Some Excitation Independent Bounds for the Total Active Reflection Coefficient of Antenna Arrays

The total active reflection coefficient (TARC) is a key parameter in the characterization of multiport antenna arrays used in multiple-input multiple-output (MIMO) platforms. The TARC dependency on scattering parameters and port excitations is expressed using matrix
$p$
- and Frobenius norms. It is demonstrated how coupling and excitation effects can be separated from each other by mathematical manipulations of the TARC equation and represented in the form of inequalities. The derived inequalities are used to determine excitation-independent upper bounds or approximations for the TARC. It is shown that the proposed bounds can be used to define proper margins for efficient prediction of the TARC variations without a priori information about port excitations. In addition to the
$p$
- and Frobenius norm inequalities, the extrema of the TARC as a function of the port phases are analytically calculated using the Hessian matrix. Finally, the derived bounds are verified and compared numerically via simulating different antenna arrays in both random and uniform configurations with different port excitations corresponding to different beamsteering directions. It is shown that the TARC values are fluctuating very close to the tightest proposed bounds which are defined based on the Frobenius and 2-norm of the array
$S$
-matrix.