Degrees of Freedom Region of the MIMO $2 \times 2$ Interference Network With General Message Sets
We establish the degrees of freedom (DoF) region for the multiple-input multiple-output (MIMO) two-transmitter, and two-receiver (<inline-formula> <tex-math notation="LaTeX">$2 \times 2$ </tex-math></inline-formula>) interference network with a general message set consisting of nine messages, one for each pair of a subset of transmitters at which that message is known and a subset of receivers where that message is desired. An outer bound on the general nine-message <inline-formula> <tex-math notation="LaTeX">$2 \times 2$ </tex-math></inline-formula> interference network is obtained and it is shown to be tight, thereby establishing the DoF region for the most general antenna setting wherein the four nodes have an arbitrary number of antennas each. The DoF-optimal scheme is applicable to the MIMO <inline-formula> <tex-math notation="LaTeX">$2 \times 2$ </tex-math></inline-formula> interference network with constant channel coefficients, and hence, <italic>a fortiori</italic>, to time/frequency varying channel scenarios. In particular, a linear precoding scheme is proposed that can achieve all the DoF tuples in the DoF region. In it, the precise roles played by transmit zero-forcing, interference alignment, random beamforming, symbol extensions, and asymmetric complex signaling (ACS) are delineated. For instance, we identify a class of antenna settings, in which ACS is required to achieve the fractional-valued corner points. Evidently, the DoF regions of all previously unknown cases of the <inline-formula> <tex-math notation="LaTeX">$2 \times 2$ </tex-math></inline-formula> interference network with a subset of the nine-messages are newly established as special cases of the general result of this paper. For instance, the DoF region of the well-known four-message (and even three-message) MIMO <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula> channel is newly established. This problem had remained open despite previous studies, which had found inner and outer bounds that were not tight in general. Hence, the DoF regions of all special cases obtained from the general DoF region of the nine-message <inline-formula> <tex-math notation="LaTeX">$2\times 2$ </tex-math></inline-formula> interference network of this paper that include at least three of the four <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula> channel messages are new, among many others. This paper sheds light on how the same physical <inline-formula> <tex-math notation="LaTeX">$2 \times 2$ </tex-math></inline-formula> interference network could be used by a suitable choice of message sets to take most advantage of the channel resource in a flexible and efficient manner.