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Robust power allocation for twotier heterogeneous networks under channel uncertainties
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 224 (2018)
Abstract
In this paper, the tradeoff among system sum energy consumption and robustness is studied. In this regard, a robust power allocation problem is formulated for a twotier heterogeneous network with uplink transmission mode and consideration of imperfect channel state information. The objective is to minimize the total transmit power of femtocell users (FUs), while the interference to macrocell user receiver is limited to a predefined interference level, the transmit power of each FU transmitter is kept within their power budgets, and the actual signaltointerferenceplusnoise ratio of each femtocell receiver is above a minimum threshold. Considering the uncertainties of the interference links from FUs to macrocell base stations and forward transmission links of each FU, the robust power allocation problem is formulated as a semiinfinite programming problem (SIPP). By the worstcase approach, the SIPP is transformed into a convex optimization problem solved by the Lagrange dual decomposition method. Moreover, the feasible regions of constraints, computational complexity, and sensitivity degree of the proposed robust algorithm are also analyzed. Simulation results investigate the impact of channel uncertainties and the superiority of the proposed algorithm by comparing with nonrobust algorithm.
Introduction
With the rapid increase of mobile data, more than 50% phone calls and 70% data services take place in indoor environment [1]. However, traditional homogeneous cellular networks cannot meet this requirement. Femtocell enabled in macrocell networks consists of a new heterogeneous cellular network which can satisfy the requirement of the increasing wireless data services due to lowpower consumption and flexible deployment of femtocell users [2]. In HetNets, there are usually two types of users: FUs and MUs. On the one hand, FUs considered as lowpower nodes utilize the same spectrum resource with MUs and improve indoor area coverage so that the spectrum efficiency and system capacity of communication system can be improved heavily. On the other hand, crosstier interference from femtocell networks to macrocell networks and the interference from MBS must be carefully controlled. Therefore, PA is a key technique for guaranteeing the QoS of users in HetNets.
Since PA can mitigate mutual interference of multiusers, ensure the QoS of each UE and improve system overall throughput, it has been considered as an effective method to achieve resource allocation in HetNets. In [3], for OFDMA femtocell networks, with consideration of FUs’ fairness in each femtocell and protection of MUs, a PA algorithm is proposed via distributed FoschiniMiljanic power update technology. Similarly, in [4], a PA scheme with consideration of femtocell clustering is investigated based on branchandbound algorithm and the simplex algorithm to enhance data rate of FUs and alleviate the interference to MUs in macrocellfemtocell HetNets. A distributed utilitybased SINR strategy for femtocell networks in [5] is investigated to reduce the crosstier interference from femtocell networks to macrocell networks. But maximum transmission power limitation of each user is ignored in this paper. These PA schemes have an efficient performance in mitigating the interference between MUs and femtocell users under perfect CSI. However, in practical systems, perfect CSI is hard to be accurately acquired because of the effect of channel fading and feedback delays. Therefore, PA under imperfect CSI should be considered ahead of time in practical transmission system of HetNets.
Currently, to improve robustness of heterogeneous communication network, based on robust optimization theory, many authors have dedicated to study robust PA algorithms under channel uncertainty in twotier HetNets [6]. In [7], to enhance the robustness of system, an uplink RPA problem is investigated in twotier femtocell networks to deal with the uncertainties and protect the QoS of all users by using outage QoS constraints. Considering the same network scenario, a RPA scheme is put forward under channel uncertainties in [8] to maximize the network benefit among all users. In [9], a RPA algorithm is investigated to minimize the transmission power of FUs for energysaving, which is a formulated subject to the QoS constraints and crosstier interference constraints with the consideration of channel estimation errors. But the crosstier interference received at FUs is ignored. Moreover, only single user scenario and the probability constraints are considered in [7–9]. A robust Stackelberg game is presented to formulate the twotier uplink RPA problem to satisfy different service requirements of both FUs and MUs in [10]. However, they ignore SINR protection of FUs. To improve system capacity, in [11], a resource allocation scheme for twotier OFDMbased cognitive femtocell networks is proposed by taking the mutual interference, imperfect spectrum sensing, and channel uncertainty into account, where the energy consumption is ignored. In [12], to maximize the utilities of all users, based on hierarchical game theory, the authors propose a robust uplink PA algorithm under the consideration of probability interference constraints. For a multitier cognitive HetNet, in [13], the authors study a SIP problem to maximize the SINR of microcell users under channel uncertainties, which is converted into a geometric programming problem by using a relaxation approach. In [14], based on worstcase theory, a distributed RPA algorithm is proposed to obtain maximum rate of femtocell users in OFDMAbased femtocell networks subject to intratier and crosstier interference uncertainties. Aiming at enhancing the robustness of system, the author in [15] studies an outagebased robust optimization problem under partial CSI feedback and no CSI feedback. In [16], a RPA algorithm is proposed to minimize the total power of all users subject to outage probability constraints under timevarying wireless channels in twotier femtocell networks. However, the existing works do not deal with the channel uncertainties with the consideration MUtoFU links, interference links among FUs, and SINR requirement of FUs, simultaneously. Additionally, the feasible region of optimal power and sensitivity analysis is not considered.
Energy consumption, user performance, and robustness are the three important characteristics of each cellular network (i.e., macrocell network, femtocell network) in HetNets where the tradeoff between optimality and robustness should be also studied. To this end, by considering the channel uncertainties in SINR constraint of each FU and interference power constraint to MUs, we investigate a RPA problem in twotier HetNets under uplink transmission mode that minimizes the total transmit power of FUs. To solve the proposed problem, we transform the problem into a convex one by using bounded ellipsoidal model and worstcase approach, then the analytical solution is obtained by using Lagrange theory.
The main contributions of our paper are summarized as follows:

