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Outage probability analysis and optimal transmit power allocation for multihop full duplex relay network over Nakagamim fading channels
EURASIP Journal on Wireless Communications and Networking volumeÂ 2018, ArticleÂ number:Â 192 (2018)
Abstract
This paper investigates the outage probability and rate performance of multihop full duplex (FD) and half duplex (HD) decodeandforward (DF)based relay networks. First of all, we derive new closed form equation for the endtoend outage probability of multihop FDR system taking into account the interrelay interference (IRI), i.e., interference caused by simultaneous transmissions by the nodes in nonorthogonal frequency bands, and the residual selfinterference (RSI) present at the full duplex relay (FDR) nodes, in independent nonidentical Nakagamim fading channels. We also derive an approximate expression for the outage probability which is found to be highly accurate. Further, we provide an asymptotic expression as well, for Rayleigh fading channels. Furthermore, we derive exact and approximate expression for the outage probability of multihop spectral efficient HDR network that employ twophase relaying. We then consider optimal power allocation (OPA) for multihop FDR and HDR networks that maximizes the endtoend transmission rate with individual peak power constraint at the nodes. Since the optimization problem is nonconvex, we develop an efficient, lowcomplex and fastconverging iterative algorithm for power allocation based on sequential convex programming. Secondly, we consider OPA for multihop FDR and HDR networks that minimizes the endtoend outage probability with individual peak power constraints at the nodes. We devise geometric programming (GP) to obtain the OPA vector. The results demonstrate that OPA can significantly improve the outage and endtoend rate performance of both FDR and HDR networks as compared to uniform power allocation policy. The results from the analytical model are validated by extensive Monte Carlo simulations.
1 Introduction
In multihop wireless relay networks, communication between the source node and destination node is facilitated with the help of a number of relay nodes in cascade [1, 2]. In conventional multihop relay networks, the relay nodes operate in half duplex mode. Recently, extensive investigations have been reported on the use of full duplex relaying (FDR) instead of half duplex relaying (HDR) [3]. The full duplex relay nodes are capable of performing reception and transmission concurrently using the same frequency band, resulting in improved spectral efficiency as compared to HDR systems. However, FDR systems suffer from selfinterference (also known as echo or loop interference), arising due to imperfect isolation between the transmitting and receiving antennas of the relay, which can be mitigated by adopting effective antenna isolation and interference cancellation techniques. In spite of the advancements made in the design of such techniques, studies have shown that FDR systems are always affected by the presence of residual selfinterference (RSI). On the other hand, HDR system does not suffer from RSI; however, the spectral efficiency is reduced significantly [3â€“6].
The endtoend performance of multihop HDR networks has been studied extensively in the literature, e.g., [1, 2, 7â€“11]. Recently, the outage probability of a decodeandforward (DF) FDR system was analyzed by a few researchers [4, 12â€“15]. In [4], the authors analyzed the outage probability of dualhop FDR system assuming Rayleigh fading environment. This model was extended to a multihop FDR system in [12], while the authors of [13] have considered a dualhop FDR system in Nakagamim fading channel. In [14, 15], the authors considered multihop FDR system in Nakagamim fading channels; however, closedform equation for the outage probability was not provided. From the detailed survey of related work, it is observed that an exact closedform analytical expression for the outage probability of multihop FDR system has not appeared in the literature so far considering Nakagami fading channels. Further, it is observed that there are no results available for asymptotic outage probability considering Rayleigh/Nakagami fading channels.
On the other hand, a few researchers have addressed the optimal power allocation (OPA) problem for FDRbased dualhop and multihop networks [16â€“22]. In [16], the authors have proposed a hybrid full duplex/half duplex scheme for amplifyandforward (AF) dualhop network with transmit power adaption at the relay that maximizes the instantaneous and average spectral efficiency. In [17], OPA was derived for AF dualhop FDR that maximizes the transmission rate of the network. In [18], OPA was derived under total and individual power constraints for dualhop DFbased FDR system. Likewise, dualhop FDR systems were considered in [19, 20] as well. In [21], OPA that maximizes the endtoend throughput under a total power constraint was considered for a multihop FDR system while [22] considered OPA that minimizes the weighted sum of transmit powers under an outage probability constraint. However, work in [21] ignored the presence of RSI in the network while [22] assumed that both RSI and interrelay interference (IRI) are negligible.
The first objective of this paper is to obtain closedform equation for the outage probability of a multihop DFbased FDR network by assuming the links to experience independent Nakagamim fading. We also derive an approximate expression for the outage probability which is more concise as compared to the exact expression. We validate the accuracy of the derived approximate expression through numerical and simulation studies. Further, we provide an asymptotic expression for the outage probability considering special case of Rayleigh fading channels. We then investigate OPA for multihop FDR system that maximizes the endtoend instantaneous transmission rate with constraints on maximum transmit power at each node. We observe that the OPA problem is nonsmooth and nonconvex; accordingly, an efficient, lowcomplex, iterative power allocation algorithm based on sequential convex programming (SCP) has been devised to solve the OPA problem. Further, we provide the proof for the convergence of the proposed algorithm to a KarushKuhnTucker (KKT) point of the original problem, which guarantees the achievability of local optimal solution to the problem. We also make a comparison of the proposed algorithm with that of bisectionbased power allocation algorithm and establish that SCPbased power allocation algorithm converges faster than the bisectionbased algorithm. Furthermore, we consider OPA for multihop FDR system that minimizes the endtoend outage probability with constraints on maximum transmit power at each node. We establish that the problem can be transformed to a standard geometric programming problem (GPP) and the OPA can be determined by solving the GPP. Moreover, we derive analytical equation for the outage probability of spectral efficient multihop HDR system described in [23] and then consider OPA for multihop HDR system as well. We describe the procedure to find the optimal transmit power for both FDR and HDR systems. Extensive simulation results are used to establish that the proposed power allocation algorithm converges very fast and is highly effective for solving the optimization problem considered in the paper. The results further demonstrate that OPA can significantly improve the endtoend outage performance and transmission rate of both HDR and FDR systems as compared to uniform power allocation (UPA) policy.
Rest of the paper is organized as follows. Section 2 describes the system model. Theoretical derivation of the endtoend outage probability of the multihop FDR/HDR systems are described in Section 3. Section 4 considers the OPA for FDR/HDR systems and describes the iterative algorithm to solve the optimization problem. The results are presented in Section 5 and the paper is concluded in Section 6.
2 System model
Multihop relaying is a promising technique to improve the coverage and throughput performance of wireless networks. In this case, intermediate relay nodes carry information from source node to destination node. Recently, this concept has been proposed for cooperative wireless networks [24] and cellular networks [25, 26]. In relayassisted cellular networks, deployment of relay nodes can provide improved coverage for base stations and can enhance the signal quality experienced by cell edge user.
