Distributed Cluster Formation and Power-Bandwidth Allocation for Imperfect NOMA in DL-HetNets

12/02/2018 ∙ by Abdulkadir Celik, et al. ∙ King Abdullah University of Science and Technology 0

In this paper, we consider an non-ideal successive interference cancellation (SIC) receiver based imperfect non-orthogonal multiple access (NOMA) schemes whose performance is limited by three factors: 1) Power disparity & sensitivity constraints (PDSCs), 2) Intra-cluster interference (ICRI), and 3) Intercell-interference (ICI). By quantifying the residual interference with a fractional error factor (FEF), we show that NOMA cannot always perform better than orthogonal multiple access (OMA) especially under certain receiver sensitivity and FEF levels. Assuming the existence of an offline/online ICI management scheme, the proposed solution accounts for the ICI which is shown to deteriorate the NOMA performance particularly when it becomes significant compared to the ICRI. Then, a distributed cluster formation (CF) and power-bandwidth allocation (PBA) approach are proposed for downlink (DL) heterogeneous networks (HetNets) operating on the imperfect NOMA. We develop a hierarchically distributed solution methodology where BSs independently form clusters and distributively determine the power-bandwidth allowance of each cluster. A generic CF scheme is obtained by creating a multi-partite graph (MPG) via partitioning user equipments (UEs) with respect to their channel gains since NOMA performance is primarily determined by the channel gain disparity of cluster members. A sequential weighted bi-partite matching method is proposed for solving the resulted weighted multi-partite matching problem. Thereafter, we present a hierarchically distributed PBA approach which consists of the primary master, secondary masters, and slave problems...

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I Introduction

Ultra-Dense networks have been considered to be a promising solution for the fifth generation (5G) networks as network densification has the ability to boost network coverage and capacity while reducing operational and capital expenditures [1]. However, traditional HetNets dedicate radio resources to a certain user equipment (UE) either in time or frequency domains, i.e., orthogonal multiple access (OMA), where the number of served UEs at a given time instant is strictly limited by the availability of the radio resources. Considering the expected explosive number of devices, required massive connectivity necessitates more spectrum efficient access schemes with extended coverage.

In this regard, non-orthogonal multiple access (NOMA) has recently attracted attention by permitting to share the same radio resources among a set of UEs [2], which is also referred to as a NOMA cluster. In particular, NOMA has the capability of providing a higher spectral efficiency while supporting a large number of UEs over the same radio resource. Employing successive interference cancellation (SIC), power domain NOMA can serve multiple UEs at different power levels by ensuring that some UEs can cancel some others’ interference out before decoding their own signal. In order to differentiate the desired signal from noise and undecoded signals, the SIC receivers require the disparity of received power levels with a hardware sensitivity gap [3], which is referred to as power disparity and sensitivity constraints

(PDSCs). Moreover, SIC receivers can still observe some residual interference after cancellation due to the propagation of detection and estimation errors, which is often quantified with a

fractional error factor [4]. In such a case, the ICRI is mainly because of the uncancellable interference and residual interference due to SIC inefficiency. Hence, performance gain achieved by NOMA is primarily limited by imperfections and constraints of SIC receivers and power control policy. Furthermore, cluster formation strategy is an inherently crucial aspect to maximize the benefit offered by NOMA as it is shown that NOMA gain is determined by channel gain discrepancy of cluster members [5]. Due to its combinatorial nature, CF is a challenging task to accomplish especially in HetNets and necessitates a fast yet high-performance clustering methods.

In NOMA based downlink (DL) heterogeneous networks (HetNets), all clusters compete for a commonly shared bandwidth whereas clusters within a certain cell contend for the available power of the serving base station (BS). On the other hand, members of a cluster have to share the total power allocated by the BS to their cluster. Taking all these different entities and inter-dependencies into consideration, a centralized CF and PBA scheme require an excessive amount of message passing and coordination among the BSs. To overcome such a communication overhead, it is desirable to have a distributed CF and PBA approach where BSs independently form their own clusters and decide on power and bandwidth allowances, which is the main focus of this paper.

