A Mathematical Theory of Communication

19 Band limited ensembles of functions

1. State the sampling theorem as given by Shannon (Theorem 13).
   A function with no frequencies above \(W\) is completely determined by its values at points \(\frac{1}{2W}\) seconds apart: \(f(t) = \sum_n X_n \frac{\sin \pi(2Wt - n)}{\pi(2Wt - n)}\) with \(X_n = f\!\left(\frac{n}{2W}\right)\).
   • Shannon, 1948, A Mathematical Theory of Communication (Bell System Technical Journal), §19
2. Why do band-limited signals of duration \(T\) correspond to points in a \(2TW\)-dimensional space?
   By the sampling theorem a signal of band \(W\) and duration \(T\) is fixed by \(2TW\) samples, which serve as coordinates; an ensemble of such signals is then a probability distribution in that space.
   • Shannon, 1948, A Mathematical Theory of Communication (Bell System Technical Journal), §19