I am currently working on developing a strategy to measure a time-varying signal and would appreciate some input. In my signal acquisition process, the acquired signal $a(t)$ can be thought of as the convolution of the "true" signal, $y(t)$, with a kernel, $k(\tau)$.
\begin{equation} a(t) = \int y(\tau) k(t-\tau)\, \mathrm{d}\tau \end{equation}
The kernel I'm investigating exhibits the rise and fall-off characteristics of a cosine function, where the rise-time and fall-off time are determined by specific system parameters. However, I have the flexibility to introduce an arbitrary delay after the rise-period. This delay I can vary on a time-scale much shorter than the rise time. During this delay, the kernel "just continues" to aquire signal. Furthermore, I can sweep the kernel over the signal.
Currently, I'm considering two naive approaches. The first approach involves sweeping a minimal kernel over the signal, while the second approach entails increasing the waiting time. The challenge with the first approach is that the achievable time-resolution is largely limited by the slower rise and fall-off periods of the kernel. On the other hand, the second approach requires numerical differentiation in signal processing, which can be problematic.
Intuitively, I believe there could be benefits if we combine both approaches. However, I'm unsure how to effectively perform data analysis in such a scenario. I have the feeling that this problem must have been solved elsewhere.
I hope I have provided enough context, and I look forward to hearing your suggestions and insights.
Thank you!