I-a Related Works

Recent efforts on power domain NOMA can be exemplified as follows: In [3], authors formed clusters based on channel gain ordering and derive closed-form power allocations for a given cluster power and bandwidth pair. The impact of UE selection/clustering is investigated in [5] for a two-UE DL-NOMA system with fixed and cognitive radio inspired power allocation schemes. The work in [6] addressed max-min fair UE clustering problem using three different sub-optimal approaches. Authors of [7] iteratively built clusters where each iteration jointly optimize beam-forming and power allocation for given clusters. Another work considered beam-forming and power allocation of a multiuser multiple-input-multiple-output (MIMO) NOMA system where two-UE clusters are formed from high and low channel gain UEs with the consideration of channel gain correlations [8]. We investigate cluster formation and resource allocation problems for DL and UL HetNets in [9] and [10], respectively.

Joint power and channel allocation for the NOMA system are addressed in [11] wherein a near optimal solution was proposed by combining Lagrangian duality and dynamic programming. By using Lyapunov optimization framework, the short and long-term network utility is maximized by joint data rate and power control in [12]. In [13], authors study the problem of resource optimization, mode selection and power allocation in wireless cellular networks under the assumption of full-duplex NOMA capability and queue stability constraints. Sun et. al. considered joint power and subcarrier allocation for full-duplex multi-carrier (MC) NOMA systems for UL and DL transmission of a single BS [14]. MC-NOMA is also studied in [15] where authors jointly design the power and rate allocation, user scheduling, and successive SIC decoding policy to minimize the power consumption. In [16], the power is controlled to achieve the different objective for given channel allocations. Sub-Channel assignment, power allocation, and user scheduling are addressed by formulating the sub-channel assignment problem as equivalent to a many-to-many two-sided user-subchannel matching game [17]. By only utilizing a single scalar, an -fairness approach is developed to achieve different UE fairness levels in [18]. In[19], authors investigated resource allocation for hybrid NOMA system subject to proportional rate constraints. The work in [20] focused on resource allocation in energy-cooperation enabled two-tier NOMA HetNets with energy harvesting BSs. Authors of [21] allocated spectrum and power using a many-to-one matching game and sequential convex programming, respectively.

Since it can cause severe performance degradation, a practical design of NOMA must account for real-life imperfections which can be a result of imperfect channel state information (CSI), residual interference due to the FEF, or PDSCs of SIC receivers. In [22]

, authors investigate the impact of partial CSI on the performance of the NOMA networks. They first consider an imperfect CSI model where the BS and UEs have an estimate of the channel and a priori knowledge of the variance of the estimation error. Analytical and numerical findings demonstrated that the average sum rate of NOMA systems can always outperform conventional OMA. Based on the second order statistics, authors show that NOMA is still superior to conventional OMA. In opportunistic one-bit feedback has been used for NOMA in

[23]

where a closed-form expression for the common outage probability is derived along with the optimal diversity gains under short and long-term power constraints. As discussed in

[24], the SIC receiver performance is mainly determined by the tradeoff between computational complexity and error propagation during the interference cancellation (IC) process. Albeit their significant contributions, to the best of authors’ knowledge, none of the aforementioned works consider the residual interference due to the SIC error propagation. Excluding [3], these works also do not take the PDSCs and cluster formation design into account. Furthermore, challenges of the HetNet environment is only addressed in [20, 21] where authors do not consider a distributed approach. To the best of our knowledge, this paper is the first work to consider an imperfect NOMA with residual interference and to develop a distributed CF and PBA for DL-HetNets.

I-B Main Contributions

Our main contributions can be summarized as follows:

  • In practice, constraints and imperfections of SIC receivers constitute a limiting factor on the achievable gain by NOMA. This work is the first to consider the impacts of residual interference on NOMA performance due to the non-ideality of the SIC receivers. Although our work is not aimed at proposing an ICI management scheme, the proposed power allocation method is also capable of accounting for the leftover ICI from any offline/online ICI management scheme. Obtained results demonstrate that NOMA cannot always perform better than the OMA under certain FEF levels and receiver sensitivity values. It is also shown that being agnostic to the ICI can severely degrade the performance especially when it becomes significant in comparison with the ICRI, which clearly indicates the necessity for an effective ICI management scheme.

