Optimum Ordering for Coded V-BLAST

Transcription

1 Optmum Orderng for Coded V-BLAST Alan Urarte Toboso Thess submtted to the Faculty of Graduate and Postdoctoral Studes n partal fulfllment of the requrements for the degree of Master of Appled Scence n Electrcal and Computer Engneerng. Unversty of Ottawa 0 Alan Urarte Toboso, Ottawa, Canada, 0

2 Abstract The optmum orderng strateges for the coded V-BLAST system wth capacty achevng temporal codes on each stream are studed n ths thess. Mathematcal representatons of the optmum detecton orderng strateges for the coded V-BLAST under nstantaneous rate allocaton (IRA), unform power/rate allocaton (URA), nstantaneous power allocaton (IPA) and nstantaneous power/rate allocaton (IPRA) are derved. For two transmt antennas, t s shown that the optmum detecton strateges are based on the per-stream before-processng channel gans. Based on approxmatons of the per-stream capacty equaton, closed-form expressons of the optmal orderng strategy under the IRA at low and hgh sgnal to nose rato (SNR) are derved. Necessary optmalty condtons under the IRA are gven. Thresholds for the low, ntermedate and hgh SNR regmes n the -Tx-antenna system under the IPRA are determned, and the SNR gan of the orderng s studed for each regme. Performances of smple suboptmal orderng strateges are analysed, some of whch perform very close to the optmum one.

3 Table of Contents. Introducton..... Motvaton..... Contrbutons of the thess Thess outlne Lterature revew MIMO systems and channel capacty BLAST archtecture Reducng complexty of the V-BLAST Performance analyss and mprovement of the V-BLAST V-BLAST under channel estmaton errors The coded V-BLAST Summary The V-BLAST algorthm Transmsson strateges for MIMO communcatons Recever archtectures for MIMO communcatons Successve nterference cancellaton (SIC) V-BLAST: Channel model and assumptons The V-BLAST archtecture usng per-stream codng Optmum orderng n the uncoded V-BLAST Summary Instantaneous Rate Allocaton (IRA) Optmum orderng under the IRA... 9

4 4... General case Two Tx antennas General necessary optmalty condtons Optmal orderng strateges at low and hgh SNR Suboptmal orderngs Summary Unform Power and Rate Allocaton (URA) Optmum orderng under the URA General case Two Tx antennas Summary Non-unform power allocaton Instantaneous Power Allocaton (IPA) Optmum orderng under the IPA Suboptmum orderng Instantaneous Power and Rate Allocaton (IPRA) Optmum orderng under the IPRA Suboptmum orderng Analytcal boundares of the SNR regmes for two Tx antennas Summary SNR gan of orderng Two Tx antennas Two Tx antennas: valdaton of the results Three transmt antennas: an example v

5 7.3. Summary Lnk to Multple Access Wreless Channels Concluson Summary of the thess Future research References Appxes... 0 Appx A: Waterfllng algorthm... 0 Appx B: Optmum orderng under the IPRA (m=) Appx C: Matlab codes... 3 v

6 Table of Fgures Fgure : Number of MIMO publcatons snce ts dscovery Fgure : BLAST archtecture Fgure 3: Pctoral representaton of the V-BLAST archtecture when codng s used at each stream Fgure 4: Geometrc llustraton of the nterference nullng step Fgure 5: Upper bound gven by (4.7) (dark grey) and numercal lower bound (lght gray) of N (n percent) VS. # of Tx antennas Fgure 6: Complexty comparson between the exhaustve search of the optmum orderng and the analyss va the necessary optmalty condtons Fgure 7: Normalzed hstogram of orderngs that satsfy the necessary optmalty condtons for the (4x4) system Fgure 8: Normalzed hstogram of orderngs that satsfy the necessary optmalty condtons for the (5x5) system Fgure 9: Emprcal CDFs of the Max. Sum Ch. Gans detecton orderng, the optmum detecton orderng and the unordered detecton for the (5x5) system; SNR=-0dB; 0 4 channel realzatons Fgure 0: Emprcal CDFs of the Max. Sum Ch. Gans detecton orderng, the optmum detecton orderng and the unordered detecton for the (4x4) system; SNR=-0dB; 0 4 channel realzatons Fgure : Emprcal Pout vs. SNR of the optmum detecton orderng and the unordered detecton for the (5x5) system at a hgh target rate based on 0 4 channel realzatons Fgure : Emprcal CDFs of the nverse orderng, the MSCLS orderng, the optmum orderng and the unordered detecton for the (5x5) system; SNR=-0dB; 0 4 channel realzatons Fgure 3: Emprcal CDFs of the nverse orderng, the MSCLS orderng, the optmum orderng and the unordered detecton for the (4x4) system; SNR=0dB; 0 4 channel realzatons Fgure 4: Emprcal CDFs of the unprojected detecton orderng, the optmum detecton orderng and the unordered detecton for the (3x3) system; SNR=-0dB; 0 4 channel realzatons v

7 Fgure 5: Emprcal CDFs of the unprojected detecton orderng, the optmum detecton orderng and the unordered detecton for the (5x5) system; SNR=-0dB; 0 4 channel realzatons Fgure 6: Emprcal CDFs of the Foschn detecton orderng, the optmum detecton orderng and the unordered detecton under the URA for the (3x3) system; SNR=-0dB; 0 5 channel realzatons Fgure 7: Emprcal CDFs of the Foschn detecton orderng, the optmum detecton orderng and the unordered detecton under the URA for the (3x3) system; SNR=0dB; 0 5 channel realzatons Fgure 8: Tx sde archtecture of the coded V-BLAST wth non-unform power allocaton. The total normalzed power s m m Fgure 9: Emprcal CDFs of the Foschn detecton orderng, the optmum detecton orderng and the unordered detecton under the IPA for the (3x3) system; SNR=0dB; 0 5 channel realzatons Fgure 0: Emprcal CDFs of the Foschn detecton orderng, the optmum detecton orderng and the unordered detecton under IPA for the (4x4) system; SNR=0dB; 0 5 channel realzatons Fgure : Emprcal CDFs of the nverse orderng and the optmum orderng under WF for the (3x3) system; SNR=0dB; 0 4 channel realzatons... 7 Fgure : Emprcal CDFs of the nverse orderng and the optmum orderng under WF for the (4x4) system; SNR=-0dB; 0 4 channel realzatons Fgure 3: Emprcal CDFs of the nverse orderng and the optmum orderng under WF for the (4x4) system; SNR=0dB; 0 4 channel realzatons Fgure 4: Number of actve streams vs. SNR for a gven channel realzaton for the (x) system under the IPRA Fgure 5: Defnton of the SNR gan of orderng Fgure 6: SNR gan of orderng for the (x) system Fgure 7: SNR gan vs. SNR; numercal and analytcal solutons Fgure 8: SNR of orderng (3x3) Fgure 9: Uplnk wth sngle Tx antenna at each user and multple Rx antennas at the base staton Fgure 30: Pctoral representaton of the waterfllng algorthm v

8 Lst of Acronyms Acronym APA APRA ARA AWGN BER BLER BPSK CDF CSI D-BLAST FWF GSO IPA IPRA IRA ISTI MAC MIMO MISO ML MMSE M-PSK M-QAM MRC NC EGC QoS Rx SER SIC SIMO SISO SNIR SNR SVD TBER Tx Meanng average power allocaton average power/rate allocaton average rate allocaton addtve whte Gaussan nose bt error rate block error rate bnary phase-shft keyng cumulatve dstrbuton functon channel state nformaton Dagonal Bell Labs Layered Space-Tme fractonal waterfllng Gram-Schmtt Orthogonalzaton nstantaneous power allocaton nstantaneous power/rate allocaton nstantaneous rate allocaton nter-stream nterference multple access channel multple-nput multple output multple-nput sngle-output maxmum lkelhood mnmum mean-square error M-ary phase-shft keyng M-ary quadrature ampltude modulaton maxmum rato combnng noncoherent equal gan combnng qualty of servce recever symbol error rate successve nterference cancellaton sngle-nput multple-output sngle-nput sngle-output sgnal to nose plus nterference rato sgnal to nose rato sngular value decomposton total bt error rate transmtter v

9 Acronym URA V-BLAST WF ZF Meanng unform power and rate allocaton Vertcal Bell Labs Layered Space-Tme waterfllng zero-forcng x

10 Lst of Symbols and Notatons Notaton Meanng T transposton + conjugate transposton orthogonal projecton v Eucldean norm of vector v V matrx V T v -th component of vector v v, v,..., v m v -th column of matrx V v, v,..., v m projecton of the -th column of matrx v V v, v,..., v m orthogonal to the subspace spanned by the m rght-sde columns Symbol Meanng Frst appearance m number of transmt antennas Secton. n number of receve antennas Secton. q [,,..., ] T Tx vector (3.) q q q m r [,,..., ] T Rx vector (3.) r r r n H h,..., h m MIMO channel matrx (3.) crcularly symmetrc addtve whte Gaussan ξ CN(0, 0I) nose vector wth..d. entres (3.) 0 nose power (3.) P projecton matrx orthogonal to the spatal sgnatures of the m yet to be detected symbols Secton 3.5 correlated nose vector after the nterference ξ CN(0, P 0 ) nullng step (3.4) w optmum ZF weghts (3.5) optmum orderng (3.8) ' scalar nose after applyng the optmum ZF CN(0, ) (4.) weghts average SNR at each receve antenna (4.) 0 0 P out outage probablty (4.4) C capacty under the IRA (4.7) IRA C system capacty under orderng (4.7) x

11 Symbol Meanng Frst appearance C capacty under the IRA (4.7) h N IRA j projecton of h orthogonal to h j where s the angle between both vectors maxmum number of orderngs that can satsfy the necessary optmalty condtons (4.7) (4.7) C URA capacty under the URA (5.) dagonal matrx whch entres represent the Λ dag,..., m squared root of the power assgned to each (6.) stream C capacty under the IPA (6.6) IPA harmonc mean per-stream power gan for a g g m (6.7) gven orderng C capacty under the IPRA (6.7) IPRA C capacty under the waterfllng (6.7) WF C FWF capacty under the fractonal waterfllng water level (from the WF algorthm) for a gven orderng (6.9) * optmum power allocaton provded by the WF 0 h algorthm for a gven orderng (6.9) G SNR gan of the optmum orderng procedure (7.) x

12 Acknowledgments Ths thess would not have been possble wthout the support, gudance and patent of my drect supervsor Dr. S. Loyka and co-supervsor Dr. Francos Gagnon. I would also lke to thank my brother J. A. Urarte for hs specal help, advces and for always beng there no matter tme or day of the week. My grattude also goes to my famly for runnng always by my sde despte beng separated by thousands of klometers. Specal thanks to my dad for beng the wnner of many battles, you are an nspraton for me. My deep thankfulness to my mom, brother, wfe and grandmother for ther uncondtonal support and love through all ths process. x

13 . Introducton.. Motvaton The Multple-nput multple-output (MIMO) communcaton archtecture has been wdely studed durng the last 5 years due to the fact that t provdes very hgh spectral effcences that cannot be attaned by conventonal technques []-[4]. However, ths ncrease n the spectral effcences s accompaned by a sgnfcant growth n the system complexty. Vertcal Bell Labs Layered Space-Tme (V-BLAST) was proposed by Foschn [3] as a low complexty MIMO scheme able to acheve a substantal porton of the total MIMO capacty gven that the multpath envronment s rch enough. In the V-BLAST archtecture, multple data streams are transmtted over the multple transmt (Tx) antennas smultaneously, whch are detected at the recever usng successve nterference cancelaton to acheve good system performance at moderate complexty. The order at whch the streams are detected affects the V-BLAST performance. Unordered V-BLAST has been commonly used to study the performance of ths archtecture durng the last years; optmzaton strateges that help enhance ts performance have been proposed n [4], [3], and [3]. Meanwhle, ordered V-BLAST represents a challenge to analytcal examnaton due to the ncreased complexty added by the orderng procedure. The optmum orderng procedure for the uncoded V-BLAST n Raylegh fadng channels was proposed by Foschn n [3]. The stream detecton order s organzed accordng to ther after processng SNRs n the decreasng order,.e. at each step the remanng stream wth hghest after processng SNR s detected frst and then ts contrbuton s subtracted

14 from the receved vector for next detecton steps. The optmalty of ths orderng strategy s based on the fact that t mnmzes the total error probablty of the system. On the other hand, a closed-form analyss of the optmum detecton orderng for the coded V-BLAST has not been settled yet. A closed-form analyss of the optmum detecton orderng for the coded V-BLAST s provded n ths thess. The optmum orderngs under the IRA, the URA, the IPA and the IPRA are studed. Any optmzaton strategy n coded systems must target the outage probablty. Snce the nstantaneous optmzatons of the outage probablty and the capacty acheve the same lowest value of the outage probablty n the coded V-BLAST [3], the optmzaton of the detecton orderng s studed from the system capacty pont of vew... Contrbutons of the thess The man contrbutons of ths thess are as follows: Dervaton of the optmal orderng strateges n the coded V-BLAST under the IRA, the URA, the IPA and the IPRA. Comprehensve closed-form analyss of the optmal orderng strateges under the IRA, the URA, the IPA and the IPRA when usng two Tx antennas. Closed-form expressons of the optmal orderng n the coded V-BLAST under the IRA at low and hgh SNR based on approxmatons of the per-stream capacty equaton. Dervaton of necessary optmalty condtons for an orderng strategy n the coded V-BLAST under the IRA.