2.1 Multihop FDR system
A schematic diagram of the DFbased multihop FDR network is shown in Fig. 1a. Here, F_{0} is the source node, F_{N+1} is the destination node, and the relay nodes are represented as F_{1},F_{2}â€¦F_{N}. Assume that F_{0} transmits signal x_{0} to the relay F_{1} and let y_{1} be the received signal at F_{1}. Since DF relaying is assumed, after decoding y_{1},F_{1} transmits x_{1}, which is a delayed version of x_{0}. This process is continued till the signal reaches F_{N+1}. Simultaneous transmission and reception by the FDR nodes in the same frequency band leads to selfinterference at the relay nodes, which is caused by insufficient isolation between transmitter and receiver sections of the relay node. We assume the relay nodes to use two directional antennas, one for transmission and the other for reception. The directional antennas provide isolation between the transmission and reception for the selfinterference channel, thereby reducing the selfinterference.
Assume that y_{j} and x_{j} respectively represent the signals received and transmitted by F_{j}. Now, y_{j} consists of the following: (i) actual informationbearing signal sent from F_{jâˆ’1}, (ii) RSI component present at F_{j}, and (iii) IRI from other relays F_{i}, i=0,1,...,N, iâ‰ {jâˆ’1,j} [27]. Let the channel gain between F_{i} and F_{j} be h_{i,j}, {i,jâˆˆ(0,1....N+1)}. We consider block fading channel where the channel gains remain constant in each transmission block (i.e., each block contains n number of symbols) and change independently in the next block. We assume h_{i,j} to follow Nakagamim distribution with shape parameter m_{i,j} and mean power E[h_{i,j}^{2}]=Ï€_{i,j}. Accordingly, h_{i,j}^{2} is a Gamma random variable with shape parameter m_{i,j} and scale parameter \(\frac {\pi _{{ {i,j}}}}{m_{ {i,j}}}\). The probability distribution function (PDF) of h_{i,j} is given by [28]
where Î“(.) is the Gamma function. As reported in [4, 5, 12â€“20], we consider that the RSI channel at relay F_{j}, (after employing all known practical interference isolation and cancellation techniques), can be modeled as a fading channel. Let h_{j,j} be the fading coefficient corresponding to the RSI channel at F_{j}. We assume h_{j,j} (j=0,1,..N) to follow Nakagamim fading model, since it can span a wide range of fading scenarios that include both Rayleigh as well as Rician.
The equation for the received signal y_{j} at F_{j} for a given time instant is given by:
where n_{j} is the additive white Gaussian noise component at F_{j} with power \(\sigma ^{2}_{j}\). Notice that (2) considers the signals that arrive at F_{j} from F_{i} {iâ‰ jâˆ’1}, as interference, i.e., it is assumed that F_{j} does not implement any sort of diversity techniques as reported in [29].
2.2 Multihop HDR system
The conventional Khop HDR systems require K orthogonal time slots for communication which degrades the spectral efficiency. For fair comparison of FDR network against equivalent HDR network, the spectral efficient HDR system proposed in [23] is considered, which uses twophase relaying and is illustrated in Fig 1b. Here, nodes F_{0}, F_{2}, F_{4}... transmit in the first time slot while all other nodes are in the receive mode. In the second time slot, nodes F_{1}, F_{3}, F_{5}..., transmit while the rest of the nodes stay in the receive mode.
Let \(y^{0}_{j}\) and \(x^{0}_{j}\) respectively be the received and transmitted signals at a given time instance at node F_{j}. Now, \(y^{0}_{j}\) consists of two components: (i) actual information bearing signal sent from F_{jâˆ’1} and (ii) IRI from other relays F_{p}, where p=2i+q, \(i = 0, 1, \dots \left \lfloor {{\frac {Nq}{2}}} \right \rfloor \) and q represents the remainder when jâˆ’1 is divided by 2. Accordingly, the received signal \(y^{0}_{j}\) at F_{j} can be written as:
In Section 3.2, we investigate the outage probability and rate performance of multihop spectral efficient HDR network and make an extensive comparison study among multihop FDR/HDR systems. Even though the spectral efficient HDR has been proposed in [23], the optimal power allocation, outage, and rate performance were not evaluated in the literature so far.
3 Derivation of outage probability
This section describes the analytical model for finding the outage probability of multihop FDR/HDR sytem considering Nakagmim fading scenario.
3.1 Outage probability of multihop FDR system
Let P = (P_{0}, P_{1}, â€¦, P_{N})^{T} be the transmit power vector, where P_{i} represents the transmission power of relay node F_{i}. Let Î“_{j}(P) be the signaltointerferenceplusnoise ratio at F_{j}; j=1,...,N+1. Now, Î“_{j}(P) is given by
The interference power terms in (4) include both RSI as well as IRI. Let us define \(X_{{i},{j}} = \frac {P_{i} h_{{i},{j}}^{2} }{\sigma ^{2}_{j}},\{ {i,j} \in (0, 1....N+1)\}\). Since the channel coefficients h_{i,j}^{2} have Gamma PDF, the random variables {X_{i,j}} also have Gamma PDF with shape parameters m_{i,j} and scale parameters Î²_{i,j}. The PDF of X_{i,j} can be determined using the results of random variable transformation and is given by [28]:
where \(\beta _{i,j} = \frac {\pi _{{i},{j}}}{m_{i,j}}\ \frac {P_{i}} {\sigma ^{2}_{j}} \). Now, (4) can be written as follows:
Assuming complex Gaussian inputs, unit bandwidth, and that the nodes adopt a continuous rate scheme, the instantaneous transmission rate of the link between F_{jâˆ’1} and F_{j} is given by:
In addition, the endtoend instantaneous rate of the source to destination link for DF relaying is the minimum of the instantaneous transmission rates of the N+1 links and is given by [27]
The endtoend outage probability of multihop DF FDR system is defined as the probability that endtoend instantaneous rate is less than the given target rate r and is given by
where Pr(.) denotes probability and Î“_{T} is the SINR threshold, which is related to the link target transmission rate, r, through the relation Î“_{T}=2^{r}âˆ’1 [4]. The jth term in (9) is given by
Let \(Y_{j} =\sum _{\substack {i=0 \\ i\neq j1}}^{N} X_{{i},{j}} \) and \(W_{j} = \frac {X_{{j1},{j}}}{\Gamma _{T}}\). Further, let \(f_{Y_{j}}(y)\) and \(f_{W_{j}}(w)\) respectively be the PDFs of Y_{j} and W_{j}. Now, (10) becomes
Using (5), the PDF of W_{j} can be written as follows:
3.1.1 Exact expression for outage probability