  • After formulating a centralized CF and PBA as an mixed-integer non-linear programming (MINLP) problem, we develop a distributed solution methodology where BSs independently form clusters and determine the power-bandwidth allowance of each cluster. Noting that existing solutions have contended to basic NOMA clusters of size two, a generic CF scheme is obtained by creating a

    multi-partite graph (MPG) via partitioning UEs with respect to their channel gains. A sequential weighted bi-partite matching (WBM) method is proposed for solving the resulted weighted multi-partite matching (WMM) based CF problem in cubic order. If edges are merely weighted by the channel gain disparity of UEs, it is proven that the complexity of solving the WMM can even be reduced to quasi-linear order without executing any matching algorithm. Obtained results show that proposed CF delivers a performance very close to the exhaustive centralized solution with a significantly reduced processing load.

  • By employing primal and dual decomposition methods, we propose a hierarchically distributed PBA approach which consists of a primary master problem, secondary master problems, and slave problems. For a given cluster power and bandwidth allowance, optimal power allocations and Lagrange multipliers of imperfect NOMA are derived in closed-form subject to PDSCs and QoS constraints. Closed-form solutions are used by secondary master problems to update total power allowance of clusters. Based on achieved cluster utilities, the primary master problem iteratively updates the bandwidth allowances to maximize the network utility and broadcasts updated bandwidth allocations to the BSs. Finally, we show that proposed algorithm greatly reduces the communication overhead and investigate the NOMA performance in comparison with OMA under different BS density, traffic offloading bias factor, UE density, and cluster size scenarios for DL-HetNets.

Table of Notations
Not. Description
Set of BSs, where is the MBS.
Set of UEs, where is the set of UEs of BS
Set of all clusters, where is the set of clusters.
Set of users belong to cluster of BS, , .
BS transmission power, i.e., for MBS/SBSs.
Noise power spectral density.
Total available bandwidth for all UEs.
Binary variable for cluster membership relations.
Total power fraction allocated for clusters,
Power allocation variable, , .
Bandwidth allocated to UEs within , , .
SINR of UE.
SIC error factor of UE.
Receiver sensitivity og UE.
Composite channel gain from BS to UE.
Achievable capacity of UE which requires
Affordable number of cancellations.
Number of clusters of BS, .
Set of UE partitions of BS, .
Edge weights for multi-partite graphs.
Lagrange function related to .
Lag. multiplier related to cluster power consumption.
Lag. multiplier related to QoS constraint of UE
Lag. multiplier related to PDSC of UE.
Step size for updates of distributed algorithm.
Composite parameter defined as .
Composite parameter defined as .
Composite parameter defined as .
TABLE I: Table of Notations

I-C Notations and Paper Organization

Throughout the paper, sets and their cardinality are denoted with calligraphic and regular uppercase letters (e.g.,

), respectively. Vectors and matrices are represented in lowercase and uppercase boldfaces (e.g.,

and ), respectively. Superscripts , , and are used for indexing BSs/cells, clusters, and UEs, respectively. The optimal values of variables are always marked with superscript , e.g., and .

The remainder of the paper is organized as follows: Section II introduces the system model along with constraints and imperfections of SIC. Section III first formulates the optimal CF and PBA problem. Section IV and Section V address proposed CF and distributed PBA methods, respectively. Numerical results are presented in Section VI and Section VII concludes the paper with a few remarks.

Fig. 1: Illustration of clustering and traffic offloading in DL-HetNets.

Ii System Model

Ii-a Network Model

We consider DL transmission of a 2-tiered HetNet where each tier represents a particular cell class, i.e., tier-1 consists of a single macrocell and tier-2 comprises of smallcells as shown in Fig. 1. Denoting the number of small BSs (SBSs) as , the index set of all BSs is represented by where and represent the macro BS (MBS) and SBSs, respectively. We note that the terms BS, cell and their indices are used interchangeably throughout the paper. Maximum transmission powers of BSs are generically denoted as which equals to and for the MBS and SBSs, respectively. Furthermore, index set of all UEs is given as where is the set of UEs associated with BS. UE-BS association is based on received signal strength (RSS) information with a certain offloading bias factor [26, 25]. A simple traffic offloading example is shown in Fig. 1 where yellow-colored circles represent the offloading regions corresponding to different bias factors and blue-colored stars represent the UEs offloaded from the MBS. Likewise, the set of all clusters are denoted as where is the set of clusters of BS. Hence, is partitioned into clusters such that symbolizes the set of UEs within cluster , that is, and . Cluster is allowed to utilize portion of the entire DL bandwidth, , . Since each cluster has its own dedicated bandwidth, total number of clusters and bandwidth proportions are equivalent. Also noting that all clusters of cell share the DL transmission power of the BS, power fraction allocated for is defined as , .

