Open Access

Average bit error probability for the λ-MRC detector under Rayleigh fading

  • Mitchell Omar Calderon Inga1 and
  • Gustavo Fraidenraich1Email author
EURASIP Journal on Wireless Communications and Networking20112011:166

https://doi.org/10.1186/1687-1499-2011-166

Received: 11 February 2011

Accepted: 10 November 2011

Published: 10 November 2011

Abstract

In this paper, an exact expression for the average bit error probability was obtained for the λ-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 cooperationVirtual MIMOBit error probabilityRayleigh fading

I. Introduction

Diversity techniques have been widely accepted as one of effective ways of combat multipath fading in wireless communications [1], in particular spatial diversity is specially effective at mitigating these multipath situation. However, in many wireless applications, the use of multiple antennas is not practical due to size and cost limitations of the terminals. 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 theory work on relay channels [2], [3]. Cooperative networks achieve diversity gain by allowing the users to cooperate, and thus, each wireless user is assumed to 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, assuming a suboptimal receiver called λ-MRC, the bit error probability was computed assuming a fixed channel. This kind of receiver combines the signal from the first period of transmission with the signal transmitted jointly by the both users in the second period of transmission. The variable λ [0,1] establishes the degree of confidence in the bits estimated by the partner. For situations where the inter-user channel presents favorable conditions, the variable λ should be close to unity; on the other hand, for very severe channels conditions, the parameter λ should tend to zero. Unfortunately, the bit error probability was computed only for 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 that 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 Y0(t), Y1(t), and Y2(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) is the signal transmitted by user i under power constraint P i , for i = 1, 2, and Z i (t) are white zero-mean Gaussian noise random processes with spectral height N i 2 for i = 0, 1, 2, and the fading coefficients K ij are Rayleigh distributed with E K i j 2 = 2 α i j 2 . We also assume that the BS can track perfectly the variations in K10 and K20, user 1 can track K21 and user 2 can track K12.

The system proposed in [6] is based on a conventional code division multiple access (CDMA) system and divides the transmission 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 j ( i ) is user j's i th bit, b ^ j ( i ) is the partner's estimate of user j's i th bit, and c j (t) is user j's spreading code. The parameters a ij represent the power allocation scheme, and they must maintain an average power constraint that can be expressed as
1 L L n a 11 2 + L c a 12 2 + a 13 2 + a 14 2 = P 1 1 L L n a 21 2 + L c a 22 2 + a 13 2 + a 14 2 = P 2
(4)
X 1 ( t ) = a 11 b 1 ( i ) c 1 ( t ) , i = 1 , 2 , . . . , L n a 12 b 1 ( L n + 1 + i ) 2 c 1 ( t ) , i = L n + 1 , L n + 3 , . . . , L - 1 a 13 b 1 ( L n + i ) 2 c 1 ( t ) + a 14 b ^ 2 L n + i 2 c 2 ( t ) , i = L n + 2 , L n + 4 , . . . , L X 2 ( t ) = a 21 b 2 ( i ) c 2 ( t ) , i = 1 , 2 , . . . , L n a 22 b 1 ( L n + 1 + i ) 2 c 2 ( t ) , i = L n + 1 , L n + 3 , . . . , L - 1 a 23 b ^ 1 ( L n + i ) 2 c 1 ( t ) + a 24 b 2 L n + i 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 periods are dedicated to cooperation: odd periods for transmitting 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 2Lc cooperative periods, we have a distinction between "odd" and "even" periods. During the "odds" periods, each user sends only their own bit, which is received and detected by the partner as well as by the BS.

The partner's hard estimate of b1 is given by b ^ 1 = sign 1 N c c 1 T 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 2 T c , T c is the chip period, and N 2 2 is the spectral height of Z2(t).

The BS forms a soft decision statistic by calculating
y odd = 1 N c c 1 T Y 0 odd
(7)

where Y 0 odd = K 10 X 1 + K 20 X 2 + Z 0 odd .

During the "even" periods, each user send a cooperative signal to BS according to Y 0 even = K 10 X 1 + K 20 X 2 + Z 0 even , and the BS extracts a soft decision statistic by calculating
y even = 1 N c c 1 T Y 0 even
(8)
The combined statistics at BS for user 1 is therefore given by
y odd = K 10 a 12 b 1 + n odd y even = K 10 a 13 b 1 + K 20 a 23 b ^ 1 + n even
(9)

where nodd and neven are statistically independent and both distributed according to a Gaussian distribution N ( 0 , σ 0 2 N c ) .

