Dear Tamer,

The formula you provide for Rk is meaningless until you specify the value of t. Maybe you mean Rk = R(T)/T (where T, as you said, is the sampling time)?

Anyway, your formula seems to imply a linear growth of the noise covariance with time. This is not inherent to the Kalman filter, but to the assumptions you make on the time evolution of noise. Different stochastic models lead to different relations between R(t) and Rk. In your case, it seems that the noise is modelled as a diffusion process (a continuous-time version of a random walk, something known as a Wiener process when the differential steps are Gaussian). In this kind of processes, the error grows as the square root of time, and hence its variance (or covariance) grows linearly with time. Thus, R(t) = R1*t for some R1 (note that R1 equals R(t) at t = 1). If you now take T (your sampling time) as the time unit, you get Rk = R1. Thus R(t) = Rk*t, where t is given in multiples of T. In particular, Rk = R(T)/T. Actually Rk = R(t)/t for any t (which allows to regard Rk as the time derivative of R(t)).

If, instead of a diffusion process, you modelled noise as a stationary Gauss-Markov process, for instance, then you'd have that R(t) would not depend on time (hence R(t) = Rk regardless of the value of t). So, you see that the relation between Rk and R(t) depends on you underlying stochastic model (the fact that you're using a Kalman filter has nothing to do with it). The choice of the stochastic model depends on the physical process you're trying to model. Making the right choice for your filter is not an easy task, and is often the key to success.

By the way, the same considerations apply to the process noise of the Kalman filter, not only to the measurement noise.

Hope this helps...

To compute the discrete form of the measurement noise matrix (often denoted as RkR_kRk​ in Kalman filtering or estimation problems), you need to understand the context in which this matrix is used.