RESEARCH Open Access Average bit error probability for the l-MRC detector under Rayleigh fading Mitchell Omar Calderon Inga and Gustavo Fraidenraich * Abstract In this paper, an exact expression for the average bit error probability was obtained for the l-MRC detector, proposed in Sendonaris et al. (IEEE Trans Commun 51: 1927-1938, IEEE Trans. Commun 51: 1939-1948), under Rayleigh fading channel. In addition, a very accurate approximation was obtained to calculate the average bit error probability for any power allocation scheme. Our expressions allow to investigate the possible gains and situations where cooperation can be beneficial. Keywords: User cooperation, Virtual MIMO, Bit error probability, Rayleigh fading I. Introduction Diversity t echniques have been widely accepted as one of effective ways of combat multipath fading in w ireless communications [1], in particular spatial diversity is spe- cially effective at mitigating these multipath situation. However, in many wireless applications, the use of mul- tiple antennas is not practi cal due to size and cost lim- itations of the termi nals. One possible way to have diversity without increasing the number of antennas is through the use of cooperative diversity. Cooperative diversity has root in classical information the ory work on relay channels [2], [3]. Cooperative net- works achieve diversity gain by allowing the users to cooperate, and thus, each wirelessuserisassumedto transmit data as well as act as a cooperative agent for another user [4], [5]. The first implementation strategy for cooperation was introduced in [1], [6], where the achievable rate region, outage probability, and coverage area were analyzed. In this pioneering work, assumin g a suboptimal recei- ver called l-MRC, the bit e rror probability was com- puted assuming a fixed channel. This kind of receiver combines the signal from the first period of transmis- sion with the signal transmitted jointly by the both users in the second period of transmission. The variable lÎ[0,1] establishes the degree of confidence in the bits estimate d by the partner. For situations where th e inter- user channel presents favorable condi tio ns, the variable l should be close to unity; on the other h and, for very severe channels conditions, the parameter l should tend to zero. Unfortunately, the bit error probability was computed only f or a fixed channel and remained open for the situation where all the fading coefficients are Rayleigh distributed. In this paper, an exact and approximate expression is computed for the average bit error probability assuming a Rayleigh fading for the inter-user channel and for the direct channel between users and base station (BS). II. System Model This section summarizes the system model t hat was employed in [1], [6]. A. System Model The channel model used in [6] can be mathematically expressed as Y 0 ( t ) = K 10 X 1 ( t ) + K 20 X 2 ( t ) + Z 0 ( t ) (1) Y 1 ( t ) = K 21 X 2 ( t ) + Z 1 ( t ) (2) Y 2 ( t ) = K 12 X 1 ( t ) + Z 2 ( t ) (3) where Y 0 (t), Y 1 (t), and Y 2 (t) are the baseband models of the received signal at the BS, user 1, and user 2, respectively, during one symbol period. Also, X i (t)isthe signal transmitted by user i under power constraint P i , for i =1,2,andZ i (t) are white zero-mean Gaussian noise random processes with spectral height N i /2 for i * Correspondence: gf@decom.fee.unicamp.br Department of Communications, University of Campinas, Campinas, Brazil Inga and Fraidenraich EURASIP Journal on Wireless Communications