Interference Alignment in Virtualized Heterogeneous Cellular ...

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2016.2562741, IEEE Transactions on Vehicular Technology IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. XX, NO. XX, XXX 2016

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Interference Alignment in Virtualized Heterogeneous Cellular Networks with Imperfect Channel State Information (CSI) Kan Wang, Hongyan Li, Member, IEEE, F. Richard Yu, Senior Member, IEEE, Wenchao Wei and Long Suo

Abstract—Wireless network virtualization is a promising technique for future wireless networks. In this work, different from traditional virtualization approaches by means of resource isolation at the subchannel or time-slot level, we propose a novel framework of heterogeneous cellular network virtualization combined with interference alignment (IA) technology, which utilizes IA to cancel the mutual interference, by aligning the interference from other transmitters into a lower dimensional subspace at each receiver. In this framework, we formulate the virtual resource allocation as a joint virtualization and IA problem, considering the gain not only from interference mitigation introduced by IA but also from the sum rate improvement brought by virtualization. In addition, to reduce the computational complexity, with the recent advances in discrete stochastic approximation (DSA), we propose a two-step algorithm to solve the formulated problem. The basic principle is to design IA schemes for each feasible association combination, then traverse the association space to search for the optimal association combination with the maximum sum rate. Extensive simulations are conducted with different system parameters to show the effectiveness of the proposed scheme. Index Terms—Heterogeneous cellular networks, wireless network virtualization, interference alignment, discrete stochastic approximation.

Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. This work was supported in part by the National Science Foundation under Grant 91338115, Grant 61231008, Grant 61372089 and Grant 61301168; by the National S&T Major Project under Grant 2015ZX03002006; by the Fundamental Research Funds for the Central Universities under Grant WRYB142208 and Grant JB140117; by the Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT0852; by the 111 Project under Grant B08038; by the Shanghai Academy of Spaceflight Technology (SAST) under Grant 201454; by the Natural Sciences and Engineering Research Council of Canada (NSERC); and by the China Scholarship Council. Kan Wang, Hongyan Li and Long Suo are with the State Key Laboratory of Integrated Service Networks, School of Telecommunications Engineering, Xidian University, Xi’an, Shaanxi 710071, China (e-mail: [email protected]; [email protected]; [email protected]). Hongyan Li is the corresponding author. F. Richard Yu is with the Depart. of Systems and Computer Eng., Carleton University, Ottawa, ON, Canada (e-mail: [email protected]). Wenchao Wei is with Shanghai Aerospace Electronic Technology Institute, Shanghai, China (e-mail: [email protected]).

I. I NTRODUCTION Due to the ever-increasing demand to support high data rate services, wireless network virtualization has been proposed as a promising solution for future wireless networks [1], [2]. Using network virtualization, wireless services and applications can be decoupled from the physical infrastructure, so that diverse services can dynamically share the same infrastructure, thus maximizing the infrastructure’s utilization. Meanwhile, since multiple virtual networks can be created and assigned to different mobile virtual network operators (MVNOs) by abstracting and slicing substrate resources, the same physical infrastructure can be shared by multiple MVNOs, thus facilitating the reduction of capital expenses (CapEx) and operation expenses (OpEx) of physical networks operated by infrastructure providers (InPs) [3]. Furthermore, MVNOs can provide some specific over-the-top services (e.g., video and gaming) to facilitate the attraction of more subscribers to legacy mobile network operators (MNOs) [4], while InPs can benefit more by leasing the isolated virtualized networks to MVNOs [4]. Another promising technology in wireless networks is interference alignment (IA), which can be leveraged to eliminate interference in interference networks [5]. Different from conventional interference mitigation techniques (e.g., orthogonal spectrum allocation or iterative water-filling) [6], with IAbased transmissions, each user is able to access one half of the spectrum and degrees of freedoms (DoFs) free from interference [7]. The basic principle behind IA is to align the interference signals to a reduced dimensional subspace of the received signal space at each receiver, so that an interferencefree orthogonal subspace is available for the data transmission [8], [9]. While most previous works focus on applying IA in multiple input multiple output (MIMO) interference channels, the authors of [7], [9]–[11] showed that IA can also be leveraged in MIMO broadcast channels or MIMO downlink heterogeneous networks. Compared with conventional Kuser interference channels, the desired signal and intra-cell interference received by each user undergo the same channel condition in cellular networks [12], thus resulting in alternative IA schemes. In general, wireless network virtualization is accomplished via resource isolation at the time level, frequency level, timefrequency level or even at the hardware level [1], [13]. In this work, we utilize IA to realize the virtualization of heterogeneous cellular networks, namely, isolating virtual slices in the spatial domain. The motivation of this work is based on the following observations. On one hand, with IA, multiple

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base stations (BSs) can simultaneously transmit data at the same frequency free of interference, thus enabling the sharing of the same infrastructure by multiple MVNOs. Meanwhile, the heterogeneity of BSs in heterogeneous cellular networks can provide users with more candidates to associate with, allowing for more flexibility in virtualization. On the other hand, network virtualization facilitates the IA design likewise. For instance, the topology of networks can significantly affect the result of IA [7], and the process of IA is oblivious to the performance of the desired signals [14]. Virtualization can overcome this drawback by implementing the optimal association combination with the maximum data rates when IA is applied. Nevertheless, the integration of network virtualization with IA poses significant challenges. Firstly, it is non-trivial to resolve this joint virtualization and IA optimization problem, which involves both discrete and continuous variables and thus has high computational complexity. Secondly, perfect channel state information (CSI) is hardly available in the hypervisor (the entity responsible for virtualization) due to estimation error and transmission delay, which could significantly degrade the performance of IA [15]. In particular, the performance analysis of IA under imperfect CSI is of great interests and has been paid special attention in the literature [14]– [16]. The unique nature of virtualized heterogeneous cellular networks with imperfect CSI necessitates us to resort to a two-step optimization approach, which designs IA schemes for each possible association combination, then traverses the association space to seek for the optimal combination with the maximum sum rate. In this paper, we study IA schemes in virtualized heterogeneous cellular networks with imperfect CSI. The distinctive features of this paper are as follows. • We propose a novel framework of cellular network virtualization combined with IA technology, which exploits the IA schemes to enable the virtualization of substrate networks. In the proposed framework, with IA schemes, substrate resources can be abstracted and sliced into multiple virtual networks, which will be leased by multiple MVNOs. • We formulate the virtual resource allocation as a joint virtualization and IA problem, considering the gain not only from interference mitigation introduced by IA but also from the rate improvement brought by multiple optional association combinations. In addition, imperfect CSI is taken into account. • To reduce the computational complexity, we propose a two-step scheme to solve the formulated problem, based on recent advances in discrete stochastic approximation (DSA) algorithms. The basic principle behind our proposed algorithm is to design IA schemes for each feasible association combination, then traverse the association space to seek for the optimal combination with the maximum sum rate or minimum leakage interference power. • Extensive simulations are conducted with different system parameters to show the effectiveness of the proposed scheme. It is shown that, with the proposed framework of