Ii-B Imperfections and Constraints of SIC

The SIC receiver first decodes the stronger interferences and then subtracts them from the broadcasted signal until they obtain the desired signal. Accordingly, the received interference strengths are required to be sufficiently higher in comparison to the intended signal for a successful IC process. Accordingly, BSs broadcast the superposed signals with low power level for high channel gain UEs and high power level signals for low channel gain UEs. In this case, the highest channel gain UE can cancel all the interference while being allocated to the lowest power level. On the other hand, the lowest channel gain UE cannot cancel any interference while being allocated to the highest power level. In such a way that the performance of the entire cluster is enhanced in a fair manner.

To be more specific, let us now focus on cluster of BS, where is a binary indicator for the cluster membership and cluster members are sorted in the descending order of the channel gains, , without loss of generality. Such clusters are demonstrated in Fig. 1 with green circles around the UEs, for example, first cluster of the MBS is . As per SIC principles, NOMA allocates transmission power weights as , hence, normalized received power at UE is given as

(1)

where normalization is with respect to BS transmission power , is the higher rank decoding order set, and is the lower rank decoding order set for UE. UE can only cancel the interference induced by higher rank members, while interference from lower rank members cannot be decoded as they are weaker than the desired signal. Furthermore, the hardware sensitivity of the SIC receivers requires a minimum signal-to-interference-plus-noise-ratio (SINR) to distinguish intended signals from noise. Therefore, power disparity & sensitivity constraints (PDSCs) can be expressed in linear scale as [3]

(2)

where denotes the hardware sensitivity. The intuition behind (2) is that during the IC process of UE, receiver of observes undecoded signals of UE, , as noise. Moreover, a non-ideal SIC observes some residual interference due to the error propagation which is caused by detection and estimation errors [24]. Accordingly, a generic SINR representation of the imperfect SIC receiver is given as

(3)

where the first two terms of the denominator characterize the ICRI, is the fractional error factor (i.e., can be regarded as the SIC efficiency) which determines the residual interference after the IC, , is ICI generated by UEs located at different macrocell coverage area and cannot be canceled by the SIC receiver 111Notice that a dominant ICI can decrease the SIC efficiency (i.e., increase ) due to the failure in the decoding process., is the entire DL bandwidth, and

is the thermal noise power spectral density. We must note that the ICI can have a significantly negative impact on the NOMA performance in Het-Nets. Indeed, it can be handled by adopting the existing interference management schemes of traditional OMA networks, which can be classified as offline (e.g., fractional frequency reuse, sectorization or spatial reuse) and online (silencing, dynamic spectrum access, cooperative beamforming, cooperative multi-point transmission,etc.) techniques

[27, 28]. Although the remainder of the paper factors the ICI in the optimal power allocation scheme, our contributions do not include developing ICI management schemes. Hence, the capacity achievable by UE is given by

(4)

Even if this work assumes a perfect CSI estimation, uncertainty about CSI is crucial from two aspects: First, power allocations obtained based on imperfect CIS can considerably degrade the NOMA performance due to the increasing impact of intra-cluster interference. Second, although small and fast fading component may not have a significant effect on the channel gain order, large-scale fading should still be estimated with sufficient accuracy to reduce the impacts of CSI estimation errors on the clustering strategy222Even though it is out of this paper’s scope, extending the proposed methods to a robust optimization framework is necessary to account for CSI estimation errors..

Iii Cluster Formation and Power Bandwidth Allocation

In a HetNet, determining optimal values for integer-valued cluster numbers/sizes and binary valued UE-cluster associations yields high computational complexity. Furthermore, highly non-convex nature of PBA problem induces an extra complexity to achieve optimal performance. A generic CF policy may consider all UEs as potential candidates for all clusters, which has a high time complexity in the order of for an exhaustive CF solution. Since clusters can be formed among UEs associated with different BSs, such an approach also necessitates a low-latency and robust coordination among the BSs in order to decide on clusters and optimal transmission power levels, which requires the exchange of channel gains and decoded signals to perform SIC locally and naturally yields a high communication overhead. However, we consider a more practical scenario where BSs independently form clusters among their UEs such that the order of time complexity can be reduced to . In this way, a distributed CF and PBA approach can be developed as each BS is responsible for CF and PBA of its own UEs. In what follows, we first formulate a centralized joint CF and PBA problem and then present the proposed distributed solution.

