www.semargroup.org, www.ijsetr.com ISSN 2319-8885 Vol.03,Issue.17 August-2014, Pages:3542-3548 Implementation of MIMO Multi-Cell Broadcast Channels Based on Interference Alignment Techniques B.SANTHOSHA KUMARI 1, P.GANESH BABU 2 1 PG Scholar, Dept of CSP, Loyola Institute of Technology & Management, JNTUK, Andhrapradesh, India, E-mail: santhoshtenali@gmail.com. 2 Asst Prof, Dept of ECE, Loyola Institute of Technology & Management, JNTUK, Andhrapradesh, India, E-mail: ganeshbabu.pantangi@gmail.com. Abstract: The Interference Alignment (IA) is a promising technique to efficiently mitigate interference and to enhance capacity of a wireless communication network. This paper proposes an interference alignment scheme for a network with multiple cells and multiple multiple-input and multiple-output (MIMO) users under a Gaussian interference broadcast channel (IFBC) scenario. We first extend a grouping method already known in the literature to a multiple-cells scenario and jointly design transmits and receiver beam forming vectors using a closed form expression without iterative computation. Then we propose a new approach using the principle of multiple access channel (MAC) - broadcast channel (BC) duality to perform interference alignment while maximizing capacity of users in each cell. The algorithm in its dual form is solved using interior point methods. We show that the proposed approach outperforms the extension of the grouping method in terms of capacity and base station complexity. Finally, a rate balancing technique is introduced to maintain fairness among users. Keywords: MIMO, Cellular Network, Interference Alignment (IA), Beam Forming. I. INTRODUCTION Interference mitigation techniques have become an important part of wireless network design. An interference alignment (IA) technique has been proposed recently in [2] as an efficient capacity achieving scheme at high signal to-noise ratio (SNR) regime. The fundamental concept of interference alignment is to align the interference signals in a particular subspace at each receiver so that an interference free orthogonal subspace can be solely allocated for data transmission. Since the work of [2], interference alignment techniques have attracted significant research interests and various algorithms have been proposed and analyzed, for example, multiple-input multiple-output (MIMO) interference network [3] [5], X network [6] [8], and cellular network [9], [10]. A. Related Work Initially, the interference alignment has been proposed to achieve optimal degrees of freedom (DoF) in a single-input single-output (SISO) interference channel (IC) [2]. It was shown that interference alignment can achieve the optimal DoF of K/2, in a K-user time varying interference channel. In [3], the authors provided examples of iterative algorithms that utilize the reciprocity of wireless networks to achieve interference alignment with only local channel knowledge at each node. The work in [11] proposed the alignment of multiuser interference at each receiver based on a carefully constructed signal structure, which was referred to as interference alignment in signal space. For the interference alignment in signal space, transmit pre-coding technique is used to align the multi-user interferences in the same interference space which is orthogonal to the desired signal space at each receiver. Furthermore, the authors in [4] provided an inner bound and an outer-bound for the total number of degrees of freedom for the K user MIMO Gaussian time varying interference channels with M antennas at each transmitter and N antennas at each receiver. For the case of K user M N MIMO interference channel, authors in [4] showed that the total number of degrees of freedom is equal to min(m,n)k if K R and min(m,n) R/R+1K if K > R, where R = max(m,n)/min(m,n). An interference alignment scheme was provided in [12] for the deterministic channel model of the K user interference channel. Multi-cell and multi-user downlink transmission schemes have been actively discussed for future generation cellular networks. The idea is to maximize the network capacity by efficiently mitigating interference. Authors in [9] proposed an interference alignment based scheme for cellular networks, namely subspace interference alignment. This is based on aligning interferences onto multi-dimensional subspace (instead of one dimension) for simultaneous alignments at multiple non-intended base stations (BS). In the multi-cell MIMO Gaussian interfering broadcast channels (MIMO-IFBC), each BS supports multiple users within its cell. Therefore there exist two kinds of interference namely Copyright @ 2014 SEMAR GROUPS TECHNICAL SOCIETY. All rights reserved.