15 Dervaton of SNR thresholds that separate the low, the ntermedate and the hgh SNR regmes n the coded V-BLAST wth two transmt antennas under the IPRA. Defnton and closed-form analyss of the SNR gan of orderng n the coded V- BLAST wth two transmt antennas under the IPRA..3. Thess outlne The man goal of ths thess s the closed-form analyss of the optmum stream detecton orderng n the coded zero-forcng V-BAST. Chapter gves a revew of the relevant research carred out n the wreless MIMO feld, devotng specal attenton to the BLAST archtecture. Chapter 3 ntroduces the channel model used along the thess and descrbes the V-BLAST algorthm. Chapters 4 and 5 provde the closed-form analyss of the optmal detecton orderng under the IRA and the URA respectvely. The optmal detecton orderngs under the IPA and the IPRA are nvestgated n Chapter 6. Chapter 7 defnes the SNR gan of orderng and gves a detaled analytcal breakdown to the SNR gan of orderng when usng two Tx antennas under the IPRA. Chapter 8 exts the results obtaned from the pont-to-pont perspectve to the multuser communcatons vewpont. Fnally, Chapter 9 states the concluson down from the results presented n the thess and outlnes areas for future research. 3

16 . Lterature revew.. MIMO systems and channel capacty Desgnng wreless communcatons systems wth hgh spectral effcences and hgh qualty of servce (QoS) represents a sgnfcant engneerng challenge. Several strateges have been used to ncrease the data rate that can be sent through a channel wth arbtrary small error probablty,.e. the channel capacty. One approach conssts of ncreasng the bandwdth so that more bts can be transmtted to the medum per unt tme (ncreasng the data rate n bts/sec). However, bandwdth s a very lmted and expensve resource and n addton, ncreasng the bandwdth does not ncrease the spectral effcency (n bts/sec/hz) of the system. Another alternatve s to ncrease the transmt power (P t ) snce spectral effcency s an ncreasng functon of P t, nevertheless most communcaton systems are power lmted due to nterference and/or human health concerns. Due to the multpath characterstc of the wreless propagaton channel, multple copes of the transmtted sgnal arrve at the recever at dfferent moments of tme. Ths combnaton of sgnals wth a phase dfference at the recever causes abrupt varatons n the receved sgnal power (P r ) or n the receved SNR, whch s known as fadng. Fadng creates outage events and further lmts the capacty of the wreless channel; however t can be mtgated usng dversty technques. Dversty technques use ndepent channels to s dfferent replcas of the desred sgnal to the recever [33]. Dversty technques can operate n tme, frequency and/or space domans. The two representatons of space dversty technques are transmt dversty where multple transmt antennas are used (also known as multple-nput sngle-output or MISO systems) and receve (Rx) dversty where multple receve antennas are used (also 4

17 known as sngle-nput multple-output or SIMO systems). In both cases t s requred for antennas to be placed suffcently far apart so that the channel gans between dfferent antenna pars fade ndepently. These systems provde a dversty gan that s reflected n the exponent of SNR n the error probablty equaton. If the error probablty of a system s expressed as a P (.) e where 0 s the average SNR, then the dversty gan or dversty order of the system s L L 0 and a s the SNR gan. The error probablty decreases as the L-th power of SNR, correspondng to a slope of L n the error probablty curve (n db scale). MISO and SIMO systems offer hgher capacty than sngle-nput sngle-output (SISO) systems wthout ncreasng power or bandwdth; however the capacty ncreases logarthmcally wth the number of transmt or receve antennas respectvely []. In , the multple-nput multple-output (MIMO) wreless system archtecture was proposed by Foschn [] and Telatar [] as a spectrally effcent way of communcaton. The key dea of MIMO systems s to transmt multple data streams usng a set of Tx antennas and to use multple Rx antennas and approprate sgnal processng to recover them. It was shown n [] that wth MIMO the capacty of the wreless (uncorrelated) propagaton channel ncreases lnearly wth mn m, n where m and n are the number of Tx and Rx antennas respectvely. Expermental results n [3] showed that the MIMO capacty was more than 0 tmes hgher than that of the SISO system for the same bandwdth and total transmt power constrant. These poneerng works generated a great nterest n the 5

18 area of MIMO systems around the word. The number of publcatons n ths area has been enormous snce ts dscovery n (Fgure shows the detals). Year Fgure : Number of MIMO publcatons snce ts dscovery... BLAST archtecture The BLAST archtecture, ntroduced by Foschn [] n 996, s a low complexty transcever archtecture to communcate over the MIMO wreless channel. Although suboptmal, t s able to attan a sgnfcant fracton of the theoretcal MIMO capacty over the rch-scatterng wreless channel. BLAST mples ndepent transmsson of streams at the transmtter sde and successve nterference cancellaton (SIC) at the recever. In the BLAST archtecture (see Fgure ), ndepent data streams are transmtted at the same tme and frequency usng a set of Tx antennas. At the recever, the detecton of a gven stream ncludes three man procedures: Ths data ncludes all publshed paper contanng all the keywords MIMO, wreless, channel, spacetme, communcatons returned by the Google Scholar search engne, for each year. 6

19 nterference cancellaton from already detected streams, nterference nullng of yet-to-be-detected streams, optmal orderng Under ths scenaro, each data stream s decoded ndepently after nullng the nterference generated by the yet-to-be-detected streams and after cancelng the nterference from the already detected ones. Durng the nterference cancellaton procedure the stream s reencoded and ts contrbuton s subtracted from the Rx vector. The order at whch the streams are detected affects the general performance of the BLAST archtecture and hence an orderng procedure s necessary as well. The orgnal detecton algorthm for the BLAST archtecture s the combnaton of lnear nullng and SIC. The nullng vectors can be generated by the zero-forcng (ZF) or the mnmum mean-square error (MMSE) crteron, thus the correspondng algorthm s generally called ZF-SIC or MMSE-SIC algorthm. In the orgnal transmsson process for the BLAST archtecture proposed n [], nstead of assgnng each of the ndepent streams to a specfc antenna, the btstream/antenna assocaton s perodcally cycled,.e. each stream s dspersed dagonally across antennas and tme. The BLAST archtecture under ths transmsson strategy s known as Dagonal BLAST (D-BLAST). The complexty of the D-BLAST may be too hgh for practcal systems. In 998, V-BLAST (Vertcal BLAST) was ntroduced n [3] as a low complexty wreless communcaton archtecture. In the V-BLAST, the layerng s horzontal, meanng that all the symbols of a certan stream are transmtted through the same antenna (one stream per antenna). It was shown n [4] that ths archtecture s able to acheve very hgh spectral 7

20 effcences e.g. spectral effcences n the order of 0 40 bts/sec/hz n an ndoor propagaton envronment at realstc SNR s and error rates. More detals of the V-BLAST algorthm wll be dscussed n the next chapter. Fgure : BLAST archtecture. V-BLAST has nterested researchers from all over the word due to ts enormous potental. Many papers have been publshed addressng dfferent ssues snce ts dscovery. Sgnfcant research effort has been made not only to reduce V-BLAST complexty but also to mprove ts performance. The V-BLAST performance under real condtons envronments, where channel estmatons errors occur, has also been a feld of wde research..3. Reducng complexty of the V-BLAST The complexty of the V-BLAST algorthm les on two prncpal factors: the detecton strategy used and the optmal orderng procedure. Addressng these concerns Wa et al. proposed n [5] the replacement of the optmal decodng order by a suboptmal one, based on the pseudo-nverse of the channel matrx, and the utlzaton of Gram-Schmtt Orthogonalzaton (GSO) to compute the pseudo-nverse n fndng the weght vectors n the 8

21 orgnal V-BLAST. They obtaned a 7% reducton of the total number of arthmetc operatons for a x8 system n slow fadng channel. A recursve MMSE-SIC algorthm was presented n [7]. The MMSE nullng vectors and the optmal detecton order were calculated from the prevous computatonal results va smple recursve pseudo-nverse formulas. The complexty of the proposed algorthm was shown to be lower than that proposed n [6], where another fast recursve algorthm was presented usng the Sherman-Morrson formula and the prncple of parttoned matrces. The reducton n complexty n terms of multplcatons, addtons and floatng-pont operatons was more evdent for a practcal (small) number of transmt antennas. Effcent detecton algorthms utlzng the QR decomposton of the channel matrx were proposed n [9]-[3]. An algorthm that jontly calculates an optmzed detecton order and the QR decomposton of the channel matrx was proposed n [0]-[] (MMSE Sorted QR Decomposton). Hassb [9] proposed a square-root MIMO detecton algorthm based on detectng frst the symbol assocated wth the maxmum dagonal entry n the R matrx after the QR decomposton of the channel matrx. Ths algorthm was mproved by Zhu et al. [] by a 36%, reducton n the number of multplcatons and addtons. A new mprovement of the already mproved square-root algorthm was proposed n [3] based on a fast algorthm for nverse Cholesky factorzaton. These detecton algorthms based on QR decomposton reduce the number of matrx nverson and represent an nterestng alternatve. QR decomposton s the decomposton of a matrx H nto a product H=QR of an orthogonal matrx Q and an upper trangular matrx R. 9

22 Based on the standard MMSE V-BLAST algorthm, a reduced complexty detecton algorthm (RC-MMSE-SIC) was proposed n [8]. The man dea of the proposed algorthm s to detect the streams whose sgnal to nterference plus nose ratos (SINRs) are above a certan threshold nstead of detectng only the stream wth largest SINR n each detecton step as proposed by the orgnal V-BLAST algorthm. The algorthm also makes use of the GSO to compute the pseudo-nverse n fndng the weght vectors. The scheme, although suboptmal, decreases the computatonal complexty of the standard MMSE V-BLAST..4. Performance analyss and mprovement of the V-BLAST Sgnfcant research effort has been made to analyse and mprove the V-BLAST s performance. In [4], a geometrcally-based analytcal approach to the performance analyss of the ZF V-BLAST algorthm was presented. It was shown that wthout optmal orderng and under uncorrelated Raylegh fadng channel, the dversty order at the -th processng step s n m, where n and m represent the number of receve and transmt antennas respectvely. Outage probabltes and average bt error rates (BERs) expressons were derved for the specfc case of xn (two transmt and n receve antennas) systems when optmal orderng s mplemented. Moreover t was shown that the effect of the optmal orderng n xn systems s a SNR gan of 3 db at the frst detecton step and no dversty gan s attaned. However, the use of noncoherent equal gan combnng (NC EGC) after the nterference nullng (orthogonal projecton) was assumed n the analyss, whch s not optmum, and the after-projecton nose correlaton was gnored. Furthermore, the error propagaton was dsregarded. 0

23 In [6], an analytcal performance evaluaton of the unordered ZF V-BLAST n Raylegh fadng channels was made, ths tme the optmum maxmum rato combnng (MRC) was employed after the nterference projecton whle takng nto account the after projecton nose correlaton. The MRC weghts provde the best performance n terms of the output SNR, and hence, the BER. It was demonstrated that the optmum MRC weghts nclude the projectons and are orthogonal to each other resultng n the after-combnng nose components to be ndepent at each step; then closed-form expressons for the nstantaneous BER at each step were derved. Average BER expressons were also obtaned based on the facts that the nstantaneous SNR at each step are ndepent of each other and that the nter-stream nterference (ISTI) s Gaussan for a Raylegh ndepent and dentcally dstrbuted (..d.) channel. Exact BER expressons, takng nto account the error propagaton, were obtaned for bnary phase-shft keyng (BPSK) showng that whle the error propagaton affects dramatcally hgher detecton steps (resultng n the dversty order beng equal to n m at each step), the frst detecton step s not affected. As the frst stage domnates the error performance (t has the lowest dversty order when error propagaton s gnored), t was concluded that the error propagaton has only a mnor effect on the total average BER. Whle the assumpton of no orderng allowed makng such an nsghtful evaluaton, t lmted the results obtaned. In [8], an analytcal approach to the analyss of the xn V-BLAST was presented and the results were shown to be consstent wth those n [4]. In [9], prevous work n [4] was exted, the authors evaluated the outage and error rate performance of the ordered ZF V-BLAST wth more than two transmt antennas n..d Raylegh fadng channels usng a geometrcally-based framework. Based on a number of bounds on the outage probablty,

24 accurate closed-form approxmatons to the average block error rate (BLER 3 ) and the total bt error rate (TBER 4 ) were derved. It was shown that for m transmt antennas, the effect of the optmal orderng s an m -fold SNR gan at the frst step, but no dversty gan s obtaned. Ths work also exted the analyss made n [5] and [7] where from a dverstyorder-based analyss t was demonstrated that the dversty order s not affected by the orderng procedure. The rgorous mathematcal proof of the m-fold SNR gan of orderng was derved n [5]. The analytcal approaches to V-BLAST provde sgnfcant nsght nto the algorthm performance and ts bottlenecks creatng the base for optmzaton. In the V-BLAST algorthm, lower detecton steps enjoy lower dversty order (gnorng the error propagaton) lmtng the system performance. In ths sense, a wdely used technque for mprovng the error performance of the uncoded V-BLAST s to use a non-unform power allocaton that reduces the error rates at early stages. Ether nstantaneous/average BLER or nstantaneous/average TBER have been used as the objectve functons to mnmze n the problem of fndng the optmum power allocaton. In [], the optmum transmt power allocaton was numercally obtaned for xn V-BLAST usng the nstantaneous BLER as the objectve functon. An approxmaton of the transmt power vector that mnmzes the nstantaneous TBER (gnorng the error propagaton) was derved n [], [3]. Meanwhle n [0], closed-form expressons for the optmum power allocatons of the uncoded ZF V-BLAST and MMSE V-BLAST, usng the nstantaneous TBER and accountng for error propagaton, were derved based on a number 3 It s defned n [9] as the probablty of havng at least one error n the detected transmt symbol vector. 4 It s defned n [9] as the error rate of the output stream to whch all the ndvdual sub-streams are merged after the detecton.

25 of approxmatons. It was also shown n [0] (va smulatons), that the error propagaton does not have a sgnfcant mpact n the performance of the optmzed systems. In [4], compact closed-form approxmatons for the optmum power allocatons, based on the average BLER and TBER were obtaned. It was demonstrated that the SNR gan of the optmum power allocaton cannot exceed the number of transmt antennas and that the optmzaton based on the TBER results n the same performance as the one based on the total BLER. The latter was shown to be more sutable for analytcal technques snce t does not requre explct characterzaton of the error propagaton effect..5. V-BLAST under channel estmaton errors Most V-BLAST detecton algorthms.e. ZF-SIC or MMSE-SIC and optmzaton technques are based on perfect channel knowledge beng avalable at the recever. However, perfect channel knowledge s never avalable a pror. In practce, the channel has to be estmated. Ths can be done, for example, by transmttng plot symbols that are known n advance at the recever. As the system performance deps on the qualty of the channel estmate, extensve research has been carred out to study the mpact of channel estmatons errors on V-BLAST archtecture. In [6] the performance of the uncoded V-BLAST under channel estmaton errors was analyzed. The performance was examned through a perturbaton analyss. The perturbaton of the channel matrx was approxmated by an addtonal nose term added to the orgnal unperturbed system suggestng that under channel estmaton error the V-BLAST system wll suffer from addtonal system nose. A tght error floor was derved as a result of the equvalent system nose, whch s a combnaton of the channel estmaton errors and the 3

26 addtve whte Gaussan nose at the recever. It was shown va smulatons that the MMSE s not more robust than the ZF recever under channel estmaton errors. Smulatons also showed that ZF-SIC s a more robust opton able to tolerate about twce the amount of channel estmaton errors as the ZF recever. However, the authors dd not consder the effect of channel estmaton error n the ZF recever caused by computng the pseudonverse of the naccurate channel estmate. Furthermore, they focused on the approxmaton of the beforeprocessng SNR. An error-propagaton analyss of the uncoded ZF V-BLAST wth channel estmaton errors was carred out n [7]. A tght upper bound for the average symbol error rate (SER) was derved. Furthermore, t was ponted out that the V-BLAST processng wth channelestmaton errors and the detecton order based on perfect channel estmates produce no sgnfcant change n the system performance (n terms of the average SER). The effect of channel estmaton errors on the performance of MIMO ZF recevers n uncorrelated Raylegh flat fadng channels was nvestgated n [8]. Ths tme the focus was on the approxmaton of the after-processng SNR dstrbuton. By modelng the estmaton error as ndepent complex Gaussan random varables, tght approxmatons for both the after-processng SNR dstrbuton and bt error rate (BER) for MIMO ZF recevers wth M- QAM and M-PSK modulatons were derved n closed-form. Besdes the prevously mentoned error floor, t was found that the BER under channel estmatons error s an ncreasng functon of the number of Tx antennas. An analytcal method to derve the average SER of the sgnals detected at each step n ZF V-BLAST was presented n [30]. The method accounts for both error propagaton and channel estmaton errors. It was shown that ZF V-BLAST s more senstve to channel 4

27 estmaton errors at hgh SNR, and that the effects of naccurate channel estmatons have more sgnfcant mpact n later detecton steps when optmal order s employed. Nevertheless, at hgh SNR channel estmaton errors are less probable to occur. Some researchers have proposed modfcatons to the orgnal V-BLAST algorthm n order to mprove ts performance under channel estmaton errors. For example a robust symbol detecton orderng method for the uncoded ZF V-BLAST was proposed n [9] (robust n the sense to be less senstve to channel estmaton errors). The proposed orderng was shown to optmze the average post-detecton SINR over the channel estmaton errors and to organze the detecton order by decodng the symbol correspondng to the best average SINR frst. Although the authors clamed that the proposed algorthm s capable of achevng global performance optmzaton, ths scheme does not take nto account the error propagaton due to the SIC. The error propagaton s present n the uncoded V-BLAST and hence t lmts the overall system performance. However, n the coded V-BLAST the use of capacty achevng temporal codes at each stream allows the per-stream transmsson rates to match the correspondng capactes, so there are no errors when the streams are not n outage, and hence no error propagaton n-between the streams. Ths thess s manly devoted to the study of the optmum orderng strateges n the coded ZF V-BLAST. Gven our focus, the use of capacty achevng temporal codes at each stream and perfect channel estmaton at the recever wll be assumed. 5

