 Research
 Open Access
Joint subcarrier pairing and resource allocation for cognitive networks with adaptive relaying
 Hamza Soury^{1}Email author,
 Faouzi Bader^{2},
 Musbah Shaat^{3} and
 MohamedSlim Alouini^{1}
https://doi.org/10.1186/168714992013259
© Soury et al.; licensee Springer. 2013
 Received: 11 May 2013
 Accepted: 21 October 2013
 Published: 9 November 2013
Abstract
Relayed transmission in a cognitive radio (CR) environment could be used to increase the coverage and capacity of communication system that benefits already from the efficient management of the spectrum developed by CR. Furthermore, there are many types of cooperative communications, including decodeandforward (DAF) and amplifyandforward (AAF). In this paper, these techniques are combined in an adaptive mode to benefit from its forwarding advantages; this mode is called adaptive relaying protocol (ARP). Moreover, this work focuses on the joint power allocation in a cognitive radio system in a cooperative mode that operates ARP in multicarrier mode. The multicarrier scenario is used in an orthogonal frequency division multiplexing (OFDM) mode, and the problem is formulated to maximize the endtoend rate by searching the best power allocation at the transmitters. This work includes, besides the ARP model, a subcarrier pairing strategy that allows the relays to switch to the best subcarrier pairs to increase the throughput. The optimization problem is formulated and solved under the interference and power budget constraints using the subgradient algorithm. The simulation results confirm the efficiency of the proposed adaptive relaying protocol in comparison to other relaying techniques. The results show also the consequence of the choice of the pairing strategy.
Keywords
 Cognitive radio
 Cooperative communication
 Amplifyandforward
 Decodeandforward
 Power allocation
 Pairing
 Adaptive relaying protocol
1 Introduction
The growth of technology has affected directly modern communication systems. This expansion can be observed when a comparison is made between the earlier systems with some bits per second as a communication rate and the 300 Mbps already considered in the longterm evolution (LTE) wireless communication systems. The growth of the data rate in wireless standards and services was accompanied by a rise in applications and costumers which implies a strong increase in the demand for the limited frequency spectrum. This means that the actual available spectrum resource may not be able to respond to the emerging and future technology demands.
In current systems, the frequency allocation, the type of service, the maximum transmission powers, and the time duration of the licenses are managed by governmental agencies, which apply the ‘commandandcontrol’ allocation model by assigning a fixed frequency block for each communication service. This scheme is statistic and inflexible in spectrum management which leads, as shown by practical measurements, to inefficient use of the provided spectrum because licensed users are not necessarily using the allocated portion of spectrum at all times or over all the spatial locations, and at the same time prevent other users from accessing the unused spectrum.
Cognitive radio (CR) can manage the spectrum utilization by detecting spectrum holes and avoiding the occupied spectrum using the available part of the spectrum. In fact, the spectrum utilization can be improved by allowing the secondary users (SUs) to use the vacant channels left by the licensed users (PUs) [1]. Such systems have to distribute their limited resources among the SUs in order to maximize the capacity without causing harmful interference to the PUs (see, e.g., [2, 3]). Since OFDM is widely used in various wireless system and shows a high spectral efficiency and flexibility, it is often recommended for cognitive radio systems [4].
To increase coverage and achievable capacity of the communication system, relays (R) are used to transfer the information from the cognitive source (CS) to the destination (D) when the direct link is not available [5] (in some cases even if a direct link exists, the relays are used to improve the performance of the communication systems). The resource allocation problem for the noncognitive OFDM based relay system has been widely studied [6, 7]. In [8], a cooperative scheme with decodeandforward technique is combined with the cognitive radio to produce a communication system with high performance and higher coverage area. Note that in cognitive cooperative communication systems, both transmitters, namely source and relay, have to be aware about the interference threshold tolerated by the PU.
In cooperative communication systems, the most known relaying techniques are amplifyandforward (AAF) [9] and decodeandforward (DAF) [10]. In the AAF case, the R amplifies the received signal from the source (S) by some fixed factor, then forward it to D. However, the relay using the DAF strategy decodes ‘perfectly’ the received signal from S and then encodes it again (with the same code known by S and D) and forwards it to D. Note that these procedures are done at each subcarrier. The disadvantages of these two techniques of relaying come with the fact that: (1) AAF relaying can amplify the noise coming form the (SR) link, which degrades the signal quality, and (2) DAF relaying causes a propagation of error in case of uncorrect decoding of the information symbols.
