3438IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 8, AUGUST 2008For example, if . . . At receiver (53) aswe get.andare chosen. . . At receiver where . . . (54) where (55)To clarify the notation further, consider the case where . consists of exactly one element, i.e., Assuming . The set consists of all column vectors of the form where all take values 0, 1. and can be veri ed to have and elements, respectively. are chosen using (52). Clearly, forNow, for(56)(57) , the identity matrix. We now Note that choose and so that they satisfy the relations in (53)–(54) and then use equations in (52) to . Thus, our goal is to nd madetermine and so that tricesfor all Letbe the. column vector. . .We need to choose for . The sets of column vectors of and where be equal to the sets andcolumn vectors are chosen toThus, the interference alignment conditions (52)–(54) are satis ed. Through interference alignment, we have now ensured that the dimension of the interference is small enough. We now need to verify that the components of the desired signal are linearly independent of the components of the interference so that the signal stream can be completely decoded by zero-forcing the interference. Consider the received signal vectors at receiver 1. vectors . The desired signal arrives along the As enforced by (52), the interference vectors from transmitare perfectly aligned with the interference from ters transmitter 2 and therefore, all interference arrives along the vectors . In order to prove that there are interference free dimensions it suf ces to show that the columns -dimensional matrix of the square, (58) are linearly independent almost surely. Multiplying the above matrix with and substituting for and , we get a matrix whose th row has entries of the formsandwhereand and are drawn independently from a continuous distribution. The same iterative argument as in Section IV-B can be used. For instance, expanding the corresponding determinantAuthorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 01,2010 at 01:21:44 UTC from IEEE Xplore. Restrictions apply.