We proposed a RPA algorithm based on energy minimization for the uplink of a HetNet with one macrocell and multiple femtocells by considering all channel uncertainties. Our motivations behind this system model are (a) multiple overlapped femtocell network is a more practical and promising candidate to improve system throughput and spectrum efficiency; (b) with considering all possible channel uncertainties, the robustness of system can be improved where both transmission links among different femtocells and transmission links in macrocell can be guaranteed at the same time.

We used a simple method to transform the NPhard problem into a convex one. Also the feasible regions of optimal PA problem and the proposed RPA problem are given.

Then, we addressed the complexity and sensitivity degree of the algorithm and obtained the analytical relationship between overall energy consumption and uncertain parameters. The simulation parts demonstrated the effectiveness of the proposed algorithm.
The rest of the paper is given as follows. Section 2 presents the methods of this study. The system model is given in Section 3 and transformation process of the designed RPA problem is presented in Section 4. Section 5 proposes a RPA algorithm based on the above deterministic model. And the performance analysis is given in Section 6. The simulation results are presented in Section 7. Finally, the conclusion is given in Section 8.
Methods
Considering system energy consumption and transmission robustness of a HetNet with one macrocell with multiple femtocells, this study presented a power minimization scheme subject to all channel uncertainties. After network initialization is accomplished, our proposed RPA algorithm at each FU transmitter is used to adjust the corresponding transmit power to achieve total power consumption minimization under the constraints of interference power of MUs and SINR requirement of each FU. Due to instability of wireless channel, we considered all channel uncertainties and converted the nominal problem into a deterministic one based on worstcase principle. Then, the optimal solution can be obtained by utilizing Lagrange dual decomposition theory. The RPA algorithm can be accomplished by the following steps: (1) at FU’s receiver, it estimates the forward channel gains and obtains the estimated direct channel gain values. Determine the error upper bound according to the robustness requirement of system and the accuracy of channel estimation algorithm. Then, the related system parameters (e.g., estimated channel gains, background noise power) are fed back to its transmitter. (2) Data fusion center at FU’s BS collects the tolerable interference power levels, determines the minimum value, and broadcasts to all transmitters in femtocells. (3) Based on these system parameters and its own robustness requirement, each transmitter adjusts the transmit power by the designed RPA algorithm.
System model
We consider an uplink transmission model of twotier HetNets with one macrocell and multiple femtocells as shown in Fig. 1, where one MBS serves L MUs and K FBSs. Each FBS serves M FUs. Define the set of FUs as ∀i,j∈{1,2,⋯,M}, the set of FBSs as ∀k∈{1,2,⋯,K}, and the set of MUs as ∀l∈{1,2,⋯,L}. We suppose that both users and femtocell base stations are randomly distributed in the coverage area. For the sake of clarity, Table 1 gives the summary of the notations which are adopted in this paper.
In HetNets, femtocells share the same frequency resources with macrocells [1]. To protect the basic QoS of MUs, we need to limit the interference power caused from femtocell networks to macrocell networks under a certain allowable range [17]. Therefore, we consider a global interference constraint at the FU side, i.e.,
According to information theory, the received SINR at FBS over link i can be formulated as
where the first term of denominator denotes the interference power from neighboring FUs (i.e., intratier interference). The second part of denominator is the interference power from macrocell networks (i.e., crosstier interference).
