Optimum Receive Antenna Selection Minimizing Error Probability

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1 Optimum Receive Antenna Selection Minimizing Error Probability Sang Wu Kim Eun Yong Kim Abstract The optimum receive antenna selection combining rule that minimizes the bit error probability is presented. It is derived from a general relationship between the bit error probability and the log-likelihood ratio (LLR), and selects the receive antenna providing the largest LLR magnitude. The optimum selection combining rule is applied to single transmit (Tx) antenna and space-time block code (STBC) systems, with N R receive (Rx) antennas, and the bit error probability is derived for BPSK signaling in Rayleigh flat fading channels. A suboptimum selection combining rule based on noncoherent envelope detection is also presented and analyzed. For STBC systems, the optimum generalized selection combining in which M( N R) Rx antennas providing the largest LLR magnitude are selected and combined is presented. Index Terms Selection combining, log-likelihood ratio, diversity, space-time block code, multiple antenna, Rayleigh fading channel. I. INTRODUCTION The radio link is characterized by multi-path fading which causes the link quality to vary with time, frequency, and space. Diversity techniques are known to be very effective in mitigating multi-path fading, and multiple-transmit multiplereceive antennas are known to provide an increased diversity order [],[2]. However, the implementation of such multipletransmit multiple-receive antenna systems requires multiple RF chains and A/D converters, much of the hardware and power consumption lies. A promising approach for reducing hardware complexity and power consumption, while retaining a reasonably high performance, is to employ some form of selection combining [],[4],[5]. In this paper, we present the optimum receive antenna selection combining rule that minimizes the bit error probability in interference-limited systems. The optimum selection combining rule, derived from a general relationship between the bit error probability and the log-likelihood ratio (LLR), is to select the antenna providing the largest LLR magnitude. We apply this rule to a single transmit (Tx) antenna system and space-time block code (STBC) system [6],[7], with N R receive (Rx) antennas, and derive the bit error probability for BPSK signaling in Rayleigh flat fading channels. Also, we present a suboptimum selection combining rule based on a noncoherent envelope detection. We compare the bit error probabilities of optimum and suboptimum selection combining rules with those of traditional combining rules such as signal-to-interference ratio (SIR)-based selection combining and maximum ratio combining (MRC) [],[8]. We find that Sang Wu Kim sangkim@ieee.org Eun Yong Kim eykim@bada.kaist.ac.kr the optimum selection combining provides a gain of -.5dB over the SIR-based selection combining, and is only.-.6db away from MRC that requires multiple RF chains. The envelope-based suboptimum selection combining provides a gain of.8-2.db over SIR-based selection combining, when selecting one out of four Rx antennas. For STBC systems, we present the optimum and suboptimum generalized selection combining rules in which M( N R ) Rx antennas providing the largest LLR magnitude are selected and combined. The remainder of this paper is organized as follows. In Section II, we present the system model. In Section III, we present a general relationship between the bit error probability and LLR, and present the optimum selection combining rule that minimizes the bit error probability. In Section IV, we consider a wireless channel with one Tx antenna and N R Rx antennas, and derive the bit error probability with optimum and suboptimum antenna selection combining rules. In Section V, we consider a STBC system with two Tx antennas and N R Rx antennas and derive the bit error probability with optimum and suboptimum antenna selection rules. In Section VI, we consider a STBC system with two Tx antennas and N R