I'm quite sure it's a dumb question, but I'm struggling with it.
I have a modelization in which the state/value of a variable is updated every $2u$. This is the discrete step I'm using.
The variable will experience an exponential decay with a known time constant $\tau$ and from an known initial value $A$. At each time step, I'm looking for an efficient way to compute the new value of this variable.
Thus, I was trying to figure out by how much the value changed in $2u$, i.e. what is the $\Delta$? I cannot find the correct equation that model the $\Delta$ in a discrete exponential decay.
My current best guess is:
$$\Delta = A\cdot\left(1-\exp\left(-\frac1\tau\right)\right) $$
which comes from the equation:
$Y = A\cdot(1-r)^x$ with $x$ the number of time step and $r$ the decay rate.
i.e. if I take $r = \exp\left(-\frac1{RC}\right)$ and then look to $Y_{N+1}-Y_N$, I get the $\Delta$ equation above.
However, It does not capture the $2$ us time step, i.e. it should have a larger delta with a larger discrete step.
I do not know how to insert proper equation of this Stack Exhange forum, so I apologize for the inconvenient reading, and would gladly have an example given in the comment/edit so I can properly edit the question.
Thanks 🙂
Best Answer
So your variable is $$A(t)=A\exp(-t/\tau)$$ and $$A(t+2u)=A\exp(-(t+2u)/\tau)=A\exp(-t/\tau)\exp(-2u /\tau)=A(t) \exp(-2u /\tau)$$ so the variable should be multiplied by $\exp(-2u /\tau)$ every "step" of size $2u$.