Robust power allocation for massive connections underlaying cognitive radio networks with channel uncertainty

Consider a CRN as shown in Fig. 1. There is a PU and massive SCs which utilize the same spectrum as the PU in the underlay mode. We denote \( \mathcal {N}= \left \{ {1,2, \cdots,N} \right \}\) as the set of all SCs. A connection consists of a S-Tx and a secondary receiver (S-Rx) and so does the PU. The SCs are supposed to be randomly distributed around the PU in the CRN. In addition, let p _PU and p _i represent the transmit power of the PU and the ith SC, respectively. Also, let p _max denote the maximum transmit power of the S-Tx, I _max the maximum interference allowed by the SC, and \(I_{_{\max }}^{p}\) the maximum interference allowed by the PU. Moreover, vector p collects {p _i }.

where h ₁ and h ₂ are two independent Gaussian variables which follow the distribution of N∼(0,δ ²).

The channel gain g _PU≥0 between the primary transmitter (P-Tx) and P-Rx is assumed to have been easily acquired by some conventional channel estimation techniques [22]. Similarly, the channel gain g _i,j between the ith S-Tx and the jth S-Rx is also known accurately, i.e., the S-Tx has perfect channel state information [20]. Because SC and PU rarely have any cooperation during the process of their transmissions, the channel gain \(g_{i}^{\text {PU}}\) between them can be hardly estimated precisely unlike the channel gain between SCs.

where the channel gain \(g_{i}^{\text {PU}}\) is an independent and identically distributed exponential random variable with mean θ and ε is the outage probability.

Besides, in this CRN system, there is the interference from the P-Tx to S-Rx. Set \({g_{\text {PU}}^{i}}\) as the channel gain between the P-Tx and S-Rx, which is perfectly known. Figure 2 reflects the system performance including the interference from the P-Tx to S-Rx. In the scenario of this paper, we assume that the P-Tx is far away from SCs and there is massive mutual interference between massive SCs which is much more than the interference from P-Tx. Based on the above phenomenon and explanations, we ignore the interference from the P-Tx when focusing on the optimization of the SCs’ capacity.

In (10), the goal of the optimal problem \((\mathcal {P})\) is to find the optimal power allocation method to maximize the total throughput capacity of the system. For the sake of the description convenience, we regard \((\mathcal {P})\) as the original problem. Among the three constraints, the first one is to restrict the transmit power of S-Tx, the second one ensures the coexistence of the communication between the SCs, and the final one focuses on the protection for the PU. However, since the objective function is non-linear and it is not even convex, it will become very complicated if we directly solve the optimization problem of the above form. Additionally, the mutual influence actually exists between multiple variables. Therefore, not only does the problem become tricky but also the complexity will rise steeply as the number of the variables increases. Paper [23, 24] suggest that a large number of iterations is needed in order to tackle with this problem. Because of the structure of the function, this problem cannot directly use the methods proposed by [25, 26]. As a result, intractability and unavailability of the uncertain channel gain along with SCs’ interference make the non-linear and non-convex expression mathematically difficult to handle. In addition, there is not yet any satisfactory solution to the open issue so far. In the following part, we try to introduce a feasible way to tackle with the difficulty, which takes advantage of the convex optimization theory.

where all of f ₀,⋯,f _q are convex functions, f _i (x) is the constraint of inequality, and \({h_{j}}(x) = {a_{j}^{T}}x - {b_{j}}\) is the constraint of equality [27]. But as it is shown in problem (10), neither optimization objective nor the constraints is convex function. The convex optimization cannot be used straightly. So, we first investigate the primal objective function and approximate it with a series of convex ones.

It is can be seen that −f ₀(p ^(k)) provides a lower bound. Maximizing function (12) is equivalent to maximizing the initial function. Through the above analysis, the original problem is non-convex, but we can approximate it with convex functions.

Nevertheless, for constraint C3, the uncertainty of the channel between S-Tx and P-Rx makes C3 intractable, i.e., C3 has no idea to satisfy the requirement of a convex function. As a result, we take different attitudes towards the three constraints. For the first two, we can acquire the goal through above steps (17). As to C3, we have to study its transformation as the following description.

