Selection of optimal transmit power in multi-hop underlay cognitive full-duplex relay networks
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Abstract
This paper investigates optimization of underlay multi-hop cognitive full-duplex relay (CogFDR) networks in independent non-identically distributed Rayleigh fading channels. First of all, analytical expressions for the outage probability experienced in the secondary network is formulated, taking into account (i) residual self-interference (RSI) arising due to full-duplex operation, (ii) inter-relay interference (IRI) arising due to frequency re-use, and (iii) interference generated by the primary transmitter on the secondary network. Optimal power allocation (OPA) that either minimizes the end-to-end outage probability or maximizes the end-to-end instantaneous rate is investigated with constraints on total available power in the secondary network and tolerable interference power at the primary receiver. The OPA vector for the outage minimization problem is obtained by solving an equivalent geometric programming problem (GPP) and that for the rate maximization problem is obtained by applying the rate balancing criterion for each hop. Extensive performance evaluations conducted with the help of Monte Carlo simulations show that transmit power optimization can improve the end-to-end rate and outage probability performance of multi-hop CogFDR network in comparison to equal power allocation on average (EPA).
Keywords
Full-duplex Multi-hop Cognitive Underlay Power allocation Geometric programmingAbbreviations
- AF
Amplify-and-forward
- CogFDR
Cognitive-based FDR
- CogHDR
Cognitive-based HDR
- CR
Cognitive radio
- CRN
Cognitive relay network
- DF
Decode-and-forward
- EPA
Equal power allocation on average
- OPA
Optimal power allocation
- FD
Full-duplex
- FDR
Full-duplex relaying
- HD
Half-duplex
- HDR
Half-duplex relaying
- ICSI
Instantaneous channel state information
- IRI
Inter-relay interference
- PR
Primary receiver
- PT
Primary transmitter
- RSI
Residual self-interference
- SCSI
Statistical channel state information
- SR; Secondary receiver; ST
Secondary transmitter
1 Introduction
Cognitive radio (CR) networks improve the spectrum utilization efficiency by allowing secondary (unlicensed) users to make use of the primary (licensed) user spectrum without degrading the communication performance of the primary network [1, 2]. In general, a CR network can operate either in the overlay or in the underlay mode. In the former, the secondary users are allowed to access the spectrum occupied by the primary if the primary user is inactive. In the underlay mode, the secondary users can coexist with the primary user by sharing its frequency band, provided the interference caused by the secondary node transmissions on the primary network, lies within a predefined threshold value. However, this leads to serious constraints on maximum transmit power that the secondary nodes can choose for their operation. To improve the coverage and capacity performance of the secondary network, relay nodes can be deployed to assist the secondary transmitter (ST) in relaying information to the secondary receiver (SR). Such networks that combine the features of cooperative relaying and cognitive radio are known as cognitive relay networks (CRNs) [3].
In CRNs, the relay nodes can operate either in half-duplex (HD) or in full-duplex (FD) mode. In half-duplex relaying (HDR), two orthogonal time or frequency channels are allotted for the relay transmission and reception. Whereas, full-duplex relaying (FDR) allows concurrent reception and retransmission using the same frequency band. Even though FDR leads to two-fold improvement in spectral efficiency, it causes self-interference (SI) because of the coupling between the relay node’s transmitter and receiver circuits [4]. Recently several proposals for cancellation of SI have appeared in the literature [5, 6]. However, even with advanced SI cancellation methods, a level of residual self-interference (RSI) remains that significantly deteriorates the performance of FD systems [7, 8, 9]. In a FD-based CRN, the relay nodes in the secondary network operate in the FD mode [10]. The CRN based on FD would differ in performance in comparison to that of conventional HD network, due to the presence of RSI. This paper considers a cognitive FD relay (CogFDR) network in underlay mode, with a multi-hop secondary network, i.e., the secondary network comprises of a secondary transmitter (ST), a secondary receiver (SR), and N FD relay nodes connecting ST and SR. In underlay CRNs, even though the secondary users are allowed to coexist by sharing the spectrum allotted to the primary, they need to choose the transmission powers such that the interference induced at the primary receiver (PR) does not go beyond the maximum tolerable limit. This significantly affects the outage and rate performance of the secondary user in the CogFDR network. Further, the performance is affected by the amount of RSI as well, which is related to the transmission power of the relay nodes and the power gain of the corresponding channel. The major objective of this paper is to develop a model for finding the secondary users’ outage probability in a multi-hop CogFDR system and to investigate transmit power optimization for improving the secondary users’ rate and outage probability performance.