Here, we derive the exact expression for the outage probability of multihop FDR network. Since Y_{j} is the sum of independent Gamma random variables, for integer values of Nakagami parameter m_{i,j} (iâ‰ jâˆ’1, i=0....N), the exact PDF of Y_{j}, i.e., \(f_{Y_{j}}(y)\) is given by [30]:
where the weights Îž(i,k,{m_{i,j}},{Î²_{i,j}}) can be found using the recursive formula given by [30]:
Substituting (13) in (11) and interchanging the order of summation and integration, Pr(Î“_{j}(P)â‰¥Î“_{T}) can be written as follows:
Now, substituting (5) and (12) in (16) and after some mathematical simplifications, (16) yields the following:
The final equation for the outage probability is obtained by solving (17) and substituting the result in (9) and is given by (refer to Appendix 1 for the detailed derivation):
3.1.2 Approximate expression for outage probability
Since the exact outage probability given by (18) contains weights Îž(i,k) which are to be computed recursively, outage probability calculation is computationally expensive. To reduce the computational complexity, we derive an approximate simplified expression based on WelchSatterwaite approximation for the sum of Gamma random variables [31]. In Section 5 of this paper, we validate the accuracy of the approximate expression through numerical illustrations. Notice that finding the outage probability expression requires \(f_{Y_{j}}(y)\), which is the PDF of the sum of N independent Gamma random variables. According to [31], sum of N independent Gammadistributed random variables can be approximated by a Gamma random variable. Accordingly, the PDF of the approximated Gamma random variable \(\tilde {Y}_{j}\), \({f}_{\tilde {Y}_{j}}(y)\) can be written as follows:
where \({m}_{j} = \frac {\left ({\sum \nolimits }_{\substack {i=0 \\ i\neq j1}}^{N} m_{i, j} \beta _{i,j}\right)^{2}}{{\sum \nolimits }_{\substack {i=0 \\ i\neq j1}}^{N}m_{i, j} \beta ^{2}_{i,j}}\) and \({\beta }_{j} =\frac {{\sum \nolimits }_{\substack {i=0 \\ i\neq j1}}^{N} m_{i, j} \beta ^{2}_{i,j}}{{\sum \nolimits }_{\substack {i=0 \\ i\neq j1}}^{N}m_{i, j} \beta _{i,j}} \)are the shape and scale parameters of \(\tilde {Y}_{j}\). An approximate expression for the probability Pr(Î“_{j}â‰¥Î“_{T}) can be determined by substituting \(f_{W_{j}}(w)\) given by (12) and \({f}_{\tilde {Y}_{j}}(y)\) given by (19) in (11), i.e., \({f}_{\tilde {Y}_{j}}(y)\) is used instead of \({f}_{Y_{j}}(y)\) in (11). After rearranging the relevant terms, the following expression can be obtained for this probability, i.e.,
Notice that the double integral term in (20) is similar to that given in (17). This integral can be solved using an approach similar to the one followed for the derivation of (18), described in Appendix 1. By following the steps laid out in Appendix 1, the following expression can be obtained:
Now, substituting (21) in (9), we get the approximate closedform expression for the outage probability of multihop FDR system as follows:
The simplified outage probability expression for the special case of Rayleigh fading can be obtained by substituting m_{i,j}=1 âˆ€ i,j in (22) and is given as follows:
where \(\hat {m_{j}} = \frac {\left ({\sum \nolimits }_{\substack {i=0 \\ i\neq j1}}^{N} \hat {\beta }_{i,j}\right)^{2}}{{\sum \nolimits }_{\substack {i=0 \\ i\neq j1}}^{N} \hat {\beta }^{2}_{i,j}}\), \({\hat {\beta }}_{j} =\frac {{\sum \nolimits }_{\substack {i=0 \\ i\neq j1}}^{N} \hat {\beta }^{2}_{i,j}}{{\sum \nolimits }_{\substack {i=0 \\ i\neq j1}}^{N}\hat {\beta }_{i,j}} \)and \(\hat {\beta }_{i, j} = \pi _{i,j} \left (\frac {P_{i}}{\sigma ^{2}_{j}}\right)\).
The amount of RSI power at a FDR node is directly proportional to the transmit power at the relay node. In the high transmit power region, the amount of RSI and IRI would be very high. This leads to an outage probability floor for FDR networks in the high transmit power region, i.e., outage probability remains constant as the transmit power increases. An expression for asymptotic outage probability would be useful to theoretically verify the existence of outage floor behavior in mutihop FDR networks. Recall that Rayleigh fading, which is applicable for a rich multipath environment, is a special case of Nakagami fading [28]. So asymptotic outage results for Rayleigh fading are relevant under certain channel conditions. Accordingly, we derive the asymptotic outage probability of multihop FDR system under Rayleigh fading, which can be obtained by letting P_{i}â†’âˆž in (23). This implies \(\hat {\beta }_{i, j} \to \infty \) and \( \hat {\beta }_{j} / \hat {\beta }_{j1, j} << 1\). By using the binomial linear approximation (1+x)^{âˆ’n}â‰ˆ1âˆ’nxâ‰ˆ exp(âˆ’nx) for small x and substituting \(\hat {\beta }_{j} = {\left ({\sum \nolimits }_{\substack {i=0 \\ i\neq j1}}^{N} \hat {\beta }_{i,j}\right)}/{ \hat {m_{j}}}\) in (23), the asymptotic outage probability is given as follows:
Now, (24) can be used to describe the outage probability floor behavior which occurs when the links are unbalanced, i.e., when \(\hat {\beta }_{b1, b} << \hat {\beta }_{j1, j} \ \forall j, j\neq b\), the product of exponentials terms in (24) becomes very low due to the fact that the desired signal over the bth link becomes very poor and it cannot be improved by increasing the transmit power, which leads to outage floor behavior. Similarly, when the interference at a node is very high as compared to the other nodes, it also leads to outage floor. However, the product term in (24) takes the higher possible numerical value when all the links are balanced and it provides minimum value for the outage probability. Since \(\hat {\beta }_{i, j}\) depends on the transmit powers of the nodes, optimal allocation of power among the nodes results in balanced links which leads to improved outage probability performance.
3.2 Outage probability of multihop HDR system
For the spectral efficient multihop HDR system described in [23], the SINR at relay node F_{j}, \(\Gamma ^{HDR}_{j} (\boldsymbol {P})\) is given by
The achievable endtoend instantaneous rate of the multihop HDR system is the minimum of the achievable instantaneous rate over each link [27] and is given by
where \(R^{HDR}_{j}(\boldsymbol {P})\) is the achievable instantaneous rate of link j, j=1,2,â€¦,N+1 and is given as follows:
The fraction 1/2 in (27) is due to the fact that spectral efficient multihop HDR requires two orthogonal time slots for transmission and reception [4, 12].
