# The math behind minimum controller sample rate compared to actual error frequency spectrum

It seems intuitive to me that, in a digital system, a system sampling the error rate "too slowly" will fail to stabilize the system.

Is there a theory/set of metrics/equation I can use to represent this in the frequency or time domain?

i.e.

$$e(t) : \text{Actual error (not sampled error) with respect to time} \\ e(s) : \text{Laplace or frequency domain representation of error signal} \\ F_s : \text{Rate at which error is sampled}$$

If $$e(t)$$ has frequency components much higher than $$F_s$$ (or maybe if $$F_s < 2\text{max}[e(s)]$$ ), is it possible to properly control the system?

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Commented Nov 19, 2021 at 2:10

If you do not filter out (or filter down) the high frequency components of $$e(t)$$, they will be "folded back" into the desired spectrum through the process of "aliasing".