I get it, about poles and zeroes, when we talk in terms of the transfer function.
But, if the transfer function is the ratio of the output to the input, then if the input signal is zero, then the gain should go to infinity, right?
Am I missing something?
The short answer, from Matlab
>> 0/0
ans = NaN
The gain is undefined if the input is zero because the output is also zero.
The long answer with some digressions:
If you look at a Control / Linear Systems Textbook, one should see that a system is defined as linear if it is BOTH zero state linear AND zero input linear. In most Signal Processing Textbooks, only the zero state linearity is stated which seems to be a contradiction but really isn't. Trying to Control a NonCausal system would seem to be kind-of futile, so $t=0$ is where things begin (mathematically expressed by the one sided Laplace Transforms ) but one has to account for initial conditions of physical objects like capacitor voltages and inductor currents so you can have output at $t=0$ with input $u(t)=0$, but the gain is still causal input to output.
In Signal Processing, you can record a signal and process forward and/or/both in-reverse time, so the most general expressions have lower integrals $t=-\infty$. The entire history of components such as capacitor voltages and inductor currents are realized as zero state conditions at $t=-\infty$ and the full input $u(t)$ history from $t=-\infty$
A transfer function is a property of a circuit or a black-box network with reference to its input and output ports. Causality requires that there is no output signal before there is an input signal.
