The Z Transformation

The desire to use digital computers in automatic control loops created the need for methods with which to analyze systems that are partly continuous and partly discrete. Since the methods of network theory could he applied to the analysis of the continuous part of such a hybrid system, it was natural that such methods should be extended to include the discrete case. This resulted in the z transform introduced by Raggazini and Zadeh. There is today an extensive literature devoted to the z transform." 3 ' 4 However, the fundamental assumption of the z transform derivation is that the process of instantaneous sampling is equivalent to the amplitude modulation of a train of unit impulses by the "sampled" function. Hut the unit impulse as commonly defined has infinite height and zero width, and the process of amplitude modulating such a function is not intuitively clear. While it is true that such a process may be considered as an approximation to the behavior of a linear network with an amplifier and sampling switch, "impulse sampling" bears no simple relation to the manner in which the digital computer operates. The digital computer, in the type of real time operation typical of control system applications, works with sequences of numbers which represent a continuous function evaluated at particular instance of time. Since these numbers must of necessity be finite, "impulse sampling" is not an obvious mathematical model for describing the working of the computer. It is the intention of this paper to define the problem from the point of view of operations within the computer and to develop a rigorous and appealing derivation of the z transform.