Reduced-Complexity Baseband Compensation of Joint Tx/Rx I/Q Imbalance in Mobile MIMO-OFDM

Transcription

1 1 Redced-Compexiy Baseband Compensaion of Join Tx/Rx / mbaance in Mobie MMO-OFDM Baachande Naasimhan, Sden Membe, EEE, Sdhasan Naayanan, Sden Membe, EEE, Haing Minn, Senio Membe, EEE, and Naofa A-Dhahi, Feow, EEE Absac Diec-convesion mipe-inp mipe-op ohogona feqency division mipexing (MMO-OFDM ansceives enjoy high daa aes and eiabiiy a pacica impemenaion compexiy Howeve, anaog fon-end impaimens sch as / imbaance and high mobiiy eqiemens of nexgeneaion boadband wieess sandads es in pefomanceimiing ine-caie inefeence (C n his pape, we sdy he effecs of C de o hese impaimens fo OFDM wih space feqency bock codes and spaia mipexing, deive a geneaized inea mode and popose a non-ieaive edced-compexiy digia baseband join compensaion scheme Fhemoe, we pesen a pio scheme fo join esimaion of he channe and he / imbaance paamees and evaae is pefomance hogh simaions O poposed scheme is effecive in esimaing and compensaing fo feqency-independen and feqencydependen ansmi and eceive / imbaances even in he pesence of a esida feqency offse ndex Tems OFDM, MMO, / imbaance, MMSE, Eqaizaion, Mobiiy, C NTRODUCTON To povide high daa aes a pacica ansceive compexiy, viay a boadband wieess sysems se OFDM, as eviden fom sandads sch as Wodwide neopeabiiy fo Micowave Access (WiMAX, Long Tem Evoion (LTE, Eopean Digia Video Boadcasing-Teesia (DVB-T and Mobie Boadband Wieess Access (MBWA Howeve, OFDM is vey sensiive o synchonizaion eos and Radio Feqency (RF fon-end impaimens Moeove, mobie boadband wieess sandads eqie high mobiiy sppo, eg, 10 km/h in WiMAX, 50 km/h in MBWA, and 500 km/h in LTE fo some feqencies MMO-OFDM sysems povide hge capaciy impovemens b hey empoy mipe RF ansceive fon-ends inceasing sysem cos and compexiy Diec convesion ansceives, eieving he compexiy and cos isses, ae paicay aacive fo MMO, b hey sffe fom sevea RF impaimens sch as in-phase/qadae (/ imbaance and caie feqency offse (CFO / imbaance cod be feqency-independen (F o feqency-dependen (FD and cases inefeence beween image feqency componens Anohe fom of C among sb-caies is cased by CFO o he The ahos ae wih he Depamen of Eecica Engineeing, Univesiy of Texas a Daas, MSEC33, Richadson, TX, E-mai {bxn06000,sxn06100,haingminn,adhahi}@daased This wok is sppoed by a gif fom Reseach in Moion nc This wok was pesened in pa a CSS 009 eaive moion beween he ansmie and eceive These C componens case high eo foos, hence seveey imiing he pefomance of MMO-OFDM sysems Theefoe, / imbaance is even moe impoan an isse in MMO han in singe-inp singe-op (SSO sysems Moeove, he RF fon-ends of diffeen anennas may no have he same chaaceisics, which inceases he nmbe of paamees o be esimaed and in his pape we povide a geneaized scheme fo hei esimaion and compensaion n 1, we deived a geneaized mode fo FD join ansmie/eceive (Tx/Rx / imbaance in a SSO-OFDM sysem nde high mobiiy and pesened ow-compexiy channe esimaion and compensaion agoihms n he ieae, 9 addess he isse of / imbaance in MMO-OFDM sysems None of hese papes addesses he isse of mobiiy Whie 10 addesses he isse of / imbaance aong wih mobiiy and CFO, he pape consides ony SSO sysems Refeences 4 anayze and compensae / imbaance in space ime bock coding (STBC-OFDM schemes which assme he channe o be consan ove wo o moe OFDM symbos This assmpion does no hod fo highy mobie OFDM sysems The second cass of mipe-anenna echniqes fo mobie MMO-OFDM sysems is spaia mipexing (SM-OFDM, sdied in 5 8 Whie 5 deas wih he F / imbaance case, 7 modes FD imbaance ony a he eceive side Ahogh 8 accons fo FD / imbaance on boh sides as we as CFO, i does no accon fo mobiiy which dasicay impacs sysem design n his pape, we popose he se of space feqency bock coding (SFBC fo mobie OFDM sysems wih FD / imbaance a boh he ansmie and eceive Since he C de o mobiiy ess in inefeence wihin an SFBC code bock, we popose a spe-bock sce fo he Aamoi SFBC The spe-bock sce enses he ohogonaiy of he Aamoi bock in spie of he impaimens Hence, deecion can be pefomed a easonabe compexiy whie si eaizing spaia and mipah divesiy gains Fhemoe, we compae SFBC-OFDM and SM-OFDM sysems in he pesence of high mobiiy and / imbaance n boh cases, we se owcompexiy non-ieaive inea MMSE decoding becase C effecs es in a high-dimensiona signa veco and, hence, maximm-ikeihood (ML mehods ae pohibiivey compex Moeove, we show ha boh o compensaion and channe esimaion schemes ae qie obs o esida CFO The es of his pape is oganized as foows n Secion,

