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

- Mitchell Omar Calderon Inga
^{1}and - Gustavo Fraidenraich
^{1}Email author

**2011**:166

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

© Inga and Fraidenraich; licensee Springer. 2011

**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

## 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

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*) 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 ${\mathcal{N}}_{i}\u22152$ for *i* = 0, 1, 2, and the fading coefficients *K*_{
ij
}are Rayleigh distributed with $E\left[{K}_{ij}^{2}\right]=2{\alpha}_{ij}^{2}$. We also assume that the BS can track perfectly the variations in *K*_{10} and *K*_{20}, user 1 can track *K*_{21} and user 2 can track *K*_{12}.

*L*symbols and cooperation time of 2

*L*

_{ c }symbols, the transmitted signals can be expressed as shown in (5), where

*L*

_{ n }=

*L*-2

*L*

_{ c }, ${b}_{j}^{\left(i\right)}$ is user

*j*'s

*i*th bit, ${\widehat{b}}_{j}^{\left(i\right)}$ 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

In the first *L*_{
n
}= *L* - 2*L*_{
c
}symbol periods, each user transmits its own bits to the BS. The remaining 2*L*_{
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 2*L*_{c} 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.

*b*

_{1}is given by ${\widehat{b}}_{1}=\mathsf{\text{sign}}\left(\left(1\u2215{N}_{c}\right){c}_{1}^{T}{Y}_{2}\right)$, resulting in a probability of bit error equals to

where *Q* (·) is the Gaussian error integral, *N*_{
c
}is the CDMA spreading gain, ${\sigma}_{2}^{2}={\mathcal{N}}_{2}\u2215\left(2{T}_{c}\right)$, *T*_{
c
}is the chip period, and ${\mathcal{N}}_{2}\u22152$ is the spectral height of *Z*_{2}(*t*).

where ${Y}_{0}^{\mathsf{\text{odd}}}={K}_{10}{X}_{1}+{K}_{20}{X}_{2}+{Z}_{0}^{\mathsf{\text{odd}}}$.

where *n*_{odd} and *n*_{even} are statistically independent and both distributed according to a Gaussian distribution $\mathcal{N}\left(0,{\sigma}_{0}^{2}\u2215{N}_{c}\right)$.

where ${v}_{\lambda}={\left[{K}_{10}{a}_{12}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\lambda \left({K}_{10}{a}_{13}+{K}_{20}{a}_{23}\right)\right]}^{T},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{v}_{1}={\left[{K}_{10}{a}_{12}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\lambda \left({K}_{10}{a}_{13}+{K}_{20}{a}_{23}\right)\right]}^{T}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\sqrt{{N}_{c}}\u2215{\sigma}_{0}$ and ${v}_{2}={\left[{K}_{10}{a}_{12}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left({K}_{10}{a}_{13}-{K}_{20}{a}_{23}\right)\right]}^{T}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\sqrt{{N}_{c}}\u2215{\sigma}_{0}$.

## 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

_{12}is the average signal-to-noise ratio, defined as

*U*

_{1}and

*U*

_{2}, respectively, as

*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), its support will be all the real line. Taking this into account, we can rewrite (11) as

*U*

_{1}and

*U*

_{2}[7]. Thus, we have to evaluate the integral

*P*

_{ ef }, we have to know the distribution of

*U*

_{1}and

*U*

_{2}, thus to facilitate the calculations, 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}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

*K*

_{10}×

*K*

_{20}plane where

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

*K*

_{10}and

*K*

_{20}are independent Rayleigh distribution with parameters

*α*

_{10}and

*α*

_{20}, respectively, we have

*U*

_{1}easily as

and unfortunately, it is not possible to evaluate (28) in a closed-form solution.

*U*

_{1}. Note the excellent agreement between them showing the correctness of our formulation.

*U*

_{2}. Note that in this case, the region of integration, ${D}_{{u}_{2}}$, will be given by

where $\Re \left(\cdot \right)$ denotes the real part of a number, and *R*_{1} and *R*_{2} are described in the Appendix.

*U*

_{2}in order to validate our formulation.

*U*

_{1}and

*U*

_{2}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, $\mathsf{\text{SNR}}=\frac{P}{{\sigma}_{0}^{2}}$. According to Section II-B, we consider three symbols periods, each period with an average power of

*P*. Also, for simplicity, we consider that $E\left[{K}_{10}^{2}\right],\phantom{\rule{2.77695pt}{0ex}}E\left[{K}_{20}^{2}\right]$ and $E\left[{K}_{12}^{2}\right]$ are identical.

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

*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 λ = 1,

*σ*

_{0}= 1 and

*a*

_{12}=

*a*

_{13}=

*a*

_{23}= 1. Therefore, the new random variables are given by

*K*

_{10}×

*K*

_{20}where ${N}_{c}\left({k}_{10}^{2}+{\left({k}_{10}+{k}_{20}\right)}^{2}\right)\le {u}_{1}^{\prime}$, it can be seen that ${D}_{{u}^{\prime}1}$ 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 ${D}_{{u}^{\prime}1}$. For this reason, we consider a simplified version of ${D}_{{u}^{\prime}1}$, as being the area of a circle expressed as ${k}_{10}^{2}+{k}_{20}^{2}\le {u}_{1}^{\prime}$. This simplification can be applied since a circle corresponds to a particular case of the general ellipse. Hence

*K*

_{10}and

*K*

_{20}are independent Rayleigh distributed with parameters

*α*

_{1}and

*α*

_{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

*γ*

_{1}is the mean of

*U*

_{1}given in (14)

*U*

_{1}given in (14) for two different values of λ 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 λ. Although only these two cases are presented here, many other cases were compared and the approximation still remains very good.

*U*

_{2}. The region of the

*K*

_{10}×

*K*

_{20}plane where ${U}_{2}^{\prime}\le {u}_{2}^{\prime}$ is similar to (29). Note that the range of ${{U}_{2}}^{\prime}$ varies from $-\infty \le {{u}_{2}}^{\prime}\le \infty $, discarding all the distributions with positive support. In order to observe the behavior of the PDF of ${{U}_{2}}^{\prime}$, 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

*U*

_{2}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

*U*

_{2}.

All these calculations lead to the approximate bit error probability for the λ-MRC detector as shown in (41), where *γ*_{12} is given in (13), *γ*_{1} is given in (40), *γ*_{2} is given in (43), and *ν*_{
2
}is given in (44).

*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 approximation given in (41). We can observe that both curves are almost the same, validating our approximation.

*α*

_{10}=

*α*

_{20}= 1 and

*α*

_{12}= 0.8.

_{12}. 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,

*α*

_{12}. The direct channel parameters

*α*

_{10}=

*α*

_{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 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, *a*_{12}, *a*_{13}, and *a*_{23}. 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

## Declarations

## Authors’ Affiliations

## References

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## Copyright

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.