Power Allocation in Two-Way Relay Networks with MABC DF Protocol and No Instantaneous Channel State Information

Abstract

In this paper we examine a two way relay network with multiple access broadcast, decode and forward protocol. A new power allocation scheme is proposed which is based on minimizing the outage probability. The outage probability is defined as the event in which the data rate of the different nodes lie outside the achievable rate region. For the first time, a closed-form mathematical expression is derived for the outage probability of such system, in the more general case than previous attempts. The only limiting constraint that is enforced on the problem is that the relay power is less than the power of the end main nodes (terminals). The obtained expression is then considered as the cost function of an optimization problem, in order to be minimized by allocating appropriate power to the nodes. The proposed scheme for power allocation needs no instantaneous channel coefficient estimation, but rather needs only statistical Channel State Information. The correctness of the obtained expression for the outage probability is verified by Monte Carlo simulations and the performance of the proposed scheme for power allocation is analyzed numerically. We have demonstrated that this technique can provide up to 1 dB gain in outage probability for average SNR values.

Appendix: Determining the Probability Regions

In this section, we explore the probability regions for Γ _i s in (15). They are regions in the X–Y plane ($X = P_{a} \frac{{\left| {h_{a} } \right|^{2} }}{{\sigma^{2} }}$ and $Y = P_{b} \frac{{\left| {h_{b} } \right|^{2} }}{{\sigma^{2} }}$ as explained in Sect. 3.1) where integrating the probability distribution function (in this article f _X (x) f _y (y)) over that region results in the Γ _i .

1.1 Probability Region For Γ ₁

We can simplify Γ ₁ in (14) as

$$\varGamma_{1} = pr\left\{ {R_{1} \le X,R_{2} \le Y,\frac{X}{{\lambda_{3} }} < Y < \lambda_{4} X,R_{3} \le X + Y} \right\}.$$

(22)

As explained in Sect. 3, we assume that P _r  ≤ P _a , P _b and Consequently λ ₃, λ ₄ ≤ 1. This results in Γ ₁ = 0, since $\lambda_{4} X \le \frac{X}{{\lambda_{3} }}$ and the condition $\frac{X}{{\lambda_{3} }} < Y < \lambda_{4} X$ in (22) will not hold.

1.2 Probability Region For Γ ₂

Considering λ ₃, λ ₄ ≤ 1, Γ ₂ in (14) can be simplified as

$$\begin{aligned} \varGamma_{2} & = pr\left\{ {\hbox{max} \left( {R_{1} ,\frac{{R_{2} }}{{\lambda_{4} }}} \right) \le X \le \hbox{min} \left( {\lambda_{3} Y,\frac{Y}{{\lambda_{4} }}} \right),R_{3} \le X + Y} \right\} \\ & = pr\left\{ {\hbox{max} \left( {R_{1} ,\frac{{R_{2} }}{{\lambda_{4} }}} \right) \le X \le \lambda_{3} Y,R_{3} \le X + Y} \right\}. \\ \end{aligned}$$

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S ₂ (the probability region for Γ ₂) is either the shaded region in Fig. 12a $\left( \hbox {if} \, \hbox{max} \left( {R_{1} ,\frac{{R_{2} }}{{\lambda_{4} }}} \right) < \frac{{\lambda_{3} }}{{1 + \lambda_{3} }}R_{3} \right)$ or Fig. 12b $\left( \hbox{if} \, \hbox{max} \left( {R_{1} ,\frac{{R_{2} }}{{\lambda_{4} }}} \right) > \frac{{\lambda_{3} }}{{1 + \lambda_{3} }}R_{3} \right)$.

Fig. 12

The probability region for Γ ₂; a $\hbox{max} \left( {R_{1} ,\frac{{R_{2} }}{{\lambda_{4} }}} \right) < \frac{{\lambda_{3} }}{{1 + \lambda_{3} }}R_{3}$; b $\hbox{max} \left( {R_{1} ,\frac{{R_{2} }}{{\lambda_{4} }}} \right) > \frac{{\lambda_{3} }}{{1 + \lambda_{3} }}R_{3}$

1.3 Probability Region For Γ ₃

Γ ₃ in (14) can be reformulated as

$$\begin{aligned} \varGamma_{3} & = pr\left\{ {\hbox{max} \left( {R_{2} ,\frac{{R_{1} }}{{\lambda_{3} }}} \right) \le Y \le \hbox{min} \left( {\lambda_{4} X,\frac{X}{{\lambda_{3} }}} \right),R_{3} \le X + Y} \right\} \\ & = pr\left\{ {\hbox{max} \left( {R_{2} ,\frac{{R_{1} }}{{\lambda_{3} }}} \right) \le Y \le \lambda_{4} X,R_{3} \le X + Y} \right\}; \\ \end{aligned}$$

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where the constraints λ ₃, λ ₄ ≤ 1 have been used in the second equality. S ₃, the probability region for Γ ₃ has been depicted in Fig. 13 for both $\frac{{R_{1} }}{{\lambda_{3} }} < \frac{{\lambda_{4} }}{{1 + \lambda_{4} }}R_{3}$ and $\frac{{R_{1} }}{{\lambda_{3} }} > \frac{{\lambda_{4} }}{{1 + \lambda_{4} }}R_{3}$.

