Efficient Algorithm for Detecting Layered Space-Time Codes

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1 4TH INTERNATIONAL ITG CONFERENCE ON SOURCE AND CHANNEL CODING, BERLIN, JANUARY Effcent Algorthm for Detectng Layered Space-Tme Codes D. Wübben, J. Rnas, R. Böhnke, V. Kühn and K.D. Kammeyer Department of Communcatons Engneerng Unversty of Bremen, Germany Emal: {wuebben,rnas,boehnke,kuehn,kammeyer}@ant.un-bremen.de Abstract Layered space-tme codes have been desgned to explot the capacty advantage of multple antenna systems n Raylegh fadng envronments. In ths paper, we present a new effcent detecton algorthm based on a sorted QR decomposton. It only needs a fracton of computatonal effort compared to the standard detecton algorthm requrng multple calculatons of the pseudo nverse of the channel matrx. The derved algorthm s not restrcted to layered space-tme archtectures, but can generally be used for detecton n vector channel systems. Keywords Layered space-tme codes, dversty, MIMO system, wreless communcaton. I. Introducton In a Raylegh fadng envronment, multple antenna systems provde an enormous ncrease n capacty compared to sngle antenna systems [1]. Consequently, multple-nput multple-output (MIMO) systems are predestned for hgh data rate wreless communcatons. Space-Tme codes are desgned to explot ths hgh capacty by usng space as a second dmenson of codng [2, 3]. Layered space-tme (LST) codes are a specal knd of space-tme codes wth the advantage of a feasble decodng complexty. The orgnal D-BLAST (Dagonal Bell Labs Layered Space Tme) archtecture proposed by Foschn [4] uses a dagonally layered codng structure n whch code blocks are dspersed across dagonals n space-tme. Thereby, an averaged channel whch s the same for all layers s acheved and the probablty of deep fades s reduced. Due to the dagonal arrangement of the code blocks, D-BLAST s not feasble for real tme mplementatons. A smplfed verson was proposed n [5] and s known as V-BLAST (Vertcal BLAST). It assocates each layer wth a specfc transmt antenna wch leads to an easer detecton and decodng process. For detectng the layers, the multple calculaton of the pseudo nverse of the channel matrx s necessary. In order to sgnfcantly reduce the computatonal effort of detecton, we ntroduce a new and very effcent way of detectng layered space-tme codes. Ths work was supported n part by the German mnstry of educaton and research (BMBF) wthn the project HyEff (project no. 01 BU 153) and the German natonal scence foundaton (DFG) wthn the project AKOM (project #Ka 841/9-1) Ths approach utlzes an adjusted QR decomposton (QRD) by sortng the detecton sequence due to exchangng the columns of the channel matrx. Ths new algorthm s compared to V-BLAST by smulaton results and by an estmaton of the computatonal effort. The remander of ths paper s organzed as follows. In secton II, the MIMO system and the LST archtecture are descrbed. In secton III, V-BLAST and a QRD based algorthm for detectng LST archtectures are revewed. The new approach s ntroduced n secton IV and the performance of both detecton algorthms are compared n secton V. In secton VI forward correcton codng s added n each layer and the computatonal effort of both detectng algorthms s compared n secton VII. A summary and concludng remarks can be found n secton VIII. II. System descrpton We consder the mult antenna system wth n T transmt and n R n T receve antennas shown n Fg. 1. The data s demultplexed n n T data substreams of equal length (called layers). These substreams are mapped nto M-PSK or M-QAM symbols c 1,..., c nt. Alternatvely, a forward error correcton (FEC) code can be used to encode the data substreams before mappng. We wll nvestgate the applcaton of FEC code n secton VI and assume uncoded symbols untl then. The substreams are organzed n frames of length L and are transmtted over the n T antennas at the same tme. The system s equal to V-BLAST proposed n [4] and denoted as Layered Space-Tme (LST) archtecture n [6]. Transmtter Layer1 Layer2 LayernT c 1 c 2 c n T H x 1 x 2 x n R Detector Recever Fg. 1. Model of a MIMO system wth n T transmt and n R receve antennas In order to descrbe the MIMO system, one tme slot of the tme-dscrete baseband model of the rec.

