Driven by an explosive increase of wireless data traffic, dense small base stations with low power are deployed in a macro cell to increase the network capacity and accommodate more user terminals -. Such a network architecture is termed as the heterogeneous cellular network, where a macro cell typically coexists with various small cells, e.g., femto cells, pico cells and micro cells . In heterogeneous cellular networks, a number of economical small base stations (SBSs) are generally deployed around a relatively expensive macro base station (MBS), which can expand the network capacity and coverage at a reduced expense , . Meanwhile, there are also some challenging issues to be addressed for heterogeneous cellular networks, including the user association, resource allocation and interference management -. For example, the authors of  studied energy efficiency and load balancing for the user association problem, and proposed an energy conservation algorithm for energy-efficient transmissions. In , a load-balancing scheme was presented for addressing the association between subscribers and available wireless local area networks (WLANs).
Typically, both the orthogonal and non-orthogonal spectrum sharing may be considered for heterogeneous cellular networks, where a number of small cells are in the range of macro cells . In the orthogonal spectrum sharing mechanism, a given spectrum band of heterogeneous networks is first divided into a series of orthogonal sub-bands, which are then allocated to the macro cells and small cells, respectively. In other words, the macro cells and small cells are allowed to exclusively occupy their respective sub-bands, which is referred to as orthogonal spectrum sharing (OSS). By contrast, the non-orthogonal spectrum sharing (NOSS) enables SBSs and MBSs to simultaneously access the same spectrum, also called non-orthogonal multiple access (NOMA) in , . As discussed in existing literature -, the NOSS can achieve a higher spectral efficiency than OSS, since the same spectrum resources are simultaneously accessed by both the macro cell and small cell in heterogeneous cellular networks. However, this also results in mutual interference between the macro cell and small cell, which may degrade the quality-of-service (QoS) of heterogeneous networks.
To this end, an interference management scheme was presented in  for the sake of alleviating the adverse impact of mutual interference on intended users. Moreover, macroscopic and microscopic control schemes were proposed in  for the purpose of mitigating intercell interference in heterogeneous networks. It is noted that the broadcast nature of radio propagation and open architecture of heterogeneous networks make legitimate wireless transmissions of both macro cells and small cells become extremely vulnerable to eavesdropping attacks . Therefore, it is particularly important and necessary to explore the transmission security for heterogeneous cellular networks. Differing from traditional encryption technology, which only provides the computational security, physical-layer security (PLS) is emerging as a promising means of guaranteeing perfect secrecy by exploiting physical characteristics of the communications medium , . In the past decades, extensive research efforts have been devoted to improving PLS with the aid of cooperative relaying -, multiple-input multiple-output (MIMO) -, beamforming , , and millimeter-wave techniques , .
Until now, an increasing research attention has been paid to PLS for NOSS systems in heterogeneous cellular networks -. To be specific, the PLS for large-scale non-orthogonal multiple access networks was studied in  by using stochastic geometry and closed-form expressions for the secrecy outage probability were derived for both single-antenna and multiple-antenna scenarios. In , an effect of beamforming and artificial noise generation on the PLS of large-scale spectrum sharing networks was examined in terms of an average secrecy rate and secrecy outage probability. Later on, an access threshold based secrecy mobile association scheme was proposed in  for improving the secrecy throughput of heterogeneous networks. Moreover, the author of  investigated the PLS for orthogonal spectrum sharing systems with the aid of joint multiuser scheduling and friendly jammer selection. It was found in  that no extra secrecy diversity gain is achievable through the use of an opportunistic friendly jammer selection. In addition, the PLS benefit of heterogeneous ultra-dense networks was studied in  by jointly considering the caching and wireless energy harvesting. More recently, an interference-canceled underlay spectrum-sharing (IC-USS) scheme, called IC-NOSS throughout this paper, was proposed in  for the sake of enhancing the PLS of heterogeneous cellular networks, where closed-form expressions for outage probability and intercept probability are derived.
This paper differs from  in the following aspects. First, we examine the secrecy outage probability of IC-NOSS scheme in this paper, which is different from , where closed-form expressions for intercept probability and outage probability are derived from a security-reliability tradeoff (SRT) perspective. Mathematically speaking, it is more challenging to obtain a closed-form expression for secrecy outage probability than that for the intercept probability and outage probability. Second, this paper characterizes the secrecy diversity of IC-NOSS through an asymptotic secrecy outage analysis in the high signal-to-noise ratio (SNR) region, however no secrecy diversity is provided in . The main contributions of this paper are summarized as follows. First, we derive closed-form expressions for an overall secrecy outage probability of the IC-NOSS, interference-limited NOSS (IL-NOSS) and OSS schemes by jointly taking into account both the macro-cell and small-cell transmissions. Second, the secrecy diversity analysis of these three schemes is carried out by characterizing an asymptotic behavior of the secrecy outage probability in the high SNR region. Finally, numerical results show that the overall secrecy outage probability of IC-NOSS is substantially smaller than that of the OSS and IL-NOSS methods.