Iii-a Centralized CF and PBA

The centralized CF and PBA problem can be formulated as in where we make the following assumptions: 1) A UE can be associated with exactly one cluster at a time and 2) The cluster size is determined by ensuring that the highest channel gain UE can cancel all other cluster members. In particular, the first assumption enables us to decompose the entire problem into sub-problems and develop a hierarchically distributed PBA scheme. On the other hand, BS locally allocates a certain power fraction of its available DL transmission power to its own clusters, .

In , ensures that a UE is assigned to only one cluster and puts an upper bound on the number of UEs within a cluster by which is determined based on the affordable number of ICs by UEs that is a design parameter and denoted by . While limits the power consumption of to the fraction , restricts the total amount of clusters’ power weights to available BS power. Likewise, constrains the sum of cluster bandwidths to the available total bandwidth, . introduces the QoS requirements where and is the data rate demand of UE333 is obtained from 4 which reduces a single term due to the .. Since clusters are assumed to have dedicated spectrum, cluster index also represents the subcarrier indexes. present PDSCs where . Finally, variable domains are defined in where the power allocation for UE on cluster is set to zero () if UE.

is apparently a MINLP problem which requires impractical time complexity even for moderate sizes of HetNets. As a fast yet high performance suboptimal solution methodology is of the essence to employ NOMA in practice, we develop a solution methodology by decoupling this hard problem into CF and PBA subproblems, which is desirable to obtain a tractable solution methodology and also durable as it is already shown that the NOMA performance gain is primarily determined by channel gain disparity of the cluster members [5].

Iii-B Distributed CF and PBA

Let us first classify the type of network resources based on the following hierarchical relationship: The bandwidth proportion is a primary-global resource in which BSs compete for their clusters. Therefore, and are primary-global coupling variable and complicating constraint among the BSs, respectively. Likewise, as each BS has its own power source, the fraction of total transmission power is a secondary-global resource in which clusters contend for their members. Hence, and are secondary-global coupling variable and complicating constraint for BS clusters, respectively. On the other hand, is a local resource needed by cluster members UE to satisfy PDSCs and QoS constraints. Thus, and are secondary-global coupling variable and complicating constraint for cluster members, respectively.

Fig. 2: Schematic illustration of the proposed hierarchically distributed solution.

Exploiting this hierarchical relation and decomposability of the problem, CF and PBA problem can be solved in a distributed manner as shown in Fig. 2 where each BS can form its own clusters by decoupling binary clustering variable from power-bandwidth variables because the achievable performance gain of NOMA has shown to be mainly determined by the channel gain disparity of the cluster members [5]. Thereafter, decomposition methods is applied to given cluster formations in order to conclude the optimal power and bandwidth portions in a distributed manner as follows: A central unit (preferably the MBS) is responsible for primary-master problem which decides on bandwidth portions as per the utility achieved by each cluster. Depending upon the bandwidth allocated by , , BS is responsible for its own secondary-master problem in order to determine the power fraction for clusters of BS, . Finally, is further divided into slave problems which maximize the total cluster utility for a certain bandwidth and power fraction given by primary and secondary masters, respectively.

Iv Weighted Multi-Partite Matching Based CF

In our proposed solution, BS independently forms clusters among UEs in a distributed fashion. Therefore, let us omit the cluster indices and consider the CF of a single BS without loss of generality. Denoting the affordable number of ICs as , the maximum size of clusters is limited by as the highest channel gain UE within a cluster can cancel interference of at most UEs. From multi-user detection theory it is known that the capacity region of NOMA improves with the channel gain disparity of users. Since the performance gap between NOMA and OMA schemes are primarily determined by the channel gain disparity of cluster members [5], the proposed clustering method tries to maximize overall channel gain disparity.

