A multivariable bounded-reality criterion

The multivariable approach to the synthesis of networks composed of a finite number of uniform lossless transmission lines, commensurable or incommensurable, and lumped passive elements is verified by showing that the multivariable rational matrix, W(λ0,λ1, .. ,λn) is bounded real in (n + 1) complex variables if and only if W is bounded real in p after substituting, λ0 = α0p + β0 and λi = tanh (αip + βi) for λi (1⩽i⩽n) where all the α's and β's are nonnegative and arbitrary, except for being not simultaneously zero in like indexed pairs. Consequences of this result are discussed.