Adaptive relaying or Adaptive Relaying Protocol (ARP), as named in [6], is one of the proposed solutions that benefits from the advantages of DAF and AAF, and aims to minimize the disadvantages of these two relaying techniques. In [7, 11], the relay can execute AAF and DAF, and there is a technique based on the signaltonoiseratio (SNR) which triggers the switching between the AAF and DAF strategies. It assumes that at high SNR (for an SNR above some SNR thresholds), the relay can decode perfectly, so it is better to operate with DAF; for low SNRs (below a certain threshold), when it is harder to decode correctly, it is preferable to use the AAF to avoid propagation errors.
The objective of this paper is to provide an efficient procedure to integrate the adaptive relaying technique in a CRbased environment for a joint optimization of the choice of pairing strategy and the power allocation at the transmitters (S and R) to reach high capacity, without causing harmful interference to the primary user. The proposed solution goes through an algorithm based on the dual problem and subgradient method [12–14]. For simplicity, we begin by selecting the subcarrier and assume that the relay uses the same subcarrier for receiving (from S) and for transmission (to D). We also consider other type of pairing selection, like random subcarrier selection and optimal subcarrier selection from SR to RD and compare the performances of these different schemes.
1.1 Related work
Many works were done to solve the power allocation problem for the cooperative cognitive radio communication systems. In [15], Zou et al. study the spectrum sensing of the cognitive system using a cooperative relay cognitive radio system. Their work focuses on the trade off between the spectrum sensing and SU transmission. The use of relays is presented also in [16], where the authors study the outage probability of the SU when DAF is used. In [17], a cooperative scheme has been used with DAF relaying. [17] deals also with a multiple relay systems and relay selection strategy. The AAF and DAF are both used in [18], where the system decides to use one of these schemes according to the known CSI. If the relay operates in AAF mode, it will amplify the received signal. However, for the DAF mode, if decoding is unsuccessful, the relay will remain silent. Otherwise, the relay reencodes the decoded data and transmits it to the destination. As we will explain later, we use a different scheme in our work focus on the cognitive context, and this yields another optimization problem. In cite [19], a similar scheme was presented using the subcarrier pairing technique and AAF forwarding strategy only. On the other hand, similar scheme were presented in [20–22] (with relay selection and subcarrier pairing); these works focused on power allocation of a relaying system in cognitive radio scenario using only the DAF technique without considering the AAF technique. The contribution of the present work is the use of an adaptive scheme of relaying based on switching between both techniques, i.e., AAF and DAF.
1.2 Summary of contribution
This paper proposes a new adaptive relaying protocol based in the AAF and DAF modes. In this protocol, the relays are able to perform the AAF and the DAF according to the capability of the relay to decode successfully the signal. Note that the decision of switching, between both modes, is based not only on the channel information but also on the received SNR which will add more complexity on the optimization problem to maximize the total rate subject to power and interference constraints; the ARP is described in [6], but it is not used for cognitive radio as it is in this paper. Also, the optimization problem to maximize the total rate is missing in [6]. This paper deals also, in a second part, with the subcarrier pairing problem using the ARP model which differs from the multirelays especially in the interference computations. Finally, this paper uses the dual problem and subgradient algorithm to solve numerically the optimization problem.
1.3 Outline of the paper
The remainder of the paper is organized as follows. In Section 2, we present the system model and the matching subcarrier problem with a proposed algorithm. The proposed solution is illustrated by some selected numerical results to compare the performance over the different types of relaying (AAF, DAF, and ARP). In Section 3, we investigate the pairing problem by including the pairing parameters within the optimization problem studied in Section 2. More specifically, the same algorithm is used with some modifications to find the best subcarrier distribution, and we end the section with some simulation results showing the difference between the pairing techniques used in this work. Finally, we conclude this work in Section 4, with a summary of the main results.
2 Nearoptimal algorithm for subcarrier matching scheme
In this section, we focus on the simple case in which the power allocation of a cognitive system with one relay system using a matching pairing strategy is adopted. In particular, we assume that the relay forwards the signal over the same received subcarrier and study the difference in performance between the three types of relaying schemes introduced in the previous section.