2 4TH INTERNATIONAL ITG CONFERENCE ON SOURCE AND CHANNEL CODING, BERLIN, JANUARY MIMO system s nvestgated. Let c = (c 1 c 2... c nt ) T denote the vector of transmtted symbols, then the correspondng receved sgnal vector x = (x 1 x 2... x nr ) T s calculated by x = H c + ν. (1) In equaton (1), ν = (ν 1 ν 2... ν nr ) T depcts the vector of nose terms at the n R recevng antennas, assumng uncorrelated whte gaussan nose of varance /2 per dmenson for all antennas. The transmtted symbols are normalzed so that the average receved energy per bt s one. The n R n T channel matrx h 1,1... h 1,nT H = (2) h nr,1... h nr,n T contans..d. complex fadng gans h j, descrbng the tap gans between transmt antenna and receve antenna j. Column of H s denoted by h and represents the sngle-nput multple-output (SIMO) channel between transmt antenna and the n R receve antennas. We assume a statc flat-fadng envronment,.e. the channel matrx H s constant over a frame and changes ndependently from frame to frame. The dstnct fadng gans are assumed to be uncorrelated and are perfectly known by the recever. III. Detectng Layered Space-Tme Codes In ths paragraph, two dfferent detecton algorthms for the LST archtecture are descrbed. Frst, the standard detecton algorthm proposed by Bell- Labs [5] and known as V-BLAST s depcted. Shu and Kahn utlzed the QR decomposton of the channel matrx for the detecton of the layers to derve error bounds for V-BLAST and D-BLAST n [6]. They presumed the knowledge of the best detecton sequence, but dd not dscuss the problem of an effcent assortng algorthm, whch s done n secton IV of ths paper. A. BLAST-Algorthm It s obvous from equaton (1) that the receved sgnals are a lnear combnaton of the n T transmtted sgnals. The optmum way of recoverng the n T sgnals at the recever would be maxmum-lkelhood detecton, whch s not feasble due to the enormous complexty. In [4] and [5], Foschn et al. proposed a successve nterference cancellaton technque whch nulls the nterferer by lnearly weghtng the receved sgnal vector wth a zero-forcng (ZF) nullng vector. In every detecton step, all sgnals but one are regarded as nterferer. By applyng the nullng vector to nterference cancellaton, the nfluence of these sgnals s nulled out, the target sgnal s detected and subsequently subtracted from the receved sgnal vector (Interference Cancellaton). For detectng sgnal, the nullng vector w has to be orthogonal to columns h l, l of the channel matrx. The condton 1 { w T 1 l = h l = (3) 0 l s fulflled by the -th row of the Moore-Penrose pseudo-nverse G := H + := ( H H H ) 1 H H, (4) of the channel matrx H. Wth g () denotng row of G, the receved sgnal vector x s lnearly weghted wth the nullng vector w T = g () and the result y = w T x = g () (H c + ν) (5) = c + ν. s used as a decson statstc for the -th substream where ν = g () ν denotes the actual nose. By applyng the quantzaton operaton Q[ ] approprate to the sgnal constellaton, sgnal can be estmated: ĉ = Q[y ]. (6) The nterference caused by the detected sgnal ĉ s now subtracted from the receved sgnal vector x x +1 = x h ĉ (7) and the correspondng column n the channel matrx s set to zero. The ndexed varables (H,G,x ) denote from now on the specfc varables n detecton step, begnnng wth the assgnment (H 1 = H, G 1 = G, x 1 = x) n the frst step. Usng the nomenclature ntroduced n [5], H +1 := H descrbes the nullng of column of the channel matrx H and corresponds to an equvalent system wth n T transmt and n R receve antennas. Thus, the pseudo nverse of ths reduced channel matrx H +1 s used to calculate the nullng vector for detectng layer + 1. The order of detectng affects the error probablty of the algorthm [5]. The sequence S = {k 1, k 2,..., k nt } s defned as a permutaton of the numbers 1, 2,..., n T to depct a specfc detecton sequel. Thus the values y k1, y k2,..., y knt are fltered one by one, the transmtted sgnals ĉ k1, ĉ k2,..., ĉ knt are estmated and the nterference s cancelled step by step accordng to equatons (5) to (7). In order to derve the mnmum total error probablty, t s optmal always to choose and detect the layer wth the largest post detecton sgnalto-nose rato [5]: E { c k 2} SNR k = E { n k 2 } w k 2 1 g (k ) 2. (8) 1 The transpose and conjugate transpose (Hermtan) of x are denoted by x T and x H, respectvely.

