IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 1, JANUARY 2014

Energy-Efficient Opportunistic Interference Alignment Jangho Yoon, Student Member, IEEE, Won-Yong Shin, Member, IEEE, and Hwang Soo Lee, Member, IEEE Abstract—We introduce an energy-efficient distributed opportunistic interference alignment (OIA) scheme that greatly improves the sum-rates in multiple-cell uplink networks while reducing the transmit power consumption compared to the conventional OIA scheme. In the proposed scheme, each user employs optimal transmit vector design and power control strategy in the sense of minimizing the amount of generated interference to other-cell base stations while satisfying a given required signal quality. Our main result indicates that owing to the reduced interference level, the proposed OIA method attains larger sum-rates than those of OIA with no power control (i.e., with full transmit power) for almost all signal-to-noise ratio regions, thus resulting in improved energy efficiency. Index Terms—Energy efficiency, opportunistic interference alignment (OIA), power control, transmit vector design.

I. I NTRODUCTION

I

NTERFERENCE management is a crucial problem in communications. Over the past decade, there has been a great deal of research to characterize the asymptotic capacity of interference channels using the simple notion of degreesof-freedom (DoF), also known as multiplexing gain. Recently, interference alignment (IA) [1]–[5] was proposed as a novel approach to fundamentally solving the interference problem when there are multiple communication pairs. It was shown that the IA scheme in [1] can achieve the optimal DoF, which are equal to K/2, in the K-user interference channel with time-varying channel coefficients. The underlying idea of [1] has led to interference management schemes based on IA in various wireless network environments: multipleinput multiple-output (MIMO) interference networks [2], [3], X networks [4], and cellular networks [5]. In this work, we consider the interfering multiple-access channel (IMAC), i.e., the multiple-cell uplink network. Besides the IA scheme of [5], termed subspace IA, that achieves the optimal DoF for the network, the concept of opportunistic IA (OIA) [6] was recently introduced for the single-input multiple-output IMAC with time-invariant channel coefficients. The OIA scheme in [6] was shown to asymptotically achieve full DoF provided that the number of mobile stations (MSs) satisfies a certain scaling condition. In [7], the work [6] was extended to the MIMO IMAC model by employing two different types of pre-processing techniques: antenna

Manuscript received August 25, 2013. The associate editor coordinating the review of this letter and approving it for publication was D.-A. Toumpakaris. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2012R1A1A2004947) and the Ministry of Science, ICT & Future Planning (MSIP) (2012R1A1A1044151). J. Yoon and H. S. Lee are with the Department of Electrical Engineering, KAIST, Daejeon 305-701, Republic of Korea (e-mail: [email protected], [email protected]). H. S. Lee is the corresponding author. W.-Y. Shin is with the Department of Computer Science and Engineering, Dankook University, Yongin 448-701, Republic of Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/LCOMM.2013.111513.131938

selection-based OIA and singular value decomposition (SVD)based OIA. However, the studies in [6], [7] have not taken into account energy efficiency (or greenness), which is a growing issue in the telecommunication community. Especially for the IMAC model, due to the limited lifetime of mobile devices, the design of low-power network protocols and wireless transmission techniques is an important challenge. In this letter, based on the existing OIA framework, we propose a new energy-efficient OIA protocol for MIMO multiple-cell uplink networks, i.e., MIMO IMACs, which leads to significant performance enhancement on the sum-rates even with reduced transmit power consumption, resulting in higher energy efficiency compared to the conventional OIA scheme [7]. We first introduce a modified version of leakage of interference (LIF), which quantifies the energy efficiency as well as the induced interference. We then propose a joint optimal beamforming-power control strategy performed at each MS in the sense of minimizing the LIF while satisfying a given required signal quality in a distributed fashion, which does not involve joint processing among all communication links. In addition, we study two special cases for which the computational complexity required to find the optimal transmit vector is reduced significantly. To verify that our scheme outperforms conventional OIA with no power control (i.e., with full transmit power), computer simulations are performed—the achievable sum-rates are evaluated, and the energy-normalized throughput as our greenness measure is also shown. The main contributions of this letter are summarized as follows: Modification of the conventional LIF Design of a joint beamforming-power control algorithm • Case studies for the reduced computational complexity The rest of this letter is organized as follows. In Section II, we describe the system and channel models. In Section III, the proposed energy-efficient OIA protocol with transmit vector design and power control is presented. Numerical results of the OIA scheme are provided in Section IV. Finally, we summarize the paper with some concluding remarks in Section V. Throughout this paper, the superscript H denotes the conjugate transpose of a matrix, In is the identity matrix of size n×n, C is the field of complex numbers, E[·] is the expectation operator, and · is the L2 -norm of a vector. • •

