What is the smallest sampling points that can recover the signal?

Question

Propose that we have a signal with a model of $$ f_{4at0} = v_0(1 - \epsilon + \epsilon \exp(-c_l(t_{t2d}-t_0))) $$

$f_{4at0}$ is dependent variable while $t_{t2d}$is independent variable,others are parameters.

It is a representation of the relationship of biological material concentration-time that is inside the blood serum.Since this material cannot be assayed real time in the human body, the blood samples have to be collected from patients.It's really inconvenient to obtain blood samples from human.So, my question is what's the lower limit of sampling points? And how to schedule the sampling points?

ps: here is the matlab code that I used .I want to use the "Nyquist Sampling Principle",but I don't know how.

cl=6.2;

epsilon=0.95;

t0=0.3625;

delta=0.1;

v0=1000000;

f4at0=v0*(1-epsilon+epsilon*exp(-cl*(tt2d-t0)));

syms tt2d;

f4=v0*(1-epsilon+epsilon*exp(-cl*(tt2d-t0)));

fourier(f4,tt2d)

Output:

ans =

100000*pi*dirac(-tt2d) + 950000*transform::fourier(exp((31*tt2d)/5 + 899/400), -tt2d, -tt2d)

I don't know how to understand transform::fourier(exp((31*tt2d)/5 + 899/400), -tt2d, -tt2d).

I don't know how to calculate T since there is also a Dirac function. I don't know if there are other principles apart from the "Nyquist Sampling Principle" that can be used here.

Answer 1

This depends if you want to do sampling or just model identification. Model identification means "figuring out the numbers in your model from actually measured data". If the model is good and the data is not noisy you only need a few points. The minimum is equal to the number of unknown model parameters. Let's say you don't know v0, cl, epsilon and t0. In this case you need at least four points to identify those. Your model isn't linear so there is some math involved, however it's not overly complicated and very workable. If the measurements are noisy it would be good to take more points and do a least square error fit. The number of points depends on the amount of noise and precision required for the results.

You can also sample the sequence but you need to consider sampling theorem. Your time sequence represents a first order lowpass filter. The cutoff frequency is determined only through your model parameter "cl" and is simply cl divided by 2 pi. At this frequency the spectrum is at -3 dB, at ten times this frequency it's at -20dB and at a 100 times this frequency it's -40 dB etc. Since it never gets all the way to zero you have to live with some amount of aliasing. Again it boils down to how much error you can/want to tolerate.