User-Centric Interference Nulling in Downlink Multi-Antenna Heterogeneous Networks
Heterogeneous networks (HetNets) have strong interference due to spectrum reuse. This affects the signal-to-interference ratio (SIR) of each user, and hence is one of the limiting factors of network performance. However, in previous works, interference management approaches in HetNets are mainly based on interference level, and thus cannot effectively utilize the limited resource to improve network performance. In this paper, we propose a user-centric interference nulling (IN) scheme in downlink two-tier HetNets to improve network performance by improving each user’s SIR. This scheme has three design parameters: the maximum degree of freedom for IN (i.e., maximum IN DoF), and the IN thresholds for the macro and pico users, respectively. Using tools from stochastic geometry, we first obtain a tractable expression of the coverage (equivalently outage) probability. Then, we characterize the asymptotic behavior of the outage probability in the high reliability regime. The asymptotic results show that the maximum IN DoF can affect the order gain of the asymptotic outage probability, while the IN thresholds only affect the coefficient of the asymptotic outage probability. Moreover, we show that the IN scheme can linearly improve the outage performance, and characterize the optimal maximum IN DoF which minimizes the asymptotic outage probability.
The modern wireless networks have seen a significant growth of high data rate applications. The conventional cellular solution, which comprises of high power base stations (BSs), cannot scale with the increasing data rate demand. One solution is the deployment of low power small cell BSs overlaid with conventional large power macro-BSs, so called heterogeneous networks (HetNets). HetNets are capable of aggressively reusing existing spectrum assets to support high data rate applications. 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 , a cooperation strategy among a fixed number of strongest BSs of each user is investigated for HetNets. Reference  proposes another interference cooperation strategy among BSs whose long-term average received powers (RP) are above a threshold. However, both cooperation strategies do not take into account the strength of each user’s desired signal, and thus cannot effectively improve each user’s SIR. Moreover, both  and  only consider single-antenna BSs.
Deploying multiple antennas at each BS in HetNets can further improve network performance. With multiple antennas, besides boosting the desired signal to each user, more effective interference management techniques can be implemented. Interference coordination strategies using multiple antennas have been extensively investigated in single-tier cellular networks (see [3, 4] and the references therein). However, there are very limited works on the analysis of interference management techniques in multi-antenna HetNets. In , the authors consider a HetNet with a single multi-antenna macro-BS and multiple small-BSs, where the multiple antennas at the macro-BS are used for serving its scheduled users as well as mitigating the interference to the receivers in the small cells using different interference coordination schemes. These schemes are analyzed and shown to improve the performance of the HetNet. However, since only one macro-BS is considered, the analytical results obtained in  cannot reflect the macro-tier interference, and thus cannot offer accurate insights for practical HetNets. In , a fixed number of BSs form a cluster, and adopt an interference coordination scheme in downlink multi-antenna HetNets, where the BSs mitigate interference to users in the cluster. Bounds of the coverage probability are derived based on an unrealistic assumption that the BSs in each cluster are the strongest BSs of all the users in this cluster. This scheme cannot effectively improve each user’s SIR, as it does not consider each user’s desired signal strength. Recently, a novel user-centric IN scheme, which takes account of each user’s desired signal strength and interference level, is proposed and analyzed for single-tier cellular networks with multi-antenna BSs in . However, in , the maximum degree of freedom for IN (IN DoF) at each BS is not adjustable. This scheme thus cannot properly exploit the limited resource in single-tier cellular networks. Moreover, directly applying the approach in  to HetNets cannot fully exploit different properties of the macro and pico users in HetNets.
