I have an input rectangle pulse, \$f(t) = rect(t/\tau) \$ and a transmission line whose transfer function depends on frequency. The transfer function of the t.l. is the transmitted wave divided by the incoming wave: \$(1+\Gamma)/((e^{j\gamma l}) + \Gamma e^{-j\gamma l})\$.

\$\Gamma\$ is the reflection coefficient. \$\Gamma = \left(Z_L-Z_0\right)/\left(Z_L+Z_0\right)\$

where \$ Z_0=\sqrt{\left(R^\prime+jwL^\prime\right)/\left(G^\prime+jwC^\prime\right)}\$ is the characteristic impedance of the line, \$Z_L \$ is a constant

\$\gamma\$ is the propagation constant. \$ \gamma=\sqrt{\left(R^\prime\ +\ jwL^\prime\right)\left(G^\prime+jwC^\prime\right)}\$

The transfer function \$H(s)\$ is now a complex function where \$s=j\omega\$. I am looking to plot \$g(t) \$ which would be the inverse fourier transform of \$F(s)H(s) \$. I also know how to obtain the Fourier series coefficients for \$f(t) \$, but I am not sure what to do with these.

How should I proceed to obtain the output function in the time domain?

  • \$\begingroup\$ Treat the input as a step change and multiply the Laplace T.F. by \$\dfrac{1}{s}\$ then do the reverse Laplace using tables and there you have it. \$\endgroup\$ – Andy aka Mar 13 '20 at 8:08
  • \$\begingroup\$ Fourier series applies to periodic signals. This is not the case here. \$\endgroup\$ – Chu Mar 13 '20 at 20:53
  • \$\begingroup\$ So what do you suggest @Chu \$\endgroup\$ – Hector Mar 13 '20 at 21:58
  • \$\begingroup\$ @Andyaka I don't think I can simplify this to the point where I can use a table \$\endgroup\$ – Hector Mar 13 '20 at 22:08
  • \$\begingroup\$ How good at partial fractions are you? \$\endgroup\$ – Andy aka Mar 13 '20 at 22:48

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