Who Invented the Z Transform

"The techniques of the z-transform method are not new, for they can be actually traced back as early as 1730 when DeMoivre introduced the concept of the 'generating function' (which is actually identical to the z-transform) to probability theory."

Jury, E. I., Theory and Application of the z-Transform Method, (c) 1964, John Wiley & Sons, New York, Page 1.

References at the end of Chapter 1 show:

[5] Helm, H. A., "The z-Transformation," B.S.T. Journal, Vo 38, No. 1, 1956, pp 177-196 [32] Lago, G. V., "Additions to z-Transformation Theory for Sampled-Data Systems," Trans. AIEE Vol. 74, Part II, 1955, pp. 403-408

His name actually _ends_ with z: Witold Hurewicz, in 1947. It was named in 1952 by a sampled-data control group at Columbia University, one of who's grad student members taught a course at CCNY a year or two later that I audited and promptly forgot.

Jerry

Complex variables have been known since the time of Cauchy in the 18th century - remember all those contour integrals? You mean for engineering applications I presume? Also sampled systems were known in stats by Whittaker in the 1920s I think in Edinburgh who also discovered the sampling theorem.I also heard (in this newsgroup) that it was Prof Zadeh who coined the term z-transform though he did not name it Z after his own name.

Sanctus

Wasn't there this Russian ... ?

:-)

There was a Russian who also discovered the sampling theorem.

Sanctus

see.. wrote:

Neat! Thanks Jul.

[-Rick-]

Nyquist -- Swedish -- 1927

Kotelnikov -- Russian

Zhukovskii (Joukowski) -- Russian In mathematics today the conformal mapping of the complex plane z z + 1/z is called the Joukowski transformation. This gave Zhukovskii [2]:- ... a means of designing aerofoils using conformal mappings and the techniques of complex variables. Those Joukowski aerofoils were actually used on some aircraft, and today these techniques provide a mathematically rigorous reference solution to which modern approaches to aerofoil design can be compared for validation.

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Complex variables are a bit removed from difference equations. The analysis of sampled data was cumbersome until Hurewitz simplified it. Like many other simple ways to look at something, you can find lots of antecedents once it is isolated. (Special relativity is implicit in Maxwell's Treatise. Maxwell masked it with "aether".)

Jerry

Hurewicz was Polish, but born under the Czar's rule.

From

"Witold Hurewicz's father was an industrialist. Witold attended school in a Russian controlled Poland but with World War I beginning before he had begun secondary school, ..."

From

"A method for solving linear, constant-coefficient difference equations by Laplace transforms was introduced to graduate engineering students by Gardner and Barnes in the early 1940s. They applied their procedure, which was based on jump functions, to ladder networks, transmission lines, and applications involving Bessel functions. This approach is quite complicated and in a separate attempt to simplify matters, a transform of a sampled signal or sequence was defined in 1947 by W. Hurewicz as

[not reproduced here]

which was later denoted in 1952 as a "z transform" by a sampled-data control group at Columbia University led by professor John R. Raggazini and including L.A. Zadeh, E.I. Jury, R.E. Kalman, J.E. Bertram, B. Friedland, and G.F. Franklin.

"The Hurewicz equation is not expressed in the same way as the z transform we have introduced -- it is one-sided, and it is expressed as a function of the sampled data sequence f rather than the complex number z -- but the relationship is clear, and the applications were similar from the beginning. So perhaps the z transform should really be called the "Hurewicz transform".

"In any case, it is presumably not an accident that the z transform was invented at about the same time as digital computers."

Make of this what you will. I first heard if Hurewicz in 1953 or so, from an instructor who joked that symbolic logic is just "Booleshit". Don't confuse him with Hurwitz.

Jerry

What about Tustin in England, didn't he do a lot of work in this area?

fred.

Wasn't he the bilinear guy?

Jerry

Jerry Avins wrote in news:1qednboPe9cB snipped-for-privacy@rcn.net:

I think the Z transform was discovered by Zefram Cochran; the guy who will invent the warp drive.

So what did Shannon do?

Sanctus

Hi,

It was I who asked Julius Kasuma (by the way, what's happened to Julius?) to ask Prof. Zadeh if the z-transform (created by Zadeh and his colleagues at Columbia University years ago) was named after Zadeh himself.

Julius was a grad student at Berkeley at the time of my question and Prof. Zadeh was also at Berkeley.

Julius approached the professor & according to Julius, Prof. Zadeh said they used the letter "z" because that letter wasn't typically used for anything else (like "e" for voltage, "i" for current, "R" for resistance, etc.) in electral engineering.

Interesting huh?

See Ya', [-Rick-]

...

Impedance? OK, that's capitalized.

Jerry

He formalized the already known results (due to Whitaker, Kotelnikov, Hartley, and Nyquist) and mentions at the front of one of his papers on sampling that these ideas are known to anyone skilled in the art. But he is the one that connects them together in a formal sense. Also he goes on to define entropy (as applied to information and not thermodynamics) and then provides both the noiseless and noisy coding theorems, both of which are completely new.

Clay

"A Mathematical Theory of Communication" A must read for DSPers

His complete bibliography:

I got into this thread late: but it wasn't Laplace that invented the S-transform: it was Olover Heavaside. It's sometimes called the Heaviside Transform.

Pure mathematicians rejected his work because of a lack of proof to their expected standards but engineers used it becuase it worked. Engineers still use Heaviside Functions albeit modified to conform more to the pure mathematicians Laplace transform.

Hilbert Transforms were ofcourse developed by Hilbert and the Dirac delta by Dirac. Hilbert incidently had developed relativity before Einstein: Einstein who had read Hilberts papers simply 'gazzumped' him to publication and has been reaping the publicity since.

The "delta function" came before Dirac, but it was little known. He popularized the concept and showed how useful it can be to applications in quantum mechanics.

Really? Which aspects of relativity theory did David Hilbert invent?

Clay

Most detailed histories of almost any scientific activity show that there was someone who did some piece of work and another who later realized that it had a much broader scope. The recognition tends to go to the worker who realizes the importance.

All of the Maxwell equations were know before Maxwell but they had not been collected into a system to provide a common example of this.

Einstein's special relativity used the known Lorentz transformation.