New Uplink Opportunistic Interference Alignment: An Active Alignment Approach

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1 New Uplink Opportunistic Interference Alignment: An Active Alignment Approach Hui Gao, Johann Leithon, Chau Yuen, and Himal A. Suraweera Singapore University of Technology and Design, Dover Drive, Singapore Department of Electrical & Computer Engineering, National University of Singapore, Engineering Drive, Singapore, {hui_gao, yuenchau, Abstract In this paper, we propose two new opportunistic interference alignment OIA schemes to mitigate interference in the K-cell uplink interference channel. Unlike the existing schemes that basically rely on the channel randomness to achieve asymptotical IA for all cells, the proposed schemes employ an active alignment approach to increase the possibility for perfect partial IA for one cell within the network. Specifically, each user adopts the active alignment transmit beamforg such that the user-generated interference is perfectly aligned along the preferred reference interference direction at one of the basestations BS in other cells. With this approach, our schemes increase the chance to obtain degrees-of-freedom gain even with a small number of users. In addition, several new user selection schemes are proposed, and the BS receiver is optimized to enhance performance. Extensive simulations are conducted, and the results show that the proposed schemes outperform the state of the art under the considered scenarios. I. INTRODUCTION Interference alignment IA has been widely considered as a potent solution to improve the degrees-of-freedom DoF performance in typical interference channels ICs [] []. The principle of IA is to confine all interferences to a subspace with imal dimensions at the receiver so that the available dimensions for the intended signals can be maximized to achieve high capacity. However, the promise granted by the traditional IA is based on the assumption of global channel state information CSI at all nodes in the network, which requires significant system overhead for CSI. In order to reduce the CSI overhead, limited-feedback IA and distributed IA schemes have been proposed in [] [] and [], respectively. Nevertheless, the size or power of the feedback signal needs to scale fast to achieve DoF gain with limited-feedback IA [] [] while complicated iterative computations are required with distributed IA []. With the aim for a more practical implementation with low complexity, opportunistic IA OIA has been proposed for the time-division duplexing TDD downlink [] [] and uplink [] [] K-cell interference channel. The key idea of OIA is to harvest the multiuser diversity to construct asymptotically perfect IA when the number of users is large, without explicit CSI feedback and iterative computation. Based on local CSI and random for downlink or predefined for uplink reference interference direction, each user calculates and reports the performance-relevant metric to facilitate user selection within the network. For the downlink OIA, [], [] proposed a chordal distance based user selection with random base-station BS beamforg. This scheme is further improved in [] by taking the useful signal into account. The downlink OIA schemes proposed in the literature cannot be directly extended to the uplink case. When each user is only equipped with one antenna [], [], uplink OIA solely relies on the channel randomness and system requires a large number of users to achieve an acceptable performance. Such constraints may limit the practical implementation of OIA. It has been recently shown that deploying the multiple antennas at the user side is beneficial for uplink OIA, since transmit beamforg TB can improve system performance []. Inspired by this observation, [], [] proposed a Joint TB and user selection to Minimize the Sum of the Leakage Interference JMinSLI, and [] proposed a Joint TB and user selection to Maximize the Signal to Generated Interference and Noise Ratio JMaxSGINR. It is shown that [] [] significantly improve the uplink OIA performance as compared with [], []. However, [] [] may still suffer from two potential shortcogs: the local optimizations do not guarantee interference-free conditions at any BS in the network, which indicates that the network DoF gain is zero with limited number of users; the simple zero-forcing ZF BS receiver is sub-optimal due to imperfect alignment. In this paper, we propose two new uplink OIA schemes, which partially overcome the limitations of the existing work in the current literature. In particular, we propose an active alignment approach to facilitate OIA with integrated solutions for user TB, user selection and BS receiver. In contrast to [] [] that achieve asymptotical overall IA in a pure opportunistic fashion when the number of users scales, the proposed OIA is featured by active alignment TB which achieves partial IA with nonzero probability. Specially, each user adopts a TB such that the generated interference is perfectly aligned with the reference interference direction of one BS in other cells. Such a TB approach shows the user s intention for partial IA, therefore, it is termed as active alignment to distinguish from the pure opportunistic approach in [] []. We consider two TB methods, namely the Cooperative Active-alignment TB CATB and the Distributed Active-alignment TB DATB. The CATB assumes very simple network coordination so