The endtoend outage probability of multihop DF HDR system is defined as the probability that endtoend instantaneous rate is less than the given target rate r and is given by
Here, the threshold SINR (Î³_{T}) for HDR systems is given as Î³_{T}=2^{2r}âˆ’1, where r is the target rate of the FDR system. Notice that (28) is similar to (9) with \(\Gamma ^{HDR}_{j} (\boldsymbol {P})\) given by (25). By adopting an approach similar to multihop FDR systems, it can be shown that the exact outage probability of the spectral efficient multihop HDR system in the presence of independent nonidentical Nakagamim fading channels is given by the following equation:
where the coefficients Îž(p,k) are computed recursively as given by (14) and (15). To get a simplified expression for the outage probability, we use the WelchSatterwaite approximation for the sum of N independent Gamma random variables [31] as used in Section IIIA for the multihop FDR system. By adopting a similar approach, the following approximate expression can be obtained for the outage probability of multihop HDR system in independent nonidentical Nakagamim fading channels:
where \(\tilde {m}_{j} = \frac {\left ({\sum \nolimits }_{\substack {i=0 \\ p\neq j1}}^{\left \lfloor {{\frac {Nq}{2}}}\right \rfloor } m_{p, j} \beta _{p,j}\right)^{2}}{{\sum \nolimits }_{\substack {i=0 \\ p\neq j1}}^{\left \lfloor {{\frac {Nq}{2}}}\right \rfloor }m_{p, j} \beta ^{2}_{p,j}}\), \(\tilde {\beta }_{j} =\frac {{\sum \nolimits }_{\substack {i=0 \\ p\neq j1}}^{\left \lfloor {{\frac {Nq}{2}}}\right \rfloor } m_{p, j} \beta ^{2}_{p,j}}{{\sum \nolimits }_{\substack {i=0 \\ p\neq j1}}^{N}m_{p, j} \beta _{p,j}} \). The approximate outage probability expression for the Rayleigh fading case is obtained by substituting m_{i,j}=1 âˆ€ i,j in (30) and is given by
Similar to (24), the asymptotic outage probability of multihop HDR system for the Rayleigh fading case can be obtained as follows:
where \(\bar {m}_{j} = \frac {\left ({\sum \nolimits }_{\substack {i=0 \\ p\neq j1}}^{\left \lfloor {{\frac {Nq}{2}}}\right \rfloor } \bar {\beta }_{p,j}\right)^{2}}{{\sum \nolimits }_{\substack {i=0 \\ p\neq j1}}^{\left \lfloor {{\frac {Nq}{2}}}\right \rfloor }\bar {\beta }^{2}_{p,j}}\), \(\bar {\beta }_{j} =\frac {{\sum \nolimits }_{\substack {i=0 \\ p\neq j1}}^{\left \lfloor {{\frac {Nq}{2}}}\right \rfloor } \bar {\beta }^{2}_{p,j}}{{\sum \nolimits }_{\substack {i=0 \\ p\neq j1}}^{N} \bar {\beta }_{p,j}} \)and \(\bar {\beta }_{i, j} = \pi _{i,j} \left (\frac {P_{i}}{\sigma ^{2}_{j}}\right)\).
4 Optimal power allocation
In this section, we address two power allocation problems that optimize the performance of multihop FDR and HDR systems. The first problem investigates the maximization of endtoend rate, while the second problem investigates the minimization of endtoend outage probability of multihop FDR and HDR systems, with constraints on maximum transmit power at the nodes. Note that rate maximization requires the knowledge of complete instantaneous channel state information (ICSI) at the nodes and outage probability minimization requires the availability of statistical channel state information (SCSI) alone.
4.1 Optimal power allocation for multihop FDR system
Here, we formulate two OPA problems for multihop FDR system: (i) for maximizing the endtoend instantaneous rate and (ii) for minimizing the endtoend outage probability. We establish that the rate maximization problem is nonconvex. We then propose an iterative power allocation algorithm based on sequential convex programming (SCP) for finding the OPA vector that meets the objective. On the other hand, we transform the outage probability minimization into a standard geometric programming problem (GPP) and establish that the optimal solution can be obtained by solving the GPP.
4.1.1 Endtoend instantaneous rate maximization
Here, we formulate the OPA problem that maximizes the endtoend instantaneous rate of the system. Since the objective function (i.e., rate) is nonconcave, we show that the optimization problem is nonconvex and derive the optimality conditions for the problem. We then propose an iterative power allocation algorithm based on SCP for finding the OPA vector that meets the objective.
The objective of the OPA problem is to maximize the endtoend instantaneous rate subject to the constraint on maximum transmit power (P_{i,max}) at each node F_{i}. This problem can be formulated as
Since the objective function in (33a) is a nonsmooth and nonconcave function of P [32], the optimization problem () is nonconvex and it is not possible to find the global optimal solution. However, the nonsmooth optimization problem can be converted to a smooth optimization problem and the optimality condition can be derived by writing the hypograph form [32] for the problem () as follows:
Theorem 1
The endtoend instantaneous rate of the multihop FDR system is maximized (i.e., optimization problems () and () have optimal solution), if and only if the instantaneous transmission rates over all the links of the network are equal, i.e.,
where Î¶^{âˆ—} is the maximum achievable endtoend rate.
Proof
See Appendix 2. â–¡
Theorem 1 depicts the necessary condition for any solution of () to be optimal and any feasible solution of () which violates (35) only be a feasible solution which is not optimal. To find OPA vector P^{âˆ—} that achieves the maximum rate, we have to solve the system of N+1 nonlinear equations (35) with N+2 unknowns (P^{âˆ—}, Î¶^{âˆ—}), which is very difficult especially when N is large. Thus, we turn our attention towards developing an efficient iterative algorithm to solve ().
Sequential convex programming (SCP) is an efficient lowcomplex and fastconverging algorithm to solve nonconvex problems [32]. Accordingly, we devise an iterative algorithm based on SCP to determine the OPA for (). SCP finds the local optimal solution of a difficult problem, through solving a sequence of approximate convex problems. However, it is not possible to apply SCP directly to (), since it is not possible to find the approximate convex problem for (). Accordingly, we use the transformed smooth optimization problem () for applying SCP. Notice that () and () are equivalent in the sense of optimality. The optimization problem () can further be transformed by writing R_{j}(P) as difference of two concave functions [33, 34]. This transformation makes it easier to determine the approximate convex problem of () at each iteration of the SCP [35]. Define the functions f and g as:
Now, the functions f and g are concave functions of P [32] and R_{j}(P) can be written as f_{j}(P)âˆ’g_{j}(P). Accordingly, the optimization problem () can be recasted as
Let P^{n} be the power allocation vector at step n of the iterative algorithm. The firstorder Taylor series approximation \(\tilde {g_{j}}(\boldsymbol {P},\boldsymbol {P}^{n})\) of g_{j}(P) at P^{n} is given as follows:
where the gradient of g_{j}(P) at P is represented as âˆ‡g_{j}(P) and is expressed as follows:
Here, e_{j} is the N+1 dimensional column vector, e_{j}(jâˆ’1)=0 and e_{j}(i) = \(\frac {h_{{i},{j}}^{2}}{\ln 2}\), iâ‰ jâˆ’1. Further, \(\tilde {g_{j}}(\boldsymbol {P},\boldsymbol {P}^{n})\) has the following properties with g_{j}(P).