2 γ (1 λ (kλ (k, (g(k, + φ (kφ ( g (k,, λ (kφ (g(k, + φ (kλ ( g (k, λ (k φ ( g (k, + φ ( λ (g(k,, λ ( λ ( g (k, + φ (k φ (g(k, γ ( λ (kφ (k, ( g(k, + φ (kλ (g (k,, λ (kλ ( g(k, + φ (kφ (g (k (1, λ (kλ (g (k, + φ (k φ ( g(k,, λ (k φ (g (k, + φ (k λ ( g(k, we eview he SSO join / imbaance and mobiiy mode in 1 n Secion, we discss he design of SFBC and SM encoding schemes and deive hei inp-op modes n Secion V, we pesen he inea digia baseband MMSE compensaion scheme, and in Secion V, we discss he pioaided channe esimaion schemes We exend he mode o incde CFO effecs in Secion V n Secion V, we pesen echniqes o edce compaiona compexiy Finay, we discss simaion ess in in Secion V Noaion A ime-domain qaniies { x(, x, X} have a ba wheeas feqency-domain qaniies {x(f, x, X} do no Vecos { x, x} ae epesened by owe-case and maices { X, X} by ppe-case bodface ees ( H, ( T and ( denoe especivey he hemiian, anspose and conjgae opeaions F denoes he niay DFT maix Linea convoion is denoed by whie cica convoion modo-n is by N 0 m n is he a-zeo m n maix The opeao diag( when appied o a maix gives a veco conaining he diagona eemens of he maix and when i acs on a veco yieds a diagona maix whose diagona eemens ae he eemens of he veco The opeao ( denoes he ace of a maix (n s s ( n Seqence o mpse Tain Seqence o mpse Tain n n s ( n ( nt s ( n ( nt mbaance Fie ( ( mbaance Fie LPF s ( mbaance Fie h ( ( ( h ( LPF ( s ( LPF h ( h ( LPF ( mbaance Fie a cos( f a cos( f f ( ( c a sin( f c g p a sin( f c f c (, Channe AWGN w p ( ( y p Fig 1 Bock diagam of he join Tx/Rx / imbaance, esida CFO and mobiiy mode SYSTEM MODEL AND ASSUMPTONS F / imbaance is cased by osciao ampide and phase mismaches a he mixe Le α be he ampide mismach and θ be he phase mismach a he ansmie We define he foowing / imbaance paamees ( µ cos θ j α sin θ (, ν α cos θ j sin θ FD / imbaance is cased by he non-idea and mismached feqency esponses of he LPF s on he and banches, whose feqency esponses ae ξ (k and ξ (k, especivey, fo k 0,, N 1 whee N is he OFDM FFT size Define λ (k 1 ( (µ + ν ξ (k + (µ ν ξ (k φ (k 1 ( (µ + ν ξ (k (µ ν ξ (k x p ( n a simia manne, we can define he eceive-side paamees α, θ, µ, ν, ξ (k, ξ (k and λ (k 1 ( (µ + ν ξ (k + (µ ν ξ (k φ (k 1 ( (ν + µ ξ (k + (ν µ ξ (k A of hese paamees have been abeed in he bock diagam in Fig 1 Now, e he ansmied signa on sb-caie k be s(k and he ime-vaying channe be chaaceized by g(k, whee his feqency-domain qaniy epesens he inefeence de o sb-caie a sb-caie k Fo a k, hese C ems ae coeced ino he maix G Fhemoe, define he maix Λ Diag(λ (0,, λ(n 1 and he maices Φ, Λ, Φ coesponding o qaniies φ (k, λ (k, φ (k As we showed in 1, he eceived signa is given by z (Λ GΛ + Φ G # Φ # s+(λ GΦ + Φ G # Λ # s # +v Ψ (1 Ψ ( ( whee s s(0,, s(n 1 T, z z(0,, z(n 1 T and v v(0,, v(n 1 T ae he ansmied, eceived and noise vecos, especivey Hee, he k-h eemen of b # is he conjgae of he (N k eemen of b and he (n, k eemen of B # is he conjgae of he (N n, N k eemen of B f we wie down expessions fo each individa em z(k in (, we can see ha he effec of / imbaance is o case inefeence beween a sb-caie k and is image k N k afe being weighed by / imbaance paamees We showed in 1 ha fo each sb-caie pai (k, k k 1,, N 1, we can wie z(k N 1 0 ( γ (1 (k, s( + γ ( (k, s( + v(k (3 whee z(k z(k, z (k T, s( s(, s ( T, v(k v(k, v (k T and he γ ( (k, s ae as defined in (1 We showed in 1 ha he em γ ( (k, can be ignoed Moeove, he effec of C de o mobiiy can be consideed significan ony on a few, say D, adjacen sb-caies Wih hese appoximaions, (3 can be simpified as z(k D D γ (1 (k, s( + v(k (4 Fo beviy, we define γ (1 g a (k, (g b (k, (k, g b (k, (g a (k, SYSTEM MODELS FOR MMO TRANSMSSON A Aamoi SFBC We focs on he Aamoi SFBC fo wo ansmi anennas de o is popaiy and is wide-spead adopion in san-