2 that all users in the network are able to cast overlapping interference shadows to one BS in the network. Therefore, the interference-free BS achieves DoF gain for its cell as well as for the network. The DATB, on the other hand, adopts local optimizations with TB selection. Due to active-alignment TB, the network has nonzero probability to enjoy interferencefree BS as well as the network DoF gain. Moreover, we propose several TB and user selection criteria. Specifically, we consider the angle-based and the strength-based user TB selections, which characterize the degree of alignment and the strength of signals, respectively. Finally, we propose BS receiver to maximize the signal-to-interference plus noise ratio SINR. Extensive simulations are conducted, which show that the proposed schemes outperform the existing ones [] [] especially when the number of interfering cells is large. II. SYSTEM MODEL AND EXISTING SCHEMES A. System Model We consider the K-cell TDD uplink interference network as shown in Fig.. In the i-th cell, i [, K], there are one N R -antenna BS BS i and N users {UE } N i= with N T antennas each. The reciprocal channel between UE and BS i is denoted as H [k] CN R N T, k [, K], where the entries of H [k] are independent and identically distributed i.i.d. as CN,. Following [] [], we assume that BS i knows the CSI of the K selected users from all K cells, and each user has the CSI to all the K BSs. In order to exploit the multiuser diversity, it is assumed that only one user is selected per cell for a single-stream uplink transmission. We denote the index of the selected user in the i-th cell as n i while the schemes to detere n i will be presented later. In addition, each BS has predefined an normalized interference space direction q [i] C NR and an signal space direction s [i] C NR to facilitate uplink OIA where q [i] s [i] =. It is assumed that { q [i]} K can be i detered offline and is known to all the users in this network. The original idea of uplink OIA is to choose the users which generate imum interference to other cells while the selection criteria is inspired by the IA principle and relevant to { q [i]} K [], []. Moreover, when N i T, each user can use TB w C N T, w =, to enhance performance, and each BS uses a linear receiver r [i] C NR, r [i] =, to detect the signal. Similar to [] [], the antenna configuration is assumed as N R N T < K, therefore, neither the BS nor the user is able to fully cancel the interference. Notation: A, A and [A] i,j denote the conjugate transpose, inverse of matrix and the i, j th entry of A. λ n A, B, u n A, B denote the n-th largest generalized eigenvalue and the generalized eigenvector corresponding to λ n A, B. σ n A and v n A denote the n-th largest singular value and the right-singular vector corresponding to σ n A, respectively. a is the Euclidean-norm of a vector a. a = a denotes normalization operation a on vector a. a, b = cos a b denotes the acute angle between a b vectors a and b. C represents the set of complex numbers. CN µ, ε denotes a complex Gaussian variable with mean µ and variance ε. E { } is the expectation operator. The integer set {,,...K} is abbreviated as [, K]. BS UE n * UE, UE, N, Useful signal BS K UE n * K UE K U, UE K N Interference Fig.. K-cell uplink interference network, where each cell has one N R - antenna BS and N users with N T antennas each. For the i-th cell, i [, K], a user UE n i is selected for uplink transmission. Regarding uplink transmission, BS i receives interfering signals when detecting the target symbol x n i as P K x n i = r [i] h [i] n k x n k + n [i], k= where x n i is the decision variable of x n i, x n k is the Gaussian information { source symbol of UE n k with an normalized power E x n } k =, n [i] C NR is the Gaussian noise vector with elements following i.i.d. CN, σ, and P is the transmit power, h [i] H [i] n k w n k n k = C NR, is the effective channel of UE seen by BS i. Based on, we define the SINR of BS i as ρ r [i] h [i] γ [i] n = n i h[i] n i r [i] r [i] ρ, K k=,k i h[i] n k h[i] n k + I N R r [i] where ρ = P σ is transmit SNR, and the instantaneous rate of BS i is given as R [i] n = log + γ [i] n. Based on, we define the network DoF gain as DoF n = K i= DoF [k] n = K lim ρ i= R [i] n ρ log ρ, which indicates that the DoF of the network is nonzero as long as one BS achieves DoF gain, i.e., DoF n >, if DoF [i] n >, i [, K]. It is noted n in to represents a user selection scheme. In addition, the design for TB w and BS receiver r [i] will also influence system performance. In the following subsection, we will briefly review the relevant schemes with emphasis on their designs for w, n and r [i]. B. Review on Existing Schemes The work in [] [] jointly optimize TB and choose user with JMinSLI and JMax-SGINR. In addition, schemes in [] [] adopt the ZF regarding q [i] receiver r [i] = s [i].

