Joint Transmitter-Reciever Optimization for Multiple Input Multiple Output (MIMO) Systems

Transcription

1 Join Transmier-Reciever Opimizaion for Muliple Inpu Muliple Oupu (MIMO Sysems eun Chul WANG and wang Bo (Ed LEE School of Elecrical Engineering, Seoul Naional Universiy, OREA Absrac Muliple ransmi (Tx and muliple receive (Rx anennas sysems, referred o as muliple inpu muliple oupu (MIMO sysems, have been proposed o achieve higher daa raes in wireless communicaion sysems. In his paper, we invesigae join opimizaion of ransmier and receiver for MIMO sysem when he channel informaion is available a boh ransmier and receiver. We discuss he problem of designing Tx and Rx filer based on minimum bi error probabiliy crierion. We firs derive he opimum Tx and Rx filer for he special case of wo inpu daa symbols. Based on his opimum filer, we propose a Tx and Rx filer opimizaion for he general case of inpu daa symbols. The performance of he proposed Tx and Rx filer is compared wih ha of he convenional minimum mean-squared error (MMSE filer. Performance analysis shows ha he proposed filer provides significan improvemen over he MMSE filer in bi error probabiliy. I. INTRODUCTION Muliple ransmi (Tx and muliple receive (Rx anennas, referred o as muliple inpu muliple oupu (MIMO, are nown o significanly improve he capaciy of wireless communicaion sysems []-[4]. In paricular, i is well nown ha he achievable capaciy of MIMO lins increases almos linearly wih he number of anennas in rich scaering environmens. Due o his poenially significan capaciy improvemen, MIMO sysems are promising for high daa rae service of he nex generaion wireless communicaion sysems. A MIMO sysem can be classified according o wheher or no he channel informaion is available a eiher he ransmier or he receiver. When he channel informaion is nown a boh ransmier and receiver, a join ransmierreceiver opimizaion may be used o improve he performance of MIMO sysems, and may offer a beer performance compared o ransmier or receiver opimizaion only. Since a MIMO sysem is an inerference-limied sysem, several join ransmier-receiver opimizaion echniques using he Tx and Rx filer have been proposed. In [5], he opimum Tx and Rx filer ha minimizes he mean-squared error was derived for a sricly band-limied sysem, and a more complee analysis based on a frequency domain analysis appeared in [6] and [7]. In [8], an opimum Tx and Rx filer was derived using he minimum mean-squared error (MMSE crierion, wih which he sum of mean-squared errors beween he inpu daa symbols and he esimaed daa symbols is minimized. In [9], he auhors generalized heir previous wor [8], and proposed he Tx and Rx filer ha minimizes he weighed sum of mean-squared errors. All he wors in [5]-[9] have been conduced based on he minimum mean-squared error (MMSE crierion. owever, since he MMSE crierion is no direcly relaed o bi error probabiliy, i does no ensure he bes bi error probabiliy (BEP performance. Recenly, in [0], a minimum bi error probabiliy (BEP crierion for a single inpu single oupu (SISO sysem is invesigaed, and was shown o yield significan performance improvemen over he convenional MMSE crierion. In his paper, we invesigae he join opimizaion of Tx and Rx filer for a muliple inpu muliple oupu (MIMO sysem based on a minimum BEP crierion insead of MMSE crierion. We derive he opimum Tx and Rx filer ha minimizes he bi error probabiliy for he special case of wo inpu daa symbols. Based on his opimum filer, we propose a Tx and Rx filer opimizaion for he general case of inpu daa symbols. II. MULTIPLE INPUT MULTIPLE OUTPUT (MIMO SYSTEM A muliple inpu muliple oupu (MIMO sysem wih N T ransmi (Tx anennas and N R receive (Rx anennas is considered. A he ransmier, muliple daa symbols are consruced using he same modulaion scheme, and passed hrough he Tx filer. These filered daa symbols are simulaneously ransmied