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Tight performance bounds for twoway opportunistic amplifyandforward wireless relaying networks with TDBC protocols
EURASIP Journal on Wireless Communications and Networking volume 2011, Article number: 192 (2011)
Abstract
Based on amplifyandforward network coding (AFNC) protocol, the outage probability and ergodic capacity of twoway network coding opportunistic relaying (TWORAFNC) systems are investigated as well as the corresponding closedform solutions. For the TWORAFNC systems, it is investigated under two scenarios, namely, the TWORAFNC systems without direct link (TWORAFNCNodir) and the TWORAFNC systems with direct link (TWORAFNCDir). First, we investigate TWORAFNCNodir systems by employing the approximate analysis in high SNR, and obtain closedform solutions to the cumulative distribution function (CDF) and the probability density function (PDF) of the instantaneous endtoend signaltonoise ratio (SNR) with very simple expressions. The derived simple expressions are given by defining an equivalent variable ω_{ eqk } (θ), 2 ≤ θ ≤ 3. When θ = 2, the derived results are the tight lower bounds to CDF and PDF. The sequent simulation demonstrates that the derived tight lower bounds are also very effective over the entire SNR region though which the results are derived in high SNR approximation. Then, with the derived tight closedform lower bound solutions (θ = 2) in TWORAFNCNodir systems, we investigate TWORAFNCDir systems as well as the overall comparison of the outage probability and the ergodic capacity between the two system models. The comparison analysis performed over path loss model basis shows that, in urban environment, due to utilizing the direct link the TWORAFNCDir outperform considerably the TWORAFNodir systems. However, when the value of path loss exponent is relatively large, the achievable gain is very small and the direct link can be omitted. In this case, the TWORAFNCDir model can be displaced by TWORAFNCNodir model having lower implementation complexity.
1. Introduction
Cooperative diversity is an overwhelming research topic in wireless networks [1–6]. The notion of cooperative communications is to enable transmit and receive cooperation at user level by forming virtual multipleinputmultipleoutput (MIMO) system, so that the overall performance including power efficiency and communications reliability can be improved greatly. However, due to the halfduplex constraint in practical systems, the advantages of the cooperative diversity come at the expense of both the spectral efficiency and the implementation complexity. Especially, in multirelay wireless network with K relays, to achieve the full diversity order K + 1 orthogonal wireless channels (slot or frequency) are required, which incurs in the enhancement of implementation complexity as the perfect time synchronization among the relays.
In order to reduce the implementation complexity and to improve the spectrum efficiency but still realize the potential benefits of multirelay cooperation, based on the network coding (NC) techniques [7] and the opportunistic relaying (OR) techniques [8], the twoway network coding opportunistic relaying (TWORNC) has emerged as a promising solution [9], and instantly become one of research hot topics in wireless network fields [10–17]. The basic idea of the TWORNC is that, in multirelay twoway systems, a round of signals exchange between two sources consists of two phases, namely, access phase (AC phase) and broadcast phase (BC phase). Notice that, according to the different transmission protocols employed, time division broadcast (TDBC) and multiaccess broadcast (MABC), the AC phase can include one or two subslots while the BC phase only does one slot. In AC phase, two sources transmit their signals to relays while the relays are listening state. After receiving the signals from both sources, the best relay selection is performed based on a predefined criterion, which results in that only a best relay is selected for NCing the received signals and broadcasting the NCed signal in BC phase. Thus, after performing NC on the received signals, the selected best relay simultaneously broadcasts the NCed signal to two sources. After receiving the broadcasted signal from the best relay, each source can remove the selfinterference from the received signal by taking its own transmitted signal as prior. Thus, the two sources can obtain the wished signal. Obviously, the TWORNC perfectly integrates the NC and OR techniques and possesses the advantages of the two techniques. The TWORNC systems hold the improvement not only on the spectral efficiency improved by as much as 33 or 50% due to the employing of NC [7], but also on the implementation complexity decreased because the perfect time synchronization among the relays is no longer performed.