In this paper, we consider the downlink two-tier multi-antenna HetNets and propose a user-centric IN scheme to improve network performance by improving each user’s SIR. In this scheme, each macro (pico) user first sends an IN request to a macro-BS111Note that, compared to pico-BSs, macro-BSs usually cause stronger interference due to larger transmit power, and each macro-BS normally has a larger number of transmit antennas (i.e., a better capability of performing spatial cancellation). 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 an IN threshold for macro (pico) users. Then, each macro-BS utilizes zero-forcing beamforming (ZFBF) precoder to avoid interference to at most users which send IN requests to it as well as boost the desired signal to its scheduled user. This scheme has three design parameters: the maximum IN DoF , and the IN thresholds for the macro and pico users, respectively. In general, the investigation of interference management techniques in multi-antenna HetNets is very challenging, mainly due to i) the statistical dependence among macro-BSs and pico-BSs , ii) the complex distribution of desired signal using multi-antenna communication schemes, and iii) the complicated interference distribution caused by interference management techniques (e.g., beamforming). In this paper, by adopting appropriate approximations and utilizing tools from stochastic geometry, we first obtain a tractable expression of the coverage (equivalently outage) probability. Then, we characterize the asymptotic behavior of the outage probability in the high reliability regime. The asymptotic results show that the maximum IN DoF can affect the order gain of the asymptotic outage probability, while the IN thresholds only affect the coefficient of the asymptotic outage probability. Moreover, we show that the IN scheme can linearly improve the outage performance, and characterize the optimal maximum IN DoF which minimizes the asymptotic outage probability. The analytical results obtained in this paper provide valuable design insights for practical HetNets.
Ii System Model
Ii-a Downlink Two-Tier Heterogeneous Networks
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 the macro-BSs and the pico-BSs are spatially distributed as two independent homogeneous Poisson point processes (PPPs) and with densities and , respectively. The locations of the 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. 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. We 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-B User Association
We assume open access . User (denoted as ) is associated with the BS which provides the maximum long-term average RP 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 RP 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 RP 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 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-C 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 calculated in a similar way. We investigate the coverage probability of , which is defined as the probability that the SIR of is larger than a threshold , and can be mathematically written as
where is the SIR threshold.
Iii Interference Nulling Scheme
In this section, we first elaborate on a user-centric IN scheme to avoid interference from some macro-BSs which generate dominant interference. Then, we obtain some distributions related to this scheme.
Iii-a IN Scheme Description
First, we refer to a macro-BS which causes the SIIR at a 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, 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 macro-BS as the potential IN users of macro-BS (in this time slot). Consider a particular time slot. Let denote the number of the potential IN users of macro-BS . Consider two cases in the following. i) If , macro-BS makes use of at most () DoF to perform IN to some of its potential IN users. Note that is another design parameter of this IN scheme. 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, if , 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). ii) If , 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 , macro-BS utilizes the low-complexity ZFBF precoder to serve its scheduled user and simultaneously perform IN to its IN users. Specifically, denote , where333The notation means that is distributed as . 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 and the ZFBF vector at macro-BS is designed to be , where is the first column of . ii) If , 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 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 non-IN scheme can be included in the IN scheme as a special case by letting and .444All the analytical results in this paper hold for 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 by555The index of the typical user and its serving BS is .
where is the channel vector between and its serving BS , is the distance between and , is the beamforming vector at , with , and . Here, , , and , where is the channel vector between BS in the th tier and , is the beamforming vector at BS in the th tier, and .
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). We first calculate the probability mass function (p.m.f.) of the number of the potential IN users of ’s serving macro-BS (when ), 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.666Note that approximating the scheduled users as a homogeneous PPP has been considered in existing papers (see e.g., ). Moreover, simulation results in Section IV-A will verify the accuracy of this approximation. Then, we have the p.m.f. of as follows:
The p.m.f. of is given by
Here, () are given as follows:
See Appendix -A.
Next, we calculate the p.m.f. of based on Lemma 1, which is given as follows:
The p.m.f. of is given by
Let denote the probability that a randomly selected (according to the uniform distribution) potential IN macro-BS of selects for IN. Note that the event that sends IN requests and the event that all the other scheduled users send IN requests are dependent. For analytical tractability, we approximate these two events as independent events. Then, we have , which can be calculated as follows:
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 depend on the locations of these macro-BSs and are thus dependent). For analytical tractability, we assume that different potential IN macro-BSs of select for IN independently. Using independent thinning, can be approximated by a homogeneous PPP with density , where .