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virtualization combined with IA technology, the sum-rate performance can be improved significantly. The rest of this article is organized as follows: In Section II, we describe the proposed framework and system model for downlink cellular heterogeneous networks. The formulation of joint virtualization and IA optimization is also presented. In Section III, we introduce a necessary condition derived from the IA feasibility for a fixed virtualization combination, which facilitates the reduction of computational complexity. In Section IV, we specify the CSI mismatch model in cellular networks, followed by the noisy version of two IA schemes, namely, Min-WLI algorithm with imperfect CSI and MaxSINR algorithm with imperfect CSI, respectively. In Section V, we design DSA-based algorithms to traverse the virtualization combination space, in the context of both static and time-varying channels. In Section VI, we provide and discuss the simulation results. We conclude this work in Section VII with future work. Notation: Bold upper and lower case letters denote matrices and vectors, respectively; (·) −1 denotes the matrix inversion, (·)† denotes the pseudo-inverse of matrix, (·) H denotes matrix conjugate transpose, I dmi denotes a dmi × dmi identity matrix; E(·) denotes the expectation operator, v d (·) denotes the eigenvector of matrix corresponding to the d-th smallest eigenvalue, Tr(·) denotes the trace operator, and · denotes the Euclidean 2-norm of vector. II. S YSTEM M ODEL In this section, we first propose a cellular network virtualization framework with IA technology, then introduce the problem formulation where multiple macro base stations (MBSs) and small base stations (SBSs) coexist to serve multiple users. A. IA-based Cellular Network Virtualization Framework In virtualized cellular networks, a hypervisor is responsible for virtualizing BSs to slices, then associating users from different MVNOs with virtualized slices [3]. In general, the abstraction of substrate networks to virtual slices is realized by means of resource isolation at the subchannel or time-slot level, or even at the hardware level [1], [13]. In addition, wireless network virtualization enables recent development in software-defined wireless networks [17]–[20]. Different from traditional approaches, in this paper, we utilize spatial IA [6], [12] to cancel the mutual interference, namely, isolating slices in the spatial domain. 1 By aligning the interference from other transmitters into a lower dimensional subspace at each receiver, IA can obtain the the optimal capacity scaling with respect to transmit signal-to-noise ratio (SNR) [9]. In addition, the IA technique leaves the remaining interference-free subspace for the transmission of desired data streams, thereby enabling the sharing of same infrastructure by multiple links [21]. Compared to the isolation at the 1 Although

IA can also be realized in frequency and time domains, exponentially long symbol extensions are in general required, which is hard to satisfy in real-world networks. Therefore, we focus on the MIMO channels without symbol extensions in our work.



SBS → user 2. Therefore, IA schemes are necessary to align both intra-cell and inter-cell interference into the subspace at the receiver so as to realize virtualization.

MVNO 1

MVNO 2

Virtual MBS 1 Virtual SBS 1

Virtual MBS 2 Virtual SBS 2

MVNO layer

User 1

User 2

User 3

User 4

Inter-cell interference

MBS Desired signal

SBS

Intra-cell interference

User 1

User 3

Hypervisor and virtualized elements

User 2

User 4

Virtual network controller Virtual resource manager

Hypervisor

Virtual MBS 1 Virtual MBS 2

Network virtualization Virtual SBS 1 Virtual SBS 2

Substrate networks layer

SBS MBS User 1

User 2

MVNO 1

User 3

3

User 4

MVNO 2

Fig. 1: IA-enabled cellular network virtualization framework.

spectrum level [22], [23], the spatial IA could achieve the higher spectrum efficiency yet at the cost of more antennas, since the MIMO technology is exploited to enable the isolation of multiple MVNOs on the same frequency band. Above all, in the process of virtualization, the hypervisor needs to perform association, finding out the set of users served by each BS. Meanwhile, IA is implemented to cancel the mutual interference among BSs. Therefore, a joint virtualization and IA optimization is confronted by the hypervisor in the proposed framework. The proposed framework of cellular network virtualization combined with IA technology is illustrated in Fig. 1. The substrate networks are owned by one InP, but can be virtualized to multiple virtual networks for MVNOs to lease. As depicted in Fig. 1, for instance, there exist one MBS and SBS in physical networks, two users (i.e., user 1 and user 2) subscribed to MVNO 1 and two users subscribed to MVNO 2 (i.e., user 3 and user 4). First, the MBS is mapped into virtual MBS 1 and virtual MBS 2. Similarly, virtual SBS 1 and virtual SBS 2 are extracted from the same SBS. Then, the virtualized elements are aggregated and sliced into two virtual networks by the hypervisor. These two virtual networks are leased by MVNO 1 and MVNO 2, respectively [24]. Note here that, in reality, the hypervisor assembles three functions, namely, network virtualization, virtual network controller as well as virtual resource manager [25]. The embedding of virtual networks onto physical networks is realized by associating users 1 and 3 with MBS, while associating users 2 and 4 with SBS simultaneously. Taking the link MBS → user 1 as an example, it will cause intra-cell interference to the link MBS → user 2, and inter-cell interference to both links SBS → user 1 and

B. Problem Formulation We consider a cellular network where multiple heterogeneous BSs aim to serve multiple users. Denote by G b and Gs the set of MBSs and SBSs, respectively. In addition, let G = Gb Gs = {1, . . . , G} and I = {1, . . . , I} be the set of all BSs and users, respectively. It is assumed that all G BSs are managed by one InP, whereas I users can be subscribed to a total of M MVNOs, the set of which is denoted by M = {1, . . . M }. Denoteby Im the set ofusers subscribed to MVNO m, namely, I = m Im and Im In = ∅, ∀m = n. For ease of notation, we refer to the i-th user of m-th MVNO as user (m, i). User (m, i) is equipped with N mi antennas to receive dmi desired data streams, while BS g has M g antennas to send data streams. It is assumed that all BSs share the same frequency to transmit data streams. Let H mi,g be the channel matrix between BS g and user (m, i), and it is assumed that all elements in Hmi,g are independent and identically distributed (i.i.d.) variables drawn from a complex Gaussian distribution CN (0, 1). In the heterogeneous cellular networks, the hypervisor is responsible for virtualizing G BSs, then associating I users from M different MVNOs with virtualized slices. In other words, the hypervisor finds out the set of users served by each BS g ∈ G. To identify the user set that BS g serves, we introduce the association matrix X = [x mi,g ] such that xmi,g = 1 if user (m, i) is connected to BS g and 0 otherwise. Considering that eachuser can only be associated with one BS, the condition that g∈G xmi,g ≤ 1, ∀(m, i) ∈ I must be satisfied. For a specific BS g, it can serve multiple users belonging to different MVNOs, thus enabling the radio resource and infrastructure sharing among MVNOs. The transmit signal of BS g is a linear combination of encoded information symbol: xmi,g Vmi,g smi , (1) ag = m,i

where smi ∼ CN (0, Idmi ) is the information symbol of user (m, i) [9], and Vmi,g ∈ CMg ×dmi is the precoder matrix intended for user (m, i) by BS g. The received signal at user (m, i) is given by [26] [27] as √ ymi = ρmi,g Hmi,g ag + nmi g∈G