Rx antennas and present optimum and suboptimum generalized selection combining rules. In Section VII, numerical results are provided along with discussions. In Section VIII, conclusion is made. II. SYSTEM MODEL In this paper, we consider the reverse link (mobile to base) of a digital mobile radio communication system. The system is assumed to be interference-limited. The base station consists of an N R element antenna array, and the antenna element is selected on a symbol-by-symbol basis. The modulation is chosen to be binary phase-shift-keying (BPSK) with coherent detection in two channel models. In Section IV, we consider a wireless communication system with one Tx antenna and N R Rx antennas (Fig.). The received low-pass equivalent signal y i,giventheith Rx antenna selected, is K y i h i x + h k,i x k () k h i and h k,i are channel gains between the Tx antenna and the ith Rx antenna for the desired and the kth interfering users, respectively, and K is the number of interfering users. We assume that h i and h k,i are Rayleigh distributed with E{ h i 2 } E{ h i,k 2 } and that the phases of h i and h k,i are uniformly distributed over [,2π]. x and x k, representing the desired and the kth interfering signals, respectively, are //$7. (C) 2 IEEE 44

2 + E s or E s with probability /2, E s is the transmit energy per symbol. We assume that the interference term in (), denoted hereafter as n i, has a Gaussian distribution with mean zero and variance I /2 per dimension. The Gaussian assumption follows from the Cramer s central limit theorem []. In Sections V and VI, we consider a space-time block code (STBC) system with two transmit antennas and N R receive antennas (Fig.2). The received low-pass equivalent signals r i and r 2i from the ith antenna at time t and t + T, respectively, are given by [6] r i h i x + h 2i x 2 + n i (2) r 2i h i x 2 + h 2i x + n 2i () h ji is the channel gain between jth Tx antenna and ith Rx antenna, and n i and n 2i, i, 2,..., N are i.i.d. Gaussian random variables with mean zero and variance I /2 per dimension. x and x 2 are + E s /2 or E s /2 with probability /2. Then, the combiner outputs y i and y 2i are given by y i h ir i + h 2i r 2i (4) ( h i 2 + h 2i 2 )x + h in i + h 2i n 2i (5) y 2i h 2ir i h i r 2i (6) ( h i 2 + h 2i 2 )x 2 h i n 2i + h 2in i. (7) We assume that the channel is quasi-static flat fading, i.e. the channel gains {h i } and {h ji } remain constant over several symbols. We also assume that the channel gains are i.i.d., and are known at the receiver. III. OPTIMUM SELECTION COMBINING Consider a channel the channel output y i,giventhe ith Rx antenna is selected, is given by y i h i x + n i, (8) h i is the channel gain between the Tx antenna and ith Rx antenna, x is + E s or E s with probability /2, and n i is a white Gaussian noise with mean zero and variance I /2 per dimension. Then, the log-likelihood ratio (LLR) Λ i for x is given by Λ i ln P (x + E s h i,y i ) P (x E s h i,y i ) (9) 4 E s Re{h i y i }. () I }{{} h i 2 x+re{h i ni} The sign of Λ i is the hard decision value, and the magnitude of Λ i represents the reliability of hard decision. In general, the bit error probability P e,i and the LLR Λ i is related by P e,i +e Λi () A derivation of () is provided in Appendix I. Since P e,i decreases with increasing Λ i, the optimum receive antenna selection combining rule that minimizes the bit error probability is to select the antenna providing the largest Λ i. Then, the minimum average bit error probability is [ ] P e,opt E Λmax (2) +e Λmax Λ max max Λ i. () i N R It should be noted that averaging Λ i in () over n i and taking the absolute value yields 4 h i 2 E s /I, which is the SIR at diversity branch i. Therefore, selecting the antenna providing the largest SIR or equivalently the largest h i 2 (assuming that the average interference power is a constant across all antennas) takes the average interference power into account. However, the optimum selection combining rule exploits the interference term Re{h i n i} in () by selecting the antenna for which signs of