3 JMinSLI in [], [] define the SLI of UE as L w = K k=,k i s [k] H [k] w. A singular value decomposition SVD based approach [] is proposed to imize L w and the optimal w is w SVD = u NT G, [ ] T where G = g [],..., g[i ], g[i+] [k]..., g[k] and g := T s [k] H [k] C N T. The imum value of η is given by η = σ N T G i,j and the optimal user in the i-th cell is selected according to n JMinSLI i = arg η, where n JMinSLI represents user selection with JMinSLI. JMaxSGINR in [] defines the SGINR of UE as µ w = w A w w B w, where A = ρ H [i] s[i] s [i] H [k] and B = ρ K k=,k i H[k] s[k] s [k] H [k] + I N R. According to Rayleigh- Ritz theorem [] the optimal w is solved by the generalized eigenvalue decomposition GEVD approach as w GEVD = v A, B, and the maximum value of µ is given as µ = λ A, B. Based on µ, the optimal user in the i-th cell is selected as n JMaxSGINR i = arg µ. Although JMinSLI and JMax-SGINR outperform the original uplink OIA with single-antenna users [], [], they basically rely on the randomness of channel to achieve asymptotical IA when the number of user scales. However, with a limited and fixed number of users, the network DoF gain is not achievable [], [] due to the imperfect alignment of OIA, i.e., DoF [i] n =, i [, K]. Therefore, the system throughput will get saturated in the high SNR regime. In addition, JMinSLI and JMax-SGINR assume the ZF receiver s [i] for simplicity. However, the ZF receiver may not be optimal due to the imperfect alignment of OIA as well. III. NEW UPLINK OIA WITH ACTIVE ALIGNMENT In this section, we present new uplink OIA schemes with active alignment. The new designs aim to improve the possibility to achieve the network DoF gain. In addition, we propose a BS receiver which optimizes receive SINR. A. Transmit Beamforg Design with Active Alignment In order to increase the chance of obtaining a network DoF gain, we will design the TB with active alignment without assumption of a large number of users. In particular, each user in the i-th cell adopts the following TB w A k = H [k] H [k] H[k] q [k], n [, N], where k [, K]/{i}. With TB w A k, the generated interference of UE will be perfectly aligned with the reference interference direction q [k] of BS k in the other cells. Such operation reflects the user s intention for active alignment with TB, and also increases the chance to achieve the DoF gain. Based on this idea, we develop two solutions for TB: CATB with limited multicell cooperation and DATB without multicell cooperation. CATB: The key component with CATB is the network alignment pattern, which indicates all users in the network to design their TB vectors. Specifically, we introduce a set of alignment patterns as {p k } K k=, where p k is a vector of K elements and p k is defined as [p k ] i [,K]/{k} = k and [p k ] k [, K]/{k}. The i-th element of p k indicates that the users in the i-th cell will use the TB w A [p k] i. For example, if p is used, all users in the network, except the users in the nd cell, will align their interferences to q [] at BS with w A. The users in the nd cell will align to any other BS, except BS. By doing so, we achieve perfect IA at BS. Without loss of generality, let us focus on p k. Since all the users in the i-th cell, i [, K]/{k}, will cast overlapping interference shadows at BS k, BS k achieves the DoF gain for the k-th cell as well as the whole network cf.. However, if the network always uses the same pattern p k, it will cause unfairness among the cells, since BS k always enjoys interference-free transmission and other BSs are interferencelimited. To address the fairness problem, the network alignment patterns {p k } can be used in a round-robin fashion at different transmission instances, therefore, each cell enjoys the equal chance for interference-free transmission. The roundrobin using of {p k } requires network collaboration. DATB: While CATB requires some sort of collaboration in detering the network alignment pattern though the collaboration is imal, we propose DATB which enables distributed TB optimization by the user. Specifically, UE will choose a preferred TB vector from the K TB candidates { w A k}, and the selection criteria to k [,K]/{i} detere the TB of UE as well as the transmitting user UE n i of the i-th cell will be discussed next. B. TB and User Selection Criteria In this subsection, we introduce several selection criteria. These criteria are applicable to select both TB and user with DATB, they are also applicable to select user with CATB. We mainly discuss two types of criteria which can be