II. S YSTEM AND C HANNEL M ODELS We consider the MIMO IMAC model [7] to describe realistic multiple antenna cellular networks. Suppose that there are K cells, each of which has N MSs. We also assume that each MS is equipped with L transmit antennas and that each cell is covered by one BS with M receive antennas. Under the model, each BS in a cell is interested only in traffic demands of users in its cell. Hkg,s ∈ CM×L indicates the channel matrix between BS

c 2013 IEEE 1089-7798/13$31.00

YOON et al.: ENERGY-EFFICIENT OPPORTUNISTIC INTERFERENCE ALIGNMENT

k and the sth MS in cell g.1 We assume a block fading channel model, where the channel is constant during a transmission block and changes independently between consecutive transmission blocks. The channel matrices are assumed to be Rayleigh, whose elements follow an independent complex Gaussian distribution C(0, 1). We assume that each selected MS transmits a single data stream at a time. Let Φg = {φg (1), · · · , φg (S)} denote the set of the MSs who are given the opportunity to transmit in the gth cell, where φg (s) ∈ {1, · · · , N }, φg (i) = φg (j) for i = j, and S ∈ {1, · · · , M } indicates the number of active MSs and is assumed to be the same for all cells. When S symbols are transmitted by the MSs belonging to Φg in the gth cell using transmit beamforming vectors {vg,φg (1) , · · · , vg,φg (S) } ∈ CL×1 , the received signal rg ∈ CM×1 at BS g is given by rg =

Hgg,s vg,s mg,s

s∈Φg

+

K k=1 k=g

Hgk,s vk,s mk,s + ng , (1)

s∈Φk

where mg,s ∈ C is the transmit message of MS s in cell g and ng ∈ CM×1 denotes the independent and identically distributed (i.i.d.) and circularly symmetric complex additive white Gaussian noise (AWGN) vector whose elements have zero mean and unit variance. We assume that each transmit beamforming vector is normalized to ||vg,s ||2 = 1 and that each MS has the transmit power constraint pg,s 2 E[|mg,s | ] ≤ η. It is assumed that each BS uses a simple zeroforcing (ZF) decoder, which is based on the received intra-cell channel links and that interference from other cells is treated as noise in the decoding process.

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the amount of misaligned or generated interference when it transmits a data stream with power pg,s , which is given by LIFg,s (pg,s , vg,s ) = pg,s vH (2) g,s Cg,s vg,s , K k where Cg,s = k=1,k=g Hkg,sH Wk WH k Hg,s , which indicates the sum of the covariance matrices of the generated interference. Note that the term pg,s in (2) can be determined in a distributed manner without any coordination between MSs. Under the MIMO IMAC model, unlike the conventional OIA scheme that performs the eigen-beamforming vector design at each MS in the sense of minimizing the LIF value in (2) with full transmit power [7], we consider a joint beamformingpower control strategy that can further reduce interference while maintaining a pre-defined desired signal quality. We now describe our distributed OIA protocol that comprises both transmit vector design and power control performed at each MS, which consists of the following four steps: • Step 1: Each BS randomly chooses and broadcasts an S-dimensional receive space with pilot signaling. • Step 2: Suppose that the required received power level of the desired signal, ρ, is known a priori at each MS. Each MS finds the transmit power and vector terms {pg,s , vg,s } such that its LIF be minimized based on its local channel state information in a distributed manner regardless the transmit power of the other MSs. The optimization problem at MS s belonging to cell g, which differs from the conventional one with no constraint (3b) [7], is given as follows: {ˆ pg,s , ˆvg,s } = argminpg,s ,vg,s LIFg,s (pg,s , vg,s )