How to test and verify the constraint (19)? The key lies in the distribution of X. To determine the distribution explicitly, we use Gaussian approximation based on the following Lemma 1 [28].

Due to the above Lemma 1, the Lyapunovcondition should be satisfied as follows to apply the Lyapunov’s (CLT):

where r _i is the third-order center distance of X _i , i.e., E[(X _i −m _i )³] and m _i and \({\sigma _{i}^{2}}\) are the mean and variance of the exponential distributed random variable X _i , respectively. We can easily check this condition [20].

where F _N (·) is the cumulative distribution function (CDF) of a normal distribution with mean m and variance σ ² and \(\text {erfc}(z) = \frac {2}{{\sqrt \pi }}\int _{z}^{\infty } {{e^{- {t^{2}}}}dt}\). For (24), we assume that \({f_{3}}\left (\mathbf {p} \right) = \frac {1}{2}\text {erfc}(\frac {{I_{\max }^{p} - m}}{{\sqrt 2 \sigma }}) - \varepsilon \). It is obvious that it is not a standard convex function at all which needs separate treatment. Inspired by the scheme proposed in [20], once a power allocation is given from the first two constraints, we can check whether it satisfies the outage constraint (24) or not.

If we acquire an optimal solution p from \((\mathcal {SP}1)\), we will examine whether it meets or not by substituting into \((\mathcal {SP}2)\). In another word, the input of \((\mathcal {SP}2)\) is the output of \((\mathcal {SP}1)\).

If there is no lower bound of the Lagrangian, the dual function value is −∞.

In the above section, we transfer the constraint C3 into a tractable form and approximate the original problem with convex functions. As is mentioned, once a power allocation is given, we can check whether it satisfies the outage constraint C3 or not. So our power allocation algorithm consists of two steps. Firstly, we apply the convex optimization to getting over the problem \((\mathcal {SP}1)\) based on the interior method. Note that the optimal solution must satisfy the constraints C1 and C2. Then, we should check the optimal solution whether satisfies the outage constraint C3 or not. If it does, the optimal solution of problem \((\mathcal {SP}1)\) is also the optimal solution of the original problem; otherwise, all the transmit power decreases by a step γ until the constraint C3 is satisfied.

where the inequality constraints must meet the condition strictly. Since \((\mathcal {SP}1)\) is convex as is mentioned above and we can learn from Slater’s theorem that the Lagrangian dual function (29) satisfies the strong duality, so it also satisfies the following important theorem [27] which is its sufficient and necessary condition.

The above formulation is only one of the KKT conditions. According to the Lemma 2 mentioned above, we can ensure that the optimal duality gap is zero. So when p ^∗ is optimal, we can attain the dual optimum as well as λ ^∗,ν ^∗. Nevertheless, we can hardly tackle with the KKT conditions analytically of the intractable problem in this paper. As a result, we introduce the interior point method to deal with the problem together with the KKT conditions.

It can be seen that the non-convex optimization problem \((\mathcal {P})\) has already been approximated by \((\mathcal {P}1)\) in which only the third constraint is not convex. Then, we research for the existence of an optimal solution. Moreover, the optimality conditions (KKT conditions) are captured. In this section, we attempt to develop a quick and effective iterative algorithm. The barrier method is introduced to search for the global optimal solution throughout the previous analysis. As is well known, the barrier method is a representative technique of the class of interior point methods. It usually contains a logarithmic barrier function and the Newton method for minimization without constraints [21]. The advantage of the barrier method is that it is simple to implement and it can surely outperform other methods in performance.

Newton algorithm is a universal descent method, where the iteration consists of damping Newton stage and quadratic convergence stage, i.e., pure Newton stage. The trail of steps is the direction of searching in Newton algorithm. The backtracking line search is applied to the iteration, which is a kind of non-accurate line search approach. Newton algorithm plays a role as the internal iteration of the interior point method. In terms of searching the feasible solution, it works simply and effectively [27].

where \(\phi \left (\mathbf {p} \right) = - \sum \limits _{i = 1}^{N} {\log \left ({-{f_{1,i}}\left (\mathbf {p} \right) - {f_{2,i}}\left (\mathbf {p}\right)}\right)} \) is called the logarithmic barrier function of problem \((\mathcal {SP}1)\). Because −(1/t) log(−u) is convex, differentiable, and increasing when u<0, Newton algorithm can be applied to searching for the minimum of (35) within p _i ∈[0,p _max]. Why do we select Newton algorithm? As is mentioned above, it uses the second derivation information when computing the descent direction. It exhibits the quite quick convergence rates. As [29] suggests, the approximation quality will improve apparently as the parameter t grows.

