# what is the best way of sampling an exponential decay?

Some signal is decribed by a formula

$$S(t)=S_0 e^{-tC}\text.$$

I have given $t_\text{min},t_\text{max}$ and number of points. What is the optimal way of sampling such a signal? How to choose the proper values of $t$?

• You didn't specify an error/noise model, so all answers can only be based on the "pure" function as you specified it above. Also, you didn't define what "best" should mean, so every answer will make some assumption over what that means. You should probably edit your question to clarify! – Marcus Müller Feb 22 '17 at 15:41
• $C$ is in units of frequency. so the sample rate, $\tfrac1T = \tfrac{N-1}{t_\text{max}-t_\text{min}}$ (where $N$ is the number of points), should be much higher than $C$. – robert bristow-johnson Jan 5 '18 at 22:57

If you know that this is the only signal you're observing, than literally any two sampling points from $[t_\text{min};t_\text{max}]$ will do.

This is a problem involving a bijective function with two unknowns (assuming $C\in \mathbb R$). So, two observation will, as always, do.

To minimize the error, you'd choose sampling times that are as far apart as possible, considering that your function is strictly monotonous.

• This is an NMR signal, so each measurement takes some time. I want to fit the curve as well as possible with limited number of points. That is why i want to sample the data more densely at the beginning. – Karol Borkowski Feb 22 '17 at 15:26
• So your point is? Note that you haven't specified any error model, so all I can say here applies to your exact function as you posted it! – Marcus Müller Feb 22 '17 at 15:38

This post is old, and Marcus's answer is correct but, based on Karol's comment: minimize the Cramér–Rao lower bound for each parameter.

I found this paper by Jones and Co-workers useful:

https://doi.org/10.1006/jmrb.1996.0151

(1) Jones, J. A.; Hodgkinson, P.; Barker, A. L.; Hore, P. J. Optimal Sampling Strategies for the Measurement of Spin-Spin Relaxation Times. J. Magn. Reson. Ser. B. 1996, 113 (1), 25–34.