  This paper is concerned with a filtering problem for a class of nonlinear
quantum stochastic systems with multichannel nondemolition measurements. The
system-observation dynamics are governed by a Markovian Hudson-Parthasarathy
quantum stochastic differential equation driven by quantum Wiener processes of
bosonic fields in vacuum state. The Hamiltonian and system-field coupling
operators, as functions of the system variables, are represented in a Weyl
quantization form. Using the Wigner-Moyal phase-space framework, we obtain a
stochastic integro-differential equation for the posterior quasi-characteristic
function (QCF) of the system conditioned on the measurements. This equation is
a spatial Fourier domain representation of the Belavkin-Kushner-Stratonovich
stochastic master equation driven by the innovation process associated with the
measurements. We also discuss a more specific form of the posterior QCF
dynamics in the case of linear system-field coupling and outline a Gaussian
approximation of the posterior quantum state.