3 4TH INTERNATIONAL ITG CONFERENCE ON SOURCE AND CHANNEL CODING, BERLIN, JANUARY Consequently, t s optmal to choose the row g (k) of G wth mnmal norm and thus detect the assocated sgnal c k n detecton step. The whole detecton algorthm s shown n Fg. 2. (1) for = 1,..., n T (2) G = H + (3) k = arg mn j g (j) 2 (4) wk T = g (k) (5) y k = wk T x (6) ĉ k = Q[y k ] (7) x +1 = x h k ĉ k (8) H +1 = H k (9) end Fg. 2. V-BLAST algorthm for detectng layered space-tme sgnals B. QR decomposton of the channel matrx In [6], Shu and Kahn used the QR decomposton of the channel matrx H to derve bounds for the error probablty of LST codes. Therefore, the n R n T channel matrx H H = Q R, (9) s factorzed nto the untary n R n T matrx Q and the upper trangular n T n T matrx R. By denotng the column of H by h and column of Q by q, the decomposton n equaton (9) s descrbed columnwse by r 1,1... r 1,nT (h 1... h nt ) = (q 1... q nt ) r nt,n T (10) By multplyng equaton (1) from the left wth the Hermtan matrx of Q, a n T 1 modfed receved sgnal vector y = Q H x = R c + η. (11) s created from the n R 1 receved sgnal vector x. Snce Q s untary, the statstcal propertes of the nose term η = Q H ν reman unchanged. Element k of vector y becomes wth the nterference term y k = r k,k c k + η k + d k (12) d k = n T =k+1 r k, c. (13) Thus, y k depends on the weghted transmt sgnal r k,k c k, the nose η k and the nterference term d k. Snce R s upper trangular, d k s ndependent of the upper layer sgnals c 1,..., c k 1 and hence the lowest layer (transmt sgnal c nt ) s descrbed by y nt = r nt,n T c nt + η nt. (14) Then, the decson statstc y nt s ndependent of the remanng transmt sgnals and can be used to estmate ĉ nt [ ] ynt ĉ nt = Q (15) r nt,n T by applyng the quantzaton operaton Q[ ]. For detectng layer n T 1, the nterference term r nt 1,n T ĉ nt s elmnated n the modfed receved sgnal y nt 1 = r nt 1,n T 1 c nt 1+r nt 1,n T c nt +η nt 1. (16) Consequently, an nterference free decson statstc to estmate c nt 1 s obtaned under the assumpton ĉ nt = c nt. Detectng layer k = n T 1,..., 1 takes place n an equvalent way. Wth prevous decsons ĉ k+1,..., ĉ nt, the nterference term ˆd k s calculated and cancelled out n the modfed receved sgnal y k. Assumng that all prevous decsons are correct ( ˆd k = d k ), the value z k = y k ˆd k = r k,k c k + η k (17) s free of nterference and thus t can be used to detect c k wth ĉ k = Q[z k /r k,k ]. As already stated, the order of detecton s crucal for the error probablty of the LST system due to the rsk of error propagaton [5]. When usng the QR decomposton for detecton, the sequence of detecton s acheved by permutatng the elements of c and the correspondng columns of H and thereby results n dfferent matrces Q and R. The optmum R maxmzes SNR k = E { c k 2} r k,k 2 E { n k 2 } r k,k 2 (18) n each step of the detecton process (corresponds to the maxmzaton of r k,k for k = n T,..., 1) and can be found by performng O(n 2 T /2) QR decompostons of permutatons of H [7]. In order to reduce the computatonal effort of fndng a detecton sequence, we derve a suboptmal but less complex algorthm for sortng n the next secton. IV. Sorted QR decomposton In ths secton, a new and very effcent approach that comes close to the error performance of V- BLAST s ntroduced. It s bascally an extenson of the modfed Gram-Schmdt algorthm [8] by orderng the columns of H n each orthogonalzaton step. In order to descrbe ths new algorthm, we frst revew the modfed Gram-Schmdt algorthm wthout sortng. In the subsequent the motvaton for the sorted approach and the descrpton of the sorted QR decomposton are presented.