To guarantee transmission qualities of each FU, i.e., the received SINR at each FBS (i.e., \(\gamma _{i}^{k}\)) should be bigger than a minimum SINR threshold, which is given as
where \(\gamma _{i}^{k,\text {min}}\) denotes the minimum SINR of the ith FU in the kth femtocell.
Considering the limitation of battery capacity of FUs, the transmit power of each FU is bounded, and we have the following constraint,
where \(p_{i}^{k,\text {max}}\) denotes the maximum transmission power of the ith FU in femtocell network k.
In order to better analyze the impact of interference from femtocells to macrocells, we define the outage probability of MUs as follows,
where P(m) denotes the outage probability of mth MURx, and I_{ac} denotes the actual interference from femtocells to macrocells (i.e., \(I_{\text {ac}}= {\sum \nolimits }_{k} {{\sum \nolimits }_{i} {p_{i}^{*k}G_{i}^{k}} }\), where \(p_{i}^{*k}\) denotes the optimal transmit power of FBS). When I_{ac}<I_{th}, there is no outage; otherwise, the actual outage can be calculated by \(\frac {I_{\text {ac}}I_{\text {th}}}{I_{\text {th}}}~\times ~100\%\).
To improve system capacity and spectrum efficiency, we formulate the following total transmit power minimization problem of FUs for uplink transmission model of twotier HetNets, i.e.,
Nominal optimization problem (P1)
where Ω_{n} denotes the feasible region of P1 (i.e., nonrobust optimization problem). To achieve these goals, we should discuss Ω_{n} when system information is exactly obtained. Obviously, when I_{th} is extremely small, the feasible solution may not exist since FUs are very close to the MURx. On the one hand, FUs cannot be allowed to transmit high power in order to guarantee MU’s QoS. On the other hand, FUs need to improve their transmission power for their SINR requirement. Hence, we analyze the feasible case for satisfying the QoS of both FUs and MUs.
Remark 1
Let \(\textbf {p}^{l}=[p_{1},...,p_{L}]^{T}, \textbf {p}^{k,max}=\left [p_{1}^{k,max},...,\right. \left.p_{M}^{k,max}\right ]^{T}\), \(\textbf {m}=\left [\gamma _{1}^{k,min}\sigma _{1}^{k}/h_{1}^{k},...,\gamma _{M}^{k,min}\sigma _{M}^{k}/h_{M}^{k}\right ]^{T}\) and \(\textbf {g}=[g_{ij}]=\left [\gamma _{i}^{k,min}g_{l}^{k}/h_{i}^{k}\right ]\). h is the M×M intratier channel gain matrix with \(\left [{{h_{ij}}}\right ] = \left \{ \begin {array}{l} {{h_{j}^{k}} \left /\right. {h_{i}^{k}}}\;\;if\;j \ne i\\ 0\;\;\;\;\;\;\;\;\;\;if\;j = i \end {array} \right.\). F is a M×M gain matrix of FUs whose elements are \(\begin {array}{l} \textbf {F}=[F_{ij}]=\left \{ \begin {array}{l}\begin {aligned} &\gamma _{i}^{k,min}h_{ij} &if \;\;j\neq i,\\ &0 &if \;\;j = i. \end {aligned}\end {array} \right. \end {array}\). From constraint (3), we have p^{k,min}=(I−F)^{−1}(gp^{l}+m), where \(\textbf {p}^{k,min}=\left [p_{1}^{k,min},...,p_{M}^{k,min}\right ]^{T}\) denotes the minimum transmission power of FUs in the kth femtocell, and I is a M×M unit matrix. The P1 is feasible if and only if the following conditions hold [18]:
where \(\textbf {p}^{k}=[p_{1}^{k},...,p_{M}^{k}]^{T}\) is the feasible solution of P1, ρ(F)is the spectral radius of F[19] and \(\textbf {G}^{k}=\left [G_{i}^{k},...,G_{M}^{k}\right ]\) denotes channel gain vector between FUTxs and MURxs.
If channel gains in C_{1} and C_{2} can be perfectly known, P1 can be proved to be a convex optimization problem, which is easily solved under the feasible region Ω_{n} by the existing scheme, such as [20]. However, in practical dynamic communication environment, channel gains are actually uncertain that can influence system performance. For example, channel uncertainties between FUTxs and MURxs may bring the harmful interference to MUs, even cause in outage. Therefore, it is necessary to study RPA problem.
Robust power allocation model
In this section, the uncertainties of channel gains in P1 are considered and we use bounded ellipsoidal uncertainty sets to model them. Then, the SIP problem is transformed into a deterministic convex problem based on the CauchySchwartz inequality theory and worstcase approach.
Models of channel uncertainties
