Ztransformácia

Z-transformácia, also referred to as the Z-transform or discrete-time Z-transform, is a mathematical tool used to analyze discrete-time signals and linear time-invariant systems by mapping a sequence x[n] into a complex function X(z). It is especially helpful for solving difference equations and for designing digital filters.

Formally, the bilateral Z-transform is defined as X(z) = sum_{n=-∞}^{∞} x[n] z^{-n}, where z is a complex variable. A unilateral (one-sided) version is X(z) = sum_{n=0}^{∞} x[n] z^{-n}, commonly used for causal systems. The choice between bilateral and unilateral depends on the problem context.

The convergence of these series holds only in a region of the complex plane called the region of convergence (ROC). The ROC is determined by the sequence’s growth and its poles; for causal sequences it typically extends to infinity outside a outermost pole, for anti-causal sequences inside an innermost pole, and for two-sided sequences it forms an annulus between poles. The DTFT, a frequency-domain representation, is obtained by evaluating X(z) on the unit circle z = e^{jω}, provided the ROC includes |z| = 1.

Inversion retrieves x[n] from X(z) using contour integration or partial fraction expansion, given a known ROC. Key properties include linearity, time-shifting (a shift in time corresponds to a multiplicative factor z^{-k} in X(z)), and the convolution-multiplication relationship (the product of Z-transforms corresponds to a discrete-time convolution in x[n]).

Applications span digital signal processing and control, including spectral analysis, filter design, and the solution of difference equations. A simple example: x[n] = a^n u[n] has X(z) = 1 / (1 − a z^{-1}) with ROC |z| > |a|.