=

√ ρmi,g Hmi,g ( xnj,g Vnj,g snj ) + nmi , ∀m, i,

g∈G

n,j

(2) where nmi ∼ CN (0, σ 2 INmi ) is the additive white Gaussian α is the large-scale noise (AWGN) vector, and ρ mi,g = rmi,g channel fading between BS g and user (m, i) with distance rmi,g and path-loss exponent α [28]. Note here that, as in [21], the channel gains are dependent on both small-scale fading (i.e., Hmi,g ) and large-scale fading (i.e., ρ mi,g ). Since user (m, i) can only be exclusively served by one BS, we assume that g is exactly the BS that serves user (m, i), i.e.,



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xmi,g = 1 and xmi,g = 0, ∀g = g, g ∈ G. Henceforth, (2) can be written as √ ymi = ρmi,g Hmi,g Vmi,g smi +nmi

i.i.d. and drawn from a continuous probability distribution [6] (also refereed to as generic in [12]). The establishments of conditions (6a)-(6b) are based on the fact that the association matrix X is predetermined. In desired signal our work, nevertheless, the set of users that one BS serves √ + ρmi,g Hmi,g ( xnj,g Vnj,g snj ) needs to be optimized firstly, namely, one BS may connect (n,j)=(m,i) with none, one user or multiple users. Since each user can connect to any BS, there are totally possible G I association intra-cell interference √ + ρmi,g Hmi,g ( xnj,g Vnj,g snj ), ∀m, i, combinations. Intuitively, the set of association combinations can be classified into two cases: g =g (n,j)=(m,i) • Infeasible IA solutions: There exist no feasible solutions, inter-cell interference (3) i.e., we cannot solve the precoder and decorrelator mawhere we denote (n, j) = (m, i) as any user different from trices for some combinations (e.g., too many users are (m, i), i.e., j = i or n = m. connected to the same BS or BSs do not have sufficient At the receiver side, by aligning the interference (composed antennas to transmit independent data streams.). of both intra- and inter-cell interference) into the null space • Feasible IA solutions: Both intra- and inter-cell interferof the decorrelator matrix U mi ∈ CNmi ×dmi , IA enables the ence can be aligned into the interference subspace and sharing of frequency band by multiple transmissions. Each nullified by decorrelator matrices at each receiver side. user decodes its desired signal through U mi as follows [28]: The feasibility of IA in downlink cellular networks has been ˜ mi = UH y (4) extensively researched in the literature [10], [11], [28]–[31]. In mi ymi . [28], by utilizing the partial connectivity in heterogeneous netThen, achievable rate of user (m, i) can be described as works, the optimal number of users that can be accommodated Rmi = xmi,g log det(Idmi + (σn2 Idmi for the maximization of DoFs was discussed. The authors of g∈G [11] gave a necessary condition in symmetric cellular networks in terms of the number of users, cells as well as antennas. + ρmi,g UH H V nj,g mi mi,g For the asymmetric configuration, the necessary condition was g ∈G (n,j) provided in [30]. In [10], the numbers of users and antennas H H −1 · Pnj,g Vnj,g Hmi,g Umi ) ρmi,g at BSs essential for the proposed IA scheme was derived. In H H · UH mi Hmi,g Vmi,g Pmi,g Vmi,g Hmi,g Umi ), ∀m, i, [31], by exploiting the vector space strategy, the sufficient (5) and necessary conditions were respectively provided for the where Pmi,g = diag(Pg,1 , Pg,2 , . . . Pg,dmi ) is the power symmetric three-user configuration with identical number of allocation of BS g for information symbol vector s mi , and dimensions, transmit antennas and receive antennas, as well Pg,k is the power allocated to the k-th information symbol as the fully symmetric configuration (i.e., the same number of user (m, i). Without loss of generality, it is assumed that of antennas at both transmitters and receivers) but with an Pg,k = Pg , ∀1 ≤ k ≤ dmi . It should be noted here that in arbitrary number of users per cell. general the MBS has higher transmit power per stream than The authors of [12] proposed three necessary conditions the SBS, i.e., Pg > Pg , ∀g ∈ Gb , g ∈ Gs . (Theorem 2 therein) for the asymmetric cellular networks. In this paper, we only concentrate on the configuration where III. F EASIBILITY C ONDITIONS dmi = 1, ∀m, i, and that with more data streams per user will Assuming that (m, i) is connected to BS g, one commonly be incorporated in the future work. Then (6a)-(6c) reduce to used method in cellular networks is to find out the solution of Vmi,g and Umi such that the following conditions must be (7a) uH mi Hmi,g vmi,g > 0, satisfied H umi Hmi,g vnj,g = 0, ∀(n, j) = (m, i), (7b) rank(UH (6a) mi Hmi,g Vmi,g ) = dmi , H (7c) umi Hmi,g vnj,g = 0, ∀g = g, (n, j) = (m, i), UH (6b) mi Hmi,g Vnj,g = 0dmi ×dmi , ∀(n, j) = (m, i), UH mi Hmi,g Vnj,g = 0dmi ×dmi , ∀g = g, (n, j) = (m, i). (6c)

Condition (6b) guarantees that the intra-cell interference caused by user (m, i)’s associated BS can be aligned in its interference subspace, and condition (6c) ensures the cancellation of inter-cell interference at user (m, i). Condition (6a) is a rank constraint to ensure the dimension of H mi,g Vmi,g is dmi and the signal subspace is independent of interference subspace [28]. Note that, condition (6a) is almost guaranteed once (6b)-(6c) are met, provided that all entries of H mi,g are

where vmi,g ∈ CMg ×1 and umi ∈ CNmi ×1 are the precoder and decorrelator vectors per user, respectively. To satisfy (7) in conjunction with the cellular network virtualization, we extend the necessary conditions in [12] with a modification as follows: Theorem 1. In the virtualization combination space, the necessary conditions for the existence of a feasible IA solution are



Mg ≥

ami,g , ∀g ∈ G, Nmi ≥ 1, ∀(m, i) ∈ I, (8a)

(m,i)∈I

g :(g,g )∈GI

+

(Mg −

ami,g )

(m,i)∈I

ami,g

(8b)

(m,i)∈I

(Nmi − 1)

(

A. Imperfect CSI

ami,g )|Kg |, ∀GI ⊆ J ,

(g,g )∈GI (m,i)∈I

where J = {(g, g )|g, g ∈ G, g = g }, GI can be any ¯ g = {(m, i)|ami,g = 1}. arbitrary subset of J and K g ⊆ K Proof: The proof of Theorem 1 is derived from the Theorem 2 in [12], which states that for cellular networks with configuration g∈G (Mg × i∈K¯ g (Ni , d)), when both Mg and Ni are divisible by d, two necessary conditions for IA feasibility are given as ¯ g |d, Ni ≥ d, ∀i, g, Mg ≥ |K ¯ g |d)|K ¯ g | (Mg − |K g :(g,g )∈GI

+

g:(g,g )∈G

≥

I

(Ni − d)

effect of imperfect CSI, precoder and decorrelator matrices are designed to align the interference in a subspace at each user. Then, DSA is applied to optimize the association combination, searching over the solution space X = [x mi,g ]. In this section, we first present the IA schemes, followed by association optimization in Section V.

