- Research
- Open Access

# Distributed resource allocation with imperfect spectrum sensing information and channel uncertainty in cognitive femtocell networks

- Xiaoge Huang†
^{1}Email author, - Lin Shi†
^{1}, - Chenlu Zhang
^{2}, - Dongyu Zhang
^{1}and - Qianbin Chen
^{1}

**2017**:201

https://doi.org/10.1186/s13638-017-0985-1

© The Author(s) 2017

**Received:**25 July 2017**Accepted:**9 November 2017**Published:**28 November 2017

## Abstract

To meet the ever-increasing demands of mobile traffic, femtocells are considered as one of the promising solutions. In this paper, we study a sensing-based resource allocation scenario in cognitive femtocell networks and present an efficient distributed imperfect-spectrum-sensing-based resource allocation (DIRA) algorithm while considering the channel uncertainty to maximize the total data rate of cognitive femtocell networks by jointly optimizing both subchannel assignment and power allocation taking into account the influence of the sensing accuracy. However, the general optimization problem turns out to be a mixed integer programming problem. In order to make it tractable, the original optimization problem is divided into two sub-optimization problems,namely,optimal subchannel allocation and optimal power allocation. Specifically, the proposed distributed fairness-based subchannel allocation (DFSA) algorithm guarantees fairness by introducing channel condition difference and satisfaction degree as the indicators of subchannel allocation. Additionally, optimal power allocation with the consideration of imperfect spectrum sensing and interference uncertainty is performed using the proposed chance-constrained power optimization (CPO) algorithm. Bernstein’s approximation is conducted to make the chance constraint tractable. Simulation results illustrate that the distributed imperfect-spectrum-sensing-based resource allocation (DIRA) algorithm can provide considerable fairness among femtocells and at the same time maximize the total data rate of the cognitive femtocell network.

## Keywords

- Cognitive femtocell
- Resource allocation
- Imperfect spectrum sensing
- Interference uncertainty

## 1 Introduction

To accommodate with this ever-increasing demand for mobile data transmission, the mobile network operators (MNOs) is facing with urgent requirement of seeking for new technologies to enhance the capacity by 1000 times [1]. In this context, small cell deployment has been viewed as one of the most effective and cost-efficient solutions [2]. Small cell is an umbrella term for low-powered radio access nodes with a range of 10 m up to several hundred meters. It can be generally categorized into femtocells, picocells, or microcells according to their coverage range in ascending order. As several studies show more than 70% of data traffic occurs indoors [3], this has led to increasing interest in femtocells, which is known as home base station. The low-power, short-range, easy plug-and-play, and self-organization features of femtocells benefit both the users and operators.

Embedding femtocells in traditional cellular system helps offloading the overloaded traffic in macrocells, expanding coverage, and boosting network capacity. However, the scarcity of the available wireless spectrum resource becomes a challenging issue in the development of wireless communication technologies [4], which urges MNOs to optimally utilize the bandwidth in order to obtain the maximization of network capacity. In this case, dedicated-channel deployment of femtocells is no longer preferable from the operator’s perspective because of radio resource shortage and implementation difficulty. Compared with dedicated-channel deployment, co-channel deployment is more attractive due to easy implementation and more efficient utilization of wireless spectrum. However, co-channel deployment might introduce severe cross-tier interference between macrocells and femtocells where they share the same spectrum [5], especially when they are close to each other [6], and co-tier interference between co-channel deployed femtocells in dense deployment scenario. Therefore, interference mitigation in such two-tier networks needs to be tackled. Moreover, the random deployment, restricted/closed access of femtocells, and no coordination between two tiers turn this problem into a hard nut to crack [7]. Facing with the above challenges, cognitive radio has emerged as a potential technology which enables radio devices to monitor the radio environment and dynamically adjust transmission parameters according to the sensing result. A femtocell base station (FBS) equipped with cognitive function will be capable of identifying and choosing the suitable subchannel that provides the least harmful influence to others.

