UserCentric Interference Nulling in Downlink MultiAntenna Heterogeneous Networks
Abstract
In heterogeneous networks (HetNets), strong interference due to spectrum reuse affects each user’s signaltointerference ratio (SIR), and hence is one limiting factor of network performance. In this paper, we propose a usercentric interference nulling (IN) scheme in a downlink largescale HetNet to improve coverage/outage probability by improving each user’s SIR. This IN scheme utilizes at most maximum IN degree of freedom (DoF) at each macroBS to avoid interference to uniformly selected macro (pico) users with signaltoindividualinterference ratio (SIIR) below a macro (pico) IN threshold, where the maximum IN DoF and the two IN thresholds are three design parameters. Using tools from stochastic geometry, we first obtain a tractable expression of the coverage (equivalently outage) probability. Then, we analyze the asymptotic coverage/outage probability in the low and high SIR threshold regimes. The analytical results indicate that the maximum IN DoF can affect the order gain of the outage probability in the low SIR threshold regime, but cannot affect the order gain of the coverage probability in the high SIR threshold regime. Moreover, we characterize the optimal maximum IN DoF which optimizes the asymptotic coverage/outage probability. The optimization results reveal that the IN scheme can linearly improve the outage probability in the low SIR threshold regime, but cannot improve the coverage probability in the high SIR threshold regime. Finally, numerical results show that the proposed scheme can achieve good gains in coverage/outage probability over a maximum ratio beamforming scheme and a usercentric almost blank subframes (ABS) scheme.
Heterogeneous networks, multiple antennas, intertier interference coordination, stochastic geometry, optimization.
I Introduction
Heterogenous wireless networks (HetNets), i.e., the deployment of low power small cell base stations (BSs) overlaid with conventional large power macroBSs, provide a powerful approach to meet the massive growth in traffic demands by aggressively reusing existing spectrum assets [1, 2]. However, spectrum reuse in HetNets causes strong interference. This affects the signaltointerference ratio (SIR) of each user, and hence is one of the limiting factors of network performance. Interference management techniques are thus desirable in HetNets[3]. One such technique is interference cooperation. For example, in [4, 5, 6], different interference cooperation strategies are considered and their performances are analyzed for largescale HetNets under random models in which the locations of BSs and users are spatially distributed as independent homogeneous Poisson point processes (PPPs)[7, 8]. However, in [4, 5, 6], the cooperation clusters are formed to favor a typical user located at the origin of the network (referred to as the typical user) only, and hence, the analytical performance of the typical user is better than the actual network performance (of all the users). In addition, [4, 5, 6] only consider singleantenna BSs. Orthogonalizing the time or frequency resource allocated to macro cells and small cells can also mitigate interference in HetNets. One such technique is almost blank subframes (ABS) in 3GPP LTE [9]. In ABS, the time or frequency resource is partitioned, whereby offloaded users and the other users are served using different portions of the resource in HetNets with offloading. The performance of ABS in largescale HetNets with offloading is analyzed in [9] using tools from stochastic geometry. Note that ABS focuses on mitigating the interference of offloaded users, and [9] only considers singleantenna BSs.
Deploying multiple antennas at each BS in HetNets can further improve network performance. With multiple antennas, besides boosting signals to desired users, more effective interference management techniques can be implemented [10, 11, 12, 13, 14, 15]. For example, in [10, 11, 12, 13], the authors consider HetNets with a single multiantenna macroBS and multiple multiantenna smallBSs, where the multiple antennas at the macroBS are used for serving its scheduled users as well as mitigating the interference to some small cell users using different interference coordination schemes. These schemes are analyzed and shown to improve the network performance. In particular, [13] also considers multiple antennas at each user, and proposes an opportunistic interference alignment scheme to design the transmit and receive beamformers to mitigate interference. Each small BS is assumed to have a different nearest victim small user, and victim user selection is not considered. Note that since only one macroBS is considered in [10, 11, 12, 13], the analytical results obtained in [10, 11, 12, 13] cannot reflect the macrotier interference, and thus may not offer accurate insights for practical HetNets. In [14, 15], largescale multiantenna HetNets are considered. Specifically, [14] considers offloading, and proposes an interference nulling (IN) scheme where some degree of freedom at each macroBS can be used for avoiding its interference to some of its offloaded users. The rate coverage probability is analyzed and optimized by optimizing the amount of degree of freedom (DoF) for interference nulling. However, the IN scheme proposed in [14] only improves the performance of scheduled offloaded users, and scheduled offloaded users are selected by the corresponding macroBS for interference nulling with equal probability. Hence, the IN scheme proposed in [14] may not effectively improve the overall rate coverage probability. In [15], a fixed number of BSs which provide the strongest average received power for the typical user form a cluster, and adopt an interference coordination scheme where the BSs in each cluster mitigate interference to users in this cluster. The coverage probability is analyzed based on the assumption that the BSs in each cluster are the strongest BSs of all the users in this cluster.
