The Laplace transform - Steven W. Smith Book - impulse response cancellation method

Question

While studying the Laplace transform using Steven W. Smith Book I found something uncomprehending. In the 32th chapter - The Laplace Transform, page 590, last paragraph describes the cancelling phenomena when an impulse response is cancelled using an exponentially weighted sinusoid (see picture below). When cancelling occurs then we are dealing with zero or pole at the s-plane. What is not clear for are the products of the probing waveform and impulse response examples (3rd column in the figure below):

a) Decreasing with time: how it can be said that $p(t) \times h(t)$ is finite?
b) Exact cancellation (zero): how it can be said that $p(t) \times h(t)$ is zero?
c) Too slow of increase: how it can be said that $p(t) \times h(t)$ is finite?
d) Exact cancellation (pole): how it can be said that $p(t) \times h(t)$ is infinite?
e) Too fast of increase: how it can be said that $p(t) \times h(t)$ is undefinied?

I would be glad if someone could explain me what is the connection between $p(t) \times h(t)$ shape and if it is pole or zero.

Answer 1

Peter's comment is correct, it's about the integral of the product $p(t)h(t)$:

$$I=\int_{-\infty}^{\infty}p(t)h(t)dt\tag{1}$$

The impulse response $h(t)$ has the following form:

$$h(t)=\delta(t)+c_1\, e^{\sigma t}\cos(\omega_0t),\qquad t\ge 0\tag{2}$$

with some constant $c_1$ and some $\sigma<0$.

From what I understand, the function $p(t)$ must look like

$$p(t)=c_2\,e^{\sigma_pt}\cos(\omega_0t)\tag{3}$$

with some constant $c_2$. I can't be sure but it seems likely that $c_1=c_2$.

Now we consider $5$ cases:

1. $\sigma_p<0$
2. $\sigma_p=0$
3. $\sigma_p>0$ and $\sigma+\sigma_p<0$
4. $\sigma_p=-\sigma$
5. $\sigma_p>0$ and $\sigma+\sigma_p>0$

In the first three cases we obtain for the product $p(t)h(t)$ a decaying exponential times $\cos^2(\omega_0t)$, plus a Dirac impulse, the integral of which is finite in all cases. In the second case, the exponent equals $\sigma$, and it appears that the constants can be chosen in such a way that the value of the integral can be made zero. I don't see any clear explanation of this in the book chapter, but I might be missing something.

In cases $4$ and $5$, the exponential is constant ($4$) or growing indefinitely ($5$), hence the integral diverges in both cases.