Currently, there are a few literatures contributing to TWORNC systems [9–17]. In the existing works, based on the fact that whether the direct link between two sources is utilized, the TWORNC systems can be grouped into two kinds. The first is the TWORNC systems in which there is no direct link, which is referred to as TWORNCNodir model in the work. On the contrary, in the second TWORNC systems where the direct link is utilized perfectly, which is called the TWORNCDir model. Oechtering and Boche [9] first investigated the TWORNCNodir systems under the MABC transmission protocol. In the work, the employed NC scheme was superposition network coding [4–6]. For the MABC TWORNCNodir systems, the updated contribution can be found in [10], where the decodeandforward NC (DFNC) was employed. By comparing the achievable maximum diversity gains of two TWORDFNCNodir sytems where the maxmin criterion and maxmum sumrate criterion were adopted, respectively, authors have presented a intelligent switching selection criterion which switches between maxmin and maximum sumrate criterions according to a certain threshold of signaltonoise ratio (SNR). In [11], with the amplifyandforward NC (AFNC) protocol, authors first obtained the endtoend rate expressions, R_{12} and R_{21}. Then, with the aim of maximizing the sum rate of R_{12} and R_{21}, i.e., max(R_{12} and R_{21}), the maximum sumrate criterion has been proposed. In [12], besides the investigation of the TWORNCNodir system based on the conventional single best relay selection criterion, authors have studied the TWORNCNodir system where a double best relay selection criterion was employed. In [13], with the aim of minimzing pairwise error probability, authors have investigated the TWORDFNCNodir systems.
At the same time, compared with the decodeandforward TWOR [14], TWORAFNC is one of the most attractive protocols due to its operational simplicity. Motivated by its practical implementation potential, the TWORAFNC systems have been studied in [15–17] under different channels and assumptions. In these studies, the employed system models are still TWORAFNCNodir. With the Rayleigh channels, in [16] authors have pointed out that both the instantaneous endtoend SNRs considered as independence random variables (RVs) in [15] are dependent mutually, and have presented the exact closedform expression for outage probability in integral form as well as the lower bound. A more overall investigation of the TWORAFNCNodir has been presented in [17].
Observing the above summarization, we can find that, in existing works, the TWORNCNodir systems have been investigated widely. For the one with direct link, TWORNCDir, there is no open report yet. However, in practical implementation systems, such direct link between two sources is always existent. In halfduplex systems, the employed transmission protocol should be TDBC when the direct link is exploited. The reliability of TWORNC systems can be further improved when such practical existence direct transmission exploited. For the investigation of TWORNCDir systems, the canonical argument line [18] is that we must first obtain the statistical description of the instantaneous endtoend SNRs for the systems without direct link, which includes cumulative distribution function (CDF), probability density function (PDF), moment generating function (MGF), etc. Besides this, the ones of direct link gains are also required. With these obtained statistical results of the system without direct link, we can investigate the TWORNCDir systems by employing some complicated mathematic manipulations such as (inverse) Laplace transformation, integral, etc. This mathematic manipulation requires that the statistical descriptions of the endtoend SNRs for the systems without direct link should be given with simple closedform expressions. However, observing the results given in [16, 17], we find that it is difficult to obtain the simple closedform expressions of these statistics even if the lower bound expression (4) in [16] is considered. Motivated by these observations, the work contributes to a comprehensive comparison investigation for the TWORAFNC. We will first obtain the closedform tight lower bound statistical descriptions for the TWORAFNCNodir, and then by using the derived tight lower bound we will study performance of the TWORAFNCDir systems.
2. TWORAFNC system model