and Networking 2011, 2011:166 http://jwcn.eurasipjournals.com/content/2011/1/166 © 2011 Inga and Fraidenraich; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribu tion, and reproduction in any medium, provided the original work is properly cited. = 0, 1, 2, and the fading coe fficients K ij are Rayleigh dis- tributed with E  K 2 ij  =2α 2 ij . We als o assume that the BS can track perfectly the variations in K 10 and K 20 ,user1 can track K 21 and user 2 can track K 12 . The system proposed in [6] is based on a conventional code division multiple access (CDMA) system and divides the transmiss ion into two parts: the first without cooperation and the second with cooperation. For a given coherence time of L symbols and cooperation time of 2L c symbols, the transmitted signals can be expressed as shown in (5), where L n = L-2L c , b (i ) j is user j’s ith bit, ˆ b (i ) j is the partner’s estimate of user j’s ith bit, and c j (t)isuserj’ s spreading code. The parameters a ij represent the power a llocation scheme, and they must maintain an average power constraint that can be expressed as 1 L  L n a 2 11 + L c  a 2 12 + a 2 13 + a 2 14  = P 1 1 L  L n a 2 21 + L c  a 2 22 + a 2 13 + a 2 14  = P 2 (4) X 1 (t)= ⎧ ⎪ ⎨ ⎪ ⎩ a 11 b (i) 1 c 1 (t), i = 1, 2, , L n a 12 b (L n +1+i)/2 1 c 1 (t), i = L n +1,L n + 3, , L − 1 a 13 b (L n +i)/2 1 c 1 (t)+a 14 ˆ b (L n+i )/2 2 c 2 (t), i = L n +2,L n + 4, , L X 2 (t)= ⎧ ⎪ ⎨ ⎪ ⎩ a 21 b (i) 2 c 2 (t), i = 1, 2, , L n a 22 b (L n +1+i)/2 1 c 2 (t), i = L n +1,L n + 3, , L − 1 a 23 ˆ b (L n +i)/2 1 c 1 (t)+a 24 b (L n+i )/2 2 c 2 (t), i = L n +2,L n + 4, , L (5) In the first L n = L -2L c symbol periods, each user transmits its own bits to the BS. The remaining 2L c per- iods are dedicated to cooperation: odd periods for trans- mitting its bits to both the partner and the BS; even periods for transmitting a linear combination of its own bit and the partner’s bit estimate. B. Error Calculations 1) Error Rate for Cooperative Periods: During the 2L c cooperative periods, we have a distinction between “odd” and “even” periods. During the “ odds” periods, each user sends only their own bit, w hich is rec eived and detected by the partner as well as by the BS. The partner’ shardestimateofb 1 is given by ˆ b 1 =sign  1/N c  c T 1 Y 2  , resulting in a probability of bit error equals to P e 12 = Q  K 12 a 12 √ N c σ 2  (6) where Q (·) is the Gaussian error integral, N c is the CDMA spreading gain, σ 2 2 = N 2 / ( 2T c ) , T c is the chip period, and N 2 / 2 is the spectral height of Z 2 (t). The BS forms a soft decision statistic by calculating y odd = 1 N c c T 1 Y od d 0 (7) where Y odd 0 = K 10 X 1 + K 20 X 2 + Z od d 0 . During the “even ” periods, each user send a coopera- tive signal to BS according to Y even 0 = K 10 X 1 + K 20 X 2 + Z eve n 0 , and the BS extracts a soft decision statistic by calculating y even = 1 N c c T 1 Y eve n 0 (8) The c ombined statistics at BS for user 1 is therefo re given by y odd = K 10 a 12 b 1 + n odd y even = K 10 a 13 b 1 + K 20 a 23 ˆ b 1 + n even (9) where n odd and n even are statistically independent and both distributed according to a Gaussian distribution N (0, σ 2 0 /N c ) . The optimal detector shown in [1] is rather complex and does not have a closed-form expression for t he resulting bit error probability. Thus, they consider the following suboptimum detector ˆ b 1 =sign([K 10 a 12 λ(K 10 a 13 +K 20 a 23 )]y ) (10) where y = [y odd y even ] T √ N c / σ 0 and lÎ[0,1]. They call this suboptimum detector