Join Opimizaion of Rae Allocaion and BLAST Ordering o Minimize Ouage Probabiliy Arumugam Kannan, Badri Varadarajan and John R. Barry School of Elecrical and Compuer Engineering Georgia Insiue of Technology, Alana, GA 30332-0250 USA {aru, badri, barry}@ece.gaech.edu Absrac We consider a wireless communicaion sysem over a muliple-inpu muliple-oupu Rayleigh-fading channel wih successive-cancellaion deecion. The ouage probabiliy of such a sysem is srongly dependen on wo choices: he order in which he layers are deeced, and he rae-allocaion sraegy a he ransmier. We propose he rae-alized ordering algorihm, a generalizaion of he BLAST ordering algorihm ha is shown o minimize ouage probabiliy. We furher opimize he allocaion of rae and energy a he ransmier, for a variey of receiver ordering sraegies. Finally, we joinly opimize he receiver ordering and ransmier rae and energy allocaions. Our main conclusion is ha, for a wide range of daa raes and SNR, he ouage probabiliy is minimized by a combinaion of rae-alized ordering and a parially uniform rae and energy (PURE) allocaion sraegy. The joinly opimum sysem ouperforms he Bell Labs layered space-ime (BLAST) archiecure by 5 db a 8 b/s/hz and an ouage probabiliy of 0 3, when operaing over a 4-inpu 4-oupu Rayleigh-fading channel. Also, he joinly opimum sysem shows an improvemen of.5 db over a recenly proposed combinaion of opimum allocaion and fixed ordering. I. INTRODUCTION The use of muliple ransmi and receive anennas in wireless communicaion sysems offers dramaic muliplexing and diversiy gains, enabling high specral efficiencies and low error raes over wireless fading channels. The ransmier and receiver should be appropriaely designed o exploi he gains offered by muliple-inpu, muliple-oupu (MIMO) sysems. Furher, sysem design should aim for low compuaional complexiy. The BLAST archiecure [][2] achieves a high muliplexing gain by ransmiing independen daa sreams, possibly coded, from each ransmi anenna. The BLAST receiver employs he ordered successive cancellaion (SC) decoding algorihm, which maximizes he minimum SNR among all sages of decoding. The SC receiver is much less complex han he opimum join ML decoder, bu i does no fully exploi he diversiy gain offered by he MIMO channel. Consequenly, he BLAST sysem suffers from high error raes, even a high SNR. This research was suppored in par by Naional Science Foundaion grans CCR-0082329 and CCR-02565. The error performance of he SC decoder can be improved by rae and energy allocaion across he various anennas. Ideally, he opimum allocaion is obained hrough a feedback pah from receiver o ransmier, where eiher he channel iself or he opimum allocaion, compued by he receiver, is fed back [3][4][5]. Transmier allocaion sraegies based on receiver feedback are effecive, bu such feedback is no always available in pracical sysems. Insead of relying on feedback, one can allocae raes and energies based on he saisics [6][7] of he random channel, which may be known o he ransmier even if he insananeous channel is no. The loading sraegy ha minimizes he error probabiliy, for he fixed case where he receiver performs no ordering, was obained in [6]. The ouage capaciy of such a sysem was analyzed in [8]. A sraegy which selecs a subse of he ransmi anennas based on channel saisics was shown [9][0] considerably improve he performance of he convenional BLAST sysem. However, he joinly opimal ransmier receiver srucure has no ye been explored. In his work, we aim o minimize ouage probabiliy. The ouage probabiliy is a lower bound on he word error rae achievable by a coded MIMO sysem. The bound is igh when he ouer error-correcing ouer code is powerful, in he sense ha i approaches capaciy over an AWGN channel. In his paper, we propose he rae-alized ordering algorihm, which minimizes