Fig. 13

The probability region for Γ ₃; a $\frac{{R_{1} }}{{\lambda_{3} }} < \frac{{\lambda_{4} }}{{1 + \lambda_{4} }}R_{3}$; b $\frac{{R_{1} }}{{\lambda_{3} }} > \frac{{\lambda_{4} }}{{1 + \lambda_{4} }}R_{3}$

1.4 Probability Region For Γ ₄

Simplifying Γ ₄ in (14) results in

$$\varGamma_{4} = pr\left\{ {\frac{{R_{1} }}{{\lambda_{3} }} \le Y,\frac{{R_{2} }}{{\lambda_{4} }} \le X,\lambda_{4} X < Y < \frac{X}{{\lambda_{3} }},R_{3} \le X + Y} \right\}.$$

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Depending on the quantities $\frac{{R_{1} }}{{\lambda_{3} }}$ and $\frac{{R_{2} }}{{\lambda_{4} }}$, different 10 types of probability region for Γ ₄ will arise which are depicted in Fig. 14.

Fig. 14

The probability region for Γ ₄; a $\frac{{R_{2} }}{{\lambda_{4} }} < \frac{{\lambda_{3} }}{{1 + \lambda_{3} }}R_{3} ,\frac{{R_{1} }}{{\lambda_{3} }} \le \frac{{\lambda_{4} }}{{1 + \lambda_{4} }}R_{3}$; b $\frac{{R_{2} }}{{\lambda_{4} }} \le \frac{{\lambda_{3} }}{{1 + \lambda_{3} }}R_{3} ,\frac{{\lambda_{4} }}{{1 + \lambda_{4} }}R_{3} < \frac{{R_{1} }}{{\lambda_{3} }} \le \frac{1}{{1 + \lambda_{3} }}R_{3}$; c $\frac{{R_{2} }}{{\lambda_{4} }} \le \frac{{\lambda_{3} }}{{1 + \lambda_{3} }}R_{3} ,\frac{1}{{1 + \lambda_{3} }}R_{3} < \frac{{R_{1} }}{{\lambda_{3} }}$; d $\frac{{\lambda_{3} }}{{1 + \lambda_{3} }}R_{3} < \frac{{R_{2} }}{{\lambda_{4} }} \le \frac{1}{{1 + \lambda_{4} }}R_{3} ,\frac{{R_{1} }}{{\lambda_{3} }} \le \frac{{\lambda_{4} }}{{1 + \lambda_{4} }}R_{3}$; e $\frac{{\lambda_{3} }}{{1 + \lambda_{3} }}R_{3} < \frac{{R_{2} }}{{\lambda_{4} }} \le \frac{1}{{1 + \lambda_{4} }}R_{3} ,\frac{{\lambda_{4} }}{{1 + \lambda_{4} }}R_{3} < \frac{{R_{1} }}{{\lambda_{3} }} \le R_{3} - \frac{{R_{2} }}{{\lambda_{4} }}$; f $\frac{{\lambda_{3} }}{{1 + \lambda_{3} }}R_{3} < \frac{{R_{2} }}{{\lambda_{4} }} \le \frac{1}{{1 + \lambda_{4} }}R_{3} ,R_{3} - \frac{{R_{2} }}{{\lambda_{4} }} < \frac{{R_{1} }}{{\lambda_{3} }} \le \frac{1}{{\lambda_{3} \lambda_{4} }}R_{2}$; g $\frac{{\lambda_{3} }}{{1 + \lambda_{3} }}R_{3} < \frac{{R_{2} }}{{\lambda_{4} }} \le \frac{1}{{1 + \lambda_{4} }}R_{3} ,\frac{1}{{\lambda_{3} \lambda_{4} }}R_{2} < \frac{{R_{1} }}{{\lambda_{3} }}$; h $\frac{1}{{1 + \lambda_{4} }}R_{3} < \frac{{R_{2} }}{{\lambda_{4} }},\frac{{R_{1} }}{{\lambda_{3} }} \le R_{2}$; i $\frac{1}{{1 + \lambda_{4} }}R_{3} < \frac{{R_{2} }}{{\lambda_{4} }},R_{2} < \frac{{R_{1} }}{{\lambda_{3} }} \le \frac{1}{{\lambda_{3} \lambda_{4} }}R_{2}$; j $\frac{1}{{1 + \lambda_{4} }}R_{3} < \frac{{R_{2} }}{{\lambda_{4} }},\frac{1}{{\lambda_{3} \lambda_{4} }}R_{2} < \frac{{R_{1} }}{{\lambda_{3} }}$