Iv Coverage Probability Analysis
In this section, we investigate the coverage probability in the general and small SIR threshold regimes, respectively.
Iv-a General SIR Threshold Regime
As discussed in Section II-B, the typical user is in one of two disjoint user sets: and . By utilizing tools from stochastic geometry and the preliminary results obtained in Section III-B, we have the following theorem:
Theorem 1 (Coverage Probability)
Under design parameters , and , we have
coverage probability of a macro-user: , given in (9) at the top of the next page,
coverage probability of a pico-user: , given in (10) at the top of the next page,
Here, and () are given in (11) and (12) at the top of the next page, respectively, where777 denotes the Laplace transform of the aggregated interference . and are given in (13) and (14) at the top of the next page (with and ), respectively. Moreover, (), , and .
See Appendix -C.
Theorem 1 allows us to easily evaluate the coverage probability in a numerical way. Fig. 2(a) plots the coverage probability versus the IN thresholds and . We see from Fig. 2(a) that the “Analytical” curves, which are plotted using in Theorem 1, are reasonably close to the “Monte Carlo” curves (the error is no larger than ), although Theorem 1 is obtained based on some approximations (cf. Section III-B). Compared to the non-IN scheme, we observe that the IN scheme can achieve a good coverage probability gain (up to ). Moreover, we see that the coverage probability depends on both and . Specifically, when () is small, increasing () increases the chance of macro-BSs performing IN to near users; while when () is large, increasing () reduces the chance of macro-BSs performing IN to near users, as the resource is wasted in performing IN to users far away. Further, we see that the coverage probability can be improved by jointly adjusting and .
Iv-B Small SIR Threshold Regime
To obtain more insights on the impact of the design parameters, in this part, we investigate the asymptotic behavior of the IN scheme in the high reliability regime where .888Note that small is a practical regime. For example, in current standards (e.g., 3GPP LTE), is very small (e.g., -7 dB) . In addition, in wide-band systems (e.g., CDMA and UWB), is usually small .
Iv-B1 Asymptotic Outage Probability Analysis
In this part, we analyze the asymptotic outage probability when . First, we define the order gain of the outage probability :
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, i.e.,
for most of the schemes using multiple antennas in the 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 have999 means .
outage probability of a macro-user: ,
outage probability of a pico-user: ,
overall outage probability: , where
Here, is given in (2) at the top of the next page with
Moreover, decreases with .
See Appendix -D.
Note that the IN scheme has three design parameters: the maximum IN DoF (i.e., ) and the IN thresholds (i.e., and ). 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, the maximum IN DoF can affect the order gain, while the IN thresholds 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. Note that in this paper, IN is only performed at macro-BSs, and is the upper bound of 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. 2(b) plots the outage probability versus SIR threshold for different design parameters. From Fig. 2(b), we clearly see that the outage probability curves with the same maximum IN DoF have the same slope (indicating the same order gain). For the two outage probability curves with the same but different and , we observe a shift between these two curves (indicating different coefficients). Therefore, Fig. 2(b) verifies Theorem 2, and shows that the result in Theorem 2 can effectively reflect the outage probability for small .
Iv-B2 Optimization of Maximum IN DoF
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 which maximizes the order gain, and the optimal maximum IN DoF which minimizes the asymptotic outage probability, respectively.
From Theorem 2, we see that the order gain of the asymptotic outage probability is . Thus, we also denote it as . First, we introduce the optimal design parameter which maximizes , i.e.,
Thus, the optimal order gain of the asymptotic outage probability is . Next, we introduce the optimal design parameter which minimizes the outage probability, i.e.,
The properties of and are given by:
Lemma 4 (Optimality Properties)
When , we have for all ;
, such that for all , we have101010As is independent of , we also write it as .
See Appendix -E.
Result in Lemma 4 shows that is independent of . Thus, the IN scheme does not provide order-wise performance improvement compared to the non-IN scheme. Note that the impact of