as the l -MRC. The probabil- ity of bit error for this detector is given by P e 1 =(1− P e 12 )Q ⎛ ⎜ ⎝ v T λ v 1  v T λ v λ ⎞ ⎟ ⎠ + P e 12 Q ⎛ ⎜ ⎝ v T λ v 2  v T λ v λ ⎞ ⎟ ⎠ (11) where v λ = [K 10 a 12 λ ( K 10 a 13 + K 20 a 23 ) ] T , v 1 = [K 10 a 12 λ ( K 10 a 13 + K 20 a 23 ) ] T √ N c / σ 0 and v 2 = [K 10 a 12 ( K 10 a 13 − K 20 a 23 ) ] T √ N c / σ 0 . III. Rayleigh fading calculations The expression presented in (11) is only valid for a fixed (time-invariant) channel, that is, the fading coefficients K ij are fixed. The aim of this paper is to obtain an expression for the bit error probability when the fading coefficients vary according to a Rayleigh distribution. A. Bit Error Probability The bit error probability associated with the signal from user 1, at user 2, for a fixed gain is described in (6). Now assuming a nonstatic situation, the average bit error probability can be computed averaging (6) with respect to a Rayleigh distribution Inga and Fraidenraich EURASIP Journal on Wireless Communications and Networking 2011, 2011:166 http://jwcn.eurasipjournals.com/content/2011/1/166 Page 2 of 10 ¯ P e 12 = E[P e 12 ]= 1 2  1 −  γ 12 2+γ 12  (12) where g 12 is the average signal-to-noise ratio, defined as γ 12 = 2a 2 12 α 2 12 N c σ 2 2 (13) From (11), we can define two random variables U 1 and U 2 , respectively, as v T λ v 1  v T λ v λ =  U 1 =  ( K 10 a 12 ) 2 + λ ( K 10 a 13 + K 20 a 23 ) 2  √ N c  ( K 10 a 12 ) 2 + λ 2 ( K 10 a 13 + K 20 a 23 ) 2  σ 0 (14) v T λ v 2  v T λ v λ =  U 2 =  ( K 10 a 12 ) 2 + λ  ( K 10 a 13 ) 2 − ( K 20 a 23 ) 2  √ N c  ( K 10 a 12 ) 2 + λ 2 ( K 10 a 13 + K 20 a 23 ) 2  σ 0 (15) since K 10 and K 20 are Rayleigh distributed, the support of (14) will be always greater than zero. On the other hand, since we have negative values in the numerator of (15), i ts support will be all the real line. Taking this into account, we can rewrite (11) as P e 1 =  1 − P e 12  Q   U 1  + P e 12 Q ( U 2 ) (16) To obtain the error probability, we must average P e 1 , over the probability density function (PDF) of U 1 and U 2 [7]. Thus, we have to evaluate the integral P e f =  1 − ¯ P e 12  ∞  0 Q  √ u 1  f u 1 (u 1 )du 1 + ¯ P e 12 ∞  − ∞ Q(u 2 )f u 2 (u 2 )du 2 (17) In orde r to calculate P ef ,wehavetoknowthedistri- bution of U 1 and U 2 , thus to faci litate the calcul ations, we assume an equal power allocation situation, where a 12 = a 13 = a 23 = a. With this assumption the random variables U 1 and U 2 will be simplified to U 1 = a 2  K 2 10 + λ ( K 10 + K 20 ) 2  N c  K 2 10 + λ 2 ( K 10 + K 20 ) 2  σ 2 0 (18) U 2 = a  K 2 10 + λ  K 2 10 + K 2 20  √ N c   K 2 10 + λ 2 ( K 10 + K 20 ) 2  σ 0 (19) Since U 1 depends on K 10 and K 20 , it is possible to write the cumulative distribution function (CDF) and the PDF of U 1 , respectively, as F u 1 (u 1 )= ∫  k 10 ,k 20 ∈D u 1 f u 1 ( k 10 , k 20 ) dk 10 dk2 0 (20) f u 1 ( u 1 ) = dF u 1 ( u 1 ) du 1 (21) In this case, D u 1 is the region of the K 10 × K 20 plane where a 2  k 2 10 + λ ( k 10 + k 20 ) 2  N c  k 2 10 + λ 2 ( k 10 + k 20 ) 2  σ 2 0 ≤ u 1 (22) Note that this region is very similar to a rotated ellipse but not exactly an ellipse. Since K 10 and K 20 are in dependent Rayleigh distribu- tion with parameters a 10 and a 20 , respectively, we have F u 1 ( u 1 ) = a(u 1 )  k 1 0 =0 b(u 1 )  k 2 0 =0 f k 10 k 20  k 10, k 20  dk 20 dk 1 0 (23) where a ( u 1 ) = 1 λ 1  u 1 λ 2 A 1 (24) b ( u 1 ) =  2A 1  B 1 −  2A 1 k 2 10 − u 1  λ  − 2A 1 k 10 λ 2 2A 1 λ (25) A 1 = a 2 N c σ 2 0 (26) B 1 = λ  u 1  u 1 λ 2 − 