4 4TH INTERNATIONAL ITG CONFERENCE ON SOURCE AND CHANNEL CODING, BERLIN, JANUARY A. Modfed Gram-Schmdt The Gram-Schmdt algorthm computes matrx R of the QR decomposton lne by lne from top to bottom and matrx Q columnwse from left to rght [8]. Startng wth the assgnment Q := H = (h 1,..., h nt ), the followng operatons are executed n every step = 1,..., n T : Assgn norm of column vector q to the dagonal element r, of the upper trangular matrx R (r, = q ) and subsequently scale q to length one (q = q /r, ). Orthogonalze columns q l, < l n T, wth regard to q,.e. subtract the component parallel to q. The component n the drecton of q s equal to the projecton r,l = q H q l and the orthogonal part s calculated by q l = q l r,l q. In every step the vectors q 1,..., q form an orthonormal bass of the vector space spanned by h 1,..., h and the vectors q l, < l n T, contan the components of the correspondng h l orthogonal to ths vector space. The dagonal element r, denotes the length of h orthogonal to q 1,..., q 1 or h 1,..., h 1, respectvely. Furthermore, the coeffcents r,l specfy the component of h l, < l n T, n the drecton of q. B. Sorted Gram-Schmdt From the explanaton n secton IV-A t s obvous that the modfed Gram-Schmdt process calculates the dagonal elements from r 1,1 to r nt,n T. As already stated, t would be optmal to maxmze r k,k by permutatng the rows of Q n every detecton step,.e from r nt,n T to r 1,1. Thus, the optmal detecton sequence S OPT maxmzes SNR k n every detectng step k, k = n T,..., 1. Unfortunately, the search for S OPT s very costly, because t requres O(n 2 T /2) QR decompostons. The sorted Gram-Schmdt process (Sorted QR Decomposton, SQRD) proposed here searches for the detecton sequence S that acheves small SNR k n the upper layers. Consequently, the absolut values of the dagonal elements r k,k n the upper left area of the trangular matrx R are small. Thus, a detecton fault caused by the lttle sgnal-to-nose rato SNR k nfluences only few layers 1,..., k 1 through error propagaton. In order to llustrate the functonalty, one orthogonalzaton step of the SQRD algorthm s explaned n detal. The frst 1 elements of the sequence S are already calculated and therefore the vectors q 1,..., q 1 are fxed. The orderng of the remanng columns s varable and s determned by the orderng rule n order to force small sgnal-tonose ratos for the upper layers. Therefore, the column wth mnmal norm s chosen from the vectors q,..., q nt and denoted wth q k. The correspondng h k has the smallest component orthogonal to the space spanned by q 1,..., q 1, wch leads to the smallest r, of the possble permutatons n step and thereby the smallest SNR. The only change to the modfed Gram-Schmdt algorthm s the reorderng of the columns of Q. In every decomposton step the column q,..., q nt wth the mnmal length orthogonal to the already spanned vector space q 1,..., q 1 s chosen. The whole algorthm for the sgnal detecton s shown n Fg. 3. It conssts of a decomposton part (lne (1) to (11)) and a detecton part (lne (12) to (17)). In the decomposton part, the orderng s done n lne (3) and (4) and provdes the permutaton vector S, the orthogonal matrx Q and the upper trangular matrx R. In the detecton part, the receved sgnal vector s sorted accordng to the permutaton S, and the modfed receved sgnal vector y s calculated (lne (12)). The followng lnes (13) to (17) represent the teratve detecton process descrbed n secton III-B. SQRD Algorthm (1) R = 0, Q = H, S = (1,..., n T ) (2) for = 1,..., n T (3) k = arg mn q l 2 l=,...,n T (4) exchange col. and k n Q, R, S (5) r, = q (6) q = q /r, (7) for l = + 1,..., n T (8) r,l = q H q l (9) q l = q l r,l q (10) end (11) end Sgnal Detecton (12) y = Q H x (13) for k = n T,..., 1 (14) ˆdk = n T =k+1 r k, ĉ (15) z k = y k ˆd k (16) ĉ k = Q[z k /r k,k ] (17) end (18) Permutate ĉ accordng to S Fg. 3. SQRD algorthm and sgnal detecton of layered space-tme codes V. Performance Analyss The performance of the proposed SQRD detecton algorthm and the standard LST detecton algorthm (V-BLAST, [5]) was compared by means of Monte Carlo smulatons for several scenaros. Fg. 4 shows the bt error rates (BER) for an uncoded transmsson of QPSK symbols n a system wth n T = 8 and n R = 12 antennas. The teratve methods unsorted QR decomposton, SQRD and V-BLAST acheves a performance enhancement n comparson to the smple multplcaton wth the

