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.