How Is a Signal Changed After Sampling?

Question

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?

Answer 1

This is to explain what actually sampling is and why and how it's done.

Here I've generated a simple continuous-like sinusoidal signal with frequency fm = 10kHz. In order to make it appear as a continuous signal when plotting, a sampling rate of fs=500kHz is used

Pretending the above-generated signal as a “continuous” signal, we would like to convert the signal to the discrete-time equivalent by sampling. By Nyquist Shannon Theorem, the signal has to be sampled at at-least fs=2*fm=20 kHz. Let’s sample the signal at f{s1}=30kHz and then at f{s1}=50kHz for illustration.

CODE

fs=500e3; %Very high sampling rate 500 kHz
f=10e3; %Frequency of sinusoid
nCyl=5; %generate five cycles of sinusoid
t=0:1/fs:nCyl*1/f; %time index
x=cos(2*pi*f*t);

plot(t,x)
title('Continuous sinusoidal signal');
xlabel('Time(s)');
ylabel('Amplitude');

For Sampling-

fs1=30e3; %30kHz sampling rate
t1=0:1/fs1:nCyl*1/f; %time index
x1=cos(2*pi*f*t1);

fs2=50e3; %50kHz sampling rate
t2=0:1/fs2:nCyl*1/f; %time index
x2=cos(2*pi*f*t2);

subplot(2,1,1);
plot(t,x);
hold on;
stem(t1,x1);
subplot(2,1,2);
plot(t,x);
hold on;
stem(t2,x2);

Answer 2

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.

Answer 3

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.