28 .6. The coded V-BLAST Whle the studes mentoned above deal wth the uncoded V-BLAST, uncoded systems are rare and most modern communcaton systems use codng [33]. Average and nstantaneous optmzaton of power and rate allocaton for the coded V-BLAST have been studed n [3] and [3]. The performance metrcs n these systems are the outage probablty and the outage capacty. In [3], an analyss and performance evaluaton of three average optmzaton strateges targetng the outage probablty under the total power constrant was carred out. The three optmzaton strateges were: average power allocaton (APA), motvated by the fact that many practcal system use power control; average rate allocaton (ARA), sutable for varable-rate system usng dentcal and fxed power amplfers to smplfy the RF part of the system; and jontly average power and rate allocaton (APRA), whch s sutable for varable-rate varable-power systems. It was shown that the APA offers an SNR gan (upper-bounded by the number of transmt antennas), but the dversty order of the system remans unchanged. Ths s the same result obtaned for the uncoded V-BLAST n [4]. The ARA ncreases the system dversty order (the dversty orders at each stream are equal) and hence s more effcent than the APA. The APRA only offers a power gan over the ARA, but no dversty gan. The same study was made for the case of nstantaneous optmzaton n [3]. In ths case the three optmzaton strateges studed were: nstantaneous power allocaton (IPA), nstantaneous rate allocaton (IRA), and jontly nstantaneous power and rate allocaton (IPRA). It was demonstrated that the maxmzaton of the nstantaneous system capacty (va the IPRA) also mnmzes the outage probablty and, hence, both problems are equvalent under arbtrary fadng dstrbuton. It was also proven that the conventonal waterfllng (WF) algorthm s not optmal for V-BLAST. Instead the fractonal 6

29 waterfllng (FWF) algorthm was proposed and shown to maxmze the V-BLAST capacty va the IPRA. Furthermore t was demonstrated that ths algorthm attans the full MIMO channel dversty n the low outage regme. An optmum nstantaneous power allocaton to maxmze the system capacty of a mult-stream transmsson under unform power and rate allocaton was also presented. Ether n the presence of uncoded or coded V-BLAST t s a general agreement that the nstantaneous optmzaton offers better performance than the average one, but at the cost of ncreasng the system complexty due to the necessary feedback and power reallocaton for every channel realzaton. The results obtaned n [3] and [3] are very nsghtful and wll be wdely used here. However, they are lmted because the optmal orderng was not consdered. The am of ths thess s precsely to help fll ths gap by studyng optmal orderng strateges n the coded V- BLAST..6. Summary MIMO s one of the most mportant technologcal dscoveres n the wreless communcaton feld. MIMO systems offer theoretcal transmsson rates over the wreless propagaton channel never magned before. However, the hgh complexty assocated wth MIMO technology s the man lmtaton for some applcatons. V-BLAST s a transcever archtecture desgned to attan a sgnfcant porton of the theoretcal capacty offered by the MIMO wreless propagaton channel at a relatvely low complexty. Substantal research efforts have been made to reduce the V-BLAST complexty and to analyse/mprove ts performance under both deal and realstc condtons. A lterature 7

30 revew outlnng the most sgnfcant research about ths archtecture has been provded n ths chapter. Whle several papers dealng wth the detecton orderng n the uncoded V- BLAST have been publshed, lttle s known about the detecton orderng when codng s used. In order to shed lght on ths ssue, ths thess studes the optmum detecton orderng n the coded V-BLAST when capacty achevng temporal codes are used at each stream. 8

31 3. The V-BLAST algorthm 3.. Transmsson strateges for MIMO communcatons Modern wreless communcatons systems demand hgh data rate accompaned wth hgh relablty.e. low error probablty. MIMO system has shown to offer very hgh spectral effcences n rch scatterng envronments, ts capacty scales lneally wth the mnmum number of antenna elements []-[]. When the channel matrx s perfectly known at the transmtter, the full capacty of the wreless MIMO channel can be acheved by transmttng ndepent streams n the drectons of the rght sngular vectors of the channel matrx and assgnng the powers followng the well-known waterfllng algorthm. Under the prevous approach, perfect channel state nformaton (CSI) must be avalable at the transmtter. Also a sngular value decomposton (SVD) of the channel matrx, whch s a very complex and tme consumng operaton, s requred. Furthermore, n the case of fadng channels the channel state changes constantly and the SVD of the channel matrx has to be executed for every realzaton of the channel overwhelmng the system. In practcal systems, a smple and effcent approach s to s ndepent data streams through the dfferent transmt antennas. Ths transmsson strategy does not requre the knowledge of the channel matrx at the transmtter sde. Nevertheless, by havng ths nformaton at the transmtter, nstantaneous power/rate allocaton can be used n order to mprove the system performance. Ths transmsson strategy s also able to acheve the wreless MIMO channel capacty when approprate sgnal processng s used for the detecton of each stream at the recever. 9

32 3.. Recever archtectures for MIMO communcatons Due to the nature of the wreless propagaton channel, n MIMO systems a mxture of the sgnals transmtted from each transmt antenna mpnges over each receve antenna. The objectve of the recever s to recover the sgnals from each transmt antenna n a relable and computatonally effectve way. The capacty of the wreless MIMO channel can be attaned by jontly decodng the receved data streams usng the Maxmum Lkelhood (ML) recever. However, the complexty of ths method grows exponentally wth the number of streams makng t unfeasble for systems wth a hgh number of transmt antennas. The complexty of the ML recever can be reduced usng lnear recevers as MMSE and ZF recevers. These recever archtectures use lnear operatons to convert the problem of jont decodng of the data streams nto one of ndvdual decodng of the data streams [33]. The MMSE recever optmally trades off fghtng nter-stream nterference and sotropc Gaussan nose. It maxmzes the output SINR for any value of SNR. The MMSE recever can be used for the detecton of each stream separately. In the detecton of a gven stream t can be nterpreted as a recever that frst whtens the spatally-colored nose (represented by the sum of the nter-stream nterference and the sotropc Gaussan nose) and then apples MRC for the case of whte Gaussan nose to maxmze the output SNR [33]. The ZF recever focuses on completely nullng out the nter-stream nterference dsregardng the presence of nose. Ths s done through the multplcaton by a projecton matrx whch s orthogonal to the subspace spanned by the spatal sgnatures of the yet to be detected streams. Then the demodulaton of the gven stream can be performed match 0

33 flterng to the projected channel gan vector. Ths recever maxmzes the output SNR subject to the constrant of nullng out the nterference from other streams Successve nterference cancellaton (SIC) The performance of the MMSE and ZF recevers can be mproved by the successve cancellaton of the already detected streams. Once a data stream s successfully recovered, ts contrbuton can be subtracted from the receved vector and next detected streams wll not face the nterference caused by the already detected one. Ths combnaton of MMSE or ZF recevers wth SIC s precsely the detecton algorthm used for the V-BLAST archtecture. The correspondng algorthm s known as ZF- SIC (or ZF V-BLAST) f the ZF recever s used or as MMSE-SIC (or MMSE V-BLAST) f the MMSE recever s employed. The MMSE-SIC s able to acheve the full capacty of the wreless MIMO channel whle the ZF-SIC can attan a substantal porton of t at a lower complexty. Due to ts lower complexty and treatable equatons, the ZF-SIC wll be used to evaluate the optmal detecton orderng of the coded V-BLAST V-BLAST: Channel model and assumptons The baseband MIMO channel model employed n the thess s: r Hq ξ (3.) where q [ q, q,..., q ] T m and r [ r, r,..., r ] T n are the transmtted and receved sgnal vectors respectvely, H s the nxm ( n Rx and m Tx antennas) channel matrx n m wth ts, j th entry representng the complex channel gan from transmt antenna j to receve antenna ; as Raylegh fadng channel s assumed, H s modeled wth ndepent,

34 dentcally dstrbuted (..d.) crcularly symmetrc standard complex Gaussan entres, denoted as h CN(0,). ξ s the crcularly symmetrc addtve whte Gaussan nose vector wth..d. entres.e. where j ξ CN(0, I). 0 The column-wse representaton of the channel matrx H s gven by H h h,..., m h s a column vector contanng the channel gans from the -th Tx antenna to all Rx antennas. In fact h s a crcularly symmetrc standard complex Gaussan random vector.e. h CN (0, I). The system model n (3.) can also be represented as m r h q ξ (3.) Other assumptons are as follows. A flat fadng envronment s assumed, where the channel remans constant durng a frame of nformaton bts but t may vary from frame to frame.e. slow block fadng channel. The channel can be perfectly tracked by the recever (no channel estmaton error) and, n the cases where feedback s requred to the transmtter, the feedback sesson s executed wthout errors. ZF-SIC s assumed for sgnal detecton. Capacty-achevng temporal codes are used for each stream n the V-BLAST so that the perstream rates match the correspondng capactes and there are no errors when streams are not n outage and, therefore, no error propagaton n-between the streams. The transmtted sgnal, nose and channel gans are ndepent of each other and there s no performance degradaton due to synchronzaton and tmng errors The V-BLAST archtecture usng per-stream codng In the coded V-BLAST archtecture (see Fgure 3) the ncomng bt stream s demultplexed nto m data streams. These streams are then encoded usng capacty achevng temporal Gaussan codes and transmtted n parallel at the same tme and frequency usng a

35 set of m Tx antennas. At each receve antenna the sgnals nterfere wth each other due to the effect of the wreless propagaton channel. By usng ZF-SIC, an effcent sgnal processng procedure mplemented at the recever sde, the nterference (caused by the other streams) at each receve antenna s elmnated. Ths procedure transforms the wreless propagaton channel nto a set of vrtually ndepent sub-channels [6]. The ZF V-BLAST algorthm has three man steps: () nterference cancellaton, () nterference nullng and (3) optmal orderng. Fgure 3: Pctoral representaton of the V-BLAST archtecture when codng s used at each stream. For a better understandng of the algorthm, the nterference cancellaton and nullng steps are dscussed frst for a gven orderng, and then the system model s exted to the case where an optmal orderng s employed. Interference cancellaton: At the -th step (when the -th Tx symbol s detected), the nterference from the already receved symbols can be subtracted from the receved 3

36 vector based on ther estmatons qˆ, qˆ ˆ,..., q and on the knowledge of the channel matrx at the recever : r ' r h qˆ (3.3) j j j Interference nullng: After the nterference cancellaton step, the nterference from yet to be detected symbols can be nulled out projectng the receved vector at ths step orthogonal to the subspace spanned by the yet to be detected symbols [6]. Ths s accomplshed by multplyng the receved vector by a matrx P orthogonal to the spatal sgnatures of the m yet to be detected symbols 5.e. P H [ h... h m]. Fgure 4 llustrates the geometrcal representaton of the nterference nullng step. Fgure 4: Geometrc llustraton of the nterference nullng step. After the nterference nullng step, we are n presence of a vector channel under correlated nose: r h ξ (3.4) q where h Ph, ξ Pξ and ξ has the followng dstrbuton ξ CN(0, P ). The 0 correlaton matrx of the after-projecton nose ξ follows from the sotropc property of the 5 - P I H H H H where H h,..., h 3 m 4

37 unprojected nose ξ and the followng property of the projecton matrx: P P P P. It was demonstrated n [6] that when applyng MRC to the case of correlated nose, the nose correlaton does not affect the output SNR and t s the same as n the case of..d. nose after the projecton. Moreover, t was shown that the optmum MRC weghts already contan the projecton matrx, and they can be called optmum ZF weghts because they cancel nterference and maxmze the output SNR. The optmum ZF weght vector for detectng the -th stream s gven by w h h (3.5) and the output nstantaneous SNR (condtonal on no error at prevous steps) s h 0 (3.6) Optmal orderng: The order n whch transmtted symbols are detected s optmzed to mnmze the total error probablty of the system. To change the symbol detecton order s equvalent to re-orderng the columns of the channel matrx. The detecton orderng s defned by:,,..., m where s the number of the stream whch s detected frst. For example, for a channel matrx gven by H = h,h, orderng, H ' = h,h s equvalent to orderng the columns of the channel matrx as. Assumng that the columns of the channel matrx H are ordered followng the optmal orderng procedure.e. H = h,h,...,h ' ' ' ' m ', the system model can be expressed as r h q ξ (3.7) After applyng the optmum ZF weghts at the -th step the nstantaneous afterprocessng SNR (condtonal on no error at prevous steps) s gven by 0 ' h ' /, 5

38 where h ' s the projecton of h ' orthogonal to the subspace spanned by the m remanng columns of the channel H'. The after-processng nstantaneous SNR at the -th step s ch-squared dstrbuted wth n m degrees of freedom.e. ' ' / 0 x( nm) h, offerng a dversty order of n m at each step. The dversty order ncreases from step to step wth the frst step havng the lowest dversty order and the last step havng the hghest [9] Optmum orderng n the uncoded V-BLAST In the case of uncoded systems, the optmal order of stream processng s organzed accordng to ther after processng SNR n a decreasng order [3], that s, at each step the stream wth the hghest SNR at the output of the ZF detector wll be detected frst, and then ts contrbuton s subtracted from the receved vector. Ths process s repeated wth the next strongest stream, among the remanng undetected ones. It was shown n [3] that ths orderng strategy maxmzes the mnmum after-processng per-stream channel gan,.e. * arg max mn h ( ) (3.8) It can be shown that, when usng two Tx antennas, ths optmum orderng mples the detecton of the stream wth hghest before detecton channel gan frst;.e. the column wth hghest norm s placed frst n the optmum orderng of the channel matrx columns [4]. Note: The above detecton orderng was proposed by Foschn n [3]. Due to ths fact, t wll be referred to as Foschn orderng n ths thess. Uncoded communcaton systems are not wdely used; most modern communcaton systems use codng [33]. Therefore, t s mportant to dscuss the optmal orderng for the coded V-BLAST. Ths ssue s covered n the next chapters. 6

39 3.7. Summary The detecton algorthm used for the V-BLAST archtecture s a combnaton of MMSE or ZF recever wth SIC. The ZF-VBLAST s a very attractve detecton algorthm due to ts potental and smplcty. The algorthm has three man procedures: nterference cancellaton, nterference nullng and optmal orderng. The optmum ZF weghts null out nterference and maxmze the output SNR. 7

40 4. Instantaneous Rate Allocaton (IRA) The optmum detecton orderng for the coded V-BLAST under nstantaneous rate allocaton (IRA) s studed n ths chapter. Under the IRA, the per-stream rates can be adjusted to match the per-stream channel capacty due to the use of capacty achevng codes. Furthermore, the power allocaton s unform across the streams,.e. the normalzed power assgned to each stream s equal to one. After the nterference cancellaton and nullng steps, the equvalent scalar channel of the -th stream s: r h q, CN(0, ) (4.) ' ' out 0 Equaton (4.) follows from applyng the optmum ZF weght vector (3.5) to the vector channel n (3.4). The per-stream rate equals the per-stream capacty: 0 C ln h [nat/s/hz] (4.) where s the average SNR at each Rx antenna. The total capacty of the system s 0 0 equal to the sum of the capactes of all the streams, m C C (4.3) The system s n outage when the total capacty s less than the system target rate mr and the outage probablty s: Pout PC mr P C mr (4.4) From (4.4) t can be seen that an optmzaton strategy that mproves the performance of the coded V-BLAST must target the outage probablty or the total system capacty. 8

