Z transform

(mathematics) The z-transform of a sequence whose general term is ƒ_n is the sum of a series whose general term is ƒ_nz^-n, where z is a complex variable; n runs over the positive integers for a one-sided transform, over all the integers for a two-sided transform.

The preferred operational-calculus tool for analysis and design of discrete-time systems. (It should not be confused with the z transformation.) The role of the z transform with regard to discrete-time systems is similar to that of the Laplace transform for continuous systems. In fact, the Laplace transform is a specialized case of the z transform. The z transform is by far the more insightful tool, and the Laplace transform is just the limiting case of the z transform in a practical as well as a conceptual way. See also Control systems; Digital filter; Laplace transform; Linear system analysis.

It is useful to consider a band-limited real continuous signal, x(t), with no significant amount of energy above a frequency, fc. This signal is sampled at uniformly spaced intervals of time, T, 2T, 3T, …, nT, …, where the sampling interval, T, and the sampling frequency, f_s, are reciprocals of one another, and where f_s is greater than 2f_c, a condition necessary for unambiguous interpretation of the sampled signal. If the common shorthand notation x(nT) = x_n is used, the definition of the z transform of x(t) is given by the equation

The coefficient of z^−p is therefore the value of the pth sample of the time signal. This gives a great deal of physical insight to the use of the z transform, a feature not shared by the Laplace transform. The Laplace transform can be found by evaluating the limit of the z transform as T approaches zero. See also Analog-to-digital converter; Information theory.