Frequency Scaling of a Linear Time-Invariant Network by a Time-varying Function

Abstract

Given a linear time-invariant RLC network, with input x(t) and output y(t) , then the well-known frequency scaling theorem states that multiplication of all L 's and C 's by some constant a^{-1} is equivalent to changing the input to ax(at) and the output to ay(at) . We show here that when the multiplier is a time-varying function a^{-1}(t) , the equivalent result is to change the input from x(t) to a(\gamma^{-1}(t))x(\gamma^{-1}(t)) and the output from y(t) to a(\gamma^{-1}(t))y(\gamma^{-1}(t)) where \gamma(t)= \int_{0}^{t}\frac{d \tau}{a(t)} . Some illustrative examples are \footnote[1]{given}. (1)In this correspondence a^{-1} means \frac{1}{a} a^{-1}(t)= 1/a(t) ; but u^{-1}(t), \gamma^{-1}(t), are inverse functions.