hrough N T ransmi anennas, and received a N R receive anennas. A he receiver, each received signal is corruped by an addiive whie Gaussian noise, and processed wih he Rx filer. A baseband equivalen MIMO sysem model is shown in Figure. The ransmied signal may be expressed in vecor form as x = PTd ( where x = [x x x N T ] T denoes he ransmied signal vecor, and he superscrip T denoes he ranspose. d =[d d d ] T denoes he daa symbol vecor, where denoes he number of daa symbols, which is assumed o be =min{n T, N R }. Alhough he proposed opimizaion can be applied o any modulaion, we assume for analyical simpliciy ha d s are recangular M-ary quadraure ampliude modulaed (M- QAM daa symbols wih E[dd ] = I, where E[ ] is he expecaion operaion, he superscrip denoes he conjugae ranspose, and I n denoes he n n ideniy marix. In his paper, we consider an equal power ransmission, and he ransmi power is assumed o be he same as P for all daa symbols. Noe ha he oal Tx power is P. ATxfilerT consiss of Tx weigh vecors, and may be expressed as T =[ ]where =[,, NT, ] T denoes a Tx weigh vecor for he daa symbol d. Noe ha as a consequence of applying he Tx weigh vecor, he ransmi power may increase or decrease. To mainain he ransmi power, we normalize he Tx weigh vecor such ha N T i, =. ( i= We assume ha he ransmied signal experiences frequency-fla Rayleigh fading for all N T N R ransmi-receive anenna pairs. Each channel response is assumed o vary slowly enough o be consan over daa symbol duraion. The channel responses may be inegraed o a channel marix as =[h i,j ]whereh i,j denoes he channel response from he jh Tx anenna o he ih Rx anenna. The channel responses h i,j s

2 are assumed o be independen and idenically disribued (i.i.d. zero-mean circular complex Gaussian random variables wih uni variance. The channel marix is assumedobeperfeclynownabohheransmierand receiver. The received signal vecor y =[y y y N R ] T may be expressed as y = x+ n (3 where n =[n n n N R ] T is an addiive whie Gaussian noise (AWGN vecor, whose elemens are i.i.d. zero-mean circular complex Gaussian random variables wih variance of σ 0. A he receiver, he received signals pass hrough he Rx filer R, which consiss of Rx weigh vecors: R =[r r r ],wherer is an N R receive weigh vecor for he daa symbol d. We assume ha decision order is in accordance wih daa index; d is firsly decided, and hen d is secondly decided, and so on. A successive inerference cancellaion is employed o remove he inerference; when we decide d,he inerference from he - decided symbols is cancelled ou. For analyical simpliciy, we assume ha he - decisions used in cancellaion are error free. Thus, he decision variable for d may be expressed as ˆ d = Pr d + Pr i id = + i + r n (4 where he firs erm denoes a desired signal, and he second and hird erms are he inerference from he undecided symbols and AWGN, respecively. III. MINIMUM BEP TX AND RX FILTER OPTIMIZATION In his secion, we invesigae a join Tx and Rx filer opimizaion. We adop he minimum bi error probabiliy as an opimizaion crierion. A join Tx and Rx filer opimizaion problem is formulaed in subsecion III-A. The Rx filer opimizaion is considered in subsecion III-B where he Rx filer is expressed in erms of Tx filer. Using his Rx filer expression, an opimum Rx filer may be obained afer finding he opimum Tx filer. In subsecion III-C, hetx filer opimizaion is considered, and an opimum Tx filer is derived for he special case of =. A. Problem Formulaion Based on he cenral limi heorem [], he inerference can be approximaed as a Gaussian disribued random variable for large. Using his Gaussian approximaion, he bi error probabiliy (BEP may be wrien as [] 4 ( M 3 BEP = BEP Q ( M (5 = = log M where Q(x is a Gaussian ail inegral [], and BEP and, respecively, denoe he bi error probabiliy and he signal o noise plus inerference raio ( for he h symbol d.from(4, may