3436IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 8, AUGUST 2008where the pre-log factor of denotes the degrees of freedom is also found achieved by each user. The sum-rate to be the capacity by a converse argument that is nearly identical to the converse for the phase-alignment example. APPENDIX II CONVERSE FOR LEMMA 1 We present the proof for the case that all nodes are equipped antennas. In this case the statement of the lemma bewith comesis strictly positive with probability . Here refers to are the smallest eigenvalue of matrix . mutually independent and jointly Gaussian. Consider any reliable coding scheme for this interference channel, spanning channel uses. We use the notation to indicate the vector of values taken by variable for . Starting from Fano’s inequality, we have(31) and eliminate messages We consider the case , leaving us with a two-user MIMO interference channel. The following converse holds for both time-varying and constant channel coef cients. The channel input-output equations are written equivalently as: (32) (33) With probability one the channel matrices are invertible. So we can equivalently write (34) (44)(43)(35) where (36) (45) (37) Since the capacity of the interference channel depends only on the noise marginals, we assume without loss of generality that (38) (39) where (40) (41) where the last step follows from the known result that the sum capacity of a multiple access channel with an andegrees of tenna receiver can only contribute at most . Similarly, for any freedom. Thus, we have we obtain . Finally, adding up all the outerbounds in (4), we obtain the converse statement for the degrees of freedom of the user interference antennas at each node channel with (48) APPENDIX III ACHIEVABILITY FOR THEOREM 1 FOR ARBITRARY Let . We show that lies in the degrees of freedom (46) (47)(42) andAuthorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 01,2010 at 01:21:44 UTC from IEEE Xplore. Restrictions apply.