Ztransformations

The Z-transform is a mathematical tool used in discrete-time signal processing to map a sequence into a complex frequency domain. For a discrete-time sequence x[n], the bilateral Z-transform is X(z) = sum_{n=-∞}^{∞} x[n] z^{-n}, where z is a complex number. A unilateral Z-transform, often used for causal systems, is X(z) = sum_{n=0}^{∞} x[n] z^{-n}. The Z-transform provides a convenient representation of linear, time-invariant difference equations and enables analysis using algebraic methods in the Z-domain.

Convergence and region of convergence. The series converge only in a region of the complex plane known as the region of convergence (ROC). The ROC depends on x[n] and is bounded by poles. For causal sequences, the ROC lies outside the outermost pole; for anti-causal sequences, the ROC lies inside the innermost pole; two-sided sequences produce an annulus between poles.

Inverse transform and methods. The inverse Z-transform reconstructs x[n] from X(z). It can be obtained by contour integration, power-series expansion, or via tables and partial-fraction decomposition. In practice, tables and the residue theorem are commonly used.

Properties. The Z-transform is linear. Time-domain shifts correspond to multiplicative factors of z: Z{x[n - n0]} = z^{-n0} X(z). Z{a^n x[n]} = X(a z). Time reversal, x[-n], maps to X(1/z). Convolution in time becomes multiplication in the Z-domain: Z{x * y} = X(z) Y(z).

Poles, zeros, and stability. The poles of X(z) come from the denominator, and zeros from the numerator. For causal, stable systems, the ROC lies outside the outermost pole and includes the unit circle (|z| = 1).

Applications. The Z-transform underpins analysis and design of digital filters, solution of difference equations, spectral analysis, and systems identification in discrete-time signal processing.