I've come across that the matched $z$-transform maps poles of the $s$-plane design to locations in the $z$-plane. My question is, what is the $s$-plane and what does this mean? I'm aware that the variable $s$ can often be used to denote the complex plane but the fact that this has been used in conjunction with the $z$ variable has confused me slightly.
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The $s$-plane is the complex plane associated with the Laplace transform, i.e., with transfer functions of continuous-time systems, whereas the $z$-plane is the complex plane associated with the $\mathcal{Z}$-transform, i.e., with transfer functions of discrete-time systems. The matched $Z$-transform is one way of transforming continuous-time systems to discrete-time systems.
Take a look at this answer for more information about transformations from continuous-time to discrete-time.
