Generalized degrees of freedom of the (noisy) X channel

We provide outer bounds on the capacity region of the two-user Gaussian X channel, i.e. a generalization of the two-user interference channel where there is an independent message from each transmitter to each receiver. We identify the conditions under which operating the X channel as the interference channel is optimal from the perspective of generalized degrees of freedom (GDOF) and sum capacity. Specifically, we first extend the bound on the sum rate of the interference channel obtained by Etkin, Tse, and Wang in to the X channel. This bound provides insights into the operating regimes in which two channels have the same GDOF. We then extend the noisy interference capacity characterization previously obtained for the interference channel to the X channel. Therefore, we show that the X channel associated with noisy (very weak) interference channel has the same sum capacity as the noisy interference channel.