(8c)

g:(g,g )∈GI (m,i)∈Kg

≥

5

(9a) (9b) (9c)

i∈Kg

¯ g ||Kg |d, ∀GI ⊆ J . |K

(g,g )∈GI

In reality, (9a) ensures the linearly independence among data streams, and (9b) indicates that the number of independent variables must be no less than that of equations. Considering that (9a) and (9b) are based on predetermined associations between users and BSs (i.e., without virtualization), different association combinations needs to be ¯ g in (9a) and (9b) with taken into account. By replacing K (m,i)∈I ami,g and making d = 1, the proof is straightforward. As mentioned above, some associations lead to feasible IA solutions while others just the opposite. Therefore, Theorem 1 can play a significant role when performing association optimization to enable virtualization. Simulation results also demonstrated that if the necessary conditions in Theorem 1 are met, then IA solutions can be found via the numerical algorithm to be discussed later. IV. IA S CHEMES U NDER I MPERFECT CSI AND THE A SSUMPTION OF P REDEFINED A SSOCIATION It is observed in (5) that the maximization of (m,i)∈I Rmi is a joint virtualization and IA optimization problem, which introduces variables x mi,g , Umi and Vmi,g . However, due to the introduction of association matrix X = [x mi,g ], existing design schemes cannot be directly applied to align and cancel interference. Inspired by [32], to address this non-trivial challenge, we employ a two-step optimization to reduce the computational complexity. First, given predetermined association combination (i.e., X is fixed) and by statistically analyzing the

CSI is forwarded to the hypervisor by substrate networks to perform joint virtualization and IA optimization. Ideally, given predefined virtualization result, CSI between users and BSs is available in the hypervisor, enabling the computation of precoder and decorrelator matrices. Nevertheless, CSI observed by the virtual resource manager (residing in the hypervisor) is typically imperfect for the following reasons. First, unavoidable measurement error gives rise to inaccurate CSI estimation [15], [16]. Second, the transmission delay between substrate networks and hypervisor always makes the estimation lag behind the actual network states. Consequently, it is necessary to incorporate the effect of imperfect CSI on IA performance. The imperfect CSI estimation can be modeled as ˆ = H + E, H

(10)

ˆ is the observed CSI in the hypervisor, H is the actual where H CSI matrix and E represents the estimation error. Note that, all coefficients in E are i.i.d. random variables drawn from CN (0, η), and E is independent of H. In reality, the observed CSI available in the hypervisor is ˆ rather than H the inaccurate estimation of H, and thus H should be employed in IA conditions. Assuming that user (m, i) is served by BS g and as in [15], IA conditions should be modified as ˆ ˆ mi,g | > 0, ∀m, i, g, |ˆ uH mi Hmi,g v (11) H ˆ ˆ mi Hmi,g v ˆ nj,g = 0, ∀(n, j) = (m, i) or g = g, u ˆ mi and v ˆ mi,g are the calculated precoder and decorrewhere u ˆ Obviously, this will result in imperfect lator vectors using H. alignment at the receiver in real networks, namely, ˆnj,g > 0, ∀(n, j) = (m, i) or g = g, (12) ˆH u mi Hmi,g v thus degrading the sum rate and giving rise to residual leakage interference. Therefore, adaptive IA schemes incorporating the effect of imperfect CSI need to be designed. The authors of [6] exploited the channel reciprocity and proposed an elaborate methodology, by which the precoder and decorrelator matrices are updated iteratively to achieve IA. The authors of [15] extended the two algorithms (Min-WLI and Max-SINR) in [6] to scenarios where estimation error is inevitable but follows the Gaussian distribution. Since the IA design given virtualization can be considered as one subroutine of our proposed two-step optimization, we outline the MinWLI and Max-SINR algorithm with CSI uncertainty (which have been analyzed in detail in [15]) briefly in the following.



B. Min-WLI Algorithm for Cellular Networks Under Imperfect CSI Different from the standard Min-WLI algorithm where perfect CSI is exploited to compute interference plus noise covariance matrices, when it comes to Min-WLI algorithm under imperfect CSI, the effect of CSI uncertainty must be taken into account. As in [15], we first rewrite (10) as the expression for actual CSI as follows, H=

1 ˆ H + W, 1+η

Algorithm 1 Min-WLI algorithm with CSI uncertainty ˜ g : Mg × |K ˜ HV ˜g = ¯ g |, V Start with random matrices V g I|K¯ g | . 2: Start iterations. 3: Compute the inter-cell interference covariance matrix E[Qmi ] at the receivers in the original networks as in (14). 4: The decorrelator vector of each receiver is derived from the eigenvector corresponding to the smallest eigenvalue: 1:

(13)

where all coefficients in W are i.i.d. random variables drawn η ). Then by replacing H in standard Minfrom CN (0, 1+η WLI algorithm (see [15] and references therein) with (13) and exploiting the statistical information with respect to H, we can obtain the expectation version of interference matrices as follows: Pg ˆ mi,g V ˜ g V ˜ gH H ˆH ρmi,g H mi,g (1 + η)2 g =g Pg η ¯ g |INmi , + ρmi,g |K 1+η

6

E[Qmi ] =

(14)

g =g

umi = v1 [E[Qmi ]].

(16)

Compute the inter-cell interference covariance matrix at each BS in the reciprocal networks as in (15). ˜g = 6: The precoder matrix at BS g is designed as V ¯ vd [E[Bg ]], for d = 1, . . . , |Kg |. 7: Continue until convergence. 8: Compute the cascaded matrix to null all intra-cell interference as ⎡ ⎤† .. . ⎢ ⎥ ⎢ H ¯ ˜ g⎥ (17) Vg = ⎢umi Hmi,g V ⎥ . ⎣ ⎦ .. . 5:

The effective precoder matrix for BS g is obtained as ¯ g. ˜ gV Vg = V 10: Normalize Vg and umi . 9:

and E[Bg ] =

Pg 2 (1 + η)

ˆ H unj uH H ˆ ρnj,g H g,nj nj g,nj

¯ g =g (n,j)∈K g

Pg η + 1+η

ρnj,g IMg ,

¯ g =g (n,j)∈K g

(15) where Pg is the identical power allocated to each user by BS g . Note here that, (14) and (15) are expected inter-cell interference covariance matrices for original and reciprocal networks, respectively. Since both (14) and (15) are only responsible for processing inter-cell interference leakage, a second precoder preceding the fixed precoder is employed to cancel the intra-cell interference, as summarized in Algorithm 1.

C. Max-SINR Algorithm for Cellular Networks Under Imperfect CSI Instead of only minimizing the leakage interference, the IA schemes can be employed to maximize the signal-tointerference-plus-noise ratio (SINR) likewise. In this context, orthogonal precoder vectors are in general suboptimal for SINR maximization [6]; therefore, it is not necessary to assume the orthogonality of precoder vectors among users connected to the same BS. Similarly, by taking advantaging of statistical knowledge of CSI estimation error, what we are more concerned with are the expectations of interference plus

noise covariance matrices in the original and reciprocal networks, which are calculated as in (18) and (19), respectively. E[Qmi ] Pg ρmi,g = 2 (1 + η) g =g

ˆ mi,g vnj,g vH H ˆH H nj,g mi,g

¯ (n,j)∈K g

Pg η ¯ g | + σ 2 )INmi , +( ρmi,g |K 1+η g =g

(18)

and E[Bmi ] Pg = (1 + η)2 g

Pg η +( 1+η g

H ˆ ˆH ρnj,g H g,nj unj unj Hg,nj

(n,j)=(m,i)

(19)

ρnj,g + σ 2 )IMg .