inter-user interference (IUI) and inter-cell interference (ICI). To mitigate both IUI and ICI, authors proposed a zeroforcing (ZF) scheme for the IFBC with the aim of maximizing the sum rate performance in a multiple-input single-output (MISO) scenario. In, the ZF scheme for the MIMO-IFBC was extended to the case of multiple receiver antennas. Authors provided a precise expression of the spatial multiplexing gain for two mutually interfering MIMO broadcast channels using linear transceiver. Authors developed an interference alignment technique for a downlink cellular system which requires feedback only within each cell. The scheme provided substantial gain especially when the interference from a dominant interferer is significantly stronger than the remaining interference. Furthermore, for a two-cell MIMO-IFBC, the authors proposed a novel interference alignment technique for jointly designing the transmitter and receiver beam forming vectors using a closed-form expression without a need for iterative computation. It was shown both analytically and numerically that the proposed scheme achieves the optimal DoF. While characterizing performance in terms of degrees of freedom is appropriate at high SNR, for moderate SNR region, consideration should be given for maximization of capacity explicitly. The capacity optimization for a single-user MIMO Gaussian channel was first studied. Later, this work was extended to multiple users. One of the important pre-coding techniques for BCs is the dirty paper coding (DPC). The principle of DPC has been used to determine the achievable capacity region of MIMO-BC channels. The work in proved that the capacity region coincides with the DPC rate region. It was shown in that the sum rate MIMO-BC capacity is equal to the maximum sum rate of the achievable region of the BC channels for the case of two users. This result is known as MAC-BC duality, i.e., the achievable capacity region of the BC is the same as that of the dual MAC, and vice versa. In, this duality was extended to multiuser MIMO Gaussian BC channels. B. Main Contributions In this paper, we propose an interference alignment algorithm for a multi-cell multi-user MIMO-IFBC based on a hybrid interference alignment and MAC-BC duality based beam former design. We compare the proposed interference alignment scheme with an extension of a grouping scheme proposed. The extension of consists of two parts: the receiver beam forming vector design for effective ICI channel alignment, and the transmit beam forming vector design for removing ICI and IUI. Specifically, all the users in the same cell cooperatively construct the receiver beam forming vectors so as to align the effective ICI channel. Based on this ICI channel alignment, each BS is able to treat these ICI channels as one ICI channel. Therefore, each BS can remove ICI and IUI by making the transmit beam forming vectors orthogonal to the subspace spanned by the effective ICI channel and IUI channel vectors. Hence, the cost in terms of number of transmitting antennas is reduced by employing the extension of grouping algorithm. B.SANTHOSHA KUMARI, P.GANESH BABU The proposed interference alignment scheme uses the principle of MAC-BC duality to maximize capacity of each cell while performing interference alignment to users in other cells. The problem in its dual form is convex and can be solved using interior point methods. Since our method performs interference alignment only to users in other cells, and inter user interference in each cell is dealt with using a capacity maximization problem, the number of antennas required at each BS is smaller than if interference alignment is performed for every users, including users within each cell for a moderate range of SNR. Also, the simulation results demonstrate that the achievable capacity is even higher than that of the extension of the grouping method when multiple data streams are employed. The work is finally extended to include rate balancing to maintain fairness among users. TABLE I: Abbreviations Used In This Paper C. Organization and Notation The remainder of this paper is organized as follows Interference Alignment for MIMO Interfering described in Section II. In Section III, Preliminaries IA in MIMO in Section IV, Simulation results are provided in Section V and conclusions are drawn in Section V. The following notations are used in the paper. Bold upper and lower case letters denote matrices and vectors, respectively; (.) 1 denote the matrix inversion, (.) T denote the matrix transpose, (.) H denote the matrix conjugate transpose, I N t denotes an N t N t identity matrix; E [.] denotes the expectation operator; t r (.) denotes the trace of a matrix, and[x]+ denotes max(x, 0); (.) b and (.) m denotes the quantities associated with a broadcast channel and a multiple access channel, respectively. TableI lists some of the abbreviations used in the paper. II. INTERFERENCE ALIGNMENT FOR MIMO INTERFERING A. Interference Alignment Interference channels, where multiple transmit and receive user pairs communicate using the same radio resources, are a building block of wireless networks. The interference channel is a good model for communication in cellular networks, wireless local area networks, and ad-hoc networks. Conventional thinking about the interference channel is that each user pair has no information about other users in the network and therefore its optimum strategy is to be greedy and maximize its own rate. Unfortunately, the sum of the data rates achieved across all user pairs with this strategy is of the same order as the rate of a single communication link. Recent work on the interference
