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We explore the degrees of freedom of M x 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 outer bound 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 to MN/M+N-1 per orthogonal time and frequency dimension. Achievability is proved by constructing interference alignment schemes for X networks that can come arbitrarily close to the outer bound 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 outer bound is exactly achievable. [ABSTRACT FROM AUTHOR]

It is known that the capacity of parallel (multicarrier) Gaussian point-to-point, multiple access and broadcast channels can be achieved by separate encoding for each subchannel (carrier) subject to a power allocation across carriers. In this paper we show that such a separation does not apply to parallel Gaussian interference channels in general. A counter example is provided in the form of a 3 user interference channel where separate encoding can only achieve a sum capacity of 2 log(1 + 3 SNR) while the actual capacity, achieved only by joint encoding across carriers, is 3 log(1 + 2 SNR). As a byproduct of our analysis, we propose a class of multiple-access-outer bounds on the capacity of the 3 user interference channel. [ABSTRACT FROM AUTHOR]

We find the degrees of freedom of a network with S source nodes, B relay nodes, and D destination nodes, with random time-varying/frequency-selective channel coefficients and global channel knowledge at all nodes. We allow full-duplex operation at all nodes, as well as causal noise-free feedback of all received signals to all source and relay nodes. An outer bound to the capacity region of this network is obtained. Combining the outer bound with previous interference alignment based achiev-ability results, we conclude that the techniques of relays, feedback, full-duplex operation and noisy cooperation do not increase the degrees of freedom of interference and X networks. As a second contribution, we show that for a network with K full-duplex nodes and K ( K - 1) independent messages with one message from every node to each of the other K - 1 nodes, the total degrees of freedom are bounded above and below by K(K - 1)/(2K - 2) and K(K - 1)/(2K - 3) respectively. [ABSTRACT FROM AUTHOR]

An interference alignment example is constructed for the deterministic channel model of the K-user interference channel. The deterministic channel example is then translated into the Gaussian setting, creating the first known example of a fully connected Gaussian K-user interference network with single antenna nodes, real, nonzero and constant channel coefficients, and no propagation delays where the degrees of freedom outerbound is achieved. An analogy is drawn between the propagation delay based interference alignment examples and the deterministic channel model which also allows similar constructions for the two-user X channel as well. [ABSTRACT FROM AUTHOR]

For the fully connected K user wireless interference channel where the channel coefficients are time-varying and are drawn from a continuous distribution, the sum capacity is characterized as C(SNR) =K/2 log(SNR) + o(log(SNR)). Thus, the K user time-varying interference channel almost surely has K/2 degrees of freedom. Achievability is based on the idea of interference alignment. Examples are also provided of fully connected K user interference channels with constant (not time-varying) coefficients where the capacity is exactly achieved by interference alignment at all SNR values. [ABSTRACT FROM AUTHOR]