Characterization of stochastic transients and transmission media: The method of power-moments spectra
A method is presented for the mathematical characterization of stochastic transmission media and stochastic transients that may serve as inputs to either deterministic or stochastic media. The media are treated as stochastic "black boxes": that is, linear two-terminal time-invariant systems whose unit-impulse response functions are non-repeatable. Ensemble average properties of transient waveforms and impulse response functions are characterized by a new hierarchy of functional descriptors which possess simple, exact, algebraic input-response resations. These functional descriptors are called power-moments spectra—their integrals yield the moments with respect to time of the ensemble-averaged instantaneous power of the process being described. Series expansions are developed which efficiently approximate the ensemble-averaged instantaneous power of stochastic transients from these moments. Formulae are derived for generating the power-moments spectra of waveforms given either empirically or analytically. The low-order power-moments spectra of deterministic waveforms are related to the group delay and to a new definition of group dispersion; these quantities are, in turn, related to the time centroid and nominal duration of finite bandwidth waveforms. The method has general applicability to problems involving either the scattering of transient waveforms or their transmission in stochastic media.