Basic Sampling Theorem

A very important theorem of sampled-data systems is  

To obtain dynamic information about a plant from a signal that contains components out to a frequency the sampling frequency a, must be set at a rate greater than twice.  

Example 18.1. Suppose we have a signal that has components out to 100 radians per minute. We must set the sampling frequency at a rate greater than 200 radians per minute. 

This basic sampling theorem has profound implications. It says that any high-frequency components in the signal (for example, 60-cycle-per-second electrical noise) can necessitate very fast sampling, even if the basic process is quite slow. It is, therefore, always recommended that signals be analog-filtered before they are sampled. This eliminates the unimportant high-frequency components. 

To prove the sampling theorem let us consider a continuous that is a sine wave with a frequency coq and an amplitude Aq.  

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This basic difference equation - known as the Schema Theorem [holl92] - expresses the fact that the sample representation of schemas whose average fitness remains above average relative to the whole population increases exponentially over time. As it stands, however, this equation addresses only the reproduction operator, and ignores effects of both crossover and mutation. 

We conclude from this basic theorem that the sample representation of low-order schemas with above average fitness relative to the fitness of the population increases exponentially over time. ... 

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The normal distribution is perhaps the most widely used distribution because of the central limit theorem which basically states that as the sample size (n) for a random variable becomes large, the sampling distribution of the mean becomes approximately a normal distribution regardless of the distribution of the original variable. 

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