3 G p (m, n gp,1(m, a n gp,(m, a n (gp,1(m b, n (gp,(m b, n (gp,(m, a n (gp,1(m, a n gp,(m b, n gp,1(m b, n gp,1(m, b n gp,(m, b n (gp,1(m a, n (gp,(m a, n (gp,(m, b n (gp,1(m, b n gp,(m a, n gp,1(m a, n 3, (5 L s1(m-1 s1(m+1 s(m+1 s(m-1 s1(m s(m D pios -s*(m s1*(m - s*(m+1 s1*(m+1 Fig -s*(m-1 s1*(m-1 -s*(m -1 s1*(m -1 -s*(m s1*(m -s*(m +1 s1*(m +1 s1(m -1 s(m -1 s1(m s(m s1(m +1 s(m +1 -x*(m -1 x1*(m -1 x1(m-1 x1(m x(m x1(m+1 x(m+1 x(m-1 -x*(m-1 x1*(m-1 -x*(m x1*(m -x*(m+1 x1*(m+1 -x*(m x1*(m -x*(m +1 x1*(m +1 x1(m -1 x(m -1 x1(m x(m x1(m +1 x(m +1 g p, m 1( m, g p, m g p g p g p ( m, (, 1 m, m (, m, m g ( p, 1 m g p m', ', ( m', m' g ( p, 1 m m', ', ( m', m' yp(m yp(m+l+d yp(m -L-D yp(m Poposed spe-bock sce fo Aamoi SFBC-OFDM zp(m zp(m+l+d zp(m -L-D dads 1 Denoing he nmbe of eceive anennas by N P, o poposed scheme is shown in Fig ses he spe-bock sce descibed in 1 n conveniona SFBC OFDM, one Aamoi code-bock occpies wo adjacen sb-caies and in he pesence of mobiiy, i wod es in ina-code-bock inefeence which desoying he ohogonaiy and pefomance of he Aamoi code Wih o poposed spe-bock sce, his ina-code-bock inefeence is significany edced Moeove, i ess in an eegan inea mode ha cod be esimaed and eqaized wih ow compexiy Now, o dea wih / imbaance, we posiion wo spe-bocks sch ha hey occpy sb-caies ha ae exacy images of each ohe Fo exampe, if spe-bock S b occpies sb-caies k b o k b + (L + D 1, he image spe-bock S b occpies sb-caies k b (L + D 1 o k b whee k b N k b Hee k b is he index of he fis sb-caie of spebock S b, L is he nmbe of Aamoi SFBC bocks wihin he spebock and D is he nmbe of pios, so ha he engh of he spe-bock is (L + D, whee D is chosen accoding o he exen of C cased by mobiiy The spe-bocks ae aanged symmeicay wih espec o he DC sb-caie Wihin a spe-bock S b, hee ae L code-bocks and he symbos wihin hese code-bocks ae abeed as {s 1 (ms (m; m 0,, L 1} The Aamoi codebock m is fomed by ansmiing symbos s 1 (m and s (m, especivey, fom Anennas 1 and of Sb-caie (k b + m and ansmiing symbos s (m and s 1(m, especivey, fom Anennas 1 and of Sb-caie (k b +m+l+d Sbcaies (k b + L o (k b + L + D 1 ae occpied by pios This aids in sepaaing he fis-so Aamoi code symbos 1 Exensions o ohe SFBC schemes based on ohogona designs 11 ae saighfowad The sbscips 1 and ae no ansmi anenna indices Aso, hee is no expici spe-bock indexing fo convenience of noaion zp(m {s 1 (m, s (m} fom he second-so Aamoi code symbos { s (m, s 1(m}, hs esing in he eegan inea mode wih he Aamoi bock sce deived in his secion The mobiiy-indced C coefficien a Sb-caie m de o Sb-caie n is denoed by g p,q (m, n whee p and q ae he eceive and ansmie anenna indices, especivey g p,q (m, m epesens he gain a Sb-caie m shod be noed ha, even fo a significan Doppe, wih he choice of a age L, inefeence among sb-caies fom diffeen spebocks is negigibe Moeove, i is assmed ha g p,q (m, n g p,q (m + L + D, n + L + D m and (m n fixed Fo his assmpion o hod, L has o be chosen aong wih D sch ha (L + D sb-caies ie wihin he coheence bandwidh of he channe Unde his assmpion, sing (4, and aking he necessay conjgaes, we can wie he eceived symbo veco z p (m z p (m zp(m + L + D zp(m L D z p (m T beonging o spe-bocks S b and S b a eceive anenna p as z p (m m+d nm D G p (m, ns(n + v p (m (6 whee s(m s 1 (m s (m s 1(m s (m T, v p (m v p (m v p(m+l+d v p(m L D v p (m T and G p (m, n is defined in (5 The qaniies in G p (m, n ae defined as g a p,q(m, n λ,p (mλ,q (ng p,q (m, n +φ,p (mφ,q(n g p,q(m, n g b p,q(m, n λ,p(m φ,q(n g p,q(m, n +φ,p(m λ,q (ng p,q (m, n (7 Eqaion (6 fo diffeen p s can be combined ogehe o obain he foowing mode z 0(m z 1(m z NP 1(m z(m m+d nm D G 0(m, n G 1(m, n G NP 1(m, n G(m,n s(n+ v 0(m v 1(m v NP 1(m v(m (8 n (8, G(m, n epesens a bock ha accons fo a he / imbaance effecs and we denoe is size by B B c which in he Aamoi SFBC case is eqa o 4N P 4 and compised of G p (m, n s which ae, in n, composed of Aamoi bocks as shown in (5 is ineesing o see ha ahogh he sysem ode has inceased and image componens inefee wih each ohe, he Aamoi bock sce is si eained becase of he spe-bock aangemen n Secion V, we wi show how o expoi his Aamoi bock sce o achieve significan savings in compaiona compexiy B Spaia Mipexing We conside an SM-OFDM sysem wih N ansmi and N P eceive anennas wih N P N o enabe MMSE inea decoding The encoding is ivia in ha each ansmi anenna caies an independen symbo in each sb-caie Theefoe,