be calculaed as r ( ( r = (6 r Φr where F denoes he noise plus inerference power given as Φ = IN + ( ( R i i (7 Γ i= + and Γ P σ 0 denoes he ransmi signal-o-noise raio (TxSNR. A join Tx and Rx filer opimizaion considered in his paper may be expressed as ( Top, R op = argmin{bep} (8 T, R where T op and R op denoe he opimum Tx and Rx filer, respecively. B. Receive Filer Opimizaion In his subsecion, we consider he Rx filer opimizaion. I may be seen from (6 ha one receive weigh vecor affecs only one ; r affecs only. Thus, o minimize he BEP in (5, he receive weigh vecor r should maximize. According o he generalized eigenvalue problem [3], he receive weigh vecor r ha maximizes in (6 is found as r = µ Φ (9 where µ is an arbirary consan ha does no affec he.forsimpliciy,weseµ =forall. Subsiuing (9 ino (6, he may be expressed as i i = Φ =Γ( Φ i= + + i i i ( Φ (0 where he second equaliy is derived using he relaion i i i i Φ =ΓIN R Φ Φ. ( i= + + ( i Φi i Noe ha in (0 is expressed in erms of he Tx weigh vecors raher han boh Tx and Rx weigh vecors. Accordingly, BEP in (5 may be considered as a funcion of he Tx weigh vecors. Consequenly, he opimizaion in (8 may be accomplished by finding he Tx weigh vecors ha minimize he BEP cos funcion 3 J(,,, = ( = Q M where he muliplying facor ( M ( M ( 4 log in (5 is dropped since i is independen of he Tx weigh vecors. Afer finding he opimum s ha minimize (, he corresponding opimum r s can be obained from (9. Before going o he nex secion, le us define he eigenvalue decomposiion of [4]: Λ 0 V = V V = VΛV (3 0 0 V where V=[v v v ]isann T uniary marix, and v s are eigenvecors which form he bases for he range space of, and V= v v v N T is an N T (N T - uniary marix whose column vecors form he bases for he null space of.themarixλ =diag{λ, λ,, λ }isa diagonal marix, and λ denoes he h larges eigenvalue of : λ λ λ. Noe ha v is an eigenvecor corresponding o he eigenvalue λ. Noe ha when =, here are wo non-zero eigenvalues λ and λ, whose corresponding eigenvecors are v and v, respecively. C. Transmi Filer Opimizaion When = When =, here are wo Tx weigh vecors and o be opimized, and he cos funcion in ( is given as

3 (, = ( + ( M M J Q Q. (4 Noe ha is a funcion of boh and, whereas is a funcion of only. ence, for a given, may be considered as a fixed value, whereas is a funcion of. Consequenly, for a given, he Tx weigh vecor should maximize he o minimize he cos funcion J(,. According o he eigenvalue problem [3], for a given, he Tx weigh vecor ha maximizes may be expressed as ( = evec max A (5 where evec max [X] denoes he eigenvecor associaed wih he maximum eigenvalue of X, andhemarixa(x is defined as Γ( xx A( x. (6 +Γ ( x x Subsiuing = evec max [A( ], he becomes a funcion of, and may be expressed as ( = Γ λ max [A( ], where λ max [X] denoes he maximum eigenvalue of X. Obviously, he is a funcion of : ( =Γ (. Thus, he opimizaion problem in (4 may be simplified o finding ha minimizes he cos funcion ( ( M M ( ( J( = Q + Q. (7 Nowwederiveanopimum ha minimizes (7. As shown in he following Theorem, an opimum may be expressed using v and v. Theorem : When =, an opimum ha minimizes (7 lies on he range space of,andmaybeexpressedas = qv+ qv,whereqis a real number in [0, ] and v and v are he eigenvecors corresponding o wo non-zero eigenvalues of. Proof: See [7] When = qv + qv,he ( and ( may be expressed as a funcion of q [7]: Γ ( q = {( λ+ λψ( q ( λq+ λ( q (8 + λ+ λ ψ( q λq+ λ ( q 4λλ ψ( q λq+ λ( q ( ( ( ( ( λ λ ( q =Γ q+ ( q (9 where ψ(q in (8 is given as ψ(q = Γ { +Γ ( λq + λ( q }. Since and are expressed in erms of q insead of, he cos funcion in (7 may be changed as ( M ( M Jq ( = Q ( q + Q ( q. (0 Since (q is a monoonically decreasing funcion and (q is a linearly increasing funcion, he exisence of a global minimum of J(q is obvious. To find q ha minimizes J(q, several numerical mehods, such as Golden Secion