Sequential processing of uncorrelated measurements in Kalman Filter

Question

I'm starting to brush up on the Kalman Filtering I learned a couple decades ago. From what I remember, if you have a measurement vector

$$ z=H x + v $$ and the $n$ components of the measurement noise $v$ are uncorrelated with each other, then you can process the measurements sequentially in the KF equations. Is there a good reference somewhere (preferably online) that discusses why this is so? Intuitively it is understandable if each measurement $z$ depends on only one state element in $x$, but it isn't obvious if each measurement of $z$ depends on several elements of $x$.

Answer 1

I think it is evident in both cases. It follows directly from $$ P(A|B,C,D) = P(((A|B)|C)|D)$$ .i.e. $$P(x[k]\!=\!\hat{x}[k]|\!\hat{z}_1[k]\!=\!z_1[k],\!\hat{z}_2\![k]\!=\!z_2[k])\!=\! P\Bigg( \Big( x[k]\!=\!\hat{x}[k]|\!\hat{z}_1[k]\!=\!z_1[k] \Big)|\!\hat{z}_2[k]\!=\!z_2[k] \Bigg)$$ As long as you don't do a prediction update it should be fine. For nonlinear system, you won't get the same exact estimate, because you will end up with a new dynamics, input, measurement and feedback matrices on every "single-measurement" update