4 4 we se s q (m o denoe he symbo ansmied on Anenna-q of Sb-caie m Then, sing (4, we can wie he symbos eceived on Anenna p of Sb-caies m and m as foows zp (k } zp(k {{ } z p (k N 1 q0 k+d k D sq ( } s q( {{ } s q ( g a p,q (k, (g b p,q(k, gp,q(k, b (gp,q(k, a vp (k + vp(k } {{ } v p (k G p,q(k,, (9 whee he qaniies in G p,q (k, have been defined in (7 Fo N P eceive anennas, (9 geneaizes o z 0 (k z 1(k z NP 1(k whee z(k G(k, k+d G(k, k D s 0( s 1 ( s N 1( s( + G 0,0 (k, G 0,N 1(k, v 0 (k v 1(k v NP 1(k, v(k G NP 1,0(k, G NP 1,N 1(k, Eqaion (10 diffes fom an SM-OFDM sysem wih no / imbaance in ha he channe bock-maix G(k, is of size B B c N P N ahe han N P N and he op a each sb-caie aso depends on he symbos on is image sb-caie and hei neighbos V DGTAL BASEBAND COMPENSATON SCHEME Eqaions (8 and (10 ae simia excep fo he maix dimensions and he sb-bock sce n he SFBC case, we sed indices (m, n o epesen Aamoi bocks Fo convenience, hencefoh, we epesen boh of hem sing he indices (k, Now, in boh (8 and (10, i can be seen ha fo a given D, an inp symbo veco s(k eaks ino vecos s(k D o s(k + D Theefoe, we need o pocess {z(k D,, z(k,, z(k + D} o esimae s(k Wih his in mind, we define he foowing qaniies z(k D s(k D z(k z(k, s(k s(k, z(k + D s(k + D v(k D G(k D, k ṽ(k v(k, G(k G(k, k v(k + D G(k + D, k and G(k is defined in (11 and of size (D + 1B (4D + 1B c Then he inea MMSE esimae of s(k is given by (10 ŝ(k w H (k z(k (1 whee w H (k G H (k ( G(k GH (k + 1 SNR (D+1B 1 w H (k D w H (k w H (k + D and each of he w H (k s is of size shod be noed ha s(k consiss of N symbos Theefoe, he appicaion of (1 o each k ess in he esimaion of N ansmied symbos Moeove, z(k is of size (D + 1B which fo D 1 and N P N is eqa o 4 fo SFBC and 1 fo SM Howeve, he symbos fo one SFBC code bock ae spead ove wo sb-caies whie hose fo he SM code bock ae ove js one sb-caie Hence, ony six aps pe sb-caie ae needed o deec a code bock when D 1 V CHANNEL ESTMATON Eqaion (4 epesens a inea mode wheein he channe and / imbaance paamees ae combined, so ha we can pefom inea esimaion keeping he nmbe of paamees sma Thee ae fo paamees (in each γ (1 (k, o be esimaed fo each ansmi-eceive anenna pai Howeve, hee ae (D+1 ems in he smmaion in (4 and, hence, a oa of 4(D + 1 paamees Noe ha γ (1 (k, s fo k epesen he channe gain fo he sb-caie pai {k, N k} and he es of he ems ( k epesen he C Moeove, he γ (1 (k, k s ae he diagona eemens of Ψ (1 and Ψ ( and in genea, hey ae mch age han hei especive offdiagona ones Theefoe, we ignoe he non-diagona ems and esimae ony he diagona ems Nex, we descibe he pio scheme, which is vaid fo boh F and FD / imbaances Eqaion (4 invoves a sbcaie and is image Theefoe, i makes sense o assign pios in image ocaions Since hee ae fo paamees and, we assign a pai of pios on each side of he DC sbcaie fo each ansmi anenna A any pio ocaion, ony one of he anennas is excied wih pios and he ohes ae swiched off, so ha he seqences ae ohogona acoss he anennas saisfying he cieion fo opimaiy in 13 The pio paens fo SFBC and SM ae shown in Figs 3(a and 3(b, especivey n case of SFBC, he pio pai on each side is no conigos becase of he spe-bock sce A eceive anenna p, we need o esimae g p,q (k, k Assme ha we excie Anenna q wih a pio qadpe on sbcaies k 1, k, k, k 1, whee k k 1 + L + D fo SFBC and k k fo SM Then, fo sb-caie pai {k 1, k 1} zp(k1 zp(k 1 g a p,q (k 1, k 1 (gp,q(k b 1, k 1 gp,q(k b 1, k 1 (gp,q(k a 1, k 1 sq (k 1 vq (k s q(k vq (k 1, (13 and we can wie a simia eqaion fo he sb-caie pai {k, k } Now, since we assmed g p,q (k 1, k 1 g p,q (k, k, he eqaions fo he wo pais can be combined as z p (k 1 zp(k 1 z p(k zp(k } {{ } z(k 1 s q (k 1 0 s q(k s q(k 1 0 sq(k 1 s q(k 0 s q(k 0 0 s q(k 0 sq(k S(k 1 g a p,q(k 1, k 1 v q (k 1 (gp,q(k a 1, k 1 (gp,q b (k 1, vq k 1 + (k 1 v q (k gp,q(k b 1, k 1 vq (k g(k 1 (14

