# Laplace vs Fourier Transforms in Transmission Lines

Consider an electric transmission line with transient equations $$\frac{\partial v}{\partial x} =-Ri -L\frac{\partial i}{\partial t}, \qquad \frac{\partial i}{\partial x} =-Gv -C\frac{\partial v}{\partial t},$$ zero initial conditions $$v(x,0)=0, \qquad i(x,0)=0, \qquad x\in [0,L],$$ and sending end boundary conditions $$v(0,t)=v_s(t), \qquad i(0,t)=i_s(t), \qquad t\in [0,\infty).$$ I am studying wave attenuation by using the Fourier transform and ABCD matrix representation. If I were to take the inverse Fourier transform of the solution, I may not obtain the correct initial condition. If I use the Laplace transform instead, does the single-sided Laplace transform provide the same attenuation analysis by simply evaluating the complex Laplace variable on the imaginary axis? Is there another way to fix the ambiguity of the initial condition using Fourier transforms?

I hope my questions are clear. Any thoughts on this matter would be helpful. Thank you.

• I think no. This is a classical concern with Fourier analysis, especially when it is incorrectly applied to transient signals. The passionately discussed example is that of a critically sampled system at fs = 2*fmax, which is theoretically sufficient if the initial conditions are assumed correct, but practically not possible because the required initial conditions cannot be applied in a FT analysis. – P2000 Jul 6 at 21:29