4 BS BS l,, Useful signal Signal space Interference space SRA IRA BS BS, l, Interference Useful signal strength Interference strength, l, Fig.. Illustration the angle-based selection and strength-based selection. The case for UE, is presented with the trial TB vector w A, where number of interfering cells is K =. roughly divided into the angle-based and the strength-based strategies. The angle-based criteria are characterized by a straightforward description for the degree of alignment, while the strength-based criteria directly consider the interference and useful signal strength. In addition, each criterion can be further divided into the altruistic selection and the balanced selection. For the altruistic selection, system aims to imize the generated interference to other cells without considering the useful signal. On the other hand, the balanced selection takes both the generated interference and the useful signal into consideration. The angle and the strength based selection metrics are graphically illustrated in Fig. with an example. Angle-Based Criteria: The interference-reference angle IRA, i.e., the angle between the generated interference and the reference interference directions, serves as a reasonable metric to describe the degree of IA [], []. Since user UE has multiple candidate TBs in { w A k}, k [,K]/{i} the choice on k can be linked with the IRAs to address the degree of IA. To this end, we define the IRAs for UE with a candidate w A k as ] θ [k k = H [k ] wa k, q [k ], where k [, K]/{i, k}, and we further introduce the useful signal-reference angle SRA for UE with w A k as θ [i] k = H [i] wa k, q [i]. Next, the Angle-based altruistic TB selection for UE is w A k, and k is given by k = arg Θ k, k [,N]/{i} where Θ k := ] k [,K]/{i,k} θ[k IRAs with w A selection for UE is k Ab, and kab k is the sum of the k. Then, the Angle-based balanced Ab TB is given as Ξ k, k Ab = arg k [,N]/{i} where Ξ k := Θ k /θ [i] sum of the IRAs and the SRA with w A and we have w A k k is the ratio between the k. According to and w A k Ab as the TB vectors for altruistic and balance strategies, respectively. While and select the optimal TB for each user, we continue to select the best user for each cell, which employs the same selection principle as TB selection. For the altruistic selection, UE uses w A k and the transmitting user is selected according to the altruistic criterion as n i = arg Θ k, where Θ k := ] k [,K]/{i} θ[k k. Similarly, the balanced user selection is given as n Ab i = arg Ξ k, where Ξ k := Θ k /θ [i] k Ab. Strength-Based Criteria: With a similar reasoning as with the development of the angle-based criteria, we first focus on TB selection. Let us denote the leakage interference LI of UE with candidate TB vector w A k as l [k ] k = s [k ] H [k ] wa k, k [, K]/{i, k}, and the useful signal strength of UE with w A k as l [i] s k = [i] H [k ] wa k. Then, we introduce the Strength-based altruistic TB selection for UE, which results in w A k, and k is given by k = arg L k, k [,N]/{i} where L k := ] k [,K]/{i,k} l[k k is the SLI of UE with w A k. The Strength-based balanced TB selection results in w A k k = arg, and k k [,N]/{i} is given as D k, where D k := L k /l [i] k is the ratio between the SLI and the useful signal strength with w A k. After each user select the optimal TB, we proceed with user selection using the same principle, where the altruistic user selection is given as n i = arg L k, where L k := ] k [,K]/{i,k} l[k k is defined as the SLI of UE with TB w A k cf., and the balanced user selection is given as n i = arg D k, where D k := D k /l [i] k. So far we have focused on TB and user selection for DATB. In fact, for user selection with CATB, we can simply replace k, kab, kna and knb in,, and with [p k] i from, respectively. Remark: The criterion is inspired by the chordal distance based selection criterion [] and the criterion Ab is an

