Return to Answer

$\textbf{Q}(\Delta_k) = \sqrt{\Delta_k} \cdot Z$

From an implementation standpoint, this is all fairly straightforward. You'll likely have to do a fair amount of process and measurement noise fitting anyways, so to properly incorporate measurements at irregular time intervals, all you really need to do is to ensure that your process noise covariance is scaled by a factor of the square root of the difference in time between the last filter state and the current measurement time (\Delta), which should look something along the lines of:

$\textbf{Q}(\Delta) = \sqrt{\Delta} \cdot Z$

From an implementation standpoint, this is all fairly straightforward. You'll likely have to do a fair amount of process and measurement noise fitting anyways, so to properly incorporate measurements at irregular time intervals, all you really need to do is to ensure that your process noise covariance is scaled by a factor of the square root of the difference in time between the last filter state and the current measurement time ($\Delta$), which should look something along the lines of:

$\textbf{Q}(\Delta_k) = \textbf{Q}_k - \textbf{Q}_{k-1} = \sqrt{\Delta} \cdot Z$

$\textbf{P}_{k|k-1} = \textbf{F_k}\textbf{P}_{k-1|k-1}\textbf{F}_k^{T} + \textbf{Q}(\Delta_k)$.

Perhaps obvious but I'll point it out just in case, more than likely your model dynamics $\textbf{F} are also affected by a non-uniform timestep (i.e. if you have any sort of velocity / acceleration components). For simplicity this is not addressed here.

$\textbf{Q}(\Delta_k) = \sqrt{\Delta_k} \cdot Z$

$\textbf{P}_{k|k-1} = \textbf{F}_k\textbf{P}_{k-1|k-1}\textbf{F}_k^{T} + \textbf{Q}(\Delta_k)$.

or

$\textbf{P}_{k|k-1} = \textbf{F}_k\textbf{P}_{k-1|k-1}\textbf{F}_k^{T} + \sqrt{\Delta_k} \cdot Z$.

Perhaps obvious but I'll point it out just in case, more than likely your model dynamics $\textbf{F}$ are also affected by a non-uniform timestep (i.e. if you have any sort of velocity / acceleration components). For simplicity this is not addressed here.

Now, shifting for the last time back to Kalman filter notation, we note that $\textbf{Q}(\Delta)$ also represents the shift in a Wiener Process over time. We can thus write the expression for $\textbf{Q}(\Delta)$ as:

Now, shifting for the last time back to Kalman filter notation, we note that $\textbf{Q}(\Delta)$ also represents the shift in a Wiener Process over time. We can thus write the expression for $\textbf{Q}(\Delta)$ as: