*If the ROC contains the unit circle (i.e., |z| = 1) then the system is stable.*

It was later dubbed "the z-transform" by Ragazzini and Zadeh in the sampled-data control group at Columbia University in 1952.

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.

The following substitution is used: from the Z-domain to the Laplace domain.

Through the bilinear transformation, the complex s-plane (of the Laplace transform) is mapped to the complex z-plane (of the z-transform).

This extends to cases with multiple poles: the ROC will never contain poles.

In example 2, the causal system yields an ROC that includes |z| = ∞ while the anticausal system in example 3 yields an ROC that includes |z| = 0. The stability of a system can also be determined by knowing the ROC alone.In the case where the ROC is causal (see Example 2), this means the path C must encircle all of the poles of is stable, that is, when all the poles are inside the unit circle.With this contour, the inverse Z-transform simplifies to the inverse discrete-time Fourier transform, or Fourier series, of the periodic values of the Z-transform around the unit circle: The Z-transform with a finite range of n and a finite number of uniformly spaced z values can be computed efficiently via Bluestein's FFT algorithm.To learn more or modify/prevent the use of cookies, see our Cookie Policy and Privacy Policy.Find the response of the system $s(n 2)-3s(n 1) 2s(n) = \delta (n)$, when all the initial conditions are zero.For example, the location of zeros and poles move from inside the unit circle using one definition, to outside the unit circle using the other definition.where C is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC).The discrete-time Fourier transform (DTFT)—not to be confused with the discrete Fourier transform (DFT)—is a special case of such a Z-transform obtained by restricting z to lie on the unit circle.The region of convergence (ROC) is the set of points in the complex plane for which the Z-transform summation converges.Due to periodicity, there are only a finite number of unique amplitudes, which are readily computed by the much simpler discrete Fourier transform (DFT).(See DTFT; periodic data.) The bilinear transform can be used to convert continuous-time filters (represented in the Laplace domain) into discrete-time filters (represented in the Z-domain), and vice versa.

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## Chapter 6 - The Z-Transform

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