5 a N= CATB ZF Rx DATB ZF Rx JMaxSGINR ZF Rx [] JMinLIF ZF Rx [] [] b N= a CATB, Opt. Rx Ab c DATB, Opt. Rx b CATB, ZF Rx d DATB, ZF Rx Fig.. Average rate per cell with different TB designs and the predefined ZF detector. The number of interfering cells are K =. Fig.. Comparison of different selection criteria for CATB and DATB with ZF receiver and the optimized receiver, where K = and N =. extension of our downlink OIA [] to the uplink scenario. The criterion is a modification of the JMinSLI [], [] and the difference is that our TB design with active alignment, and the criterion is an enhancement to. Note that the SGINR based user selection can be also used, however, SGINR criterion involves the average power of the receiver noise as shown in. Besides the proposed schemes, there may be other selection criteria obtained by different combinations with the angle and strength based metrics []. A more comprehensive study is left as future work. C. Optimized BS Receiver In the existing uplink OIA schemes, ZF receiver is adopted for simplicity [] [], however, the projection toward the predefined signal direction s [i] is not interference-free due to imperfect IA. Based on this observation, we can reshape the SINR of BS i in as γ [i] n = r[i] C [i] n r[i] r [i] D [i] n where C [i] n = ρ H [k] n i w n iw n i H[k] n k ρ K, r[i] and D [i] n = k=,k i H[k] n k w n k w n k H[k] n k + I N R. Then the optimized receiver of BS i can be obtained by GEVD [] as r [i] opt = v C [i], D[i]. n n It is noted that the optimized BS receiver is very general and is applicable with our schemes as well as JMinSLI and JMaxSGINR schemes. IV. NUMERICAL RESULTS AND DISCUSSION In this section, we provide numerical results to validate the proposed uplink OIA schemes. We choose the average rate per cell as the overall performance metric, which is { K i= R[i] n } /K. For the basic antenna defined as R n = E configuration, we assume N T = N R = for both BSs and users. The used channel model is consistent with the description in Section II and all the simulation results are averaged over channel realizations. Test Case I: Effectiveness of TB with active alignment. We compare CATB and DATB with selection, JMinSLI [], [] and JMaxSGINR [] schemes in this test. We use the ZF receiver as [] [], and set the number of interfering cells as K =. In order to exclude the influence of user selection, we consider the scenario with N = in Fig. a, which shows that CATB and DATB performs very well in the medium-to-high SNR regime due to DoF advantages. In contrast, the SVD [], [] and GEVD [] based TB design cannot achieve DoF gain, and their performances quickly saturate even at low-to-medium SNR regime. When N =, OIA improves performance for all the schemes by harvesting multiuser diversity, and Fig. b shows that the proposed schemes outperform JMinSLI and JMaxSGINR in the medium-to-high SNR regime while maintain comparable performance at the low-to-medium SNR regime. Test Case II: Comparison among different selection schemes. We compare different selection criteria for the proposed CATB and DATB schemes with both the ZF receiver and the optimized receiver. The number of interfering cells is set as K = and the users per cell are N =. Fig. a and c present the results for the optimized receiver. It is shown that the proposed OIA performs better with the strength-based selection and the balanced selection as compared to the angle-based selection and altruistic selection. The results with the ZF receiver are shown in Fig. b and d with the similar observation as that of a and c. These results indicate that although the angle-based metrics directly characterize the degree of alignment, the signal strength based metrics, however, are more straightforward to detere the final performance metric of interests. Moreover, the strengthbased metric has include the angle cf. Fig., therefore, may provide more accurate information for selection. Test Case III: Comparison between the predefined ZF receiver and the optimized receiver. We show the effectiveness of the proposed optimized receiver for the existing schemes with both mild interference with K = and strong interference with K =. It is shown in Fig. that the optimized receiver improves not only the proposed schemes but also the JMinSLI and JMaxSGINR schemes. Moreover, the effectiveness of the optimized receiver is preserved even in the presence of strong interference. In addition, the DoF advantage of CATB scheme is manifested when interference is strong.

