Frequency Dependent Tradeoffs in Multivariable Systems

We discuss the generalized gain-phase relation developed in [1]. The role of gain is taken by a singular value, and that of phase by the phase difference between the associated singular vectors. The integral relation is analogous to that for scalar systems, except for an additional surface integral due to directionality properties. We develop a method for analyzing this surface integral and explaining its presence in intuitive terms. We then apply our results to analyse the design utility of the "phase-transfer" phenomenon noted in [1].