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For example if I have a $10 Hz$ signal and I sample it at $19 Hz$ (less than the Nyquist frequency) how can I determine the dominant frequency of the output and why?

If I then apply a lowpass filter, how will this allow for a dominant frequency of $10 Hz$ to be obtained again?

EDIT

If I know what the frequency of the sampled signal is, e.g. the sampled signal $x \left( t \right) = \sin \left(2 \pi \cdot 10 \cdot t \right)$, how can I determine the dominant frequency of the output signal in this case?

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    $\begingroup$ After our discussion below: That's aliasing, and I bet if you're familiar with the term "Nyquist frequency", it will be introduced a few paragraphs before or after that term. $\endgroup$ May 18, 2022 at 16:54
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    $\begingroup$ Does this answer your question? Properties of Discrete-time Sinusoidal Signal $\endgroup$ May 18, 2022 at 16:54
  • $\begingroup$ hmm I can’t see aliasing anywhere in my lecture notes! $\endgroup$ May 18, 2022 at 17:00
  • $\begingroup$ That would really be surprising! I don't think your lecturer would say "and if you violate the condition to sample at more than the Nyquist frequency, a kitten feels pain"; the motivation for that is exactly that: Aliasing! (do be nice to your kittens, though) $\endgroup$ May 18, 2022 at 17:03
  • $\begingroup$ I think I kind of understand the link that you sent, but don’t really get how it applies to this situation $\endgroup$ May 18, 2022 at 17:08

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For example if I have a 10Hz signal and I sample it at 19Hz (less than the Nyquist frequency) how can I determine the dominant frequency of the output and why?

not unambiguously, because you violated the necessary condition for sampling, as you say yourself.

So, not at all. Unless you know that your frequency range of interest does not include 9 Hz, you cannot know whether the signal was 10 Hz or 9 Hz; both signals would lead to exactly the same samples.

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  • $\begingroup$ Okay - what about if I am sampling a signal that I know the frequency of e.g. if I am sampling the signal x(t) = sin(10*2*pi*t) (a signal with frequency of 10Hz only)? $\endgroup$ May 18, 2022 at 16:45
  • $\begingroup$ but if you know the frequency, why would you need to detect the frequency? You just wrote it down; 10 Hz. Which is the same as 9 Hz (in the real-valued case) in $f_s=19\,\text{Hz}$ discrete time. $\endgroup$ May 18, 2022 at 16:47
  • $\begingroup$ I’m trying to work out how the maths works for the time being $\endgroup$ May 18, 2022 at 16:49
  • $\begingroup$ there's no math working here. If you say "I have a signal that I know to be 10 Hz, what is it's frequency?" then the answer is "10 Hz, as you say". After sampling, it's still 10 Hz – but that 10 Hz is mathematically indistiguishable from 9 Hz. I might simply be missing what you're trying to figure out! $\endgroup$ May 18, 2022 at 16:50
  • $\begingroup$ Well I understand that sometimes signals are sampled at less than their nyquist frequency and then passed through a lowpass filter in order to return the original signal. I’m trying to work out what happens in between sampling and being input to the lowpass filter $\endgroup$ May 18, 2022 at 16:50

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