The optimal detector shown in [1] is rather complex and does not have a closed-form expression for the resulting bit error probability. Thus, they consider the following suboptimum detector
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 λ [0,1]. They call this suboptimum detector as the λ-MRC. The probability of bit error for this detector is given by
P e 1 = ( 1 - P e 12 ) Q v λ T v 1 v λ T v λ + P e 12 Q v λ T v 2 v λ T 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
P ̄ e 12 = E [ P e 12 ] = 1 2 1 - γ 12 2 + γ 12
(12)
where γ12 is the average signal-to-noise ratio, defined as
γ 12 = 2 a 12 2 α 12 2 N c σ 2 2
(13)
From (11), we can define two random variables U1 and U2, respectively, as
v λ T v 1 v λ T 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 λ T v 2 v λ T 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 K10 and K20 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), its 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 U1 and U2[7]. Thus, we have to evaluate the integral
P e f = 1 - P ̄ e 12 0 Q u 1 f u 1 ( u 1 ) d u 1 + P ̄ e 12 - Q ( u 2 ) f u 2 ( u 2 ) d u 2
(17)
In order to calculate P ef , we have to know the distribution of U1 and U2, thus to facilitate the calculations, we assume an equal power allocation situation, where a12 = a13 = a23 = a. With this assumption the random variables U1 and U2 will be simplified to
U 1 = a 2 K 10 2 + λ K 10 + K 20 2 N c K 10 2 + λ 2 K 10 + K 20 2 σ 0 2
(18)
U 2 = a K 10 2 + λ K 10 2 + K 20 2 N c K 10 2 + λ 2 K 10 + K 20 2 σ 0
(19)
Since U1 depends on K10 and K20, it is possible to write the cumulative distribution function (CDF) and the PDF of U1, respectively, as
F u 1 ( u 1 ) = k 10 , k 20 D u 1 f u 1 k 10 , k 20 d k 10 d k 20
(20)
f u 1 u 1 = d F u 1 u 1 d u 1
(21)
In this case, D u 1 is the region of the K10 × K20 plane where
a 2 k 10 2 + λ k 10 + k 20 2 N c k 10 2 + λ 2 k 10 + k 20 2 σ 0 2 u 1
(22)

Note that this region is very similar to a rotated ellipse but not exactly an ellipse.

Since K10 and K20 are independent Rayleigh distribution with parameters α10 and α20, respectively, we have
F u 1 u 1 = k 10 = 0 a ( u 1 ) k 20 = 0 b ( u 1 ) f k 10 k 20 k 10 , k 20 d k 20 d k 10
(23)
where
a u 1 = 1 λ 1 u 1 λ 2 A 1
(24)
b u 1 = 2 A 1 B 1 - 2 A 1 k 10 2 - u 1 λ - 2 A 1 k 10 λ 2 2 A 1 λ
(25)
A 1 = a 2 N c σ 0 2
(26)
B 1 = λ u 1 u 1 λ 2 - 4 A 1 k 10 2 λ - 1
(27)
now it is possible to derive the PDF of U1 easily as
f u 1 u 1 = k 10 = 0 a u 1 b u 1 u 1 b u 1 α 20 2 e - b ( u 1 ) 2 2 α 20 2 k 10 α 10 2 e - k 10 2 2 α 10 2 d k 10
(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 U1. Note the excellent agreement between them showing the correctness of our formulation.
Figure 1

Comparison between analytical and simulated PDF for U 1

Following similar rationale, we now find the CDF and PDF of U2. Note that in this case, the region of integration, D u 2 , will be given by
a k 10 2 + λ k 10 2 - k 20 2 N c k 10 2 + 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 d k 10 d k 20 if u 2 < 0 , k 20 = 0 k 10 = 0 a u 2 f k 10 k 20 k 10 , k 20 d k 10 d k 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 d k 20 if u 2 < 0 , 0 a u 2 u 2 f k 10 k 20 a u 2 , k 20 d k 20 if u 2 0 .
(31)
where
a u 2 = 1 2 A 2 λ 1 3 R 1 + R 2 2
(32)
b u 2 = 1 2 A 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 R1 and R2 are described in the Appendix.

In the same way as in the first case, (31) cannot be obtained in a closed-form solution. Figure 2 compares the analytical and simulated PDF of U2 in order to validate our formulation.
Figure 2

Comparison between the exact and simulated PDF for U 2 .

Once that the PDFs of U1 and U2 were exactly computed, it is possible to obtain the average bit error probability 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, SNR = P σ 0 2 . According to Section II-B, we consider three symbols periods, each period with an average power of P. Also, for simplicity, we consider that E K 10 2 , E K 20 2 and E K 12 2 are identical.
Figure 3

Comparison of the exact and simulated bit error probability adopting an equal power allocation scheme with λ = 0.5.

Although (17) presents the exact solution to the average bit error probability, in some cases, the complexity to compute this expression can be prohibitive. For this reason, we found a very accurate approximation for the bit error probability presented in the sequel.