(n,j)=(m,i)

Note that, (18) is only concerned with inter-cell interference plus noise in the original networks, while (19) considers both inter- and intra-cell interference and can be interpreted as the matrix for the mapping of user (m, i), on a per-use basis, onto BS g. As in [15], the iterative algorithm is applied to maximize the SINR, as described in Algorithm 2.



Algorithm 2 Max-SINR algorithm with CSI uncertainty ¯g, g ∈ 1: Start with random unit vectors v mi,g , ∀(m, i) ∈ K G. 2: Start iterations. 3: Compute the inter-cell interference plus noise covariance matrix E[Qmi ] at the receivers in the original networks as in (18). 4: Compute the decorrelator vectors : umi =

−1

(E[Qmi ]) (E[Qmi

ˆ mi,g vmi,g H

ˆ ])−1 H

mi,g vmi,g

.

(20)

Compute the intra- and inter-cell interference plus noise covariance matrix E[B mi ] on a per-user basis in the reciprocal networks as in (19). 6: Compute the precoder vectors : 5:

vmi,g = 7:

ˆ H umi (E[Bmi ])−1 H g,mi . −1 ˆ (E[Bmi ]) HH g,mi umi

(21)

Continue until convergence.

V. D ISCRETE S TOCHASTIC A PPROXIMATION (DSA) FOR V IRTUALIZATION U NDER I MPERFECT CSI Based on the discussion in Section IV, IA schemes can be performed under the assumption of known virtualization. In this section, we concentrate on the association optimization to realize the virtualization, via which different users subscribed to different MVNOs can connect to the same BS. Note from (5) that the sum rate of networks is actually dependent on the variables umi , vmi,g as well as xmi,g . Therefore, the basic principle behind our proposed algorithm is to design IA schemes for each association combination, then traverse the association space to search for the optimal association combination with the maximum sum rate or minimum leakage interference power. As mentioned above, X = {x mi,g }I×G denotes the association indication matrix and there are totally Q = G I possible association combinations, the set of which can be represented as Φ = {X1 , X2 , . . . , XQ }. For each association combination X q ∈ Φ, hypervisor first justify the feasibility of IA utilizing Theorem 1. If there exists no feasible u mi that can nullify interference, the corresponding association combination is discarded; otherwise, the hypervisor performs IA schemes for the maximization of (m,i) Rmi , which can be as a function of H and X, namely, R(H, X) = expressed ∗ ¯ g Rmi . To search for the optimal solution X g∈G (m,i)∈K with the maximum R(H, X), a discrete stochastic optimization problem can be formulated as X∗ = arg max R(H, X). X∈Φ

(22)

The CSI available in the hypervisor at time slot t is the ˆ t . This implies that the hypervisor can estimation of H, H only acquire the noisy version of R(H, X t ), which can be ˆ t , Xt ). From the discussion in Section IV it is denoted as r(H

7

ˆ t , Xt ) is a random variable due to the straightforward that r( H t ˆ ˆ t is the estimation error in H . In addition, assuming that H unbiased estimation of H, we can come to the conclusion that ˆ t , Xt ), ∀t is a sequence of i.i.d. random variables [32]– r(H [34]. In reality, at slot t, X ∗ is picked in the solution space as the one with greatest expectation, i.e., ˆ t , Xt )], X∗ = arg max E[r(H t X ∈Φ

(23)

ˆ t , Xt )] is consistent with the where the calculation of E[r( H fact the expectations of interference covariance matrices are utilized in both Algorithm 1 and 2. One possible approach to solve (23) is the exhaustive-search algorithm, by which for each association combination the empirical average is taken as an approximation of the statistical expectation. Yet, two challenges arise. On one hand, statistical averaging necessitates a large value of sample time, which contradicts the time-variation of cellular networks. On the other hand, even though the number of BSs or users is not very large, the computational complexity will be prohibitively high. In view of these, we resort to another more efficient DSA-based alternative. DSA algorithm, with low complexity, has been extensively applied in resource allocation and antenna selection [32], [33], [35], [36]. In particular, an antenna selection-based IA scheme is proposed to maximize the sum rate of cognitive networks based on DSA algorithm [14]. Inspired by the work in [14], we first apply aggressive DSA to cellular networks with static channels, then adaptive step size DSA to time-varying channels. It should be noted here that DSA algorithm could only adapt to slowly time-varying channel conditions [33], [35]. Meanwhile, the IA schemes involved in each iteration of DSA cannot accommodate a large number of users and BSs, namely, including too many nodes when performing IA may result in a strong leakage interference due to imperfect alignment. Henceforth, the DSA algorithm is not applicable to the dynamic and dense cellular networks. Coordinated multipoint transmission (CoMP) with user-centric adaptive clustering is a promising technique to address this issue [37], and will be coved in our future work. A. DSA for Static Networks Given P (in general, P < Q ) feasible association combinations, we introduce P indicating variables to represent the virtualization results. As such, a one-to-one mapping can be established. With 1 at the p-th position and 0 elsewhere, e p is denoted as a indicating unit vector for the p-th combination (i.e., Xp ) in the solution space. Therefore, for ease of notation, we can map the virtualization sequence X t to a sequence of Dt , where Dt = ep if Xt = Xp . In addition, let π t (p) be the occupation probability for the combination X p at time slot t, which is time-variant and can be refreshed slot by slot. Therefore, a probability vector π t = [π t (1), π t (2), . . . , π t (P )] t with π (n) ∈ [0, 1] and 1≤p≤P π t (p) = 1 can be utilized to indicate the occupation state for the entire solution space. Next, we first present the DSA for static networks in Algorithm 3 and then give a proof of convergence.



Algorithm 3 Aggressive DSA for Virtualization 1:

2: 3:

4: 5: 6:

7:

8:

9: 10: 11: 12: 13: 14:

Virtualize the set of BSs and initialize the set of users subscribed to different MVNOs. t←0 Pick one feasible virtualization decision X t from Φ, and set πt (Xt ) = 1 and π t (X) = 0 for X = Xt . Xt∗ ← Xt . for t = 0, 1, . . . do ˆ mi,g for each BS Hypervisor collects estimated CSI H g → user (m, i) link via user measurement report. For the selected Xt , hypervisor executes Algorithm 1 ˆ t , Xt ). or Algorithm 2 and then obtains r( H ˜ t ∈ Φ, r(H ˆ t, X ˜ t) Given another uniformly selected X is computed likewise. ˆt , X ˆ t , Xt ) < r(H ˜ t ) then if r(H t+1 t ˜ X ←X. else Xt+1 ← Xt . end if The occupation probability vector is updated as π t+1 ← π t + v t+1 (Dt+1 − π t ),

15: 16: 17: 18: 19: 20: 21: 22:

(24)

with v t = 1t . if πt+1 (Xt∗ ) < π t+1 (Xt+1 ) then X(t+1)∗ ← Xt+1 else X(t+1)∗ ← Xt∗ end if Output Xt∗ t←t+1 end for

As in [14], [32], [38], the proof for convergence of Algorithm 3 is provided as follows. Theorem 2. Algorithm 3 converges to the global optimum, if the number of iterations (slots) is sufficiently large.