2.1 System model
2.2 Interference analysis
where d_{ i } and G_{ i } denote the spectral distance and the channel gain, respectively, between the i th subcarrier and the PU band, while Ω^{ i } is the interference factor of the i th subcarrier to the PU band [24]^{a}. Note that (2) expresses the interference in terms of the total transmit power P_{ i } of the i th subcarrier linearly, which will be used to solve the optimization problem in the next subsections.
where Ψ(f) is the PSD of PU signal, and Y_{ i } is the channel gain between the i th subcarrier and the PU signal. By completing the interference analysis of the different agents of the cognitive system, we can formulate the optimization problem before proceeding to the solution.
2.3 Capacity analysis and problem formulation
Let us first define the variables of the problem. Let $({P}_{\text{SR}}^{i};{P}_{\text{RD}}^{i})$ be the power transmitted over the i th subcarrier in the (SR;RD) link. The i th subcarrier channel gain over the (SR;RD) link is given by $({H}_{\text{SR}}^{i};{H}_{\text{RD}}^{i})$. Finally, the noise variance is assigned by ${\sigma}_{i}^{2}={\sigma}_{\text{AWGN}}^{2}+{J}_{i}$, where ${\sigma}_{\text{AWGN}}^{2}$ is the variance of the additive white Gaussian noise (AWGN), and J_{ i } is the interference introduced by the PU signal into the i th subcarrier which is evaluated using (3). This interference can be modeled as an AWGN as described in [2]. To make the analysis more clear, the noise variance σ^{2} is assumed to be the same for all subcarriers and both time slots.
2.3.1 Processing during the first time slot
where ${n}_{\text{SR}}^{i}$ is the noise between S and R with a variance ${\sigma}_{\text{SR},i}^{2}={\sigma}^{2}$, and i = {1, 2, … N} denote the i th subcarrier.
where ${\mathrm{\Omega}}_{\text{SP}}^{i}$ denotes the interference factor of the i th subcarrier to the PU band.
2.3.2 Capacity in the second time slot
where $\mathbb{E}\left[\phantom{\rule{0.3em}{0ex}}Y\right]$ denotes the expected value of the random variable Y.
This approximation is used in [26], and it is based on the assumption that the system has a high SNR for the amplified signal between the relay and the destination. It is proved in [27] that this approximation is also accurate even in the moderatelow SNR regime.
2.3.3 Total capacity
2.3.4 Optimization problem of the subcarrier matching technique
In this problem, the power constraints at each transmitter (S and R) are missed. However, when we take a look at the interference constraints, we note that the power constraint is defined indirectly. Moreover, we use the identity ${\mathrm{\Omega}}_{\text{SP}}^{i}\ge \underset{j}{\text{min}}{\mathrm{\Omega}}_{\text{SP}}^{j}\phantom{\rule{1em}{0ex}}$ to ensure the following inequality $\sum _{i=1}^{N}{P}_{\text{SR}}^{i}\le \frac{{I}_{\mathit{\text{th}}}}{\underset{j}{\text{min}}{\mathrm{\Omega}}_{\text{SP}}^{j}}$. Thus, the interference constraint implies, indirectly, a power constraint in the two time slots; even if the resulting interference is too small, the problem remains approximately the same without significant change. This analysis can help us in this chapter because the problem is relatively simple and is not a mixed integer programming problem, so the power constraint can be omitted. However, in the next section, the problem is more complex, and we should define the power constraints at the transmitters to avoid nonconvergence of the algorithm.
Under the previous assumption of perfect knowledge of the channel coefficient and the noise variance, the problem is a convex optimization problem with the parameter ${P}_{\text{RD}}^{i}$ and ${P}_{\text{SR}}^{i}$. In the next part, we solve this problem using the Lagrangian method and the KarushKuhnTucker (KKT) conditions. Moreover, using the fact that the problem is convex, the dual solution and the primal solution are the same, so the problem can be solved using the dual formulation.
