ztranszformáció

The z-transform is a mathematical tool used in digital signal processing and control theory to transform a discrete-time signal, which is a sequence of numbers, into a function of a complex variable, typically denoted by z. It is analogous to the Laplace transform used for continuous-time signals. The z-transform is defined for a discrete-time signal x[n] as the infinite series X(z) = sum from n=-infinity to infinity of x[n]z^(-n). This transformation converts a sequence of time-domain values into a function in the z-domain.

The z-transform is particularly useful for analyzing and designing discrete-time systems. By transforming a system's impulse response into the z-domain, we can analyze its properties such as stability and frequency response more easily. Operations like convolution in the time domain correspond to simple multiplication in the z-domain, which simplifies system analysis and filter design. The region of convergence (ROC) is an important concept associated with the z-transform, as it defines the range of z values for which the transform converges. The ROC provides critical information about the system's causality and stability.

The inverse z-transform allows us to convert a function in the z-domain back to its original discrete-time signal. Common methods for computing the inverse z-transform include partial fraction expansion and contour integration. The z-transform is widely applied in areas such as digital filter design, spectral analysis, and the analysis of sampled data systems.