8

irreducible and aperiodic Markov chain within solution space Φ, and spends more time on X t∗ . Next, what we need to demonstrate is that both (25) and (26) hold provided that the number of iterations is sufficient. Let μx and σx2 be the mean and variance of random variable ˆ t , Xt ), respectively. Via the empirical accumulative distrir(H bution function F t (r) = (1/t) tt =1 I{r(Ht ,Xt ) 0} P r{r(H (27) ˆ t , Xt∗ ) > 0}. ˆ t , Xt ) − r(H >P r{r(H Due to the fact that the difference of two Gaussian variables also follows a Gaussian distribution, (27) is equivalent to P r{N (μx∗ − μx , σx2∗ + σx2 ) > 0} >P r{N (μx − μx∗ , σx2 + σx2∗ ) > 0}.

(28)

ˆ t , Xt ) holds, it is ˆ t , Xt∗ ) > R(H Following from that R( H straightforward that μ x∗ − μx > 0. In addition, due to the same variance, (28) is satisfied. Meanwhile, (26) can be rewritten as ˆ t, X ˜ t ) > 0} ˆ t , Xt∗ ) − r(H P r{r(H ˆ t , Xt ) − r(H ˆ t, X ˜ t ) > 0}, >P r{r(H which is equivalent to following inequality: μx∗ − μx˜ μx − μx˜ > . 2 2 σx∗ + σx˜ σx2 + σx2˜

(29)

(30)

As in [32] and [33], extensive simulations are conducted to obtain mean values and variances and eventually (30) is also verified. Thus, both (25) and (26) hold if the number of iterations is sufficiently large. The global convergence of Algorithm 3 is proved. B. DSA for Time-Varying Channels

In static networks, with the growth of number of iterations, t π must be increasingly conservative. Consequently, the step Proof: To demonstrate the convergence of Algorithm 3, size v t = 1t is adopted, so that X t∗ will not move far away we firstly present two sufficient conditions in [39]: from X∗ . In particular, only when D t+1 − π t takes a large t t∗ t t t t t∗ ˆ t ˆ ˆ ˆ P r{r(H , X ) > r(H , X )} > P r{r(H , X ) > r(H , X )}, value that Xt∗ can be substituted by X t+1 . (25) Nevertheless, in networks with time-varying channels, the ˆ t , Xt∗ ) > r(H ˆ t, X ˜ t )} > P r{r(H ˆ t , Xt ) > r(H ˆ t, X ˜ t )}. algorithm with v t = 1 tends to stay at a local optimal P r{r(H t (26) solution and thus gets excessively conservative [4], which It is pointed out in [39] that aggressive DSA converges to cannot sufficiently explore the solution space Φ. Consequently, the global optimum if inequalities (25) and (26) are satisfied the choice of the step size will play a significantly role on the simultaneously. In reality, inequality (25) states that it is more performance of algorithm. On one hand, the faster the channel likely to move from X t to Xt∗ rather than the opposite changes, the larger v t should be. On the other hand, the closer ˜ t to Xt∗ is from X∗ , the smaller v t should be [33]. Note that, v t direction. And (26) states that it is more likely for X t∗ t move to X rather than to any other state X . In addition, takes the value between 0 and 1, and two representative cases as shown in [39] that, the sequence X t , ∀t is a homogeneous are as follows: when v t = 0, (24) reduces to π t+1 ← π t ,



which will stay at a fixed solution; when v t = 1, (24) reduces to π t+1 ← Dt , which is equivalent to the exhaustive search. In order to track the optimal step size v t∗ in time-varying networks, least-mean-square (LMS) algorithm is leveraged [32], [33]. Besides the search of X t∗ , the optimal step size v t∗ in each step t is optimized simultaneously. With respect to DSA with adaptive step size, π t can be considered as a function of v t , denoted as π v,t . Thus, its mean-square v,t derivative with respect to v t is Jv,t = ∂π ∂v t by definition, namely, π v+Δ,t − π v,t − Jv,t |2 } = 0. Δ→0 Δ Furthermore, the estimation error is denoted as lim E{|

(31)

v,t = Dt+1 − π v,t .

(32)

Then, by taking the partial of the square of (32) with respect to v t , it follows that ∂v,t 2 = −2(v,t )T Jv,t = −2(Dt+1 − π v,t )T Jv,t . (33) ∂v t Meanwhile, taking the partial of both sides of (24) with respect to v t , we can obtain J

v,t+1

=J

v,t

+

v,t

−v J

t v,t

= (1 − v )J t

v,t

+

v,t

.

(34)

The adaptive step size DSA is summarized in Algorithm 4, + where η is the learning rate, and {a} vv− is the projection of real number a into interval [v − , v + ]. Algorithm 4 Adaptive DSA for Virtualization 1: 2: 3: 4: 5: 6: 7:

Initialization, sampling and acceptance as in Algorithm 3. Adaptive filter for updating state occupation probabilities. v,t ← Dt+1 − π v,t π v,t+1 ← π v,t + v t v,t + vt+1 ← {v t + ηv,t Jv,t }vv− Jv,t+1 ← (1 − v t )Jv,t + v,t Output Xt∗ .

9

1) DSA algorithm combined with Max-SINR under imperfect CSI (denoted as DSA + Max-SINR), which traverses the association combination space based on DSA, and utilizes the Max-SINR algorithm with CSI uncertainty to evaluate the IA performance for fixed combination. 2) Exhaustive search with Max-SINR under perfect CSI (denoted as Max-SINR w. perfect CSI), which exhaustively searches the combination space and utilizes the standard Max-SINR algorithm to evaluate the IA performance for fixed combination. 3) Max-SINR w.o. virtualization, which exploits the statistical knowledge using Max-SINR algorithm with CSI uncertainty, but without the sharing of the infrastructure. 4) DSA algorithm combined with Min-WLI under imperfect CSI (denoted as DSA + Min-WLI). 5) Exhaustive search with Min-WLI under perfect CSI (denoted as Min-WLI w. perfect CSI). 6) Min-WLI w.o. virtualization. A. System Parameters In the simulation, all results are averaged over 1000 drops. We investigate a time-slotted network, where 4 MVNOs coexist and share the same infrastructure. There are totally 6 users, which can access to any MVNO with a probability of 25% and are all equipped with 4 receive antennas. Similarly, the number of transmit antennas is set as 4 for both MBS and SBS2 . In addition, the maximum number of both MBSs and SBSs is set as 10. In the system, the path loss from MBSs or SBSs to users is 128.1 + 37.6 log10 (R(km)) or 140.7 + 36.7 log10 (R(km)), respectively [28], [41]. The 3GPP SCM channel is employed; meanwhile, the shadowing fading standard deviation is 10 dB, noise power spectral is -174 dBm/HZ and noise figure is 9 dB. For measurement error, the CSI noise is assumed to be a zeromean complex Gaussian variable with the standard deviation being 5% of the expectation. B. Performance Comparison

VI. S IMULATION R ESULTS AND D ISCUSSIONS In this section, we show the performance of proposed joint virtualization and IA optimization algorithm via simulation results. We study the impact of following parameters: (1) the number of MBSs, (2) the number of SBSs, (3) the transmit SNR per stream of MBS, and (4) the transmit SNR per stream of SBS. In addition, the following two metrics are utilized to measure the performance of the proposed algorithm: (i) the sum rate and (ii) the fraction of interference in the desired signal space [6], which can be defined as fmi =