search or parabolic inerpolaion [5], may be used. Le he value of q ha minimizes J(q beq op,andhenheopimum may be expressed as = q v + q v. ( ( op op op From (5, he corresponding opimum is given as ( op = evec ( op max ( A. ( The opimum receive weigh vecors are obained from (9 by replacing and wih ( op and ( op, respecively: ( op ( op r = Φ, (3 ( op ( op = r Φ. (4 where and in F and F are also replaced wih ( op and ( op, respecively. IV. PROPOSED TX ANDRX FILTEROPTIMIZATION In his secion, we exend he resuls of Secion III o he case when he number of daa symbols is more han, and propose a new Tx and Rx filer opimizaion. The opimum Rx filer may be obained from (9 afer finding an opimum Tx filer. ence he Tx and Rx filer opimizaion may be accomplished by finding he Tx filer. In his secion, we find he Tx filer using he following observaion from Secion III: When =, wo opimum Tx weigh vecors lie on he -dimensional range space of. Based on his observaion, one may guess ha if we resric he pair of wo Tx weigh vecors o lie in he -dimensional range subspace of, hen we may obain he opimized Tx filer. Consider he pairs of wo Tx weigh vecors +m and -m, m =0,,, (-/, and he pairs of wo eigenvecors v +m and v -m, m =0,,, (-/, where x denoes he larges ineger smaller han or equal o x. We resric he wo Tx weigh vecors +m and -m o be in he -dimensional range subspace of spanned by v +m and v -m. The reason for choosing v +m and v -m as he basis of he -dimensional range subspace for +m and -m is o mae he s similar o one anoher. The eigenvecor v -m corresponds o he (m+-h smalles eigenvalue, and v +m corresponds o he (m+-h larges eigenvalue. Thus, by choosing v +m and v -m, he effec of he (m+-h smalles eigenvalue on he s may be compensaed by ha of he (m+-h larges eigenvalue, and he s become similar o one anoher. For he case of an odd, he remaining one Tx weigh (+/ is assumed o be in he -dimensional range subspace of spanned by v (+/. A vecor in he -dimensional space spanned by v +m and v -m may be expressed as a linear combinaion of v +m and v -m. Thus, he wo Tx weigh vecors +m and -m may be expressed as + m = c + mv+ m + cmv m (5 m = c + mv+ m + c mv m, (6 where c +m, c -m, c' +m and c' -m are complex coefficiens wih he consrains c +m + c -m = and c' +m + c' -m =. Since v s are he eigenvecors of ha are orhogonal o one anoher, i holds ha = 0 if ' and + ' +. This implies ha wo daa symbols whose Tx weighs are orhogonal o each oher, does no inerfere wih each oher. ence, from (0, he +m and -m are simplified o Γ ( + mm mmm m m + m + m=γ( + mmm + m, (7 +Γ ( m m m m

4 ( =Γ (8 m m m m m where m is a projecion of ono he -dimensional range subspace spanned by v +m and v -m : λ + m 0 v + m m m = [ v + m vm] 0 λ. (9 mvm Noe ha +m and -m are deermined by only wo Tx weigh vecors +m and -m. ence, he cos funcion J(,,, in ( may be simplified as J(,,, = ( J ( + m, m,wherej( +m, -m is given as m= 0 ( M ( M J(, = Q + Q. (30 + m m + m m Consequenly, he minimizaion of he cos funcion J(,,, may be accomplished by minimizing he cos funcions J( +m, -m s. Observe he similariy beween (4 and (30. ence, following he procedure in Secion III-C, we may obain he proposed Tx weigh vecors as = v v, (3 ( op ( op ( op m qm + m + qm m op ( ( op ( + m = evec max A m m (3 ( where q op m in (3 denoes q ha minimizes (0 when λ and λ in (q and (q are replaced wih λ +m and λ -m, respecively. Similarly wih A(x in(6,a m (x in (3 is defined as ( A x Γ ( m mxx m m +Γ ( x m mx m m m. (33 Wih he proposed Tx weigh vecors given in (3 and (3, he corresponding Rx weigh vecors may be obained by (9. Noe ha, when =, he proposed Tx and Rx weighs vecors correspond o he opimum Tx and Rx weigh vecors in (-(4. V. NUMERICAL RESULTS In his Secion, we presen he performance of he proposed filer. For comparison, we also presen he performance of he MMSE filer. All resuls shown in his Secion are obained