The investigation of interference management techniques in largescale singletier multiantenna cellular networks is less involved than that in largescale multiantenna HetNets, and hence has been more extensively conducted (see [16, 17, 18, 19] and the references therein). In [16, 17, 18], all the BSs are grouped into disjoint clusters. Coordination [16, 17] and cooperation [18] are performed among the BSs within each cluster to mitigate intracluster interference. Specifically, [16] and [17] design disjoint BS clustering from a transmitter’s point of view and fail to consider each user’s interference situation. The dynamic clustering proposed in [18] considers all the users’ signal and interference situations to optimize the network performance. However, it requires centralized control and may not be suitable for large networks. Recently, a novel distributed usercentric IN scheme, which takes account of each user’s desired signal strength and interference level, is proposed and analyzed for (singletier) multiantenna small cell networks in [19]. However, in [19], the maximum DoF for IN (i.e., maximum IN DoF) at each BS is not adjustable, and thus cannot properly utilize resource in small cell networks. Moreover, directly applying the scheme in [19] to HetNets cannot fully exploit different properties of macro and pico users in HetNets.
In this paper, we consider a downlink largescale twotier multiantenna HetNet and propose a usercentric IN scheme to improve the coverage probability by improving each user’s SIR. This scheme has three design parameters: the maximum IN DoF , and the IN thresholds for macro and pico users, respectively. In this scheme, each scheduled macro (pico) user first sends an IN request to a macroBS^{1}^{1}1Note that, compared to a picoBS, a macroBS usually causes stronger interference due to larger transmit power, and has a better capability of performing spatial cancellation due to a larger number of transmit antennas. Thus, it is more advisable to perform IN at macroBSs. if the power ratio of its desired signal and the interference from the macroBS, referred to as the signaltoindividualinterference ratio (SIIR), is below the IN threshold for macro (pico) users. Then, each macroBS utilizes zeroforcing beamforming (ZFBF) precoder to avoid interference to at most scheduled users which send IN requests to it as well as boost the desired signal to its scheduled user. In general, the performance analysis and optimization of interference management techniques in largescale multiantenna HetNets are very challenging, mainly due to i) the statistical dependence among macroBSs and picoBSs [10], ii) the complex distribution of a desired signal using multiantenna communication schemes, and iii) the complicated interference distribution caused by interference management techniques (e.g., beamforming). Our main contributions are summarized below. The analytical and numerical results obtained in this paper provide valuable design insights for practical HetNets.

We obtain a tractable expression of the coverage (equivalently outage) probability, by adopting appropriate approximations and utilizing tools from stochastic geometry.

We obtain the asymptotic expressions of the coverage/outage probability in the low and high SIR threshold regimes, using series expansions of special functions. The analytical results indicate that the maximum IN DoF can affect the order gain of the outage probability in the low SIR threshold regime, but cannot affect the order gain of the coverage probability in the high SIR threshold regime; the IN thresholds only affect the coefficients of the coverage/outage probability in the low and high SIR threshold regimes.

We consider the optimizations of the maximum IN DoF for given IN thresholds in the two asymptotic regimes, which are challenging integer programming problems with very complicated objective functions. By exploiting the structure of each objective function, we characterize the optimal maximum IN DoF. The optimization results reveal that the IN scheme can linearly improve the outage probability in the low SIR threshold regime, but cannot improve the coverage probability in the high SIR threshold regime.

We show that the IN scheme can achieve good gains in coverage/outage probability over a maximum ratio beamforming scheme and a usercentric ABS scheme, using numerical results.
The key notations used in the paper are listed in Table I.