(3a)

subject to III. E NERGY-E FFICIENT O PPORTUNISTIC I NTERFERENCE A LIGNMENT In this section, we describe the energy-efficient OIA protocol and specify a joint beamforming-power control algorithm. Two special cases with reduced computational complexity are also shown. A. Protocol Description We propose a distributed energy-efficient OIA method, which not only enhances the achievable sum-rates but also reduces the transmit power consumption compared to conventional OIA schemes under the MIMO IMAC model. Similar to [6], [7], the overall procedure of our protocol is based on the channel reciprocity of time-division multiplexing systems and each MS is assumed to be able to obtain accurate estimates of all received channel links via pilot signaling. Each BS broadcasts its receive subspace Wg = [wg (1), · · · , wg (S)], where wg (s) ∈ CM×1 is an orthonormal basis vector of Wg . Then each MS aligns its signal to the space orthogonal to the receive subspace of other cells (i.e., the interference subspace). However, the generated interference may not be exactly aligned for S < M or may not be nulled out for S = M due to the limited number of dimensions when L ≤ (K − 1)S. Thus, each MS computes the LIF representing 1 We assume that the pathloss gains from an MS to all BSs are the same as unity.

pg,s vH g,s Gg,s vg,s = ρ, 2

||vg,s || = 1, pg,s ≤ η,

(3b) (3c)

g Hgg,sH Wg WH g Hg,s .

where Gg,s = Note that given the constraint (3b), the received signal-to-noise ratio (SNR) at BS g equals ρ. • Step 3: The MSs who have a feasible solution set {ˆ pg,s , ˆvg,s } send the computed LIF to their cell BS. Note that there may exist some MSs who do not satisfy the two constraints (3b) and (3c) simultaneously. Those MSs do not send the LIFs. • Step 4: Each BS selects the MSs yielding the LIFs up to the Sth smallest one. Finally, the selected MSs in each cell send their packets using the optimized pre-processor (i.e., the transmit vector and power level). Each BS decodes the MSs’ signal using ZF filtering, while treating interference from other cells as noise. B. Transmit Vector Design/Power Control Algorithm In this subsection, we show how to find the optimal set {ˆvg,s , pˆg,s } at each MS. The following theorem establishes our main result. Theorem 1: For the MIMO IMAC, given the constraints (3b) and (3c), the LIF-minimized transmit vector ˆ vg,s at MS s in cell g is given by eM ((Cg,s + μg,s IL )−1 Gg,s ) if ζ(Gg,s ) ≥ ρ/η, ˆvg,s = infeasible otherwise,

32

IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 1, JANUARY 2014

where eM (·) denotes the eigenvector corresponding to the maximum eigenvalue of a matrix, indicates the maximum ζ(·) K k eigenvalue of a matrix, Cg,s = k=1,k=g Hkg,sH Wk WH k Hg,s , g Gg,s = Hgg,sH Wg WH g Hg,s , and μg,s is the Lagrange multiplier corresponding to the transmit power constraint (3c). Here, −1 −1 H ≤ η. μg,s = 0 if ρ eM (C−1 g,s Gg,s ) Gg,s eM (Cg,s Gg,s ) Otherwise, μg,s is given by the value that satisfies ρ eM ((Cg,s +μg,s IL

)−1 G

H −1 g,s ) Gg,s eM ((Cg,s +μg,s IL ) Gg,s)

= η.