B. Approximate Bit error Probability

The main problem in order to obtain a simpler expression for the bit error probability is to simplify the PDFs of U1 and U2 given, respectively, in (14) and (15). In order to obtain an approximation, the expressions (14) and (15) can be reduced when λ = 1, σ0 = 1 and a12 = a13 = a23 = 1. Therefore, the new random variables are given by
U 1 = N c K 10 2 + K 10 + K 20 2
(35)
U 2 = N c 2 K 10 2 - K 20 2 K 10 2 + K 10 + K 20 2
(36)
Considering D u 1 as the region of the plane K10 × K20 where N c k 10 2 + k 10 + k 20 2 u 1 , it can be seen that D u 1 corresponds to the area of an ellipse whose 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 being the area of a circle expressed as k 10 2 + k 20 2 u 1 . This simplification can be applied since a circle corresponds to a particular case of the general ellipse. Hence
F u 1 u 1 = k 20 = - u 1 u 1 k 10 = - u 1 - k 20 2 u 1 - k 20 2 f k 10 k 20 k 10 , k 20 d k 10 d k 20
(37)
This gives
f u 1 u 1 = k 20 = - u 1 u 1 1 2 u 1 - k 20 2 f k 10 k 20 u 1 - k 20 2 , k 20 + f k 10 k 20 - u 1 - k 20 2 , k 20 d k 20
(38)
Since K10 and K20 are independent Rayleigh distributed with parameters α1 and α2, 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 U1 will be given by
f u 1 u 1 4 u 1 γ 1 2 e - 2 u 1 γ 1
(39)
where γ1 is the mean of U1 given in (14)
γ 1 = E U 1
(40)
Figure 4 shows the comparison between our approximate PDF given in (39) and the computer simulation for the PDF of U1 given in (14) for two different values of λ keeping the same values for a12 = 1, a13 = 2, and a23 = 3. We observe that the curves are very close for both values of λ. Although only these two cases are presented here, many other cases were compared and the approximation still remains very good.
Figure 4

Comparison between the simulated pdf of (14) and our approximation given in (39).

A similar rationale can be applied in order to find a good approximation for U2. The region of the K10 × K20 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 with positive support. In order to observe the behavior of the PDF of U 2 , a large number of simulations were performed, and the Gaussian distribution proves to fit extremely well in all the cases. Therefore, assuming a Gaussian distribution, the following 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 2 π ν 2 e - ( u 2 - γ 2 ) 2 2 ν 2
(42)
where
γ 2 = E U 2
(43)
ν 2 = var U 2
(44)
Figure 5 shows the comparison between the approximate PDF given in (42) and the computer simulation for the PDF of U2 given in (15), for two different values of λ. Note that the approximation is less accurate for small values of λ, but this inaccuracy does not have a significant influence in the bit error probability. In all the cases, the approximation fits very well the exact PDF of U2.
Figure 5

Comparison between the simulated PDF of (15) and our approximation given in (42).

Using (39) and (42) into (17), it is possible to obtain a very accurate approximate bit error probability. Fortunately, both integrals can be found in a closed-form solution as
0 Q u 1 4 u 1 γ 1 2 e - 2 u 1 γ 1 d u 1 = 1 2 1 - γ 1 γ 1 + 6 γ 1 + 4 3 2
(45)
and
- Q u 2 1 2 π ν 2 e - u 2 - γ 2 2 2 ν 2 d u 2 = Q γ 2 1 + ν 2
(46)

All these calculations lead to the approximate bit error probability for the λ-MRC detector as shown in (41), where γ12 is given in (13), γ1 is given in (40), γ2 is given in (43), and ν 2 is given in (44).

Assuming an equal power allocation scheme (a12 = a13 = a23 = a), Figure 6 shows the comparison between the theoretical bit error probability presented in (17) using the exact PDFs (28) and (31) and our approximation given in (41). We can observe that both curves are almost the same, validating our approximation.
Figure 6

Comparison between exact and approximate bit error probability using an equal power allocation scheme with λ = 0.5.

Our results are quite exact for a different power allocation scheme as well. This can be seen in Figure 7, where a comparison between the exact simulated bit error probability and our approximation given in (41) was performed. In this figure, the following parameters were used α10 = α20 = 1 and α12 = 0.8.
Figure 7

Comparison between exact and approximate bit error probability for a non equal power scheme allocation λ = 0.5.

