Need help to understanding concept of particle filter I get confuse with the basic theory of particle filter for tracking single object..
What I want to ask is:


In some paper, they explained about velocity in state vector.. Are we 
randomly put number to velocity or get it from some formula? 
In pdf by eiji ota, he desrcibe system model as Xn=f(Xn-1)+Wn where Wn is noise.. Why need noise in particle filter?
And can you give me simple explanation in prediction step how the particles predict at t=1? I'm still blank in there..
The following papers are good resources:


Gordon, N. J.; Salmond, D. J. and Smith, A. F. M. (1993). "Novel approach to nonlinear/non-Gaussian Bayesian state estimation". IEE Proceedings F on Radar and Signal Processing 140 (2): 107–113. 
Arulampalam, M.S.; Maskell, S.; Gordon, N.; Clapp, T.; (2002). "A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking". IEEE Transactions on Signal Processing 50 (2): 174–188. 
Doucet, A.; Johansen, A.M.; (December 2008). "A tutorial on particle filtering and smoothing: fifteen years later". Technical report (Department of Statistics, University of British Columbia). (Available here)


I'd recommend you read at least the first one. 

A particle filter works on a model specified by 2 equations (a hidden Markov model):

1) The state equation
$x_{k+1} = f_k (x_k,w_k)$ which tells you how the state (minimal set of variables to describe a system) evolves as being driven by noise.

2) The measurement equation $y_k = h_k (x_k, v_k)$ which maps the state to what you observe. 

Given the observations $y_0, \ldots, y_n$, estimate the distribution of $x_n$ is the problem the particle filter tries to solve. Once you have the approximated distribution of $x_n$ (which is the distribution of the particles), you can estimate $x_n$ (say, by averaging the particles). 

The particle filter essentially starts with a bunch of samples (called particles), evolves the state by running each particle through the state equation and re-samples the particles based on the observation you see in order to make the distribution of particles consistent with the observations.