Due to the slow sensitivity of the function g_{j}(P) to the changes in P, the approximation given by (39) is very tight over a fairly large neighborhood of P^{n}. Further, property (i) arises due to the concavity of g_{j}(P) and it depicts that the global information can be estimated using the local information (i.e., g_{j}(P^{n}) and \(\nabla g_{j}^{T} (\boldsymbol {P}^{n})\)) ([32], chapter 3, sec 3.1.3).
Now, by plugging (39) into () and using (41), the approximated convex optimization problem can be transformed into a lower bound optimization problem to () as:
The constraints (42b) and (42c) form a convex set. Further, the constraint (42b) is smooth; hence, () is a standard convex optimization problem, which can be solved by using efficient software packages for convex optimization [36]. Algorithm 1 describes the iterative procedure to be followed for finding the OPA vector. In each iteration n, Algorithm 1 solves the approximate convex problem () and the solution obtained in iteration n (i.e., P^{n}) is used to determine the successive approximate convex problem in iteration n+1. The process continues until the condition on the convergence is satisfied. In the simulation, we observe that Algorithm 1 converges very quickly to the optimal solution.
Lemma 1
The iterative Algorithm 1 described above (i) converges and (ii) the convergence point is a KKT point for the original problem ().
Proof
See Appendix 3 â–¡
As the original problem () is nonconvex, it is not possible to prove the convergence to a global optimal point. Since the algorithm is based on SCP, which finds local optimal solutions, it may not guarantee global optimal solution to the original nonconvex problem. Even if it achieves global optimality, it cannot be proved theoretically. However, from Lemma 1, the algorithm achieves a KKT point to the original problem. The computational complexity of Algorithm 1 is proportional to \(O \left (\lceil{I^{max}  I^{0}  \frac {\epsilon }{\Delta I}}\rceil\ max\left \{{(N+1)}^{3}, S\right \}\right)\) ([32], chapter 1, sec 1.3.1), where \(I^{max}\ = \ \mathop {\max \min }\limits _{\boldsymbol {P} \quad j} \quad f_{j}(\boldsymbol {P})  g_{j}(\boldsymbol {P})\), Î”I = minn[I^{n}âˆ’I^{nâˆ’1}] and S is the cost of computing f_{j}(P) and their first and second gradients [32]. Accordingly, the proposed iterative algorithm has affordable computational complexity and the optimization problem can be solved in polynomial time which makes it suitable for practical applications.
4.1.2 Endtoend outage probability minimization
Here, we formulate the OPA problem that minimizes the endtoend outage probability of the multihop FDR system. Since the expressions for the endtoend outage probability described in Sections 3.1.1 and 3.1.2 are not in a tractable form to optimize, we use the asymptotic outage probability described in Section 3.1.2. We transform the outage probability minimization into a standard geometric programming problem (GPP) and establish that the optimal solution can be obtained by solving the GPP. Consider the expression for asymptotic outage probability of multihop FDR system under Rayleigh fading:
From (43), it is observed that minimization of \(\hat {P}^{FDR}_{out, asy}\) is equivalent to minimization of \(Q(\boldsymbol {P})~=~\log \left (1\hat {P}^{FDR}_{out, asy}\right) = {\sum \nolimits }_{j =1}^{N+1} \frac {\Gamma _{T}}{\hat {\beta }_{j1, j}} \left (1+ {{\sum \nolimits }_{\substack {i=0 \\ i\neq j1}}^{N} \hat {\beta }_{i,j}}\right) = {\sum \nolimits }_{j =1}^{N+1} \frac {\Gamma _{T}}{P_{j1}{\pi }_{j1, j}} \left (\sigma ^{2}_{j}+ {{\sum \nolimits }_{\substack {i=0 \\ i\neq j1}}^{N}P_{i} {\pi }_{i,j}}\right)\). Choosing Q(P) as the objective function, the OPA problem is formulated as:
Now, by inspecting (), we observe the following cases: (1) the objective (44a) is a posynomial function, (2) the left hand side of the inequality constraint is a posynomial function, and (3) the right hand side of the inequality constraint is monomial [37]. Now, () is a GPP in standard form; GPPs can be reformulated as convex optimization problems and can be solved efficiently using standard software package for convex optimization [36].
The power allocation considered above minimizes the asymptotic outage for Rayleigh fading case. With the help of numerical and simulation results, we have verified that the OPA minimizes the outage for Nakagami fading as well, provided m_{i,j}=mâˆ€i, j. We provide the numerical results for the optimal power allocation in Section 5.
Further, the power allocation procedures described in this section can be implemented in a centralized manner. It is assumed that either ICSI or SCSI corresponding to various links in the multihop network are available at a centralized controller (i.e., either source or destination). The nodes in the network have to collect the SCSI or ICSI and the information should be fed back to the centralized controller. For node F_{i}{i âˆˆ(1,â€¦,N+1)} to have the required information (i.e., either ICSI or SCSI), all the N+2 nodes in the network can sequentially transmit a predefined pilot message. Under the assumption of channel reciprocity and with the broadcast nature of wireless transmissions, every other node can use the predefined pilot transmission to learn the channel. After N+2 pilot transmissions, every node would learn their channel to every other node. Each node communicates the acquired information to the central entity. The central entity can run Algorithm 1, compute the OPA vector, and distribute the transmit power values among the nodes in the network [38].
4.2 Optimal power allocation for multihop HDR system
Here, we formulate OPA problems for multihop HDR system. Let \(R^{HDR}_{j}(\boldsymbol {P})\) be the instantaneous transmission rate corresponding to the link j, j=1,2,...N+1. From (27), the achievable endtoend instantaneous transmission rate of the multihop HDR system is computed using the formula given as follows:
The OPA problem for the multihop HDR that maximizes the endtoend rate is formulated as
The optimization problem () is similar to the problem (). Optimal solution for problem (), i.e., OPA for multihop HDR system, can be determined by following the procedure adopted for the multihop FDR system.
Consider the asymptotic outage probability for multihop HDR system \(\bar {P}^{HDR}_{out, asy}\). Now, minimization of \(\bar {P}^{HDR}_{out, asy}\) given by (32) is equivalent to \(Q^{HDR}(\boldsymbol {P}) = \log (1\bar {P}^{HDR}_{out, asy})\). Choosing Q^{HDR}(P) as the objective function for outage probability minimization, the OPA problem for multihop HDR system is formulated as:
The optimization problem () is similar to the problem (). Optimal solution for problem (), i.e., OPA for multihop HDR system, can be determined by following the procedure adopted for the multihop FDR system.
