This is the first of two papers concerned with the formulation of a continuous-time quantum-mechanical filter. Efforts focus on a quantum system with Hamiltonian of the formH 0+u(t)H 1, whereH 0 is the Hamiltonian of the undisturbed system,H 1 is a system observable which couples to an external classical field, andu(t) represents the time-varying signal impressed by this field. An important problem is to determine when and how the signalu(t) can be extracted from the time-development of the measured value of a suitable system observableC (invertibility problem). There exist certain quasiclassical observables such that the expected value and the measured value can be made to coincide. These are called quantum nondemolition observables. The invertibility problem is posed and solved for such observables. Since the physical quantum-mechanical system must be modelled as aninfinite-dimensional bilinear system, the domain issue for the operatorsH 0,H 1, andC becomes nontrivial. This technical matter is dealt with by invoking the concept of an analytic domain. An additional complication is that the output observableC is in general time-dependent.