2.4 Solution
For simplicity reasons and for making the mathematical notation easy to follow, we denote the following: ${P}_{\text{SR}}^{i}$ by ${P}_{1}^{i}$, ${P}_{\text{RD}}^{i}$ by ${P}_{2}^{i}$, ${\gamma}_{\text{SR}}^{i}$ by ${\gamma}_{1}^{i}$, ${\gamma}_{\text{RD}}^{i}$ by ${\gamma}_{2}^{i}$, ${\mathrm{\Omega}}_{\text{SP}}^{i}$ by ${\mathrm{\Omega}}_{1}^{i}$, and ${\mathrm{\Omega}}_{\text{RP}}^{i}$ by ${\mathrm{\Omega}}_{2}^{i}$.
2.4.1 Dual problem
From (24) and for a given set of α_{ i }, the problem can be divided into N independent problems. Thus, we divide the dual function (the Lagrangian) into N dual functions (Lagrangian), such that g_{ i } (${\mathcal{L}}_{i}$). For each subcarrier i, for given λ and μ, and according to the value of α_{ i } (which can take two values, 0 or 1), it appears that there are two cases:

Case α_{ i } = 1. In this case, the relay is working on AAF for the i th subcarrier. As we have in (23), only the terms related to the AAF approach. Hence, the dual function can be simplified as follows:$\begin{array}{ll}{g}_{i}(\mu ,\lambda )& =\underset{{P}_{1}^{i},{P}_{2}^{i}\ge 0}{\text{max}}{\mathcal{L}}_{i}\phantom{\rule{2em}{0ex}}\\ =\underset{{P}_{1}^{i},{P}_{2}^{i}\ge 0}{\text{max}}\frac{1}{2}\underset{2}{log}\left(1+\frac{{P}_{1}^{i}{\gamma}_{1}^{i}{P}_{2}^{i}{\gamma}_{2}^{i}}{{P}_{1}^{i}{\gamma}_{1}^{i}+{P}_{2}^{i}{\gamma}_{2}^{i}}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\lambda {P}_{2}^{i}{\mathrm{\Omega}}_{2}^{i}\mu {P}_{1}^{i}{\mathrm{\Omega}}_{1}^{i}.\phantom{\rule{2em}{0ex}}\end{array}$The maximum of ${\mathcal{L}}_{i}$ can be found by searching the partial derivative of ${\mathcal{L}}_{i}$ subject to ${P}_{1}^{i}$ and ${P}_{2}^{i}$ which leads to$\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\frac{\partial {\mathcal{L}}_{1}}{\partial {P}_{1}^{i}}=\frac{{\left({P}_{2}^{i}\right)}^{2}{\left({\gamma}_{2}^{i}\right)}^{2}{\gamma}_{1}^{i}}{({P}_{2}^{i}{\gamma}_{2}^{i}+{P}_{1}^{i}{\gamma}_{1}^{i})({P}_{2}^{i}{\gamma}_{2}^{i}+{P}_{1}^{i}{\gamma}_{1}^{i}+{P}_{2}^{i}{\gamma}_{2}^{i}{P}_{1}^{i}{\gamma}_{1}^{i})}\mu {\mathrm{\Omega}}_{1}^{i}$(27)$\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\frac{\partial {\mathcal{L}}_{1}}{\partial {P}_{2}^{i}}=\frac{{\left({P}_{1}^{i}\right)}^{2}{\left({\gamma}_{1}^{i}\right)}^{2}{\gamma}_{2}^{i}}{({P}_{2}^{i}{\gamma}_{2}^{i}+{P}_{1}^{i}{\gamma}_{1}^{i})({P}_{2}^{i}{\gamma}_{2}^{i}+{P}_{1}^{i}{\gamma}_{1}^{i}+{P}_{2}^{i}{\gamma}_{2}^{i}{P}_{1}^{i}{\gamma}_{1}^{i})}\lambda {\mathrm{\Omega}}_{2}^{i}.$(28)We then equal both (27) and (28) to zero. The solution of these equations leads to ${P}_{1}^{i\ast}={c}_{i}{P}_{2}^{i\ast}$, where ${c}_{i}=\sqrt{\frac{{\gamma}_{2}^{i}\lambda {\mathrm{\Omega}}_{2}^{i}}{{\gamma}_{1}^{i}\mu {\mathrm{\Omega}}_{1}^{i}}}$. Thus, the new value of ${P}_{2}^{i}$ is${P}_{2}^{i\ast}={\left[\frac{{\gamma}_{2}^{i}}{\mu {c}_{i}{\mathrm{\Omega}}_{1}^{i}({\gamma}_{2}^{i}+{c}_{i}{\gamma}_{1}^{i})}\frac{1}{{c}_{i}{\gamma}_{1}^{i}}\frac{1}{{\gamma}_{2}^{i}}\right]}^{+},$(29)
where [x]^{+} = max(0, x).

