Omnidirectional Precoding for Massive MIMO With Uniform Rectangular Array—Part II: Numerical Optimization Based Schemes

In Part II of this two-part paper, we develop novel numerical optimization approaches to design omnidirectional precoding schemes for a massive multiple-input multiple-output system equipped with a uniform rectangular array (URA). To this end, we first formulate the omnidirectional precoding design as a semidefinite program (SDP) with rank constraint. An iterative rank-reduction algorithm is then proposed to produce omnidirectional transmission solution with a minimum number of precoding vectors. It is shown that the proposed algorithm can always obtain three precoding vectors (i.e., a rank-3 precoding matrix) to generate a perfectly omnidirectional power radiation pattern for any <inline-formula><tex-math notation="LaTeX">$P \times Q$</tex-math></inline-formula> URA with <inline-formula><tex-math notation="LaTeX">$\min \lbrace P, Q\rbrace >2$</tex-math></inline-formula>, or yield an omnidirectional transmission solution with (theoretically minimum) two precoding vectors for any <inline-formula><tex-math notation="LaTeX">$P \times Q$</tex-math></inline-formula> URA with <inline-formula><tex-math notation="LaTeX">$\min \lbrace P, Q\rbrace \leq 2$</tex-math></inline-formula>. In addition, to facilitate analog precoding design, we further impose constant-modulus constraints for every entries of the precoding matrix. Through judicious (re-)formulation, we develop an efficient Newton's method, which can compute four constant-modulus precoding vectors to generate an omnidirectional power radiation pattern for any URA configuration. The numerical results demonstrate the merits of the proposed schemes.