analyically hrough Mone Carlo inegraion [5] based on 0 6 independen realizaions of he channel marix. Since Average BEP when (N T, N R =(a, b is he same as ha when (N T, N R =(b, a, we consider only he case of N T N R,in which he number of inpu symbols is equal o N R. Moreover, we consider only 4-QAM (M =4. Figure shows he Average BEP s of he proposed filer and he MMSE filer when N T =andn R =. In his figure, ABEP denoes he average BEP for he h symbol d. Noe ha for he MMSE filer, he ABEP is much worse han ABEP, and dominaes he Average BEP. The reason is ha when MMSE filer is applied, is much smaller han [8]. Unlie he MMSE filer, wo s of proposed filer become he same for high TxSNR. ence, for he proposed filer, he ABEP becomes he same wih he ABEP as TxSNR increases. Moreover, he proposed filer is designed o compensae he (m+-h smalles eigenvalue wih he (m+-h larges one. ence, he proposed filer ouperforms he MMSE filer. Figure 3 depics he ABEP s of he proposed filer and he MMSE filer when N T =4andN R = 4. Lie Figure, his figure shows ha he proposed filer ouperforms he MMSE filer. Noe ha he ABEP of he MMSE filer is limied by ABEP 4, which is associaed wih he smalles. Noe also ha ABEP is almos he same wih ABEP 4,jusasABEP and ABEP 3. A sligh difference beween he pair of ABEP and ABEP 4, and ha of ABEP and ABEP 3 is due o he inheren differences in he eigenvalues. Figure 4 depics he ABEP s of he proposed and MMSE filers for various N T.AsN T increases, he slopes of ABEP curves for boh he proposed and MMSE filers become seeper due o he increase of diversiy gain. As N T doubles, he ABEP curves for boh he proposed and MMSE filers shif oward lef more han 3 db due o he increases of beamforming and diversiy gains. These observaions confirm ha boh he proposed and MMSE filers achieve he beamforming gain and he diversiy gain. Much more, he proposed filer is shown o provide significan improvemen han he MMSE filer. For example, he proposed filer provides 6.0 db of TxSNR gain over he MMSE filer a ABEP of 0-6 when N T = 4, and.8 db gain when N T =8. Noe ha as N T increases, he performance of he MMSE filer approaches ha of he proposed filer. The reason is ha, when N R is fixed, he eigenvalues of become he same as N T increases, and he s of proposed filer become he same wih hose of MMSE filer. Figure 5 shows he effecs of N T and N R on he performance of he proposed and MMSE filers when N T = N R.Noeha, lie Figure 4, he performance of he proposed filer is improved as boh N T and N R increase. Whereas, comparing he resuls in Figure 4, he performance improvemen of he MMSE filer is no significan. Especially, he slopes of ABEP curves for he MMSE filer remain consan regardless of N T and N R. The reason is ha when N T = N R, he smalles eigenvalue λ is much smaller han he ohers: λ << λ i, i =,,, -. In his case, he smalles, which limis he performance of he MMSE filer, may be expressed as (MMSE Γ λ.whenn T = N R =, he disribuion of λ is nown as he chi-square disribuion wih degrees of freedom for all [6]. ence, he slopes of ABEP curves for he MMSE filer become he same when N T = N R.InFigure5, we also plo he simulaion resuls for he proposed filer in which he effecs of error propagaion are included. Even hough he effecs of error propagaion are ignored in analysis resuls, he analysis and simulaion resuls show a close agreemen. The reason is ha, for he proposed filer, wo daa symbols whose Tx weigh vecors lie in differen - dimensional subspaces do no inerfere wih each oher. ence he effecs of error propagaion are negligible for he proposed filer. VI. CONCLUSIONS In his paper, we have invesigaed he join opimizaion of ransmi (Tx and receive (Rx filer for a muliple inpu muliple oupu (MIMO sysem using minimum bi error probabiliy crierion. For he case of wo inpu symbols, we have derived he opimum Tx and Rx filer which minimizes he bi error probabiliy. Based on his opimum filer, we have proposed a Tx and Rx filer for arbirary number of daa symbols. The performance of he proposed filer has been compared wih he MMSE filer. I was found ha he proposed filer provides a significan improvemen over he MMSE filer in bi error probabiliy. For example, he

