Capacity Theorem for Finite Duration Symbols

Abstract

The proof of capacity theorem assumed that the symbols are of infinite time duration. Since the infinite time symbols are not meaningful and is not used in the existing technology, in this paper we decided to prove it over finite time interval. The result shows that the new capacity of a band limited channel explicitly depends on the sample rate. This new approach helps us to show that all shift keying methods do not meet the Shannon's model and therefore cannot achieve the basic capacity limits. We also show that the function modulation method can be engineered to achieve the new extended limits. To establish our theory we use the well known mathematical concept of infinite dimensionality of function space. The new capacity result is stated in a theorem.

Time–frequency‐based instantaneous frequency estimation of sparse signals from incomplete set of samples

“…2. Note that three independent samples would be enough, because a sinusoidal function, can be completely specified by three non-zero samples taken at any three points in its period [25]. However, in the practical applications, some additional samples are needed (e.g.…”

“…Hence, for an incomplete set of the samples, we can summarise two criteria which are required for the choice of the distribution and the IF estimation: The LAF ℜ should be directly related to the expected signal form, locally reducing the problem to the linear phase case. The auto‐correlation function chosen in that way will provide a locally sparse signal with one frequency component, which needs just a couple of samples for its reconstruction [25]. In the real applications, we usually can approximately assume the signals IF law based on the nature of the physical phenomena described by the signal.…”