has been wih he eseach Cene of Koea eecom, and joined he KAIS Insiue in 013, whee he is cueny a eseach Pofesso. e is aso ineesed in obia angua momenum ansmission and massive anenna echnoogies fo ceua mobie communicaion sysems. is eseach ineess ae in he aea of communicaion signa pocessing, which incudes synchonizaion, inefeence canceaion, and adapive fie design. Yong. ee S 80 M 84 SM 98 eceived he B.S. and M.S. degees fom Seou aiona Univesiy, Seou, Souh Koea, in 1978 and 1980, especivey, and he Ph.D. degee fom he Univesiy of Pennsyvania, Phiadephia, USA, in 1984, a in eecica engineeing. e was an Assisan Pofesso wih he Depamen of Eecica and Compue Engineeing, Sae Univesiy of ew Yok, Buffao, fom 1984 o Since 1989, he has been wih he Depamen of Eecica Engineeing, Koea Advanced Insiue of Science and echnoogy KAIS, whee he is cueny a Pofesso. e was he Povos fom 011 o 013, he Dean of Engineeing fom 005 o 011, and he ead of Eecica Engineeing fom 004 o 005 wih KAIS. is eseach aciviies ae in he aea of communicaion signa pocessing, which incudes new wavefom design, inefeence managemen, esouce aocaion, synchonizaion, deecion and esimaion fo muipe-inpu muipe-oupu, wieess communicaion sysems, incuding ine-of-sigh communicaions, and miimee wave communicaion sysems.