Frequency-Stable Beamforming Over Wireless MIMO Channels

The Wigner-Smith (WS) matrix provides a powerful framework for characterizing the delay properties of scattering processes in time-invariant propagation environments. When the underlying scattering matrix is unitary, the WS matrix is guaranteed to be Hermitian, with real-valued eigenvalues known as proper delay times. These eigenvalues quantify the delay experienced by narrowband wave packets shaped according to the corresponding WS eigenvectors (also known as “principal modes”). Originally introduced in quantum mechanics to describe collision-induced delays in particle scattering, the WS matrix has since found broader applications. We explore its relevance for wireless multiple-input multipleoutput (MIMO) systems using antenna arrays on both sides of the radio link. In particular, we highlight that the principal modes of the WS matrix exhibit robustness to small frequency shifts, even for the strongly non-unitary scattering matrices commonly encountered in wireless communication scenarios.