Ztransformovat

Ztransformovat is a Czech verb used in signal processing to describe applying the z-transform to a discrete-time signal or sequence. The result, typically denoted X(z), is a function defined in the complex z-domain that encodes time-domain information in a form suitable for analysis and design of systems.

The bilateral z-transform of a sequence x[n] is defined as X(z) = sum over all n of x[n] z^{-n}, with a region of convergence (ROC) in the complex plane where the series converges. The unilateral (or one-sided) z-transform, X(z) = sum from n=0 to ∞ of x[n] z^{-n}, is used for causal sequences and has its own ROC. The inverse z-transform recovers x[n] from X(z), often via contour integration or by reference to transform tables and partial fraction expansion.

Key properties include linearity, time shifting, and the convolution theorem, which states that convolution in time becomes multiplication in the z-domain. The z-transform is related to the discrete-time Fourier transform (DTFT) via evaluation on the unit circle (z = e^{jω}), and to the Laplace transform as a discrete-time counterpart with the substitution z = e^{sT} for sampling interval T.

Common applications are analysis and design of digital filters, solving linear difference equations, and system identification. Examples illustrate ROC-dependent forms, such as X(z) = 1/(1 − a z^{-1}) for a causal sequence with ROC |z| > |a|.