4A 1 k 2 10 ( λ − 1 )  (27) now it is possible to derive the PDF of U 1 easily as f u 1 ( u 1 ) = a(u 1 )  k 1 0 =0 ∂b ( u 1 ) ∂u 1 b ( u 1 ) α 2 20 e − b(u 1 ) 2 2α 2 20 k 10 α 2 10 e − k 2 10 2α 2 10 dk 1 0 (28) and unfortunately, it is not possible to evaluate (28) in a closed-form solution. In order to validate the above formulation, Figure 1 shows the analytical and simulated PDF of U 1 . N ote the excellent agreement between them showing the correct- ness of our formulation. Following similar rationale, we now find the CDF and PDF of U 2 . N ote that in this case, the region of integra- tion, D u 2 , will be given by Inga and Fraidenraich EURASIP Journal on Wireless Communications and Networking 2011, 2011:166 http://jwcn.eurasipjournals.com/content/2011/1/166 Page 3 of 10 a  k 2 10 + λ  k 2 10 − k 2 20  √ N c   k 2 10 + λ 2 ( k 10 + k 20 ) 2  σ 0 ≤ u 2 (29) leading to the following CDF and PDF, respectively, as F u 2 ( u 2 ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∞  k 20 = |u 2 | A 2 ∞  k 10 =0 f k 10 k 20 ( k 10 , k 20 ) dk 10 dk 20 if u 2 < 0 , ∞  k 20 =0 a(u 2 )  k 10 =0 f k 10 k 20 ( k 10 , k 20 ) dk 10 dk 20 if u 2 ≥ 0. (30) and f u 2 ( u 2 ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∞  |u 2 | A 2 ∂b ( u 2 ) ∂u 2 f k 10 k 20 ( b ( u 2 ) , k 20 ) dk 20 if u 2 < 0 , ∞  0 ∂a ( u 2 ) ∂u 2 f k 10 k 20 ( a ( u 2 ) , k 20 ) dk 20 if u 2 ≥ 0. (31) where a ( u 2 ) =   1 2A 2 λ 1 √ 3   R 1 +  R 2 2  (32) b ( u 2 ) =   1 2A 2 λ 1 √ 3   R 1 +  R 2 2  (33) A 2 = a √ N c σ 0 (34) where  ( · ) denotes the real part of a number, and R 1 and R 2 are described in the Appendix. Inthesamewayasinthefirstcase,(31)cannotbe obtained in a clos ed-form solution. Figure 2 compares the analytical and simulated PDF of U 2 in order to vali- date our formulation. Once that the PDFs of U 1 and U 2 were exactly com- puted, it is possible to obtai n the average bit error prob- ability by simply substituting (28) and (31) into (17). Figure 3 shows the simulation result of the bit error probability and the result of our theoretical expression given in (17), where we can observe that both curves are almost coincident. In this figure, S NR = P σ 2 0 .According to Section II-B, we consider three symbols periods, each period with an average power of P. Also, for simplicity, 0 100 200 300 400 500 600 0.000 0.002 0.004 0.006 0.008 0.010 0.012 u 1 f u 1 (u 1 ) a 12 =1, a 13 =1, a 23 =1, σ 0 =1, N c =8 Exact simulated pdf λ=0.5 Exact analytical pdf λ=0.5 Figure 1 Comparison between analytical and simulated PDF for U 1 Inga and Fraidenraich EURASIP Journal on Wireless Communications and Networking 2011, 2011:166 http://jwcn.eurasipjournals.com/content/2011/1/166 Page 4 of 10 we consider that E  K 2 10  , E  K 2 20  and E  K 2 12  are identical. Although (17) presents the exact solution to the aver- age bit error probability, in some cases, the c omplexity to compute this expression can be prohibitive. For this reason,wefoundaveryaccurateapproximationforthe bit error probability presented in the sequel. B. Approximate Bit error Probability The main problem in order to obtain a simpler expres- sion for the bit error probability is to simplify the PDFs of U 1 and U 2 given, respectively, in (14) and (15). In order to obtain an approximation, the expressions (14) and (15) can be reduced when l =1,s 0 =1anda 12 = a 13 = a 23 = 1. Therefore, the new random variables are given by U  1 = N c  K 2 10 + ( K 10 + K 20 ) 2  (35) U  2 = √ N c  2K 2 10 − K 2 20   K 2 10 + ( K 10 + K 20 ) 2 (36) Considering D