1.1 Related work
Extensive research has been reported in the literature on the performance evaluation and optimization of CRNs operating in half-duplex mode (i.e., CogHDR networks), e.g., [11, 12, 13, 14, 15]. The authors in [11] have considered relay selection for the secondary network in underlay CogHDR with secondary nodes selecting fixed transmit powers for their operation. The outage probability of an underlay decode-and-forward (DF) CogHDR network has been analyzed in [12], while one-dimensional relay location optimization problem has been addressed in [13]. Optimal relay placement in the secondary network of a CogHDR network has been addressed in [14] assuming Nakagami-m fading channels, while the work in [15] has considered transmit power optimization to maximize the achievable ergodic capacity of CogHDR network. In addition to these, many authors have attempted to analyze and improve the performance of wireless networks, see, e.g., [16, 17, 18, 19] and references therein. Resource allocation is very crucial in wireless systems. The authors in [16] have considered joint optimization of computation and communication power in multi-user massive MIMO system, which improves the energy efficiency. In [17], the authors have studied the performance of a cooperative multi-relay system where the relays harvest energy from RF signals of the source. The authors in [18] have analyzed the performance of a dual-hop FD system with multiple relays, which provide spectral efficiency and diversity gains respectively. In [19], the authors have proposed a virtual FD relaying scheme for a cooperative multi-path relay channel (MPRC) with multiple half-duplex relays.
Recently, the performance analysis and optimization of dual-hop CogFDR network has been carried out extensively by many researchers [20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. In [20], the authors have presented an analytical model to evaluate the outage probability of secondary user in dual-hop CogFDR network by considering a joint decoding scheme at the SR in which the signals arriving at the SR from ST and the relay are utilized together for the decoding purpose. Two-way FD relay spectrum sharing protocol for a dual-hop CogFDR network has been proposed in [21], and the outage probability of the network has been evaluated under the proposed protocol. The outage performance of both primary and secondary users in a CogFDR network has been analyzed in [22]. Optimal power allocation algorithms for amplify-and-forward (AF)-based CogFDR network has been analyzed in [23] assuming Rayleigh fading. Two relay selection schemes, i.e., partial relay selection and optimal relay selection, have been proposed in [24], and the outage performance of CogFDR network has been carried out assuming the availability of multiple relay nodes. Joint power control and relay selection algorithms that maximize the transmission rate of the secondary network has been addressed in [25] for a dual-hop CogFDR framework. In [26], the authors have investigated transmit power optimization problem in the context of dual-hop CogFDR network. In [27], the authors have proposed an analytical model for evaluating the outage probability of a dual-hop CogFDR network by considering various locations for the relay node, while neglecting the impact of direct transmission link from ST to SR. In [28], the authors have derived a closed form expression for the outage probability of an OFDM-based CRN with relay selection considering a dual-hop secondary network. In [29], authors have considered optimal power allocation for maximizing the transmission rate in AF-based dual-hop CogFDR network.