Implementation of MIMO Multi-Cell Broadcast Channels Based On Interference Alignment Techniques channel, however, has shown that sum rates can scale linearly with the number of users at high SNR, using a transmission strategy known as interference alignment. Interference alignment is a linear pre-coding technique that attempts to align interfering signals in time, frequency, or space. In MIMO networks, interference alignment uses the spatial dimension offered by multiple antennas for alignment. The key idea is that users coordinate their transmissions, using linear pre-coding, such that the interference signal lies in a reduced dimensional subspace at each receiver. Allowing some coordination between transmit and receive user pairs enables interference alignment. In this way, it is possible to design the transmit strategies such that the interference aligns at each receiver. From a sum rate perspective, with K user pairs, an interference alignment strategy achieves a sum throughput on the order of K/2 interference free links! Basically each user can effectively get half the system capacity. Thus unlike the conventional interference channel, there is a net sum capacity increase with the number of active user pairs. This result has special importance in cellular and ad hoc networks, showing that coordination between users can help overcome the limiting effects of interference generated by simultaneous transmission. We have been studying several aspects of interference alignment at UT, with an emphasis on its practice. Our areas of interest include algorithms for computing interference alignment solutions and more general pre-coding strategies for the interference channel, interference alignment performance in measured channels, clustering to reduce overhead in interference alignment, and analysis of interference alignment in the presence of channel estimation error. Interference management is one of the most challenging issues to improve a cell throughput in cellular networks. It was shown that the interference alignment (IA) technique achieves the optimal degrees-of-freedom (DoF) in the K-user interference channel with time-varying channel coefficients. Subsequent works have shown that the IA is also useful for other wireless networks including multiple-input multipleoutput (MIMO) interference channels and cellular networks. On the other hand, there have been some notable techniques that exploit the benefit of fading in a single cell network, obtaining multiuser diversity (MUD) gain: opportunistic scheduling, opportunistic beam forming, and random beam forming. Moreover, scenarios with achievable MUD gain have been studied in ad hoc networks, cognitive radio networks, and multi-cell downlink and uplink networks. Recently, an opportunistic interference alignment (OIA) concept which combines the IA and user scheduling was proposed for interfering multiple access channels (IMAC). OIA has been known to achieve the optimal DoF in IMAC if a certain user scaling condition is satisfied even though it operates in a distributed fashion. achieved if N = ω (SNRKM 1)1, where N, K, and M denote the number of users in a cell, total number of cells in the network, and number of transmit antennas at each BS, respectively. The authors extended the random beam forming technique, originally proposed for a single cell network into a multi-cell downlink assuming a single antenna at users. The authors obtained the same user scaling law as in the same network by using the same technique but using different derivations. In, the authors also considered the effect of multiple antennas at users on the required user scaling for the optimal DoF, i.e., N = ω (SNRKM L) where L denotes the number of receive antennas at users. In, the user scaling for given DoF in a 3-cell single-input multi-output (SIMO) downlink network is derived. In the same work, a general K- cell downlink network and multiple antennas at BSs are also taken into account. The user scaling is the same as, since all of these previous works are based on the multi-cell random beam forming technique addition, the interference decaying rate with respect to N for given SNR is characterized in conjunction with the derived user scaling law. Furthermore, simulation results show that the proposed OIA significantly outperforms the previous schemes even in practical environments. III. PRELIMINARIES IA IN MIMO In this section, we review MIMO s DoF resources for spatial multiplexing (SM) and interference cancellation (IC). We also review how IA can help reduce the number of DoFs required for IC. Table II lists the notation used in this paper. MIMO s DoF Resources for SM and IC. The number of DoFs of a node is typically assumed to be the same as the number of antennas at the node and represents the total available resources at the node for SM and IC. SM refers to the use of one or multiple DoFs (both at transmitting and receiving nodes) for data transport, with each DoF corresponding to one independent data stream. IC refers to the use of one or more DoFs to cancel interference from other nodes, with each DoF being responsible for cancelling one interfering stream. IC can be done either at a transmit node (to cancel interference to another node) or a receive node (to cancel interference from another node). For example, consider two links in Fig. 1. To transmit z 1 data streams on link (T 1, R 1 ), both nodes T 1 and R 1 need to consume z 1 DoFs for SM. Similarly, to transmit z 2 data streams on link (T 2, R 2 ); both nodes T 2 and R 2 need to consume z 2 DoFs for SM. The interference from T 2 to R 1 can be cancelled by either R 1 or T 2. If R 1 cancels this interference, it needs to consume z 2 DoFs. If T 2 cancels this interference, it needs to consume z 1 DoFs IA in MIMO. For multi-cell downlink networks, so called interfering broadcast channel (IBC), similar techniques were also proposed, it was shown that the optimal DoF of KM can be Fig.1. SM and IC in MIMO.
