Nonregenerative MIMO Relaying with Optimal Transmit Antenna Selection

Nonre generati v e MIMO Relaying with Optimal T ransmit Antenna Selecti on Ste ven W . Peters, Student Member , IEEE, and Robert W . Heath, Jr ., Senior Member , IEEE Abstract W e derive optimal SNR-based transmit antenna selection rules at the source a nd relay for the nonregene rativ e half duplex MIMO relay ch annel. While anten na selection is a subo ptimal f orm of beamfor ming, it has the advantage that the optimization is tractable and c an be implemented with only a fe w bits of feedback from the destination to the so urce and relay . W e comp are the b it error rate of optimal antenna selection at b oth the sourc e and relay to o ther p ropo sed beamf orming techn iques and propo se method s for performin g the n ecessary lim ited f eedback. I . I N T RO D U C T I O N Despite the lack of p recise knowledge of its basic theoretical beh avior and limits, relaying is beginning to find practical app lication in stan dards such as IEEE 802.16j [1]. By deploying relati vely inexpen siv e relays, service providers can reduce the number of base stations required to serve a given area, o r inc rease capac ity at the cell edge. Relaying resea rch efforts have also increase d rece ntly [2]–[7]. Ca pacity bounds for the full-duplex MIMO relay channe l were deriv ed in [2], [3]. The authors of [6] deri ve the optimal infinite-SNR div ersity-multiplexing tradeoff for the half duplex MIMO relay chan nel a nd find tha t a comp ress-and- forward strategy is optimal in this sense . Recently , practical strategies have bee n dev eloped for MIMO relaying. Both [ 4] a nd [7] de ri ve the mutual-i nformation-maximizing nonregenerati ve linea r relay for spatial multiplexing when the direct link is igno red. This letter derives the optimal transmit an tenna s election c riteria a t both source and relay ; i.e., all transmissions o ccur using the trans mit a ntenna tha t will gi ve the destination the highest post-proce ssing signal-to-noise ratio. W e consider the case whe re only a single spatial stream is to b e sent from s ource The authors are with the W ireless Netwo rking and Communications Group, Department of Electrical and Computer Engineer- ing, 1 Uni versity Station C0803, University of T exas at Austin, Austin, TX, 78712 -0240 (email: { peters, rheath } @ece.ute xas.edu, phone: (512) 471-1 190, fax: (512) 47 1-6512). EDICS: COM- { ESTI,MIMO,NETW } This work w as supp orted by the Semiconductor Research Corporation u nder contract 2007-HJ-1648 . October 9, 2018 DRAFT A CCEPTED TO IEE E SIGNAL PR OCESSING L ETTERS, J ANU AR Y 2008 2 to destination. This scenario arises when the channel is ill- conditioned (i.e., there is a do minant pa th of propagation in the source-des tination channe l), or if robustness via div ersity is preferred over throughp ut (i.e., near the c ell edge). Unlike mos t p revious practial MIMO relay results (e.g ., [4], [5], [7]) , the strate gy deri ved here is the optimal transmit antenna selection strategy when the direct link from source to relay is not ign ored. W e prove that tr ansmit antenna selection, combined with an MMSE recei ver at the d estination, ac hieves the full div ersity order of the MIMO single relay ch annel. That is, a t h igh SNR the probability of outag e decays with S NR a s quickly as is po ssible in s uch a model. Further , a ntenna selec tion req uires less feedback tha n bea mforming. Distrib uted spac e-time code s, which may also achieve the full diversity gain, not only require their own level of overhead for coordination and synch ronization, but also require the relay to be able to deco de the message transmitted by the source. Compared to rec ent results using limited feed back beamforming [8], unde r the tes ted parame ters given in the aforementioned paper , antenna