Multiple-input multiple-output (MIMO) linear systems extreme inputs/outputs

A linear structure is excited at multiple points with a stationary normal random process. The response of the structure is measured at multiple outputs. If the autospectral densities of the inputs are specified, the phase relationships between the inputs are derived that will minimize or maximize the trace of the autospectral density matrix of the outputs. If the autospectral densities of the outputs are specified, the phase relationships between the outputs that will minimize or maximize the trace of the input autospectral density matrix are derived. It is shown that other phase relationships and ordinary coherence less than one will result in a trace intermediate between these extremes. Least favorable response and some classes of critical response are special cases of the development. It is shown that the derivation for stationary random waveforms can also be applied to nonstationary random, transients, and deterministic waveforms.