5 5 G(k G(k D, k D G(k D, k D G(k D, k 0 G(k, k D G(k, k G(k, k + D 0 G(k + D, k G(k + D, k + D G(k + D, k + D, (11 We pefom eas-sqaes (LS esimaion of he channe and, hence, we eqie ha he ace (S H (k 1 S(k 1 σ p 4 fo he LS eo o be minimm 14 Hee, σ p coesponds o he powe consain on he pios n OFDM symbos wih boh daa and pio sb-caies, he pocede gives he channe esimaes ony a he pio ocaions We se a DFT based inepoaion scheme ha gives he ime-domain channe aps fom he feqency-domain channe coefficiens avaiabe a he pio ocaions Le g a p,q be he veco ha conains a avaiabe g a p,q(k 1, k 1 s wih odeed indices Then, we have diag(ψ (1 p,q F( F H F 1 FH g a p,q (15 whee F is a maix conaining he comns of he DFT maix ha coespond o he pio ocaions Simiay, we coec a he gp,q s b ogehe o obain diag(ψ ( p,q This way, we have obained a he diagona eemens To obain he nondiagona eemens, we se he mehod descibed in 15 which invoves ineay inepoaing each ap in he ime-domain acoss hee consecive OFDM symbos, and hen going back o he feqency domain o obain he maices Ψ (1 p,q and Ψ ( p,q An 1 An Spebock Spebock DC Spebock Spebock Sb 1 Sb S b' S b' 1 Whie f is he aca feqency offse in Hez, we define he moe sef qaniy f N f F s, which is he feqency offse nomaized by he sb-caie spacing Refeing o Fig 1, whee ȳ(n is he eceived baseband eqivaen signa, i can be shown ha in he pesence of CFO, he eqivaen eceived signa (wiho consideing / imbaance is no ȳ(n b jπ fn ȳ CFO (n e N ȳ(n (16 Now, if we define Ω jπ f Diag(1, e N jπ f(n 1,, e N T, hen we ge Ωȳ and in he feqency domain, Ωy, whee Ω F ΩF H Theefoe, we can ewie ( as z (Λ G Ω Λ + Φ G # Ω Φ# s +(Λ G Ω Φ + Φ G # Ω Λ# s # + v (17 whee G Ω ΩG Hee we assme ha he esida CFO confined o wihin he sb-caie spacing, ie, f < 1 Now Ω is a cican maix (since Ω is diagona and G is a banded maix Howeve, ony a few sb-diagonas on each side of he main diagona of Ω ae significan Theefoe, we can se he es of he eemens o be zeo, ie, Ω can aso be epaced by is banded appoximaion Ahogh his inceases he nmbe of sb-diagonas in he esing mode, simaion sdies show ha he compensaion schemes poposed in his pape ae qie obs o ypica esida CFO vaes as i wi be shown in Secion V This impies ha we do no have o expiciy dea wih he CFO pobem ding compensaion and aso no expiciy esimae he CFO paamee f k 1 k1 L D k k L D k' L D Feqency k' k' 1 L D k' 1 V COMPLEXTY CONSDERATONS Fig 3 Poposed pio paens (a SFBC-OFDM DC k1 k1 1 k k 1 k' 1 k' k' 1 1k' 1 Feqency (b SM-OFDM V CARRER FREUENCY OFFSET OFDM sysems ae sensiive o caie feqency offse (CFO which is navoidabe becase i is diffic o accaey synchonize wo emoe osciaos (one a he ansmie and he ohe a he eceive o he same feqency and phase The effec of his offse, denoed by f in Fig 1, is o desoy he ohogonaiy of he sb-caies hs casing C 16 The inea MMSE compensaion scheme in (1 invoves he compaion of a Gammian and maix invesion Fo D 1 wih wo ansmi and wo eceive anennas, he SM and SFBC schemes eqie maix invesions of size 1 1 and 4 4 especivey Howeve, hee ae common channe eemens in adjacen sb-caies Moeove, he Hemiian and Aamoi sce of hese maices aow fo fhe compaiona edcions, which we wod discss in deai in his secion f we se he same agmens in 17 fo compex Hemiian maices, i is easy o show ha M compex divisions and M 3 compex MAC opeaions ae eqied o inve hem The compaion of he Gammian of a compex maix of size M M c eqies MMc(M+1 compex MAC opeaions We have o dea wih maices of size (D + 1B (4D + 1B c Hee, B B c epesens he size of he bocks in maix G(m The Gammian compaion wod eqie B B s (D+ 1(4D + 1(DB + B + 1 compex MAC s and he maix invesion wod eqie (D+1 B compex divisions and (D+1 3 B 3 compex MAC s The foowing echniqes hep s edce he compaiona compexiy

6 6 A Recsive Compaion of he Gammian Fo boh SM and SFBC, he MMSE esimaion in (1 invoves he compaion of he Gammain G(m G(m G H (m fo m 0,, L 1 Noe ha hee ae sevea common eemens beween G(m and G(m + 1 G(m is a bock maix of size (D + 1 (4D + 1 bocks and each bock-ow has ony (D + 1 non-zeo bocks As he code-bock index inceases fom m o m + 1, ony one bock-ow is epaced by a new bock-ow Theefoe, ony one new bock-ow needs o be comped each ime which eqies (D+1(D+1 bock MAC s The esing bocks hemseves ae symmeic and hence (D+1(D+1 BBc(B+1 compex MAC s ae needed in comping he Gammian B Recsive Agoihm fo nvesion The ohe( expensive compaion sep is he invesion of G M (m G(m + 1 SNR Fo his, we conside wo diffeen ways of paiioning he bock-hemiian maix G(m as shown in (18 whee each epesens a bock of size B B c Conside he invesion fomae (19 and (0 3 fo G(m 18, whee (m A (m B (md 1 (mc (m and (m D (m C (ma 1 (mb (m Since G(m is Hemiian, G 1 (m is aso Hemiian and hence R (m H (m and R (m H (m Moeove, wih he paiioning in wo diffeen ways, we see ha D (m A (m + 1 Sppose ha G 1 (m has been comped, hen G 1 (m + 1 is comped sing he foowing seps 1 Compe (m by inveing 1 (m Compe D 1 (m S (m H (m (m (m 3 Now A 1 (m + 1 D 1 (m Compe G 1 (m + 1 sing he foma fo G 1 ((m + 1 in (0 A 1 (m + 1 and D 1 (m epesen common bocks in G M (m as we move fom code-bock m o m + 1 and of size D D which is a significan oveap wih he enie maix of size (D + 1 (D + 1 n wha foows, we smmaize he compexiy invoved in comping G 1 (m + 1 comp-1compaion of (m 1 (m is Hemiian and js one bock-size B This eqies B compex divisions and B3 compex MAC s comp-compaion of D 1 (m he compaion of H (m (m invoves D bock MAC s Afe his, we compe H (m (m (m and he esing maix has Hemiian bocks Theefoe, we need D(D + 3 bock MAC s comp-3compaion of 1 (m+1 we se he foma 1 (m+1 B H (m+1a 1 (m+1b (m+1 since C (m + 1 B H (m + 1 and we iize A 1 (m + 1 D 1 (m Consideing he Hemiian nae of A 1 (m + 1, his eqies D(D + 3 bock MAC opeaions comp-4compaion of (m + 1 we se he foma (m+1 A 1 (m+1b (m+1 1 (m+1, whee A 1 (m + 1B (m + 1 is avaiabe fom comp-3 The es of he compaions eqie D bock MAC s 3 The (m noaion in (19 and (0 indicaes ha he qaniies in he coesponding maices ae maix fncions of m comp-5compaion of P (m + 1 fom (0, i is no had o see ha P (m + 1 A 1 (m (m + 1 (m + 1 H (m + 1 The compaion of (m + 1 fom 1 (m + 1 is simia o comp-1 and he compaion of (m + 1 (m + 1 H (m + 1 is simia o comp-3 Compaions comp-,3,4 and 5 eqie a oa of D(6D + 11 bock MAC s, ie, D(6D + 11 B3 s compex MAC s Compaions comp-1 and 5 eqie a oa of B 3 compex MAC s and B compex divisions Theefoe, ovea we need o compe (6D + 11D + B3 compex MAC s and B compex divisions C Expoiing he Sce of he Aamoi Bock Maix Finay, we expoi an ineesing popey of a maix wih Aamoi bocks; namey, he mipicaion and invesion compexiy is popoiona o he nmbe of Aamoi bock ows/comns and no he aca size of he maix Given a maix wih Aamoi bocks, we can pefom bock Gassian eiminaion, wih each compex division epaced by an Aamoi maix invesion (which is ivia and each compex MAC opeaion epaced by Aamoi MAC opeaion Theefoe, a bock Aamoi maix of size M M cod be consideed o be effecivey of size M M which wod eqie abo (6D +11D+ M 3 16 Aamoi MAC opeaions o inve i The Aamoi bock has ony wo disinc eemens and hence one Aamoi MAC opeaion ansaes o exacy fo compex MAC s The invesion of he Aamoi bock maix eqies (6D + 11D + B3 4 compex MAC s which is haf ha of a ega compex G M (m Using simia agmens and he es of Secion V-A, i is easy o show ( ha he Gammain opeaion eqies (D + 1(D + 1 B B c B compex MAC opeaions D saion TABLE NUM OF COMPUTATONS N COMPLEX MAC OPERATONS SFBC-OFDM SM-OFDM B B c 8 4 B B c 4 4 RC BF RC BF D D We isae he compaion savings in Tabe fo he SFBC and SM schemes discssed in his pape n each case, we pesen he oa nmbe of compex MAC opeaions pe symbo sing edced-compexiy (RC and be-foce (BF appoaches ns o ha he compexiy is smae fo SM- OFDM This is becase in he pesence of / imbaance and mobiiy, he sfficien saisic fo decoding an SM code bock is avaiabe in ony (D + 1 sb-caies wheeas ha fo SFBC is avaiabe in 4(D + 1 sb-caies Moeove, he savings achieved by esoing o he RC appoach ae mch highe fo SFBC in compaison o he SM case and his is de o he Aamoi code sce

7 A (m B G M (m (m C (m D (m A Simaion Sep G 1 ( (m D 1 G 1 A ( (m 1 1 C 1 + A 1 1 V SMULATON RESULTS D 1 B 1 C A D 1 C A 1 The sysem paamees ae simia o he 51 sb-caie pofie of he 8016e mobie WiMAX sandad 19, ie, he bandwidh is 5 MHz, he samping feqency is 56 MHz and he caie feqency is 5 GHz The channe code sed is ae- 1 convoiona code (171,133 wih andom bi ineeaving We se a fame sce wih fo OFDM daa symbos wih embedded pios, a peambe and a posambe The pio ovehead aio of 1 7, which is he same as ha of he Paia Usage of Sb-Caies (PUSC mode of he WiMAX sandad, b hei ocaions ae based on he pio paen shown in Figs 3(a and 3(b Fo SFBC, he paamee L is se o be six Uness saed ohewise, he foowing seings ae sed fo he simaions The channe s powe deay pofie is he SU-3 specificaion wih mobiiy impemened sing he Jakes mode A nomaized Doppe spead of % coesponding o a vehice speed of 100 km/h is sed The F / imbaance paamees sed ae α α 05 db, θ θ 5 