selection a t both source and des tination is ab out twice a s likely t o cause bit errors as a Gra ssmannia n code book with 16 codes, which is a loss of about 1 dB a t h igh SNR. In return, antenna selec tion r equires only log 2 N S N R bits of feedbac k versus 3 log 2 N + 2 b bits in [8], where N S and N R are the number of an tennas at the source an d relay , resp ectiv ely , N is the size of the Grassmann ian codeboo k, and b is the qua ntization in bits of the SN R feedb ack req uired in [8]. This letter us es capital boldface letters to refer to matrices and lowercase boldf ace letters for column vectors. The notation k h k refers to the L2-no rm of the vector h , and H ∗ is the co mplex con jugate transpose of the ma trix H . The v ector h ( i ) refers to the i th column of the matrix H . Finally , A . = B ⇐ ⇒ lim SNR →∞ log A log SNR = − B . I I . S Y S T E M M O D E L & A N T E N N A S E L E C T I O N W e assume a single source S transmitting information to a destination D with a s ingle relay R aiding the transmiss ion. The sou rce, destination, and relay are equippe d with N S , N D , an d N R antennas , respectively . All no des operate i n ha lf-duplex mode. Unlike most p rior w ork in MIMO relaying, we do not ignor e the d ir ect link between S and D . The source S wishes to transmit the sc alar s ymbol s to D , whe re E | s | 2 = E s = SNR , E s is the av erage power constraint at bo th S and R , and σ 2 = 1 is the overall noise power at each nod e. Since the signa l-to-noise ra tio is the me tric of interest, an imbalan ce of noise ene r gy among the node s ca n be mode led in the appropriate fading parame ter for H X Y . For instance , if the relay has noise power October 9, 2018 DRAFT A CCEPTED TO IEE E SIGNAL PR OCESSING L ETTERS, J ANU AR Y 2008 3 σ 2 r , in an indep endent Rayleigh fading en vironment these definitions would chan ge the chan nel fading parameter o f the correspo nding exponential distribution from λ S R to λ S R σ 2 r . W e denote the chan nel from X to Y , X ∈ {S , R} , Y ∈ {R , D } , X 6 = Y , as H X Y , and h ( i ) X Y is the vector cha nnel from the i th transmit a ntenna at X to Y . W e also define γ ( i ) X Y =    h ( i ) X Y    2 SNR (1) to be the equi valent receiv e SNR fr om X i → Y . W e as sume the block fading mode l. In the first stage, if S trans mits s on antenna i , R receives the signal y R = h ( i ) S R s + n R , (2) where n R is the z ero-mean spatially wh ite complex Ga ussian no ise vector with covariance σ 2 I N R as observed by R . Since the relay is also transmitting on only on e of its a ntennas , it must comb ine its receiv ed vector to form a single symbol. It ca n be s hown that the optimal way to do this is to perform MRC on the signal, resulting in a scalar s R = α ( h ( i ) S R ) ∗ y R , (3) where α is the sca ling factor to ens ure R transmits at its expec ted power constraint; i.e., α 2 = 1 k h ( i ) S R k 4 + k h ( i ) S R k 2 / SNR . (4) At D , the first stage results in y D , 1 = h ( i ) S D s + n D , 1 . (5) In t he second stag e, R transmits s R to D on anten na k : y D , 2 = h ( k ) RD s R + n D , 2 . (6) The des tination now has two obse rvations containing s . T o p ut the c hanne l in standard MIMO notation, we de fine h =   h ( i ) S D k h ( i ) S R k h ( k ) RD √ k h ( i ) S R k 2 +1 / SNR   (7) n =   n D , 1 h ( k ) RD ( h ( i ) S R ) ∗ n R k h ( i ) S R k √ k h ( i ) S R k 2 +1 / SNR + n D , 2   (8) y D =   y D , 1 y D , 2   (9) October 9, 2018 DRAFT A CCEPTED TO IEE E SIGNAL PR OCESSING L ETTERS, J ANU AR Y 2008 4 γ ( i ) = γ ( i ) S D γ ( i ) S D ( γ ( i ) S R + 1) 2 + γ ( i ) S R γ ( k ) RD ( γ ( i ) S R + 1 + γ ( i ) S R + 1 + γ ( i ) S R γ ( k ) RD ) γ ( i ) S D ( γ ( i ) S R + 1) 2 + γ ( i ) S R γ ( k ) RD ( γ ( i ) S R + 1 + γ ( k ) RD ) ! (11) so tha t y D = h s + n . (10) W e assume the destination D