Finally, the overall procedure for solving the optimization problem is presented in the following algorithm.

In order to verify its effectiveness, the proposed algorithm is compared with the algorithm in paper [20] and this will be discussed further with the simulation results.

The presented results are obtained by simulating 1000 independent trials and then taking the expected value. To benchmark performance of the proposed near-optimal scheme, a simple suboptimal algorithm in paper [20] is considered. In paper [20], Son et al. investigate power allocation algorithms for OFDM-based cognitive radio systems, where the intra-system channel state information (CSI) of the SC is perfectly known and the inter-system CSI is only partially available to the S-Tx. They develop a suboptimal power allocation algorithm, which repeatedly solves a sub-problem having only a transmit-power constraint and then adjusts the available transmit power until the desired outage probability is achieved. In this paper, we regard this algorithm as a comparison with our proposed algorithm. More information can be referred to paper [20].

Figure 3 illustrates the simulation scenario for the sum throughput under the different algorithms where γ=0.2, ε=0.05. It is assumed that there are 50 SCs. We can see that the proposed algorithm can acquire higher throughput than the algorithm proposed in [20]. When each S-Tx uses the maximum transmit power, the sum throughput is highest, but the strong interference on the PU limits the transmit power. On the other hand, from the tendency of the curve in Fig. 3, it can be observed that the sum throughput derived from our proposed scheme has always outperformed the other one derived from paper [20]. This is due to the algorithm proposed in [20], which is based on the binary search, causes great waste of power resource. Hence, the proposed algorithm can perform better. In our paper, when the number of SUs becomes large, the corresponding iteration will be more. Additionally, −f ₀(p) as the approximate lower bound of the best solution updates itself after each iteration. In this case, −f ₀(p) will become more and more close to the optimal solution. Therefore, our proposed algorithm is able to obtain the better effect of the system performance.

Figure 4 compares the spectral efficiency in the cases of different step values, where the number of SCs is assumed 50. A few interesting observations can be obtained. It is shown that the rapidity of the convergence correlates closely with the value of γ. That is to say, the convergence of the sum throughput turns faster as γ becomes larger. Moreover, the faster the convergence is, the lower the accuracy is and the deviation increases when the accuracy of the transmit power becomes lower. Therefore, the value of γ is better to the optimal transmit-power if it is smaller. Notably, γ should be set within [0.05,0.25] through a lot of tests and trials. In Fig. 5, γ is set as 0.1. We examine the sum throughput by varying the outage probability ε. The number of SCs still remains 50. There are some conclusions that can be drawn from Fig. 5 including that the throughput improves as ε grows larger, and the outage probability and the interference threshold have the same impact on the sum throughput.

Figure 6 discloses the average throughput of each SC and the sum throughput in terms of different numbers of SCs. In this figure, the left vertical axis represents the average throughput of each SC and the right one is the sum throughput. As can be seen obviously, with the number of SCs increasing, the average throughput is decreasing and the sum throughput rises first and then gradually falls. When there are about 160 SCs, the up trend of the sum throughput becomes gentle since the interference constraints between SCs has offset the rise in throughput. When there are about 210 SCs, the sum throughput declines sharply. This is because when the number of SCs increases seriously, the interference between SCs is also increasing seriously, even more than the throughput. It is also observed that both of the average throughput of each SC and the sum throughput under ε=0.01 are extremely close to those under ε=0.2. In a word, the constraint of the PU becomes more and more relaxed with the number of the users increasing. On the one hand, if the number of SCs is large, the interference constraints of SCs play a dominant role in the problem. On the other hand, when there are a small number of SCs, however, the interference limitation of PU would become principal.