In practical systems, due to the effect of channel fading and feedback delays, the CSI is uncertain, which can be assumed to have a bounded uncertainty of unknown distribution. Ellipsoidal set is widely used to approximate unknown and potentially complicated uncertainty sets [21]. For example, for OFDMbased cognitive radio networks, the author in [22] proposed a worstcase robust distributed PA scheme, which employs the ellipsoidal approximate method to model the channel uncertainties. In [23], based on game theory, the author presented a robust optimization equilibrium for competitive rate maximization under bounded channel uncertainty and formulated the imperfect CSI by using ellipsoidal uncertainty sets. According to those existing literatures, it is obvious to see that the ellipsoidal approximation has the advantage of parametrically modeling complicated data sets and provides a convenient input parameter to algorithms. Furthermore, there are statistical reasons that lead to ellipsoidal uncertainty sets and also result in optimization problems with convenient analytical structures [24].
Therefore, by using ellipsoidal approximation, each uncertain parameter can be written as the sum of its nominal value and perturbation part, e.g.,
where h_{ij} is the normalized intratier interference channel gain relevant to channel gain of link i. \({\bar {h}_{ij}}\) is the nominal value of channel gain between active FURx and other FUTxs from neighbor femtocells, and Δh_{ij} is the corresponding perturbation part.
Let H_{i} represent the uncertainty set of the ith row of matrix h. We use an ellipsoid set to describe H_{i}. Additionally, we denote \(\bar {\textbf {h}}=[{\bar {h}_{ij}}]\) and \(\Delta \bar {\textbf {h}}\) = \([\Delta \bar {h}_{ij}]\). Under ellipsoid approximation, the uncertainty set of H_{i} can be written as
where \({\bar {\mathbf {h}}_{i}}\) is the ith row of \(\bar {\mathbf {h}}\), and the corresponding perturbation part as Δh_{i}, and ε_{i}≥0 is the maximum evaluated error of every row in \({\bar {\mathbf {h}}_{i}}\).
Similarly, the uncertainty relevant to the interference channel gain between FU and MURx can be written as
where \(\bar {G}_{i}^{k}\) and \(\Delta G_{i}^{k}\) represent the nominal value and the perturbation part of channel gain between FU and MURx, respectively.
Let G_{i} represent the uncertainty set of the ith column of matrix \(\textbf {G} = \left [G_{1}^{1} \cdots G_{M}^{1}; \cdots ; G_{1}^{K} \cdots G_{M}^{K}\right ]\). Denote the ith column of \(\bar {\mathbf {G}}\) and the corresponding perturbation part as \({\bar {\mathbf {G}}_{i}}\) and ΔG_{i}, respectively. The uncertainty parameter G_{i} is described by an ellipsoid set as follows
where δ_{i}≥0 is the maximum deviation of each item in \({\bar {\mathbf {G}}_{i}}\).
Furthermore, we also consider uncertainties of the normalized crosstier interference channel gains from MUTx to FURx.
where \({\bar {g}_{il}}\) is the nominal value, and Δg_{il} is the perturbation part. Let g_{i} represent the uncertainty sets of the ith row of matrix g. Denote the ith row of \(\bar {\mathbf {G}}\) as \({\bar {\mathbf {G}}_{i}}\), and the corresponding perturbation part as Δg_{i}. In this case, the uncertainty region is given as
where ω_{i}≥0 is the maximum deviation of each row in \({\bar {\mathbf {G}}_{i}}\).
Robust power allocation optimization model
Considering the channel uncertainties, the RPA problem is formulated as
Robust power allocation problem (P2)
where Ω_{r} denotes the feasible region of RPA problem. Since P2 is limited by an infinite number of constraints like sets H_{i}, G_{i}, and g_{i}, P2 is proved to be a SIP problem [10]. A feasible method to solve the SIP problem is to transform it into a deterministic robust problem by considering the worst case in the constraints of P2. In other words, we can keep the system performance under any case of estimation errors.
According to the CauchySchwartz inequality theory and worstcase approach [25], the uncertain part of C_{4} and C_{5} can be converted into
Based on (15) and (18), the RPA problem (P2) can be reformulated as follows
Worstcase power allocation problem (P3)