λ1 [E[Qmi ]] , Tr[E[Qmi ]]

(35)

where λ1 [A] denotes the smallest eigenvalue of A, and λ1 [E[Qmi ] will approach zero at user (m, i) for perfect IA. Meanwhile, for performance comparison, four other algorithms are also evaluated. All algorithms are listed as follows:

Fig. 2 shows the impact of the number of MBSs (ranging from 1 to 10) on the performance of different algorithms. In this scenario, there are 6 users, 2 additional SBSs, the transmit SNR per stream of MBSs and SBSs is 80 dB and 30 dB, respectively, and all BSs are uniformly distributed in a 5 km × 5 km square. As Fig. 2(a) shows, the sum rate increases with the growth of the number of MBSs yet with a gradually decreasing rate. That is due to the fact that a network incorporating more MBSs will introduce the diversity gain from virtualization, but the gain will become saturated when the number of MBSs is large enough. Meanwhile, the DSA + Max-SINR algorithm achieves a larger sum rate compared to DSA + Max-WLI, and can approach Max-SINR w. perfect CSI (which can be considered as the upper bound) within a 2 In our simulation work, we only distinguish MBS and SBS in terms of the transmit power, and the configuration with massive antennas on each MBS will be in our future work.



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Fig. 2: (a) The sum rate of all MVNOs and (b) the fraction of interference in the desired signal space per user versus the number of MBSs. (There are 6 users, 2 SBSs, the transmit SNR per stream for MBSs and SBSs is 80 dB and 30 dB, respectively, and all BSs are uniformly distributed in a 5 km × 5 km square.) 5% difference on average. For both Max-SINR w.o. virtualization and Min-WLI w.o. virtualization, the sum rate remains unchanged due to the lack of virtualization, namely, each user is fixedly connected to the BS allocated to its subscribed MVNO. Fig. 2(b) suggests that the fraction of interference in the desired signal space decreases with the growth of number of MBSs, and feasible IA is obtained for almost all algorithms at the arbitrary number of MBSs. Compared to DSA + MaxSINR, DSA + Max-WLI algorithm gets the less interference leakage, which approximates 0 infinitely. Nevertheless, DSA + Max-SINR can be considered as with a feasible IA solution likewise, since its upper-bound Max-SINR w.o. virtualization reaches a value between 0.001 and 0.01. In Fig. 3, we compare the behaviour of all algorithms at different transmit SNR per stream at MBS (ranging from 10 dB to 80 dB). In this setting, there are 4 users, 2 SBSs, 6 MBSs, the transmit SNR per stream at SBS is 50 dB and all BSs are uniformly distributed in a 5 km × 5 km square. As Fig. 3(a) indicates, the sum rate almost linearly increases with the growth of transmit SNR. DSA + Max-SINR achieves a larger sum rate compared to DSA + Max-WLI, and can approach Max-SINR w. perfect CSI within a 5 % difference on average. Meanwhile, due to the lack of virtualization, although both Max-SINR w.o. virtualization and Max-WLI w.o. virtualization can obtain improved performance as the transmit power increases, the increasing rate is much smaller than that of the other four algorithms combined with virtualization. Fig. 3(b) shows that the fraction of interference in the desired signal space increases with the growth of transmit power. Except for Max-SINR w.o. virtualization (reaching a fraction of 0.25 approximately), all the other algorithms can take extremely small values and thus obtain feasible IA solutions.

In particular, DSA + Min-WLI, Min-WLI w. perfect CSI as well as Min-WLI w.o. virtualization can approach 0 infinitely with the performance remaining unchanged. In Fig. 4, we compare the performance of different algorithms as the number of SBSs (ranging from 1 to 10) increases. In this scenario, there are 6 users, 1 additional MBS, the transmit SNR per stream of MBSs and SBSs is 80 dB and 50 dB, respectively, and all BSs are uniformly distributed in a 200 m × 200 m square. By comparing Fig. 4(a) with Fig. 2(a), it is observed that the sum rate can be improved considerably due to the very close distance between SBSs and users. Similarly, the sum rate grows as the number of SBSs increases with a diminishing rate. In particular, the sum rate of DSA + MaxSINR algorithm is larger than that of DSA + Max-WLI, and get very close to that of Max-SINR w. perfect CSI within a 4% difference on average. Fig. 4(b) indicates that the interference leakage reduces with the increase of number of SBSs, and perfect IA is feasible for almost all algorithms at the arbitrary number of SBSs. Fig. 5 presents the behaviour of all algorithms at different transmit SNR per stream at SBS (ranging from 5 dB to 50 dB). In this setting, there are 4 users, 4 SBSs, 1 MBS, and the transmit SNR per stream for MBS is 80 dB. All BSs are uniformly distributed in a 200 m × 200 m square. By comparing Fig. 5(a) with Fig. 3(a), it can be seen that the introduction of SBSs can effectively improve the sum rate although SBSs are in general with relatively small transmit power. Fig. 5(b) indicates that the interference leakage in the desired signal space grows as the transmit power increases. Except for Max-SINR w.o. virtualization (reaching a value of 0.20 approximately at 50 dB), all the other algorithms can obtain feasible IA solutions.



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Fig. 3: (a) The sum rate of all MVNOs and (b) the fraction of interference in the desired signal space per user versus the transmit SNR per stream at MBS. (There are 4 users, 2 SBSs, 6 MBSs, the transmit SNR per stream for SBS is 50 dB and all BSs are uniformly distributed in a 5 km × 5 km square.)

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(b)

Fig. 4: (a) The sum rate of all MVNOs and (b) the fraction of interference in the desired signal space per user versus the number of SBSs. (There are 6 users, 1 MBS, the transmit SNR per stream for MBSs and SBSs is 80 dB and 50 dB, respectively, and all BSs are uniformly distributed in a 200 m × 200 m square.) C. Convergence of the Proposed Algorithms

distributed in a 200 m × 200 m square.

In this subsection, we investigate the convergence performance of proposed schemes under static and time-varying channels, respectively, provided that there are 6 users, 8 SBSs, 1 MBS, and the transmit SNR per stream for MBS and SBS is 80 dB and 50 dB, respectively. All BSs are uniformly

Fig. 6 shows the sum rate variation with iterations, assuming that the standard deviation of CSI noise is 5% of the expectation of CSI. The results corresponding to the aggressive DSA algorithm are averaged over 1000 drops. From Fig. 6, it can be observed that the performance of the aggressive DSA



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Fig. 5: (a) The sum rate of all MVNOs and (b) the fraction of interference in the desired signal space per user versus the transmit SNR per stream at SBS. (There are 4 users, 4 SBSs, 1 MBS, the transmit SNR per stream for MBS is 80 dB and all BSs are uniformly distributed in a 200 m × 200 m square.)

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Fig. 6: The sum rate of all MVNOs under static channels.

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Fig. 7: The sum rate of all MVNOs under time-varying channels with low CSI measurement noise.

algorithm gets closer to that of the scheme with perfect CSI with the growth of number of iterations.

