# Application of the Sampling Theorem to a System

I have the following system:

$$G(s)=\frac{1}{10s^{2}-5s+2}$$

that I want to discretize. How do I apply the sampling theorem to a system so I can choose the correct sampling period?

Furthermore if I use a wrong sampling frequency how can I demonstrate that there is something wrong with the system? I know that we won't be able to recover the right analog data if something like that happens but I don't understand how I can actually "witness" it.

• Your transfer function is unstable. Apart from this, the sampling and reconstruction elements need to be inside the loop. Any sampling that applies outside the loop is merely changing the nature of the input signals and has no effect on closed loop system performance. – Chu Jul 5 '17 at 22:04
• @Chu I know...It is supposed to be unstable. – Adam Jul 5 '17 at 23:35

How do I apply the sampling theorem to a system so I can choose the correct sampling period?

You can't apply a sampling regime to a transfer function of a system because all the TF contains are placements for poles and zeroes and it contains no knowledge about the input to the system in the frequency domain.

You could make an argument that the TF is basically a low pass filter and that beyond a certain input frequency the output will be very small but there's nothing in your question that allows this judgement call to be made.

Furthermore if I use a wrong sampling frequency how can I demonstrate that there is something wrong with the system?

Engineers decide beforehand what the input frequency limits are and apply anti-alias filters to prevent significant aliasing errors so, if something comes along that is larger than expected in normal situations and might cause an aliasing error then you could compare the RMS of the anti aliasing filter output with the raw signal and make a decision based on this.

• In Matlab that it is required to add a sample time when discretizing a system without having any input in mind how do you choose it? – Adam Jul 9 '17 at 15:25
• I don't use matlab. – Andy aka Jul 9 '17 at 15:43
• What I meant is that when you discretize a system via any means (Matlab or other software) you have to pick a sample time prior to choosing an input. So there has to be a way to choose it beforehand. – Adam Jul 9 '17 at 15:49
• I can only say that you choose it based on what you know about the input. – Andy aka Jul 9 '17 at 16:21