3430IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 8, AUGUST 2008This result, in conjunction with the converse argument proves the theorem. lies in , we construct an To show that interference alignment scheme using only time slots. We symbols transmitted over collectively denote the time slots as a supersymbol. We call this the symbol extension of the channel. With the extended channel, the signal vector at the th user’s receiver can be expressed asIn this achievable scheme, receiver eliminates interference to decode . At receiver 1, by zero-forcing all desired streams are decoded after zero-forcing the interference to achieve degrees of freedom. To obtain interfer-dimensional received signal ence free dimensions from a , the dimension of the interference should be not vector more than . This can be ensured by perfectly aligning the interference from transmitters 2 and 3 as follows: (7) At the same time, receiver 2 zero-forces the interference from and . To extract interference-free dimensions from -dimensional vector, the dimension of the interference a . For instance has to be not more thanwhereis a column vector representing the symbol extension of the transmitted symbol , i.e.. . . This can be achieved by choosing Similarly and represent symbol extensions of and , respectively. is a diagonal the matrix representing the symbol extension of the channel as shown in the equation at the bottom of the page. is achievable on We show that lies in this extended channel implying that the degrees of freedom region of the original channel. is encoded at In the extended channel, message transmitter 1 into independent streams sent along vectors so that is and so that (8) where , means that the set of column vectors of matrix is a subset of the set of column vectors of matrix . Similarly, at receiver 3, we wish to choose and so to decode that (9) , and so that (7), Thus, we wish to pick vectors (8), (9) are satis ed. Note that the channel matrices have a almost surely. Since multiplying by a full rank full rank of matrix (or its inverse) does not affect the conditions represented by (7), (8), and (9), they can be equivalently expressed as (10) (11) (12) where (13) (14) (15) (16) The received signal at the th receiver can then be written as Note that is a matrix. and are matrices. Since all channel matrices are invertible, we and so that they satisfy (10)–(12) and then can choosewherecolumn vector and is a -dimensional matrix. Similarly and are each encoded into independent streams by transmitters 2 and , respectively. and 3 asis a. . ..... . .Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 01,2010 at 01:21:44 UTC from IEEE Xplore. Restrictions apply.