Consider a causal continuous-time LTI system with transfer function
where . There is a zero at , as . There is also a pole at , as blows up at .
If , then we have , and becomes the identity system. In this case, the zero is “on top” of the pole, and we say that pole-zero cancellation has occurred. Usually, this phenomenon is discussed in the -domain, but what happens in the time-domain?
We can view as the cascade connection of two causal LTI systems
Let , and be the transfer functions of and , respectively. Then, we have . Suppose we apply a Dirac delta impulse, , to the system. The output of the first filter in the cascade will then be the natural mode of
where is the Heaviside step function. If , then the natural mode decays exponentially. Recall that the inverse Laplace transform of is . Then, we have that the impulse response of the overall system is
and, since for all , we obtain
If , then the zero cancels the pole and we get at the output of . In other words, the zero at annihilates the natural mode created by the pole at . In theory, we could use pole-zero cancellation to kill an unstable natural mode. In theory…
What if the pole-zero cancellation is not “exact”? Let , where is “small”. Then the impulse response of is
If we have a stable natural mode and the “weak” exponential will eventually vanish. The closer the zero is to the pole, the “weaker” the natural mode will be, until it completely disappears when there is exact pole-zero cancellation. However, if then we have an unstable natural mode that diverges regardless of how small is.
Alternatively, as , we can view as the parallel connection of the identity system and a causal LTI system
If , then becomes the identity system. In my humble opinion, viewing as a parallel connection of two systems is not very illuminating, as the annihilation of the natural mode in the time-domain (due to differentiation) is not evident.