VII. C ONCLUSION AND F UTURE W ORK

Fig. 7 illustrates the convergence performance under timevarying channels, with the assumption that the standard deviation of CSI noise is 5% of the expectation of CSI. The fading of channels is assumed to be block fading with a block size of 200. It can be observed that the adaptive step size DSA is always capable of tracking the optimal solution given by the upper bound.

In this paper, we studied heterogeneous cellular network virtualization combined with IA technology. First, we developed a novel framework that exploits the IA schemes to enable the virtualization of substrate networks. Then, in the proposed framework, we considered the virtual resource allocation as a joint virtualization and IA optimization problem. Different from existing works, the formulation incorporates the gain



not only from interference mitigation introduced by IA but also from the rate improvement brought by virtualization. In addition, due to the inaccurate channel estimation and measurement, imperfect CSI was taken into account. To tackle this high computational complexity optimization problem, we exploited a two-step scheme to solve the formulated problem, behind which the basic principle is to design IA schemes for each feasible association combination, then traverse the association space to seek for the optimal combination with the maximum sum rate. Extensive simulation results were presented to show that MVNOs can benefit from both IA and virtualization, and the proposed schemes can achieve near-optimal performance and good convergence. Future work is in progress to extend the IA over spacial dimension in MIMO networks to schemes with symbol extensions over time/frequency dimensions. Meanwhile, the extension of virtualization scheme to dynamic and dense cellular networks is also in progress. R EFERENCES [1] C. Liang and F. R. Yu, “Wireless network virtualization: A survey, some research issues and challenges,” IEEE Commun. Surveys Tutorials, vol. 17, no. 1, pp. 358–380, Firstquarter 2015. [2] C. Liang, F. R. Yu, and X. Zhang, “Information-centric network function virtualization over 5g mobile wireless networks,” IEEE Network, vol. 29, no. 3, pp. 68–74, May 2015. [3] R. Kokku, R. Mahindra, H. Zhang, and S. Rangarajan, “NVS: A substrate for virtualizing wireless resources in cellular networks,” ACM/IEEE Trans. Networking, vol. 20, no. 5, pp. 1333–1346, Oct. 2012. [4] Y. Cai, F. R. Yu, C. Liang, B. Sun, and Q. Yan, “Software defined device-to-device (d2d) communications in virtual wireless networks with imperfect network state information (nsi),” IEEE Trans. Veh. Tech., 2015, to appear, available on-line. [5] V. Cadambe and S. Jafar, “Interference alignment and degrees of freedom of the k -user interference channel,” IEEE Trans. Inform. Theory, vol. 54, no. 8, pp. 3425–3441, Aug. 2008. [6] K. Gomadam, V. Cadambe, and S. Jafar, “A distributed numerical approach to interference alignment and applications to wireless interference networks,” IEEE Trans. Inform. Theory, vol. 57, no. 6, pp. 3309– 3322, Jun. 2011. [7] V. Ntranos, M. Maddah-Ali, and G. Caire, “Cellular interference alignment,” IEEE Trans. Inform. Theory, vol. 61, no. 3, pp. 1194–1217, Mar. 2015. [8] J. Tang and S. Lambotharan, “Interference alignment techniques for MIMO multi-cell interfering broadcast channels,” IEEE Trans. Commun., vol. 61, no. 1, pp. 164–175, Jan. 2013. [9] X. Rao and V. K. N. Lau, “Interference alignment with partial CSI feedback in MIMO cellular networks,” IEEE Trans. Signal Process., vol. 62, no. 8, pp. 2100–2110, Apr. 2014. [10] H.-H. Lee, M.-J. Kim, and Y.-C. Ko, “Transceiver design based on interference alignment in MIMO interfering broadcast channels,” IEEE Trans. Wireless Commun., vol. 13, no. 11, pp. 6474–6483, Nov. 2014. [11] B. Zhuang, R. Berry, and M. Honig, “Interference alignment in MIMO cellular networks,” in Proc. IEEE Int. Conf. Commun. Acoust., Speech, & Signal Process. (ICASSP), May 2011, pp. 3356–3359. [12] T. Liu and C. Yang, “On the feasibility of linear interference alignment for MIMO interference broadcast channels with constant coefficients,” IEEE Trans. Signal Process., vol. 61, no. 9, pp. 2178–2191, May 2013. [13] M. Kamel, L. B. Le, and A. Girard, “LTE wireless network virtualization: Dynamic slicing via flexible scheduling,” in Proc. IEEE Veh. Tech. Conf. (VTC) - Spring, Sept. 2014, pp. 1–5.

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Kan Wang received the B.S. degree in broadcasting and television engineering from Zhejiang University of Media and Communications, Hangzhou, China, in 2009. He is currently working toward the Ph.D. degree in military communications with the State Key Lab of ISN, Xidian University, Xi’an, China. From Oct. 2014 to Oct. 2015, he was also with Carleton University, Ottawa, ON, Canada, as a visiting scholar funded by China Scholarship Council (CSC). His current research interests include 5G cellular networks, resource management, and interference alignment.

Hongyan Li (M’08) received the M.S. degree in control engineering from Xi’an Jiaotong University, Xi’an, China, in 1991 and the Ph.D. degree in signal and information processing from Xidian University, Xi’an, in 2000. She is currently a Professor with the State Key Laboratory of Integrated Service Networks, Xidian University. Her research interests include wireless networking, cognitive networks, integration of heterogeneous network, and mobile ad

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F. Richard Yu (S’00-M’04-SM’08) received the PhD degree in electrical engineering from the University of British Columbia (UBC) in 2003. From 2002 to 2006, he was with Ericsson (in Lund, Sweden) and a start-up in California, USA. He joined Carleton University in 2007, where he is currently an Associate Professor. He received the IEEE Outstanding Leadership Award in 2013, Carleton Research Achievement Award in 2012, the Ontario Early Researcher Award (formerly Premiers Research Excellence Award) in 2011, the Excellent Contribution Award at IEEE/IFIP TrustCom 2010, the Leadership Opportunity Fund Award from Canada Foundation of Innovation in 2009 and the Best Paper Awards at IEEE ICC 2014, Globecom 2012, IEEE/IFIP TrustCom 2009 and Int’l Conference on Networking 2005. His research interests include cross-layer/cross-system design, security, green IT and QoS provisioning in wireless-based systems. He serves on the editorial boards of several journals, including Co-Editorin-Chief for Ad Hoc & Sensor Wireless Networks, Lead Series Editor for IEEE Transactions on Vehicular Technology, IEEE Communications Surveys & Tutorials, EURASIP Journal on Wireless Communications Networking, Wiley Journal on Security and Communication Networks, and International Journal of Wireless Communications and Networking. He has served as the Technical Program Committee (TPC) Co-Chair of numerous conferences. Dr. Yu is a registered Professional Engineer in the province of Ontario, Canada.

Wenchao Wei is currently with Shanghai Aerospace Electronic Technology Institute, Shanghai, China. His research interests include satellite data system design, space data processing, and space-based telemetry & telecontrol technology.

Long Suo received the B.S. degree in Communications Engineering and the M.S. degree in Electronics and Communications Engineering from Xidian University, Xi’an, China, in 2010 and 2013, respectively. He is currently working toward the Ph.D. degree in Communications and Information Systems with the State Key Lab of ISN, Xidian University. His current research interests include interference management and interference alignment.

hoc networks.