In addition to the above papers, a few authors have analyzed the performance of multi-hop CRN as well, i.e., a CRN with multi-hop secondary network [30, 31, 32, 33, 34, 35]. In [30], the authors have analyzed optimal power allocation that maximizes the end-to-end throughput for a CRN with multi-hop secondary network; however, RSI is ignored by assuming the intermediate relay nodes to operate in half-duplex mode. In [31, 32], the outage probability of a DF multi-hop half-duplex secondary network with power beacon-assisted energy harvesting based is analyzed. In [33], the authors have analyzed the end-to-end outage probability of a cluster-based multi-hop CRN. The authors of [34] have analyzed the throughput and end-to-end outage performance of multi-hop CRN, where the secondary relay nodes harvest energy from PU and preceding secondary relay nodes. In [35], the authors have analyzed the impact of channel information imperfection on the outage performance of multihop CRNs. However, all the above mentioned papers [30, 31, 32, 33, 34, 35] consider the relay nodes to use HDR while the focus of the current work is on multi-hop CogFDR networks. Even though FDR has been employed in the secondary network of CRNs [20, 21, 22, 23, 24, 25, 26, 27, 28, 29], majority of them consider dual-hop secondary network for their analysis of outage and other performance metrics. Finding the transmit power allocation that optimizes the performance of secondary network in multi-hop CogFDR network remains as an open problem.
1.2 Major objectives and contributions
In this paper, we focus on the outage probability and instantaneous transmission rate analysis of a multi-hop CogFDR/HDR network. We develop an analytical model for finding the outage probability of the secondary user considering the effects of both RSI and inter-relay interference (IRI), i.e., interference caused by simultaneous transmissions from neighboring nodes that operate in the same frequency band. The interference generated by the primary transmitter on the secondary network is also taken into account for the analysis. We then investigate optimal power allocation for the secondary nodes that either minimizes the end-to-end outage probability or maximizes the instantaneous rate subject to two constraints: (i) constraint imposed on the secondary node’s transmit power by the tolerable interference power threshold at PR and (ii) constraint on total transmit power available in the secondary network. We make an extensive study on the end-to-end rate and outage performance of the CogFDR network under optimal power allocation (OPA). For comparison, we report the results for CogHDR network as well. Numerical and simulation results establish that OPA can lead to significant improvement in end-to-end rate and outage probability performance of the secondary network in CogFDR system. The analytical results are validated by simulation results.
Remainder of the paper is arranged as follows. Section 2 describes the methods used in this work. Section 3 gives a detailed description of the system model for the underlay CogFDR network. In Section 4, the derivation of the outage probabilities are given. Section 5 describes the transmit power optimization by ignoring the effect of PT, while Section 6 considers the impact of PT for OPA determination. The simulation and analytical results are discussed in Section 7. Finally, the paper is concluded in Section 8.
2 Methods
In this paper, we consider a multi-hop underlay cognitive relay network with N DF relays placed between the secondary transmitter and receiver. The outage probability of the secondary network is derived considering the effect of RSI, IRI, and interference from the primary network. The outage probability minimization problem is formulated as a geometric programming problem. The transmit power allocation which maximizes end-to-end rate is determined by equating the rates over each hop. Monte Carlo simulations are done to validate the analytical results.
3 System model
where n_{j} denotes the additive white Gaussian noise at F_{j}. In addition to this, u_{j−1}, u_{j}, and u_{j+1} are the signals transmitted by F_{j−1}, F_{j}, and F_{j+1}, respectively, during the specified time instance.
4 Derivation of outage probability of the secondary user
In this section, we derive analytical expressions for the secondary user’s outage probability for the following cases: (i) by ignoring interference from PT and (ii) by considering the effect of interference from PT. We derive the outage probability expression for the CogHDR network as well.
4.1 Outage probability derivation ignoring interference from primary transmitter
In this section, we assume the PT to be located far away from the secondary network so that primary transmission does not cause any interference to the secondary network. Here, we derive the outage probability expressions for the multi-hop CogFDR and CogHDR networks.
4.1.1 Multi-hop CogFDR network
Selection of P_{j} according to (13) is known as equal power allocation on average (EPA) [26]. Notice that EPA does not optimize the system performance. In Section 5, we consider OPA for minimizing the outage probability.