### 1.1 Related works

A considerable amount of literature is available for resource allocation and interference management in cognitive femtocell system. In [8], a hybrid overlay/underlay spectrum access mechanism was proposed to improve the overall system performance of cognitive femtocell networks. The subchannel allocation problem was formulated as a coalition formation game, and a modified recursive core algorithm was proposed to achieve stable and efficient allocation. In [9], a distributed joint power control method came up with a proper solution to manage the interference in two-tier femtocell networks.

Usually, subchannel allocation and power control are jointly considered in literatures. In [10], a resource allocation scheme was proposed to maximize the total data rate of cognitive small cells without causing intolerable interference to macrocell users, in which femtocell user equipments (FUEs) could estimate the interference channel through the pilot signals broadcasted by macrocell base station (MBS). Moreover, the proposed algorithm could ensure the fairness with only a tiny reduction in throughput performance. In [11], a decentralized approach for dynamic subchannel and power allocation was considered. Reinforcement learning-based algorithm was applied to solve the uncoordinated spectrum sharing problems. The spectrum allocation was based on reinforcement learning algorithm, while the power allocation applied convex optimization algorithm which was decided by each femtocell independently. Besides, game theory is also a feasible solution which has been widely investigated in existing researches to jointly investigate the subchannel and power allocation in femtocell networks. In [12], a cooperative Nash bargaining game model was developed to study the subchannel allocation and power control problem jointly for cognitive small cell networks with the consideration of cross-tier interference mitigation, outage limitation, imperfect CSI, and fairness.

Due to the inherent hardware limitations and variable wireless environment, spectrum sensing errors are inevitable, causing interference to macrocell user equipments (MUEs) in cognitive femtocell networks [13]. Hence, owing to the imperfectness of the spectrum sensing, the traditional resource allocation algorithms might experience performance decrease. In [14], authors used the particle swarm optimization (PSO) algorithm to solve a joint uplink resource allocation problem for cognitive networks under the consideration of imperfect spectrum sensing. In [15], a multi-objective optimization problem that jointly considered the femtocell throughput maximization and transmit power minimization was formulated, subject to interference constraints on both femtocell and macrocell including the co-channel interference and adjacent channel interference constraints under spectrum sensing error probabilities. In [16], pricing and power allocation strategies were studied in a two-tier femtocell network with the aim of maximizing energy efficiency, where both perfect and imperfect spectrum sensing cases were considered.

Besides, the information uncertainty is also significantly important for the variable wireless environment, including channel state information uncertainty [17], network access state uncertainty [18], and background noise uncertainty [19]. Particularly, those kind of uncertainty problems can be presented as a probabilistic problem by relaxing the constraint into an equivalent chance constraint [20–22]. Based on the aforementioned solutions, jointly considered subchannel allocation and power allocation are a proper way to solve resource allocation problems. The objectives of resource allocation mainly focus on interference management [9, 10, 12, 13, 19–22], capacity enhancement [8, 10, 13, 15, 20–22], power efficiency improvement [15, 16, 19], and fairness [10, 12]. However, to the best of our knowledge, resource allocation for cognitive femtocell network jointly considering interference management, fairness, imperfect spectrum sensing, and interference uncertainty has not been studied in previous works.

### 1.2 Contributions

- -
Firstly, the distribution of femtocells is modeled by a hexagonal grid in the existing literature. However, the topological randomness causes the grid-based model too idealized for both macrocells and femtocells, especially when most of femtocells are installed by their subscribed users. In this paper, the cognitive femtocell network topology is based on stochastic geometry; macrocell and femtocell base stations are modeled as two independent Voronoi tessellations, where MBSs and FBSs follow Poisson point process (PPP) distribution in their own tessellation.

- -
Secondly, the reliability of the spectrum sensing is taken into consideration which will affect the interference constraint. Moreover, the femto-to-macro interference constraint under channel uncertainty can be cast as the chance constraint. Bernstein’s approximation is conducted to make the chance constraint tractable.

