In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex valued frequency-domain (the z-domain or z-plane) representation.

It can be considered a discrete-time equivalent of the Laplace transform (the s-domain or s-plane). This similarity is explored in the theory of time-scale calculus.

While the continuous-time Fourier transform is evaluated on the s-domain's vertical axis (the imaginary axis), the discrete-time Fourier transform is evaluated along the z-domain's unit circle. The s-domain's left half-plane maps to the area inside the z-domain's unit circle, while the s-domain's right half-plane maps to the area outside of the z-domain's unit circle.

In signal processing, one of the means of designing digital filters is to take analog designs, subject them to a bilinear transform which maps them from the s-domain to the z-domain, and then produce the digital filter by inspection, manipulation, or numerical approximation. Such methods tend not to be accurate except in the vicinity of the complex unity, i.e. at low frequencies.

The foundational concept now recognized as the Z-transform, which is a cornerstone in the analysis and design of digital control systems, was not entirely novel when it emerged in the mid-20th century. Its embryonic principles can be traced back to the work of the French mathematician Pierre-Simon Laplace, who is better known for the Laplace transform, a closely related mathematical technique. However, the explicit formulation and application of what we now understand as the Z-transform were significantly advanced in 1947 by Witold Hurewicz and colleagues. Their work was motivated by the challenges presented by sampled-data control systems, which were becoming increasingly relevant in the context of radar technology during that period. The Z-transform provided a systematic and effective method for solving linear difference equations with constant coefficients, which are ubiquitous in the analysis of discrete-time signals and systems.

The method was further refined and gained its official nomenclature, "the Z-transform," in 1952, thanks to the efforts of John R. Ragazzini and Lotfi A. Zadeh, who were part of the sampled-data control group at Columbia University. Their work not only solidified the mathematical framework of the Z-transform but also expanded its application scope, particularly in the field of electrical engineering and control systems.

A notable extension, known as the modified or advanced Z-transform, was later introduced by Eliahu I. Jury. Jury's work extended the applicability and robustness of the Z-transform, especially in handling initial conditions and providing a more comprehensive framework for the analysis of digital control systems. This advanced formulation has played a pivotal role in the design and stability analysis of discrete-time control systems, contributing significantly to the field of digital signal processing.

Interestingly, the conceptual underpinnings of the Z-transform intersect with a broader mathematical concept known as the method of generating functions, a powerful tool in combinatorics and probability theory. This connection was hinted at as early as 1730 by Abraham de Moivre, a pioneering figure in the development of probability theory. De Moivre utilized generating functions to solve problems in probability, laying the groundwork for what would eventually evolve into the Z-transform. From a mathematical perspective, the Z-transform can be viewed as a specific instance of a Laurent series, where the sequence of numbers under investigation is interpreted as the coefficients in the (Laurent) expansion of an analytic function. This perspective not only highlights the deep mathematical roots of the Z-transform but also illustrates its versatility and broad applicability across different branches of mathematics and engineering.