Notation  Description 

,  PPP of BSs in the th tier, PPP of users 
,  Density of PPP , density of PPP 
,  Transmit power at each BS in the th tier, number of transmit antennas at each BS in the th tier 
Path loss exponent in the th tier  
Set of macrousers (), set of picousers ()  
Distance between the typical user and its serving BS in the th tier  
Association probability of the typical user to  
Number of the potential IN users of an arbitrary macroBS  
,  SIR coverage probability, SIR threshold 
,  Maximum IN DoF, IN threshold for the th tier in the IN scheme 
Ii Network Model
We consider a twotier HetNet where a macrocell tier is overlaid with a picocell tier, as shown in Fig. 1. The locations of macroBSs and picoBSs are spatially distributed as two independent homogeneous Poisson point processes (PPPs) and with densities and , respectively. The locations of users are also distributed as an independent homogeneous PPP with density . Without loss of generality, denote the macrocell tier as the st tier and the picocell tier as the nd tier. We focus on the downlink scenario. The macroBSs and the picoBSs share the same spectrum concurrently. Each macroBS has antennas with total transmission power , each picoBS has antennas with total transmission power , and each user has a single antenna. Assume . We consider both largescale fading and smallscale fading. Specifically, due to largescale fading, transmitted signals from the th tier with distance are attenuated by a factor , where is the path loss exponent of the th tier and . For smallscale fading, we assume Rayleigh fading channels.
Iia User Association
We assume open access [4]. User (denoted as ) is associated with the BS which provides the maximum longterm average (over smallscale fading) received power among all the macroBSs and picoBSs. This associated BS is called the serving BS of user . Note that within each tier, the nearest BS to user provides the strongest longterm average received power in this tier. User is thus associated with (the nearest BS in) the th tier, if^{2}^{2}2In the user association procedure, the first antenna is normally used to transmit signal (using the total transmission power of each BS) for received power determination according to LTE standards. , where is the distance between user and its nearest BS in the th tier. We refer to the users associated with the macrocell tier as the macrousers, denoted as , and the users associated with the picocell tier as the picousers, denoted as . All the users can be partitioned into two disjoint user sets: and . After the user association, each BS schedules its associated users according to TDMA, i.e., scheduling one user in each time slot. Hence, there is no intracell interference.
IiB Performance Metric
In this paper, we study the performance of the typical user denoted as , which is a scheduled user located at the origin [20]. Since HetNets are interferencelimited, we ignore the thermal noise in the analysis of this paper. Note that the analytical results with thermal noise can be obtained in a similar way[21]. The coverage probability of is defined as the probability that the SIR of is larger than a threshold [4], i.e.,
(1) 
where is the SIR threshold. The outage probability of is defined as the probability that the SIR of is smaller than or equal to a threshold, i.e., . The coverage/outage probability provides the cumulative probability function (c.d.f.) of the random SIR over the entire network[7]. In Sections IV, V and VI, we shall analyze the coverage/outage probability in the general, low and high SIR threshold regimes, separately.
Iii Usercentric Interference Nulling Scheme
In this section, we first elaborate on a usercentric IN scheme. Then, we obtain some distributions related to this scheme.
Iiia Scheme Description
First, we refer to an interfering macroBS which causes the SIIR at scheduled user in the th tier () below threshold as a potential IN macroBS of , where . We refer to as the IN threshold for the th tier. Mathematically, interfering macroBS is a potential IN macroBS of scheduled user if , where is the distance from macroBS to . Note that and are two design parameters of the IN scheme. In each time slot, each scheduled user sends IN requests to all of its potential IN macroBSs. We refer to the scheduled users which send IN requests to interfering macroBS as the potential IN users of interfering macroBS (in this time slot). We introduce another design parameter of this IN scheme, referred to as the maximum IN DoF. Consider a particular time slot. Let denote the number of the potential IN users of interfering macroBS . Note that implies . Consider two cases in the following. i) If and , macroBS makes use of at most DoF to perform IN to some of its potential IN users. In particular, if , macroBS can perform IN to all of its potential IN users using DoF; if , macroBS randomly selects out of its potential IN users according to the uniform distribution, and perform IN to the selected users using DoF. Hence, in this case, macroBS performs IN to potential IN users (referred to as the IN users of macroBS ) using DoF (referred to as the IN DoF of macroBS ). ii) If or , macroBS does not perform IN. In this case, we let . In both cases, DoF at macroBS is used for boosting the desired signal to its scheduled user.