- -
Thirdly, in order to maximize the total data rate of femtocells under the consideration of imperfect spectrum sensing and channel uncertainty while balancing the fairness of networks, a distributed fairness-based subchannel allocation (DFSA) algorithm is implemented which uses channel condition difference as an indicator of subchannel allocation, taking the satisfaction degree of each FBS into consideration. To further increase the total capacity of femtocell network, transmit power of femtocells on each subchannel will be adapted subject to interference constraints and minimum quality of service (QoS) requirements using the chance-constrained power optimization (CPO) algorithm.

The layout of this paper is outlined as follows. Section 2 describes the network model. Section 3 formulates our optimization task with imperfect spectrum sensing and interference uncertainty. In Section 4, the DIRA algorithm is proposed. Section 5 evaluates the performance of our proposed algorithm, and numerical results are presented with discussions. Section 6 concludes this paper.

## 2 System model

### 2.1 Network topology

*d*

_{ m }and

*d*

_{ f }, respectively. The two-tier network model can be abstracted as two Voronoi tessellations as shown in Fig. 2. By construction, each user located in the intersection of two cells will associate with either the MBS or the FBS of the Voronoi cell covering that user. We denote the sets of MBSs and FBSs as \(\mathcal {M}=[1,...,M]\) and \(\mathcal {F}=[1,...,F]\). In each femtocell, there are

*K*FUEs. An orthogonal frequency-division multiple access (OFDMA) downlink system is considered, where the total bandwidth of

*B*

_{ w }is divided into

*N*subchannels.

### 2.2 Channel model

Cognitive FBSs opportunistically access the licensed subchannels belong to the macrocells. Through periodic spectrum sensing performed by the cognitive FBS, subchannels can be identified as busy or idle. In this paper, we apply overlay spectrum sharing mode between MBSs and FBSs, which means subchannels which are determined as idle can be utilized by the FBS.

*g*

_{ f,k,n }be the channel gain of the

*k*th FUE in femtocell \(f\in {\mathcal {F}}\) on subchannel \(n\in {\mathcal {N}}\). Hence, the received signal-to-interfernece-plus-noise ratio (SINR) of FUE

*k*in femtocell

*f*on subchannel

*n*is given as

*f*to FUE

*k*on subchannel

*n*;

*I*is the received interference from other FBSs, where \(I=\sum _{e=1,e\neq f}^{F}{p_{{e,k,n}}^{F} g_{{e,k,n}}}\); and

*N*

_{0}is the noise power. The data rate of FUE

*k*in femtocell

*f*on subchannel

*n*is presented as

*g*

_{ f,k,n }can be acquired accurately through traditional channel estimation techniques. However, the channel gain

*g*

_{ f,n }from FBS

*f*to the MUE on subchannel

*n*is difficult to estimate due to the lack of cooperation between MBSs and FBSs. Thus, we model

*g*

_{ f,n }as

where \(\overline {g}_{f,n}\) denotes the estimated channel gain on subchannel *n* obtained by averaging *g*
_{
f,k,n
} and \({\tilde {g}}_{f,n}\) represents the uncertain part of the channel gain.

## 3 Problem formulation with imperfect spectrum sensing

In this section, firstly, we discuss the cross-tier interference caused by femtocells, which consists of co-channel interference caused by imperfect spectrum sensing and out-of-band interference introduced by sidelobe power leakage of orthogonal frequency division modulation (OFDM) signals. Secondly, an optimization framework which aims to maximizing the total data rate of the cognitive femtocell network is formulated with the consideration of imperfect spectrum sensing and interference uncertainty.

### 3.1 Interference from imperfect spectrum sensing and out-of-band emission

To ensure the QoS performance of MUEs, the interference caused by opportunistic access of FBSs in the licensed channel should be controlled. The interference introduced to the macro-tier consists of two parts: (i) co-channel interference as a result of imperfect spectrum sensing and (ii) cognitive radio out-of-band (OOB) emission.

