Regardless of the sampling method used, by definition it captures only pieces of the message. So, how can the sampled signal be used to recover the whole message?

This question can be answered by considering the mathematical model that defines the sampled signal:

Sampled message = the sampling signal X the message

As you can see, sampling is actually the multiplication of the message with the sampling signal. And, as the sampling signal is a digital signal which is actually made up of a DC voltage and many sinewaves (the fundamental and its harmonics) the equation can be rewritten as:

Sampled message = (DC + fundamental + harmonics) X message

When the message is a simple sinewave (like in Figure 1) the equation's solution tells us that the sampled signal consists of:

• A sinewave at the same frequency as the message
• A pair of sinewaves that are the sum and difference of the fundamental and message frequencies
• Many other pairs of sinewaves that are the sum and difference of the sampling signals harmonics and the message

This ends up being a lot of sinewaves but one of them has the same frequency as the message. So, to recover the message, all that need be done is to pass the sampled signal through a low¬pass filter. As its name implies, this type of filter lets lower frequency signals through but rejects higher frequency signals.