  We consider the quantum (trajectories) filtering equation for the case when
the system is driven by Bose field inputs prepared in an arbitrary non-zero
mean Gaussian state. The a posteriori evolution of the system is conditioned by
the results of a single or double homodyne measurements. The system interacting
with the Bose field is a single cavity mode taken initially in a Gaussian
state. We show explicit solutions using the method of characteristic functions
to the filtering equations exploiting the linear Gaussian nature of the
problem.
