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Question

I'm trying to figure out the sampling rate for my ADC to sample essentially signal essentially of the form: $$y(t) = \sin(\max(t, \omega_{max})\times t) + n$$ where $n$ is noise.

Context

This signal is from a motor, where the rotational speed linearly ramps up to some $\omega_{max}$. To figure out the required sampling rate, the only method I know of is to use the Nyquist rate of 2 times the maximum frequency content. The above signal can be re-written as

$y(t) = rectangularPulse(t \epsilon [0, \omega_{max}]) \times \sin(t^2) + heaviside(t-\omega_{max}) \times \sin(\omega_{max}t)$

This isn't bandlimited since a rectangular pulse is a $\text{sinc}$ in the frequency domain. What I want to do is low pass filter the signal first as described here. But I'm at a loss as to figure out what I should set my cut off frequency to, as the fourier transform of the signal doesn't die off at any frequency (unless I'm mistaken).

I feel like this should be a classical problem, but I can't find anything online. If you know of how to solve this problem, or you could point me to where to look, it would be greatly appreciated!

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  • $\begingroup$ I don't see how your first equation corresponds to a motor ramping up. I would write it as $$y(t) = \begin{cases} \sin(r(t) t), \,\text{ $t \leq t_R$} \\ \sin(\omega_\text{max} t), \, \text{ $t > t_R$} \end{cases},$$ where $r(t)$ is a linear ramp and $t_R$ is the time it takes the motor to reach $\omega_\text{max}$. What do you think? $\endgroup$ – MBaz Oct 25 '19 at 0:23
  • $\begingroup$ That works! I think the above should be written as $min(t, $\omega_{max})$. I'll correct it to yours to be safe thanks! $\endgroup$ – YYH Oct 25 '19 at 1:12
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If your frequency increase linearly, then you have a chirp signal with linearly increasing frequency.

https://en.wikipedia.org/wiki/Chirp_spectrum

There is a closed-form solution for the Fourier transform. I'm not sure how that will help you. Why not pick a sampling frequency

$\omega_s \gt 2 \omega_{max}$

Since your highest frequency is $\omega_{max}$, no point in considering lower frequencies for your sampling rate. Is this for a control-loop algorithm? In that case, I would use a more rule-of-thumb criterion.

$\omega_s \gt 20 \omega_{max}$

This makes you less susceptible to sampling delays (among other things)

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