4.1.2 Multi-hop CogHDR network
4.2 Outage probability of secondary user under the effect of interference from the primary transmitter
In this section, we find the outage probability of the secondary network for CogFDR and CogHDR scenarios by taking into account the effect of interference from PT on the secondary network performance. Assume that PT transmits a signal x_{p} to PR with fixed power P_{PT} and data rate R_{p}. The secondary nodes re-use this time resource to transmit their signals.
4.2.1 Multi-hop CogFDR network
Now, \(P_{\mathrm {out,FDR}}^{(1)}\) is determined by substituting (23a)–(25) in (21).
4.2.2 Multi-hop CogHDR network
5 Optimal power allocation (OPA) for the secondary nodes in CogFDR network ignoring primary interference
In this section, we consider optimizing the performance of CogFDR secondary network ignoring the influence of the interference from the primary network. We consider two optimization problems for the secondary network: (i) determine the OPA to minimize the end-to-end outage probability and (ii) determine the OPA to maximize the end-to-end instantaneous rate.
5.1 Outage probability minimization in multi-hop CogFDR network
where f_{0},f_{1},...f_{m} are posynomials and h_{1},h_{2}...h_{p} are monomials. From (31), notice that the objective function is in posynomial form. The inequality constraints also have a posynomial form in the left hand side and a monomial form on the right hand side. Hence, (31) is a standard GPP which can be solved using standard convex optimization software tools [39]. Notice that average channel gains (i.e., statistical channel state information-SCSI) alone is needed for finding the OPA for problem (31). The numerical and simulation results for the outage probability are presented in Section 7.
5.2 Instantaneous rate maximization in multi-hop CogFDR network
Here, (35c) represents the peak interference power threshold for PR.
Lemma 1
In DF relaying under a total power constraint, maximizing the minimum of rates corresponding to the N+1 links in the system is equivalent to a condition where the instantaneous received SINRs over all the links become equal, i.e., \(\Gamma _{j,\text {FDR}}^{(0)}(P) = \Gamma, \forall j=1,2,...N+1\).
Proof
The maximum achievable rate R^{max}(P) in a multi-hop relay network strictly increases with increase in total transmission power P of the network. Let us assume that \(P_{j}^{*}, j=\{0,1,2,..N\}\) is the optimal power allocation for achieving a maximum rate of R^{max}(P). In contradiction to this statement, consider an arbitrary hop k connecting nodes F_{k−1} and F_{k} having an instantaneous rate over this link, i.e., R_{k} satisfying R_{k}>R^{max}(P). Since R_{k} is an increasing function of P_{k−1}; let us slightly decrease P_{k−1} such that R_{k}≥R^{max}(P) is still unaffected. As P_{k−1} decreases, the IRI from node F_{k−1} to F_{k−2} decreases which improves the received SINR at F_{k−2}, which in turn increases the rate R_{k−2}. To compensate for the increase in R_{k−2}, we can slightly reduce P_{k−3}, which in turn increases R_{k−4} and so on. Therefore, with this new power allocation (i.e., with total power less than P), R_{k}≥R^{max}(P) still holds. This contradicts the original property that R^{max}(P) is a strictly increasing function of P. This proves that the optimal rate over any given hop F_{j} cannot be greater than R^{max}, so the optimum power allocation corresponds to the condition for which the hop rates are equal, i.e., R_{k}=R^{max}(P). Since rate is given by the expression, \(R_{j}(\textbf {P})~=~\text {log}_{2} (1+\Gamma _{j,\text {FDR}}^{(0)}(\textbf {P}))\), where log_{2}(x) is a monotonically increasing function of x, we can say that optimal power allocation which maximizes the minimum of rates is achieved by equating the rates, or equivalently, by equating the SINRs of all the links in the network. □
Notice that (38a) and (38b) are polynomial equations of order N+1, which needs to be solved to get Γ. Even though it is tough to determine the closed form solutions for Γ from the above equations, we can use efficient numerical algorithms to find the solution [43]. Let Γ_{1} and Γ_{2} respectively be the solution for Γ obtained from (38a) and (38b). Now, Γ (i.e., SINR corresponding to the optimal powers) is chosen as Γ=min(Γ_{1},Γ_{2}). Once Γ is known, the optimal power values \(P_{j,\text {FDR}}^{*}\), j=0,1,...N can be determined as follows: First of all, we find \(P_{N,\text {FDR}}^{*}\) using (37a). After that, we find \(P_{j,\text {FDR}}^{*}\), j=0,1,...N−1 using the recursive relations given by (37c). Further, notice that the complexity involved in the calculation of optimal transmit power is equivalent to that of finding the real roots of two polynomial equations of order N+1. Algorithm 1 illustrates the steps involved in determining the optimal transmit powers.