5 Results and discussion
This section describes the results for the outage probability and related metrics of multihop FDR and HDR systems under the proposed power allocation algorithm. We assume a linear network, where the distance between the adjacent nodes are set as equal. Further, the normalized distance between the nodes F_{0} and F_{N+1} (normalized with respect to a reference distance d_{0}) is set equal to 10 and it is assumed to be the same regardless of the number of relay nodes, N in the network. We set \(\pi _{{i},{j}}\ = \ G d^{\eta }_{ij}\), where G is the propagation constant, Î· is the path loss exponent, and d_{ij} is the distance between relay nodes F_{i} and F_{j}. We select Î· as 3 and the target rate r=0.1 bps/Hz. The maximum available transmit power and the average power due to RSI are the same for all the nodes, i.e., P_{j,max}=P_{max} and Ï€_{j,j} = Ï€_{0} âˆ€ j. Furthermore, we assume the Nakagami parameter m_{i,j}= m and the noise power to be unity. Under UPA, it is assumed that all the nodes in the network operate at P_{max}. Extensive Monte Carlo simulations have been performed to corroborate the analytical results. In the implementation of the proposed Algorithm 1, the tolerance level Îµ has been chosen as equal to 10^{âˆ’6}.
In Fig. 2, the analytical and simulation results for outage probability of fourhop FDR/HDR system has been drawn against P_{max} assuming UPA (i.e., equal transmit power for all the nodes). As P_{max} increases, the outage probability of both FDR and HDR systems decreases. However, for higher values of P_{max}, the outage probability exhibits a floor behavior, due to the higher level of IRI. Figure 2 also shows the impact of RSI (Ï€_{0}) on the outage probability of FDR system. When the RSI is comparatively lower, FDR performs significantly better than HDR due to the fact that HDR systems need to achieve double the target rate of FDR, i.e., for a given rate r, the threshold SINR required for HDR system is higher than that of FDR [4]. However, when RSI is higher, the range of P_{max} values over which FDR performs superior than HDR decreases. For low and medium values of P_{max}, FDR continues to exhibit improved performance than HDR. However, when P_{max} is higher, FDR becomes inferior to HDR because of the RSI and higher amount of IRI in the FDR system as compared to the spectral efficient HDR system [23] considered in this paper. Figure 2 also shows the results evaluated with approximate outage probability expressions given by (22) and (30) for the multihop FDR and HDR systems respectively. It can be observed that the approximate outage probability model is highly accurate, i.e., the outage probability corresponding to the approximate model is within 5% of the exact outage probability values.
In Fig. 3, we plot the analytical and simulation results for the outage probability of fourhop FDR network under UPA and the proposed OPA for different values of m, the Nakagami fading parameter, while Fig. 4 shows the corresponding results for HDR systems. The results show that the outage probability of FDR and HDR systems improve significantly under OPA compared to UPA, especially for higher values of P_{max}. When P_{max} is very high, FDR/HDR systems with conventional UPA suffer from very high IRI whereas the proposed OPA strategy allocates the transmit power among the nodes in such a way that the effect of IRI is reduced, thereby improving the outage performance. With P_{max}=30 dB and m=1, the optimal power allocation vector (in dB) has been obtained as P^{âˆ—}â€‰=â€‰[30,28.88,25.46,23.05]; it provides approximately 30% reduction in outage probability for the FDR system as compared to that achievable under UPA. Similarly, with P_{max}=30 dB and m=1, the corresponding optimal power allocation (in dB) for the HDR system is given by P^{âˆ—}â€‰=â€‰[30,30,23.45,23.45]; it provides approximately 49% reduction in outage probability for the HDR system as compared to that achievable under UPA. Further, as m increases, the percentage reduction in outage probability has been observed to be higher, i.e., when m=2, the outage probability reduces by 69% (89%) for FDR (HDR) system under OPA, as compared to the corresponding system under UPA.
In Fig. 5, we plot the outage probability of the FDR system against the number of relays (N) in the network. Since the distance between source and destination is assumed to be constant, the mean signal power over each link depends on N. As shown, OPA always has an edge over UPA when outage performance is considered. Initially, the outage probability decreases as N is increased owing to the reduction in hop length. However, when N increases beyond a certain value, the outage probability significantly increases. This happens due to higher amount of IRI in the system as N becomes larger. From the graphs, it is also clear that an optimum value for N exists that leads to minimum outage probability and the corresponding value of N depends on P_{max}.
In the next set of simulations, we consider perfect knowledge of ICSI. The channel gain matrix H=[h_{i,j}^{2}] has been assumed as follows:
where each element h_{i,j}^{2} is generated randomly as Gamma\(\sim \left (m_{ {i,j}}, \frac {\pi _{ {i,j}}}{m_{ {i,j}}} \right)\). The power allocation vector that maximizes the endtoend instantaneous rate has been determined by using Algorithm 1.
Figure 6 illustrates the convergence of Algorithm 1 for fourhop FDR and HDR systems under ICSI. It can be seen that the proposed algorithm converges very fast and the optimal solution can be obtained with a few number of iterations. For example, when P^{0}=0.5 P_{max}, the algorithm converges within three iterations for the FDR system and the maximum endtoend instantaneous rate R(P) obtained is equal to 2.1999 bps/Hz and optimal power (in dB) allocation vector is obtained as P^{âˆ—}=[ 40,38.06,27.86,35.20]. Further, for this P^{âˆ—}, the maximum endtoend instantaneous rate has been observed to be equal for all the individual links, i.e., R_{j}(P)=2.1999 bps/Hz. This corroborates the necessary condition for optimality (35) described in Section 4. Similarly, in the case of HDR systems, the maximum endtoend instantaneous rate R^{HDR}(P) is obtained as 1.8764 bps/Hz which corresponds to the optimal power (in dB) allocation vector P^{âˆ—}â€‰=â€‰[38.91,40,25.14,32.46]. Further, the maximum instantaneous rate over each link has been observed to be equal, i.e., \(R^{HDR}_{j}(\boldsymbol {P})=1.8764\) bps/Hz (j = 1,2,3,4) which validates (35).
Figure 7 depicts the comparison of the maximum endtoend instantaneous rate that can be achieved for fourhop FDR and HDR systems as a function of P_{max} under the proposed OPA strategy. The endtoend instantaneous rate that can be obtained under UPA is also shown. The results confirm that the proposed OPA improves the endtoend instantaneous transmission rate for both FDR and HDR systems, especially for higher values of P_{max}. When P_{max}=30 dB, the percentage improvement in endtoend rate is 174% for the FDR system and 224% for the HDR system as compared to the corresponding systems with UPA. Notice that, initially, the endtoend instantaneous rate increases as P_{max} is increased; however, it remains constant for higher values of P_{max} for both FDR as well as HDR systems, owing to the higher levels of IRI arising out of the increased P_{max}.
Figure 8 compares the speed of convergence of SCP and bisectionbased algorithms (i.e., Algorithms 1 and 2 given in Appendix 4) respectively considered for the rate maximization problem in this paper. Notice that both algorithms converge to the same optimal point for HDR as well as FDR systems. However, it is observed that SCPbased algorithm converges faster than bisectionbased method for a given tolerance level Îµ. Specifically, it can be observed that the SCPbased algorithm converges within four iterations while the bisectionbased algorithm converges to the optimal point only after 22 iterations. This implies that the SCPbased OPA algorithm has lower computational complexity as compared to the bisectionbased OPA algorithm.
