ztransformaation

The z-transform is a mathematical tool used in the analysis of discrete-time signals and linear time-invariant systems. For a discrete-time sequence x[n], the bilateral z-transform is defined by X(z) = sum from n = -infinity to infinity of x[n] z^{-n}, where z is a complex variable. A unilateral variant, X(z) = sum from n = 0 to infinity of x[n] z^{-n}, is often used for causal sequences. The sums converge in a region of the complex plane known as the region of convergence (ROC). The ROC, together with the poles of X(z), determines both the transform’s existence and the time-domain behavior of the original sequence.

The z-transform has several key properties. It is linear, so the transform of a sum is the sum of the transforms. Time shifting corresponds to multiplication by a power of z: Z{x[n - k]} = z^{-k} X(z) for the bilateral transform. Convolution in time translates to multiplication in the z-domain: if y[n] = x[n] * h[n], then Y(z) = X(z) H(z) within the intersection of their ROCs. The transform is also useful for solving difference equations and for analyzing or designing digital filters, since system behavior can be studied via poles and zeros in the Z-domain.

Relation to other transforms: on the unit circle where z = e^{jω}, the bilateral z-transform becomes the discrete-time Fourier transform (DTFT). The inverse z-transform reconstructs x[n] from X(z) under appropriate ROC conditions.

Example: for a causal exponential x[n] = a^n u[n], the bilateral z-transform is X(z) = 1 / (1 − a z^{-1}) with ROC |z| > |a|. This example illustrates poles, ROC, and stability considerations.