Ztransforms

Z-transforms are a standard tool in digital signal processing for analyzing discrete-time signals and systems. The bilateral Z-transform of a sequence x[n] is X(z) = sum_{n=-∞}^{∞} x[n] z^{-n}, defined for complex z where the series converges. The unilateral Z-transform uses n ≥ 0: X(z) = sum_{n=0}^{∞} x[n] z^{-n}.

The region of convergence (ROC) is the set of z values for which the series converges. The ROC is typically an annulus or a half-plane and is essential for interpreting poles, zeros, and stability. The transform is analytic within the ROC.

For many common sequences, X(z) is a rational function. Poles are the z-values where X(z) becomes unbounded; zeros are where X(z) vanishes. The ROC, together with pole locations, determines stability and causality of the corresponding system.

Key properties include linearity; time-domain shifting corresponds to multiplication by z^{-k} (within the ROC); convolution in time becomes multiplication in the Z-domain: Z{x[n] * h[n]} = X(z) H(z).

The ROC and pole pattern relate to stability: for a causal LTI system, the ROC extends outward from the outermost pole; the system is BIBO stable if the impulse response is absolutely summable, which typically implies the unit circle lies in the ROC.

On the unit circle, z = e^{jω}, the Z-transform reduces to the discrete-time Fourier transform, giving the frequency response of the system.

Applications include solving difference equations, digital filter design, and analysis of discrete-time control systems.