Case α_{ i } = 0. For this case, the relay switches to the DAF technique at the i th subcarrier having $i\in \mathcal{D}$. The problem of the DAF relaying has been solved in [8]. We just have to know the value of ${P}_{1}^{i\ast}$ which can be obtained from the following relation ${P}_{2}^{i\ast}=\frac{{P}_{1}^{i\ast}{H}_{\text{SR}}^{i}}{{H}_{\text{RD}}^{i}}$. The solution is found to be given, in this case, by the following expression:${P}_{1}^{i\ast}={\left[\frac{1}{\mu {\mathrm{\Omega}}_{1}^{i}+\lambda \frac{{H}_{\text{SR}}^{i}}{{H}_{\text{RD}}^{i}}{\mathrm{\Omega}}_{2}^{i}}\frac{1}{{\gamma}_{1}^{i}}\right]}^{+}.$(30)
By obtaining the optimal values of the transmitted powers ${P}_{1}^{i\ast}$ and ${P}_{2}^{i\ast}$, the dual function is now a function of μ and λ. In the next subsection, we use an algorithm named subgradient algorithm [12] that proceeds to the search of the optimum values of μ and λ iteratively.
2.4.2 Subgradient method to solve the dual problem
With the obtained optimal values of primal variables (${P}_{1}^{i\ast}$, ${P}_{2}^{i\ast}$), the dual problem can be solved using the subgradient method [12–14]. In fact, our algorithm is based on the calculation of the Lagrangian multipliers λ and μ in each iteration. The decision about the type of relaying mode over each subcarrier is made using (10). The implementation procedures is described in the power allocation algorithm depicted in Algorithm 1.
Algorithm 1 Power Allocation Algorithm
The parameter δ^{(k)} appears in lines 14 and 15 of Algorithm 1, denoting the step size of the k th iteration. This algorithm is well described in [12–14], where many types of step size can be used in the subgradient algorithm. In our model, we tried different step sizes and then used the best one in terms of best performance and less complexity. In the proposed scheme, the optimal power requires (N^{2}) function evaluations for every subcarrier to be matched in the second time slot. Therefore, the complexity of the proposed algorithm is $\mathcal{O}\left(T{N}^{2}\right)$, where T is the number of iterations required for convergence. A comparison between the different schemes used in this paper is derived in Section 3.3.
2.5 Simulation results
The simulations are performed under the scenario given in Section 2.1. An OFDM system of N subcarriers (N ∈ {16, 32, 64}) at the source and destination and one relay system is assumed. The values of T_{s}, Δf, and I_{ th } are assumed to be 4 μ s, 0.3125 MHz, and 20 dBm, respectively. The channel gains are outcomes of independent Rayleigh distributed random variables with mean equal to 1.
It can be also shown that the ARP relaying protocol achieves, for the different depicted values of SNR, the best results. This can be explained by the fact that the ARP protocol is able to switch (in an adaptive way) from one relaying mode to another (AAF or DAF) using in each moment the relaying mode that achieves the best performance. In other words, the ARP tends to use the AAF relaying protocol for low values of SNR, and use the DAF for higher SNRs. Thus, ARP is able to take advantage of each relaying mode, depending on the SNR range. Figure 3 shows, finally, how the system capacity scales as function of the increase in the total number of carriers of the system.
As a general observation from Figures 3, 4, and 5, it can be shown that the ARP scheme behavior always reaches the optimal scheme for different SNR values. However, the major limitation of the proposed scheme is its complexity. Thus, a new algorithm with much less complexity is required to make a step towards possible real implementation. Further, work should focus on the development of suboptimal algorithm that achieves a near optimal performance with affordable complexity of implementation.
The goal of Section 2 is to compare achieved performances between three relaying schemes, namely, the AAF, DAF, and ARP in a cognitive radio environment, using the matching subcarrier technique. Given the superior performance of ARP, the next section focuses on the ARP scheme for a more complicated problem which includes the research of the best subcarrier pairing to maximize the capacity under interference and power constraints.
