User-Centric Interference Nulling in Downlink Multi-Antenna Heterogeneous Networks
In heterogeneous networks (HetNets), strong interference due to spectrum reuse affects each user’s signal-to-interference ratio (SIR), and hence is one limiting factor of network performance. In this paper, we propose a user-centric interference nulling (IN) scheme in a downlink large-scale 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 macro-BS to avoid interference to uniformly selected macro (pico) users with signal-to-individual-interference 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 user-centric almost blank subframes (ABS) scheme.
Heterogeneous networks, multiple antennas, inter-tier interference coordination, stochastic geometry, optimization.
Heterogenous wireless networks (HetNets), i.e., the deployment of low power small cell base stations (BSs) overlaid with conventional large power macro-BSs, 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 signal-to-interference ratio (SIR) of each user, and hence is one of the limiting factors of network performance. Interference management techniques are thus desirable in HetNets. One such technique is interference cooperation. For example, in [4, 5, 6], different interference cooperation strategies are considered and their performances are analyzed for large-scale 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 single-antenna 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 . 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 large-scale HetNets with offloading is analyzed in  using tools from stochastic geometry. Note that ABS focuses on mitigating the interference of offloaded users, and  only considers single-antenna 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 multi-antenna macro-BS and multiple multi-antenna small-BSs, where the multiple antennas at the macro-BS 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,  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 macro-BS is considered in [10, 11, 12, 13], the analytical results obtained in [10, 11, 12, 13] cannot reflect the macro-tier interference, and thus may not offer accurate insights for practical HetNets. In [14, 15], large-scale multi-antenna HetNets are considered. Specifically,  considers offloading, and proposes an interference nulling (IN) scheme where some degree of freedom at each macro-BS 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  only improves the performance of scheduled offloaded users, and scheduled offloaded users are selected by the corresponding macro-BS for interference nulling with equal probability. Hence, the IN scheme proposed in  may not effectively improve the overall rate coverage probability. In , 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 large-scale single-tier multi-antenna cellular networks is less involved than that in large-scale multi-antenna 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  are performed among the BSs within each cluster to mitigate intra-cluster interference. Specifically,  and  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  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 user-centric IN scheme, which takes account of each user’s desired signal strength and interference level, is proposed and analyzed for (single-tier) multi-antenna small cell networks in . However, in , 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  to HetNets cannot fully exploit different properties of macro and pico users in HetNets.
In this paper, we consider a downlink large-scale two-tier multi-antenna HetNet and propose a user-centric 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 macro-BS111Note that, compared to a pico-BS, a macro-BS 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 macro-BSs. if the power ratio of its desired signal and the interference from the macro-BS, referred to as the signal-to-individual-interference ratio (SIIR), is below the IN threshold for macro (pico) users. Then, each macro-BS utilizes zero-forcing 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 large-scale multi-antenna HetNets are very challenging, mainly due to i) the statistical dependence among macro-BSs and pico-BSs , ii) the complex distribution of a desired signal using multi-antenna 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 user-centric ABS scheme, using numerical results.
The key notations used in the paper are listed in Table I.
|,||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 macro-users (), set of pico-users ()|
|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 macro-BS|
|,||SIR coverage probability, SIR threshold|
|,||Maximum IN DoF, IN threshold for the th tier in the IN scheme|
Ii Network Model
We consider a two-tier HetNet where a macro-cell tier is overlaid with a pico-cell tier, as shown in Fig. 1. The locations of macro-BSs and pico-BSs 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 macro-cell tier as the st tier and the pico-cell tier as the nd tier. We focus on the downlink scenario. The macro-BSs and the pico-BSs share the same spectrum concurrently. Each macro-BS has antennas with total transmission power , each pico-BS has antennas with total transmission power , and each user has a single antenna. Assume . We consider both large-scale fading and small-scale fading. Specifically, due to large-scale 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 small-scale fading, we assume Rayleigh fading channels.
Ii-a User Association
We assume open access . User (denoted as ) is associated with the BS which provides the maximum long-term average (over small-scale fading) received power among all the macro-BSs and pico-BSs. This associated BS is called the serving BS of user . Note that within each tier, the nearest BS to user provides the strongest long-term average received power in this tier. User is thus associated with (the nearest BS in) the th tier, if222In 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 macro-cell tier as the macro-users, denoted as , and the users associated with the pico-cell tier as the pico-users, 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 intra-cell interference.
Ii-B Performance Metric
In this paper, we study the performance of the typical user denoted as , which is a scheduled user located at the origin . Since HetNets are interference-limited, 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. The coverage probability of is defined as the probability that the SIR of is larger than a threshold , i.e.,
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. In Sections IV, V and VI, we shall analyze the coverage/outage probability in the general, low and high SIR threshold regimes, separately.