5 proposed scheme provides 6.0 db TxSNR gain a a 0-6 BEP when 4-QAM is used, he number of Tx anennas N T is 4, and he number of Rx anennas N R is. REFERENCES [] E. Telaar, Capaciy of muli-anenna Gaussian channels, AT&T Bell Labs Tech. Memo., Mar [] G. J. Foschini and M. J. Gans, On limis of wireless communicaions in a fading environmen when using muliple anennas, Wireless Personal Commun., vol. 6, no. 3, pp , Mar [3] G. J. Foschini, G. D. Golden, R. A. Valenzuela, and P. W. Wolniansy, Simplified processing for high specral efficiency wireless communicaion employing muli-elemen arrays, IEEE J. Selec. Areas Commun., vol. 7, pp , Nov [4] V. Taroh, N. Seshadri, and A. R. Calderban, Space-ime codes for high daa rae wirelss communicaion: Performance crierion and code consrucion, IEEE Trans. Inform. Theory, vol. 44, pp , Mar [5] J. Salz, Digial ransmission over cross-coupled linear channels, AT&T Technical Journal, vol. 64, pp , July 985. [6] J. Yang and S. Roy, On join ransmier and receiver opimizaion for muli-inpu-muli-oupu (MIMO ransmission sysems, IEEE Trans. Commun., vol. 4, pp. 3-33, Dec [7] A. Scaglione, G. B. Giannais, and S. Barbarossa, Redundan filerban precoder and equalizer par I: Unificaion and opiamal designs, IEEE Trans. Signal Processing, vol. 47, pp , July 999. [8]. Sampah and A. Paulraj, Join ransmi and receive opimizaion for high daa rae wireless communicaions using muliple anennas, in Proc. Asilomar Conf. Signals, Sysems and Compuers, vol., pp. 5-9, 999. [9]. Sampah, P. Soica, and A. Paulraj, Generalized precoder and decoder design for MIMO channels using he weighed MMSE crierion, IEEE Trans. Commun., vol. 49, pp , Dec. 00. [0]C.-C. Yeh and J. R. Barry, Adapive minimum bi-error rae equalizaion for binary signaling, IEEE Trans. Commun., vol. 48, pp. 6-35, July 000. []J. G. Proais, Digial Communicaions, 3rd ed. New Yor: McGraw-ill, 995. []M.. Simon, S. M. inedi, and W. C. Lindsey, Digial Communicaion Techniques, Englewood Cliffs, NJ: Prenice all, 995. [3]R. A. Monzingo and T. W. Miller, Inroducion o Adapive Arrays,New Yor: John Wiley & Sons, 980. [4]G. Srang, Linear Algebra and Is Applicaions, 3rd ed. For Worh: arcour Brace & Company, 988. [5]W. Press e al., Numerical Recipes in C. nd ed. Cambridge, U..: Cambridge Univ. Press, 99. [6]A. Edelman, Eigenvalues and condiion numbers of random marices, SIAM Journal on Marix Analysis and Applicaions, vol. 9, pp , Oc [7]. C. wang, and. B. Lee, Join ransmier-receiver opimizaion for muliple inpu muliple oupu (MIMO sysems, in preparaion. d x FIGURES y ˆd Average Bi Error Probabiliy MMSE (ABEP MMSE (ABEP MMSE (ABEP Proposed (ABEP Proposed (ABEP Proposed (ABEP TxSNR (Γ indb Fig.. Average bi error probabiliy when N T =andn R =( =4. Average Bi Error Probabiliy MMSE (ABEP MMSE (ABEP MMSE (ABEP MMSE (ABEP 3 MMSE (ABEP 4 Proposed (ABEP Proposed (ABEP Proposed (ABEP Proposed (ABEP 3 Proposed (ABEP TxSNR (Γ indb Fig. 3. Average bi error probabiliy when N T =4andN R =4( =4. Average Bi Error Probabiliy MMSE Proposed (N T,N R = (3, (N T,N R = (6, (N T,N R =(, (N T,N R =(4, (N T,N R =(8, TxSNR (Γ indb Fig. 4. Average bi error probabiliy for various N T when M =4andN R =. d d T Fig.. Sysem Model. x n x NT n y n n n y NR n NR R wih Inerference Cancellaion dˆ dˆ Average Bi Error Probabiliy (N T,N R = (3,3 MMSE (Analysis Proposed (Analysis Proposed (Simulaion (N T,N R =(8,8 (N T,N R = (6,6 (N T,N R =(, (N T,N R =(4,4 (N T,N R =(8,8 (N T,N R = (6,6 (N T,N R = (3,3 (N T,N R =(4,4 (N T,N R =(, TxSNR (Γ indb Fig. 5. Average bi error probabiliy for various N T and N R when N T = N R.