Interference Alignment and Degrees of Freedom of the K-User(10)

3434IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 8, AUGUST 2008matrices, their inverses and their products are all identical to the set of column vectors of the identity matrix. For instance, . Therefore is an vectors of the form eigenvector for all channel matrices. Since lies in span , (25)–(27) imply that, span span spanantennas and constant channel coef cients. It is not with users. However, known if the result can be extended to with time-variations it is easy to nd the degrees of freedom for antennas at each node. the user interference channel with The result follows directly from Theorem 1 and is presented in the following Corollary. user interference channel Corollary 1: The time-varying antennas at each node has degrees of freedom. with Proof: The converse for Corollary 1 is already derived in Appendix 1 in (31). Achievability of Corollary 1 is also straightforward. Suppose colocated antennas at a node as a sepwe view each of the arate node. In other words we do not allow joint processing of signals obtained from the co-located antennas. Then, instead of user interference channel with antenna nodes we oba user interference channel with single antenna nodes. tain a degrees of But the result of Theorem 1 establishes that freedom are achievable on this interference channel. Thus, we degrees of freedom on the user incan also achieve antenna nodes. terference channel with Last, let us consider the most general user interference channel where each node is equipped with possibly different number of antennas. In this case also a lower bound on the degrees of freedom is directly established from the result of Theorem 1. The following Corollary states this result. user Corollary 2: The total degrees of freedom for the antennas and interference channel where transmitter has receiver has antennas is bounded below asTherefore, at receiver 1, the desired signal is not lin. Therefore, early independent with the interference completely by merely zero-forcing receiver 1 cannot decode the interference signal. Evidently, interference alignment in the degrees of manner described above cannot achieve exactly freedom on the 3 user interference channel with a single antenna at all nodes. We explore this interesting aspect of the three-user interference channel further in the context of multiple antenna nodes. degrees of freedom Our goal is to nd out if exactly antennas at each node. As shown by may be achieved with the following theorem, indeed we can achieve exactly degrees of freedom so that the capacity characterization is indeed related to the degrees of freedom as for . V. DEGREES OF FREEDOM FOR THE INTERFERENCE CHANNEL WITH MULTIPLE ANTENNA NODES A. The Three-User Interference Channel With Antennas at Each Node and Constant Channel Coef cients The three-user MIMO interference channel is interesting bedecause in this case we show that we can achieve exactly grees of freedom with constant channel matrices, i.e., time-variations are not required. This gives us a lowerbound on sum ca. Since the outerbound on pacity of we have an sum capacity is also approximation to the capacity of the three-user MIMO interferantennas at all nodes. ence channel with Theorem 3: In a three-user interference channel with antennas at each transmitter and each receiver and constant comay be characterized (almost ef cients, the sum capacity surely) as (28) The outerbound follows directly from [27] which shows that the two-user interference channel with antennas at each node and constant channel coef cients has only degrees of freedom. In the three-user case, we eliminate one message at a time to obtain . Adding inequalities up all three inequalities we obtain the converse. The proof is presented in Appendices IV and V. B. The Nodes User Interference Channel With Multiple Antenna(29) Thus, no more than half the degrees of freedom are lost on the user interference channel with multiple antenna nodes. Proof: The achievability proof is straightforward as, once again, the th transmitter receiver pair can be replaced single antenna transmitter and receiver with nodes by only allowing distributed processing of signals at each antenna and discarding the remaining antennas. Thus, we obtain an interference channel with transmitters and receivers, each equipped with only a single antenna. The achievability of degrees of freedom on this interference channel then follows from the result of Theorem 1. Note that Corollary 2 only establishes an innerbound and is not always tight. For example, consider the two-user interference channel where each transmitter has two antennas while each receiver has only 1 antenna. While Corollary 2 only shows achievability of degree of freedom for this channel, it is known that this interference channel has degrees of freedom [27]. However, Corollary 2 is interesting because it shows that interference cannot reduce the degrees of freedom of the interference channel by more than half compared to when each transmitter and receiver is able to operate without interference from other users.Theorem 3 in the preceding section shows that degrees of freedom are achievable on the three-user interference channelAuthorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 01,2010 at 01:21:44 UTC from IEEE Xplore. Restrictions apply.