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It is well known that the nonlinear filtering problem has important applications in both military and civil industries. The central problem of nonlinear filtering is to solve the Duncan-Mortensen-Zakai (DMZ) equation in real time and in a memoryless manner. In this paper, we shall extend the algorithm developed previously by S.-T. Yau and the second author to the most general setting of nonlinear filterings, where the explicit time-dependence is in the drift term, observation term, and the variance of the noises could be a matrix of functions of both time and the states. To preserve the off-line virture of the algorithm, necessary modifications are illustrated clearly. Moreover, it is shown rigorously that the approximated solution obtained by the algorithm converges to the real solution in the $L^1$ sense. And the precise error has been estimated. Finally, the numerical simulation support the feasibility and efficiency of our algorithm.

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