where
It is obvious that the above P3 is a convex problem with liner constraints. To get an insight on the solution to P3 and compare it with that of the nominal problem (i.e., P1), we need now study the feasibility region of the robust problem. According to the feasible region of nonrobust problem [i.e., (7)], we derive the robust feasible region with the following form:
where ℵ[∙] denotes the sum of matrix elements, \(\overline {\textbf {F}}\) denotes the nominal matrix of Remark 1, \(\tilde {\textbf {p}}=\left [p_{i}^{k}\right ]=\left [p_{1}^{1},...,p_{M}^{1};...;p_{1}^{K},...,p_{M}^{K}\right ]\) is the feasible solution of RPA problem (i.e., P3) and \(\tilde {\textbf {p}}^{k}\) is the kth row of matrix \(\tilde {\textbf {p}}\). \(\overline {\textbf {G}}^{k}=\left [\overline {G}_{1}^{k},...,\overline {G}_{M}^{k}\right ]\) and δ=[δ_{1},...,δ_{M}]^{T} denote the nominal crosstier channel estimates and maximum channel perturbation, respectively. Obviously, conditions (21c) are satisfied. The proof of (21a) and (21b) is given in Appendix 8.
Robust power allocation algorithm
In this section, we will propose a RPA algorithm to solve P3 by applying the decomposition theory. The Lagrange function of P3 is defined as
where \(\lambda, \left \{\mu _{i}^{k}\right \}\) and \(\left \{\xi _{i}^{k}\right \}\) are Lagrange multipliers and \(\lambda \geq 0, \mu _{i}^{k}\geq 0, \xi _{i}^{k}\geq 0\). And the dual Lagrange function is
where
and the dual optimization problem is formulated as
For any FUs, the dual decomposition method can be separated into some subproblems with parallel form. Since \(L_{i}^{k}\left (p_{i}^{k},\lambda,\mu _{i}^{k},\xi _{i}^{k}\right)\) is a convex problem with respect to \(p_{i}^{k}\). According to the KKT condition [25], the optimal transmit power \(p_{i}^{k*}\) can be calculated by \(\frac {\partial L_{i}^{k}\left (p_{i}^{k},\lambda,\mu _{i}^{k},\xi _{i}^{k}\right)}{\partial p_{i}^{k}}=0\) and the result is
Define
where \(S_{\lambda },S_{\mu _{i}^{k}},\) and \(S_{\xi _{i}^{k}}\) are the subgradients of \(\lambda,\mu _{i}^{k}\), and \(\xi _{i}^{k}\), respectively.
Update \(p_{i}^{k*}(t+1)\) and Lagrange multipliers \(\lambda,\mu _{i}^{k}\), and \(\xi _{i}^{k}\) as follows
where [x]^{+}=max{0,x}, α,β, and θ are the step sizes which are positive and t is the step time (Table 2). The outline of our proposed RPA algorithm is described in the Table 2.
Performance analysis
Computational complexity
For the specific variable (i,k), the convergence times of finding the optimal solution \(p_{i}^{k*}\) via Newton iterative approach is assumed to be t_{1} for subproblem. As the dual problems can be decomposed into M×N subproblems, the sum iteration number of total subproblems is M×N×t_{1}, for all (i,k). In addition, from (31)–(33), we need the (2MN+1) steps to update the Lagrange multipliers. The iteration number of finding the optimal variables \(\left (\lambda ^{*}, \mu _{i}^{k*}, \xi _{i}^{k*}\right)\) is assumed to be t_{2}. Hence, the complexity of our proposed algorithm can be expressed as \(\mathcal {O}((MNt_{1}+2MN+1)t_{2})\).
Sensitivity analysis
In this subsection, we use local sensitivity analysis of P3 by perturbing its constraints. For all value of \(\Delta h_{i}^{j}, \Delta G_{i}^{k}\), and \(\Delta g_{i}^{l}\), the reduction of achievable sum transmit power can be approximated as
where λ^{∗} and \(\mu _{i}^{k*}\) denote optimal Lagrange multipliers. The proof is given in Appendix 8.
Numerical results
In this section, the simulation results and performance analysis are provided to verify the efficiency and performance of our proposed algorithm. In this part, we used MATLAB 2016 software to do the simulations via core i5. In our simulation, we assumed that actual channel fading follows Rayleigh fading model; therefore, actual channel gains \(G_{i}^{k}, h_{i}^{k}, {\text {and}} g_{i}^{k}\) are followed as \(\left \{0,\frac {A}{d^{r}}\right \}\), where d is the distance from transmitter to receiver, r∈ [ 2,5] denotes the pathloss exponent, and the attenuation parameter A is frequency dependent [26]. The traditional nonrobust PA algorithm is given in [5] under perfect CSI. Due to not taking into account the channel uncertainty, our RPA algorithm has more advantages in improving network performance compared with the nonrobust PA algorithm. Other simulation parameters are given in Table 3.