The reminder of this paper is organized as follows. In Section II, we present the system model of spectrum-sharing heterogeneous cellular networks and introduce the OSS, IL-NOSS and IC-NOSS schemes. In Section III, closed-form expressions for the secrecy outage probability of these schemes are derived over Rayleigh fading channels, followed by Section IV, where the secrecy diversity analysis is conducted. Next, numerical results and discussions are provided in Section V. Finally, Section VI gives some concluding remarks.
Ii System Model
In this section, we first describe the system model of a heterogeneous cellular network, where a small cell coexists with a macro cell in the face of a common passive eavesdropper (E) which is considered to silently wiretap the confidential messages of both the macro cell and small cell. As shown in Fig. 1, in the macro cell, MBS transmits its message to a macro user (MU), while SBS communicates with a small user (SU) in the small cell. An eavesdropper is assumed to tap both the MBS-MU and SBS-SU transmissions. Notice that both the MBS and SBS are connected to a core network via fiber cables, e.g., a mobile switch center (MSC) in the global system for mobile communication (GSM), a mobility management entity (MME) in the long term evolution (LTE), which guarantees the real-time information exchange between MBS and SBS. Notice that the focus of this paper is to analyze the secrecy outage probability and secrecy diversity of a specific heterogeneous network with a single macro cell and small cell. It is not mathematically tractable to derive closed-form secrecy outage probability for some more complex heterogeneous networks e.g. with randomly distributed multi-antenna nodes. Our derived secrecy outage probability expressions can be used to facilitate the secrecy performance evaluation of such complex networks through Monte-Carlo simulations. Moreover, the aforementioned system model of Fig. 1 is also applicable to a general heterogeneous network consisting of multiple MBSs and SBSs with the aid of base station pairing and grouping. If more than one MBS-SBS pair is available, we may divide the given spectrum into multiple orthogonal sub-bands to be assigned to different MBS-SBS pairs.
The macro cell and small cell are allowed to share the same spectrum band, and both the OSS and NOSS mechanisms are considered for the heterogeneous cellular network of Fig. 1. In the OSS scheme, the spectrum band is first divided into orthogonal sub-bands, which are then allocated to the macro cell and small cell, respectively. In this way, no mutual interference occurs between the macro cell and small cell. By contrast, in the NOSS mechanism, MBS and SBS simultaneously transmit their respective confidential messages over the same spectrum band. This can enhance the spectral efficiency, which, however, comes at the cost of degrading the QoS of transmissions, since mutual interference between the macro cell and small cell happens in this case.
In order to guarantee the QoS of heterogeneous cellular networks, transmit powers of MBS and SBS, denoted by and , respectively, are constrained in the conventional IL-NOSS method for the sake of keeping the mutual interference below a tolerable level. Differing from the IL-NOSS, the IC-NOSS scheme exploits a specially-designed signal transmitted at MBS, which not only generates a certain interference to the eavesdropper, but can also cancel out the mutual interference received at MU from SBS. Following the existing literature -, the channel coefficients of MBS-MU, MBS-SU, SBS-SU, SBS-MU, MBS-E, and SBS-E are, respectively, represented by , , , , , and , which are modeled as independent Rayleigh fading processes with respective channel gains of , , , , , and , namely , , , , , and . Although only the Rayleigh fading model is considered in this paper, similar secrecy results can be obtained for other channel models, e.g., Nakagami fading , Rician fading , and - fading .
According to  and , the aforementioned channel gains are modeled as , , , , , and to show an effect of the large-scale path loss, where , , , , , are transmission distances of the MBS-MU, MBS-SU, SBS-SU, SBS-MU, MBS-E, SBS-E channels, respectively, and, moreover, , , , , , and are their respective path loss factors. Besides, , , , , , and
represent small-scale fading variances of the MBS-MU, MBS-SU, SBS-SU, SBS-MU, MBS-E and SBS-E channels, respectively. Additionally, the additive white Gaussian noise (AWGN) with a zero mean and a variance ofis encountered at any receiver of Fig. 1.
