Realization of irrational transfer functions

Presented in this brief is an approach for realizing an irrational transfer function as a linear causal discrete-time system. The approach involves finding a mapping function which transforms the given irrational transfer function into a rational one. From the mapping function are derived a pair of mapping filters. The rational transfer function is realized as a linear causal time-invariant system using well-established techniques. When this system is sandwiched between the two mapping filters, a realization of the irrational transfer function is obtained. Methods are given for implementing causal mapping filters. Several examples are included in order to illustrate the application of the approach. The first example is the realization of a causal linear phase IIR digital filter. The remaining pair of examples involve a simple generalization of two well-known rational systems: the multiple-pole filter and the digital resonator.