5 4TH INTERNATIONAL ITG CONFERENCE ON SOURCE AND CHANNEL CODING, BERLIN, JANUARY pseudo nverse of H. The strong mpact of orderng the QR decomposton s obvous and only a small dfference of approxmately 0.5 db related to V-BLAST for a BER of 10 5 s vsble for the SQRD algorthm. PS Pseudo nverse Unsorted QRD SQRD V-BLAST BER ( ) 1 ( ) 2 Layer 4 Layer 3 Layer 2 Layer BER ( N ) Fg. 6. Gene detecton n a system wth n T = 4 and n R = 4 antennas, uncoded QPSK symbols 10 5 Fg. 4. Smulaton wth n T = 8 and n R = 12 antennas, uncoded QPSK symbols, spectral effcency of 16 Bt/s/Hz Fg. 5 shows the BER of the dfferent detecton algorthms for an uncoded system wth n T = 4 and n R = 6 antennas. These results confrm the good performance of the SQRD algorthm wth the reduced calculaton complexty n mnd. Pseudo nverse Unsorted QRD SQRD V-BLAST In Fg. 6 the BER per layer are shown for a system wth n T = 4 and n R = 4 antennas when the gene detecton s used. In every detecton step k = 4,..., 1 a dversty of g d = n R k+1 s acheved. Accordng to the dversty levels, the BER of layer k decays wth ( / ) g d. Thus the BER of the upper layers decay much steeper due to the hgher dversty levels n comparson to the layer detected frst (layer 4). VI. Applyng Channel Codng In order to mprove the performance of the sngle user to user communcaton, each layer s now ndependently encoded by a channel coder. For smplcty, we used the half rate (7, 5) oct convolutonal encoder and vterb decodng as shown n Fg. 7. Fg. 8 shows the Frame Error Rate (FER) of an BER 10 4 Transmtter FEC FEC FEC c 1 c 2 c n T H x 1 x 2 x n R Detector & Decoder Recever rec Fg. 5. Smulaton wth n T = 4 and n R = 6 antennas, uncoded QPSK symbols, spectral effcency of 8 Bt/s/Hz In [9] the gene detecton process was ntroduced to nvestgate the error propagaton of V- BLAST. Ths mples real nterference suppresson for each layer, but for subsequent layers deal detecton of the sgnals of precedng layers s assumed. Thus, only correct values are subtracted to reduce the system order and consequently no error propagaton takes place. Fg. 7. Coded LST archtecture wth n T transmt and n R receve antennas, FEC n each layer uncoded and a coded system equpped wth n T = 8 and n T = 12 antennas usng the V-BLAST or the SQRD detecton algorthm, respectvely. The transmtted QPSK sgnals are organzed n frames of length L = 100 symbols ncludng tal symbols for the coded case. The fgure shows the expected performance enhancement for coded systems and agan the SQRD detecton nearly reaches the error probablty of V-BLAST. Ths statement s confrmed by the smulaton result for a system wth n T = 4 and n R = 6 antennas,

6 4TH INTERNATIONAL ITG CONFERENCE ON SOURCE AND CHANNEL CODING, BERLIN, JANUARY FER uncoded SQRD uncoded V-BLAST coded SQRD coded V-BLAST as stated n Fg 2 needs f V BLAST = 8 n 4 T + 16 n 3 T n R + 8 n 2 T n R + 18 Ln T n R (19) floatng pont operatons. Usng the same countng, the SQRD algorthm needs f SQRD = 12 n 2 T n R 2 n T n R + n T (20) +L (4 n 2 T + 8 n T n R + 2 n T ) operatons. The computatonal requrements of the V-BLAST and the SQRD algorthm are compared by the quotent ρ = f SQRD f V BLAST. (21) Fg. 8. FER of uncoded and convolutonally coded system wth n T = 8 and n R = 12 antennas, frame length L = 100, QPSK symbols shown n Fg. 9. uncoded SQRD uncoded V-BLAST coded SQRD coded V-BLAST ρ ρ lm FER PS Fg. 10. Quotent ρ of requred floatng pont operatons for SQRD and