How to solve the Fast Fourier transform?

As "L" stands for Laplace in the Laplace transform and "F" stands for Fourier in the Fourier transform, what does the letter "z" stand for in the Z-transform?

  • Probably, for "Zadeh, Lotfali Askar", an american scientist, also linked to the same transform.

  • Answer:

    The auxiliary variable ”z”, was used by Ragazzini and Zadeh in [1] to denote the sampled/digitized cousin of its continuous counterpart letter ”s”, the 'standard' independent variable in Laplace transforms. They explicitly state this in their paper (See below). Why 'z' you ask? One simple guess is: 'T' is often used for 'time', U was already in use for unit function, V- (Stumper!), w- frequency, (x-y)- ( taken up for the 2-D Cartesian spatial co-ordinates) and 'z' just sounds reasonable. My guess is that there is nothing more to it. Will be glad to be proven wrong. In any case, many leading minds in signal processing theory prefer calling it Laurent transform/series [2]. [1] J. R. Ragazzini and L. A. Zadeh (1952). "The analysis of sampled-data systems". Trans. Am. Inst. Elec. Eng. 71 (II): 225–234. [2] Graf, Urs. Applied Laplace transforms and z-transforms for scientists and engineers: a computational approach using a Mathematica package. Springer, 2004.

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Why not consult Wikipedia:The basic idea now known as the Z-transform was known to Laplace, and re-introduced in 1947 by W. Hurewicz as a tractable way to solve linear, constant-coefficient difference equations. It was later dubbed ”the z-transform” by Ragazzini and Zadeh in the sampled-data control group at Columbia University in 1952. The name ”Z-transform” may have been derived from the idea of the letter ”z” being a sampled/digitized version of the letter ”s” often used as the independent variable in Laplace transforms. This seemed appropriate since the Z-transform can be viewed as a sampled version of the Laplace transform. Another possible source is the presence of the letter ”Z” in the names of both Ragazzini and Zadeh who published the seminal paper. The naming deviates from the more commonly used scientific naming practice of associating a method or theorem with the principal investigator (i.e. Fourier, Laplace, Hartley, etc.). The modified or advanced Z-transform was later developed and popularized by E. I. Jury. The idea contained within the Z-transform is also known in mathematical literature as the method of generating functions which can be traced back as early as 1730 when it was introduced by De Moivre in conjunction with probability theory. From a mathematical view the Z-transform can also be viewed as a Laurent series where one views the sequence of numbers under consideration as the (Laurent) expansion of an analytic function (the Z-transform).

John Knight

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