When hee is FD / imbaance, he and fies ae ξ (n ξ (n δ(n + 01δ(n 1 and ξ (n ξ (n δ(n 01δ(n 1, especivey A he deeco op, sof bi-ikeihood aios ae comped which ae fhe pocessed by he deineeave and Viebi decode When pios ae sed, hey ae boosed in powe by 5 db han he daa sb-caies in accodance wih he 8016e sandad Finay, in ode o pefom he MMSE compensaion, we need o fix he vae of D + 1, he nmbe of bock diagonas in he channe maix o he nmbes of aps needed fo eqaizaion Shown in Fig 4 ae he enegies in he diagonas of he channe maix fo diffeen Doppe and CFO vaes The odinae is he pecenage of enegy conained in a given nmbe of diagonas agains he enie enegy in he channe maix The highe his vae, he bee is he banded appoximaion Fig 4 eveas ha 3 is sfficien since going fom 3 o 5 ess in a vey sma incease in he pecenage of enegy B Ress and Discssion Fig 5 depics he ess fo Aamoi SFBC-OFDM wih N P 1 pefomance cves fo SM-OFDM ae shown in Fig 6 fo N P N n each case, we pesen ess fo ncompensaed impaimens, compensaed wih pefec and esimaed channe sae infomaion (CS and he idea case of no impaimens can be seen ha he ncompensaed case exhibis a high eo foo and he poposed A (m B (m C (m D (m B D 1 P (m C 1 B D 1 (m R (m (m S (m A 1 B 1 P (m (m R (m S (m 1 (m 7, (18 (19 (0 MMSE compensaion heps edcing he eo foo by sevea odes of magnide Wih pefec CS, he pefomance comes vey cose o he case of no impaimens hogh i is no exacy hee becase we se ony (D + 1 main diagonas of he channe Fo he esimaed CS case, he pefomance is wihin 1 o db fom he case of pefec CS compensaion Moeove, in Fig 7, we have poed he mean sqae eo (MSE of channe esimaion agains SNR fo he SFBC case Nex, we compae he pefomance of SFBC and SM fo diffeen channe deay speads The pefomance of SFBC is sensiive o deay spead becase he spe-bock sce eqies he channe esponse o be consan ove L + D sb-caies When deay spead is age (ie, sma channe coheence bandwidh, L shod be sma, b i ess in high pio densiy D/(L + D, sggesing a adeoff Fo a fai compaison, we conside N P N fo boh SM and SFBC sysems Moeove, SFBC and SM empoy 16- AM and 4-AM modaion, especivey, o mach aes Sevee impaimen condiions wheein α α 05 db, θ θ 5 and a Doppe of 5% (coesponding o a vehice speed of 35 km/h wee imposed Boh schemes ae simaed nde SU-3 and SU-4 channes which have deay speads of 09 and 4 micoseconds, especivey As is eviden fom he ess in Fig 8, he Aamoi SFBC pefoms bee when he channe deay spead is sma and SM pefoms bee when he deay spead is age Now, we demonsae he obsness of o poposed compensaion schemes in he pesence of esida CFO in Fig 9, whee we pefom he compensaion and channe esimaion wiho any modificaions o he schemes descibed in Secions V and V Fo p o % CFO, we see ha boh SFBC and SM exhibi gacef degadaion even wih 3 Finay, in Fig 10, we show he ess fo a 64-AM sysem whee he compensaion gains ae eviden in his case as we is woh menioning ha we have no consideed he effec of ansmi powe ampifie (PA non-ineaiy in his wok Thee exis sevea appoaches in he ieae o dea wih he PA nonineaiy sch as pe-compensaion a he ansmie Howeve, is effec on imbaance compensaion is ncea, and join compensaion of PA non-ineaiy and / imbaance a he eceive is of pacica inees fo fe eseach X CONCLUSONS n his pape, we poposed edced-compexiy digia baseband esimaion and compensaion schemes fo SM-OFDM and SFBC-OFDM in he pesence of / imbaance and high mobiiy n each case, geneaized inea modes wee

8 % of enegy in he diagonas D, 5 diag, % Doppe D, 5 diag, 5% Doppe D1, 3 diag, % Doppe D1, 3 diag, 5% Doppe D0, 1 diag, % Doppe D0, 1 diag, 5% Doppe CFO (% of sb caie spacing FD ncomp F ncomp FD comp, es CS F comp, es CS FD comp, pef CS F comp, pef CS No impaimens E /N (db b 0 Fig 4 Enegy disibion in he diagonas of he channe maix fo diffeen Doppe vaes and esida CFO eves Fig 6 pefomance of SM-OFDM sing 16-AM wih F and FD / imbaances and mobiiy (α α 05 db, θ θ 5, ξ (n ξ (n δ(n + 01δ(n 1, ξ (n ξ (n δ(n 01δ(n 1, Doppe% of sb-caie spacing 10 FD comp F ncomp FD comp, es CS F comp, es CS FD comp, pef CS F comp, pef CS No impaimens Channe Esimaion MSE F / imb FD / imb E b /N 0 (db Fig 5 pefomance of Aamoi SFBC-OFDM sing 16-AM wih F and FD / imbaances and mobiiy (α α 05 db, θ θ 5, ξ (n ξ (n δ(n+01δ(n 1, ξ (n ξ (n δ(n 01δ(n 1, Doppe% of