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In this section, we leverage the previously presented concept of importance sampling to derive the basis for a recursive nonlinear estimator using Monte Carlo integration [4]. This type of filter is generally referred to as a recursive particle filter.

Consider the following general system model:

(36.112)

(36.113)

where x_k is the state vector at time k, f(·, ·) is the process model function at time k – 1, w_{k − 1} is the process noise vector, h(·, ·) is the observation function, and v_k is the measurement noise vector at time k. The noise vectors are assumed to be independent of each other and in time with a known density function. Note that Gaussian densities are not required or assumed.

Assuming we begin with a known posterior density, p(x_{k − 1}| ℤ_{k − 1}). If N samples are drawn from an associated proposal density,

(36.114)

With normalized weights given by

(36.115)

where κ is the normalization factor required such that the sum of weights is unity, the posterior density function is expressed by the collection of particles and weights

(36.116)

or, equivalently

(36.117)

Our goal is to estimate the posterior density at time k, p(x_k| ℤ_k), by incorporating the statistical process model and the observation at time k. The density function of interest can be written as

(36.118)

(36.119)

Assuming our proposal density can be factored:

(36.120)

the posterior particle locations can be sampled from

(36.121)

Thus, the associated particle weights at time k can be calculated in a similar fashion as Eq. 36.108:

(36.122)

Substituting Eqs. 36.119 and 36.120 into Eq. 36.122 yields

(36.123)

Note that this equation is a function of the posterior weights at time k – 1; thus, the right‐hand fraction of Eq. 36.123 can be replaced according to Eq. 36.108, which yields the final particle weight update equation from time k – 1 to time k:

(36.124)

which can be normalized such that the collection of weights sums to one, thus approximating the posterior density as

(36.125)

In this manner, the particle locations and weights can be continuously maintained and updated using a recursive estimation framework.