Since conventional HDR networks use multiphase relaying, the endtoend delay experienced is larger, which considerably affects the spectral efficiency. In addition to the delay introduced by multiphase transmission, each relay node introduces processing delay as well. However, in multihop FDR networks, since the relay nodes perform simultaneous transmission and reception, the processing delay at each node and the endtoend delay would be much lower than that of HDR networks. The higher endtoend delay will have significant influence on the spectral efficiency of multihop HDR networks, while its impact would be insignificant in the case of FDR networks.
6 Conclusions
In this paper, we have derived closedform expression for the endtoend outage probabilities of multihop FDR and HDR networks in Nakagamim fading channel. We have also derived approximate expressions for the outage probability which were observed to be highly accurate. We then considered optimal power allocation (OPA) to maximize endtoend rate with constraints on maximum transmit power at each node. It was shown that the OPA problem is nonconvex. Then, based on sequential convex programming, we have proposed an iterative algorithm to solve the optimization problem. With the help of extensive simulations, it was established that the algorithm converges very fast and optimal solution (i.e., OPA vector) can be obtained with a few number of iterations. We also considered OPA to minimize the endtoend outage probability with constraints on maximum transmit power at each node. We employed geometric programming to solve the OPA problem. Further, it was shown that the proposed OPA strategies would lead to significant improvements in endtoend outage probability and data rate performance of multihop FDR/HDR systems as compared to uniform power allocation strategy.
7 Methods/Experimental
The purpose of this study is to investigate and optimize the outage and rate performance of the multihop FDR system. The system consists of a source node, destination node, and N relay nodes. The relay nodes operate in full duplex mode. The channels between the nodes are assumed to follow Nakagamim fading, and it is assumed that the RSI channel has Nakagamim fading characteristics. Further, closedform expressions for endtoend outage probability are provided, power optimization is based on SCP and GPP is performed. The investigations are repeated for spectral efficient HDR system. Since the relay node in HDR system needs two orthogonal channels for transmission and reception, for fair comparison, the target rate of HDR system is chosen as twice that of FDR system.
8 Appendix 1: Derivation of (18)
Assume that the double integral term in (17) is represented as Î¸, which can be written as follows:
Using incomplete Gamma function [40], we get
With the help of series expansion for the incomplete Gamma function and by using binomial expansion of (1+y)^{n}, we get
Now, Î¸ given by (50) can be written as follows by using the definition of Gamma function:
Notice that here M=m_{i,j}, \(a = \frac {1}{\beta _{i, j}}\), N=m_{jâˆ’1,j} and \(b = \frac {\Gamma _{T}}{\beta _{j1, j}}\). By substituting the resulting expression for Î¸ in (17), and after mathematical simplification, we get the following:
9 Appendix 2: Proof of Theorem 1
Let Î¶^{âˆ—} be the maximum achievable endtoend rate. By contradiction, assume that R_{k}(P)>Î¶^{âˆ—} for some k. Now, strictly decrease P_{kâˆ’1} by Î´_{kâˆ’1} such that R_{k}(P)>Î¶^{âˆ—}. Decrementing P_{kâˆ’1} results in reduction of interference at F_{j},jâ‰ k. Therefore, there exists a positive Îµ_{j} such that R_{j}(P)â‰¥Î¶^{âˆ—}+Îµ_{j}>Î¶^{âˆ—}, âˆ€ j. This contradicts with the assumption that Î¶^{âˆ—} is optimal. Hence, at the optimal point, instantaneous rate of all the links of the multihop FDR network are equal.
10 Appendix 3: Proof of Lemma 1
(i) The constraint set of () is compact and hence closed and bounded. From Algorithm 1, we have the sequence {I^{n}} where
In (53), the first inequality arises due to the fact that P^{n+1} is an optimal solution of () in n+1th step, thus the obtained value of f_{j}(P^{n+1}) â‰¥f_{j}(P^{n}). Further, the second inequality follows from the concavity of g_{j}(P).
Now, the sequence I^{n} is monotonic increasing sequence on a compact set and hence bounded. By Cauchy theorem, the sequence {I^{n}} converges.
(ii) The convergence of Algorithm 1 implies I^{n} =I^{n+1} in the limit. From (i) Algorithm 1 converges, then the convergence point is a KKT solution. Hence, P^{n+1} is a KKT point of ().
The Lagrangian dual for () can be written as
From (i) above, at the optimal point P^{n+1}=P^{n}. Now, the KKT conditions at the optimal point (P^{n+1},Î¶^{n+1}) and using (41) implies
Now, the conditions (55)â€“(58) are same as the KKT optimality conditions for the problem (). Thus, P^{n+1} is the KKT point of the optimization problem (). We complete the proof.
11 Appendix 4: Power allocation based on bisection method
In this section, we investigate an iterative algorithm for OPA based on bisection method. Though the optimization problem () is nonconvex in the original form, it can be transformed to a quasiconcave optimization problem [32, 39]. Since, log(.) is a monotonic increasing function, () is equivalent to the following optimization problem:
This is known as generalized fractional programming, where Î“_{j}(P) is a linear fractional function and hence it is a quasiconcave function ([32], chapter 3, sec 3.4.1); further minimum of quasiconcave functions is also a quasiconcave function [[32], chapter 3, sec 3.4.4]. Accordingly, () is a quasiconcave optimization problem and it can further be transformed by introducing an auxiliary variable Ï„, as follows:
Now, () is a quasiconcave optimization problem, whose solution can be determined by solving sequence of concave feasibility problems with the help of bisection search on Ï„. Algorithm 2 presents the bisection method to determine the OPA vector.
Algorithm 2 converges in log(Îµ_{0}/Îµ) iterations, each iteration involved in solving a concave feasibility problem () for a given value of Ï„. Accordingly, the algorithm has the computational complexity proportional to O(log(Îµ_{0}/Îµ) max{(N+1)^{3}}) ([32], chapter 4, Section 4.2).
Abbreviations
 AF:

Amplifyandforward
 DF:

Decodeandforward
 FD:

Full duplex
 FDR:

Full duplex relaying
 HD:

Half duplex
 HDR:

Half duplex relaying
 ICSI:

Instantaneous channel state information
 IRI:

Interrelay interference
 KKT:

KarushKuhnTucker
 OPA:

Optimal power allocation
 RSI:

Residual selfinterference
 SCSI:

Statistical channel state information
 UPA:

Uniform power allocation
References
MO Hasna, MS Alouini, Outage probability of multihop transmission over Nakagami fading channels. IEEE Commun. Lett.7(5), 216â€“218 (2003).
GK Karagiannidis, TA Tsiftsis, RK Mallik, Bounds for multihop relayed communications in Nakagamim fading. IEEE Trans. Commun.54(1), 18â€“22 (2006).
G Liu, et al, Inband fullduplex relaying: a survey, research issues and challenges. IEEE Commun. Surv. Tutor. 17(2), 500â€“524 (2015).
T Kwon, et al, Optimal duplex mode for DF relay in terms of outage probability. IEEE Trans. Veh. Technol. 59(7), 18â€“22 (2010).