Figure 2 presents the transmission power of each FU under multiuser scenarios, such as M=3, and channel uncertainties are ε_{i}, and ω_{i} are supposed to be 1×10^{−3}, and δ_{i} is assumed to be 10% of \(\bar G_{i}^{k}\). As can be seen in Fig. 2, with the increasing iteration numbers, the transmit power increases and tends to be converging to a stable value when the iteration number is about six, which demonstrates the perfect convergence performance of our proposed algorithm. In addition, transmit power is restricted by the maximum value \(p_{i}^{k,\text {max}}\), which shows the proposed algorithm is feasible. As a result, it satisfies the maximum power constraint (4).
To demonstrate the effectiveness of our proposed algorithm in term of QoS protection of both FUs and MUs, we also give the comparison of performance between the proposed RPA algorithm and traditional nonrobust PA algorithm [5].
Figure 3 shows comparison between our RPA algorithm and nonrobust PA algorithm in terms of SINR. It is obvious that the SINR of each FU under our proposed algorithm exceeds the minimum SINR value with considering the estimation errors, whereas the nonrobust PA algorithm cannot guarantee SINR requirements of all FUs, which will lead to a communication outage. Due to the effects of channel fading and feedback delays, FUs cannot respond in time by using traditional PA algorithm so that the QoS of each FU is hard to guarantee. Moreover, it indicates that RPA algorithm can always ensure the normal communication of FUs. Therefore, the robustness of our proposed RPA algorithm is better than the traditional PA algorithm without consideration of channel uncertainties.
Figure 4 gives comparison of the interference power received at MBS between our proposed robust algorithm and the nonrobust algorithm. As shown in Fig. 4, the interference power introduced to the MBS with considering channel uncertainties is always under the interference power threshold, whereas the actual received interference power at MBS under nonrobust PA algorithm exceeds the tolerable region. It can be explained that the MUs may experience severe performance degradation. Therefore, QoS of MUs are not guaranteed without the consideration of estimation errors and an outage event happens.
Figure 5 provides comparison of total energy consumption under our proposed RPA algorithm and traditional nonrobust PA algorithm. From Fig. 5, the total power consumption of FUs in both nonrobust algorithm and RPA algorithm increase with the increasing number of iteration and converge to a stable value; however, the total transmit power of RPA algorithm is higher than that of nonrobust PA algorithm. From Figs. 3, 4, and 5, we can get a conclusion that our RPA algorithm can well protect the QoS of MUs at the expense of energy consumption.
Considering imperfection of actual CSI, in order to demonstrate the superiority of our proposed RPA algorithm under different channel uncertainty (i.e., δ_{i},ε_{i},ω_{i}) clearly, we give the satisfaction probability of MURx and SINR performance of FUs in Figs. 6, 7, and 8.
Figure 6 shows that the satisfaction probability of MURx using two different algorithms can be presented subject to different channel estimation errors \(\Delta G_{i}^{k}\). It is clear that the satisfaction probability of MURx under the nonrobust PA algorithm is rapidly declining as the increasing interference channel uncertainty δ_{i}. Whereas the system with our RPA algorithm can cope with this problem. This is because the RPA algorithm is adaptive that can adjust \(p_{i}^{k}\) according to channel perturbation δ_{i}. In addition, bigger interference threshold I_{th} of MURx can increase the feasible region of transmit power \(p_{i}^{k}\) and then decrease the outage probability of MURx. While satisfaction probability of MU using nonrobust PA algorithm is lower than that of MU using our RPA algorithm, in other words, nonrobust PA algorithm can increase the outage probability of MUs. Hence, it enables the proposed RPA algorithm to protect the normal communication of MUs.