Accordingly, we develop a matching theory based clustering algorithm by partitioning the UE index set of BS, , into disjoint channel gain levels, i.e.,

(5)

where , is the number of clusters/sub-carriers of as a function of the cluster size and . That is, intra-partition and inter-partition channel gains are in descending order (i.e., ), and the lowest channel gain within is higher than all channel gains within . Notice that the partitioning in (5) requires sort of channel gains and has therefore a time complexity of . Pictorially, one can also think of partitions as UEs falling into non-overlapping spatial ring zones in a free-space path loss channel model as shown in Fig. 1.

Fig. 3: Weighted multi-partite matching representation for CF.

The partitioning yields a multi-partite graph (MPG) as depicted in Fig. 3 where edge weight from element of to element of is denoted as . Note that solving weighted multi-partite matching (WMM) is still a hard problem. Hence, we propose a sequential WMM scheme where sequence is modeled as a weighted bi-partite matching (WBM) between partitions and , . In Fig. 3, for example, the sequential formation of first cluster is highlighted in red colors. Note that proposed WMM is general enough to apply different clustering purposes by just changing the edge weight design. Each matching sequence of WMM is in the form of rectangular assignment problem (RAP) which is generally solved by Munkres Algorithm with cubic time complexity [29]. Accordingly, overall complexity of the proposed sequential WMM based CF is given as whereas exhaustive solution of the original problem has an exponential time complexity, i.e., .

For an exemplary edge weight design, let us consider edge weight matrix whose elements are channel gain of corresponding users in the MPG. For example, if user of partition is UE then . Accordingly, we design the edge weight from element of to element of as follows

(6)

which makes each WBM sequence favors for new cluster members that maximize the sum of channel gain disparity. Furthermore, Lemma 1 shows that the previous generic time complexity, , can even be reduced to simple sorting and indexing operations of partitioning in (5), i.e., .

Lemma 1.

When edge weights of the MPG are merely determined by channel gains as in (6), sequential WMM always forms cluster by user of each partition.

Proof.

We first remind that the intra-partition and inter-partition channel gains are in descending order (i.e., ), and the lowest channel gain within is higher than all channel gains within . At WBM sequence , therefore, column and row entries of edge weight matrix are monotonically increasing and decreasing, respectively. That is, if and if . Accordingly, Munkres Algorithm always matches element of partition with the element of partition , as it yields the highest weight. Therefore, sequential WMM always forms cluster by member of each partition. ∎

 

(7)
(8)
(9)
(10)

V Hierarchically Distributed PBA

In this section, we develop a hierarchical decomposition method in order to obtain a distributed power and bandwidth allocation technique for DL-HetNets. Decomposability of network utility maximization problems provides us with the most appropriate distributed solution methodology and can modularize control and resource allocation in HetNets as it consists of hierarchical network entities [30], i.e., MBSs, SBS, and clusters in our case. Primal decomposition is naturally applicable to master problems where “virtualization” or “slicing” of the resources are carried out by dividing the total resource for each of the cluster competing for the resource [31]. Alternatively, we employ dual decomposition method for the slave problems to obtain optimal primal and dual variables in closed-form which is used by master problems to update power and bandwidth allocation of clusters.

V-a Closed-Form Solutions for Slave Problems

The slave problems can be formulated as in where we omit the BS and cluster indices for the sake of simplicity. Remarking that depends on bandwidth portion and power fractions given by master problems, its constraints , , and correspond to constraints , , and , respectively.

In order to derive the optimal closed-form expressions for primal and dual variables, we first apply dual decomposition method to the slave problems. Accordingly, Lagrangian function of can be obtained as in (IV) where , , and are Lagrange multipliers. Taking derivatives of Lagrangian function w.r.t. , , , and , Karush-Kuhn-Tucker (KKT) conditions can be obtained as in (IV)-(10). Given that some regularity conditions are satisfied, KKT conditions are first-order necessary conditions for a solution in nonlinear programming to be optimal. If linearity constraint qualification is held, i.e., all equality and inequality constraints are affine functions, no other regularity condition is required. This is indeed the case for the slave problems since all constraints are affine functions of power weights.

K Optimal Power Allocation Necessary Conditions
2
3
TABLE II: Optimal power allocations along with the corresponding necessary conditions.
Lemma 2.

Given that necessary conditions are satisfied, closed-form optimal power allocation of UE and UE () are given in (11) and (12)-(13), respectively.

(11)