We consider TWORAFNC quasistatic reciprocal Rayleigh fading channels consisting of two source nodes, S_{1} and S_{2}, and K relay nodes, R_{1},...,R_{ K } , where K is the maximum number of the achievable relays. All the nodes are equipped with single antenna and operate in halfduplex mode. The channel coefficient between S_{ i } and R_{ k } is denoted by h_{ ik } , i = 1,2, k = 1,2,...,K, and modeled as zeromean complex Gaussian RV with variances ω_{ ik } . The one of the direct link between the two sources is denoted by βh_{0}, where β = 1 denotes that there is direct link, and β = 0 does that there is no direct link. The variance of the direct link coefficient is ω_{0}. For simplicity, we assume that all the channels' coefficients capture the path loss, shadow fading, and frequency nonselective fading due to the rich scattering environment as well as the transmitting power for signal, and at each node the received signals are affected by symmetric Gaussian additive noise with identical variance σ^{2} = 1. That is to say, all nodes transmit signals with unit transmitting power P = 1. Thus, the instantaneous squared channel strengths obey exponential distributions with hazard rates 1/ω_{1k}, 1/ω_{2k}, and 1/ω_{0}, respectively, denoted by γ_{1k}= h_{1k}^{2} ~Υ(1/ω_{1k}), γ_{2k}= h_{2k}^{2} ~Υ(1/ω_{2k}), and γ_{0} = h_{0}^{2} ~Υ(1/ω_{0}). Due to the TDBC transmission protocol employed, the total time of a round for the information exchange between two sources is divided into three slots. In the first two slots, both sources transmit their signals x_{1} and x_{2} whereas the relays are in listening state. Thus, the received signal by relay R_{ k } is given by y_{ k } = h_{1k}x_{1}+h_{2k}x_{2}+n_{ k } . In the last slot, based on the AFNC protocol the relay broadcasts y_{ k } to both sources. The received signals by sources S_{1} and S_{2} are, respectively, given by y_{1} = G_{ k }h_{1k}y_{ k } +βh_{0}x_{2}+n_{1k}, y_{2} = G_{ k }h_{2k}y_{ k } +βh_{0}x_{1}+n_{2k}, where
Assuming the maximum ratio combining employed, we can obtain the instantaneous equivalent endtoend SNRs as follows
where γ_{2k 1}and γ_{1k 2}are the receiving SNRs at S_{1} and S_{2}, respectively, and
where a = 1. Letting γ(k) = min(γ_{1}(k),γ_{2}(k)), we have the best relay selection criterion
This leads to the equivalent instantaneous endtoend SNR of the best relay for the TWORAFNC systems is expressed as
3. PDF and CDF to the equivalent instantaneous endtoend SNR
In this section, we investigate the statistical characteristic of the TWORAFNC scheme. According to the statement in Section 1, the statistical description of the TWORAFNCNodir systems is required, firstly. Then, one of TWORAFNCDir systems can be obtained readily. Thus, we first present the statistical description of the TWORAFNCNodir systems.
3.1. Statistical description for TWORAFNCNodir systems
For the TWORAFNCNodir channels, we have β = 0 in (5). With the order statistics [19], the CDF F_{γ(k)}(γ) of γ(k) = min(γ_{1}(k),γ_{2}(k)) is given by
With the similar approach as the one employed in [17], the difference γ_{1}(k)γ_{2}(k) is given by
By comparing the values of γ_{1k}, and γ_{2k}, we have
Thus, by substituting (3) into (8), the second part of the righthand side of (6) can be rewritten into two parts [17]
We first consider the part P_{1}(k). Obviously, to obtain the closedform solution to P_{1}(k), the condition \frac{\gamma \left(2{\gamma}_{1k}+a\right)}{{\gamma}_{1k}\gamma}\le {\gamma}_{1k} is required [16]. The condition can be rewritten as (γ_{1k})^{2}(3γ)γ_{1k}γa ≥ 0. With the consideration γ_{1k}≥ 0, we have the condition {\gamma}_{1k}\ge M=\left(3\gamma +\sqrt{9{\gamma}^{2}+4\gamma a}\right)\u22152. This yields that the component P_{1}(k) can be expressed as
where E_{γ 1k}(.) is the expectation operation with respect to γ_{1k}. Due to γ_{2k}= h_{2k}^{2} ~Υ(1/ω_{2k}), the PDF of γ_{2k}is {f}_{{\gamma}_{2k}}\left(\gamma \right)=\frac{1}{{\omega}_{2k}}{e}^{\frac{\gamma}{{\omega}_{2k}}}. This yields that (10) is rewritten as
Similarly, since the PDF ofγ_{1k}is {f}_{{\gamma}_{1k}}\left(\gamma \right)=\frac{1}{{\omega}_{1k}}{e}^{\frac{\gamma}{{\omega}_{1k}}} and {\gamma}_{1,k}\ge M=\left(3\gamma +\sqrt{9{\gamma}^{2}+4\gamma a}\right)\u22152, we have
In (12), we rewrite the expression of P_{11}(k) as
Letting t = xy and using some manipulations, this leads to
To obtain the closedform expression of P_{11}(k), we rewrite Equation (14) as
where Q=\left(\gamma +\sqrt{9{\gamma}^{2}+4\gamma a}\right)\u22152 and L = 2γ(γ+a/ 2). For the first part A in (15), the closedform solution can be readily obtained. By using the equation 3.471.9 in [20], we have
where K_{1}(z) is the modified Bessel function. As stated in [21], in high SNR the modified Bessel function can be approximated with K_{1}(z)~1/z. Thus, for (16), with high SNR approximation we have A~1. For the second component B given in (15), using the both approximation e^{t}~1t and x+y~2\sqrt{xy} we have
In high SNR, we have α = σ^{2}/P~0 [17]. This leads to M~3γ, Q~2γ, and L~2γ^{2}~0. Thus, we have
By using the symmetry between γ_{1}(k) and γ_{2}(k) given in (3) and substituting A, B, (15), (12), and (9) into (6), we have the approximate CDF F_{γ(k)}(γ) of γ(k)
Observing the expression (19) and substituting 1\frac{2}{{\omega}_{1k}}\gamma ~{e}^{\frac{2}{{\omega}_{1k}}\gamma}, 1\frac{2}{{\omega}_{2k}}\gamma ~{e}^{\frac{2}{{\omega}_{2k}}\gamma}, we have
