Remarks on the modulating function method for impulse response identification

This note concerns the applications of the modulating function method to the input-output relation y(t)= \int\liminf{0}\limsup{t}\phi(t-s)u(s)ds . The problem is to determine the coefficients c k in the differential equation \sum\liminf{k=0}\limsup{N}c_{k} \phi^{(k)}(t)=0 from integral transforms of y(t) and u(t) over a finite interval [ 0, T ]. The main objective is to present a device which obtains the c k 's via the inversion of a system of N linear equations, whereas the straightforward approach leads to a system involving twice as many equations.