u  1 as the region of the plane K 10 × K 20 where N c  k 2 10 + ( k 10 + k 20 ) 2  ≤ u  1 , it can be seen that D u  1 corresponds to the area of an ellipse w hose center is in the origin (0, 0). Unfortunately, the evaluation of the integral (20) is rather complex for the domain D u  1 . For this reason, we consider a simplified version of D u  1 , as bei ng the area of a circle expressed as k 2 1 0 + k 2 2 0 ≤ u  1 . This simplification can be applied since a circle corre- sponds to a particular case of the general ellipse. Hence F u  1  u  1  = √ u  1  k 20 =− √ u  1 √ u  1 −k 2 20  k 10 =− √ u  1 −k 2 20 f k 10 k 20 ( k 10 , k 20 ) dk 10 dk 2 0 (37) This gives f u  1  u  1  = √ u  1  k 20 =− √ u  1 1 2  u  1 − k 2 20  f k 10 k 20   u  1 − k 2 20 , k 20  + f k 10 k 20  −  u  1 − k 2 20 , k 20  dk 20 (38) −15 −10 −5 0 5 10 15 20 25 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 u 2 f u 2 (u 2 ) a 12 =1, a 13 =1, a 23 =1, σ 0 =1, N c =8 Exact simulated pdf λ=0.5 Exact analytical pdf λ=0.5 Figure 2 Comparison between the exact and simulated PDF for U 2 . Inga and Fraidenraich EURASIP Journal on Wireless Communications and Networking 2011, 2011:166 http://jwcn.eurasipjournals.com/content/2011/1/166 Page 5 of 10 Since K 10 and K 20 are in dependent Rayleigh distribu- ted with paramete rs a 1 and a 2 , respectively, the PDF of U’ 1 given in (38) will result in a chi-square probability distribution with four degrees of freedom [8]. Therefore, our approximation of U 1 will be given by f u 1 ( u 1 ) ≈ 4u 1 γ 2 1 e −2u 1 /γ 1 (39) where g 1 is the mean of U 1 given in (14) γ 1 = E [ U 1 ] (40) Figure 4 shows the comparison between our approxi- mate PDF given in (39) and the computer simulation for the PDF of U 1 givenin(14)fortwodifferentvalues of l keeping the same values for a 12 =1,a 13 =2,and a 23 = 3. We observe that the curves are very close for both values of l . Although only these two cases are pre- sented here, many other cases were compared and the approximation still remains very good. A s imilar rationale can be applied in order to find a good appr oximation for U 2 . The region of the K 10 × K 20 plane where U  2 ≤ u  2 is similar to (29). Note that the range of U 2  varies from −∞ ≤ u 2  ≤ ∞ , discarding all the distributions wit h positive support. In order to observe the behavior of the PDF of U 2  ,alargenumber of simulations were perfo rmed, and the Gaussian distri- bution proves to fit extremely well in all the cases. Therefore, assuming a Gaussian distribution, the follow- ing can be written P e f ≈ 1 4  1+  γ 12 2+γ 12  1 − √ γ 1 ( γ 1 +6 ) ( γ 1 +4 ) 3/2  + 1 2  1 −  γ 12 2+γ 12  Q  γ 2 √ 1+v 2  (41) f u 2 ( u 2 ) ≈ 1 √ 2πν 2 e − (u 2 −γ 2 ) 2 2ν 2 (42) where γ 2 = E [ U 2 ] (43) ν 2 =var ( U 2 ) (44) Figure 5 shows the comparison between the approxi- mate PDF given in (42) and the computer simulation for the PDF of U 2 given in ( 15), for two different values of l. Note t hat the approximation is l ess accurate for small values of l, but this inaccuracy does not have a significant influence in the bit error probability. In all −5 0 5 10 15 20 25 10 −5 10 −4 10 −3 10 −2 10 −1 SNR (dB) Bit error probability a 12 =a, a 13 =a, a 23 =a, σ 0 =1, N c =8 Simulation Exact Figure 3 Comparison of the exact and simulated bit error probability adopting an equal power allocation scheme with l = 0.5. Inga and Fraidenraich EURASIP Journal on Wireless Communications and Networking 2011, 2011:166 http://jwcn.eurasipjournals.com/content/2011/1/166 Page 6 of 10 the cases, the