Letting \frac{1}{{\omega}_{eqk}\left(\theta \right)}=\theta \left(\frac{1}{{\omega}_{2k}}+\frac{1}{{\omega}_{1k}}\right), and {F}_{\gamma \left(k\right)}\left(\gamma ,\theta \right)=1{e}^{\theta \left(\frac{1}{{\omega}_{2k}}+\frac{1}{{\omega}_{1k}}\right)\gamma}=1{e}^{\frac{1}{{\omega}_{eqk}\left(\theta \right)}\gamma}, Equation (20) is bounded by
where 2 ≤ θ ≤ 3. By combining (4) and using the order statistics, we have the approximate CDF F_{γ(b)}(γ, θ) of γ(b).
With the very simple expression for the CDF F_{γ(b)}(γ, θ) of γ(b), the PDF f_{γ(b)}(γ, θ) of γ(b) can be readily obtained by taking the derivative of F_{γ(b)}(γ, θ) with respect to γ, and is given approximately by
where b_{1},...,b_{ k } a binary sequence is whose elements assume the value of zero or one.
3.2. Statistical description for TWORAFNCDir systems
In the above section, the CDF and PDF of γ(b) are achieved with very simple closedform expressions, which make it is easy to investigate TWORAFNCDir systems. In (5), since that γ(b) and γ_{0} are independent, the MGF of γ_{ T } (b) = γ(b)+γ_{0} is given by {M}_{{\gamma}_{T}}\left(s\right)={M}_{{\gamma}_{\left(b\right)}}\left(s\right).{M}_{{\gamma}_{0}}\left(s\right). Using the definition of MGF given by M_{ γ } (s) = E(e^{sy} ), where E(x) is the expectation operation, we have the MGF of γ_{0} given by
Similarly, with (23) and the definition of MGF we have the MGF of γ(b) given by
Thus, with (24) and (25) we have the MGF {M}_{{\gamma}_{T}}\left(s\right) given
To find the PDF of γ_{ T } , we use the inverse Laplace transform defined by {f}_{{\gamma}_{T}}\left(\gamma ,\theta \right)={\ud50f}^{1}\left\{{M}_{{\gamma}_{T}}\left(s,\theta \right)\right\}, where {\ud50f}^{1}\left\{x\right\} is the inverse Laplace transform operator. This leads to
Taking the integral of (27) with respect to γ, the approximate CDF of γ_{ T } is given by
3.3. Ergodic capacity for TWORAFNC systems
The capacity of the TWORAFNCNodir systems is defined as C(γ(b)) = log(1+γ(b)). This leads to the ergodic capacity [22]
Substituting f_{γ(b)}(γ) given in (23) into (29) and following 4.337.1 in [20], we have the ergodic capacity of the considered TWORAFNCNodir systems as follows
where ε_{1}(m) is the exponential integral function defined by {\epsilon}_{1}\left(m\right)={\int}_{1}^{\infty}\frac{{e}^{tm}}{t}dt.
Similar to (30), we can obtain the ergodic capacity of TWORAFNCDir systems.
3.4. Outage probabilities for TWORAFNC systems
The outage probability of TWOAFNC systems is defined as the probability that the instantaneous endtoend SNR falls bellow a certain predefined threshold μ_{0}. For TWORAFNCNodir and TWORAFNCDir systems, the outage probabilities are, respectively, given by
4. Simulation analysis
With the above investigation, the simulations are presented in this section. In the simulation, we employ a path loss model [23], which includes path loss, show fading, and frequency nonselective fading because the rich scattering environment is given by {h}_{mn}=c\u2215\sqrt{{d}_{mn}^{p}}, where h_{ mn } and d_{ mn } , respectively, denote the link gain and the distance from node m to n, c is attenuation constant, and p is the path loss exponent. In general, in urban or suburban environment, the path loss exponent is a little greater than 3. Without loss of generality, we assume that the distance between two sources is normalized to 1, the normalized distance from S_{1} to relay R_{ k } is d_{1k}, and the one from S_{2} to R_{ k } is d_{2k}. It is also assumed that all the relays are close together and the interrelay distances are enough small. This assumption is commonly used in the context of cooperative diversity systems and guarantees equivalent average variances: ω_{1k}and ω_{2k}for links S_{1}  > R_{ k } and S_{2}  > R_{ k } . Thus, we have ω_{0} = c^{2}, ω_{1k}= ω_{0}(d_{1k})^{p}, and ω_{2k}= ω_{0}(1d_{1k})^{p}, where k = 1,...,K, K is the maximum of the achievable relays.
By taking ω_{0} = 0.33, K = 6, the path loss exponent p = 3, and spectrum efficiency R_{0} = 1, in Figure 1, we first consider the TWORAFNCNodir systems (without direct link) and compare the outage performance between the derived results and the one obtained in [16] under symmetric channels (d_{1k}= 0.5). At the same time, we also present the actual accurate simulation results denoted by "o" in Figure 1. From the presented results, we can find clearly that the derived result is the tight lower bound when θ = 2, and is the upper bound when θ = 3. In practice, θ = 3 denotes the high SNR approximate case where the two instantaneous endtoend SNRs are independent [15]. For the lower bound, in low SNR region, the derived result and the one (4) given in [16] enjoy approximately equal outage probability. In large high SNR region, the derived result is closer to the actual simulation and is tighter lower bound than the one (4) in [16]. Moreover, the gap