sb-caie spacing SNR (db Fig 7 Channe esimaion MSE fo SFBC wih F and FD / imbaances and mobiiy(α α 05 db, θ θ 5, ξ (n ξ (n δ(n + 01δ(n 1, ξ (n ξ (n δ(n 01δ(n 1, Doppe% of sb-caie spacing deived fo feqency-dependen / imbaance a boh he ansmie and eceive in he pesence of high mobiiy Channe esimaion was pefomed by designing ohogona pios in sch a way ha he / imbaance and channe paamees can be joiny esimaed fo each ansmi-eceive anenna pai sepaaey The channe esimaion and compensaion schemes emain nchanged iespecive of whehe hee is / imbaance a he ansmie side ony, eceive side ony o boh and whehe he imbaance is feqencydependen o independen Using simaions fo he WiMAX envionmen, we showed ha if ef ncompensaed fo, / imbaance and mobiiy es in high eo foos and o compensaion schemes ae effecive in edcing hese eo foos significany Moeove, o high-mobiiy simaions showed ha SFBC-OFDM opefoms SM-OFDM in he pesence of / imbaance wih inea MMSE decoding nde ow channe deay speads, wheeas nde high deay speads, SM pefoms bee a he same pio densiy Finay, we showed how o edce he compaiona compexiy of he inea MMSE eceives by expoiing he specia sce of he maices invoved Fo Aamoi SFBC-OFDM, addiiona edcions in compexiy ae achievabe de o he Aamoi bock sce, howeve, SM-OFDM has owe ovea inea MMSE decoding compexiy REFERENCES 1 B Naasimhan, D Wang, S Naayanan, H Minn, and N A-Dhahi, Digia compensaion of feqency-dependen join Tx/Rx / imbaance in OFDM sysems nde high mobiiy, EEE J Se Topics Signa Pocess, vo 3, no 3, pp , Jne 009 A Taigha and A H Sayed, MMO OFDM eceives fo sysems wih imbaances, EEE Tans Signa Pocess, vo 53, no 9, pp , Sep Y Zo, M Vakama, and M Renfos, Anaysis and compensaion of ansmie and eceive / imbaances in space-ime coded mianenna OFDM sysems, EURASP Jona onwieess Commnicaions and Newoking, vo 008, no 39105, D Tand and M Moonen, STBC MMO OFDM sysems wih impemenaion impaimens, Poc of EEE VTC 008-fa, pp 1 5, Sep H Kamaa, K Sakagchi, and K Aaki, An effecive imbaance compensaion schmeme fo MMO OFDM commnicaion sysem, in EEE CC, Jne 006, pp T C W Schenk, P F M Smdes, and E R Feddes, Esimaion and compensaion of Tx and Rx imbaance in OFDM-based MMO sysems, in EEE Radio Wieess Symp, Janay 006, pp 15 18

9 9 SFBC age deay spead SM sma deay spead SM age deay spead SFBC sma deay spead Uncomp Comp es CS Comp pef CS No impaimens E /N (db b E /N (db b 0 Fig 8 pefomance compaison of SFBC-OFDM (16-AM and SM- OFDM (4-AM nde diffeen channe deay speads (α α 1 db, θ θ 10, Doppe5% of sb-caie spacing Fig 10 Pefomance of 64-AM SFBC-OFDM scheme wih channe esimaion (F / imbaance α α 05 db, θ θ 5, ξ (n ξ (n δ(n + 01δ(n 1, ξ (n ξ (n δ(n 01δ(n 1, Doppe% of sb-caie spacing, CFO1% 10 SM 5% CFO SM % CFO SM 0% CFO SFBC 5% CFO SFBC % CFO SFBC 0% CFO 16 P H Moose, A echniqe fo ohogona feqency division mipexing feqency offse coecion, EEE Tans Commn, vo 4, no 10, pp , Oc G Sang, Linea Ageba and is Appicaions, 3d ed Haco Bace Jovanovich, nc, T Kaiah, A H Sayed, and B Hassibi, Linea Esimaion Penice Ha PTR, EEE Sd 8016e-005 and EEE Sd /Co1-005, 8 Feb E b /N 0 (db Fig 9 Pefomance of 16-AM SFBC and 16-AM SM-OFDM scheme wih CFO wih channe esimaion (F / imbaance α α 05 db, θ θ 5, Doppe% of sb-caie spacing 7, Esimaion and compensaion of feqency seecive Tx and Rx imbaance in OFDM-based MMO sysems, in EEE CC, Jne 006, pp D Tand and M Moonen, Compensaion of RF impaimens in MMO-OFDM sysems, in Poc of EEE CASSP, Api 008, pp Y-H Chng and S-M Phoong, Join esimaion of / imbaance and channe esponse fo MMO OFDM sysem, in EUSPCO, 007, pp Bahmi and M Moonen, Feqency domain imbaance and caie feqency offse compensaion fo OFDM ove doby seecive channes, in Poc of Eopean Signa Pocessing Conf, Sep 006, pp V Taokh, H Jafakhani, and A R Cadebank, Space-ime bock codes fom ohogona designs, EEE Tans nf Theoy, vo 45, no 5, pp , Jy S L, B Naasimhan, and N A-Dhahi, Redced-compexiy C miigaion fo mobie SFBC-OFDM wih appicaion o DVB-H, in EEE WCNC, Mach 008, pp H Minn and N A-Dhahi, Opima aining signas fo MMO OFDM channe esimaion, EEE Tans Wieess Commn, vo 5, no 5, pp , May S Kay, Fndamenas of saisica signa pocessing esimaion heoy Penice Ha PTR, Y Mosofi and D C Cox, C miigaion fo pio-aided OFDM mobie sysems, EEE Tans Wieess Commn, vo 4, pp , Ma 005