The final approximate expression allows us to determine the optimal value for λ in each case. As stated in [1], when the BS believes that the inter-user channel is "perfect", then λ = 1 and the optimal detector turns out to be the maximal ratio combining [7]. As the inter-user channel becomes more unreliable, i.e., as P e 12 increases, the value of the best λ decreases toward to zero. In order to demonstrate this behavior, Figure 8 shows the optimized λ* versus the inter-user channel parameter α12. This curve was obtained using computational optimization techniques that minimizes our approximate bit error probability (41) with respect to λ for each value of the inter-user channel parameter, α12. The direct channel parameters α10 = α20 = 1 were kept constant, and the equal power allocation scheme (a12 = a13 = a23 = 1) was adopted.
Figure 8

Optimized λ* versus α 12 .

IV. Conclusions

In this paper, an exact and approximate expression for the average bit error probability under Rayleigh fading for the λ-MRC presented in [1] was obtained.

The exact expression was obtained under the condition of an equal power allocation scheme. The expression was validated through simulations showing a perfect agreement between exact and simulated curves.

In order to reduce the complexity of the exact expression, a very accurate approximation was presented as well. The approximate expression is valid for any values of λ, a12, a13, and a23. The expression has been validated by simulation for a variety of values showing a small difference between the exact and approximate curves.

Both expression can be very important in many situations where the performance of a cooperative system employing CDMA should be evaluated.

Appendix

R 1 = 2 M 1 + M 2 + M 3 M 2 R 2 = 24 k 20 λ 2 μ 2 2 A 2 λ 1 3 R 1 + 8 M 1 - 2 M 2 - 2 M 3 M 2 M 1 = 2 A 2 2 λ λ 1 k 20 2 + λ 2 u 2 2 M 2 = E + 2 G + A 2 k 20 λ λ 1 u 2 2 27 F 3 M 3 = 16 A 2 4 λ 2 λ 1 2 k 20 4 - 4 A 2 2 λ 2 λ 3 + 5 λ 2 + 2 λ - 1 u 2 2 k 20 2 + λ 2 2 u 2 4 E = - λ 2 3 u 2 6 + 6 A 2 K 20 2 λ λ 5 u 2 4 - 24 A 2 K 20 4 λ 2 λ 1 2 λ 3 u 2 2 F = - 16 A 2 k 20 6 λ 2 λ 1 2 λ 4 + 8 A 2 k 20 4 λ λ 7 u 2 2 - A 2 k 20 2 λ 6 u 2 4 + λ 2 3 u 2 6 G = 32 A 2 k 20 6 λ 3 λ 1 3 λ 1 = λ + 1 λ 2 = λ 2 + 1 λ 3 = 2 λ 2 + 3 λ - 1 λ 4 = 5 λ 2 + 2 λ + 1 λ 5 = 2 λ 5 + 5 λ 4 - 5 λ 3 - 14 λ 2 - 7 λ - 1 λ 6 = 13 λ 6 + 28 λ 5 - 34 λ 3 - 12 λ 2 - 8 λ + 1 λ 7 = 7 λ 5 + 22 λ 4 + 17 λ 3 + 3 λ 2 - 1

Declarations

Authors’ Affiliations

(1)
Department of Communications, University of Campinas

References

  1. Sendonaris A, Erkip E, Aazhang B: User cooperation diversity Part II: Implementation aspects and perfomance analysis. IEEE Trans Commun 2003, 51(11):1939-1948. 10.1109/TCOMM.2003.819238View ArticleGoogle Scholar
  2. van der Meulen EC: Three-terminal communication channels. Adv Appl Probab 1971, 3(1):120-154. 10.2307/1426331MathSciNetView ArticleGoogle Scholar
  3. Cover TM, El Gamal A: Capacity theorem for the relay channel. IEEE Trans Inform Theory 1979, 25(5):572-584. 10.1109/TIT.1979.1056084MathSciNetView ArticleGoogle Scholar
  4. Nosratinia A, Hunter TE: A Hedayat, Cooperative communication in wireless networks. IEEE Commun Mag 2004, 42(10):74-80. 10.1109/MCOM.2004.1341264View ArticleGoogle Scholar
  5. Laneman JN, Tse DNC, Wornell GW: Cooperative diversity in wireless networks: Efficient protocols and outage behavior. IEEE Trans Inform Theory 2004, 50(12):3062-3080. 10.1109/TIT.2004.838089MathSciNetView ArticleGoogle Scholar
  6. Sendonaris A, Erkip E, Aazhang B: User cooperation diversity Part I: System description. IEEE Trans Commun 2003, 51(11):1927-1938. 10.1109/TCOMM.2003.818096View ArticleGoogle Scholar
  7. Proakis JG: Digital Communications. 4th edition. McGraw-Hill, New York; 2001.Google Scholar
  8. A Papoulis S: Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes. 4th edition. McGraw-Hill; 2002.Google Scholar

Copyright

© Inga and Fraidenraich; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. 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, distribution, and reproduction in any medium, provided the original work is properly cited.