I'm reading 'Discrete Time Control Systems' book by Ogata and came across a few statements (specifically, (3-1) and (3-2)) which I have not been able to understand.
It is said that any continuous signal can be sampled and the output represented as $$y(t) = \sum_{n=- \infty}^{+\infty}x(nT)\delta(t-nT) $$
Now taking laplace transform $$\begin{align} Y(s) &= \sum_{n=- \infty}^{+\infty}x(nT)\mathscr{L}\{\delta(t-nT)\} \\ &= \sum_{n=- \infty}^{+\infty}x(nT)e^{-nTs} \\ \end{align}$$
Now I have a confusion:
Is the $\delta(t)$ function
- the dirac delta function, so that $\mathscr{L}\{\delta(t-nT)\} = e^{-nTs} $ but then the signal representation makes no sense as there is infinite amplitude in the output signal at multiples of $nT$
- or is it the unit impulse function (value $1$ at $t=0$ and value $0$ everywhere else) in which case how exactly has $Y(s)$ been evaluated?