Editing Z-transform (section)

== History ==
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.<ref name="kanasewich">
{{cite book|url=https://books.google.com/books?id=k8SSLy-FYagC&q=inauthor%3AKanasewich++poles+stability&pg=PA249|title=Time Sequence Analysis in Geophysics|author=E. R. Kanasewich|publisher=University of Alberta|year=1981|isbn=978-0-88864-074-1|pages=186, 249}}</ref><ref>{{cite book  | title = Time sequence analysis in geophysics  | edition = 3rd  | author = E. R. Kanasewich  | publisher = University of Alberta  | year = 1981  | isbn = 978-0-88864-074-1  | pages = 185–186  | url = https://books.google.com/books?id=k8SSLy-FYagC&pg=PA185}}</ref>

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.<ref>{{cite journal |last1=Ragazzini |first1=J. R. |last2=Zadeh |first2=L. A. |title=The analysis of sampled-data systems |journal=Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry |date=1952 |volume=71 |issue=5 |pages=225–234 |doi=10.1109/TAI.1952.6371274|s2cid=51674188 }}</ref><ref>{{cite book  | title = Digital control systems implementation and computational techniques  | author = Cornelius T. Leondes  | publisher = Academic Press  | year = 1996| isbn = 978-0-12-012779-5  | page = 123  | url = https://books.google.com/books?id=aQbk3uidEJoC&pg=PA123  }}</ref>

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.<ref>
{{cite book
 | title = Sampled-Data Control Systems
 | author = Eliahu Ibrahim Jury
 | publisher = John Wiley & Sons
 | year = 1958
 }}</ref>{{r|JuryBook}}

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.{{r|JuryBook}}