The z-transform is useful for the manipulation of discrete data sequences and has acquired a new significance in the formulation and analysis of discrete-time systems. It is used extensively today in the areas of applied mathematics, digital signal processing, control theory, population science, economics. These discrete models are solved with difference equations in a manner that is analogous to solving continuous models with differential equations. The role played by the z-transform in the solution of difference equations corresponds to that played by the Laplace transforms in the solution of differential equations.
The function notation for sequences is used in the study and application of z-transforms. Consider a function defined for that is sampled at times , where is the sampling period (or rate). We can write the sample as a sequence using the notation . Without loss of generality we will set and consider real sequences such as, . The definition of the z-transform involves an infinite series of the reciprocals
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