I am developing a Particle Filter, and would like to understand some of the maths/ideas behind them more fully. For example, i understand that the particle filter is supposed to approximate the posterior pdf at each step. But i do not understand if it does this via particle weights, or via the distribution of the particles themselves...? Many of the equations seem to show it using the weights, i.e. the pdf is approximated by the sum of the weights each shifted to the correct state value by the dirac delta. Yet at other times the weights of all the particles are normalized, so i can only assume at this point that it is the actual distribution of the particles themselves that is being used...?
Anyone shed any light?