A comparison of discrete time system representations based on delta and w operators

Traditionally shift operators have been used to represent discrete time systems. However they have the drawbacks of often introducing unstable zeros into the discretization of a minimum phase continuous time system and not giving convergence to the continuous time behaviour as the sampling time tends to zero. The delta operator has been shown to perform better in both these respects, and many authors consider that it should replace the shift operator. However there are also a number of other possibilities. The question then arises of whether the delta operator is indeed the most appropriate operator to be used in the representation of discrete time systems. In this work the performance of δ-operator system representations is compared with that of w-operator ones. It is shown that both operators are suitable to be used in systems representations and superior to the commonly used shift operator, but that, on balance, the δ operator is preferable to the w operator