*n*is a binary event denoted by \(S^{n}_{0}(\tilde {x}_{n}=0)\) and \(S^{n}_{1}(\tilde {x}_{n}=1)\), where \(\tilde {x}_{n}=0\) or 1 identifies that subchannel

*n*is sensed to be idle or busy. Similarly, the actual state of the subchannel

*n*can be denoted as \(H^{n}_{0}(x_{n}=0)\) and \(H^{n}_{1}(x_{n}=1)\), where

*x*

_{ n }represents the actual status of subchannel

*n*, with

*x*

_{ n }=0 or 1 indicating that the subchannel

*n*is vacant or occupied. Generally, there are four types of spectrum sensing probabilities: (i) idle channel detection probability: \(p^{n}_{nd}=Pr\{S^{n}_{0}|H^{n}_{0}\}\); (ii) false alarm probability: \(p^{n}_{f}=Pr\{S^{n}_{1}|H^{n}_{0}\}\); (iii) miss detection probability, \(p^{n}_{m}=Pr\{S^{n}_{0}|H^{n}_{1}\}\); and (iv) detection probability, \(p^{n}_{d}=Pr\{S^{n}_{1}|H^{n}_{1}\}\). Among the above sensing cases, false alarm and miss detection are considered as sensing errors. Since FBSs make access decisions based on the results of spectrum sensing, leading to four possible cases of spectrum sensing, which are given in Table 1, where the sensing results of femtocells are considered as priori probabilities.

Possibilities of spectrum sensing results for MUEs

Sensing state | Actual state | Probability | |
---|---|---|---|

Case 1 | \(S^{n}_{0}\) | \(H^{n}_{0}\) | \(P_{1,n}=Pr\{H^{n}_{0}|S^{n}_{0}\}\) |

Case 2 | \(S^{n}_{1}\) | \(H^{n}_{0}\) | \(P_{2,n}=Pr\{H^{n}_{0}|S^{n}_{1}\}\) |

Case 3 | \(S^{n}_{0}\) | \(H^{n}_{1}\) | \(P_{3,n}=Pr\{H^{n}_{1}|S^{n}_{0}\}\) |

Case 4 | \(S^{n}_{1}\) | \(H^{n}_{1}\) | \(P_{4,n}=Pr\{H^{n}_{1}|S^{n}_{1}\}\) |

Among the possible cases shown in Table 1, subchannel *n* is occupied by the MBSs only in cases 3 and 4. In case 3, the spectrum sensing result of subchannel *n* is vacant; however, the actual state of *n* is busy, resulting in great cross-tier interference from FBS to MUE on subchannel *n* since FBS has no idea that this subchannel is not vacant as detected. In case 4, although the detection made by FBS is correct, MUE on subchannel *n* can still suffer from OOB emissions due to sidelobe power leakage.

*P*

_{3,n }and

*P*

_{4,n }using Bayes’ theorem. Thus, we have

where \(p^{n}_{s}\) represents the occupation probability of MBS on subchannel *n*.

*j*by the femtocell transmission on subchannel

*n*, with unit transmit power, is given as

where \(\varphi (f)=T_{s}{\left [\frac {\sin \left (\pi f T_{s}\right)}{\pi f T_{s}}\right ]}^{2}\) represents the power spectral density (PSD) of an OFDM signal; *T*
_{
s
} is the duration of an OFDM signal.

*s*with unit transmit power as

*ε*∈(0,1) to guarantee the interference constraint in probability. Hence, the interference constraint can be written as a chance constraint as follows:

where *I*
_{thre,s
} is the interference limitation from FBSs to MUE *s*.

### 3.2 General optimization framework

where *τ*
_{
f,k,n
}=1 or 0 indicates whether subchannel *n* is allocated to FUE *k* in femtocell *f* or not, \(p^{F}_{\text {max}}\) is the maximum transmit power of a FBS, and \(R^{0}_{f,k}\) is the minimum QoS requirement of FUE *k* in femtcoell *f*.

In our optimization problem, C1–C2 are power constraints indicating the transmit power of a FBS should not exceed the maximum transmit power \(p^{F}_{\text {max}}\). C3 is the minimum QoS requirement for FUEs. C4–C5 are constraints of subchannel allocation, representing that a subchannel *n* can not be assigned to two different FUEs in the same femtocell. C6 is the cross-tier interference constraint represented by chance constraint, taking both imperfect spectrum sensing and interference uncertainty into consideration. Via C6, the cross-tier interference from FBSs to MUE *s* will be limited below the threshold *I*
_{thre,s
} with a probability not less than 1−*ε*. In practice, it is difficult to acquire channel information accurately; thus, it is more reasonable to present the interference constraint as a chance constraint. According to the above analysis, C4 and C5 are integer constraints, which, as a result, lead the optimization problem to be a mixed chance-constrained integer programming problem.