TABLE II: Notation B.SANTHOSHA KUMARI, P.GANESH BABU streams on link (T 2 ; R 2 ), and 1 data stream on link (T 3 ; R 3 ). Denote u k i as the transmit vector for the k th data stream s k i on link (T i ; R i ) and H j i as the channel matrix between T i and R j. Fig.2. An illustration of IA at node R 4. When IA is not employed, R 4 needs to consume 5 DoFs to cancel the interference from transmitters T 1, T 2, and T 3. Since there are only 3 DoFs available at R 4, it is not possible to cancel all 5 interfering streams, let alone to receive any data stream from T 4. But when IA is used (see Fig. 2), we can align the 5 interfering data streams into 2 dimensions, which can be cancelled by R 4 with 2 DoFs. Therefore, R 4 still has 1 DoF remaining, allowing it to receive 1 data stream from T 4. IV.SIMULATION RESULTS In this section, we compare the performance of the proposed hybrid interference alignment and MAC-BC duality based scheme with the extension of the grouping method in terms of the sum rate. We focus on two system configurations (N t, N r, K, L, d s ) = (5, 3, 2, 3, 1) and (N t, N r, K, L, d s ) = (10, 6, 2, 3, 2). For these configurations, the proposed scheme is able to achieve 6 degrees of freedom and 12 degrees of freedom, respectively. To be specific, to achieve a degrees of freedom of 6, the extension of the grouping scheme requires N t = 5 antennas for the example of IA in MIMO in the context of MIMO, IA refers to the construction of transmit data streams so that (i) they overlap at receivers where they are considered as interfering streams and (ii) they are resolvable at their intended receivers (not to be overlapped by either interfering streams or other data streams). The construction of transmit data streams is equivalent to the design of transmit vector (weights) for each data stream at each transmitter. Since the interfering streams are overlapped at a receiver, one can use fewer numbers of DoFs to cancel these interfering streams. As a result, the DoF resources consumed for IC will be reduced and thus more DoF resources become available for data transport. We use the following example to illustrate the benefits of IA in MIMO networks. Consider the 4-link network shown in Fig. 2. A solid line with arrow represents directed link while a dashed line with arrow represents directed interference. Assume that each node is equipped with three antennas. Suppose that there are 2 data streams on link (T 1 ; R1), 2 data TABLE III: Proposed Interference Alignment Scheme Using Mac-Bc Duality and Data Rate Balancing
Implementation of MIMO Multi-Cell Broadcast Channels Based On Interference Alignment Techniques (N t, N r, K, L, d s ) = (5, 3, 2, 3, 1), but the conventional zeroforcing beam forming scheme requires N t = 6 antennas for the case of (N t, N r, K, L, d s ) = (6, 3, 2, 3, 1). Moreover, for a general case with K users, L cells, and d s data stream per user, the required minimum number of transmit antennas N t for the proposed hybrid interference alignment and MAC-BC duality based scheme is [K(L 2)+2] d s. Therefore, the minimum number of antennas for each BS employing the proposed hybrid interference alignment and MAC-BC duality based scheme is 4 and 8. Let us explain this in more detail. For the case that (N t, N r, K, L, d s ) = (4, 3, 2, 3, 1), and as an example, when cell l is considered, as we employ 4 antennas at BS l, the base matrix T l for cell l as shown in a column vector, e.g., T l = [t 1,l]. Thus the duality optimization problem for this case is a pure power allocation problem. The total power constraint P l is varied from 1 to 100. The noise variances at the users have been set to unity. First we consider the case that only a single data stream is transmitted for each user. We have three cells, each containing two users with three receiver antennas. Fig. 3 depicts the sum rate versus SNR of the proposed algorithm and compares it with the grouping method. As seen, the proposed interference alignment scheme using the MAC-BC duality with five transmit antennas outperforms the extension of the grouping method with five transmit antennas. We then considered the same system but with two data streams transmitted for each user. For the proposed interference alignment scheme using MAC-BC duality, we considered both 8 and 10 transmit antennas for each base station. Fig. 4 depicts the sum rate versus SNR of the proposed algorithm and compares it with the grouping method but with 10 transmit antennas. As seen, the proposed interference alignment scheme, even with 8 transmit antennas, outperforms the extension of the grouping method with 10 transmit antennas. The reason is that although only eight antennas are employed at the BS, the sum rate is maximized using the virtual beam forming matrices Q m [k,l], hence it outperforms the perfect interference alignment algorithm due to increase subspace dimension for mitigating intra-cell interference. Fig.3. The achievable rates for the proposed hybrid interference alignment scheme and comparison to the extension of the grouping method (DoF = 6). Finally, we show the proposed interference alignment algorithm can also balance all the users data rate in each cell. All the elements of the data rate balancing vector ρ were set to one. Fig. 5 and Fig. 6 depict the convergence of the data rate for the two users against the adaptation of the Lagrangian multiplier μ k as explained. All the users attain equal data rate. The data rate without rate balancing constraints is also shown. The total sum rate in this case is 7.5 bits/s/hz and 14.6 bits/s/hz for DoF = 6 and DoF = 12 respectively. With the rate balancing constraints, Fig.4. The achievable rates for the proposed hybrid interference alignment scheme with different number of antennas setting (DoF = 12). Fig.5. Rate balancing of the proposed hybrid interference alignment algorithm using the MAC-BC duality (DoF = 6).
