A digital tool employing the mathematical z-transform converts a discrete-time signal, represented as a sequence of values, into a complex frequency-domain representation. For example, a sequence representing the amplitude of a sound wave sampled at regular intervals can be transformed into a function showing how its energy is distributed across different frequencies. This process is analogous to the Fourier transform for continuous signals.
This conversion facilitates analysis and manipulation of discrete-time systems, such as digital filters and control systems. It simplifies operations like convolution and allows for stability analysis using tools like the root locus. Developed in the mid-20th century, this mathematical framework is fundamental to modern digital signal processing and control theory, underpinning technologies ranging from audio processing to industrial automation.