41 4.. Optmum orderng under the IRA 4... General case The next proposton states the optmum detecton orderng strategy for the coded V- BLAST under the IRA. Proposton : In the coded V-BLAST, under the IRA wth capacty achevng temporal codes at each stream, the optmum detecton orderng maxmzes the nstantaneous sum capacty of the system, where * m arg mn P arg max C arg max C (4.5) out IRA P, C and C ln out 0 h are the outage probablty, the total system capacty and the per-stream capacty expressed as a functon of the detecton orderng, and h s the projecton of h orthogonal to the subspace spanned by the m remanng columns of the channel matrx under orderng. Proof: The frst equalty n (4.5) follows because the optmal detecton must mnmze the outage probablty, the second equalty follows from the fact that the nstantaneous optmzatons of the capacty and outage probablty acheve the same lowest value of the outage probablty (the proof of ths statement can be found n [3]). The thrd equalty follows after notng that under the IRA the V-BLAST total capacty s equal to the sum capacty Two Tx antennas When usng two Tx antennas, an optmum detecton order can be establshed based on the per-stream before-processng channel gans as stated n the followng theorem. 9

42 Theorem : The optmum detecton orderng for the coded V-BLAST wth two Tx and n Rx antennas under the IRA s to detect the stream wth hghest before-detecton channel gan at the last step 6 : C, ff h h (4.6) arg max The only f part n (4.6) s true when, where s the angle between the two vector columns of the channel matrx. When and/or h h any orderng s optmum. Proof: For the case of two Tx antennas, the channel matrx s gven by there are two possble orderngs:, and, H h h,. The total capactes of these orderngs are: CIRA CIRA ln 0 ln 0 h h (4.7) ln 0 ln 0 h h (4.8) where h j refers to the projecton of h orthogonal to h j and 0 s the same as n (4.). Note that at the second step unprojected channel norms have been used snce at ths step the stream assocated wth the frst column of the channel has already been detected and ts nterference was subtracted at the nterference cancellaton step. Therefore, the second stream s receved wthout nterference. To prove the f part, assume that orderng s optmum, such that and the followng chan of nequaltes holds: C IRA C (4.9) IRA 6 Ths orderng s opposte of that for xn uncoded V-BLAST. 30

43 sn sn h h h h (4.0) sn sn h h h h (4.) sn sn h h (4.) h h (4.3) Inequalty (4.0) follows from substtutng (4.7) and (4.8) nto (4.9), the geometrc representaton of the nterference nullng step presented n [4],.e. h j h sn (4.4) and the use of the followng logarthmc dentty: m m ln x ln x (4.5) Inequaltes (4.), (4.) and (4.3) result from some straghtforward mathematcal manpulatons. The only f part s proved by notng that the same chan of nequaltes holds n the reverse drecton.e. startng n (4.3) and ng n (4.9). Equaton (4.6) follows. Note that f and/or h h, the nequalty (4.0) becomes an equalty showng that any orderng s optmum under these crcumstances General necessary optmalty condtons The aforementoned strategy s optmal for two Tx antennas. Also note that ths strategy s SNR ndepent,.e. as long as the stream wth maxmum before-detecton channel gan s detected at the last step, the capacty of the system s maxmzed, ndepently of the SNR value. However, Monte-Carlo smulatons show that for the case of m the optmal detecton orderng becomes SNR depent. 3

44 Observaton : The optmum detecton orderng n the mxn coded V-BLAST m under the IRA s SNR depent. In general, there s not a sngle detecton orderng that s optmum at all SNR values. A numercal example supportng Observaton The optmum detecton orderngs for the 3x3 system under the IRA were obtaned for two dfferent SNR values (-0dB and 0dB) gven a Raylegh fadng channel realzaton. The capactes of these orderngs at each SNR value are shown below. The channel realzaton analysed s H Table : Performance of the optmal detecton orderngs under the IRA at SNR = -0dB and SNR = 0dB for the gven channel realzaton. IRA at SNR0dB C nats/sec/hz IRA at SNR 0dB nats/sec/hz CIRA at SNR 0dB *,3, IRA at SNR0dB *,, The example clearly shows the SNR depency of the optmum orderng under the IRA when m. From Observaton t can be argued that, n general, a SNR ndepent detecton orderng strategy (based only on H) cannot be optmum for all SNR values. The strategy for m s to calculate the capacty of all possble orderngs and select the one that wth the maxmum capacty. Ths ncreases the complexty and tme 3

45 consumpton of the system due to many projectons. Note that for m streams there are m! possble detecton orders a quantty that ncreases very fast wth m. Based on the xn result, there are some necessary condtons (SNR ndepent) that must be satsfed by an optmum orderng. By explotng these optmalty condtons the number of possble orderngs to evaluate capacty, and hence the complexty of the system, can be smplfed. condtons 7 : Proposton : An optmum channel orderng must satsfy the followng necessary h h (4.6) m where h and h are the projectons of vectors h and h orthogonal to the sub-space spanned by h,... h. m Proof: Consder the followng two orderngs:,...,,,..., m (4.7),...,,,..., m (4.8) Note that the dfference between these orderngs s that the order at whch two adjacent streams are detected has been swapped. Orderng s optmum provded that If C C C IRA h h IRA IRA, and C (4.9) IRA cannot be the optmum orderng because there s another orderng that offers a hgher capacty. Ths can be shown based on the followng reasonng: the frst and last m streams of each orderng are the same, hence the streams assocated wth those columns have the same contrbuton n the total capacty of 7 Proposton holds gven that all h are of dfferent length and non-orthogonal to each other. In the case where some h are of equal length and/or orthogonal to each other, any orderng among them s optmum. 33

46 both orderngs. The dfference n capacty s determned by the detecton orderng of the two swapped streams. The problem s reduced to the xn case where f h h, from Theorem the total capactes are compared as: C C Example for m=3 IRA. IRA In order to gan a better understandng of the necessary optmalty condtons the case of three Tx antennas s dscussed below. Assume that the detecton orderng *,,3 s. optmum,.e. arg max m C,,3 Let us consder another orderng:,3,.the dfference n capacty between these orderngs s gven by the two last steps because the frst stream s the same for both orderngs and,3 3, h h. Ths reduces the problem to the orderngs are *,3 and 3, xn case where the. From Theorem t s known that the optmum strategy s to detect the stream assocated to the hghest unprojected channel gan at the last step, hence f *,,3 provdes the optmum detecton orderng then h h 3. Now let us consder the detecton orderng offered by,,3 and compare ts performance wth that of the detecton orderng gven by *,,3. It can be seen that the last stream s the same for both orderngs. Gven that at frst and second steps the nterference produced by the thrd stream must be projected out, the problem s reduced to the xn where the channel matrces under each orderng are gven by H h h and 3 3 * 3 3 H h h. Agan Theorem states that the channel contanng the column wth hghest unprojected norm at last poston offers the hghest capacty, hence f *,,3 offers the optmum detecton orderng then h 3 h 3. It can be concluded that:,,3 f h h3 and h 3 h 3. The same analyss can be exted to the case of arbtrary m. 34

47 Proposton 3: Three mportant conclusons follow from the necessary optmalty condtons. Gven that all followng holds. h are of dfferent length and non-orthogonal to each other, the For a gven order that meets the necessary optmalty condtons, swappng two consecutve columns results n a new order that offers a lower capacty. A channel matrx under the optmum detecton orderng wll never contan the column wth mnmum norm at the last poston. A channel matrx under the optmum detecton orderng wll never contan the column wth maxmum norm at the second last poston. There s, n general, more than one orderng that meet the necessary condtons for optmalty. The best and worst case scenaros are consdered below.e. the mnmum and maxmum number of orderngs that can meet the condtons. Best case scenaro There are cases when only one orderng meets the necessary optmalty condtons, the followng proposton dscusses such scenaro. Proposton 4: If the followng propertes of the channel matrx apply, h h... hm, h h j j (4.0) where h and h j are the projectons of vectors h and h j orthogonal to the sub-space spanned by the set of all remanng columns (not contanng h and h j ), then only one order can be optmum and hence t s SNR ndepent.e. t s optmum for all SNR values. Second part of (4.0) means that for any two columns, the column wth hgher ndex has a hgher projected norm (projecton orthogonal to the sub-space spanned by the set of columns not contanng the selected ones). 35

48 As an example let us consder agan the case of three Tx antennas. The number of possble detecton orderngs s 3! 6. Each possble detecton orderng s represented by a dfferent orderng of the columns of the channel matrx. The channel matrx under the orderng,,3 s gven by 3 propertes hold: H h h h. Let us assume that the followng h h h (4.) 3 h h (4.) 3 3 h h (4.3) 3 h h (4.4) 3 where h j s the projecton of the -th column orthogonal to the j-th column of the channel matrx. Note that (4.) - (4.4) are explctly the propertes establshed n (4.0) for the case of m 3,.e. three Tx antennas. Based on the necessary optmalty condtons, t can be seen that from property (4.) the orderngs,3,, 3,3, and 4 3,, can be elmnated; and from propertes (4.) and (4.3) the orderngs 5,,3 and 6 3,, can be elmnated respectvely. Therefore, only the orderng can be optmum. Worst case scenaro It was already shown that n the best case scenaro only one orderng meets the necessary optmalty condtons. However, to gan more nsghts nto these necessary optmalty condtons t s mportant to consder also the worst case scenaro. The worst case scenaro can be defned as the maxmum number of orderngs that can meet the necessary optmalty condtons for a gven number of Tx antennas. For a gven number of Tx antennas m, the number of necessary optmalty condtons s equal to m -. Each of the necessary optmalty condtons can elmnate (on 36

49 ts own) half of the total number of combnatons, but as they are not ndepent of each other, when appled all at the same tme, dfferent condtons may elmnate the same combnatons and hence t s dffcult to obtan an analytcal result for the maxmum number of combnatons that are not elmnated after applyng all the condtons. In lght of ths, the approach used here conssted of nvestgatng the upper and lower bounds of ths value. In ths eavour we rely on the followng defnton: Defnton : Indepent necessary optmalty condtons are defned as to those that are able to elmnate half of the combnatons that were not elmnated after applyng the prevous condtons. Now, based on the number of ndepent necessary optmalty condtons the maxmum number of orderngs that can satsfy the necessary condtons can be bounded as stated n the followng proposton: Proposton 5: The maxmum number of possble combnatons N that can meet the necessary optmalty condtons s bounded as follows m!! m m m N (4.5) where m s the number of Tx antennas, m s the number of necessary optmalty condtons and m s the number of ndepent necessary optmalty condtons. Proof: Each of the necessary optmalty condtons can elmnate (on ts own) half of the total m! number of combnatons. By applyng only one condton, the number of orderngs that are not elmnated s m!. If now a condton whch s ndepent of the prevous one s appled, the number of remanng orderngs s m!. The frst nequalty n (4.5) follows from assumng that all the m necessary optmalty condtons are 37

50 ndepent of each other. Ths a lower bound to N because all the necessary condtons are not ndepent. Based on many specal cases, the number of ndepent necessary condtons s m. The second nequalty n (4.5) s the total number of orderngs that are not elmnated takng nto account only the ndepent condtons. Ths s clearly an upper bound because the number of orderngs that are not elmnated by consderng only the ndepent necessary condtons cannot be greater than the number of orderngs that are not removed f all condtons are consdered. Fgure 5: Upper bound gven by (4.7) (dark grey) and numercal lower bound (lght gray) of N (n percent) VS. # of Tx antennas. Unfortunately, numercal smulatons show that these analytcal bounds for N are not very tght for large values of m. Based on a combnatoral Matlab code, t was found what we beleve s the exact value of N for the m 3, 4,...,8 cases. However, as the mathematcal proof does not exst we can thnk of t as a numercal lower bound (tghter than the analytcal lower bound n (4.5)). The graph n Fgure 5 shows both the numercal lower bound and the analytcal upper bound for N (expressed n percent) as a functon of the 38

51 number of Tx antennas. From both the graph presented n Fgure 5 and Table the followng observaton can be made: Observaton : The maxmum number of detecton orderngs that satsfy the necessary optmalty condtons represents only a small percentage of all the possble detecton orderngs, and ths percentage decreases very fast as the number of Tx antennas s ncreased. Table llustrates the total number of possble detecton orderngs for a gven number of Tx antennas and the number of detecton orderngs that satsfy the optmalty condtons based on the proposed bounds. It can be seen how the detecton orderngs satsfyng the condtons drop from a 50% for m 3 to just 6.5% for m 8 (based on the upper bound). Note that based on the numercal lower bound, the detecton orderngs that satsfy the necessary optmalty condtons represent just the 3.4% of the total for the m 8 case. Table : Number of orderngs remanng. m Total comb. Numercal lower bound Analytcal upper bound Orders remanng % Orders remanng % Fgure 6 llustrates the reducton n complexty 8 offered by the analyss va the necessary optmalty condtons (n the worst case scenaro) as compared to the exhaustve 8 n terms of the number of orderngs whch capactes need to be evaluated 39

52 search of the optmum orderng. It can be seen that although the complexty ncreases wth the number of Tx antennas n both cases, the reducton n complexty by usng the necessary optmalty condtons also ncreases wth m. Fgure 6: Complexty comparson between the exhaustve search of the optmum orderng and the analyss va the necessary optmalty condtons. Best and worst case scenaros (Numercal valdatons) Monte-Carlo smulatons were run n order to valdate the prevous results. Fgure 7 shows the results of the numercal smulaton for the 4x4 coded V-BLAST under the IRA based on 0 6..d Raylegh fadng channel realzatons. It can be seen from the graph that n a 4x4 system, there s a relatvely hgh probablty 0.7 of havng one detecton orderng satsfyng the necessary optmalty condtons. The graph also shows that n the worst case only fve detecton orderngs meet these condtons and ths event takes place wth very low 40

53 5 probablty 3.x0. Note that the worst case shown n the smulaton agrees exactly wth the numercal lower bound of N for the case of four Tx antennas. System: 4x4 Channel Realzatons: 0 6 Best case: Prob: 0.7 Worst case: 5 Prob: 3.x0-5 Fgure 7: Normalzed hstogram of orderngs that satsfy the necessary optmalty condtons for the (4x4) system. It s mportant to menton that for the case of m 5 the numercal lower bound could not be acheved va Monte-Carlo smulatons (see Fgure 8), suggestng that as the number of Tx antennas s ncreased the worst case scenaro s less probable. Therefore, the followng observaton can be made: Observaton 3: The worst case scenaro takes place wth very low probablty and the probablty of occurrence decreases as the number of Tx antennas s ncreased. 4

54 System: 5x5 Channel Realzatons: 0 6 Best case: Prob: 0.5 Worst case: Prob: 0-6 Fgure 8: Normalzed hstogram of orderngs that satsfy the necessary optmalty condtons for the (5x5) system Optmal orderng strateges at low and hgh SNR Approxmated closed-form expressons for the optmal detecton orderng strateges at low and hgh SNR are provded n ths secton. The expressons are obtaned based on approxmatons of the per-stream capacty equaton. Low SNR approxmaton The operatng SNR of cellular systems wth unversal frequency reuse s typcally very low, e.g. CDMA systems where the nterference s consdered to be part of nose. As V- BLAST can be also used n multple access channels t s mportant to consder the optmum orderng strategy at low SNR. Proposton 6: At low SNR the optmum detecton orderng maxmzes the sum of the after-processng channel gans Proof: At low average SNR.e. 0, for a gven channel realzaton, the perstream rate and the total capacty can be approxmated respectvely as: 4

