The Sampling Theorem

The sampling theorem states that if samples from a signal are taken at a sufficient speed, the signal can be reconstructed from those samples.
A sample is just a value at a given time.
To sample a signal meants to take values at regular intervals.  Thse intervals are the periods between samples.  When you divide one by the period, you get the sampling frequency or sample rate.  
The Nyquist Criterion states that to correctly reconstruct a signal, the sample rate must be at least as large as twice that of the highest frequency present in a band-limited signal.
Band-limited means that above a certain frequency, the signal's Fourier transform is zero.
Aliasing occurs when a sampling frequency is too low and duplicates of the original signal appear in the frequency domain.  This can cause distortion in the re-creation of the signal.
Sampling of a continuous time signal makes it possible to deal with the signal as if it was a discrete signal.  Many applications today do this to create a continous appearance without taking infinite samples; a camera takes only a specific number of samples to form a picture, MP3's take samples at a certain rate to reduce the amount of information needed for one song, TV video is done at a frame rate (sample rate) that fools people watching into thinking it is continuous.

To sample a signal, we need a sampling function that will give us the values only at the points we would like.  The most common is the impulse-train function.  This function produces impulse functions on integer multiples of a period T.  It is defined as

To find the sampled signal, this impulse-train function is multiplied by the function to be sampled.
Below we will look at some sampled signals and simple ways of reconstruction.

Example: Sampling Rate

From the above graphics, it can be seen that a greater sampling frequency gives a better sampled signal.  When the frequency of the sampling signal is low, it is next to impossible to recognize the original signal.  As the sampling frequency increases, the original signal becomes more and more distinguished.

Example: Simple Signal Reconstruction using Linear Interpolation

Here we reconstruct the signal using Linear Interpolation.  This method is done by connecting the adjacent dots from the sampled signal.  When the sampling frequency is low, the signal can not be reconstructed correctly (in most cases).  When the sampling frequency is high enough, the signal can be reconstructed without loss.  

Notice how the signal with the highest sampling frequency looks nearly identical to the original.  There are many more complex methods of signal reconstruction, but this is the easiest to do.

Aliasing

Here is an example adapted from work by Lucas May on aliasing and the frequency domain.