Factorization properties of the optimal signaling distribution of multi-dimensional QAM constellations
In this work we study the properties of the optimal Probability Mass Function (PMF) of a discrete input to a general Multiple Input Multiple Output (MIMO) channel. We prove that when the input constellation is constructed as a Cartesian product of 1-dimensional constellations, the optimal PMF factorizes into the product of the marginal 1D PMFs. This confirms the conjecture made in [1], which allows for optimizing the input PMF efficiently when the rank of the MIMO channel grows. The proof is built upon the iterative Blahut-Arimoto algorithm. We show that if the initial PMF is factorized, the PMF on each successive step is also factorized. Since the algorithm converges to the optimal PMF, it must therefore also be factorized.