Time domain characteristics of rational systems with scale-invariant frequency response

The class of rational systems characterized by a magnitude response which is scale invariant for a specific scale change, or equivalently γ-homogeneous rational systems, are useful in modeling power-law processes. Such systems can be constructed by cascading frequency-scaled replicas of a prototype rational function which satisfies certain conditions. In this communication, we study the time domain characteristics of such systems. We show that, in the case of degree-1 or degree-2 prototypes, the magnitudes of the partial fraction expansion coefficients constitute a geometric sequence. Furthermore, in the degree-2 case, the angles of the partial fraction expansion coefficients are equal. Using these properties, we demonstrate that the impulse response of a e, γ-homogeneous rational system is essentially a linear combination of dilations of a prototype waveform and therefore exhibits a wavelet-like decomposition.