The variance of energy estimates for the product model

Article type: Research Article

Authors: Smallwood, David

Affiliations: Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185-0553, USA. Tel.: +1 505 844 3205; Fax: +1 505 844 0078; E-mail: [email protected]

Abstract: A product model, in which {x(t)}, is the product of a slowly varying random window, {w(t)}, and a stationary random process, {g(t)}, is defined. A single realization of the process will be defined as x(t). This is slightly different from the usual definition of the product model where the window is typically defined as deterministic. An estimate of the energy (the zero order temporal moment, only in special cases is this physical energy) of the random process, {x(t)}, is defined as m_0=∫_{-∞}^{∞}|x(t)|^2dt=∫_{-∞}^{∞}|w(t)g(t)|^2dt Relationships for the mean and variance of the energy estimates, m_0, are then developed. It is shown that for many cases the uncertainty (4π times the product of rms duration, D_t, and rms bandwidth, D_f) is approximately the inverse of the normalized variance of the energy. The uncertainty is a quantitative measure of the expected error in the energy estimate. If a transient has a significant random component, a small uncertainty parameter implies large error in the energy estimate. Attempts to resolve a time/frequency spectrum near the uncertainty limits of a transient with a significant random component will result in large errors in the spectral estimates.

Journal: Shock and Vibration, vol. 10, no. 4, pp. 211-221, 2003