# How to find the Fourier series of an output given input and transfer function

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?

• 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. – Andy aka Mar 13 '20 at 8:08
• Fourier series applies to periodic signals. This is not the case here. – Chu Mar 13 '20 at 20:53
• So what do you suggest @Chu – Hector Mar 13 '20 at 21:58
• @Andyaka I don't think I can simplify this to the point where I can use a table – Hector Mar 13 '20 at 22:08
• How good at partial fractions are you? – Andy aka Mar 13 '20 at 22:48