Confusing delta notation for multivariate expectation

Question

I'm seeing this odd notation describing process error covariance in an extended Kalman filter over a time interval $(t, t')$.

It says that

$$ Q(t) = \left[ \begin{matrix} Q_1(t) & 0_{3\times3} \\ 0_{3\times3} & Q_2(t) \end{matrix} \right] $$ with $$ E\{\eta_i(t)\eta_j^T(t')\} = \delta_{ij} \delta(t-t')Q_i(t) \textrm{ for }i,j=1,2. $$

It seems like there are two types of delta functions here, first a Kronecker delta and then a Dirac delta, and that this is a confusing way to say that $Q(t)$ is a diagonal matrix where the Kronecker delta has been shifted by $t'$ (but what in the world does that mean?). How do I evaluate this at a specific $t-t'$?

Answer 1

What is means is that when you plug in $t'$ then you get $\delta(t'-t')=\delta(0)=1$. If you plug in anything else, then $t-t'\ne 0$ so that $\delta(t-t')=0$.

It seems like there are two types of delta functions here, first a Kronecker delta and then a Dirac delta

Remember, the $\delta$ function takes on the value $1$ at $0$ and $0$ everywhere else, so it's just a matter of sussing out where the argument of the function is $0$ and when it isn't.

I don't think you really mean Dirac $\delta$, though, that's a distribution that takes a function and returns its value at $0$, which wouldn't make sense in this context, at least not as-stated. Even with the bad definition of it as "infinite" at $0$ and $0$ elsewhere, that would give you a value which is infinity or $0$ rather than a number.

It seems like [...] this is a confusing way to say that $Q(t)$ is a diagonal matrix where the Kronecker delta has been shifted by $t'$ (but what in the world does that mean?)

Yes, and the values on the diagonal are given by the $Q_i(t')$.

How do I evaluate this at a specific $t-t'$?

$E\{\eta_i(t)\eta_j^T(t')\}=0$ when $t\ne t'$ or $i\ne j$ and when $t=t'$ and $i=j$ we have

$$E\{\eta_i(t')\eta_i^T(t')\}=Q_i(t').$$