(4) For the case where ζ(Gg,s ) ≥ ρ/η, the optimal transmit power pˆg,s is given by −1 H −1 if μg,s = 0, ρ(eM (C−1 g,s Gg,s ) Gg,s eM (Cg,s Gg,s )) pˆg,s = η otherwise. Proof: Refer to Appendix A. It is not straightforward to derive a closed-form expression for μg,s except when μg,s = 0 is feasible. Instead, the optimal vg,s can be found by a stepwise algorithm, where μg,s μg,s and ˆ gets increased by a small constant δμ > 0 iteratively when δμ > δth for a given threshold δth > 0 so that the solution of (4) with respect to μg,s be closely approximated, as described in Algorithm 1.

C. Discussion In this subsection, we consider the following two cases: S = 1 and (K − 1)S L. For these cases, we show that the computational complexity required to find the optimal transmit vector is reduced significantly compared to the general case. 1) S = 1: In this per-cell single-user transmission scenario, the optimal transmit vector can simply be found without full eigenvector decomposition. Due to the fact that the sum of all the eigenvalues of a square matrix is given by the trace of the matrix [9], when the rank is 1, the only non-zero eigenvalue is the same as the trace of the matrix. Hence, the term eM (·) in Theorem 1 becomes eM ((Cg,s +μg,s IL )−1Gg,s) = qn (Cg,s +μg,s IL −1/λg,sGg,s), where qn (·) is a normalized null vector of a matrix and λg,s = tr((Cg,s + μg,s IL )−1 Gg,s ).

30 OIA with power control, S=1 Conventional OIA [7], S=1 OIA with power control, S=2 Conventional OIA [7], S=2

25 Achievable Sum−rates [bps/Hz]

Algorithm 1 00: Initialize μg,s = 0, δµ > 0, and δth > 0 01: Compute Cg,s , Gg,s , and ζ(Gg,s ) 02: if ζ(Gg,s ) < ρ/η 03: Set LIFg,s (pg,s , vg,s ) to infinity. 04: else 05: Compute ug,s = eM (C−1 g,s Gg,s ). −1 ≤ η 06: if ρ(uH g,s Gg,s ug,s ) −1 ˆ 07: vg,s = ug,s , pˆg,s = ρ(uH g,s Gg,s ug,s ) 08: Compute LIFg,s (ˆ pg,s , ˆ vg,s ). 09: else 10: while (δµ > δth ) 11: μg,s = μg,s + δµ , ug,s = eM ((Cg,s + μg,s IL )−1 Gg,s ) −1 < η 12: if ρ(uH g,s Gg,s ug,s ) 13: μg,s = μg,s − δµ , δµ = δµ /2 14: end if 15: end while ˆ 16: vg,s = ug,s , pˆg,s = η 17: Compute LIFg,s (ˆ pg,s , ˆ vg,s ). 18: end if 19: end if

20

15

10

5

0

0

5

10

15

20 25 30 Received SNR [dB]

35

40

45

50

Fig. 1. The achievable sum-rates with respect to the SNR. The system with K = 3, M = 2, L = 2, and N = 100 is considered.

2) (K − 1)S L: In this case, since Cg,s follows a Wishart distribution W(IL , (K − 1)S), the inverse matrix of Cg,s is approximated by C−1 g,s 1/((K − 1)S − L − 1)IL + X [10], where X ∈ CL×L is a random matrix whose (i, j)th element has zero mean and variance [(2/((K − 1)S − L − 1) + 1 + δij ] 2 . = σij ((K −1)S −L)((K −1)S −L−1)((K −1)S −L−3) Here, δij denotes the Kronecker delta, which indicates that X is close to the zero matrix if (K − 1)S L. Then the optimal transmit vector asymptotically approaches ˆvg,s eM (Gg,s ) by setting μg,s = 0, which leads to the minimum transmit power. Note that in this case, no computation of the matrix Cg,s is required, thus resulting in remarkably reduced computational complexity. IV. N UMERICAL E VALUATION In this section, to verify the performance of our proposed OIA scheme, we perform computer simulations by evaluating the achievable throughput. For comparison, we also evaluate the performance of the SVD-based OIA algorithm operating with full transmit power, which is the best achievable scheme shown in [7]. The simulation parameters are K = 3, M = 2, and L = 2. The transmit power constraint, η, and the required received power level of the desired signal, ρ, are set to η = ρ = SNR under the unit noise variance assumption. As illustrated in Fig. 1, the sum-rates of both OIA schemes are evaluated for N = 100 according to the received SNRs (in dB scale). It is shown that even with a smaller transmit power, our OIA scheme outperforms the conventional one with no power control for almost all SNR regions. This comes from the fact that the MS selection is performed based on our new LIF metric, with which the power term enables each MS to further reduce generated interference. It is also seen that there is a crossover between the two sum-rate curves for S = 1 and 2, depending on the system parameters K, M , L, and N . In addition, it is worthy to show the energy-normalized throughput [8], which is defined as the ratio of the sum-rates to the total energy consumption, to compare the energy efficiency of the two OIA schemes and a distributed beamformingpower control scheme using iterative transmit-receive vector

YOON et al.: ENERGY-EFFICIENT OPPORTUNISTIC INTERFERENCE ALIGNMENT

fact that Cg,s is a positive-definite Hermitian matrix, the Lagrangian Ł(xg,s , λg,s , μg,s ) is convex [11]. Thus, the xg,s satisfying the Karush-Kuhn-Tucker (KKT) conditions for the above optimization problem is globally optimal, and the KKT conditions are given as follows:

0.2

Energy Normalized Throughput [bps/Hz/W]

0.18 0.16 0.14

OIA with power control, S=1 Conventional OIA [7], S=1 Scheme in [3], S=1 OIA with power control, S=2 Conventional OIA [7], S=2 Scheme in [3], S=2

∂Ł(xg,s ,λg,s ,μg,s) = (Cg,s +μg,s IL) xg,s −λg,sGg,sxg,s = 0L , (5a) ∂xg,s xH g,s Gg,s xg,s = ρ, (5b)

0.12 0.1 0.08

μg,s (xH g,s xg,s − η) = 0, μg,s ≥ 0, (5c)

0.06 0.04 0.02 0 1 10

33

2

3

10 10 Number of MSs per cell

4

10

Fig. 2. The energy-normalized throughput with respect to N . The system with K = 3, M = 2, L = 2, and SNR = 20dB is considered.

update and on-off control of beams [3]. In Fig. 2, the energynormalized throughput of the three given schemes is evaluated for SNR = 20dB for increasing N . The result indicates that as N increases, the energy-normalized throughput becomes larger owing to the reduced LIF, leading to increased multiuser diversity. It is obvious that the proposed scheme outperforms the conventional OIA and the scheme in [3] for all volumes of N . Another interesting observation is that with power control the energy-normalized throughput for S = 2 becomes higher than that for S = 1 when N increases beyond 5 × 102 . V. C ONCLUSION An energy-efficient OIA protocol with transmit power control was proposed for the MIMO IMAC, which can operate in a distributed manner without any additional feedback/feedforward overhead, compared to the conventional OIA scheme with no power control. More specifically, the optimal set of transmit vector and power was explicitly analyzed. It was shown that the proposed OIA method attains much higher energy-normalized throughput than the conventional OIA for given practical system parameters. The two cases S = 1 and (K − 1)S L that greatly reduce the computational complexity were also examined. A PPENDIX A. Proof of Theorem 1 First consider the existence of an infeasible solution. To satisfy the given constraint (3b), the transmit power pg,s at the sth MS in cell g should be at least ρ/ζ(Gg,s ). If ζ(Gg,s ) < ρ/η, then the two given constraints (3b) and (3c) cannot be satisfied simultaneously, which means that a feasible solution does not exist. Let us turn to the case where ζ(Gg,s ) ≥ ρ/η. Denoting √ xg,s = pg,s vg,s the transmit signal at the sth MS in cell g, the Lagrangian function Ł(xg,s , λg,s , μg,s ) can be written as H Ł (xg,s , λg,s , μg,s ) = xH g,s Cg,s xg,s + λg,s (−xg,s Gg,s xg,s + ρ)