5.2.1 Three-hop CogFDR network : a case study
Let Γ_{1} and Γ_{2} be the solution corresponding to (44a) and (44b), respectively. Then Γ is chosen as Γ=min(Γ_{1},Γ_{2}). Substituting Γ in (42a)–(42c), we can find the optimal powers \(P_{i,\text {FDR}}^{*},\{i=0,1,2\}\).
6 Optimal power allocation (OPA) for the secondary network with primary interference
In this section, we determine the OPA that either minimizes the end-to-end outage probability or maximizes the end-to-end instantaneous rate of the secondary network under the influence of interference from PT.
6.1 Outage probability minimization in multi-hop CogFDR network with primary interference
Now, (47) is a standard GPP, which can be solved using Convex optimization software tools [39].
6.2 Rate maximization in multi-hop CogFDR network with primary interference
The polynomial Eqs. (51a) and (51b) can be solved to find Γ_{0}. Let Γ_{01} and Γ_{02} be the solution for Γ_{0} obtained from (51a) and (51b), respectively. Then, Γ_{0}=min(Γ_{01},Γ_{02}). The knowledge of Γ_{0} and ICSI will enable us to find \(P_{j,\text {FDR}}^{*}\) using (50).
7 Results and discussion
Here, we present the theoretical and simulation results for the outage and rate performance of multi-hop CogFDR/HDR networks considering the proposed OPA and the conventional EPA strategies. We consider a three-hop secondary network where the secondary nodes F_{j}, j=0,1,2,3 are assumed to be located on a straight line at (−1.5,0), (− 0.5,0), (0.5,0), and (1.5,0), respectively. The positions of PT and PR are assumed to be at (−1.5,1) and (− 0.5,1), respectively. The RSI at the secondary relays are assumed as equal to − 40 dB (unless otherwise specified). We select the path loss exponent (n) to be equal to four for all the links in the network. The channel gains are defined as \(\lambda _{i,j}=G D_{i,j}^{-n}\); here, G=1 and the distances D_{i,j} are calculated from the respective location coordinates. The target rate is chosen as r=0.1 bits/sec/Hz. The distance between ST and SR is fixed as equal to 3 km regardless of the number of relays in the network. The analytical findings are verified by extensive Monte Carlo simulations. Each point in the simulation result is obtained by averaging the results of 10^{5} simulation runs. In all figures, analytical plots are identified by lines and simulation plots by markers. We consider the EPA scheme as well, for which the transmit powers P_{j},j=1,2,...N corresponding to the secondary nodes in CogFDR network are set as given by (13).
Figure 7 depicts the outage probability comparison plots with respect to target rate variations. With increasing target rate, the outage performance is badly affected, since the SINR threshold is also increased. However, OPA leads to performance improvement in outage. At P/N_{0}=20 dB, the performance improvement in outage of CogFDR system is 21.6% as compared to the equivalent EPA case.
