Ztranszformáltnak

The Z-transform is a mathematical tool used in digital signal processing and control theory to transform a discrete-time sequence into a continuous-time function of a complex variable. This transformation is analogous to the Laplace transform used for continuous-time signals. The Z-transform is particularly useful for analyzing and designing digital filters, solving linear constant-coefficient difference equations, and understanding the behavior of discrete-time systems.

The unilateral Z-transform of a discrete-time sequence x[n] is defined as X(z) = sum_{n=0}^{infinity} x[n] z^{-n}, where z is a complex variable. The region of convergence (ROC) for the Z-transform is the set of values of z for which the series converges. The ROC is crucial as it uniquely determines the original sequence. The inverse Z-transform allows us to recover the original discrete-time sequence from its Z-transform.

The Z-transform provides a frequency-domain representation of discrete-time signals. Properties like linearity, time shifting, and convolution simplify the analysis of complex systems. For example, the convolution of two sequences in the time domain corresponds to simple multiplication of their Z-transforms in the z-domain. This property is fundamental to filter design. The Z-transform is a powerful technique for understanding the stability and performance of discrete-time systems.