M Duarte, C Dick, A Sabharwal, Experimentdriven characterization of fullduplex wireless systems. IEEE Trans. Wirel. Commun. 11(12), 4296â€“4307 (2012).
E Everett, D Dash, C Dick, A Sabharwal, in Proc. 49th AACCCC. Selfinterference cancellation in multihop fullduplex networks via structured signaling (IEEEMonticello, 2011), pp. 1619â€“1626.
K Dhaka, RK Mallik, R Schober, in Proc. ICC. Optimal power allocation in a multihop decodeandforward communication system (IEEEOttawa, 2012), pp. 4458â€“4462.
SS Ikki, S Aissa, Multihop wireless relaying systems in the presence of cochannel interferences: performance analysis and design optimization. IEEE Trans. Veh. Technol. 61(2), 566â€“573 (2012).
TP Do, I Song, YH Kim, Optimization of power allocation and relay position in multihop relay networks over generalized fading channels. IEEE Trans. Veh. Technol. 64(11), 5031â€“5041 (2015).
H Shen, W Xu, C Zhao, QoS constrained optimization for multiantenna AF relaying with multiple eavesdroppers. IEEE Signal Process. Lett.22(12), 2224â€“2228 (2015).
H Shen, W Xu, C Zhao, A semiclosed form solution to MIMO relaying optimization with sourcedestination link. IEEE Signal Process. Lett.23(2), 247â€“251 (2016).
TKr Baranwal, DS Michalopoulos, R Schober, Outage analysis of multihop fullduplex relaying. IEEE Commun. Lett.17(1), 63â€“66 (2013).
PK Sharma, P Garg, Performance analysis of full duplex decodeandforward cooperative relaying over Nakagamim fading channels, Trans. Emerg. Telecommun. Technol. 25(9), 905â€“913 (2014).
S Han, L Chen, W Meng, C Li, in Proc. ICCC. Outage probability of multihop fullduplex DF relay system over Nakagamim fading channels (IEEEShenzhen, 2015), pp. 1â€“6.
R Katla, AV Babu, in Proc. INDICON. Outage performance of multihop full duplex relaying system over Nakagamim fading channels (IEEEBangalore, 2016), pp. 1â€“6.
T Riihonen, S Werner, R Wichman, Hybrid fullduplex/halfduplex relaying with transmit power adaptation. IEEE Trans. Wireless Commun.10(9), 3074â€“3085 (2011).
LJ Rodriguez, NH Tran, T LeNgoc, Optimal power allocation and capacity of fullduplex AF relaying under residual selfinterference. IEEE Wireless Commun. Lett.3(2), 233â€“236 (2014).
B Yu, L Yang, X Cheng, R Cao, Power and location optimization for fullduplex decodeandforward relaying. IEEE Trans. Commun.63(12), 4743â€“4753 (2015).
S Li, Y Kun, M Zhou, J Wu, L Song, Y Li, H Li, Fullduplex amplifyandforward relaying: power and location Optimization. IEEE Trans. Veh. Technol. 60(9), 8458â€“8468 (2017).
S Li, et al, in Proc. 19th WPMC. Joint power allocation and location optimization for fullduplex amplifyandforward relay networks (IEEEShenzhen, 2016), pp. 250â€“255.
D Rajan, Optimum number of hops in linear multihop wireless networks. Int. J. Adv. Eng. Sci. Appl. Math. 5(1), 32â€“42 (2013).
B Mahboobi, M Ardebilipour, Joint power allocation and routing in fullduplex relay network: an outage probability approach. IEEE Commun. Lett.17(8), 1497â€“1500 (2013).
G Farhadi, JM Cioffi, Spectral efficient multihop relaying based on alternate transmission. IEEE Trans. Wireless Commun.10(11), 3601â€“3606 (2011).
JN Laneman, DNC Tse, GW Wornell, Cooperative diversity in wireless networks: efficient protocols and outage behavior. IEEE Trans. Inf. Theory. 50:, 3062â€“3080 (2004).
R Pabst, et al, Relaybased deployment concepts for wireless and mobile broadband radio. IEEE Commun. Magazine. 42(9), 80â€“89 (2004).
Y Yang, H Hu, J Xu, G Mao, Relay technologies for WiMax and LTEadvanced mobile systems. IEEE Commun. Magazine.47(10), 100â€“105 (2009).
M Sikora, JN Laneman, M Haenggi, DJ Costello, TE Fuja, Bandwidth and powerefficient routing in linear wireless networks. IEEE Trans. Inf. Theory. 52(6), 2624â€“2633 (2006).
A Goldsmith, Wireless Communications (Cambridge University Press, California, 2005).
DP Moya Osorio, EE Bentez Olivo, H Alves, JCS Santos Filho, M Latvaaho, Exploiting the direct link in fullduplex amplifyandforward relaying networks. IEEE Signal Process. Lett. 22(10), 1766â€“1770 (2015).
GK Karagiannidis, NC Sagias, TA Tsiftsis, Closedform statistics for the sum of squared Nakagamim variates and its applications. IEEE Trans. Commun.54(8), 1353â€“1359 (2006).
FE Satterthwaite, An approximate distribution of estimates of variance components. Biom. Bull.2(6), 110â€“114 (1946).
S Boyd, L Vandenberghe, Convex optimization (Cambridge University Press, New York, 2004).
H Tuy, Convex analysis and global optimization (Kluwer Academic, Hanoi, 1998).
HH Kha, HD Tuan, HH Nguyen, Fast global optimal power allocation in wireless networks by local d.c.programming. IEEE Trans. Wireless Commun. 11(2), 510â€“515 (2012).
A Beck, A BenTal, L Tetruashvili, A sequential parametric convex approximation method with applications to nonconvex truss topology design problems. J. Glob Optim. 47(1), 29â€“51 (2010).
M Grant, S Boyd, CVX: Matlab software for disciplined convex programming, version 1.21 (2010). Available: http://cvxr.com/cvx.
S Boyd, SJ Kim, L Vandenberghe, A Hassibi, A tutorial on geometric programming. Optim. Eng.8(1), 67â€“127 (2007).
D Mishra, S De, CF Chiasserini, Joint optimization schemes for cooperative wireless information and power transfer over rician channels. IEEE Trans. Commun.64(2), 554â€“571 (2016).
ZQ Luo, S Zhang, Dynamic spectrum management: complexity and duality. IEEE J. Sel. Top. Sign. Process. 2(1), 57â€“73 (2008).
IS Gradshteyn, IM Ryzhik, A Jeffrey, Table of Integrals, Series and Products, 6th edition (Academic press, 2000).
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Katla, R., Babu, A.V. Outage probability analysis and optimal transmit power allocation for multihop full duplex relay network over Nakagamim fading channels. J Wireless Com Network 2018, 192 (2018). https://doi.org/10.1186/s1363801812071
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DOI: https://doi.org/10.1186/s1363801812071