With sampling time $T$, and a continuous measuring model: $$ \begin{align} y(t) &= Cx(t)+v(t) \\ v(t) & \sim \text{N}(0,R_c) \end{align} $$ we can change it into a practical discrete one, which is $$ \begin{align} y_k &= Cx_k+v_k \\ v_k & \sim \text{N}(0,R) \end{align} $$ and $$ \mathbf{R=\frac{R_c}{T}} \tag{$*$}\label{a} $$ how to get the $\eqref{a}$ ?
I don't quit understand this because for a sampling process, we say $$ v_k=v(kT)=v(t) $$ so $v_k$ is as same as $v(t)$ in a particular time $t$. so $v_k$ and $v(t)$ have the same distribution, leading $$\text{Cov}(v_k)=\text{Cov}(v(t))$$
that means $R=R_c$.
Or using equation $\eqref{a}$ to promise the discrte Kalman filter to have the same conclusion with the continuous one?
EDIT: equation $\eqref{a}$ is used in the MIT lecture.