each user attains 3.05 bits/s/hz and 6 bits/s/hz for DoF = 6 and DoF = 12 respectively. Hence the total rate is less than that of the scheme that does not use rate balancing constraint. However, the rate balancing constraint ensures fairness among users. Finally, we provide the probability density function for the number of iterations required for the proposed interference alignment algorithm using the MAC- BC duality. As shown in Fig.7, the algorithm converges with 21 iterations most of the time. Fig.6. Rate balancing of the proposed hybrid interference alignment algorithm using the MAC-BC duality (DoF = 12). Fig.7. Probability density functions for the number of iterations of the proposed hybrid interference alignment algorithm using the MAC-BC duality. V. CONCLUSION We proposed an interference alignment scheme for a network with multiple cells and MIMO users under a Gaussian interference broadcast channel scenario. We first extended the grouping method in the multi-cell scenario to jointly design the transmitter and receiver beam forming vectors using a closed form expression without a need for iterative computation. The grouping method can ensure no B.SANTHOSHA KUMARI, P.GANESH BABU ICI and IUI at each user s receiver while reducing both the number of antennas and the complexity at the base station as compared to the conventional zero-forcing beam forming scheme. Then we proposed a hybrid interference alignment scheme based on the principle of MACBC duality. This proposed scheme removes the ICI using interference alignment while maximizes the total capacity of the corresponding cell using MAC-BC duality. Since, interference alignment is not perform explicitly to all users in the network, but the users within each cell are dealt with using capacity maximization, the number of transmit antennas required is generally lower than the existing grouping method. Finally, a hybrid rate balancing and interference alignment technique was introduced to maintain fairness among users. Hence the proposed technique is able to maximize the data rate while balancing the rate achieved by each user. The simulation results demonstrated the performance of the algorithms for various SNR values. VI. REFERENCES [1] Jie Tang, Student Member, IEEE, and Sangarapillai Lambotharan, Senior Member, IEEE, Interference Alignment Techniques for MIMO Multi-Cell Interfering Broadcast Channels, IEEE Transactions on Communications, Vol. 61, No. 1, January 2013. [2] V. R. Cadambe and S. A. Jafar, Interference alignment and degrees of freedom of the K-user interference channel, IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3425 3441, Aug. 2008. [3] K. Go madam, V. R. Cadambe, and S. A. Jafar, Approaching the capacity of wireless networks through distributed interference alignment, preprint. Available: http://arxiv.org/abs/0803.3816. [4] T. Gou and S. A. Jafar, Degrees of freedom of the K- user M NMIMO interference channel, IEEE Trans. Inf. Theory, vol. 56, no. 12, pp. 6040 6057, Dec. 2010. [5] C. M. Yetis, T. Gou, S. A. Jafar, and A. H. Kayran, On feasibility of interference alignment in MIMO interference networks, IEEE Trans. Inf. Theory, vol. 58, no. 9, pp. 4771 4782, Sep. 2010. [6] S. A. Jafar and S. Shamai (Shitz), Degrees of freedom region of the MIMO X channel, IEEE Trans. Inf. Theory, vol. 54, no. 1, pp. 151 170, Jan. 2008. [7] M. Maddah-Ali, A. Motahari, and A. Khandani, Communication over MIMO X channel: signaling and performance analysis, Univ. California, Berkeley, CA, Tech. Rep. 2007. [8] Communication over MIMO X channels: interference alignment, decomposition, and performance analysis, IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3457 3470, Aug. 2008. [9] C. Suh and D. Tse, Interference alignment for cellular networks, in Proc. 2008 Annual Allerton Conf. Commun., Control, Comput. [10] V. Nagarajan and B. Ramamurthi, Distributed cooperative pre-coder selection for interference alignment, IEEE Trans. Veh. Technol., vol. 59, no. 9, pp. 4368 4376, Nov. 2010. [11] R. Etkin and E. Ordentlich, On the degrees-of-freedom of the Kuser Gaussian interference channel is discontinuous
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