he ouage probabiliy for any given ransmier loading sraegy. Second, we obain he ransmier rae and energy allocaion ha minimizes he ouage probabiliy wih rae-alized ordering. Our main conclusion is ha, for a wide range of daa raes and SNR, he ouage probabiliy is minimized by a parially uniform rae and energy (PURE) allocaion sraegy, which disribues he available rae and energy uniformly over a fracion of he available ransmi anennas. This paper is organized as follows. In Secion II, we describe he channel model, and he sysem archiecure. In Secion III, we review he BLAST ordering algorihm, propose our new rae-alized variaion and prove is opimaliy. In Secion IV, we discuss he problem of opimum rae allocaion for each receiver ordering algorihm. The analysis is suppored by simulaion resuls in Secion V. Finally, Secion VI summarizes our conclusions.
II. SYSTEM MODEL AND BACKGROUND We consider a MIMO sysem wih ransmi and r receive anennas, wih he assumpion ha r. In keeping wih he BLAST archiecure [], independen daa sreams are ransmied on each ransmi anenna. Before ransmission, each daa sream is encoded by an error-correcing code ha is designed o approach capaciy on an AWGN channel. The i-h daa sream carries an informaion rae of R i b/s/hz wih an average energy of E i. The oal daa rae is R = R i, and he i = average ransmi energy is E = E i. i = We assume ha he channel is linear, fla-fading and quasisaic, so ha he received vecor a he k-h signaling inerval is: y k = Hx k + n k. () The elemens of he r noise vecor n k are independen, circularly symmeric Gaussian random variables wih zero mean and variance N 0 so ha E[n k n l *] = δ k l N 0 I r, where A* denoes conjugae ranspose of A. The r channel marix H is assumed o be a random Rayleigh fading marix, is enries being independen, circularly symmeric complex Gaussian random variables wih zero mean and uni variance. The receiver knows H, bu he ransmier has no informaion abou H. Under hese assumpions, he SNR per receive anenna is given by S = E/N 0. In his paper, we focus on receivers which employ he successive cancellaion (SC) decoding algorihm []. SC decoders decode one daa sream a a ime, subracing ou he esimaed conribuion of previously decoded daa sreams, and nulling ou inerference from undecoded sreams. An imporan degree of freedom in he design of SC decoders is he choice of he order in which he sreams are decoded. Le i j denoe he index of he symbol decoded in he ( i j-h sage. To decode all he symbols { j ) xˆ } in he i j -h daa k sream, he SC decoder cancels off he esimaed conribuion from he previously deeced daa sreams o obain ( j r ) k = y k h il xˆ k, (2) l = where h i denoes he i-h column of he channel marix H. If previous decisions are correc, hen ( j {r ) k } conains conribuions only from he sream of ineres i j, and inerference from he undecoded sreams. To null ou he inerference, he SC decoder uses he nulling vecor w j, defined as he firs row of he Moore-Penrose inverse of he marix [h ij, h ij +,, h i ] []. Using he nulling vecor, he SC decoder obains he j-h decision sream, d ( j ) k = w j *r ( j ) k. Assuming perfec decision feedback, he channel model reduces o ( i j ) x k j ( i l ) d k ( j ) = + w j *n k. (3) The equivalen channel (3) is an AWGN channel wih noise variance N 0 w j 2 ( i. The esimaes { j ) } of he i h xˆ j daa sream ( j are obained from {d ) k k }. We now proceed o quanify he error probabiliy of he SC decoder, as a funcion of {R i }, {E i } and he decision ordering vecor i =[i, i 2,, i ] T. For convenience, define he inverse ordering vecor j =[j, j 2,, j ] T such ha q = j iq for q =, 2,,. The effecive channel corresponding o he i-h symbol has a noise variance equal o N 0 /γ i, where γ i is he SNR scaling facor given by γ i = / w 2 ji. Recall ha he i-h daa sream has an average