3 Subcarrier pairing for adaptive relaying protocol
As it was mentioned in the previous section, there are many types of pairing techniques to switch the subcarriers from the first link to the second link. It has been shown that the pairing strategy has an important impact on the resulting capacity. Therefore, in order to reach maximum capacity, with limited resources, we should carefully choose the pairing technique. One solution is to introduce the subcarrier pairing in the final optimization problem in order to find the optimum pairing distribution that maximizes the capacity without increasing the complexity too much.
3.1 System architecture
The same OFDM cooperative system described in Section 2.1 is used in this section with some modification. In fact, a SU is present in the same coverage area of the PU and can communicate through the PU spectrum without causing harmful interference to the adjacent PUs. We assume the absence of a direct link between S and D. Thus, the SU is reaching the destination using the ARP technique of relaying through one relay R. It is assumed that the data is multiplexed into OFDM with several subcarriers whose total number is equal to N. Thus, the used spectrum by the CS is divided into N subcarriers, each having a Δf bandwidth. Both S and R can transmit over the PU spectrum and interfere with its signal without exceeding the maximum interference power tolerated by PU, I_{ th }. As mentioned before, the source and the relay transmit in two different time slots in a way that the link (SR) is active at the first time slot, while the link (RD) is active in the second time slot.
Also, the interference introduced by the PU is modeled as AWGN with a variance J_{ i }. In what follows, the noise variance is denoted by ${\sigma}_{i}^{2}={\sigma}^{2}$, and it is assumed to be the same for all subcarriers and in both time slots.
3.2 Problem formulation
3.2.1 Total capacity
3.2.2 Optimization problem
where ${\mathrm{\Omega}}_{1}^{k}$ and ${\mathrm{\Omega}}_{2}^{l}$ are the interference factors in each slot.
P_{S} and P_{R} are the available power budgets in the source and the relay, respectively. The instantaneous fading gains are assumed to be perfectly known. The channel gains can be estimated using classical channel estimation techniques.
3.3 Optimal power allocation
Moreover, for a given values of the different dual variables, we get two cases depending on the value of the variable α_{ l }:

Case 1 : The pair (k, l) is used for amplify and forward, i.e., α_{ l } = 1. Assume (k, l) to be a valid subcarrier pair; the optimal power allocation can be evaluated by solving the following subproblem for every (k, l) assignment:$\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\underset{{P}_{1}^{k},{P}_{2}^{l}}{\text{max}}{\mathcal{D}}_{\text{AF}}({P}_{1}^{k},{P}_{2}^{l})\phantom{\rule{1em}{0ex}}s.t.\phantom{\rule{1em}{0ex}}{P}_{1}^{k}\ge 0,\phantom{\rule{1em}{0ex}}{P}_{2}^{l}\ge 0.$(46)Hence, we obtain the optimal power by equating$\frac{\partial {\mathcal{D}}_{\text{AF}}({P}_{1}^{k},{P}_{2}^{l})}{\partial {P}_{1}^{k}}=\frac{\partial {\mathcal{D}}_{\text{AF}}({P}_{1}^{k},{P}_{2}^{l})}{\partial {P}_{2}^{l}}=0.$(47)The optimal power in (46) can be expressed as follows:$\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left\{\begin{array}{l}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{P}_{1}^{k\ast}={\left[\frac{{\gamma}_{1}^{k}}{({\gamma}_{1}^{k}+{c}_{k,l}{\gamma}_{2}^{l}){c}_{k,l}(\theta +\mu {\mathrm{\Omega}}_{2}^{l})}\frac{1}{{\gamma}_{1}^{k}}\frac{1}{{c}_{k,l}{\gamma}_{2}^{l}}\right]}^{+}\\ {P}_{2}^{l\ast}={c}_{k,l}{P}_{1}^{k},\end{array}\right.$(48)
where ${c}_{k,l}=\sqrt{\frac{{\gamma}_{1}^{k}(\beta +\lambda {\mathrm{\Omega}}_{1}^{k})}{{\gamma}_{2}^{l}(\theta +\mu {\mathrm{\Omega}}_{2}^{l})}}$. Hence, the power variable in (43) can be eliminated by substituting the optimal power allocation found in (48). Then, the dual function can be easily found by searching the optimal pair (k, l) that maximizes the dual function.