approximation fits very well the exact PDF of U 2 . Using (39) and (42) into (17), it is possible to obtain a very accurate approximate bit error probability. Fortu- nately, both integrals can be found in a closed-form solution as ∞  0 Q  √ u 1  4u 1 γ 2 1 e −2u 1 /γ 1d u 1 = 1 2  1 − √ γ 1 ( γ 1 +6 ) ( γ 1 +4 ) 3/2  (45) and ∞  − ∞ Q ( u 2 ) 1 √ 2πν 2 e − (u 2 −γ 2 ) 2 2ν 2 du 2 = Q  γ 2 √ 1+ν 2  (46) All these calculations lead to the approximate bit error probability for the l -MRC detector as shown in (41), where g 12 is given in (13), g 1 is given in (40), g 2 is given in (43), and ν 2 is given in (44). Assuming an equal power allocation scheme ( a 12 = a 13 = a 23 = a), Figure 6 shows the comparison between the theoretical bit error probability presented in (17) using the exact PDFs (28) and (31) and our approxima- tiongivenin(41).Wecanobservethatbothcurvesare almost the same, validating our approximation. Our results are q uite exact for a different power allo- cation scheme as well. This can be seen in Figure 7, where a comparison betw een the exact simulated bit error probability and our approximation given in (41) was performed. In t his figure, the following parameters were used a 10 = a 20 = 1 and a 12 = 0.8. The final approximate expression allows us to deter- mine the optimal value for l in each case. As stated in [1], when the BS believes that the inter-user channel is “perfect”, then l = 1 and the optimal detector turns out to be the maximal ratio combining [7]. As the inter-user channel becomes m ore unreliable, i.e., as P e 12 increases, the value of the best l decr eases toward to zero. In order to demonstrate this behavior, Figure 8 shows the optimized l* versus the inter-user channel parameter a 12 . This curve was obtained using computational opti- mization techniques that min imizes our ap proximate bit error probability (4 1) with respect to l for each value of the inter-user channel parameter, a 12 .Thedirect 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0.000 0.001 0.001 0.002 0.002 0.003 0.003 u′ 1 f u′ 1 (u′ 1 ) a 12 =1, a 13 =2, a 23 =3, σ 0 =1, N c =8 Exact simulated pdf λ=0.3 Approximate pdfλ=0.3 Exact simulated pdf λ=0.8 Approximate pdfλ=0.8 Figure 4 Comparison between the simulated pdf of (14) and our approximation given in (39). Inga and Fraidenraich EURASIP Journal on Wireless Communications and Networking 2011, 2011:166 http://jwcn.eurasipjournals.com/content/2011/1/166 Page 7 of 10 −30 −20 −10 0 10 20 30 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 u′ 2 f u′ 2 (u′ 2 ) a 12 =1, a 13 =2, a 23 =3, σ 0 =1, N c =8 Exact simulated pdf λ=0.01 Approximate pdfλ=0.01 Exact simulated pdf λ=0.9 Approximate pdfλ=0.9 Figure 5 Comparison between the simulated PDF of (15) and our approximation given in (42). −5 0 5 10 15 20 25 30 10 −5 10 −4 10 −3 10 −2 10 −1 SNR (dB) Bit error probability a 12 =a, a 13 =a, a 23 =a, σ 0 =1, N c =8 Exact Approximation Figure 6 Comparison between exact and approximate bit error probability using an equal power allocation scheme with l = 0.5. Inga and Fraidenraich EURASIP Journal on Wireless Communications and Networking 2011, 2011:166 http://jwcn.eurasipjournals.com/content/2011/1/166 Page 8 of 10 −5 0 5 10 15 20 25 10 −4 10 −3 10 −2 10 −1 SNR (dB) Bit error probability a 12 =0.7a, a 13 =1.3a, a 23 =a, σ 0 =1, α 10 =α 20 =1, α 12 =0.8, N c =8 Simulation Approximation Figure 7 Comparison between exact and approximate bit error probability for a non equal power scheme allocation l = 0.5. 