55 C 0 0 ln h h [nat/s/hz] (4.6) m C C IRA 0 and the optmum detecton orderng can be expressed as: m h (4.7) m m max = max C 0 h (4.8) Equaton (4.6) and (4.7) result from the approxmaton ln x x for small x. Equaton (4.8) follows from the fact that the optmum detecton orderng maxmzes the system capacty. Fgure 9: Emprcal CDFs of the Max. Sum Ch. Gans detecton orderng, the optmum detecton orderng and the unordered detecton for the (5x5) system; SNR=-0dB; 0 4 channel realzatons. In order to valdate proposton 6, extensve Monte-Carlo smulatons were undertaken. Fgures 9-0 show the emprcal cumulatve dstrbuton functons (CDFs) of the 43

56 optmum detecton orderng, the detecton orderng that maxmzes the after-processng channel gans and the unordered detecton for 5x5 and 4x4 V-BLAST systems respectvely. These CDFs are based on based on 0 4..d. Raylegh fadng channel realzatons. It can be seen that the detecton orderng that maxmzes the after-processng channel gans s n fact optmum at low SNR. Fgure 0: Emprcal CDFs of the Max. Sum Ch. Gans detecton orderng, the optmum detecton orderng and the unordered detecton for the (4x4) system; SNR=-0dB; 0 4 channel realzatons. Hgh SNR approxmaton The hgh SNR optmum detecton orderng can be found n a smlar way as t was found at low SNR. A key queston n ths context s whether the optmum detecton orderng offers a sgnfcant advantage at hgh SNR? Ths ssue s nvestgated below. Proposton 7: At hgh SNR all channel orders provde approxmately the same sum capacty, hence the optmal orderng does not offer a sgnfcant mprovement to the system. 44

57 Proof: At hgh SNR.e. 0, for a gven channel realzaton, the per-stream rate and the total capacty can be approxmated respectvely as: ln C ln 0 0 h h (4.9) m m CIRA C 0 h (4.30) mln ln and the optmum detecton orderng can be expressed as: m m max C = mln 0 max ln h + 0 ln HH mln (4.3) Equaton (4.9) follows from the approxmaton ln x ln x (4.30) s obtaned from the logarthmc denttes (4.5) and ln m x mln x for large x. Equaton (4.3) The frst equalty n (4.3) follows from the facts that the optmum detecton orderng maxmzes the system capacty whle the second equalty follows from m + h HH (4.33) Proof of equaton (4.33) can be found n [34]. Monte-Carlo smulatons were carred out n order to valdate the analytcal results supportng Proposton 7. Fgure shows the emprcal curve of outage probablty as a functon of SNR of the optmum detecton orderng and the unordered detecton for the 5x5 V-BLAST system based on 0 3..d. Raylegh fadng channel realzatons for a target rate of 40 nats/sec/hz. It can be seen that both detecton orderng strateges offer the same performance. 45

58 R = 40 nats/sec/hz Fgure : Emprcal Pout vs. SNR of the optmum detecton orderng and the unordered detecton for the (5x5) system at a hgh target rate based on 0 4 channel realzatons. 4.. Suboptmal orderngs In ths secton suboptmal orderngs that perform very close to the optmum one are proposed. Inverse orderng Ths orderng strategy follows the same prncple as the optmum orderng strategy for uncoded systems proposed by Foschn n [3], but the stream wth hghest afterprocessng SNR s detected last (as a gudng prncple). The algorthm s as follows, - Select the stream wth largest h and detect. - Remove the detected stream, select the one wth the largest h and detect. 3- Repeat step untl fnsh. In ths way, the last column of the channel matrx s the one wth maxmum norm. Note that at the last step of the V-BLAST algorthm t s not necessary to project out any nterference. 46

59 The second last column wll be the one wth hghest projected norm (projecton orthogonal to the column wth hghest norm), and so on up to the frst column. It s evdent that, under ths orderng, the necessary optmalty condtons are always satsfed. Ths s a SNR ndepent orderng strategy (just based on H ). It s known from Observaton that for m 3 the optmum detecton orderng s SNR depent. Hence, n general, the nverse orderng cannot be optmum. However, as llustrated n Fgures -3, ths orderng performs very close to the optmum one. Max Sum Capacty of the Last Two Steps (MSCLS) orderng Based on the fact that the total capacty of the system s domnated by the last two steps of the V-BLAST algorthm, the MSCLS orderng s also proposed. Under ths orderng strategy, the last column of the channel matrx s selected such that the capacty of the last two steps of the V-BLAST s maxmzed. The remanng columns are ordered followng the nverse orderng crtera. In ths case the necessary optmalty condtons are also satsfed. Frst notce that the last column s selected based on the capacty offered by the last two steps, and from Theorem the last two columns wll satsfy hm h m. Second, as the rest of the channel matrx are ordered followng the nverse orderng crtera, the remanng m condtons are also satsfed. Note that ths orderng strategy s SNR depent. The last column of the channel matrx s chosen based on the capacty equaton of the two last steps (the capacty equaton s SNR depent) and the remanng columns are ordered depng on the last one. Despte ths, n general, t cannot be the optmum. Ths s because the fact that selectng the last column such that the capacty of the last two steps s maxmzed does not guarantee that the total system capacty s maxmzed as well. However, the capacty offered by ths orderng s 47

60 also very close to the optmum one at all SNR values. The performance of ths detecton orderng strategy s llustrated n Fgures -3. Fgure : Emprcal CDFs of the nverse orderng, the MSCLS orderng, the optmum orderng and the unordered detecton for the (5x5) system; SNR=-0dB; 0 4 channel realzatons. Fgures -3 show the emprcal CDF curves, based on 0 4..d. Raylegh fadng channel realzatons, of the proposed suboptmal detecton orderngs, the optmum detecton orderng, and the unordered detecton for the 5x5 and the 4x4 coded V-BLAST systems respectvely at dfferent SNR values. It s seen that both suboptmal detecton orderngs offer an almost optmum performance. 48

61 Fgure 3: Emprcal CDFs of the nverse orderng, the MSCLS orderng, the optmum orderng and the unordered detecton for the (4x4) system; SNR=0dB; 0 4 channel realzatons. Unprojected orderng In ths detecton orderng strategy the streams are detected based on ther respectve unprojected channel gans n an ncreasng order. The stream assocated to the channel column wth mnmum norm s detected frst and the one assocated to the channel column wth maxmum norm s detected last. Ths s the smplest strategy snce t s not necessary to make any projecton n order to set up the stream detecton order. From Theorem the unprojected detecton orderng strategy s optmal for the x n coded V-BLAST under the IRA and Monte-Carlo smulatons show that t offers a good performance at all SNR values for small number of Tx antennas.e. m 3, 4,5 (see Fgures 4-5). However, ts performance degrades when the number of Tx antennas s ncreased. 49

62 Fgure 4: Emprcal CDFs of the unprojected detecton orderng, the optmum detecton orderng and the unordered detecton for the (3x3) system; SNR=-0dB; 0 4 channel realzatons. Fgure 5: Emprcal CDFs of the unprojected detecton orderng, the optmum detecton orderng and the unordered detecton for the (5x5) system; SNR=-0dB; 0 4 channel realzatons. 50

63 4.3. Summary The optmum detecton orderng for the coded V-BLAST under the IRA was studed. Closed-form approxmatons were derved for the optmum orderng at low and hgh SNR. The optmum orderng maxmzes the sum of the after-processng channel gans at low SNR whle all orderngs offer approxmately the same performance at hgh SNR. For the case of two Tx antennas, the optmum orderng can be establshed based on the before-processng channel gans: the stream wth hghest before-processng channel gan must be detected last. For the case of m, the optmum orderng becomes SNR depent. Based on the xn, SNR ndepent necessary optmalty condtons were derved for arbtrary number of Tx antennas. A small percentage of orderngs satsfy the necessary condtons for optmalty and ths percentage decreases wth the number of Tx antennas. The mnmum number of orderngs that can satsfy the necessary optmalty condtons s one and ths scenaro takes place wth hgh probablty. The maxmum number of orderngs that can satsfy these condtons s stll an open problem. In ths chapter, lower and upper bounds of ths maxmum were determned. Smulatons demonstrate that the numercal lower bound s reached wth very low probablty. Suboptmum detecton orderngs were proposed. Both the nverse and the MSCLS orderngs offer an almost optmum performance for arbtrary number of Tx antennas. The unprojected orderng s the smplest strategy and ts performance s very good for a small number of Tx antennas. However, ts performance degrades wth the number of Tx antennas. 5

64 5. Unform Power and Rate Allocaton (URA) In the prevous chapter, an nstantaneous rate allocaton at each stream was assumed.e. the use of a non-unform transmsson rate among the streams where the per-stream rate s adjusted to match the per-stream capacty for each channel realzaton. The system wth the unform power and rate allocaton (URA) among the streams s consdered n ths chapter. Ths scenaro takes place, for example, when the same modulaton/codng s used at each stream to smplfy the system desgn. In the presence of the URA, the system s n outage when at least one stream cannot support the target rate R. Hence, the Tx rate at each stream s equal to the capacty of the weakest stream n order to attan relable communcaton. The total capacty of the system s: where where CURA C mn s the capacty of the weakest stream and s expressed as mc (5.) mn mn 0 C mn C ln mn h (5.) C s as n (4.). The second equalty n (5.) follows from the fact that ncreasng functon of x. ln x s an 5.. Optmum orderng under the URA 5... General case We are now n a poston to establsh the optmum orderng strategy for the coded V- BLAST under the URA. Proposton 8: The optmum orderng n the coded V-BLAST under the URA wth capacty achevng temporal codes at each stream s gven by: 5

65 * arg max mn C arg max mn h (5.3) where C and h are as n (4.5). Proof: The system capacty under the URA can be expressed as a functon of the detecton orderng, mn ln mn URA C m C m h 0 (5.4) and the followng set of equatons for the optmum detecton orderng * holds * arg max mn C ( ) (5.5) arg max m log mn h 0 (5.6) arg max mn h (5.7) Equaton (5.5) follows from the fact that the optmum detecton orderng must maxmze the nstantaneous system capacty. Equaton (5.6) results from substtutng (5.4) nto (5.5) and equaton (5.7) follows from the fact that ln x s an ncreasng functon of x. The optmum orderng under the URA can be nterpreted as the one that maxmzes the after-processng channel gan (SNR) of the weakest stream. From equaton (5.3) three more propostons follow. Proposton 9: In the coded V-BLAST under the URA and usng capacty achevng temporal codes at each stream, the optmum detecton orderng s SNR ndepent,.e. t deps on H but not on 0. Hence, for a gven channel realzaton, an optmum detecton orderng remans optmum for all SNR values. Proposton 0: In the coded V-BLAST under the URA and usng capacty achevng temporal codes at each stream, there s, n general, more than one optmum order.e. dfferent orders may offer the same maxmum capacty. 53

66 The man dea under Proposton 0 s that the optmum orderng strategy deps only on the after-processng channel gan of the weakest stream. The columns to the left or rght of the column offerng the mnmum stream capacty (n the optmum orderng) can be reordered and, as long as the mnmum stream remans the same, the orderngs generated as a result of the columns permutaton process acheve the same maxmum capacty and hence they are optmal. One smple example s when H s dagonal, n ths case any orderng s optmum. Proposton 0 s also true for non-dagonal H as proved below. arbtrary m. Proof: Proposton 0 s proved for the case of m 3 and t can be easly exted to Let us consder the case of three Tx antennas m 3 can be assumed that the orderng,,3 s optmum,.e.. Wthout loss of generalty t arg max mn C ( ),,3 (5.8) where * means that the frst step of ths orderng has the mnmum capacty.e. where mn C, C, C C h h, h (5.9) 3,3 3 3 C s referrng to the capacty of the -th stream of the V-BLAST algorthm. Now where assume orderng,3, The capactes of both orderngs are equal,.e. mn C, C, C C h h, h (5.0) URA and hence both orderngs are optmal. C 3 3, 3 ( ) C ( ) 3log h (5.) URA,3 0 54

67 Proposton : Snce Foschn orderng maxmzes the mnmum stream channel gan, t s optmum for the coded V-BLAST under the URA. Two llustratve examples of ths statement can be seen n Fgures 6-7. Fgure 6: Emprcal CDFs of the Foschn detecton orderng, the optmum detecton orderng and the unordered detecton under the URA for the (3x3) system; SNR=-0dB; 0 5 channel realzatons. Numercal smulatons were undertaken to valdate proposton. Fgures 6-7 show the emprcal CDFs, based on 0 5..d. Raylegh fadng channel realzatons, of the Foschn detecton orderng, the optmum detecton orderng, and the unordered detecton for the 3x3 V-BLAST system under the URA at the SNR values of 0dB and 0dB respectvely. It s shown both the optmalty of the Foschn detecton orderng and the mprovement offered by the optmum orderng process n the coded V-BLAST under the URA. 55

68 Fgure 7: Emprcal CDFs of the Foschn detecton orderng, the optmum detecton orderng and the unordered detecton under the URA for the (3x3) system; SNR=0dB; 0 5 channel realzatons Two Tx antennas Smlarly to the case of the IRA, the optmum detecton orderng strategy under the URA for the case of two Tx antennas can be establshed based on the per-stream beforeprocessng channel gans as stated n the followng theorem. Theorem : The optmum detecton orderng for the coded V-BLAST wth two Tx and n Rx antennas under the URA s to detect the stream wth hghest before-detecton channel gan frst: arg max mn C ( ), ff h h for 0 (5.) The only f part n (5.) s true when. When and/or h h any orderng s optmum. 56

69 Proof: For the case of two Tx antennas, the channel matrx s gven by H h h. To prove the f part, wthout loss of generalty, t can be further assumed that h h. There are two possble orderngs:, and, CURA CURA ln mn, 0 ln mn, 0. The total capactes are: h h (5.3) h h (5.4) Equatons (5.3) and (5.4) follow from (5.4) and the approprate substtuton of the channel gans depng on the specfc orderng. The followng chan of equaltes and nequaltes holds: sn sn h h h h h (5.5) h h mn h, h h mn h, h (5.6) URA C C (5.7) URA Both equaltes n (5.5) follow from the use of the geometrcal representaton (4.4). The frst and thrd nequaltes n (5.5) hold because sn 9 and the second nequalty result from h h. The equalty and the nequalty n (5.6) follow mmedately from (5.5) and fnally (5.7) follows from (5.3), (5.4) and (5.6). To prove the only f part, assume that C C URA Now there are two optons. The frst opton s: so that URA mn h, h mn h, h (5.8) 9 Assumng that. 57

70 mn h, h h (5.9) so that h h h (5.0) where the frst nequalty n (5.0) results from (5.9) and the second nequalty follows because the projecton of a vector cannot ncrease ts norm. The second opton s: mn h, h h (5.) and thus the followng nequaltes hold: h mn h, h mn h, h (5.) h h (5.3) sn sn h h (5.4) h h (5.5) The nequalty n (5.) results from (5.) and (5.8). Inequalty (5.3) follows from (5.) and h h. Inequalty (5.4) follows from the geometrcal representaton (4.4). Equaton (5.) follows. Note that f and/or h h the nequalty n (5.6) becomes an equalty and thus any orderng s optmum. If 0, the column vectors of the channel matrx are parallel. Therefore, the length of the projected vector s zero and thus the transmsson strategy s to keep only one actve stream (the stream wth hghest beforedetecton channel gan). 5.. Summary The optmum detecton orderng for the coded V-BLAST under the URA was studed. The optmum detecton orderng s SNR ndepent and t maxmzes the mnmum 58

71 after-processng stream gan. It was shown that the Foschn orderng s optmum for arbtrary number of Tx antennas; however, several orderngs may offer the same optmum performance when m 3,.e. the optmum orderng s not unque. For the case of two Tx antennas, the optmum orderng can be establshed based on the before-processng channel gans: the stream wth hghest before-processng channel gan must be detected frst. 59

