It is assumed that you have noise u, with known mean and covariance, which is as well known as the state x with covariance P and it can be differentiated similarly as with respect to the state x. The differentiation in EKF is done in the mean of the noise (or state).

Often noise is additive i.e f(x,u) = g(x) + u, in which case V is an identity matrix, but for example if f(x,u) = g(x) + 2u (u is scalar), differentiating with respect to u leads to V=2. 

Dear Matti,

Thank you for the reply.

I have considered measurement noise as additive and there is no process noise.

i.e  xk = g(xk-1, uk-1)  

      z=h(x) + v. 

So it justifies to take V as identity matrix ?

Since I am considering the noise covariance constant rather than it varying at every time step, it implies V will be a constant matrix. Am I understanding correct?

Yes you are understanding correct: For model h(x)+v, V is an identity matrix, and when you are having constant measurement noise covariance then VkRkVkT = R (constant)

The derivatives in the EKF are with respect to the state variables. All that is happening is that the nonlinear model is being linearised around the current state estimate, so it's just a first order Taylor series for the dynamical and measurement equations. The measurement covariance is transformed using JRJ^T, where J is a Jacobian matrix (derivatives of the measurement function with respect to the state). The J that gets used is actually J(xhat_k|k)  ie evaluate at the most recent state estimate from the EKF.

Dear Graham,

Thank you for the answer. It is very helpful.

This is a good question.

In addition to the incisive answers already given, consider

R. van der Merwe, Sigma-point Kalman filters for probabilistic inference in dynamic space models, Ph.D. thesis, Oregon Health & Science University, 2004:

http://www.cslu.ogi.edu/publications/ps/merwe04.pdf

See Section 2.3.2, starting on page 27, for an introduction to the extended Kalman filter (EKF).   You may also find the unscented Kalman filter (UKF) introduced by Jilier in the mid-90s, helpful.  The UKF does not require the linearisation steps needed for the EKF and its performance has been shown to be superior to the EKF.

The UKF is introduced in Section 2.3.3, starting on page 30.   See, also, Section 5.5, starting on page 102, on conditions for uniform convergence of the extended Kalman PHD filter.

Dear James Peters,

Thank you for sending the link. Once I am through with EKF, I will consider it

With Regards,