energy E i, hence he insananeous SNR of he effecive channel is E i γ i /N 0, and he insananeous capaciy is log 2 ( + E i γ i /N 0 ). Since each daa sream is assumed o have a capaciy-achieving code, i is incorrecly decoded if and only if an ouage occurs, i.e., if and only if: log 2 ( + E i γ i /N 0 ) < R i, (4) or equivalenly if and only if γ i is less han /, where SNR i is he rae-alized SNR of he i-h daa sream, as defined by Forney []: SNR i E i N 0 SNR i = ----------------. (5) 2 R i For noaional convenience, we define =. The rae-alized SNR characerizes he error performance of he sysem beer han jus he SNR, since i capures he effec of he daa rae. The overall SC decoder is a bank of parallel scalar decoders for each sream. If all daa sreams are ouage-free, he SC decoder is also error-free. However, if any of he sreams is in ouage, he SC decoder is in ouage, and hence has a non-zero probabiliy of frame error. Consequenly, he frame-error rae of he coded sysem is upper-bounded by he ouage probabiliy SNR i P o (i, { }) = Pr γ ji < ----. (6) ρ i = i In he following secions, we will discuss problem of minimizing he ouage probabiliy by he opimal choice of {R i }, {E i } and he ordering vecor i. III. RECEIVER DESIGN: CHOICE OF ORDERING ALGORITHM For every insance of he channel H, he receiver uses an ordering algorihm o compue i, which in urn deermines he SNR scaling facors { γ i }. Averaging over H, he ordering algorihm deermines he probabiliy disribuion of he SNR scaling vecor Γ =[γ, γ 2, γ ] T, and hence he ouage probabiliy. In his secion, we discuss hree possible ordering algorihms and derive he ouage probabiliy for each.
III-A. Fixed Ordering The simples ordering algorihm is fixed ordering, where he sreams are decoded simply in he increasing order of heir index, i.e., i = j = [, 2,, ] T, irrespecive of H. In his case, i is well known [2] ha he SNR scaling facors { γ i } are independen. Thus, he ouage probabiliy (6) reduces o P fixed ({ }) = Pr[γ i ---- ]. (7) i = Furher, from [2], γ i has a χ 2 -disribuion wih 2(r + i) degrees of freedom, hence r + i Pr[γ i ] = exp -----. (8) l! -- ---- l l = 0 ρ i Subsiuing (8) in (7) gives a closed form expression for he ouage probabiliy of fixed ordering. III-B. BLAST Ordering The BLAST ordering algorihm [] can be summarized as follows. Given H, he firs sream o be decoded, i, is chosen as he one wih he nulling vecor of leas magniude, i.e., he maximum pos-nulling SNR. The nex sream, i 2, is chosen o maximize γ 2, among he remaining choices, and so on. I was shown in [] ha his greedy ordering algorihm is also globally opimum, as saed below. Remark. For sages j =, 2, he BLAST ordering algorihm chooses i j so as o achieve he maximum value of γ j among he j + possibiliies. In he process, i also maximizes he minimum of SNR scaling facors, namely min(γ, γ 2,, γ ). The SNR scaling facors γ, γ 2,, γ produced by BLAST ordering are no muually independen. Therefore, obaining a closed form expression for he densiy funcion of Γ is an open problem. However, he following properies hold for he ordering vecor i and he SNR scaling vecor Γ. Theorem. For a Rayleigh fading channel, he ordering vecor i and he SNR scaling vecor Γ produced by he BLAST ordering algorihm are independen. Furher, i is uniformly disribued over he se of all permuaions of [, 2,, ] T. In Secion IV-B, we will use he above facs o derive a simple expression for he union bound on he ouage probabiliy (6). III-C. Rae-Normalized Ordering In his secion, we propose he rae-alized (RN) ordering algorihm which minimizes he ouage probabiliy for any given ransmier rae allocaion. Firs, noe from (6) ha an ouage occurs if and only if min( γ j j ) across all he decoding sages is less han uniy. From his observaion, we sae he following lemma. Lemma 3.. To minimize he