Stochastic broadening of signals in distorsionless transmission lines

The paper presents a theory of a stochastic continuous transmission line in which the series inductanceLΔ1+l(x)], series resistanceRΔ1+r(x)], shunt capacitanceCΔ1+c(x)], and shunt conductanceGΔ1+g(x)] are defined as Gaussian random functions. (The continuous line is considered as a limiting case of a lumped transmision line.) The non-negative random functionsL(x),R(x),C(x), andG(x) are chosen as delta-correlated, i.e. their correlation function is of the formΘδ(x′ −x″) whereΘ is a 4×4 positive definedx-independent matrix. Propagation of a signal of Gaussian shape is analyzed. A special attention is devoted to the so-called distorsionless lines defined by the deterministic conditionR/L=G/C. As a consequence of the stochasticity of the functionsl(x),r(x),c(x), andg(x), transmitted signals do become distorted: they become broadened. An explicit formula for this broadening is derived. This work has been supported by the Slovak Grant Agency VEGA under contract No. 1/4319/97.