## 4 Distributed imperfect-spectrum-sensing-based resource allocation

The optimization problem in (9) is a mixed chance-constrained integer programming problem which is computationally complex to address. To make it tractable, we divide the original problem into two sub-optimization problems and solve them in two steps. Firstly, in order to remove the integer constraints of the optimization problem, we address the subchannel allocation problem using DFSA algorithm with the consideration of fairness among FBSs. In this way, the mixed chance-constrained integer programming problem is simplified as a chance-constrained programming problem without integer variables. Secondly, optimal power allocation to the subchannels is achieved by CPO algorithm. Bernstein’s approximation is applied to transform the chance constraint into a convex constraint, and the Lagrangian dual algorithm is used to solve the convex power optimization problem.

### 4.1 Subchannel allocation by DFSA algorithm

We propose a distributed fairness-based subchannel allocation algorithm to assign the subchannels with unit power allocation. The aim of the DFSA scheme is to achieve a fair subchannel allocation, while maintaining considerable data rate of the whole network.

*f*, the channel condition

*c*

_{ f,n }of subchannel

*n*is considered as the average estimated channel gain from FBS

*f*to all FUEs associated with this FBS, given by

*f*sorts its sensing results, namely, the available subchannels in a descending order with respect to the

*c*

_{ f,n }, as shown in Table 2. By calculating the channel condition difference between certain subchannel and the next subchannel ranked in CCT, we can obtain the channel condition difference table (CCDT), as shown in Table 3.

Channel condition table (CCT) for FBS *f*

Rank | Subchannel ID | Channel condition |
---|---|---|

1 |
| \(c^{f}_{1}=c_{f,n}\) |

2 |
| \(c^{f}_{2}=c_{f,m}\) |

⋯ | ⋯ | ⋯ |

Channel condition difference table (CCDT) for FBS *f*

Rank | Subchannel ID | Channel condition difference |
---|---|---|

1 |
| \(d^{f}_{1,2}=c^{f}_{1}-c^{f}_{2}\) |

2 |
| \(d^{f}_{2,3}=c^{f}_{2}-c^{f}_{3}\) |

⋯ | ⋯ | ⋯ |

The DFSA algorithm is based on CCDT instead of CCT. In the CCT-based subchannel allocation scheme, FBS *f* may suffer large quality degradation if FBS *l* accesses *f*’s first rank subchannel, although there are not too much differences between the second rank and first rank subchannel of FBS *l*, leaving FBS *f* with no choice but to access the second rank subchannel with relatively worse channel condition. However, our DFSA algorithm is based on CCDT; the FBS with larger first rank difference value may have the prior chance to access its first rank subchannel, which decrease the risk of suffering performance degradation due to the loss of the preferred subchannel. By taking channel condition difference as the indicator of subchannel allocation, the fairness between FBSs based on the average data rate can be guaranteed.

*f*under spectrum sensing result and \(N^{f}_{D}\) is the number of subchannels desired by FBS

*f*. The actual subchannel access will take place once the whole subchannel selection procedure is finished; otherwise, there exists an available subchannel list to be accessed by FBS

*f*. \(N^{f}_{to}\) is the number of subchannels for FBSs to access, which is initialized as 0 at the beginning of the algorithm. counter

^{ f }represents the number of subchannels still required by FBS

*f*, given by

Subchannel requirement table (SRT) for FBS *f*

Number of total available subchannels | \(N^{f}_{A}\) |

Number of desired subchannels | \(N^{f}_{D}\) |

Number of subchannels to access | \(N^{f}_{to}\) |

Number of subchannels still needed | counter |

Number of remained available subchannels | \(N^{f}_{R}\) |

Satisfaction degree | SD |

*f*, which is initialized as \(N^{f}_{A}\) and also changes during the allocation procedure. SD

^{ f }is the satisfaction degree of FBS

*f*, given by

Furthermore, with the consideration of satisfaction degree, the fairness in the number of subchannels allocated to FBSs can be guaranteed.

