Starred transform
In applied mathematics, the starred transform, or star transform, is a discrete-time variation of the Laplace transform, so-named because of the asterisk or "star" in the customary notation of the sampled signals. The transform is an operator of a continuous-time function x ( t ) {\displaystyle x(t)} , which is transformed to a function X ∗ ( s ) {\displaystyle X^{*}(s)} in the following manner: X ∗ ( s ) = L [ x ( t ) ⋅ δ T ( t ) ] = L [ x ∗ ( t ) ] , {\displaystyle {\begin{aligned}X^{*}(s)={\mathcal {L}}[x(t)\cdot \delta _{T}(t)]={\mathcal {L}}[x^{*}(t)],\end{aligned}}} where δ T ( t ) {\displaystyle \delta _{T}(t)} is a Dirac comb function, with period of time T. The starred transform is a convenient mathematical abstraction that represents the Laplace transform of an impulse sampled function x ∗ ( t ) {\displaystyle x^{*}(t)} , which is the output of an ideal sampler, whose input is a continuous function, x ( t ) {\displaystyle x(t)} .