6 a K= b K= CATB Opt. Rx CATB ZF Rx DATB Opt. Rx DATB ZF Rx JMaxSGINR Opt. Rx JMaxSGINR ZF Rx [] JMinSLI Opt. Rx JMinSLI ZF Rx [] [] a SNR= db CATB Opt. Rx DATB Opt. Rx JMaxSGINR Opt. Rx JMaxSGINR ZF Rx [] JMinSLI Opt. Rx JMinSLI ZF Rx [] [] b SNR= db Fig.. Comparision for the predefined ZF receiver and the optimized receiver with mild interference K = and strong interference K =, where N =. Fig.. User scaling performance with strong interference K =. a SNR= db CATB Opt. Rx DATB Opt. Rx JMaxSGINR Opt. Rx JMaxSGINR ZF RX [] JMinLIF Opt. Rx JMinLIF ZF Rx [] [] b SNR= db Fig.. User scaling performance with mild interference K =. Test Case IV: User Scaling Performance. Fig. and Fig. present comparisons between the proposed and state of the art schemes. We use the selection and the optimized receiver for the proposed schemes, and consider the original JMinSLI and JMaxSGINR with ZF receiver as in [] [] and also their enhanced versions with the optimized receiver. We fix the SNR as db and db to serve as the representatives of the medium and high SNR regimes. With the mild interference K =, Fig. shows that the proposed schemes achieve comparable performance as that of the enhanced JMaxSGINR scheme with optimized receiver in the medium SNR regime, and outperform [] [] and their enhanced versions under any SNR with N [, ]. Moreover, when the network experiences a strong interference K =, as shown in Fig., CATB shows significant gain due to its DoF advantage. V. CONCLUSION In this paper, two new uplink OIA schemes have been proposed with the active alignment transmit beamforg designs, where both cooperative and distributed solutions have been considered with the angle-based and the strength-based selection criteria. The optimized BS receiver has been derived. Under the considered scenarios, it is shown that the proposed schemes outperform the existing schemes and their enhanced versions in the medium-to-high SNR regime while maintaining comparable performance in the low-to-medium SNR regime. It is also shown that the strength-based selection is more effective with the proposed OIA schemes. REFERENCES [] S. Jafar and S. Shamai, Degrees of freedom region of the MIMO X channel, IEEE Trans. Inf. Theory, vol., pp., Jan.. [] V. Cadambe and S. Jafar, Interference alignment and degrees of freedom of the K user interference channel, IEEE Trans. Inf. Theory, vol., pp., Aug.. [] C. Suh and D. Tse, Interference alignment for cellular networks, in Proc. Allerton Conf., Monticello, IL, Sept., pp.. [] R. Krishnamachari and M. Varanasi, Interference alignment under limited feedback for MIMO interference channels, in Proc. IEEE ISIT, Austin, TX, June, pp.. [] J.-S. Kim, S.-H. Moon, S.-R. Lee, and I. Lee, A new channel quantization strategy for MIMO interference alignment with limited feedback, IEEE Trans. Wireless Commun., vol., pp., Jan.. [] O. Ayach and R. W. Heath Jr., Interference alignment with analog channel state feedback, IEEE Trans. Wireless Commun., vol., pp., Feb.. [] K. Gomadam, V. Cadambe, and S. Jafar, A distributed numerical approach to interference alignment and applications to wireless interference networks, IEEE Trans. Inf. Theory, vol., pp., June. [] J. H. Lee, W. Choi, and D. J. Love, On the optimality of opportunistic interference alignment in -transmitter MIMO interference channels, vol. abs/.,, Available: [] J. H. Lee and W. Choi, Opportunistic interference aligned user selection in multiuser MIMO interference channels, in Proc. IEEE GLOBECOM, Miami, FL, Dec., pp.. [] J. Leithon, C. Yuen, H. A. Suraweera, and H. Gao, A new opportunistic interference alignment scheme and performance comparison of MIMO interference alignment with limited feedback, in Proc. IEEE GLOBE- COM Workshop on Multicell Cooperation, Anaheim, CA, Dec, pp.. [] B. C. Jung and W.-Y. Shin, Opportunistic interference alignment for interference-limited cellular TDD uplink, IEEE Commu. Lett., vol., pp., Feb.. [] 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., pp., July. [] L. Wang, Q. Li, S. Li, and J. Chen, A general algorithm for uplink opportunistic interference alignment in cellular network, in IEEE GLOBECOM Workshop on Multicell Cooperation, Houston, TX, Dec., pp.. [] H. J. Yang, W.-Y. Shin, B. C. Jung, and A. Paulraj, Opportunistic interference alignment for MIMO IMAC: Effect of user scaling over degrees-of-freedom, in Proc. IEEE ISIT, Cambridge, MA, July, pp.. [] S.-H. Hur, B. C. Jung, and B. Rao, Sum rate enhancement by maximizing SGINR in an opportunistic interference alignment scheme, in Proc. Asilomar Conf., Pacific Grove, CA, Nov., pp.. [] R. Horn and C. Johnson, Matrix analysis. New York: Cambridge University Press,.