Now, we introduce the precoding vectors at macroBSs in the IN scheme. Consider two cases in the following. i) If and , macroBS utilizes the lowcomplexity ZFBF precoder to serve its scheduled user and simultaneously perform IN to its IN users. Specifically, denote , where denotes the channel vector between macroBS and its scheduled user, and denotes the channel vector between macroBS and its th IN user . The ZFBF precoding matrix at macroBS is designed to be the pseudoinverse of , i.e., and the ZFBF vector at macroBS is designed to be , where is the first column of [22]. ii) If or , macroBS uses the maximal ratio transmission (MRT) precoder to serve its scheduled user, which is a special case of the ZFBF precoder introduced for and , and can be readily obtained from it by letting , i.e., . Next, we introduce the precoding vectors at picoBSs. Each picoBS utilizes the MRT precoder to serve its scheduled user. Specifically, the beamforming vector at picoBS is , where denotes the channel vector between picoBS and its scheduled user. Note that the simple beamforming scheme (without interference management) can be included in the IN scheme as a special case by letting and/or . Note that all the analytical results in this paper hold for and/or .
Let denote the channel vector between and its serving BS , denote the distance between and BS in the th tier, denote the distance between and , and denote the beamforming vector at , with (i.e., ), and [23, Lemma 1]. Here, the notation means that is distributed as . Let denote the channel vector between and BS in the th tier, and denote the beamforming vector at BS in the th tier, with (i.e., ) [23, Lemma 1]. Let denote the symbol sent from BS in the th tier to its scheduled user satisfying . Let denote the potential IN macroBSs of which do not select it for IN. Let denote the interfering macroBSs of which are not its potential IN macroBSs. Let denote the interfering picoBSs of . As in [9, 7], we assume that all macroBSs and picoBSs are active. We now discuss the received signal of .

MacroUser: The received signal of the typical user is
(2) Note that and .

PicoUser: The received signal of the typical user is
(3) Note that and .
We now obtain the SIR of the typical user. Under the above IN scheme, experiences three types of interference: 1) residual aggregated interference from its potential IN macroBSs which do not select for IN, 2) aggregated interference from interfering macroBSs which are not its potential IN macroBSs, and 3) aggregated interference from all interfering picoBSs . Specifically, the SIR of the typical user is given by
(4) 
where
IiiB Preliminary Results
In this part, we evaluate some distributions related to the IN scheme, which will be used to calculate the coverage probability in (1). These distributions are based on approximations, the accuracy of which will be verified in Section IV. We first calculate the probability mass function (p.m.f.) of the number of the potential IN users of an arbitrary (chosen uniformly at random) macroBS, denoted as . The p.m.f. of depends on the point processes formed by the scheduled macro and pico users, which are related to but not PPPs [24]. For analytical tractability, we approximate the scheduled macro and pico users as two independent PPPs with densities and , respectively. Note that approximating the scheduled users as a homogeneous PPP has been considered in existing papers (see e.g., [24]). Then, we have the p.m.f. of as follows.
Lemma 1
The p.m.f. of is given by
(5) 
where with
(6) 
Here, the p.d.f.s of (the distance between and its serving BS ) () are given as follows [25, Lemma 4]:
(7)  
(8) 
where () are given by
(9)  
(10) 
See Appendix A.
Note that represents the average number of IN requests of the scheduled users received by an arbitrary macroBS, and represents the average number of IN requests of the scheduled users in the th tier received by an arbitrary macroBS. From (6), we can easily see that and increase with and . From (5), we know that approximately follows the Poisson distribution with mean .
Next, we calculate the p.m.f. of the number of the IN users of an arbitrary (chosen uniformly at random) macroBS based on Lemma 1.
Lemma 2
The p.m.f. of is given by
Now, we calculate the probability that an arbitrary (chosen uniformly at random) potential IN macroBS of selects for IN, referred to as the IN probability and denoted as , based on Lemma 1.
Lemma 3
The IN probability is given by
See Appendix B.
Note that different potential IN macroBSs of selects for IN dependently (as the numbers of the potential IN users of these macroBSs are dependent). For analytical tractability, we assume that different potential IN macroBSs of select for IN independently. Using independent thinning, ’s potential IN macroBSs which do not select for IN can be approximated by a homogeneous PPP with density , where .
Iv Coverage Probability–General SIR Threshold Regime
In this section, we investigate the coverage probability in the general SIR threshold regime. By total probability theorem and the preliminary results obtained in Section IIIB (under some approximations), we have the following theorem.
