Parametrizations in the Delta operator

Abstract

So far in this book we have followed common practice by describing discrete time transfer functions with the usual shift operator z. This operator is easy to implement since it consists simply of a shift. However, a disadvantage of the shift operator is that it does not at all approximate the continuous time operator d(.)/dt. Heuristically, it might be suspected that a better correspondence is obtained between continuous and discrete time representations if the shift operator is replaced by a difference operator that is more like a derivative. Such operator, nowadays often called delta operator, is defined as\(\delta = \frac{{z - 1}}{{{T_s}}}\), where T s is the sampling period. The delta operator approximates the time derivative and converges to it when the sampling period tends towards zero.