Interference Alignment and Degrees of Freedom of the K-User(16)

3440IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 8, AUGUST 2008(72)matrix . The probability of a random of the random of being collinear with is zero. rotation (and scaling) Using a similar argument, we can show that matrices The above equations imply that and span span(66) (67)(68) (69) (70)have a full rank of almost surely and therefore receivers 2 streams of and using zeroand 3 can decode the interference free transmissions per forcing. Thus, a total channel-use are achievable with probability and the proof is complete. APPENDIX V PROOF OF THEOREM 3 FORwhereODDProof: Consider a two time-slot symbol extension of the channel, with the same chanel coef cients over the two symbols. It can be expressed asand and are block-diagonal matrices representing the symbol extension of and , respectively. be the eigen vectors of . Then, we pick Let to be (71)where is a vector that represents the two symbol symbol symbol , i.e. extension of the transmittedcase, and are then determined by As in the even using (68)–(70). Now, we need the desired signal to be linearly independent of the interference at all the receivers. At receiver 1, the desired linear independence condition boils down to span where and is the two-symbol diis an matrix. agonal extension of . Notice that The linear independence condition is equivalent to saying that matrix are indeall the columns of the following pendent as shown in (72) at the top of the page. We now argue that the probability of the columns of the above matrix being denote the linearly dependent is zero. Let columns of the above matrix. Suppose the columns are linearly dependent, thenwhere is an vector representing the vector transmitted at time slot by transmitter . Similarly and represent the two symbol extensions of the received symbol and the noise vector respectively at receiver . is block diagonal matrix representing the extension a of the channelWe will now show lies in the degrees of freedom region of this extended channel channel with an achievable degrees of freedom scheme, implying that that a total of are achievable over the original channel. Transmitter transfor receiver using independently encoded mits message , i.e. streams over vectorsLetwhere is a matrix and is a vector independent streams. The following three inrepresenting terference alignment equations ensure that the dimension of the at receivers 1, 2, and 3 interference is equal to (65)Now, there are two possibilities . This implies that either one of the fol1) lowing sets of vectors is linearly dependent. Note that both sets are can be expressed as the union of eigen vectors of a) A set of b) A random transformation of this set.Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 01,2010 at 01:21:44 UTC from IEEE Xplore. Restrictions apply.