Starred transform

Remarks

The impulse-sampled signal is $x$ times a Dirac comb of period $T$, restricted to $t \ge 0$. If $X(z)$ is the unilateral Z-transform of the samples $x_n = x(nT)$, then $X^*(s) = X(e^{sT})$. Example: for $x(t) = e^{-at}$ with $a \in \C$, $X^*(s) = \sum_{n \ge 0} e^{-n T (s + a)} = 1/(1 - e^{-T(s + a)})$ for $\operatorname{Re}(s + a) > 0$. Non-example: for $x(t) = e^{t^2}$ the terms $e^{n^2 T^2 - nTs}$ do not tend to $0$ for any $s$, so the series converges nowhere and $x$ has no starred transform. $X^*$ is periodic in the imaginary direction: $X^*(s + 2\pi i/T) = X^*(s)$, since $e^{-nT \cdot 2\pi i/T} = 1$.