How to get likelihood of a bearing estimation?

Hi,

I am reading a paper on particle filter on bearing estimation. The system
 model is:

x_k = P*x_k-1 + w_k

P is a 4X4 matrix. x_k is the position and speed of a 2-D of the object
 position (a, a^, b, b^).

The measurement z_k is:

z_k=tan^-1(b_k/a_k) + v_k

w_k and v_k are Gaussian noise.

On particle filter, it needs likelihood to calculate posterior probability:

p(z_k|x_k)  


z_k is a scalar, while x_k is a vector. The pdf of p(z_k|x_k) would be very
complicated analytically. I think I may be in the wrong direction on this 
problem. The original problem was described at page 4 of this link:

http://www.ece.iastate.edu/~namrata/EE520/Gordonnovelapproach.pdf 


Could you tell me how to deal with likelihood p(z_k|x_k) in particle filter?
Thanks,