now applies a linea r filter w to y D to obtain a n estimate of s . Although suboptimal, we w ill s ee later that in so me cas es the destination may wish to apply MRC ( w = h ) on y D , and doing s o would res ult in the po st-processing signal-to-noise ratio γ ( i ) of (11) at the top of the page. In this for m, it is easy to see that, if n γ ( i ) S R < γ ( i ) S D o \ n γ ( k ) RD > γ ( i ) S D ( γ ( i ) S R + 1) / ( γ ( i ) S D − γ ( i ) S R ) o , (12) then γ ( i ) < γ ( i ) S D and relaying is worsening performanc e. This occurs when the SNR from R to D is very good relati ve to the others, a nd the SN R from S to R is worse than the d irect SNR. Effecti vely , the R to D channe l is do minating the rec eiv ed signal, b ut it consists of mos tly noise relati ve to the direct signal. Recall that MRC is only optimal when the observations contain the same noise variance [9]. Because o f the a mplified noise at R , this is not the cas e here. In this case , one can show that the op timal receive filter in the minimum mean-squ ared error (MMSE) se nse is w = R − 1 y D R y D s , (13) where R y D = E { y D y ∗ D } a nd R y D s = E { y D s ∗ } . The pos t-processing SNR is then γ ( i ) = γ ( i ) S D + γ ( i ) S R γ ( k ) RD γ ( i ) S R + γ ( k ) RD + 1 . (14) Note that this requires the destination to have kno wledge of k h S R k . If this is n ot pos sible, suboptimal MRC resulting in the SNR of (11 ) may be us ed ins tead, whic h req uires less training. A me thod for obtaining t his CSI is presented in Section II I. Note that in (14), for fixed γ ( i ) S D and γ ( i ) S R , γ ( i ) is maximized wh en γ ( k ) RD is maximized. Thus, the antenna selection at the relay is independen t of the selec tion at the sou rce, and we can substitute the index of the optimal relay transmit antenna k o in for k in all subs equen t equations. The same cannot be said of the regular MRC equation (11). Finally , we note that antenna selection at the relay is su boptimal, and the optimal strategy in this case is intuiti ve; since the SNR exp ression (14) is the addition of the independ ent SNR terms for the p arallel October 9, 2018 DRAFT A CCEPTED TO IEE E SIGNAL PR OCESSING L ETTERS, J ANU AR Y 2008 5 channe ls to the de stination from the source, the relay should a pply a filter that ma ximizes the S NR to the destination. One can show that this filter is W = v (1) ( h ( i o ) S R ) ∗ , whe re v (1) is the right singular vector of H RD correspond ing to its largest singular value, a nd i o is the index of the source antenn a tha t maximizes (14). I ntuiti vely , W is the combination of a receiv e filter matched to H S R and a transmit beamforming vector matche d to H RD . Impl ementing this filter would require perfect knowledge of H RD at the relay and an SVD operation. All of ou r results hold w ith this optimal s trategy , with k h ( k o ) RD k 2 replaced with λ RD = σ 2 RD , the squ are of the largest singu lar value of H RD . I I I . T R A I N I N G A N D L I M I T E D F E E D BA C K W e now discus s how cha nnel s tate information might be obtained in the cha nnel of interest so tha t a reliable antenn a s election s trategy ma y be implemented. All three cha nnels ne ed to be estimated at their res pectiv e receivers; this can b e a ccomplished using previously studied MIMO training methods. Only knowledge of the link SNRs (i.e., γ ( i ) X Y ’ s) is required for transmit anten na se lection. T herefore a low comp lexity s ignal, such a s a s hort narrowband tone, may be us ed for es timating SNR to choo se an a ntenna to train from. This is first sen t from R to D from each relay an tenna. D then fee ds b ack which anten na R should use to transmit, and, from this antenn a, a training sequenc e su itable for channel estimation is s ent to the destination. The source repeats this p rocess with its transmit