As it turns out, except for a special case discussed in the following paragraph, it is impossible to satisfy the sampling theorem in practice. People can only work with sampled data that are finite in duration. The process of converting a function of an unlimited duration into a function of a finite duration can be modeled simply by multiplying the unlimited function with a “gating function” that is valued as 1 for some interval and 0 elsewhere. Unfortunately, this function itself has frequency components that extend to infinity. Thus, the very act of limiting the duration of a band-limited function causes it to cease being band-limited (Young and Renswoude 1988).
There is one special case of significant importance in which a function of ...
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