Once the DFSA algorithm starts, FBSs share the CCDT and SRT with other FBSs in the same macrocell via broadcating. As a result, FBSs in the same macrocell have the knowledge of CCDTs and SRTs of each other. With this information, each FBS can perform the subchannel allocation process individually without causing collisions on the same subchannel.

In the proposed DFSA algorithm, only one subchannel can be selected by a certain FBS each time. The detail procedure of DFSA algorithm is shown in algorithm 1. The CCT, CCDT, and SRT of FBS *f* is denoted by CCT_{
f
}, CCDT_{
f
}, and SRT_{
f
}, respectively. Besides, the table update procedure is shown in algorithm 2.

### 4.2 Power allocation for femtocells by chance-constrained power optimization

*τ*

_{ f,k,n }in the original optimization problem in (9) can be substituted by the subchannel allocation result. To further maximize the total data rate of cognitive femtocell network, we optimize the transmit power under the constraints of power, QoS requirement, and cross-tire interference. Thus, the optimization problem can be written as

where \(\tau ^{*}_{{f,k,n}}\) is the subchannel allocation achieved by employing the DFSA algorithm in the previous subsection.

However, the optimization problem in (13) is still intractable due to the non-convex chance constraint in C6. To achieve a convex feasible set of C6, a convex approximation of C6 should be performed, which can be achieved by using the Bernstein approximation method [25, 27].

#### 4.2.1 Bernstein’s approximation of the chance constraint

- 1)
**p**is a vector of decision parameters. - 2)
{

*ζ*_{ n }} is the set of random variables, with marginal distribution denoted by {*π*_{ n }}. - 3)
{

*π*_{ n }} belongs to a given family of probability distribution with bounded support of [−1,1], which means*ζ*_{ n }varies in the range of [−1,1].

*t*>0,

**p**) [27]. The Bernstein approximation of (14) can be formulated as

**p**satisfies the chance constraint if it satisifies Eq. (17). This approximation is tractable if {

*Λ*

_{ n }(

*y*)} can be evaluated efficiently [25]. Consider a case of

*Λ*

_{ n }(

*y*) when

*σ*

_{ n }≥0 are constants. By choosing appropriate \(\mu ^{-}_{n}\), \(\mu ^{+}_{n}\), and

*σ*

_{ n }and replacing

*Λ*

_{ n }in (17) with its upper bound given in (18), Eq. (17) can be bounded by

which is a safe conservative approximation of Eq. (14). Ben-Tal and Nemirovski [25] give some examples of the value of \(\mu ^{-}_{n}\), \(\mu ^{+}_{n}\), and *σ*
_{
n
}≥0 based on some prior knowledge (e.g., support, unimodality, and symmetry) of the distributions.

*f*

_{0}(

**p**) and

*f*

_{ n }(

**p**) into (20), noting that \(p^{F}_{{f,k,n}}\geq 0\), then we obtain

where \(\gamma ^{s}_{{f,k,n}}\triangleq \mu ^{+}_{n}\alpha ^{s}_{{f,k,n}}+\beta ^{s}_{{f,k,n}}\). Therefore, the chance constraint C6 in Eq. (13) is transformed into a convex constraint.

#### 4.2.2 Optimal power allocation for femtocells

*F*×

*N*subproblems, which can be solved iteratively. Accordingly, we have

*f*to FUE

*k*on subchannel

*n*can be expressed as

where \([x]^{+}\triangleq \max (0,x)\); thus, the boundary constraint C1 in (25) is contained in Eq. (35).

*λ*

_{ f,k },

*ν*

_{ f,k }, and

*δ*, given by

where \(\varepsilon ^{(t)}_{1}\), \(\varepsilon ^{(t)}_{2}\), and \(\varepsilon ^{(t)}_{3}\) are step sizes and *t* is the iteration number.