Specifically, in our IC-NOSS scheme, SBS transmits a signal to SU with a normalized weight , where . Meanwhile, MBS transmits its information-bearing signal at a power of and a specially-designed signal at a power of , where and denotes an average power of the specially-designed signal in the range of . In this way, the total average transmit power of and is constrained to . Hence, we can express the received signal at MU as
where represents an AWGN encountered at MU. In order to neutralize the interference received at MU, the following equation should be satisfied, from which a special signal and its average power as well as a normalized weight are given by
where and are the channel phases of MBS-MU and SBS-MU, respectively. One can observe from (2)-(4) that the eavesdropper’s channel state information (CSI) of and is not needed, while , , , , and are required at MBS and SBS in carrying out the design of . In general, the CSIs of and
can be obtained at MU through channel estimation and then fed back to MBS and SBS over a multicast channel. The statistical information ofand may be readily estimated by accumulating the instantaneous CSIs of and . Moreover, MBS and SBS are often connected to the core network (e.g., a MSC in GSM, a MME in LTE, etc.) via fiber cables, through which the reliable information exchange of and can be achieved between MBS and SBS. Thus, both and can be acquired for designing at MBS through the core network. As discussed in , the message is typically initiated by another user terminal of cellular networks and sent via the core network first to SBS which then forwards to SU through its air interface. In other words, when the core network sends to SBS along with the power information , the same copies of and can be received and stored at MBS simultaneously. This guarantees the synchronization between MBS and SBS and no significant amount of extra time delay incurred at MBS in obtaining and compared to SBS, regardless of the latency of the core network. It is of particular interest to examine the impact of channel estimation errors and feedback delay on the secrecy performance of our IC-NOSS scheme, which is considered for future work.
From (2) and (3), an instantaneous transmit power for the special signal is obtained as
Substituting , and from (2)-(4) into (1) yields
from which the MBS-MU channel capacity relying on our IC-NOSS scheme is given by
where and . Let denote a ratio of the SNR of small cell to the SNR of macro cell (i.e., ), called the small-to-macro ratio (SMR) for short, where . Noting that as given by (3) should be in the range of , we can obtain the following inequality
which should be satisfied such that the mutual interference can be completely canceled out at MU. Also, the received signal at SU can be written as
where , and are given by (2)-(4), respectively. Treating and as interference, we can obtain the SBS-SU channel capacity as
where . Meanwhile, the signal transmissions of MBS and SBS may be overheard by the eavesdropper and the corresponding received signal is given by
where represents an AWGN encountered at the eavesdropper. It can be seen from the preceding equation that the eavesdropper may decode the confidential signals of and with or without the successive interference cancelation (SIC). In this paper, we assume that no SIC is adopted by the eavesdropper for decoding and . Although the SIC can be used for improving the achievable data rate of wiretap channel, it can also be employed by the legitimate SU for enhancing the data rate of SBS-SU transmissions. It can be expected that the use of SIC has a limited impact on the secrecy rate performance of our heterogeneous networks. Hence, by treating and as interference, the MBS-E channel capacity is obtained from (10) as
Similarly, the SBS-E channel capacity is given by
which completes the signal model of IC-NOSS scheme.
Iii Secrecy Outage Probability Analysis
In this section, we derive closed-form expressions for the secrecy outage probability of the OSS, IL-NOSS and IC-NOSS schemes over Rayleigh fading channels. As known, when the secrecy capacity of data transmissions denoted by falls below a predefined secrecy rate , perfect secrecy is not achievable and a secrecy outage event is considered to occur in this case. Thus, the probability of occurrence of a secrecy outage event (referred to as secrecy outage probability) can be expressed as
where and represent the secrecy capacity and secrecy rate, respectively.
