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Recent results establish the optimality of interference alignment to approach the Shannon capacity of interference networks at high SNR. However, the extent to which interference can be aligned over a finite number of signalling dimensions remains unknown. Another important concern for interference alignment schemes is the requirement of global channel knowledge. In this work we provide examples of iterative algorithms that utilize the reciprocity of wireless networks to achieve interference alignment with only local channel knowledge at each node. These algorithms also provide numerical insights into the feasibility of interference alignment that are not yet available in theory.

We explore the degrees of freedom of M × N user wireless X networks, i.e. networks of M transmitters and N receivers where every transmitter has an independent message for every receiver. We derive a general outerbound on the degrees of freedom region of these networks. When all nodes have a single antenna and all channel coefficients vary in time or frequency, we show that the total number of degrees of freedom of the X network is equal toper orthogonal time and frequency dimension. Achievability is proved by constructing interference alignment schemes for X networks that can come arbitrarily close to the outerbound on degrees of freedom. For the case where either M = 2 or N = 2 we find that the degrees of freedom characterization also provides a capacity approximation that is accurate to within O(1). For these cases the degrees of freedom outerbound is exactly achievable.

It has been conjectured by Høst-Madsen and Nosratinia that complex Gaussian interference channels with constant channel coefficients have only one degree-of-freedom regardless of the number of users. While several examples are known of constant channels that achieve more than 1 degree of freedom, these special cases only span a subset of measure zero. In other words, for almost all channel coefficient values, it is not known if more than 1 degree-of-freedom is achievable. In this paper, we settle the Høst-Madsen-Nosratinia conjecture in the negative. We show that at least 1.2 degrees-of-freedom are achievable for all values of complex channel coefficients except for a subset of measure zero. For the class of linear beamforming and interference alignment schemes considered in this paper, it is also shown that 1.2 is the maximum number of degrees of freedom achievable on the complex Gaussian 3 user interference channel with constant channel coefficients, for almost all values of channel coefficients. To establish the achievability of 1.2 degrees of freedom we introduce the novel idea of asymmetric complex signaling -i.e., the inputs are chosen to be complex but not circularly symmetric. It is shown that unlike Gaussian point-to-point, multiple-access and broadcast channels where circularly symmetric complex Gaussian inputs are optimal, for interference channels optimal inputs are in general asymmetric. With asymmetric complex signaling, we also show that the 2 user complex Gaussian X channel with constant channel coefficients achieves the outer bound of 4/3 degreesof-freedom, i.e., the assumption of time-variations/frequency-selectivity used in prior work to establish the same result, is not needed. * The ordering of authors is alphabetical. arXiv:0904.0274v1 [cs.IT] 1 Apr 2009 1. Signal Vector Space Alignment Schemes -Linear transmitter precoding and receiver combining operations are used to transform the interference channel into multiple non-interfering