x and Re{h i n i} are identical and h i is large, thereby providing the largest LLR magnitude. As a result, the performance is governed by the peak, as opposed to the mean, channel condition. The bit error probability resulting from the optimum selection combining provides a lower bound on the bit error probability of any selection combining rule. In contrast with the maximum ratio combining (MRC) that yields N R i Λ i, the optimum selection combining selects the best component among N R components used in MRC. In what follows, we apply the optimum selection combining rule to two wireless communication systems. IV. SINGLE TX ANTENNA We consider a wireless communication system with one Tx antenna and N R Rx antennas (Fig.). The channel output given the ith antenna is selected is given by y i h i x + n i. (4) Then, it follows from () that the LLR, given the ith Rx antenna selected, is Λ i 4 E s [ h i 2 x + Re{h i n i }]. I (5) If we let Es Z i h i 2 x + Re{h i n i } I (6) then the pdf of Z i can be shown to be (derived in Appendix II) f Zi (z) +γ +γ [e ] 2( +)z + e 2( )z γ2 + γ (7) γ E[ h i 2 ]E s /I (8) Ēs/I (9) is the average received signal-to-interference ratio (SIR) per diversity branch. The cdf of Z i is then given by F Zi (z) f Zi (u)du (2) e 2( +γ +)z 2(γ ++ γ 2 + γ) + e 2( +γ )z 2(γ + γ 2 + γ).(2) 442

3 If we let Z max max {Z i } (22) i N R and assume that Z,Z 2,..., Z NR are i.i.d., then the pdf of Z max is given by f Zmax (z) N R [F Zi (z)] NR f Zi (z). (2) Therefore, it follows from (2) and (2) and the relationship Λ max 4Z max that the bit error probability with the optimum antenna selection is given by P e,opt +e 4z f Z max (z)dz. (24) Envelope-based Selection In this subsection we present a simple suboptimum selection combining rule based on noncoherent envelope detection. First, we note that Re{h i y i } h i y i (25) h i y i. (26) Since noncoherent envelope detection of the received RF signal yields y i, we propose a suboptimum selection combining rule that selects the antenna element providing the maximum of { h y, h 2 y 2,..., h NR y NR }. It will be called envelope-based selection combining. If we let { h () y (), h (2) y (2),..., h (NR) y (NR) } be an ordered set such that h () y () h (2) y (2)... h (NR) y (NR), then the log-likelihood ratio Λ env for deciding x with envelope-based selection combining is Λ env 4 E s Re{h () I y ()}. (27) We decide that Es was transmitted if Λ env >, and otherwise, decide E s was transmitted. Then, it follows from () and (27) that the average bit error probability with the envelope-based selection combining is given by [ ] P e,env E Λenv. (28) +e Λenv SIR-based Selection Since the SIR, given that the ith antenna is selected, is h i 2 E s /I, the antenna providing the largest SIR is the one providing the largest h i 2.Ifwelet G max max { h i 2 } (29) i N R then the bit error probability, when the antenna providing the largest SIR is selected, is given by P e,snr Q( 2gE s /I )f Gmax (g)dg () f Gmax (g) is the pdf of G max given by f Gmax (g) N R ( e g ) NR e g, () assuming that { h i 2 } are i.i.d. with pdf f hi 2(g) e g. Maximum Ratio Combining The bit error probability with maximum ratio combining of N R independent paths is given by [8] N R P e,mrc [( µ)/2] NR k ( NR +k k ) [( + µ)/2] k γ µ +γ and γ is the average received SIR per diversity branch. (2) () V. SPACE-TIME BLOCK CODE In this section, we consider a space-time block code (STBC) with two Tx antennas and N R Rx antennas (Fig.2). The baseband combiner output y i for data x,giventheith Rx antenna is selected, is y i ( h i 2 + h 2i 2 )x + η i (4) η i h in i + h 2i n 2i (5) is a conditional Gaussian random variable with mean zero and variance ( h i 2 + h 2i 2 )I /2 per dimension. Then, for agiven{h ji }, y i is a Gaussian random variable with mean ( h i 2 + h 2i 2 )x and variance ( h i 2 + h 2i 2 )I /2 per dimension. Therefore, the log-likelihood ratio (LLR) Λ i for data x,given{h ji } and y i,is Λ i ln P (x + E s /2 {h ji },y i ) P (x (6) E s /2 {h ji },y i ) 4 E s /2 Re{y i } (7) I }{{} A ix +Re{η i} 2 A i h ji 2. (8) j If we let 2Es Z i ( h i 2 + h 2i 2 )x +Re{h I in i +h 2i n 2i} (9) then the pdf of Z i can be shown to be (derived in Appendix III) f Zi (z) ( + z +2/γ) [e z( +2/γ+) + e z( +2/γ ) ]. (4) If we assume that Z,Z 2,..., Z NR are i.i.d., then the pdf of Z max max i NR {Z i } is given by f Zmax (z) N R [F Zi (z)] NR f Zi (z), (4) F Zi (z) is the cdf of Z i. Therefore, it follows from (2) and (4) and the relationship Λ max 2Z max that the bit error The LLR for other information symbols is exactly the same. 44

