  Particle filters (PFs), which are successful methods for approximating the
solution of the filtering problem, can be divided into two types: weighted and
unweighted PFs. It is well known that weighted PFs suffer from the weight
degeneracy and curse of dimensionality. To sidestep these issues, unweighted
PFs have been gaining attention, though they have their own challenges. The
existing literature on these types of PFs is based on distinct approaches. In
order to establish a connection, we put forward a framework that unifies
weighted and unweighted PFs in the continuous-time filtering problem. We show
that the stochastic dynamics of a particle system described by a pair process,
representing particles and their importance weights, should satisfy two
necessary conditions in order for its distribution to match the solution of the
Kushner--Stratonovich equation. In particular, we demonstrate that the
bootstrap particle filter (BPF), which relies on importance sampling, and the
feedback particle filter (FPF), which is an unweighted PF based on optimal
control, arise as special cases from a broad class and that there is a smooth
transition between the two. The freedom in designing the PF dynamics opens up
potential ways to address the existing issues in the aforementioned algorithms,
namely weight degeneracy in the BPF and gain estimation in the FPF.