Iii User-centric Interference Nulling Scheme
In this section, we first elaborate on a user-centric IN scheme. Then, we obtain some distributions related to this scheme.
Iii-a Scheme Description
First, we refer to an interfering macro-BS which causes the SIIR at scheduled user in the th tier () below threshold as a potential IN macro-BS of , where . We refer to as the IN threshold for the th tier. Mathematically, interfering macro-BS is a potential IN macro-BS of scheduled user if , where is the distance from macro-BS 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 macro-BSs. We refer to the scheduled users which send IN requests to interfering macro-BS as the potential IN users of interfering macro-BS (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 macro-BS . Note that implies . Consider two cases in the following. i) If and , macro-BS makes use of at most DoF to perform IN to some of its potential IN users. In particular, if , macro-BS can perform IN to all of its potential IN users using DoF; if , macro-BS 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, macro-BS performs IN to potential IN users (referred to as the IN users of macro-BS ) using DoF (referred to as the IN DoF of macro-BS ). ii) If or , macro-BS does not perform IN. In this case, we let . In both cases, DoF at macro-BS is used for boosting the desired signal to its scheduled user.
Now, we introduce the precoding vectors at macro-BSs in the IN scheme. Consider two cases in the following. i) If and , macro-BS utilizes the low-complexity ZFBF precoder to serve its scheduled user and simultaneously perform IN to its IN users. Specifically, denote , where denotes the channel vector between macro-BS and its scheduled user, and denotes the channel vector between macro-BS and its th IN user . The ZFBF precoding matrix at macro-BS is designed to be the pseudo-inverse of , i.e., and the ZFBF vector at macro-BS is designed to be , where is the first column of . ii) If or , macro-BS 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 pico-BSs. Each pico-BS utilizes the MRT precoder to serve its scheduled user. Specifically, the beamforming vector at pico-BS is , where denotes the channel vector between pico-BS 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 macro-BSs of which do not select it for IN. Let denote the interfering macro-BSs of which are not its potential IN macro-BSs. Let denote the interfering pico-BSs of . As in [9, 7], we assume that all macro-BSs and pico-BSs are active. We now discuss the received signal of .
Macro-User: The received signal of the typical user is
Note that and .
Pico-User: The received signal of the typical user is
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 macro-BSs which do not select for IN, 2) aggregated interference from interfering macro-BSs which are not its potential IN macro-BSs, and 3) aggregated interference from all interfering pico-BSs . Specifically, the SIR of the typical user is given by
Iii-B 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) macro-BS, 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 . 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., ). Then, we have the p.m.f. of as follows.
The p.m.f. of is given by
Here, the p.d.f.s of (the distance between and its serving BS ) () are given as follows [25, Lemma 4]:
where () are given by
See Appendix -A.
Note that represents the average number of IN requests of the scheduled users received by an arbitrary macro-BS, and represents the average number of IN requests of the scheduled users in the th tier received by an arbitrary macro-BS. 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) macro-BS based on Lemma 1.
The p.m.f. of is given by
Now, we calculate the probability that an arbitrary (chosen uniformly at random) potential IN macro-BS of selects for IN, referred to as the IN probability and denoted as , based on Lemma 1.
The IN probability is given by
See Appendix -B.
Note that different potential IN macro-BSs of selects for IN dependently (as the numbers of the potential IN users of these macro-BSs are dependent). For analytical tractability, we assume that different potential IN macro-BSs of select for IN independently. Using independent thinning, ’s potential IN macro-BSs 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 III-B (under some approximations), we have the following theorem.
Theorem 1 (Coverage Probability)
Under design parameters , and , we have: 1) coverage probability of a macro-user: , given in (11); 2) coverage probability of a pico-user: , 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 III-B). 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 .333The 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.
V-a Asymptotic Outage Probability Analysis
In this part, we analyze the asymptotic outage probability of the IN scheme when . First, as in , we define the order gain of the outage probability (in interference-limited systems), i.e., the exponent of the outage probability as the SIR threshold decreases to 0:444Note that this definition is analogous to the standard diversity order gain of error probability in noise-limited systems, i.e., the exponent of error probability as the (mean) signal-to-noise ratio (SNR) increases to infinity.
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  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 multi-antenna HetNets, which are presented as follows.
Theorem 2 (Asymptotic Outage Probability)
Under design parameters , and , when , we have:555 means . 1) outage probability of a macro-user: ; 2) outage probability of a pico-user: ; 3) overall outage probability: , where
Here, is given in (2) with and if and ; and , otherwise. Moreover, decreases with .
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 macro-user outage probability. On the other hand, in this paper, IN is only performed at macro-BSs, 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 single-tier cellular networks where the DoF for IN in the ZFBF precoder is deterministic .
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.
V-B 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