Interference Alignment and Degrees of Freedom of the K-User(17)

CADAMBE AND JAFAR: INTERFERENCE ALIGNMENT AND DEGREES OF FREEDOM OF THE-USER INTERFERENCE CHANNEL34412)An argument along the same lines as the even case leads to the conclusion that the probability of the union of the two sets listed above being linearly dependent in a -dimensional space is zero. or . This implies that span span span spanspan Also, spanspanspanspanNote that and are -dimensional spaces. (The is handled case where their dimensions are less than in the rst part). Also, and are drawn from completely has a different set of vectors. Therefore, the union of rank of almost surely. Equivalently, span span has a dimension of almost surely. Since the set is drawn from an eigen vector that does not exist in either or , the probability of the 2-D space span intersecting with the -dimensional is zero. For example, if , let indispace cate the line formed by the intersection of the two planes and . The probability that line lies in the plane formed by . Thus, the probability that the desired signal lies in the span of the interference is zero at receiver 1. Similarly, it can be argued that the desired signal is independent of the interference at receivers 2 and 3 almost surely. Therefore, is achievable over the two-symbol extended degrees of freedom are achievable channel. Thus antenna at over the 3 user interference channel with each transmitting and receiving node. ACKNOWLEDGMENT The authors would like to acknowledge many helpful discussions with Prof. Shlomo Shamai that have guided this work. REFERENCES[1] A. B. Carleial, “A case where interference does not reduce capacity,” IEEE Trans. Inf. Theory, vol. 21, pp. 569–570, 1975. [2] A. B. Carleial, “Interference channels,” IEEE Trans. Inf. Theory, vol. 24, no. 1, pp. 60–70, 1978. [3] T. Han and K. Kobayashi, “A new achievable rate region for the interference channel,” IEEE Trans. Inf. Theory, vol. 27, pp. 49–60, Jan. 1981.[4] H. Sato, “The capacity of the Gaussian interference channel under strong interference,” IEEE Trans. Inf. Theory, vol. 27, pp. 786–788, Nov. 1981. [5] M. Costa, “On the Gaussian interference channel,” IEEE Trans. Inf. Theory, vol. 31, pp. 607–615, Sep. 1985. [6] G. Kramer, “Feedback strategies for white Gaussian interference networks,” IEEE Trans. Inf. Theory, vol. 48, pp. 1423–1438, Jun. 2002. [7] I. Sason, “On the achievable rate regions for the Gaussian interference channel,” IEEE Trans. Inf. Theory, vol. 50, pp. 1345–1356, Jun. 2004. [8] S. Vishwanath and S. Jafar, “On the capacity of vector Gaussian interference channels,” in Proc. IEEE Inf. Theory Workshop, Oct. 2004, pp. 365–369. [9] H. Chong, M. Motani, H. Garg, and H. El Gamal, “On the Han-Kobayashi region for the interference channel,” IEEE Trans. Inf. Theory, submitted for publication. [10] R. Etkin, D. Tse, and H. Wang, “Gaussian interference channel capacity to within one bit,” IEEE Trans. Inf. Theory, Feb. 2007, submitted for publication. [11] M. Charafeddine, A. Sezgin, and A. Paulraj, “Rate region frontiers for n user interference channel with interference as noise,” in Proc. 45th Annu. Allerton Conf. Commun., Contr. Comput., Oct. 2007. [12] C. Rao and B. Hassibi, “Gaussian interference channel at low SNR,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Jul. 2004. [13] A. Motahari and A. Khandani, “Capacity bounds for the Gaussian interference channel,” IEEE Trans. Inf. Theory, submitted for publication. [14] X. Shang, G. Kramer, and B. Chen, “A new outer bound and the noisy-interference sum-rate capacity for Gaussian interference channels,” IEEE Trans. Inf. Theory, Dec. 2007, submitted for publication. [15] D. Tuninetti, “Progresses on Gaussian interference channels with and without generalized feedback,” in Proc. 2008 Inf. Theory Appl. Workshop, San Diego, CA, Jan. 2008, Univ. of California. [16] S. Yang and D. Tuninetti, “A new achievable region for interference channel with generalized feedback,” in Proc. 42nd Annu. Conf. Inf. Sci. Syst. (CISS), Mar. 2008. [17] V. Annapureddy and V. Veeravalli, “Gaussian interference networks: Sum capacity in the low interference regime and new outer bounds on the capacity region,” IEEE Trans. Inf. Theory, Feb. 2008, submitted for publication. [18] A. Host-Madsen and A. Nosratinia, “The multiplexing gain of wireless networks,” in Proc. ISIT, 2005. [19] S. Jafar and S. Shamai, “Degrees of freedom region for the MIMO X channel,” arXiv:cs.IT/0607099v3, May 2007. [20] V. Cadambe and S. Jafar, “Multiple access outerbounds and the inseparability of parallel interference channels,” IEEE Trans. Inf. Theory, Jul. 2007, submitted for publication. [21] G. Bresler, A. Parekh, and D. Tse, “Approximate capacity of the many-to-one interference channel,” in Proc. Allerton Conf., Sep. 2007. [22] V. Cadambe, S. Jafar, and S. Shamai, “Interference alignment on the deterministic channel and application to Gaussian networks,” arxiv:0711.2547. [23] M. Maddah-Ali, A. Motahari, and A. Khandani, “Signaling over MIMO multi-base systems—Combination of multi-access and broadcast schemes,” in Proc. ISIT, 2006, pp. 2104–2108. [24] S. Jafar, Degrees of Freedom on the MIMO X Channel-Optimality of the MMK Scheme Tech. Report, arXiv:cs.IT/0607099v2, Sep. 2006. [25] H. Weingarten, S. Shamai, and G. Kramer, “On the compound MIMO broadcast channel,” in Proc. Annu. Inf. Theory Appl. Workshop, San Diego, CA, Jan. 2007, Univ. of California. [26] M. Maddah-Ali, A. Motahari, and A. Khandani, Communication Over X Channel: Signaling and Performance Analysis Unive. Waterloo, Waterloo, ON, Canada, Tech. Rep. UW-ECE-2006-27, Dec. 2006. [27] S. Jafar and M. Fakhereddin, “Degrees of freedom for the mimo interference channel,” IEEE Trans. Inf. Theory, vol. 53, pp. 2637–2642, Jul. 2007. [28] K. Gomadam, V. Cadambe, and S. Jafar, “Approaching the capacity of wireless networks through distributed interference alignment,” presented at the GLOBECOM 2008, Mar. 2008. [29] B. Nazer and M. Gastpar, “The case for structured random codes in network communication theorems,” in Proc. IEEE Inf. Theory Workshop, Sep. 2007.Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 01,2010 at 01:21:44 UTC from IEEE Xplore. Restrictions apply.