An introduction to the Z-transform and its application to sampled-data systems

This Data Item ESDU 86037 is an addition to the Dynamics Sub-series. The representation of a continuous signal by a sequence of equally spaced pulses each of which provides the value of the continuous signal at the particular sampling instant is introduced. The relationship of such a sequence to the Z-transform is established, and properties of the transform are given. The Z-transform is related to the Laplace transform and the relationships between poles and zeroes in the complex s-plane to those in the complex z-plane are provided. Using the Z-transform the relationship between the sampled input and output of a system element, the pulse transfer function, is found and the method of combining such functions for individual elements to obtain a pulse transfer function, is found and the method of combining such functions for individual elements of cascaded elements in open or closed loop form is illustrated. The delayed Z-transform and multirate sampling, which allow signal information to be obtained between sampling instants, are described. The use of data hold filters for reconstitution of the continuous signal is explained, and methods of obtaining the original sampled signal by inverting the Z-transform are given and illustrated.