+ μg,s (xH g,s xg,s − η), where λg,s and μg,s are the Lagrange multipliers corresponding to constraints (3b) and (3c), respectively. Due to the

where 0L ∈ CL×1 is the zero vector. The first KKT condition in (5a) can be transformed into an eigenvalue problem by multiplying by the term (Cg,s + μg,s IL )−1 . Then, it follows that ˆxg,s = pˆg,s eM ((Cg,s + μg,s IL )−1 Gg,s ), (6) where pˆg,s is a power scaling factor. The next step is to find pg,s . The second condition in (5b) is the same as (3b), which indicates the required received signal level of the desired signal at each BS. We now focus on the third condition in (5c), which yields two cases μg,s = 0 or ||xg,s ||2 = η. When vg,s = pˆg,s ˆ μg,s = 0, it follows that ˆxg,s = −1 −1 −1 −1 H ρ eM (Cg,s Gg,s ) Gg,s eM (Cg,s Gg,s ) eM (Cg,s Gg,s ) from ˆg,s = (5a) and (5b), where ˆvg,s = eM (C−1 g,s Gg,s ) and p −1 −1 −1 H ρ eM (Cg,s Gg,s ) Gg,s eM (Cg,s Gg,s ) , which are feasible if pˆg,s ≤ η. When μg,s = 0 is infeasible, we obtain pˆg,s = η using (5c). Then, using (6) in (5b) results in (4). Here, the left-hand side of (4) is continuous with respect to μg,s and approaches ρ/ζ(Gg,s ) (≤ η) as μg,s tends to infinity. Hence, there exists a positive μg,s ∈ (0, ∞) satisfying (4), which completes the proof. R EFERENCES [1] V. Cadambe and S. 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. [2] T. Gou and S. A. Jafar, “Degrees of freedom of the K user M × N MIMO interference channel,” IEEE Trans. Inf. Theory, vol. 56, no. 12, pp. 6040–6057, Dec. 2010. [3] Z. K. M. Ho, M. Kaynia, and D. Gesbert, “Distributed power control and beamforming on MIMO interference channels,” in Proc. 2010 European Wireless Conference. [4] 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. [5] C. Suh and D. Tse, “Interference alignment for cellular networks,” in Proc. 2008 Allerton Conf. Commun., Control, and Computing. [6] B. C. Jung, D. Park, and W.-Y. Shin, “Opportunistic interference mitigation achieves optimal degrees-of-freedom in wireless multi-cell uplink networks,” IEEE Trans. Commun., vol. 60, no. 7, pp. 1935–1944, July 2012. [7] H. J. Yang, W.-Y. Shin, B. C. Jung, and A. Paulraj, “Opportunistic interference alignment for MIMO interfering multiple-access channels,” IEEE Trans. Wireless Commun., vol. 12, no. 5, pp. 2180–2192, May 2013. [8] W.-Y. Shin, H. Yi, and V. Tarokh, “Energy-efficient base-station topologies for green cellular networks,” in Proc. 2013 IEEE Consumer Commun. Netw. Conf., pp. 91–96. [9] G. Strang, Linear Algebra and Its Applications. The Pearson Education Inc., 2005. [10] L. R. Haff, “An identity for the Wishart distribution with applications,” J. Multivariate Analysis, vol. 9, no. 4, pp. 531–544, Dec. 1979. [11] E. K. P. Chong and S. H. Zak, An Introduction to Optimization. John Wiley & Sons, Inc., 2008.