Wavelet-like structure of rational models for power-law processes

Scale invariant rational systems are useful for modeling of 1/f processes which exhibit a power-law spectral density over a finite band. In this paper, we show that the impulse response of a scale-invariant rational system can essentially be expressed as a linear combination of dilations of a protoype waveform in the form of a damped complex exponential. Hence, scale-invariant rational systems exhibit a discrete wavelet-like structure where the term wavelet-like refers to the fact that there are no translations of the prototype and that the prototype does not satisfy the admissibility condition required of a wavelet. We also point out that this wavelet-like structure can be viewed as a deterministic version of the wavelet-based models for nearly-1/f processes.