antenna s, with the relay forwarding its rece iv ed signal on its optimal anten na. This way , the destination can estimate the SNR between the s ource and relay to perform MMSE comb ination a s de scribed e arlier . The de stination finds (14) for ea ch source a ntenna, then fee ds ba ck to the source the index of the antenna that resulted in the lar gest γ ( i ) . The source then transmits a training se quence from this an tenna, which does not ne ed to be forwarded by the relay . Th is process requires log ( N S N D ) bits of feed back, two time slots of training, and N R + 2 N S time s lots for SNR estimation. Minimizing the time required for SNR estimation is thus important for this feedb ack s trategy . I V . D I V E R S I T Y A N A L Y S I S Antenna s election is use d to exploit the di versity ga in av ailable in the c hanne l. Using (14 ) we now show that this strategy achieves full di versity ga in. W e fi rst give a n upper bou nd on the d i versity order of the h alf-duplex MIMO relay channe l when the source an d des tination transmit orthogonally in eq ual time slots. Y uksel and Erkip [6] hav e derived this result for a rbitrary time sha ring whe n the sou rce is allowed to trans mit in the second time slot, so this result is a specia l case of the ir deriv ation. This deriv ation is included here to prove that our added restrictions (i.e., equal transmission times, source silent in the October 9, 2018 DRAFT A CCEPTED TO IEE E SIGNAL PR OCESSING L ETTERS, J ANU AR Y 2008 6 second ti me slot) do not de crease the maximum diversity order of the channel. W e first defi ne I B C = I ( S ; Y R , Y D , 1 ) (15) I 1 = I ( S ; Y D , 1 ) (16) I 2 = I ( S R ; Y D , 2 ) (17) I M AC = I 1 + I 2 , (18) where S is the rando m variable co rresponding to the transmitted signal from the sourc e, Y R is the receiv ed signal at the relay , Y D ,n is the rec eiv ed signal at the destination in the n th time slot, and S R is the transmitted signal a t the relay . Using equations (27) and (28) in [6] with t = 0 . 5 a nd the source not transmitting in the second time slot, I ( S ; Y D ) ≤ 0 . 5 min { I B C , I M AC } . (19) Now we can bound the probability of outage for a fixed I 0 as P out = Pr { I ( S ; Y D ) < I 0 } ≥ Pr { 0 . 5 min { I B C , I M AC } < I 0 } . (20) The event where the minimum of two variables is less tha n a constan t is equiv alent to the union of the ev ents that each of the v ariables is less than the constan t. Defining P out,B C = Pr ( I B C < 2 I 0 ) , and similarly for P out,M AC , we can wri te P out ≥ Pr  { I B C < 2 I 0 } [ { I M AC < 2 I 0 }  (21) = P out,B C + P out,M AC − Pr  { I B C < 2 I 0 } \ { I M AC < 2 I 0 }  . (22) Recall from (18) that I M AC is the sum of two nonnegative ran dom vari ables. Suc h a sum is a lw ays less than o r equa l to twice the ma ximum of the two random variables. Then, by making the co debook for S R independ ent from that of S , and defining P out,I 1 = Pr( I 1 < I 0 ) a nd P out,I 2 similarly , P out ≥ P out,B C + Pr { max { I 1 , I 2 } < I 0 } − Pr  { I B C < 2 I 0 } \ { I M AC < 2 I 0 }  (23) ≥ P out,B C + P out,I 1 P out,I 2 − Pr  { I B C < 2 I 0 } \ { I M AC < 2 I 0 }  . (24) October 9, 2018 DRAFT A CCEPTED TO IEE E SIGNAL PR OCESSING L ETTERS, J ANU AR Y 2008 7 Con versely , the sum of I 1 and I 2 is al ways g reater than the ma ximum of the tw o. Also, note fr om (15) and (16) that I B C ≥ I 1 so tha t P out ≥ P out,B C + P out,I 1 P out,I 2 − Pr  { I B C < 2 I 0 } \ { max { I 1 , I 2 } < 2 I 0 }  = P out,B C + P out,I 1 P out,I 2 − Pr { max { I B C , I 2 } < 2 I 0 } . (25) Finally , again ass uming indepe ndent chann els on all links, P out ≥ P out,B C + P out,I 1 P out,I 2 − P out,B C Pr { I 2 < 2 I 0 } . (26) From MI MO information the ory we know that (se e [6], Sec. III and IV) Pr { I B C < c } . = N S ( N R + N D ) (27) Pr { I 1 < c } . = N S N D (28) Pr { I 2 < c } . = N R N D , (29) for all c ∈ R . Thus, the last t erm in (26) will decay as N S N R + N S N D + N R N D with log SNR and is thus irrelev ant to the diversity analysis. The first term will decay as N S ( N R + N D ) , while the sec ond term d ecays as N D ( N S + N R ) , s o tha t P out ˙ ≤ N S N D + N R min { N S , N D } . (30) W e now d eri ve a lower bound on the diversit y order of optimal a ntenna s election in fla t i.i.d. Rayleigh fading by using (14). First we define γ ( i,k o ) S RD = γ ( i ) S R γ ( k o ) RD / ( γ ( i ) S R + γ ( k o ) RD + 1) . (31) Since we choose the source transmit anten na that maximizes the SNR γ at the destination, P out = Pr { γ < γ 0 } = Pr { max i { γ ( i ) } < γ 0 } = Pr { max i { γ ( i ) S D + γ ( i,k o ) S RD } < γ 0 } . (32) October 9, 2018 DRAFT A CCEPTED TO IEE E SIGNAL PR OCESSING L ETTERS, J ANU AR Y 2008 8 As be fore, the sum of two rand om variables i s greater than the max imum of the two. P out ≤ Pr { max i { max { γ ( i ) S D , γ ( i,k o ) S RD }} < γ 0 } = Pr { max i { γ ( i ) S D , γ ( i,k o ) S RD } < γ 0 } . (33) Since each cha nnel is mutually ind epende nt of the others, and the channel from each source anten na to the destination is also indepen dent from the others , we define P out,S D = Pr( γ (1) S D < γ 0 ) , thu s P out ≤ Pr { max i { γ ( i ) S D } < γ 0 } Pr { max i { γ ( i,k o ) S RD } < γ 0 } = ( P out,S D ) N S Pr { max i { γ ( i,k o ) S RD } < γ 0 } . (34) Now define γ M ,i = min { γ ( i ) S R , γ ( k o ) RD } . (35) If γ M ,i ≥ 1 , then γ ( i,k o ) S RD > γ M ,i / 3 . Otherwise, γ ( i,k o ) S RD > ( γ M ,i ) 2 / 3 . In either c ase, since γ 0 is arbitrary , we let γ 0 > 1 / 3 and proceed 1 P out < ( P out,S D ) N S Pr { max i { γ M ,i } < 3 γ 0 } . (36) W e can a gain split up the minimum e vent into a union: P out < ( P out,S D ) N S × Pr( { max i { γ ( i ) S R } < 3 γ 0 } [ { max i { γ ( k o ) RD } < 3 γ 0 } ) = ( P out,S D ) N S ×  Pr { max i { γ ( i ) S R } < 3 γ 0 } + Pr { γ ( k o ) RD < 3 γ 0 } − Pr { max i { γ ( i ) S R } < 3 γ 0 } Pr { γ ( k o ) RD < 3 γ 0 }  . (37) Again, since the channels between each source transmit antenn a and the re lay a re indepen dent, we de fine P out,S R = Pr( γ (1) S R < 3 γ 0 ) and P out,RD = Pr( γ ( k o ) RD < 3 γ 0 ) , a nd P out < ( P out,S D ) N S ×  ( P out,S R ) N S + P out,RD − ( P out,S R ) N S P out,RD  , (38) 1 Since P out is monotone i ncreasing with increasing γ 0 , no loss in generality occurs by assuming γ 0 > 1 / 3 . For example, let γ L < 1 / 3 . Then Pr( γ < γ L ) < Pr( γ < γ 0 ) . Thus, if P out . = d , then Pr( γ < γ L ) ˙ ≥ d . October 9, 2018 DRAFT A CCEPTED TO IEE E SIGNAL PR OCESSING L ETTERS, J ANU AR Y 2008 9 where aga in the last term will decay muc h quicker than the othe rs and c an be ignored. The first term, after multipli cation, will decay as N S N D + N S N R , while the secon d term decays as N S N D + N R N D . Thus, P out ˙ ≥ N S N D + N R min { N S , N D } . (39) Combining (39) and (30) we se e that the proposed antenna selec tion achiev es the full di versity gain in the channel. V . S I M U L A T I O N R E S U LT S W e pres ent a simple s imulation to co mpare to a rec ent res ult on limited feedbac k beamforming [8]. For each case shown, we simulate the relay ch annel with N S = N R = N D = 3 using BPSK mod ulation and an i.i.d. Rayleigh c hanne l a t eac h link. Bit error rate (BER) is the metr ic of interest. Figu re 2 gi ves the res ults for E { γ ( i ) S R } = E { γ ( k ) RD } = 2 dB for various E { γ ( i ) S D } . No te that this graph corresponds exactly to Fig. 9 in [8], and we ha ve included their results for a Grass mannian code book with more than 2 0 bits of feedb ack. Using antenna se lection at both S an d R requires 4 bits in this c ase and resu lts in a los s of approximately 1 dB at high