Z-transform X(z) = Σ_n x[n] z^{-n}: discrete-time analogue of Laplace

The (unilateral) Z-transform X(z) = Σ_{n=0}^∞ x[n] z^{-n} maps a causal sequence (x[n])_{n≥0} to a complex function X(z) on a region |z| > R_min, the 'region of convergence' (ROC). Key properties: linearity; shift-by-one Z{x[n − 1]} =…