V-BLAST wth n T = 8 and n R = 12 antennas varyng frame length L L Fg. 9. FER of uncoded and convolutonally coded system wth n T = 4 and n R = 6 antennas, frame length L = 100, QPSK symbols VII. Computatonal Effort The computatonal requrements of the proposed SQRD algorthm and V-BLAST are compared n ths secton. Therefore, the floatng pont operatons of these algorthms are specfed accordng to the system varables n T, n R and L, wth L denotng the frame length (.e. number of symbols transmtted wthn one layer). Real valued addtons, multplcatons and dvsons are equally counted as one flop to obtan a sngle value for the computatonal effort. Wth these assumptons, the V-BLAST algorthm Fg. 10 depcts the computatonal advantage of the SQRD algorthm over the V-BLAST algorthm for a system wth n T = 8 and n R = 12 antennas. Wth an ncreasng number L of symbols per layer the quotent ρ saturates to a lmt value. Ths value can be calculated by ρ lm = lm L ρ = 4 n T + 8 n R n R (22) and s shown n Fg. 10 as a horzontal lne. Furthermore, the computatonal demands of V- BLAST and SQRD concernng dfferent numbers of antennas are compared. A system wth L = 100 symbols per layer and a varyng number of transmt and receve antennas, wth n T = n R, s nvestgated. Fg. 11 shows the ncreasng computatonal advantage of the SQRD compared the V-BLAST algorthm for ncreasng number of antennas. Thus, the SQRD dramatcally reduces the computatonal requrements for systems wth larger amount of antennas.

7 4TH INTERNATIONAL ITG CONFERENCE ON SOURCE AND CHANNEL CODING, BERLIN, JANUARY ρ Effcency Wreless Communcatons Emplyng Mult- Element Arrays, IEEE Journal on Selected Areas n Commununcatons, vol. 17, no. 11, pp , November [8] G. Strang, Lnear Algebra and ts Applcatons, Harcout Brace Jovanovch College Publshers, Orlando, Florda, thrd edton, [9] S. Bäro, G. Bauch, A. Pavlc, and A. Semmler, Improvng BLAST Performance usng Space-Tme Block Codes and Turbo Decodng, n IEEE Proceedngs of Globecomm, San Francsco, CA, November n T = n R Fg. 11. Quotent ρ of requred floatng pont operatons for SQRD and V-BLAST wth L = 100 varyng n T = n R VIII. Summary and Conclusons We have descrbed a new detecton algorthm for LST codes. The algorthm s based on the Gram- Schmdt algorthm for QR decomposton and requres less computatonal effort n comparson to the standard detecton algorthm wth only small degradaton n error performance. We presented smulaton results for several scenaros and analytcally demonstrated the computatonal advantage of the proposed SQRD algorthm. Snce the core of our algorthm conssts of a sorted knd of QR decomposton, the derved algorthm s not restrcted to layered space-tme archtectures. It can generally be used to detect vector channel systems. References [1] E. Telatar, Capacty of Mult-antenna Gaussan Channels, AT & T-Bell Labs Internal Tech. Memo, June [2] S. M. Alamout, A Smple Transmt Dversty Technque for Wreless Communcatons, IEEE Journal on Selected Areas n Commununcatons, vol. 16, no. 8, pp , October [3] V. Tarokh, N. Seshadr, and A. R. Calderbank, Space- Tme Codes for Hgh Rate Wreless Communcaton: Performance Crteron and Code Constructon, IEEE Transactons on Informaton Theory, vol. 44, no. 2, pp , March [4] G. J. Foschn, Layered Space-Tme Archtecture for Wreless Communcaton n a Fadng Envronment when Usng Multple Antennas, Bell Labs Techncal Journal, vol. 1, no. 2, pp , Autumn [5] P. W. Wolnansky, G. J. Foschn, G. D. Golden, and R. A. Valenzuela, V-BLAST: An Archtecture for Realzng Very Hgh Rates Over the Rch-Scatterng Wreless Channel, n IEEE Proceedngs of ISSSE-98, Psa, Italy, 29. September [6] D. Shu and J.M. Kahn, Layered Space-Tme Codes for Wreless Communcatons usng Multple Transmt Antennas, n IEEE Proceedngs of Internatonal Conference on Communcatons (ICC 99), Vancouver, B.C., June [7] G. J. Foschn, G. D. Golden, A. Valenzela, and P. W. Wolnansky, Smplfed Processng for Hgh Spectral