72 6. Non-unform power allocaton Unform power allocaton at each stream was assumed n the precedng analyss,.e. the normalzed power assgned to each stream s equal to one. It was demonstrated n [3] that by allocatng dfferent powers to each Tx antenna the coded V-BLAST performance s mproved. Ths chapter studes the optmal orderng strateges for the coded V-BLAST under the nstantaneous power allocaton (unform rate) and under the nstantaneous power/rate allocaton. The followng fgure provdes the block dagram of the system. Fgure 8: Tx sde archtecture of the coded V-BLAST wth non-unform power allocaton. m The total normalzed power s m. The system model s gven by: r HΛq ξ h q ξ (6.) where q, r, ξ, and H are the same as n (3.). Λ s a dagonal matrx whose entres represent the squared root of the power assgned to each stream. After the nterference cancellaton and nullng steps, the equvalent scalar channel of the -th stream s: 60

73 r h q, CN(0, ) (6.) ' ' out 0 Gven that capacty achevng temporal codes are used at each stream, each stream can support a rate equal to ts nstantaneous capacty gven by where h and 0 are the same as n (4.) and 0 C ln h [nat/s/hz] (6.3) s the power allocated to stream. Furthermore, the power allocaton can be chosen n an optmum way. Any optmum nstantaneous power allocaton must target the outage probablty or the total capacty of the system. 6.. Instantaneous Power Allocaton (IPA) The optmum nstantaneous power allocaton strategy, subject to the total power constrant, when per-stream rates are equal to the mnmum per-stream capacty s dscussed n ths secton. Based on the prevous specfcaton, the per-stream rate under the IPA s gven by: mn mn ln 0 R C h (6.4) Then for a gven power allocaton vector α the system capacty under the IPA s: mn ln 0 C α m h (6.5) and the optmzaton problem for a gven order can be formulated as: C m max mn ln h IPA α s.t. m, 0 0 (6.6) The soluton to ths optmzaton problem s the channel nverson [3]. Under the channel nverson strategy, more power s allocated to the stream wth smaller nstantaneous 6

74 channel gan. Ths power allocaton s gven by where h and g orderng, g (6.7) g g s the harmonc mean per-stream power gan for a gven g m and the maxmum capacty for a gven orderng s: g (6.8) C IPA 0 mln g, g 0 0 otherwse (6.9) The explct dervatons to obtan (6.7) and (6.9) can be found n [3] Optmum orderng under the IPA General case We are now n a poston to establsh the optmum orderng strategy for the coded V- BLAST under the IPA. Proposton : The optmum orderng n the coded V-BLAST under the IPA wth capacty achevng temporal codes at each stream s gven by: Proof: The followng set of equaltes proves (6.0): * arg max g (6.0) * arg max C IPA (6.) arg max mln g 0 (6.) arg max g (6.3) 6

75 The equalty n (6.) reflects the fact that the optmum detecton orderng must maxmze the total system capacty. Equalty (6.) follows from substtutng (6.9) nto (6.) and equalty (6.3) s because ln x s an ncreasng functon of x. Note that the optmum harmonc mean per-stream power gan g * deps only on H, not on SNR. Proposton 3 follows. max g (6.4) Proposton 3: In the coded V-BLAST under the IPA and capacty-achevng temporal codes at each stream, the optmum orderng deps only on H and t s SNR ndepent. Two Tx antennas When usng two Tx antennas, the detecton orderng under the IPA can be establshed based on the per-stream before-processng channel gans as stated n the followng theorem. Theorem 3: The optmum detecton orderng for the coded V-BLAST wth two Tx and n Rx antennas under the IPA s to detect the stream wth hgher before-detecton channel gan frst 0 : arg max g, ff h h for 0 (6.5) The only f part n (6.5) s true when. When and/or h h any orderng s optmum. Proof: For the case of two Tx antennas, the channel matrx s gven by H h h. To prove the f part, wthout loss of generalty, t can be further assumed that h h. 0 ths s the Foschn orderng. 63

76 There are two possble orderngs:, and, power gans for these orderngs are. The harmonc mean per-stream g g h h = h h h h h h h h h h (6.6) (6.7) where h j refers to the projecton of h orthogonal to follow from evaluatng (6.8) for the case of h j. Equatons (6.6) and (6.7) m under the orderngs gven by and, respectvely. From Proposton, s optmum provded that and the set of nequaltes follows: g g, (6.8) h h h h sn sn sn sn h h h h (6.9) sn sn h h h h (6.0) sn sn h h (6.) h h (6.) Inequalty (6.9) follows from substtutng (6.6) and (6.7) nto (6.8) and the use of the geometrcal representaton (4.4).e. h j h sn where s the angle between h and h j. Inequaltes (6.0), (6.) and (6.) are the result of some straghtforward mathematcal manpulatons. It s evdent that the same chan of nequaltes holds n the reverse drecton.e. startng n (6.) and ng n (6.8), so the only f part s also proved and (6.5) follows. Note that f and/or h h the nequalty (6.9) becomes an equalty 64

77 showng that any orderng s optmum. If 0, the column vectors of the channel matrx are parallel. Therefore, the length of the projected vector s zero and thus the transmsson strategy s to keep only one actve stream (the stream wth hghest before-detecton channel gan) Suboptmum orderng Fgure 9: Emprcal CDFs of the Foschn detecton orderng, the optmum detecton orderng and the unordered detecton under the IPA for the (3x3) system; SNR=0dB; 0 5 channel realzatons. The optmalty of the Foschn orderng for the coded V-BLAST under the URA was demonstrated n the prevous chapter. Numercal smulatons show that, although ths orderng strategy s not optmum, n general, for the coded V-BLAST under the IPA when m 3, t performs very close to the optmum one. Its performance s shown n Fgures 9-0 where the emprcal CDF graphs of the optmum detecton orderng, the Foschn detecton 65

78 orderng and the unordered detecton are llustrated for dfferent values of m and SNR. In both cases, the smulatons are based on 0 5..d Raylegh fadng channel realzatons. Fgure 0: Emprcal CDFs of the Foschn detecton orderng, the optmum detecton orderng and the unordered detecton under IPA for the (4x4) system; SNR=0dB; 0 5 channel realzatons. 6. Instantaneous Power and Rate Allocaton (IPRA) The optmum nstantaneous power allocaton strategy, subject to the total power constrant, when the per-stream rates match the per-stream capactes s dscussed n ths secton. Based on the prevous dscusson, the per-streams rate under the IPRA s gven by: R C ln h 0 [nat/s/hz] (6.3) The total system capacty s m, ln h 0 (6.4) C C C and the optmzaton problem for a gven order can be formulated as follows: 66

79 C max ln h IPRA α s.t. m, 0 0 (6.5) It was demonstrated n [3] that the well-known waterfllng (WF) algorthm does not offer the optmal soluton to ths problem n general. The WF algorthm s optmum when the channel gans are ndepent of the allocated power. But the nterference nullng step of the V-BLAST algorthm forces the channel gans to dep on the allocated power. If the power assgned to stream s equal to zero, ths stream s not actve and t s not necessary to project out ts nterference from lower level streams. Therefore, turnng off stream affects the gan of lower level streams.e. h... h. Ths results n a non-convex problem. The soluton to ths problem s the fractonal waterfllng algorthm (FWF) [3]. The orgnal non-convex problem s splt nto m convex sub-problems, one for each set of nactve transmtters, for whch the soluton s the WF algorthm. The optmum power allocaton s gven by the power allocaton vector correspondng to the set wth the hghest capacty. The FWF algorthm can be summarzed n three man steps: - Take m sets of nactve streams (the frst stream s always actve because turnng t off does not affect the rest of streams gans). - Apply WF to each set. 3- Take the power allocaton vector correspondng to the set wth the hghest capacty. The complete proof of the optmalty of the FWF algorthm for the coded V-BLAST under the IPRA can be found n [3]. 67

80 6... Optmum orderng under the IPRA General case In order to obtan the optmum orderng strategy based on the FWF approach t s necessary to repeat the three steps mentoned above m! tmes ( m s the number of Tx antennas).e. apply FWF to each possble orderng and select the orderng wth the maxmum capacty. Ths results n a very complex and lengthy process. However, extensve numercal smulatons show the followng: Observaton 4: The optmum orderng under the FWF s also optmum under the WF, and t offers the same value of the capacty : C (6.6) WF FWF C (6.7) WF WF FWF FWF From Observaton 4, t can be argued that there s no loss of optmalty when consderng conventonal WF to fnd the optmum orderng n the coded V-BLAST under the IPRA. Ths s formally stated n the followng proposton: Proposton 4: The optmum orderng n the coded V-BLAST under the IPRA wth capacty achevng temporal codes at each stream s gven by: * h 0 (6.8) * arg max ln where the optmum power allocaton s gven by the conventonal WF algorthm.e. * 0 h (6.9) the analytcal proof s not avalable. 68

81 where s the water level for a gven order. s calculated from the total power constrant. More nformaton about the WF algorthm can be found n Appx A. Two Tx antennas Note: The dervatons n the followng secton make use of the results for the WF algorthm derved n Appx A. Under the IPRA when the channel gan of a gven stream s too weak to be transmtted, that stream s turned off by assgnng zero power to t. The problem of fndng the optmum orderng under the IPRA for the case of two Tx antennas can be dvded nto three sub-problems based on three SNR regmes. Low SNR regme: Both orderngs have one actve stream Intermedate SNR regme: The optmum orderng has one actve stream and the suboptmum has two actve streams. Hgh SNR regme: Both orderngs have two actve streams. The case where the suboptmum orderng has one actve stream and the optmum has two actve streams never takes place. Ths follows because, from the WF algorthm, the SNR threshold at whch the suboptmum orderng must start operatng wth two actve streams s lower than the SNR threshold correspondng to the optmum orderng. Ths statement s analytcally proved n the next secton. Workng wth the capacty equatons for each orderng at each SNR regme ndepently, the detecton orderng can be establshed based on the per-stream beforeprocessng channel gans as stated n the followng theorem. 69

82 Theorem 4: The optmum detecton orderng for the coded V-BLAST wth two Tx and n Rx antennas under the IPRA s to detect the stream wth hghest before-detecton channel gan at the last step: * h 0 h h (6.30) arg max ln, ff The only f part n (6.30) s true when. When and/or h h any orderng s optmum. Proof: See Appx B. We are now ready to compare the optmal orderng strateges for the coded V- BLAST under the IRA and the IPRA. Proposton 5: The optmal orderngs n the coded V-BLAST wth two Tx antennas under the IRA and under the IPRA are the same,.e. the stream wth hghest before detecton channel gan s detected last. Observaton 5: The optmal orderngs n the coded V-BLAST wth m Tx antennas m under the IRA and under the IPRA 3 are not, n general, the same. Proposton 5 s a drect consequence of Theorems and 4 whle Observaton 5 s supported by extensve numercal smulatons. Some numercal smulatons supportng Observaton 5 The results of some numercal smulatons supportng Observaton 5 are descrbed below. The optmal detecton orderngs for the 3x3 coded V-BLAST under the IRA and the WF are shown for three Raylegh fadng channel realzatons at dfferent SNR values and the opposte of Foschn orderng. 3 va WF. 70

83 ( -0dB, 0dB and 0dB). The capactes of these orderngs were calculated followng both optmzaton strateges and are llustrated n the tables. The examples clearly show that there are cases where the optmum detecton orderng under the IRA s suboptmum under the WF and vce versa. The channel realzaton analysed at SNR 0dB s H Table 3: Performances of the optmal orderngs under the IRA and the WF for both optmzaton strateges for the gven channel realzaton at SNR = 0dB. C nats/sec/hz C nats/sec/hz IRA *,, IRA * 3,, WF WF The channel realzaton analysed at SNR 0dB s H Table 4: Performances of the optmal orderngs under the IRA and the WF for both optmzaton strateges for the gven channel realzaton at SNR = 0dB. C nats/sec/hz C nats/sec/hz IRA *,, IRA * 3,, WF WF 7

84 The channel realzaton analysed at SNR 0dB s H Table 5: Performances of the optmal orderngs under the IRA and the WF for both optmzaton strateges for the gven channel realzaton at SNR 0dB. C nats/sec/hz C nats/sec/hz IRA *,, IRA * 3,, WF WF 6... Suboptmum orderng Fgure : Emprcal CDFs of the nverse orderng and the optmum orderng under WF for the (3x3) system; SNR=0dB; 0 4 channel realzatons. 7

85 In secton 4., t was shown that the nverse orderng performs very close to the optmum one for the coded V-BLAST under the IRA. In ths secton ts performance s evaluated under the WF, the FWF and compared wth the optmum orderng under IPRA usng smulatons. Fgures -3 show the emprcal CDF curves, based on 0 4..d. Raylegh fadng channel realzatons, of the Inverse detecton orderng and the optmum one for the 3x3 and 4x4 coded systems respectvely under the IPRA at dfferent values of SNR. The WF power allocaton s used n both orderngs. The almost optmum performance of the nverse orderng s evdent. Fgure : Emprcal CDFs of the nverse orderng and the optmum orderng under WF for the (4x4) system; SNR=-0dB; 0 4 channel realzatons. Please note that the WF s not, n general, the optmum power/rate allocaton for the nverse orderng (Observaton 4 s only vald for the optmum orderng), the optmum 73

86 power/rate allocaton for the nverse orderng s the FWF. However, ths orderng offers the same value of capacty under the WF and the FWF wth very hgh probablty. To show ths, the capacty of the nverse orderng was also evaluated under the FWF (for the same channel realzatons and parameters used n Fgures, and 3) and the results are llustrated n Table 6. Fgure 3: Emprcal CDFs of the nverse orderng and the optmum orderng under WF for the (4x4) system; SNR=0dB; 0 4 channel realzatons. Table 6: The nverse orderng achevng the same capacty under the WF and the FWF wth hgh probablty. Pr CWF CFWF Pr CWF CFWF The same parameters used n Fgure The same parameters used n Fgure The same parameters used n Fgure x0-3 74

87 6..3. Analytcal boundares of the SNR regmes for two Tx antennas The case of two Tx antennas under the IPRA can be studed n three dfferent SNR regmes: low SNR, ntermedate SNR and hgh SNR. The objectve of ths secton s to derve the analytcal expressons of the SNR thresholds defnng the boundares of each of these regmes. Low SNR regme In the low SNR regme, we have the two possble orderngs operatng wth only one actve stream,.e. the optmum IPRA strategy to maxmze the transmsson rate s to transmt wth one antenna n both cases. It s a property of the WF that, for a gven orderng, only one stream s actve at low SNR (see Appx A). Therefore, as long as the SNR s suffcently low both orderngs wll have only one actve stream. The SNR threshold defnng ths regme s shown n the followng proposton: Proposton 6: The low SNR regme ncludes all SNR values satsfyng 0 g g (6.3) where g h and and, sn. Proof: For the case of two Tx antennas, the two possble orderngs are:,. Wthout loss of generalty, t can be further assumed that g g. Then, from Theorem 4, s the optmum orderng. Ths orderng wll operate wth only one actve stream provded that mn g, g g (6.3) The nequalty (6.3) follows from the per-stream power allocaton formula n the WF algorthm (see Appx A) and the equalty follows from g g and the fact that the 75

88 projecton of a vector cannot ncrease ts norm.e.. s the water level correspondng to gven by 0 g g (6.33) The complete dervaton to obtan (6.33) can be found n Appx A. Substtutng (6.33) nto (6.3), the condton under whch wll operate wth only one actve stream can be wrtten as g g g 0 0 (6.34) and after some mathematcal manpulatons, (6.34) can be expressed as 0 g g g g (6.35) Orderng wll operate wth only one actve stream for all SNR values satsfyng (6.35). The same reasonng apples to the second orderng: Based on the WF per-stream power allocaton formula, wll operate wth only one actve stream provded that mn g, g 0 0 (6.36) where s the correspondng water level gven by 0 g g (6.37) Substtutng (6.37) nto (6.36) the condton under whch wll operate wth only one actve stream can be wrtten as g g mn g, g (6.38) and two possble scenaros arse. The frst scenaro takes place when 76