ouage probabiliy (6), he ordering vecor i should be chosen o ensure ha min( γ j j ) is maximized over j =, 2,,. Theorem 2. A every sage of decoding j =, 2,,, he rae-alized ordering algorihm chooses i j so as o achieve he maximum value of γ j j among he j + possibiliies. In he process, i maximizes he minimum of γ j j across all daa sreams. Proof: Le he scaled channel marix be H = HD, where D is a diagonal marix, whose j-h diagonal enry is d jj = j. If he QR decomposiion of H is given by H = QR, hen he scaled channel marix can be wrien as H = QRD = QR. I is well known [3] ha he SNR scaling facors resuling from SC decoding of H are equal o he squared diagonal enries of R. Therefore, he SNR scaling facor for he scaled channel H is simply γ j = R 2 jj d 2 jj = γ ρ j ij. Hence, he minimum of γ j is maximized by employing he usual BLAST ordering algorihm of [] on H insead of H. Combining Lemma 3. and Theorem 2, we conclude ha he rae-alized ordering algorihm minimizes he ouage probabiliy among all possible ordering algorihms. IV. TRANSMITTER DESIGN: OPTIMAL RATE AND ENERGY ALLOCATION In his secion, we discuss he ransmier opimizaion problem, namely o choose he {R i } and {E i } o minimize he ouage probabiliy a a given SNR under he consrains ha i = R i =R and i = E i =E. We begin by saing he following remark abou he opimum energy allocaion holds for all ordering algorihms. Remark 2. Suppose {R i * } and {E i * } are he rae and energy allocaions ha minimize he ouage probabiliy (6). They are relaed by: E * 2 R i * i E = --------------------------------------. (9) R 2 j * j = This is easily proved using Lagrange mulipliers. The proof is omied due o space consrains. The implicaion is ha he ransmier opimizaion is simplified o one of choosing only he daa raes {R i }, wih he opimum energies E i auomaically deermined by (9). Thus, he number of variables o be opimized is reduced from 2 o. IV-A. Fixed Ordering Transmier opimizaion for fixed ordering has already been solved [6] excep ha he relaion (9) beween he opimum energy and rae allocaions was no used. The opimum allocaion is deermined by a random search followed by consrained gradien descen over he space of all possible daa raes {R i } ha sum up o R. For example,
consider a 4-inpu, 4-oupu MIMO sysem operaing a a daa rae of R =8 b/s/hz and an SNR of S =5dB. Numerical opimizaion yields he opimum daa rae allocaion o be {R i }={0,.3,2.99,3.70}. From (9), he corresponding energy allocaion is {E i /E} = {0, 0.25, 0.36, 0.39}. Noe ha he sreams deeced laer carry a higher daa rae han sreams deeced early. This resul is inuiively saisfying because, from (7) and (8), he diversiy order of he j-h deeced sream is r + j. I is inuiively pleasing ha a higher fracion of he available bis are loaded ino sreams wih higher diversiy orders. IV-B. BLAST Ordering As saed in Secion III-B, he SNR scaling facors produced by he BLAST ordering algorihm are dependen, and no closed-form expression is known for heir probabiliy disribuion and hence he ouage probabiliy. Insead, we sugges using he union bound in (6) o perform ransmier opimizaion. From (6), i is clear ha he ouage probabiliy is bounded by P UB ({ }, j) = Pr γ ji < ----. (0) i = Each erm in he summaion can be spli ino an average over he sage j i in which he i-h sream is decoded, giving Pr γ ji < ---- = Pr(j i = k)pr[γ k < ---- j i = k]. () k = Theorem saes ha j i is uniformly disribued over {, }, hence Pr( j i = k)=/. Also from Theorem, he SNR scaling facor is independen of j i, hus condiioning on j i does no change he disribuion of γ k. Thus, () simplifies o Pr γ ji < ---- = - F k, (2) ---- k = where we define he lef-coninuous cumulaive disribuion funcions by F k ( x ) = Pr[γ k < x] for k =, 2,,. Le F( x ) denoe he average of hese disribuion funcions over he symbols. From (2), Pr[ γ ji < / ] = F(/ ). Subsiuing in (0), we ge he union bound on he ouage probabiliy o be P UB-BLAST ({ }) = F. (3) i = ρ ---- i An analyical expression for F( x ) is no known in closed form, so even he simplified union bound (3) canno be evaluaed as is. However, he funcion F( x ) can be numerically esimaed as follows. A large number of Rayleigh fading marices are generaed, he BLAST algorihm is run for each one, and he resuling { γ i } are rounded off o preseleced bins. Averaging his discree approximaion of he disribuion funcion of { γ i }, we ge a discree approximaion o F( x ). Noe ha (3) is jus he sum of he same funcion evaluaed for each of he erms {/ }. This implies ha BLAST ordering reas all he daa sreams idenically i.e., if he daa raes and energies of wo sreams i and i are equal, hen hey make he same conribuion F(/ ) o he union bound (3). From his observaion, i is emping o conclude ha he minimizing soluion is o allocae idenical daa raes and energies, R/ and E/ respecively, o all he sreams. However, his conclusion is no valid because he funcion F( x ) is no necessarily convex. For example, consider a 4-inpu, 4-oupu MIMO sysem operaing a 8 b/s/hz a an SNR of 20 db. For his sysem, we numerically esimaed F( x ) and performed a random search for he opimum rae allocaion. The uniform allocaion yielded a union bound (3) equal o 3.4334 0 2. However, he opimum allocaion was found o be he parially uniform rae allocaion {0,0,4,4}, which disribues he rae uniformly over wo of he four ransmi anennas. This allocaion yielded a union bound of.0 0 3, which is significanly lower. Based on numerical experimens, we make he following conjecure abou he opimal allocaion. Conjecure. The union bound for he BLAST-ordered SC decoder is minimized by a parially uniform rae and energy (PURE) allocaion, wih K sreams carrying a daa rae of R/K and energy of E/K, and he remaining K daa sreams carrying zero daa rae and zero energy. Hence, numerical opimizaion reduces o finding he opimal number of acive sreams K {, 2,, }. The opimum value is ypically less han a high SNR, and i decreases wih increasing SNR. This can be explained inuiively by he fac he diversiy order is he key deerminan of error performance a high SNR, and reducing he number of acive inpus implies ha he diversiy gain of he SC decoded sysem is enhanced. IV-C. Rae-Normalized Ordering We now aim o find he opimum rae and energy allocaion a he ransmier when he receiver employs he RN ordering algorihm of Secion III-C. The analyical expression for ouage probabiliy wih RN ordering is inracable. Even he union bound is inracable, because he disribuions of Γ and j for he RN ordering algorihm depend on he rae allocaions. However, based on heurisic observaions, we make he following conjecure regarding he opimum daa rae allocaion. Conjecure 2. The opimum daa rae allocaion for he rae-alized ordering algorihm is eiher he opimum allocaion for he case of fixed ordering, or a parially uniform rae and energy (PURE) allocaion, where K inpus carry a rae of R/K each, and he res carry zero daa rae.
According o Conjecure 2, one can resric he search for he opimum daa rae allocaion o + possibiliies. Of paricular ineres in Conjecure 2 is he fac ha a parially uniform allocaion is ofen he opimum soluion for RN ordering. I was discussed in Secion IV-B ha a PURE allocaion is opimum for BLAST ordering. Noe ha, wih a PURE allocaion, RN ordering amouns exacly o BLAST ordering and for his reason, he PURE allocaion is expeced o be a good soluion for RN ordering. V. N UMERICAL RESULTS In his secion we presen numerical resuls for a 4 4 MIMO sysem operaing a R =8b/s/Hz, assuming independen Rayleigh fading. We firs consider a fixed ordering [6], and quanify he benefis of opimizing he rae and energy allocaions. A S = 20 db wih a fixed