The Sampling Theorem

The only way that a computer can handle a continuous (analog) signal is by sampling it. We then have to ask ourselves, How do we do that? Easy, we record the value of the continuous signal at regular points to give us a sequence of numbers.

The next question is how often do we record (sample) the continuous signal? 20 thousand times a second (20kHz), 8 thousand times a second (8kHz), or every hour on the hour?

First, we need enough information to be able to reconstruct the signal, but if we have too much then the computer won't be able to hold all the numbers. The answer to this depends on the continuous signal.

The sampling frequency has to be at least TWICE the maximum frequency in the continous signal, that is the ABSOLUTE maximum frequency, not just the twice the maximum frequency that you are interested in. More formally this is :

The sampling frequency is also called the NYQUIST FREQUENCY, so when you here someone say that the maximum frequency is "half the Nyquist frequency", they just mean that the maximum frequency is half the sampling frequency just as the theorem says it should be.

Sampling at this rate will not result in any loss of information, but if you sample at less than this then you will not be able to reconstruct the signal as it first appeared. The reason for this is that sampling a signal is equivalent to multiplying it by a series of delta functions.

Now, multiplying two signals together in the time domain is equivalent to convolving their responces in the frequency domain. But when this happens, the frequency response of the signal becomes periodic (sometimes refered to as infinite sidebands). If the sampling frequency was high enough (greater than twice Fmax) then these sidebands keep away from each other, but if you did not have a high enough sampling frequency then the sidebands will overlap.

Here is an example when the sampling frequency was sufficent.

And here is an example when the sampling freq was too low, and resulted in loss of information.

In real life continuous signals have frequencies that are beyond any sampling frequency possible, they might even contain infinte frequencies. One way round this is to pass the signal through a low pass filter that stops any frequencies above half the sampling frequency. This is still not perfect, but is a practical method.

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