Causal and non-causal discrete-time transfer functions and their applications

Causal and non-causal discrete-time transfer functions for some discrete-time system with zero-order hold (ZOH), first-order hold (FOH) or ideal sampler are distinguished and described. In the case of ZOH or FOH the discussed notions are essential for the system with a continuous time plant described by the rational proper transfer function with the order of numerator equal to that of the denominator, while in the case of ideal sampler for the plant with the order of numerator less by one than that of the denominator. It is noted that in the case of ZOH and FOH the causal discrete-time transfer functions have an additional causal mode which increases by one the order of the model. It is shown that for description of the closed-loop systems usually the causal transfer functions are used; however, in some cases the part of the system can be described by the non-causal transfer functions.