Context: I'm trying to characterize the metastability behavior of a digital latch.
I'm modeling two cross-coupled inverters as RC circuits with negative gain. One of the inverters has a source of noise at its input (denoted $v_n(t)$) which is a gaussian white noise process.
This gives me a system of two ODE's:
$$ V_I(t) + \tau \frac{dV_I(t)}{dt}=-AV_O(t) $$ $$ V_O(t) + \tau \frac{dV_O(t)}{dt}=-A\left(V_I(t)+v_n(t)\right) $$
Substituting and re-arranging for $V_O$ yields
$$ V_O''+\frac{2}{\tau}V_O'+\frac{1-A^2}{\tau^2}V_O=\frac{-A}{\tau^2}\left(v_n+v_n'\right) $$
In the absences of this noise source, the second order homogeneous equation has a simple complementary answer $$ y_c=C_1e^{-t/\tau_1}+C_2e^{t/\tau_2} $$ where $$ \tau_1 = \frac{\tau}{A+1}, \tau_2 = \frac{\tau}{A-1} $$ For simplicity assume $A \gg 1$ so $\tau_1=\tau_2=\tau'=\frac{\tau}{A}$
In general, I'm concerned with initial conditions of the form $V_O(0)=-Av_s,V_O'(0)=0$ (i.e. some sampled, constant voltage at $V_I$ gained up to $V_O$).
This gives a solution (again approximating) of $$ V_O(t)=-A\frac{v_s}{2}\left(e^{\frac{t}{\tau'}}+e^{\frac{-t}{\tau'}}\right) $$
And finally, the quantity I'm trying to characterize is the time delay to some fixed voltage, that is $t_D$ such that $V_O(t_D)=-V_F$. $t_D$ will vary with $v_s$: $$ t_D=\tau'ln\left(\frac{2V_F}{Av_s}\right) $$
If you've stuck with me so far... Thank you!
The Problem:
How do I solve for the particular solution of: $$ V_O''+\frac{2}{\tau}V_O'+\frac{1-A^2}{\tau^2}V_O=\frac{-A}{\tau^2}\left(v_n+v_n'\right) $$ where $v_n(t)$ is a gaussian random process?
I know that derivatives of gaussian processes are also gaussian, so I can make the RHS into a single gaussian, but I need to know its parameters. Even after that, I need to solve for $V_o$, then apply solve the inverse to get a distribution of $t_D$ for a given $v_s$, which is ultimately what I'm looking for.
This is way beyond anything I learned about differential equations. Explicit solutions are greatly appreciated, as are references to the appropriate techniques!
With the equation of form:
$$ x'' + 2 \beta x' + \alpha x + \gamma = f(t) $$
You can find the solution using a convolution. To do this you find solutions to:
$$ x'' + 2 \beta x' + \alpha x + \gamma = \delta(t - t_0) $$
Where $\delta$ is the Dirac distribution and the final solution becomes:
$$ \int_{-\infty}^{\infty} x_\tau (t) f(\tau) d\tau $$
Where $x_\tau$ is the solution for delta distribution located at $\tau$. In your case this would be something like: $ x_\tau(t) = \mathbb{I}(t>\tau)[\exp(- \beta (t-\tau)\exp(i \sqrt{\alpha}(t-\tau))] $. Given that this would be a linear operator acting on gaussian process you should get something gaussian again.