4 probability with the optimum antenna selection rule is given by P e,opt +e 2z f Z max (z)dz. (42) Envelope-based Selection In this subsection we present a simple suboptimum selection combining rule based on noncoherent envelope detection. We note that Λ i 4 E s /2 I Re{y i } (4) 4 E s /2 I y i. (44) Since combining the received RF signals and detecting the envelope of the combined signal yields y i and y 2i,we propose a suboptimum selection combining rule that selects the antenna element providing the maximum of { y i },i, 2,..., N R. It will be called envelope-based selection combining. Then, the log-likelihood ratio Λ env for deciding x with envelope-based selection combining is Λ env 4 E s /2 I Re{y () } (45) y () is the baseband combiner output associated with the antenna element providing the maximum of { y i },i, 2,..., N R. We decide that E s was transmitted if Λ env >, and otherwise, decide E s was transmitted. Then, it follows from () and (45) that the average bit error probability with envelope-based selection combining is [ ] P e,env E Λenv. (46) +e Λenv SIR-based Selection The SIR, given the ith Rx antenna selected, is A i E s /(2I ). Therefore, the antenna providing the largest SIR is the one providing the largest A i.ifwelet A max max {A i }, (47) i N R then the bit error probability with SIR-based selection is given by P e,snr Q( 2aE s /(2I ))f Amax (a)da, (48) f Amax (a) is the pdf of A max given by f Amax (a) N R [ e a a l ] NR ae a (49) assuming that {A i } s are i.i.d. with f Ai (a) ae a,a. Maximum Ratio Combining STBC with two Tx antennas and N R Rx antennas provides a diversity order of 2N R. Therefore, the bit error probability with MRC at the receiver is given by (2) with N R replaced by 2N R and γ replaced by γ/2 because E s /2 is the energy per transmit antenna. l VI. OPTIMUM GENERALIZED SELECTION COMBINING In this section we consider the optimum generalized selection combining (GSC) for STBC systems. The optimum GSC for single Tx antenna case is analyzed in []. We consider selecting M out of N R Rx antennas and combining those M signals. The baseband combiner output y i,i 2,,i M for data x, given Rx antennas i,i 2,,i M are selected, is y i,i 2,,i M y im m [ ( him 2 + h 2im 2 ] )x + η im (5) m η im h i m n im + h 2im n 2i m. (5) Then, for a given {h ji }, y i,i 2,,i M is a Gaussian random variable with mean M m ( h i m 2 + h 2im 2 )x and variance M m ( h i m 2 + h 2im 2 )I /2 per dimension. Therefore, the LLR Λ i,i 2,,i M for x,given{h ji }, y i,i 2,,i M,is Λ i,i 2,,i M ln P (x + E s /2 {h ji },y i,i 2,,i M ) P (x E s /2 {h ji },y i,i 2,,i M ) (52) 4 E s I Re{y i,i 2,,i M } (5) 4 E s I [A i,i 2,,i M x + Re{η i + η i η im }] (54) A i,i 2,,i M [ h im 2 + h 2im 2 ]. (55) m It follows from () that the optimum selection combining rule that minimizes the bit error probability is to select those antennas that provide the largest Λ i,i 2,,i M. It can be shown that the bit error probability with the optimum GSC of M N R 2 antennas is identical to that of MRC, since the sign of the N R 2 optimum GSC output is the same as that of MRC. Envelope-based GSC In this subsection we present a simple suboptimum generalized selection combining rule based on noncoherent envelope detection. We note that Λ i,i 2,,i M 4 E s Re{y i,i I 2,,i M } (56) 4 E s Re{y im } I m (57) 4 E s y im. I (58) m Since combining the received RF signals and detecting the envelope of the combined signal yields y im, we propose a suboptimum generalized selection combining rule that selects M Rx antennas maximizing M m y i m. It will be called envelope-based GSC. 444