between the derived result and the actual simulation becomes smaller and smaller with the increasing SNR. However, observing the result presented in [16], we can find that the gap holds a constant, approximately. It is also observed that, for the considered symmetric channels, the result given in [16] have larger error, but the derived result matches with the simulations exactly, especially in high SNR regions. The observation indicates that the derived result is more perfect than the result (4) given in [16] when the channels are symmetric. Thus, by taking θ = 2, we investigate the tight lower bound of the outage probabilities for TWORAFNCDir systems (with direct link), and the results are denoted by "∇" in Figure 1. The result indicates that when the direct link is utilized the TWORAFNCDir can obtain approximately 3 dB gain of SNR at 10^{10} of outage probability under the symmetric channels.
Figures 2 and 3 are the comparison of the ergodic capacity versus K. Due to the lower bounds of both PDF and CDF employed, the ergodic capacity is the upper bound. From Figure 2, it is observed that when the direct link is exploited the TWORAFNC systems obtain obvious improvement on ergodic capacity. The achievable gain of ergodic capacity is decreasing with the number of relays when the number of relays is small. However, when the number of relays is relatively large, the gain of ergodic capacity is constant, approximately. Take the symmetric channels as example, at SNR = 10 dB, the gain of the upper bound of ergodic capacity is 0.6 when the number of relays is 2, but it is 0.3 when the number of relays is greater than 6.
Figure 3 is the comparison of ergodic capacity between the symmetric channels and asymmetric channels for TWORAFNC systems with and without direct link cases. It is observed that, in the both TWORAFNC systems, the ergodic capacities of symmetric channels are granter than the one of asymmetric channels. When the number of relays is small, the gain of ergodic capacity achieved by symmetric channels over the asymmetric one is increasing with K. However, when K is relatively large (K > 6), it equals to a constant, approximately.
In Figures 4 and 5, we investigate the effect of path loss exponent on the system performance. In the simulation, only the symmetric channels are considered and the values of ω_{1k}and ω_{2k}are constant, i.e., the distance d_{1k}= 0.5, ω_{1k}= ω_{2k}= c^{2} = 0.33, and ω_{0} = c^{2}×(0.5) ^{p} . This leads to that the corresponding performance of the TWORAFNCNodir systems should be constant. However, for the systems with direct link, it should be changing with the path loss exponent p. From Figures 4 and 5, we can find that the performance improvement obtained in the systems with direct are considerable when the value of path loss exponent is relatively small. For example, in urban environment (p = 3), the maximum ergodic capacity gain is 0.29 and the SNR gain is 2.5 at 10^{10} of outage probability. However, the curves of both outage probability and ergodic capacity are very close to the curves of the systems without direct link when the value of the path loss exponent is grater than 5, which yields that the performance gain obtained in TWORAFNCDir systems is very small and can be omitted. The observed result indicates that when the value of path loss exponent is relatively large, the direct link can be omitted, which yields that the complexity of systems is degraded greatly. On the contrary, we should consider the direct link. Besides this, in Figure 5(b) it is observed that the gain of ergodic capacity obtained in direct link systems is increasing with SNR when the value of SNR is small (SNR < 20 dB). When it is large (SNR ≥ 20dB), the achieved gain is a constant.
5. Conclusion
Through the high SNR approximation, we first obtain the closedform analytical solutions to CDF and PDF of the endtoend SNR for TWORAFNodir systems, and present the corresponding tight lower bound. Though the tight lower bounds are obtained in high SNR approximation, the sequent simulations demonstrate that the high SNR approximation results are also effective over the entire SNR region. Especially, the results are given with very simple closedform expressions, which are significant for the investigation of the TWORAFNC systems having direct link between two sources. With the approximate lower bounds to PDF and CDF for TWORAFNodir systems, the outage performance and the ergodic capacity for TWORAFDir systems are investigated comprehensively. The presented simulation indicates that, in urban environment, by utilizing the direct link the TWORAF systems can obtain the considerable improvement on the performance. However, when the value of path loss exponent is large relatively, the TWORAFDir model can be displaced with the TWORAFNodir model having lower implementation complexity.
Abbreviations
 AC phase:

access phase
 AFNC:

amplifyandforward NC
 BC phase:

broadcast phase
 CDF:

cumulative distribution function
 DFNC:

decodeandforward NC
 MABC:

multiaccess broadcast
 MGF:

moment generating function
 MIMO:

multipleinputmultipleoutput
 MRC:

maximum ratio combining
 NC:

network coding
 OR:

opportunistic relaying
 PDF:

probability density function
 RV:

random variable
 SNR:

signaltonoise ratio
 TDBC:

time division broadcast
 TWORNC:

twoway network coding opportunistic relaying
 TWORNCDir:

TWORNC systems where there is direct link between two sources
 TWORNCNodir:

TWORNC systems where there is no direct link between two sources.
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Acknowledgements
The study was supported by the Natural Science Foundation of China under Grant 61071090 and 61171093, by the Postgraduate Innovation Program of Scientific Research of Jiangsu Province under Grant CX10B184Z and CXZZ11_0388, and by the project 11KJA510001 and PAPD.
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Jia, X., Yang, L. & Fu, H. Tight performance bounds for twoway opportunistic amplifyandforward wireless relaying networks with TDBC protocols. J Wireless Com Network 2011, 192 (2011). https://doi.org/10.1186/168714992011192
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DOI: https://doi.org/10.1186/168714992011192