0 2 4 6 8 10 Α 12 0.0 0.2 0.4 0.6 0.8 1.0 Λ  Α 10 1, Α 20 1 Figure 8 Optimized l* versus a 12 . Inga and Fraidenraich EURASIP Journal on Wireless Communications and Networking 2011, 2011:166 http://jwcn.eurasipjournals.com/content/2011/1/166 Page 9 of 10 channel parameters a 10 = a 20 = 1 were kept constant, and the equal power allocation scheme (a 12 = a 13 = a 23 = 1) was adopted. IV. Conclusions In this paper, an exact and approximate e xpression for the average bit error probability under Rayleigh fading for the l-MRC presented in [1] was obtained. The exact expression was obtained under the condi- tion of an equal power allocation scheme. The expres- sion was validated through simulations showing a perfect agreement between exact and simulated curves. In order to reduce the complexity of the exact expres- sion, a very accurate approximation was presented as well. The approximate expression is valid for any value s of l, a 12 , a 13 , and a 23 . The expression has been validated by simulation for a variety of values showing a small dif- ference between the exact and approximate curves. Both expression can be very important in many situa- tions where the per formance of a c ooperative system employing CDMA should be evaluated. V. Competing Interests The authors declare that they h ave no competing interests. Appendix R 1 =2M 1 + M 2 + M 3 M 2 R 2 =24k 20 λ 2 μ 2 2 A 2 λ 1  3 R 1 +8M 1 − 2M 2 − 2M 3 M 2 M 1 =2A 2 2 λλ 1 k 2 20 + λ 2 u 2 2 M 2 = 3  E +2  G + A 2 k 20 λλ 1 u 2 2 √ 27F  M 3 =16A 4 2 λ 2 λ 2 1 k 4 20 − 4A 2 2 λ  2λ 3 +5λ 2 +2λ − 1  u 2 2 k 2 20 + λ 2 2 u 4 2 E = −λ 3 2 u 6 2 +6 ( A 2 K 20 ) 2 λλ 5 u 4 2 − 24 ( A 2 K 20 ) 4 λ 2 λ 2 1 λ 3 u 2 2 F = −16 ( A 2 k 20 ) 6 λ 2 λ 2 1 λ 4 +8 ( A 2 k 20 ) 4 λλ 7 u 2 2 − ( A 2 k 20 ) 2 λ 6 u 4 2 + λ 3 2 u 6 2 G =32 ( A 2 k 20 ) 6 λ 3 λ 3 1 λ 1 = λ +1 λ 2 =  λ 2 +1  λ 3 =  2λ 2 +3λ − 1  λ 4 =  5λ 2 +2λ +1  λ 5 =  2λ 5 +5λ 4 − 5λ 3 − 14λ 2 − 7λ −1  λ 6 =  13λ 6 +28λ 5 − 34λ 3 − 12λ 2 − 8λ +1  λ 7 =  7λ 5 +22λ 4 +17λ 3 +3λ 2 − 1  Received: 11 February 2011 Accepted: 10 November 2011 Published: 10 November 2011 References 1. A Sendonaris, E Erkip, B Aazhang, User cooperation diversity Part II: Implementation aspects and perfomance analysis. IEEE Trans Commun. 51(11), 1939–1948 (2003). doi:10.1109/TCOMM.2003.819238 2. 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(McGraw-Hill, 2002) doi:10.1186/1687-1499-2011-166 Cite this article as: Inga and Fraidenraich: Average bit error probability for the l-MRC detector under Rayleigh fading. EURASIP Journal on Wireless Communications and Networking 2011 2011:166. Submit your manuscript to a journal and benefi t from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the fi eld 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com Inga and Fraidenraich EURASIP Journal on Wireless Communications and Networking 2011, 2011:166 http://jwcn.eurasipjournals.com/content/2011/1/166 Page 10 of 10 . an expression for the bit error probability when the fading coefficients vary according to a Rayleigh distribution. A. Bit Error Probability The bit error probability associated with the signal. Access Average bit error probability for the l-MRC detector under Rayleigh fading Mitchell Omar Calderon Inga and Gustavo Fraidenraich * Abstract In this paper, an exact expression for the average bit. this reason,wefoundaveryaccurateapproximationforthe bit error probability presented in the sequel. B. Approximate Bit error Probability The main problem in order to obtain a simpler expres- sion for the bit error probability is to simplify the