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I can't understand sampling signal mathematically.

for example $x(t) = 2\cos(100\pi t) + \cos(300\pi t)$ and I know I should use sampling frequency $300\textrm{Hz}$ to sample this signal.

and after sampling, what is done?

How is $x(t)$ changed? and what frequency does sampled $x(t)$ have?

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    $\begingroup$ To be precise, a sampling frequency strictly greater than 300Hz should be used. $\endgroup$ – AlexTP Jun 17 '18 at 13:43
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and after sampling, what is done?

Well, nothing special, you're just getting another signal in a discrete time domain, based on your original signal in a continual domain.

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First pay attention that the Nyquist Shannon Sampling Theorem requires sampling at rate which is strictly larger than twice (Though it is for edge cases, yet this is the accurate theorem).

Sampling is 1 step to get signals to be Discrete, which means we can store them in a digital form.
Sampling is the discretization of the Time Axis and we need to apply Quantization which is discretization of the Value Axis.

Sampling Theorem states that given a sampled signal (Which is a set of the signal values at discretized grid of time samples) what are the cases it can be fully reconstructed given a single property of the signal - Band Limited Signal.

So what's changed?
You don't have the whole signal but only set of samples from it on a time grid.

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