5 SIR-based GSC Given the Rx. antenna i,i 2,,i M are selected, the SIR is proportional to A i,i 2,,i M M m [ h i m 2 + h 2im 2 ].The conventional GSC rule selects M Rx antennas providing the largest A i,i 2,,i M. It will be called SIR-based GSC. VII. NUMERICAL RESULTS AND DISCUSSION In this section, we present numerical results and discussions. Figure is a plot of average bit error probability versus Ēs/I with single transmit antenna and N R Rx antennas, Ē s is the average received energy per diversity branch and is equal to E s since we have assumed E [ h i 2].The performance curves are computed assuming an independent identically distributed (i.i.d.) slow Rayleigh fading model. We find that the diversity order (slope) depends on the number of receive antennas. For N R 4, the optimum selection combining (OSC) provides a gain of db over the SIR-based selection combining (SIR-SC), and is only.db away from MRC that requires multiple (four) RF chains. The envelope-based selection provides a gain of.8db over SIR-SC. For N R 2, the error probability of MRC can be achieved by OSC. This is because the sign of MRC output is identical to that of LLR-based selection combiner output. We can also notice that MRC of four channels (requiring four RF chains, N R 4) is outperformed by envelope-based combining of one channel (requiring one RF chain) out of eight antennas (N R 8). Figure 4 is a plot of average bit error probability versus Ē s /I with space-time block code employing two Tx antennas and N R Rx antennas. For N R 4, the OSC provides a gain of.5db over the SIR-SC, and is.6db away from MRC that requires four RF chains. The envelope-based selection combining provides a gain of 2.dB over SIR-SC. For N R 2, the error probability of MRC can be achieved by the optimum selection combining, as in single Tx antenna case. Figure 5 and 6 are plots of average bit error probability versus E s /I with generalized selection combining of M Rx antennas and STBC employing two Tx antennas and N R Rx antennas. We find that the optimum GSC and Env-GSC provide power gains of.6.db and.9.6db over SIR-GSC, respectively, for M 2and N R 4 8. Wealso find that the diversity order (slope) depends on N R, and the SIR gain increases with increasing M. VIII. CONCLUSION In this paper we presented the optimum receive antenna selection combining rule that minimizes the bit error probability. The optimum selection combining rule, derived from a general relationship between the bit error probability and the log-likelihood ratio (LLR), is to select the antenna providing the largest LLR magnitude. We applied the optimum selection combining rule to single transmit (Tx) antenna and spacetime block code (STBC) systems with N R Rx antennas, and derived the average bit error probability for BPSK signaling in a Rayleigh flat fading channel. Also, we presented a suboptimum selection combining rule based on noncoherent envelope detection and derived its average bit error probability. We compared the average bit error probabilities of optimum and suboptimum selection combining rules with those of traditional combining rules such as SIR-based selection combining and maximum ratio combining (MRC). For N R 4, the optimum selection combining provides a gain of -.5dB over the SIR-based selection combining, and is only.-.6db away from MRC that requires multiple (four) RF chains. The envelope-based selection combining provides a gain of.8-2.db over the SIR-based selection combining. The power gain increases with increasing N R. For STBC systems, we present the optimum and suboptimum generalized selection combining rules in which M( N R ) Rx antennas providing the largest LLR magnitude are selected and combined. APPENDIX I In this appendix, we show that the bit error probability, P e (R), with MAP (optimum) detection for a received observation R can be expressed P e (R) (59) +e Λ(R) P (x + R) Λ(R) ln (6) P (x R) is the log-likelihood ratio (LLR). Moreover, the relationship in (59) is true for any binary signals in any channel. Proof: It follows from (6) and P (x + R)+P (x R) that P (x + R) (6) +e Λ(R) and By definition, P (x R). (62) +eλ(r) P e (R)P (ˆx x R) (6) P (ˆx,x R)+P (ˆx,x R)(64) ˆx is the detector output. If Λ(R) >, i.e. ˆx, then Also, P e (R) P (x R)+P (ˆx R) }{{} (65) P (Λ(R)<). +eλ(r) (66) P e (R) P (ˆx x R) (67) P (ˆx,x R) P (ˆx,x R) (68) P (x R) P (ˆx R) }{{} (69) P (Λ(R)<). +eλ(r) (7) Therefore, if Λ(R) >, then P e (R). (7) +eλ(r) 445