89 0 0 0 mn g, g g (6.39) and equaton (6.38) reduces to 0 g g g g (6.40) Equaton (6.40) follows after substtutng (6.39) nto (6.38). The second scenaro takes place when mn g, g g (6.4) and equaton (6.38) reduces to 0 g g g g (6.4) Equaton (6.4) follows after substtutng (6.4) nto (6.38). From (6.40) and (6.4), orderng wll operate wth only one actve stream for all SNR values satsfyng: 0 g g g g (6.43) Based on the fact that g g g g f g g (6.44) the SNR thresholds n (6.35) and (6.43) can be compared as 4 g g g g g g g g (6.45) therefore, both orderngs wll operate wth one actve stream f 0 g g g g (6.46) And (6.3) follows. 4 Inequalty (6.45) also shows that the case where the suboptmum orderng has one actve stream and the optmum has two actve streams never takes place. 77

90 Hgh SNR regme In the hgh SNR regme, both orderngs operate wth two actve streams,.e. the optmum IPRA strategy to maxmze the transmsson rate s to transmt wth two antennas n both cases. From the WF algorthm, f the receved SNR s hgh enough n both orderngs, they wll operate wth two actve streams. The SNR threshold defnng ths regme s shown n the followng proposton. Proposton 7: The hgh SNR regme ncludes all SNR values satsfyng 0 g g (6.47) Proof: Ths proof s based on the results obtaned for the low SNR regme case. From (6.35), the optmum orderng wll operate wth two actve streams when 0 g g g g (6.48) From (6.43) the suboptmum orderng wll operate wth two actve streams when 0 g g g g (6.49) Then both orderngs wll operate wth two actve streams provded that 0 g g g g (6.50) Equaton (6.50) s a drect consequence of (6.45) and equaton (6.47) follows. Intermedate SNR regme In the ntermedate SNR regme, the optmum orderng operates wth only one actve stream whle the suboptmum orderng operates wth both streams actve,.e. n the case of the optmum IPRA strategy to maxmze the transmsson rate s to transmt wth only one 78

91 antenna whle n the case of the optmum IPRA strategy s to transmt wth both antennas. The SNR thresholds defnng ths regme are shown n the followng proposton. Proposton 8: The ntermedate SNR regme ncludes all SNR values satsfyng 0 g g g g (6.5) Proof: Equaton (6.5) follows from Propostons 6 and 7. Please note that when g g and/or, the ntermedate SNR regme does not exst. Ths can be easly verfed by substtutng these values nto (6.5). SNR regmes: an example H Fgure 4: Number of actve streams vs. SNR for a gven channel realzaton for the (x) system under the IPRA. Numercal smulatons were undertaken to valdate the analytc results of the dfferent SNR regmes thresholds. Fgure 4 plots the number of actve streams as a functon of SNR 79

92 of the x coded V-BLAST system under the IPRA for a gven Raylegh fadng channel realzaton. The performances of both the optmum and the suboptmum orderng are llustrated. It can be seen that the analytcal thresholds of the SNR regmes agree wth the numercal ones. The behavour of the number of actve streams s descrbed as follows: as long as SNR s low enough, both orderngs start transmttng wth only one antenna (low SNR regme), at SNR.6dB the system wth the suboptmal orderng swtches to two actve streams whle the system wth the optmum orderng s transmttng wth only one antenna (Intermedate SNR regme). Fnally, at SNR two streams (Hgh SNR regme)..3db both orderngs transmt wth 6.3. Summary The optmum detecton orderngs for the coded V-BLAST under the IPA and the IPRA were studed. The optmum detecton orderng under the IPA s SNR ndepent and t maxmzes the harmonc mean per-stream power gan. For the case of two Tx antennas, the optmum orderng can be establshed based on the before-processng channel gans: the stream wth hghest before-processng channel gan must be detected frst. It was shown that the Foschn orderng, although suboptmum, offers an outstandng performance for m 3. There s no loss of optmalty when consderng conventonal WF to fnd the optmum orderng n the coded V-BLAST under the IPRA. Ths s because the optmum orderng under FWF s exactly the same as the optmum orderng under WF, and t acheves the same value of capacty. For the -Tx-antenna case, the optmum orderng can be establshed based on the before-processng channel gans: the stream wth hghest before- 80

93 processng channel gan must be detected last. Analytcal SNR thresholds, whch separate the low, ntermedate and hgh SNR regmes, were derved. It was shown that the nverse orderng under the WF performs almost optmally. 8

94 7. SNR gan of orderng The SNR gan of orderng under the IPRA s dscussed n ths chapter. To explore the propertes of ths gan for two Tx antennas, we rely only on analytcal technques. However, for m, numercal smulatons are used due to the complexty of the mathematcal analyss. 0 0 Fgure 5: Defnton of the SNR gan of orderng. Recall that n the coded V-BLAST the optmzaton of capacty and outage probablty acheve the same lowest value of outage probablty. Based on ths fact, the SNR gan of orderng n the coded V-BLAST s defned as the dfference n SNR requred by the unordered V-BLAST to acheve the same capacty as the optmally ordered.e. C ( ) C ( G ) (7.) * 0 0 8

95 where the left-hand sde represents the system usng the optmum detecton orderng and the rght-hand sde represents the system usng the unordered detecton. G s the SNR gan of the optmum orderng procedure. Fgure 5 s an llustratve example of the SNR gan defnton. 7.. Two Tx antennas The man objectve of ths secton s to derve closed-form expressons for the SNR gan of orderng n the -Tx-antenna coded V-BLAST under the IPRA wth capacty achevng temporal codes. Some propostons follow from the analytcal results provdng a deep nsght nto the SNR gan offered by the optmum detecton orderng procedure. As n prevous sectons, the analyss for two Tx antennas wll be dvded nto three SNR regmes: low, ntermedate and hgh. Analytcal expressons for the SNR gan of orderng at each regme are gven below. Low SNR regme: 0 g g Proposton 9: The SNR gan of the optmum orderng procedure n the low SNR regme s gven by: G, g mn g (7.) where g h and sn. Proof: For the case of two Tx antennas, the channel matrx s gven by The two possble orderngs are:, and, be further assumed that orderng,.e. t maxmzes the system capacty. h h. Then, from Theorem 4, H h h.. Wthout loss of generalty, t can, * s the optmum 83

96 The capacty equatons for orderngs and n the low SNR regme for a gven channel realzaton are WF CWF ln g (7.3) 0 ln max, C g g (7.4) 0 Equatons (7.3) and (7.4) follow from makng the approprate substtutons of the channel gans n (.) depng on the specfc orderng. For (7.3), t has been also used the fact that g g and. From the defnton of the SNR gan, the followng set of equaltes holds: C C G (7.5) 0 0 ln g ln max g, g G (7.6) 0 0 g G max g, g (7.7) Equalty (7.5) states the formal defnton of the SNR gan of orderng based on and. Equalty (7.6) results from substtutng (7.3) and (7.4) nto (7.5). Equalty (7.7) follows after smple mathematcal manpulatons appled to (7.6). Fnally, (7.) follows from (7.7). propertes: Some conclusons arse from (7.) and are stated n the followng proposton. Proposton 0: In the low SNR regme the SNR gan of orderng has the followng It s SNR ndepent,.e. t s constant for all low SNR values. It deps on the rato g g or. If g g and/or there s no gan,.e. f the channel gans are equal and/or the channel columns are orthogonal to each other, then the SNR gan of orderng s equal to one (both orderngs offer the same capacty). 84

97 Note: The last property holds for all the SNR regmes; hence t wll not be repeated agan. Hgh SNR regme: 0 g g Proposton : The SNR gan of the optmum orderng procedure n the hgh SNR regme s expressed as: G g g g g 0 (7.8) Proof: The capacty equatons for orderngs and n the hgh SNR regme for a gven channel realzaton can be expressed as WF * * ln 0 ln 0 C g g (7.9) * * ln 0 ln 0 C g g (7.0) WF a b Equatons (7.9) and (7.0) follow from makng the approprate substtutons of the channel gans n (.4) depng on the specfc orderng. The pars, and * *, are the * * a b optmum waterfllng power allocatons for the orderngs and respectvely. Equatons (7.9) and (7.0) can be expressed n terms of the correspondng water levels as: WF ln 0 C g g (7.) WF ln 0 C g g (7.) Ths follows from makng the approprate substtutons of the channel gans and the water levels n (.7) depng on the specfc orderng. The water levels and are as n (6.33) and (6.37) respectvely. Substtutng (6.33) and (6.37) nto (7.) and (7.) respectvely, the capacty equatons can be expressed as: 85

98 C C WF WF ln A B 0 ln A A C 0 A (7.3) (7.4) where A, B and C are gven by: A gg, B g g, C g g. From the defnton of the SNR gan, the followng set of equaltes holds: C C G (7.5) 0 0 A B AG C 0 0 ln ln A A (7.6) G A 0 B C A 0 (7.7) Equalty (7.5) states the formal defnton of the SNR gan of orderng based on and. Equalty (7.6) results from substtutng (7.3) and (7.4) nto (7.5). Equalty (7.7) follows after smple mathematcal manpulatons appled to (7.6). Fnally, (7.8) follows from substtutng the orgnal expressons for A, B and C nto (7.7). propertes: Some conclusons arse from (7.8) and are stated n the followng proposton. Proposton : In the hgh SNR regme the SNR gan of orderng has the followng For gven g, g and t s a decreasng functon of SNR,.e. G 0 For gven g, g and 0 t s a decreasng functon of,.e. G For gven g, and 0 t s a decreasng functon of g,.e. G g 86

99 Intermedate SNR regme: 0 g g g g Proposton 3: The SNR gan of the optmum orderng procedure n the ntermedate SNR regme s expressed as: g0 g g G 0 gg gg (7.8) Proof: The capacty equatons for orderngs and n the ntermedate SNR regme for a gven channel realzaton are gven by (7.3) and (7.0) respectvely,.e. CWF ln g (7.9) 0 * * ln 0 ln 0 C g g (7.0) WF a b Equatons (7.9) and (7.0) follow because n the ntermedate SNR regme the optmum orderng have only one actve stream whle the sub-optmum has both streams actve. It was shown n the prevous secton that equaton (7.0) can also be expressed as n (7.4). From the defnton of the SNR gan, the followng set of equatons holds: C C G (7.) 0 0 AG C 0 ln g0 ln A (7.) G A g C A 0 0 (7.3) Equalty (7.) states the formal defnton of the SNR gan of orderng based on and. Equalty (7.) results from substtutng (7.9) and (7.4) nto (7.). Equalty (7.3) follows after smple mathematcal manpulatons appled to (7.). Fnally, (7.8) follows from substtutng the orgnal expressons for A and C nto (7.3). 87

100 Note that the SNR gan expresson s more complcated than n the prevous regmes. Proposton 4 states a concluson that arse from (7.8): Proposton 4: In the ntermedate SNR regme the SNR gan of orderng s SNR depent. Conjecture: Based on prevous propertes of the SNR gan, we conjecture that G s bounded as, G mn, g g (7.4) Note: Ths conjecture s suggested by the low and hgh SNR regmes. However, snce G may exhbt a non-monotonc behavour n the ntermedate SNR regme, we do not have a complete proof of ths result at the moment Two Tx antennas: valdaton of the results H g g,, G Fgure 6: SNR gan of orderng for the (x) system. 88

101 Numercal smulatons have been carred out to valdate the closed-form expressons for the SNR gan of orderng at each SNR regme. Fgure 6 plots the capacty as a functon of SNR for the x coded V-BLAST system under the IPRA (va WF) for a gven Raylegh fadng channel realzaton. The gap n SNR for the same value of capacty s the SNR gan offered for the optmum orderng. The vertcal lnes correspond to the analytcal expressons (6.3) and (6.47) and they separate the three SNR regmes. m*, but the gan has to be expressed as n the case of m*, because G0 th H Fgure 7: SNR gan vs. SNR; numercal and analytcal solutons. Fgure 7 shows the SNR gan of orderng (numercal and analytcal) as a functon of SNR for the example shown n Fgure 6. Ths graph helps us understand better the behavour of the SNR gan. It can be seen how the numercal and analytcal results perfectly match wth each other, valdatng the analytcal results. In the low SNR regme, the SNR 89

102 gan s constant (ndepent of the SNR) when operate wth m*, 5. In the ntermedate SNR regme, the SNR gan s stll consderably hgh and, n ths case, t s a decreasng functon of the SNR. In the hgh SNR regme the SNR gan decreases as the SNR ncreases showng that the optmal orderng does not offer a sgnfcant advantage n ths SNR regme. 7.. Three transmt antennas: an example The closed-form analyss when the number of Tx antennas s greater than two s very dffcult. Therefore, we rely on numercal smulatons to study the benefts of the optmum orderng procedure and the behavour of the SNR gan of orderng n these systems. An example s dscussed below. Fgure 8 llustrates the behavour of the capacty as a functon of SNR for the 3x3 coded V-BLAST system under the WF for a gven Raylegh fadng channel realzaton. Only four orderngs from whch some conclusons can be derved are shown 6. Some observatons follow from the graph: There s one orderng (Orderng D) that offers the worst performance at all SNR values and there s a consderable SNR gap between ths orderng and the rest. Therefore, the gan provded by the optmum orderng procedure s evdent. There are dfferent orderngs that offer the same optmum capacty, for example orderngs A and C are optmum for 0dB 0 3dB. 5 Each orderng operates wth one actve stream. 6 Two orderngs that do not provde useful nformaton are not shown to avod overloadng the graph. 90

103 The optmal orderng s, n general, SNR depent,.e. there s not, n general, one orderng that remans optmum for all SNR values. For example orderngs A and C are optmum for 0dB 0 3dB whle orderng B s optmum for 0 3dB. Fgure 8: SNR of orderng (3x3) Summary The SNR gan of orderng was defned and studed. Closed-form expressons of the SNR gan of orderng under the IPRA were derved for the case of two Tx antennas. In the low SNR regme, the gan s SNR ndepent. In the ntermedate SNR regme, t s SNR depent and n the hgh SNR regme, t s a decreasng functon of SNR. Hgh gan of orderng can be acheved n the low and ntermedate SNR regmes, whle the gan s not that large n the hgh SNR regme, approachng one as 0. 9

104 8. Lnk to Multple Access Wreless Channels Wth the ever ncreasng development of cellular systems, multuser communcatons s a topc that has ganed a lot of mportance n terms of research durng the last years. In prevous chapters, we studed the optmal orderng process for the coded V-BLAST n the context of pont-to-pont channels. In ths secton, we shft the focus to multple access channels (MAC),.e. the uplnk of a cellular network where the base staton uses multple receve antennas to dscrmnate among the users whch are n the same frequency (spacedvson multple access (SDMA)), see Fgure 9. More detals about MAC can be found n [33]. Fgure 9: Uplnk wth sngle Tx antenna at each user and multple Rx antennas at the base staton. Let us dscuss the case of a cellular wreless system where the users have only one transmt antenna (e.g. due to complexty constrants). As the users are spread out over the cell, the sgnals sent out from the transmt antennas cannot be coordnated, so each user transmts ndepent sgnals. At the base staton the sgnal of the users can be unmxed 9

