# Measuring a time-varying signal with a finite kernel

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!

• I don't see anyone commenting. This was meant to be a comment, but I don't think you have anything to lose if you see this here instead? (my account too low in pts to post an actual comment) 1. >the acquired signal a(t) can be thought of as the convolution of the "true" signal, y(t), with a kernel, k(τ) Are you sure you mean convolution, not per-sample multiplication? Because a common thing to do ("problem that has been solved elsewhere", as you put it) is to take overlapping windows (your kernels; the step size would be something larger than 1 sample (so not sliding) but less than 1/2 the len Jun 7 at 4:21
• I don't follow the problem. Your "time-varying signal" is just "signal", and you're convolving it with something. If you're trying to recover $y(t)$ from $a(t)$, why not deconvolve? Or are $y, k$ both unknown? Jun 8 at 9:29