## 5 Simulation results and discussions

*d*

_{ m }and

*d*

_{ f }in a 4000 × 4000 m scenario. MUEs and FUEs are distributed randomly in the scenario. The number of FUEs associated with an FBS is 4. The channel gains are modeled as i.i.d. exponential random variables. Shadowing effect is modeled as a log-normal variable with standard deviation 6 dB. The false alarm probability \(p^{n}_{f}\) and miss detection probability \(p^{n}_{m}\) are uniformly distributed over (0.05, 0.1) and (0.01, 0.05). The occupation probability of MBSs on subchannel

*n*is uniformly distributed over (0, 1). The Bernstein approximation parameters \(\mu ^{-}_{n}\), \(\mu ^{+}_{n}\), and

*σ*

_{ n }are chosen from [25]. The simulation parameters are given in Table 5.

System simulation parameters

Parameter | Value |
---|---|

Density of MBS distribution | 2 BS/ km |

MBS transmit power | 43 dBm |

FBS transmit power | 20 dBm |

Carrier frequency | 2 GHz |

Total bandwidth | 10 MHz |

Number of subchannels | 50 |

Thermal noise PSD | −174 dBm/Hz |

Shadowing standard deviation | 6 dB |

Pathloss from FBS to FUE (dB)[26] | 38.6+20 log10( |

*ε*in the total data rate of FBSs, where

*d*

_{ f }=12BS/ km

^{2},

*I*

_{thre,s }=−110 dBm, \(R^{0}_{f,k}=5\) kbps. As shown in Fig. 6, the total data rate of FBSs increases when the

*ε*gets larger. Obviously, a larger

*ε*relaxed the interference constraints to FBSs, which results in higher tolerance for cross-tier interference from FBSs to MUEs. It illustrates that the higher data rate of FBSs can be achieved at the cost of increasing interference to MUEs.

*d*

_{ f }=12BS/ km

^{2},

*I*

_{thre,s }=−110 dBm, and \(R^{0}_{f,k}=5\) kbps. As shown in Fig. 7, the cross-tier interference is an increasing function of

*P*

_{max}. The cross-tier interference from FBSs to MUEs in imperfect spectrum sensing case is lower than that in perfect spectrum sensing case. This is because the imperfect spectrum sensing case overestimates the cross-tier interference with the consideration of sensing errors, such as false alarm and miss detection.

*d*

_{ f }=12BS/ km

^{2},

*I*

_{thre,s }=−110 dBm, and \(R^{0}_{f,k}=5\) kbps. It can be seen that the total data rate grows with the increase of

*P*

_{max}, and the proposed DIRA scheme significantly increases the total data rate of FBSs.

## 6 Conclusions

In this paper, we investigated the resource allocation problem in cognitive femtocell networks. A joint subchannel and power allocation algorithm was proposed to maximize the total data rate of femtocells with the consideration of fairness and imperfect spectrum sensing. Particularly, a CCDT-based subchannel allocation algorithm DFSA was developed to allocate subchannels to FBSs while guaranteeing the fairness among femtocells. We introduced spectrum sensing error probabilities to capture the imperfect spectrum sensing influence and combined them with OOB emission to formulate the cross-tier interference constraint. Furthermore, due to the interference uncertainty, we formed the interference constraint as the chance constraint and implied Bernstein’s approximation to make it tractable. Finally, the optimal power allocation problem was solved by the Lagrangian dual method. Simulation results verified that our proposed algorithm can achieve fair subchannel allocation and significant data rate improvement.

## Declarations

### Acknowledgements

This work is supported by the National Natural Science Foundation of China (NSFC) (61401053) and Changjiang Scholars and Innovative Research Team in University (IRT1299).

### Authors’ contributions

XH designed the study, conception, and main algorithms. LS drafted the article and designed the simulations and theoretical certifications. CZ worked with the data analysis and prepared the manuscript. QC reviewed and edited the manuscript. DZ worked with the data analysis and encoding. All authors have made substantive intellectual contributions to this study and approved the manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

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## Authors’ Affiliations

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