# How to sample a signal in time based on a set of relaxation times?

How fine should you sample in time (and what should the range be) when calculating a spectrum based on some relaxation times ("modes" in physics, not sure about signal processing). I have some relaxation times $$\tau$$ that span several orders of magnitude and want to generate the function $$f(t) = \sum \exp(-t/\tau)$$. Is there some rule of thumb for the sampling?

The sampling theorem states that there is no loss of information if you sample a band-limited signal at more than twice the band width.

Your signal is not band limited, so sampling will always generate some amount of error. The relaxation processes are first order low-pass filters. The bandwidth is determine by the process with the smallest time constant $$\tau_{min}$$. The corresponding cutoff frequency (-3dB) will be $$f_c = \frac{1}{\tau_{min}2\pi}$$ The level falls with 10 dB per decade, so your attenuation at $$10f_c$$ will approximately be -10dB, at $$100f_c$$ it will be -20 dB, etc.

If you sample the time domain signal directly, the sampling error will manifest itself primarily as aliasing: The level and phase will be different mostly at higher frequencies. The error is largest at the Nyquist frequency.

In your case, can also sample in frequency, use a parametric model or use bilinear mapping to discretize the signal. All of these methods create different types of error, so this is basically a trade off problem.

As usual the best solution depends on the specific requirements of your application and what exactly you are planning to do with the data.

• +1 Thank you. I'm reading about the sampling theorem (new to me since my background is in a different area) and from what I see the sampling frequency should be at least $2f_{c}$. I thought the cutoff frequency should just be $1/\tau_{min}$. Can you explain how the $\pi$ comes in?
– user62149
Apr 5, 2022 at 3:37