(11)  
(12)  
(13)  
(14) 
Theorem 1 (Coverage Probability)
Under design parameters , and , we have: 1) coverage probability of a macrouser: , given in (11); 2) coverage probability of a picouser: , given in (12); 3) overall coverage probability , where () are given in (9) and (10). Here, and () are given in (13) and (14) (with and ), respectively. Moreover, () denotes the complementary incomplete beta function, , and , where denotes the set of nonnegative integers.
See Appendix C.
Fig. 2 plots the coverage probability versus the IN DoF and the SIR threshold . We see from Fig. 2 that the “Analytical” curves, which are plotted using in Theorem 1, are reasonably close to the “Monte Carlo” curves, although Theorem 1 is obtained based on some approximations (cf. Section IIIB). Please note that the approximation error shown in Fig. 2 is less than 0.022. Later, we shall consider the optimization of for given and .^{3}^{3}3The coverage probability can be further improved by jointly adjusting and . We shall consider the optimization of and in the future work.
V Asymptotic Outage Probability Analysis–Low SIR Threshold Regime
In this section, we analyze and optimize the complement of the coverage probability, i.e., the outage probability of the IN scheme in the low SIR threshold regime, i.e., . The asymptotic analysis and optimization offer important design insights for practical HetNets.
Va Asymptotic Outage Probability Analysis
In this part, we analyze the asymptotic outage probability of the IN scheme when . First, as in [26], we define the order gain of the outage probability (in interferencelimited systems), i.e., the exponent of the outage probability as the SIR threshold decreases to 0:^{4}^{4}4Note that this definition is analogous to the standard diversity order gain of error probability in noiselimited systems, i.e., the exponent of error probability as the (mean) signaltonoise ratio (SNR) increases to infinity[27][26].
(15) 
Then, we define the coefficient of the asymptotic outage probability: . Leveraging the order gain and the coefficient of the outage probability, we shall characterize the key behavior of the complex outage probability in the low SIR threshold regime.
Recently, a tractable approach has been proposed in [28] to characterize the order gain for a class of communication schemes in wireless networks which satisfy certain conditions. However, this approach does not provide tractable analytical expressions for the coefficient of the asymptotic outage probability for most of the schemes using multiple antennas in this class. By utilizing series expansion of some special functions and dominated convergence theorem, we characterize both the order gain and the coefficient of the asymptotic outage probability of the IN scheme in multiantenna HetNets, which are presented as follows.
Theorem 2 (Asymptotic Outage Probability)
Under design parameters , and , when , we have:^{5}^{5}5 means . 1) outage probability of a macrouser: ; 2) outage probability of a picouser: ; 3) overall outage probability: , where
Here, is given in (2) with and if and ; and , otherwise. Moreover, decreases with .
(16) 
See Appendix D.
From Theorem 2, we clearly see that the maximum IN DoF and the IN thresholds affect the asymptotic behavior of the outage probability in dramatically different ways. Specifically, can affect the order gain, while can only affect the coefficient. In addition, we see that affects the order gain of the asymptotic outage probability through affecting the order gain of the asymptotic macrouser outage probability. On the other hand, in this paper, IN is only performed at macroBSs, and is the upper bound on the actual DoF for IN in the ZFBF precoder (which is random due to the randomness of the network topology). Therefore, the result of the order gain in Theorem 2 extends the existing order gain result in singletier cellular networks where the DoF for IN in the ZFBF precoder is deterministic [17].
Fig. 3 plots the outage probability versus the SIR threshold in the low SIR threshold regime. We see from Fig. 3 that when the SIR threshold is small, the “Analytical” curves, which are plotted using Theorem 1, are reasonably close to the “Asymptotic” curves, which are plotted using Theorem 2. In addition, from Fig. 3, we clearly see that the outage probability curves with the same have the same slope (indicating the same order gain), and there is a shift between two outage probability curves with the same but different (indicating different coefficients). Therefore, Fig. 3 verifies Theorem 2, and shows that the asymptotic outage probability in the low SIR threshold regime provides a reasonable approximation for the outage probability when the SIR threshold is below 5 dB.
VB Asymptotic Outage Probability Optimization
From Theorem 2, we know that has a larger impact on the asymptotic outage probability than the IN thresholds. In this part, we characterize the optimal maximum IN DoF