Case 2: The pair (k, l) is used for decode and forward, i.e., α_{ l } = 0. In this case, we assume that the pair (k, l) is a valid pair that forwards by DAF technique. The following problem should be solved for each valid pair$\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\underset{{P}_{1}^{k},{P}_{2}^{l}}{\text{max}}{\mathcal{D}}_{\text{DF}}({P}_{1}^{k},l)\phantom{\rule{1em}{0ex}}s.t.\phantom{\rule{1em}{0ex}}{P}_{1}^{k}\ge 0.$(49)By differentiating the previous function over P_{1}, we obtain the optimal power allocation in this case:$\left\{\begin{array}{l}{P}_{1}^{k\ast}={\left[\frac{1}{\beta +\theta \frac{{\gamma}_{1}^{k}}{{\gamma}_{2}^{l}}+\lambda {\mathrm{\Omega}}_{1}^{k}+\mu \frac{{\gamma}_{1}^{k}}{{\gamma}_{2}^{l}}{\mathrm{\Omega}}_{2}^{l}}\frac{1}{{\gamma}_{1}^{k}}\right]}^{+}\\ {P}_{2}^{l\ast}=\frac{{\gamma}_{1}^{k}}{{\gamma}_{2}^{l}}{P}_{1}^{k}.\end{array}\right.$(50)
Like case 1, we substituted the power variable by its optimal value to get a new problem without power parameter. Therefore, the best pair (k, l) is chosen so it maximizes the dual function.
where δ^{(i)} is the step size that can be updated according to the nonsummable diminishing step size policy [12–14].
Complexity of the used algorithms
Scheme  Complexity 

Subgradient algorithm  $\mathcal{O}\left(\mathit{\text{TN}}\right)$ 
ARP with subcarrier mathcing  $\mathcal{O}\left(T{N}^{2}\right)$ 
ARP with subcarrier pairing  $\mathcal{O}\left(T{N}^{3}\right)$ 
3.4 Simulation results
According to the scenario given in Section 3.1, a multicarrier system of N=32 subcarrier and single relay is assumed. The values of the symbol duration T_{s}, Δf, and σ^{2} are assumed to be 4 μ s, 0.3125 MHz, and 10^{7}, respectively, as it was the case in Section 2.5. The channel gains are outcomes of independent Rayleigh distributed random variables with a mean equal to 1.
4 Conclusion
In this paper, we considered a near optimal power allocation algorithm for an OFDMbased system with adaptive relaying protocol using a single relay. In the first part, the problem is solved for the simple case of subcarrier matching to compare the performance of the ARP scheme to the classical AAF and DAF techniques. However, in the second part, the goal was to maximize the capacity by jointly optimizing the subcarrier pairing, the power allocation, and the relaying technique (AAF or DAF). In our framework, we assumed a limited power budget at each transmitter, and because it is a cognitive scenario, the introduced interference to the primary user was required not to exceed a predetermined tolerated threshold. The problem was formulated with the different constraints as a mixed integer programming problem. We used the dual method to solve the optimization problem iteratively using the subgradient algorithm.
Some selected simulation results confirmed the efficiency of the proposed relaying scenario (ARP), which offers better performance in comparison to the AAF and DAF techniques. These results showed also that the performance has a considerable dependence on the adopted subcarrier pairing techniques at the relay. The simulation results showed finally the effects of the interference threshold tolerated by the PU and the impact of the power budget set at the transmitters.
Endnote
^{a} This formula assumed a perfect knowledge of the CSI. The outdated CSI scenario is studied in [24], when the average interference is expressed in terms of the interference factor Ω^{ i }, the channel gain H^{ i }, and the correlation coefficient ρ ([24], Eq. (16)). This case can be integrated in our model by changing the interference factor by the new outdated CSI factor.
Declarations
Acknowledgements
This work was supported in part by King Abdullah University of Science and Technology (KAUST), the European project ACROPOLISNoE (ICT2009.1.1), the COST Action IC0902, and the NPRP grant N 52502087 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. This work is an expanded version of work presented at the International Conference on Cognitive Radio Oriented Wireless Networks (CROWNCOM 2012), Stockholm, Sweden, June 2012.
Authors’ Affiliations
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