SNR. The theoretical lower bou nd o f Figu re 2 is when the source can simultaneou sly beamform the BPSK symbols to both the relay and destination; obviously this i s an imposs ible task. The “optimal” performance curve was foun d numerically in [8] using gradient desc ent to find a local optimum. Figure 3 shows the BER of unc oded BPSK versu s E S / N 0 for a relay c hanne l with two anten nas at each node . Note that increasing E S / N 0 implies an increase in SNR a t each link (recall that n oise terms are normalized and E | s | 2 = E S = SNR ). The figure was g enerated using Monte Carlo simulations using 10 8 channe l realizations for a ccuracy at high SNR, and demons trates that antenna selection achieves the maximum d i versity order av ailable in the chan nel. V I . C O N C L U S I O N W e explored antenna se lection as a practical way of achieving the full di versity order of the nonregen- erati ve MIMO relay c hannel. It was shown to achieve this div ersity with a small SNR p enalty relativ e to Gr ass mannian code books. R E F E R E N C E S [1] Air interface for fixed and mobile br oadband w ir eless access systems—Mobile r elay specification , IEE E Std. 802.16j, 2007. October 9, 2018 DRAFT A CCEPTED TO IEE E SIGNAL PR OCESSING L ETTERS, J ANU AR Y 2008 10 [2] B. W ang, J. Zhang, and A. Host-Madsen, “On the capacity of MIMO relay channels, ” IEEE T ransa ctions on Information Theory , vol. 51, no. 1, pp. 29–43, Jan. 2005. [3] C. K. Lo, S. V ishwa nath, and R. W . Heath, Jr ., “Rate bounds for MIMO relay channels using precoding, ” in G lobal T elecommunications Conferen ce, 2005. GLOBECOM ’05. IEEE , vol. 3, Nov ./Dec. 2005. [4] O. Munoz-Medina, J. Vid al, and A. Agustin, “Linear transceiv er design i n nonreg enerativ e relays with channel state information, ” IEEE Tr ansactions on Signal Pro cessing , vol. 55, pp. 2593–260 4, June 2007. [5] Y . Fan and J. T hompson, “MIMO configuration s for r elay channels: T heory and practice, ” IEEE T ran sactions on W ir eless Communications , vol. 6, no. 5, pp. 1774–1786 , May 2007. [6] M. Y uksel and E . Er kip, “Multiple-antenna cooperati ve wireless sys tems: A div ersity–multiplexin g tradeof f pe rspectiv e, ” IEEE Tr ansactions on Information T heory , vol. 53, no. 10, pp. 3371–3393, Oct. 2007. [7] X. T ang and Y . Hua, “Optimal design of non-reg enerativ e MIMO wi reless relays, ” IEEE Tr ansactions on W ir eless Communications , vol. 6, no. 4, pp. 1398–1407 , Apr . 2007. [8] B. Khoshnev is, W . Y u, and R. Adve, “Grassmannian beamforming for MIMO amplify-and-forw ard relaying, ” preprint; av ailable at http://arxiv . org/abs/07 10.5758, Oct. 2007. [9] A. Goldsmith, W ir eless Communications . Cambridge, UK: Cambridge Press, 2005. Fig. 1. The system model used i n this letter. The source t ransmits in the first time sl ot, and the relay transmits in the second time slot. The relay is shown with separate t ransmit and receiv e antennas for con venience; this assumption is not made in the analysis. −10 −8 −6 −4 −2 0 2 10 −6 10 −5 10 −4 10 −3 10 −2 Direct link SNR (dB) BER Theoretical bound Antenna selection at source, antenna selection at relay Antenna selection at source, optimal BF at relay "Optimal" scheme from Khoshnevis et al. Grassmannian Beamforming with 16 vectors Fig. 2. BER performance for se veral MIMO amplify-and-forward beamforming str ategies. October 9, 2018 DRAFT A CCEPTED TO IEE E SIGNAL PR OCESSING L ETTERS, J ANU AR Y 2008 11 0 2 4 6 8 10 12 14 10 −10 10 −9 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 E S /N 0 (dB) P e Slope = −8 Fig. 3. BE R performance for uncoded BPS K versus E S / N 0 for i.i. d. Rayleigh fading. At high SNR, the slope of the curve approaches − N S ( N R + N D ) = − N D ( N S + N R ) = − 8 , which , as shown in Section IV , is the full div ersity ord er of t he channel. October 9, 2018 DRAFT