This question is a follow up to the gorgeous explanation of how the leaky integrator model becomes a differential equation. The link to that question and answer is below.
should this be viewed as an impulse response or step response
I have two questions related to the explanation. Hopefully Fat32 will not mind answering but anyone else is great also.
1) At the bottom of the first page, the author has 2 statements that start with "such that".
The beginning of the first one is $\delta(t) = x(t) \cdots $ and the beginning of the second one is $u(t) = x_(t) \cdots $
Clearly, the equality sign in those statements does not mean the usual equality. Could that notation be clarified because I don't follow it ?
2) At the very end of the answer, the author derives the continuous time differential equation for the leaky integrator model:
$y^{\prime}(t) + \alpha \times y(t) = x(t)$.
In the link I was originally looking at, the differential equation is discrete rather than continuous but aside from that, the same equation as that immediately above. ( proportionality constant is $1/k$ rather than $-\alpha$ ). Of course, the discrete version of above would be
$ dy_{t} = (- \alpha \times y_t + x_t ) dt $
Where the subscripting denotes the discrete unit of time for $t = 1, \ldots n$ and $dt$ denotes the width between these discrete units. This discrete version above is what Lesson 12 uses to generate the plots for how the level of the water in the bucket changes.
So my question is the following: Would this discrete equation directly above just be derived by discretization of what was derived by Fat32 or would the leaky integator model derivation need to change somewhere to justify discreteness ? Clearly, the notion-concept of $exponential declining memory goes away in the discrete version of the model so that makes me think that maybe a whole new derivation is required for justification ? If it is required, then I imagine it is somewhere but I can't find it. Thanks a lot.
Mark