Correlations of wave variables in a passive network

Abstract

We show that under certain conditions the wave variables ai(t) incident on a frequency independent N-port junction in a passive network satisfy the inner product relation:∑Nj = 1 ∑Nk = 1k ≠ j〈SijajSikak = 0 i = 1,2..,Nwhere Sij are the scattering parameters of the junction. The conditions are that ai(t) are the result of thermal excitation of the network, or of equivalent white excitation in any lossy parts of the network. A particular case of special applicability is that in which the network is a cascade of transmission lines, and in this case the N-port junction is the 2-port junction of two adjacent lines. Then the relation is〈a1(t)a2(t) 〉 = 0 The corresponding relationships for discrete time variables and the partial correlation signal processing algorithm may be viewed as particular applications.