Some restrictions of the non-causal impulse response functions

The characterization of the non-causal impulse response functions is considered. Specifically, expressions for rational, continuous, non-causal impulse response functions corresponding to different rational factorizations of the spectral density function of continuous ARMA(p, q) models are derived. Furthermore, with the restriction on the continuity condition of impulse response at the origin, certain models are found to be inadmissible representations of the non-causal impulse response functions.