Comparison of power allocation schemes for various levels of IRI δ and total transmit power P/N_{0}
δ | \({P_{j,\text {FDR}}^{*}/N_{0}}, j=\{0,1,2\}\) | |||
---|---|---|---|---|
P/N_{0}=10 dB | P/N_{0}=20 dB | |||
CogFDR-EPA | CogFDR-OPA | CogFDR-EPA | CogFDR-OPA | |
−3 dB | 3.33, 3.33, 3.33 | 4.62, 3.18, 2.19 | 33.33, 33.33, 33.33 | 64.06, 25.66, 10.27 |
−5 dB | 3.33, 3.33, 3.33 | 4.32, 3.24, 2.43 | 33.33, 33.33, 33.33 | 60.56, 27.21, 12.22 |
−7dB | 3.33, 3.33, 3.33 | 4.07, 3.28, 2.64 | 33.33, 33.33, 33.33 | 56.98, 28.63, 14.38 |
Figure 8 shows the outage probabilities of CogFDR network under EPA and OPA versus the number of relays. The RSI level at all the nodes has been assumed as equal to −40 dB. The separation between F_{0} and F_{N+1} is assumed to be constant. Accordingly, the number of hops significantly affects the outage performance of the secondary network. Initially for CogFDR, when N increases, the outage decreases owing to the reduced link distance. However, when N becomes larger, each node introduce additional RSI and IRI into the system so that the outage performance degradation happens. Notice that with increase in number of relays, the outage probability decreases monotonically in CogHDR due to reduced path loss between relay links. Further CogFDR-OPA outperforms CogFDR-EPA.
where the elements denote |h_{j,PR}|^{2}, j={0,1,2}, which are also generated from exponential random variables. The OPA vector and the corresponding instantaneous rate have been determined by following the method described in Section 5.2.
where the elements denote |h_{PT,j}|^{2}, j={1,2,3}, which are also generated from exponential random variables. Figure 14 shows the comparison of optimal end-to-end instantaneous rate of CogFDR against that of CogHDR and CogFDR-EPA. The end-to-end rate outperforms all the other cases, under OPA. Figure 15 shows that the end-to-end rate obtained under the effect of PT is 14% less than that obtained without PT (at P/N_{0}=20 dB and P_{PT}/N_{0}=10 dB).
Overall, the results demonstrate that transmit power optimization can cause notable enhancement in the instantaneous rate and outage probability performance of CogFDR systems with respect to CogFDR-EPA and CogHDR systems. However, the interference from PT reduces the instantaneous rate and increases the outage probability, and the proposed OPA scheme ensures that even under the effect of interference from PT, we obtain good outage and rate performance. The power allocation schemes can be implemented in a centralized manner. The secondary nodes must have the ICSI/SCSI of the secondary transmission links, the links from PT to F_{j} and the links from F_{j} to PR. The ICSI can be obtained with the help of ICSI feedback or pilot-aided channel estimation. Each node, after learning the channel to every other node, have to communicate the acquired information to the centralized controller, which has to run the algorithm and compute the OPA vector and distribute the power values among the nodes.
8 Conclusion
In this work, we have developed analytical models for finding the outage probability of the secondary user in an underlay multi-hop cognitive FDR system considering the impact of residual self-interference (RSI) due to full-duplex operation, inter-relay interference (IRI) arising due to frequency re-use, and interference induced by primary transmitter on the secondary nodes. To improve the outage probability and the end-to-end rate performance of the secondary user, optimal selection of powers for the secondary nodes were considered. We formulated an optimization problem in order to determine the optimal power allocation vector for the secondary nodes that would minimize the outage probability of the secondary network subject to constraints on total power available in the secondary network and also the tolerable interference power at the primary receiver. In order to maximize the end-to-end rate, a power allocation method based on SINR balancing was formulated. Monte Carlo simulations were carried out to substantiate the analytical results. The transmission rate and outage probability of the multi-hop CogFDR network improved significantly when transmit power optimization was employed, in comparison to equal power allocation method.
9 Appendix A: Derivation of (9):
10 Appendix B: Derivation of (23a):
11 Appendix C: Derivation of (27):
Now \(P_{\mathrm {out,HDR}}^{(1)}\) can be determined using (58).
Notes
Acknowledgements
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