105 through the use of multple receve antennas and approprate sgnal processng. For example when detectng the sgnal correspondng to user at the base staton, we can frst cancel the nterference from the already detected users, project out the nterference from the yet to be detected users and then apply MRC to the remanng sgnal plus correlated nose (correlated after the multplcaton by the projecton matrx). By dong ths wth each user, the multple access channel s transformed nto a set of vrtually ndepent subchannels. Ths s n fact the V-BLAST archtecture where each stream s nterpreted as a user. The system model can be expressed as n (3.).e. m r h q ξ (8.) The only dfference s that now m s the number of users n the system (analogous to the number of Tx antennas n (3.)) and h s a column vector contanng the channel gans from the -th user s Tx antenna to all Rx antennas of the base staton,.e. channel correspondng to user. h s the SIMO Snce the users are geographcally separated, ther transmt sgnals arrve n dfferent drectons to the base staton antenna array even when there s not a rch scatterng envronment, so the assumpton of a full ranked channel matrx H s usually vald [33]. Ths s n contrast to the pont-to-pont case where a rch scatterng envronment s needed snce all the transmt antennas are very close to each other. It follows then that all prevous dscussons about the stream detecton order n the pont-to-pont coded V-BLAST can be appled to the multple access wreless channels, but now nstead of stream detecton order we are dealng wth user detecton order. 93

106 Let us assume that we are n presence of a cellular network where the modulaton and codng formats at the users s can be modfed n order to acheve a rate equal to the capacty of each user channel (.e. the IRA). In the hypothetcal case where the base staton s detectng the sgnal from only two users wthn the cell, we know from Theorem that the optmum strategy s to detect last the user wth the hghest unprojected channel gan; ths s n general the user stuated closer to the base staton. Note that from Theorem 4 the same holds true f optmum nstantaneous power/rate allocaton s used at each user transmsson. Ths strategy s optmal because t maxmzes the sum rate of both users, so the total throughput of the system s maxmzed. However, from the pont of vew of the ndvdual users, t can be seen as unfar because the user wth the weakest unprojected channel gan has to deal wth hs bad channel condton and wth the nterference caused by the stronger user. Therefore, the rate at whch ths user can communcate wll be severely affected. Another ntrnsc drawback of the adaptve modulaton and codng schemes s an ncrease n the system complexty and cost, snce each user termnal must be desgned to handle dfferent codng and modulaton strateges n an adaptve way. Modern communcaton systems already mplement adaptve schemes, e.g. W-Max, W-F and LTE standards. If we are lookng for farness and lower system complexty, a cellular network where the same codng and modulaton formats are used for all users can be consdered. The URA and the IPA are sutable for such a scenaro. Under the URA all users transmt usng the same rate and power. Under the IPA, although the transmsson rate s unform among the users, the power allocated to each user s dfferent: users wth weak channel gans enjoy 94

107 hgher transmsson powers and, therefore, a better system performance s acheved 7. Agan n the hypothetcal case where the base staton s detectng the sgnal from only two users wthn the cell, ether under URA or IPA, from Theorems and 3, the optmum strategy s to detect frst the user wth the hghest unprojected channel gan.e. the user stuated closer to the base staton. In ths case prorty s gven to the weakest user who s detected at the last step and, therefore, does not have to deal wth the nterference caused by the strongest user. Ths strategy, although better n terms of farness and complexty, acheves a lower total system capacty. 7 as compared to the system under the URA. 95

108 9. Concluson 9.. Summary of the thess Based on the fact that n the coded V-BLAST the nstantaneous optmzatons of the capacty and outage probablty acheve the same lowest value of the outage probablty, the problem of fndng an optmal detecton orderng was approached from the system capacty pont of vew. It was demonstrated that the optmum detecton orderngs under the IRA and the IPRA are SNR depent whle under the URA and the IPA are SNR ndepent. It was shown that the Foschn detecton orderng s also optmum for the coded V-BLAST under the URA. Closed-form approxmatons were derved for the optmal detecton orderng at low and hgh SNR under the IRA: based on approxmatons of the per-stream capacty equaton, t was demonstrated that the optmum detecton orderng maxmzes the sum of the after detecton channel gans at low SNR and that all detecton orderng strateges offer approxmately the same performance at hgh SNR. Therefore, at hgh SNR, s not worth the complexty effort of the orderng. The optmum orderng strateges n the x n coded V-BLAST under the IRA, URA, IPA and the IPRA were establshed based on the before-processng channel gans. The analytcal proofs of the optmum orderng strateges for these systems were gven. Under the IRA and the IPRA, the best performance s acheved when the stream wth hghest beforeprocessng channel gan s detected last. Furthermore, detectng ths stream frst s the best orderng strategy under the URA and the IPA. Under the IRA, based on the x n results, necessary optmalty condtons were derved for the general case. It was shown, based on Monte-Carlo smulatons, that only a small percentage of the total possble orderngs satsfy the condtons and that ths percentage 96

109 decreases very fast wth the number of Tx antennas. In the best case scenaro, only one orderng satsfes them (ths s a very lkely scenaro). Although the problem of fndng the maxmum number of orderngs that can satsfy the necessary optmalty condtons remans open, based on the defnton of ndepent necessary condtons, ths maxmum number was lower and upper bounded. The SNR gan of orderng was defned. A closed-form analyss of the SNR gan of orderng for the case of two Tx antennas under the IPRA was provded. At low and ntermedate SNR the gan offered by the optmal orderng s sgnfcant, but at hgh SNR the orderng does not offer a sgnfcant mprovement to the system performance, and thus s not worth the complexty effort. Suboptmal detecton orderngs were proposed and ther performances were shown to be very close to the optmum one. It was shown that the results found from the pont-to-pont perspectve are also applcable to the multuser communcatons,.e. multple access channels. 9.. Future research One lmtaton of ths research s that the closed-form analyss presented s lmted to the ZF V-BLAST. One possble extenson would be to generalze these results to the MMSE V-BLAST, whch s the capacty achevng strategy. Future research s needed n some areas. Frst, no loss of optmalty by consderng the WF to obtan the optmum detecton orderng under the IPRA was observed numercally, but t lacks an analytcal proof. Second, the exact maxmum number of combnatons that can satsfy the necessary optmalty condtons under the IRA s stll an open (hard 97

110 combnatoral) problem. Thrd, the analytcal results of the SNR gan are lmted to the case of two Tx antennas, ths closed form analyss can be exted to m. Fnally, t should be ponted out that the above results have been derved under deal condtons. An mportant topc for future research s to nvestgate the mpacts of practcal condtons, such as mperfect channel estmaton. Hence, ths work s only a partal analytcal framework to the full understandng of the optmal orderng n the coded V-BLAST. 98

111 0. References MIMO channel capacty and BLAST archtecture [] I. E. Telatar, Capacty of Mult-Antenna Gaussan Channels, AT&T Bell Lab. Internal Tech. Memo., Jun. 995 (European Trans. Telecom., v. 0, N. 6, Dec. 999). [] G. J. Foschn, Layered Space-Tme Archtecture for Wreless Communcaton n a Fadng Envronment When Usng Multple Antennas, Bell Lab. Tech. J., v., N., pp. 4-59, 996. [3] G. J. Foschn, P. W. Wolnansky, G. D. Golden, and R. A. Valenzuela, Smplfed Processng for Hgh Spectral Effcency Wreless Communcaton Employng Mult-Element Arrays, IEEE J. Select. Areas Commun., vol. 7, no., pp , Nov [4] P. W. Wolnansky, G. J. Foschn, G. D. Golden, and R. A. Valenzuela, V-BLAST: An Archtecture for Achevng Very Hgh Data Rates Over the Rch-Scatterng Wreless Channel, Int. Symposum on Sgnals, Systems and Electroncs (ISSSE), Psa, Italy, 998. Reducng V-BLAST complexty [5] W. K. Wa, C.-Y. Tsu, R. S.Cheng, A Low Complexty Archtecture for the V-BLAST System, IEEE Wreless Commun. and Networkng Conf. (WCNC), Chcago, IL, USA, v., pp , Sep [6] J. Benesty, Y. Huang, and J. Chen, A fast Recursve Algorthm for Optmum Sequental Sgnal Detecton n a BLAST System, IEEE Trans. Sgnal Processng, vol. 5, no. 7, pp , July 003. [7] Z. Luo, S. Lu, M. Zhao, and Y. Lu, A Novel Fast Recursve MMSE-SIC Detecton Algorthm for V-BLAST Systems, IEEE Trans. Wreless Commun., vol. 6, no. 6, pp. 0-05, June 007. [8] J. Chen, S. Jn, and Y. Wang, Reduced Complexty MMSE-SIC Detector n V-BLAST Systems, IEEE Personal, Indoor and Moble Rado Commun. (PIMRC),pp. 5, Sep [9] B. Hassb, An Effcent Square-Root Algorthm for BLAST, n Proc. IEEE Int. Conf. Acoustcs Speech, and Sgnal Processng, (ICASSP), pp , June 000. [0] D. Wubben, R. Bohnke, J. Rnas, V. Kuhn, and K. D. Kammeyer, Effcent Algorthm for Decodng Layered Space-Tme Codes, IEEE Electroncs Letters, vol. 37, no., pp , Oct. 00. [] R. Bohnke, D. Wubben, V. Kuhn, and K. D. Kammeyer, Reduced Complexty MMSE Detecton for BLAST Archtectures, IEEE Global Telecom. Conf. (GLOBECOM), vol. 4. pp. 58 6, Dec

112 [] H. Zhu, Z. Le, and F. P. S. Chn, An Improved Square-Root Algorthm for BLAST, IEEE Sgnal Process Lett., vol., no. 9, pp , Sep [3] H. Zhu, W. Chen, B. L, and F. Gao, An Improved Square-Root Algorthm for V- BLAST Based on Effcent Inverse Cholesky Factorzaton, IEEE Trans. Wreless Commun., vol. 0, no., pp , Jan. 0. Performance analyss of the V-BLAST [4] S. Loyka and F. Gagnon, Performance Analyss of the V-BLAST Algorthm: An Analytcal Approach, IEEE Trans.Wreless Commun., vol.3, no. 4, pp , Jul [5] Y. Jang, X. Zheng, and J. L, Asymptotc Performance Analyss of V-BLAST, IEEE Global Telecom. Conf. (GLOBECOM), pp , 005. [6] S. Loyka and F. Gagnon, V-BLAST Wthout Optmal Orderng: Analytcal Performance Evaluaton for Raylegh Fadng Channels, IEEE Trans. Commun., vol. 54, no. 6, pp. 09-0, June 006. [7] H. Zhang, H. Da, and B. L. Hughes, On the Dversty-Multplexng Tradeoff for Ordered SIC Recevers Over MIMO Channels, IEEE Int. Conf. on Commun. (ICC), pp , 006. [8] R. Xu and F. C. M. Lau, A Novel Approach to Analyzng V-BLAST MIMO Systems Wth Two Transmt Antennas, IEEE Trans. Wreless Commun., vol. 6, no. 5, pp , May 007. [9] S. Loyka and F. Gagnon, On Outage and Error Rate Analyss of the Ordered V- BLAST, IEEE Trans. Wreless Commun., vol. 7, no. 0, pp , Oct [0] S. H. Nam et al., Transmt Power Allocaton for a Modfed V-BLAST System, IEEE Trans. Commun., vol. 5, no. 7, pp , July 004. [] N. Wang and S. D. Blosten, Mnmum BER Transmt Power Allocaton for MIMO Spatal Multplexng Systems, IEEE Internatonal Conference on Commun., San Dego, vol. 4, pp. 8 86, May 005. [] R. Kalbas, D. D. Falconer, and A. H. Banhashem, Optmum Power Allocaton for a V-BLAST System Wth Two Antennas at the Transmtter, IEEE Commun. Lett., vol. 9, no. 9, pp , Sept [3] N. Wang and S. D. Blosten, Mnmum BER Transmt Power Allocaton and Beamformng for Two-Input Multple-Output Spatal Multplexng Systems, IEEE Trans. Vehcular Technology, vol. 56, no., pp , March 007. [4] V. Kostna and S. Loyka, On Optmum Power Allocaton for the V-BLAST, IEEE Trans. Commun., v. 56, N. 6, pp , June

113 [5] Y Jang, M. K. Varanas and J. L, Performance Analyss of ZF and MMSE Equalzers for MIMO Systems: An In-Depth Study of the Hgh SNR Regme, IEEE Trans. Informaton Theory, v. 57, no. 4, pp , Apr. 0. V-BLAST under channel estmaton errors [6] X. Zhang and B. Ottersten, Performance Analyss of V-BLAST Structure Wth Channel Estmaton Errors, IEEE Sgnal Processng Advances n Wreless Commun. (SPAWC), pp , June 003. [7] R. Narasmhan, Error Propagaton Analyss of V-BLAST Wth Channel-Estmaton Errors, IEEE Trans. Commun., vol. 53, no., pp. 7-3, Jan [8] Ch. Wang et al., On the Performance of the MIMO Zero-Forcng Recever n the Presence of Channel Estmaton Error, IEEE Trans. Wreless Commun., vol. 6, no. 3, pp , March 007. [9] J. Chen and X. Yu, ZF V-BLAST for Imperfect MIMO Channels Usng Average Performance Optmzaton,. IEEE Int. Conf. Acoustcs Speech, and Sgnal Processng (ICASSP), pp. III 4-III-44, Apr [30] W. Peng et al., Effects of Channel Estmaton Errors on V-BLAST Detecton, IEEE Global Telecom. Conf. (GLOBECOM), pp. -5, Dec Coded V-BLAST [3] V. Kostna and S. Loyka, Optmum Power and Rate Allocaton for Coded V-BLAST: Average Optmzaton," IEEE Trans. Commun., v. 59, no. 3, pp , Mar. 0. [3] V. Kostna and S. Loyka, Optmum Power and Rate Allocaton for Coded V-BLAST: Instantaneous Optmzaton, IEEE Trans. Commun., v. 59, no. 0, pp , Oct. 0. Books [33] D. Tse and P. Vswanath, Fundamentals of Wreless Communcatons, Cambrdge Unversty Press, 005. [34] F. R. Gantmacher, The Theory of Matrces, Volume, Chelsea Publshng Company,

114 . Appxes Appx A: Waterfllng algorthm A parallel addtve whte Gaussan nose (AWGN) channel s a channel consstng n a set of non-nterferng sub-channels each of whch s corrupted by AWGN. A natural strategy for relable communcaton over a parallel AWGN channel s to allocate power to each sub-channel and use separate capacty achevng temporal codes to communcate over each of the sub-channels. The maxmum rate of relable communcaton under ths scheme s m ln 0 g (.) where m s the number of sub-channels, and g are the power and the channel gan correspondng sub-channel and 0 s the average SNR at the output of each sub-channel [33]. Furthermore the power allocaton can be chosen to maxmze the rate n (.). The optmum power allocaton strategy to transmt through a parallel AWGN channel when channel state nformaton s avalable at the transmtter s the waterfllng (WF) algorthm [33]. The power allocaton under the WF algorthm s gven by where x max x,0 constrant s met.e. 0g (.) and s the water level. s chosen such that the total power m m. The prncple of ths power allocaton strategy s to allocate more power to the stronger streams takng advantage of the better channel condtons and less or even no power to the weaker ones. Fgure 30 gves a pctoral vew of ths strategy. It s not dffcult to show that the total system capacty under the WF algorthm (gven that all streams are actve) can be expressed as a functon of the water level as C WF m ln g 0 (.3) 0

115 Fgure 30: Pctoral representaton of the waterfllng algorthm. Two actve streams When two streams are actve m* 8, the system capacty s The power allocated to the streams are gven by C ln g ln g (.4) 0 0,, g 0 (.5) and can be found from the total power constrant as follows : (.6) (.7) g g 0 where equaton (.6) states the total power constrant for the case of two actve streams. Equaton (.7) follows from substtutng the WF power allocaton (.5) nto (.6) and 8 m * s the number of actve streams. 03