- Open Access
Average bit error probability for the λ-MRC detector under Rayleigh fading
© Inga and Fraidenraich; licensee Springer. 2011
Received: 11 February 2011
Accepted: 10 November 2011
Published: 10 November 2011
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.
Diversity techniques have been widely accepted as one of effective ways of combat multipath fading in wireless communications , 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 , . 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 , . The first implementation strategy for cooperation was introduced in , , 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
A. System Model
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 for i = 0, 1, 2, and the fading coefficients K ij are Rayleigh distributed with . 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.
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.
where Q (·) is the Gaussian error integral, N c is the CDMA spreading gain, , T c is the chip period, and is the spectral height of Z2(t).
where nodd and neven are statistically independent and both distributed according to a Gaussian distribution .
where and .
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
Note that this region is very similar to a rotated ellipse but not exactly an ellipse.
and unfortunately, it is not possible to evaluate (28) in a closed-form solution.
where denotes the real part of a number, and R1 and R2 are described in the Appendix.
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
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).
In this paper, an exact and approximate expression for the average bit error probability under Rayleigh fading for the λ-MRC presented in  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.
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