The theory of optimal measurements

A. L. Shestakov ; A. V. Keller ; G. A. Sviridyuk

Journal of computational and engineering mathematics, Tome 1 (2014) no. 1, pp. 3-16

Résumé

The mathematical model (MM) of the measuring transducer (MT) is discussed. The MM is intended for restoration of deterministic signals distorted by mechanical inertia of the MT, resonances in MT's circuits and stochastic perturbations. The MM is represented by the Leontieff type system of equations, reflecting the change in the state of MT under useful signal, deterministic and stochastic perturbations; algebraic system of equations modelling observations of distorted signal; and the Showalter – Sidorov initial condition. In addition the MM of the MT includes a cost functional. The minimum point of a cost functional is a required optimal measurement. Qualitative research of the MM of the MT is conducted by the methods of the degenerate operator group's theory. Namely, the existence of the unique optimal measurement is proved. This result corresponds to input signal without stochastic perturbation. To consider stochastic perturbations it is necessary to introduce so called Nelson – Gliklikh derivative for random process. In conclusion of article observations of "noises" (random perturbation, especially "white noise") are under consideration.

Détail

Keywords: mathematical model of the measuring transducer, the Leontieff type system, the Showalter – Sidorov condition, cost functional, the Nelson – Gliklikh derivative, «white noise».