6 If Λ(R) <, we can similarly show that As a result, P e (R) P e (R). (72) +e Λ(R). (7) +e Λ(R) APPENDIX II In this Appendix, we derive (7). Let Es Y i ( h i 2 x + Re{h i n i }). (74) I Since the pdf of Z i Y i is the same whether x E s or x E s, we will assume x E s without loss of generality. Then, given h i, Y i is a Gaussian random variable with mean h i 2 γ and variance h i 2 γ/2, γ E s /I. Since the pdf of h i is Rayleigh, i.e. f hi (a) 2ae a2,the pdf of Y i is f Yi (y) πa2 γ e (y a2 γ) 2 /(a 2 γ) 2ae a2 da(75) γ( + γ) e 2y 2 y +γ (76) we used the equality [9] The cdf of Z i is F Zi (z) e ax2 b/x 2 dx 2 z Therefore, the pdf of Z i is π ab a e 2. (77) γ( + γ) e 2y 2 y +γ dy (78) ( e 2( +γ +)z ) 2(γ ++ γ 2 + γ) + ( e 2( +γ )z ) 2(γ + γ 2 + γ). (79) f Zi (z) df Z i (z) (8) dz ] +γ [e 2( +)z + e 2( +γ )z.(8) γ2 + γ APPENDIX III In this Appendix, we derive (4). Let 2Es Y i [( h i 2 + h 2i 2 )x +Re{h I in i +h 2i n 2i}] (82) Since the pdf of Z i Y i is the same whether x + E s /2 or x E s /2, we will assume x + E s /2. Then, given h i and h 2i, Y i is a Gaussian random variable with mean A i E s /I and variance A i E s /I, A i h i 2 + h 2i 2. Since the pdf of A i, f Ai (a), isae a, the pdf of Y i is f Yi (y) e y 2πγa e (y γa)2 /(2γa) f Ai (a)da (8) 2πγ ae y2 2γ a (γ/2+)a da (84) γ E s /I.Ifweletb a then 2e y f Yi (y) b 2 e y2 2γ b πγ 2 (+γ/2)b2 db (85) + y +2/γ e y +2/γ+y (86) we use the equality [9] x 2 e a/x2 bx 2 dx π 4 b ( + 2 ab)e 2 ab. (87) The cdf of Z i, F Zi (z), is F Zi (z) + + z z f Yi (y)dy (88) ( y +2/γ)e y +2/γ+y dy ( + y +2/γ)e y +2/γ+y dy (89) [ ] 2c + [2c ++c(c +)z]e (c+)z [(c + )] 2 [ 2c [2c +c(c )z]e (c )z [(c )] 2 ](9) c +2/γ. Therefore, the pdf of Z i is given by f Zi (z) df Z i (z) dz ( + z +2/γ) (9) [e z( +2/γ+) + e z( +2/γ ) ]. (92) REFERENCES [] W.C.Jakes, Microwave Mobile Communications, IEEE Press, 974. [2] G.J.Foschini and M.J.Gans, On the limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Communications, vol.6, pp.-5, 998. [] D.Gore and A.Paulraj, Space-time block coding with optimal antenna selection, Proc. of IEEE Inter. Conf. on Acoustics, Speech, and Signal Processing pp , 2. [4] R.W.Heath, S.Sandhu, and A.Paulraj, Antenna selection for spatial multiplexing systems with linear receivers, IEEE Communications Letters pp.42-44, April 2. [5] A.F.Molisch, M.Z.Win, and J.H.Winters, Capacity of MIMO systems with antenna selection, Proc. of IEEE ICC, pp , 2. [6] S. M. Alamouti, A simple transmit diversity technique for wireless communications, IEEE Jounal on Selected Areas in Communications, Vol. 6, No. 8, pp , October 998. [7] V.Tarokh, H.Jafarkhani, and A.R.Calderbank, Space-time block code from orthogonal designs, IEEE Tr. on Information Theory, Vol. 45, pp , JUly 999. [8] J.Proakis, Digital Communications, pp.78, rd Ed., McGraw-Hill, 98. [9] I.S.Gradshteyn and I.M.Ryzhik, Tables of Integrals, Series, and Products, pp.4, Eq..472, Academic Press, 98. [] H.Cramer, Random Variables and Probability Distributions, Cambridge Univ. Press, 97. [] S.W.Kim, Y.G.Kim, and M.Simon, Generalized selection combining basesd on the log-likelihood ratio, submitted to IEEE Tr. on Communications. 446

7 Fig.. Channel model for one Tx and N R Rx antennas. Fig. 4. Bit error probability versus Ēs/I with2txandn R Rx: Space-time block code. Fig. 2. Space-time block code systems with two Tx antennas and N R Rx antennas. Fig. 5. Bit error probability versus Ēs/I with2txandn R Rx: Space-time block code, GSC, M2 Fig.. Bit error probability versus Ēs/I withtxandn R Rx. Fig. 6. Bit error probability versus Ēs/I with2txandn R Rx: Space-time block code, Opt.-GSC 447