ordering, he opimal raes and energies are {0, 0, 3.63, 4.37}, and {0, 0, 0.49E, 0.5E}, respecively, which leads o an ouage probabiliy of 0.002422. In comparison, a uniform rae allocaion wih fixed ordering gives an ouage probabiliy of 0.20, abou fify imes larger. Fig. compares he error performance of hree ordering sraegies a R = 8 b/s/hz: he fixed ordering, he convenional BLAST ordering, and he rae-alized ordering. All hree sysems use he same rae allocaion {R i }={0,.3,2.99,3.70}, which is a good candidae for comparison since i minimizes he ouage probabiliy of he fixed-ordering decoder a S =5dB. I is seen from Fig. ha he rae-alized ordering ouperforms he opimized fixed ordered sysem by.5 db and BLAST ordering by 2 db a an ouage probabiliy of 0 3. Fig. 2 shows he advanage of opimizing he rae and energy allocaions a he ransmier. The lowes curve shows he error performance wih rae-alized ordering and opimized ransmier allocaions, while he nex-lowes curve shows he performance wih a fixed ordering and opimized allocaions. The rae allocaions for boh cases are calculaed anew a each SNR so as o minimize ouage probabiliy. Hence, hese curves represen he bes possible ouage probabiliy ha can be achieved. For example, he parially uniform allocaion wih K =3 is opimal for rae-alized ordering a S = 5 db, while he parially uniform allocaion wih K =2 is opimal for RN ordering a S =20dB and 25 db. Comparing he lower wo curves in Fig. 2, we see ha ransmier-opimized rae-alized ordering ouperforms ransmier-opimized fixed ordered receiver by.5 db a an ouage probabiliy of 0 3, underlining he imporance of join opimizaion of he ransmier and receiver. Also included in Fig. 2, for he sake of comparison, is he upper gray curve, which shows he performance of a convenional BLAST sysem wih uniform allocaions a he ransmier. I is seen ha he joinly opimal sysem ouperforms convenional BLAST by 5 db a an ouage probabiliy of 0 3. VI. CONCLUSIONS We sudied he BLAST sysem wih successive cancellaion decoding wih he objecive of minimizing he ouage probabiliy. We proposed he rae-alized ordered deecor and proved ha i minimizes he ouage probabiliy. We invesigaed he opimal rae and energy allocaions o minimize he ouage probabiliy for he differen varians of he SC decoder. We argued ha he parially uniform rae and UNIFORM ALLOCATION 0 0 TX-OPTIMIZED ALLOCATION BLAST ORDERING OUTAGE PROBABILITY 0 2 0 3 BLAST ORDERING RATE NORMALIZED ORDERING FIXED ORDERING OUTAGE PROBABILITY 0 2 0 3 0 4 FIXED ORDERING RATE-NORMALIZED ORDERING 0 4 0 5 20 25 SNR (db) Fig.. Comparison of ordering algorihms for {R i } = {0.00,.3, 2.98, 3.69} for a 4-inpu, 4-oupu Rayleigh fading channel a R =8 b/s/hz. 0 5 0 5 20 25 SNR (db) Fig. 2. Comparison of ransmier opimized SC decoders for a 4- inpu, 4-oupu Rayleigh fading channel a R =8 b/s/hz.
energy (PURE) allocaion is he minimizing soluion for BLAST-ordered deecion. In he case of he rae-alized ordered decoder, we proposed a rule for rae and power allocaion o minimize he ouage probabiliy based on parial analyical resuls. Simulaion resuls show ha joinly opimizing he ransmier and receiver, wihou any form of channel feedback, improves he performance of SC decoders significanly. The fac ha a PURE allocaion is ofen he opimum soluion wih RN ordering validaes he imporance of anenna selecion schemes wih SC decoding. Inuiively, anenna selecion wih SC decoding amouns o reducing he muliplexing gain of he MIMO sysem in order o improve he diversiy order. Our resuls show he imporance of achieving he righ rade-off in order o minimize he ouage probabiliy of he SC decoder. Appendix : Proof of Theorem The BLAST ordering algorihm can be viewed as a funcion Ψ(H) of he channel marix H, which oupus he pair (i, Γ). Lemma 6.. For a given channel marix H, suppose Ψ(H) = (i, Γ). Then, for all