Wigner–Smith Time Delay Matrix for Electromagnetics: Computational Aspects for Radiation and Scattering Analysis

The Wigner-Smith (WS) time-delay matrix relates a lossless and reciprocal system's scattering matrix to its frequency derivative and enables the synthesis of modes that experience well-defined group delays when interacting with the system. The elements of the WS time-delay matrix for surface scatterers and antennas comprise renormalized energy-like volume integrals involving electric and magnetic fields that arise when exciting the system via its ports. Here, direct and indirect methods for computing the WS time-delay matrix are presented. The direct method evaluates the energy-like volume integrals using surface integral operators that act on the incident electric fields and current densities for all excitations characterizing the scattering matrix. The indirect method accomplishes the same task by computing scattering parameters and their frequency derivatives. Both methods are computationally efficient and readily integrated into the existing surface integral equation codes. The proposed techniques facilitate the evaluation of frequency derivatives of antenna impedances, antenna patterns, and scatterer radar cross sections in terms of renormalized field energies derived from a single-frequency characterization of the system.