Iii-a OSS Scheme
In this subsection, we analyze an overall secrecy outage probability of the macro-cell and small-cell transmissions relying on the OSS scheme, in which the macro cell and small cell take turns to access their shared spectrum. For notational convenience, let denote a fraction of the total spectrum assigned to MBS and the remaining fraction is allocated to SBS, where . Denoting the transmit powers of MBS and SBS by and , respectively, we can obtain the channel capacity of macro-cell transmission from MBS to MU and that of small-cell transmission from SBS to SU as
where and are referred to as signal-to-noise ratios (SNRs) of MBS and SBS, respectively, and and are fading coefficients of MBS-MU and SBS-SU channels, respectively. Also, the channel capacity of macro-cell wiretap link from MBS to E and that of small-cell wiretap link from SBS to E are given by
where and are fading coefficients of MBS-E and SBS-E channels, respectively. As a result, the secrecy capacity of macro-cell transmissions is given by the difference between the capacity of MBS-MU channel and that of MBS-E wiretap channel , namely
where . Similarly, the secrecy capacity of small-cell transmissions can be written as
Using (13) and (14), we can obtain the secrecy outage probability of macro-cell transmissions as
where denotes a predefined secrecy rate of the macro-cell transmission. Substituting from (14) into (16) yields
where . Noting that andand , we have
Similarly, by letting denote a secrecy rate of the small-cell transmission, the secrecy outage probability of the small-cell SBS-SU transmission relying on the OSS scheme can be obtained from (15) as
In order to show a joint effect of the individual secrecy outage probabilities of macro cell and small cell, we define an overall secrecy outage probability as the product of the secrecy outage probability of MBS-MU transmission and that of SBS-SU transmission. Besides the aforementioned multiplied secrecy outage probability definition, another overall secrecy outage probability in the normalized sum form may be given by a mean of the individual secrecy outage probabilities. It can be observed that the normalized sum secrecy outage probability would be dominated by the higher individual secrecy outage probability, especially when the lower individual secrecy outage probability is continuously improved to be sufficiently small even without degrading the higher individual one. By contrast, any improvements or degradations of both individual secrecy outage probabilities of the MBS-MU and SBS-SU can be readily reflected in the multiplied secrecy outage probability. Moreover, the IC-NOSS scheme is mainly designed to improve the MBS-MU secrecy without affecting the SBS-SU transmission, which motivates the use of the multiplied secrecy outage probability definition in this paper. As a consequence, an overall secrecy outage probability of the OSS scheme denoted by can be written as
where and are given by (18) and (19), respectively.
Iii-B IL-NOSS Scheme
This subsection presents the secrecy outage analysis of IL-NOSS scheme, where MBS and SBS are allowed to access their shared spectrum simultaneously and mutual interference occurs between the macro cell and small cell. To this end, certain power allocation may be employed for the sake of limiting such mutual interference below a tolerable level. Considering that MBS and SBS transmit their confidential information simultaneously over the same spectrum with respective powers of and , we can obtain the MBS-MU and SBS-SU channel capacities for the IL-NOSS scheme as
where and are fading coefficients of SBS-MU and MBS-SU channels, respectively. Similarly, the corresponding MBS-E and SBS-E wiretap channel capacities for the IL-NOSS scheme, denoted by and , respectively, are given by
As a consequence, the secrecy capacity of macro-cell and small-cell transmissions relying on the IL-NOSS scheme can be respectively expressed as
where , , and are the MBS-MU, MBS-E, SBS-SU and SBS-E channel capacities of IL-NOSS scheme, respectively.
From (13) and (21), the secrecy outage probability of MBS-MU transmission relying on the IL-NOSS approach can be obtained as
Substituting from (21) into (23), we have
where . Using Appendix A, we can obtain a closed-form expression for as
where is known as the exponential integral function and other used parameters are specified in Table I.
Similarly, the secrecy outage probability of small-cell transmission can be obtained from (22) as
which can be further given by
where the used parameters are stated in Table II. Similar to (20), an overall secrecy outage probability of the IL-NOSS scheme is obtained as
where and are the individual secrecy outage probabilities of the macro cell and small cell as given by (25) and (27), respectively.
Iii-C IC-NOSS Scheme
In this subsection, we present the secrecy outage probability analysis of IC-NOSS scheme. Using (6), (9), (11) and (12), we can obtain the secrecy capacity of MBS-MU and SBS-SU transmissions relying on the IC-NOSS scheme as
where , , and are the MBS-MU, MBS-E, SBS-SU and SBS-E channel capacities for the IC-NOSS scheme, respectively.
Using (13) and (29), we can obtain the secrecy outage probability of macro cell for the IC-NOSS scheme as
which is further given by
It can be observed from (32) that deriving a general closed-form expression for is challenging. As a consequence, let us consider an asymptotic case with , for which the terms of and of (32) are high order infinitesimals and negligible, leading to
where . From Appendix B, we obtain
where is given by
wherein and . Similarly, the secrecy outage probability of small-cell transmission relying on the IC-NOSS scheme can be obtained from (30) as
wherein . It is challenging to derive a general closed-form expression for from the preceding equation of (35). To this end, we consider a special case with a sufficiently small interference gain of , which is possible in practical heterogeneous cellular systems, since a small cell may be deployed in some shadowed areas of the macro cell e.g. in-building places, underground garages, tunnels . Hence, one can obtain from (3) and (5) that both and are negligible compared to for the case of , from which (35) is given by
Noting , , , and are independent exponentially distributed random variables with respective means of , , , and , we have
where is given by