column permuaion marices, Π, Ψ(HΠ) = (Π Τ i, Γ). (4) Proof: Suppose he symbol x q corresponding o i = q was decoded in he firs sage wih channel H, he same symbol, re-labelled as i = q, where i = Π T i will be decoded in he firs sage wih he permued channel HΠ. Clearly, he value of he maximum SNR scaling facor remains unchanged for ha sage, since i corresponds o he same symbol. Similarly, proceeding hrough he sages k = {2, 3,, }, i is clear ha he SNR scaling facors remain invarian o he permuaion, and ha muliplying H by Π amouns o re-labelling he index of he symbols, as deermined by Π. I is well known ha permuing he columns of Rayleigh fading marices does no change heir disribuion. More precisely, he following resul holds. Lemma 6.2. Suppose H is a Rayleigh fading marix. Then, for all column permuaion marices Π, he random marix H = HΠ is idenical in disribuion o H, since H is circularly symmeric. From Lemma 6.2, HΠ is idenical in disribuion o H. Using Lemma 6., we arrive a he following corollary. Furher, from Corollary, noe ha join densiy funcion of (i, Γ) saisfies p(i, Γ) = p(π Τ i, Γ). Now, using Bayes rule and he fac i is uniformly disribued over! possibiliies, we obain he following expression for he join densiy funcion. p(i, Γ) = --- p(γ i) (5)! In paricular, p(i, Γ) = p(π T i, Γ) p(γ i) = p(γ Π T i) for all Π, implying ha Γ and i are independen. This proves he second claim of Theorem. VII. REFERENCES [] P. W. Wolniansky, G. J. Foschini, G. D. Golden, R. A. Valenzuela, V- BLAST: An Archiecure for Realizing Very High Daa Raes Over he Rich-Scaering Wireless Channel, Proc. ISSSE-98, Pisa, Ialy, Sep. 29, 998. [2] G. D. Golden, G. J. Foschini, R. A. Valenzuela, P. W. Wolniansky, Deecion Algorihm and Iniial Laboraory Resuls using he V- BLAST Space-Time Communicaion Archiecure, Elecronics Leers, Vol. 35, No., Jan. 7, 999, pp. 4-5. [3] S. T. Chung, A. Lozano and H. C. Huang, Approaching eigenmode BLAST channel capaciy using V-BLAST wih rae and power feedback, in Proc. VTC 200, Alanic Ciy, Oc. 200. [4] S. H. Nam and K. B. Lee, Transmi power allocaion for an exended V-BLAST sysem, Proc. IEEE PIMRC 2002, Sep. 2002, pp. 843-848. [5] R. W. Heah Jr., S. Sandhu, and A. J. Paulraj, Anenna Selecion for spaial muliplexing sysems wih linear receivers, IEEE Communicaion Leers, vol. 5, no. 4, pp 42-44, April 200. [6] N. Prasad and M. K. Varanasi, Ouage Analysis and Opimizaion for Muliaccess/V-BLAST Archiecure over MIMO Rayleigh Fading Channels, 4s Annual Alleron Conf. on Comm. Conrol, and Compu., Monicello, IL, Oc., 2003. [7] T. Guess, H. Zhang, and T. V. Kochiev, The ouage capaciy of BLAST for MIMO channels, in Proc. IEEE Inl. Conf. on Communicaions, Anchorage, Alaska, May 2003. [8] H. Zhang and T. Guess, Asympoical analysis of he ouage capaciy of rae-ailored BLAST, Proc. IEEE Globecom 2003. Vol. 4, Dec. 2003 pp. 797-80. [9] Ravi Narasimhan, Spaial Muliplexing wih ransmi anenna and consellaion selecion for correlaed MIMO fading channels, IEEE Transacions on Signal Processing, Special Issue on MIMO Wireless Communicaions, vol. 5, no., pp 2829-2838, Nov. 2003 [0] D. A. Gore, R. W. Heah Jr. and A. J. Paulraj, Saisical anenna selecion for spaial muliplexing sysems, IEEE In. Conf. On Communicaions, vol. pp 450-454, 2002. [] G. D. Forney Jr. and M. V. Eyuboglu, Combined Equalizaion and Coding using Precoding, IEEE Communicaions Magazine, Dec. 99. [2] R. J. Muirhead, Aspecs of Mulivariae Saisical Theory, John Wiley & Sons, 982. [3] J. R. Barry, E. A. Lee, D. G. Messerschmi, Digial Communicaion, Third Ediion, Kluwer Academic Publishers, 2004. Corollary. Ψ(HΠ) = (Π Τ i, Γ) is idenical in disribuion o Ψ(H) = (i, Γ). From Corollary, since Π Τ i is idenical in disribuion o i, we conclude ha i is uniformly disribued over all permuaions of [, 2,, ] T, as saed in Theorem.