Figure 7 shows the actual SINR received at FURx under intertier and crosstier channel uncertainties (i.e., ε_{i} and ω_{i}) by two algorithms. The bound of the interference perturbation is δ_{i}=0.0001. As can be seen from Fig. 7, the received SINR of FURx decreases with the increase of intertier channel perturbation ε_{i} and crosstier channel perturbation ω_{i} by our RPA algorithm and nonrobust PA algorithm. The reason is that optimal power reduction leads to the decrease of received SINR. Additionally, it is obvious that the received SINR in [5] is lower than the received SINR by using our proposed algorithm. And when ε_{i}>0.045, the received SINR in [5] cannot meet the minimum SINR requirement. What is more, received SINR of FURx declines with crosstier channel uncertainties ω_{i} increase. That is due to traditional nonrobust PA algorithm ignores channel estimation errors, and \(p_{i}^{k}\) cannot be adjusted in time under timevarying channel uncertainty.
Figure 8 presents the comparison of received SINR performance of FURx between our proposed RPA algorithm and nonrobust PA algorithm under channel uncertainties. From Fig. 8, we can see intuitively that the SINR performance of our algorithm is superior to that of nonrobust algorithm and cannot lead to the interruption of communication when channel environment is bad. In conclusion, the proposed RPA algorithm can improve the quality of communication system compared with the traditional nonrobust PA algorithm under channel uncertainties.
Conclusions
In this paper, a RPA problem is studied in uplink twotier HetNets with all possible channel uncertainties. Based on the worstcase approach, the robust resource optimization problem is converted into a convex one which is solved by using Lagrange dual method. The feasible regions and the closed analytical solution are obtained. Furthermore, performance analysis and the impact of channel uncertainties have been presented. The numerical results show that our proposed RPA algorithm is outperformance to traditional nonrobust algorithm in cases of protecting the QoS of MUs at the cost of energy loss. In our future work, we will extend the network structure for multiple macrocells and relayassisted transmission cases.
Appendix A
Proof
of condition (21) According to the discussion of Remark 1, the minimum transmission power of P3 is
where \(\boldsymbol {\Gamma } = \left [\gamma _{1}^{k,\text {min}}\omega _{1}\Psi,...,\gamma _{M}^{k,\text {min}}\omega _{M}\Psi \right ]^{T}\) denotes the perturbation part of interference, and \(\Psi = \sqrt {{\sum \nolimits }_{l} {p_{l}}^{2}}\). Additionally, Δ_{F} is a M×M matrix whose elements
According to \(\textbf {F}= \overline {\textbf {F}}+\boldsymbol {\Delta }_{F}\) and ρ(F)<1, we have
According to the definition of spectral radius and the property of Frobenius norm [27], we have
Combining with the triangle inequality [28], we have
□
Proof
of condition (21) Considering crosstier channel uncertainties between FUs and MURxs, the interference constraint condition of Ω_{r} is
According to inequality (11) and (15), we have
With that, the proof of condition (21) and (21) are completed. □
Appendix B
Based on the formula of Taylor series of the three element function, we have
where o denotes the corresponding high order infinitesimal small quantities. And, \(P^{*}\left (\bar {G}_{i}^{k},\bar {h}_{i}^{k},\bar {g}_{i}^{k}\right)\) is the optimal value for P3 without estimation errors (assuming that the estimated channel gains are equal to the actual channel gains).
Ignoring the effect of high order small variables, since P3 is convex, \(P^{*}\left (\bar G_{i}^{k}+\Delta G_{i}^{k},\bar h_{i}^{k}+\Delta h_{i}^{j},\bar g_{i}^{k} +\Delta g_{i}^{k}\right)\) is obtained from the Lagrange dual function and using the sensitivity analysis [29], we have
According to (34) and (35), we have the following expression
Abbreviations
 CSI:

Channel state information
 FBS(s):

Femtocell base station(s)
 FU(s):

Femtocell user(s)
 FURx:

FUReceiver
 FUTx:

FUTransmitter
 HetNet(s):

Heterogeneous network(s)
 KKT:

KarushKuhnTucker
 MBS:

Macrocell base station
 MU(s):

Macrocell user(s)
 MURx:

MUReceiver
 MUTx:

MUTransmitter
 OFDM:

Orthogonal frequencydivision multiplexing
 OFDMA:

Orthogonal frequencydivision multiple access
 PA:

Power allocation
 QoS:

Quality of service
 RPA:

Resource power allocation
 SINR:

Signaltointerferenceplusnoise ratio
 SIP:

Semiinfinite programming
 UE:

User equipment
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Funding
This work was supported by the National Natural Science Foundation of China (grant nos. 61601071 and 61301124), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (grant no. KJ1600412), the Municipal Natural Science Foundation of Chongqing (grant nos. CSTC2016 and JCYJA2197), the Seventeenth Open Foundation of State Key Lab of Integrated Services Networks of Xidian University (grant no. ISN1701), and the Dr. Startup Founds of Chongqing University of Posts and Telecommunications (grant no. A201612).
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YJ contributed in the conception of the study and design of the study. Furthermore, YJ and XL carried out the simulation together. XL wrote the manuscript and completed the performance analysis of our proposed algorithm with YJ’s help. YC and GQ helped to check and revise the manuscript. All authors read and approved the final manuscript.
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Xu, Y., Yu, X., Liu, Y. et al. Robust power allocation for twotier heterogeneous networks under channel uncertainties. J Wireless Com Network 2018, 224 (2018). https://doi.org/10.1186/s1363801812369
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Keywords
 Heterogenous networks
 Robust power allocation
 Channel uncertainty
 Worstcase approach