Optimum waveform signal sets with amplitude and energy constraints

It is desirable to choose the waveforms making up a signaling alphabet so that they are maximally separated one from another. This problem is considered, in the space of square-integrable functions, for signals which have finite duration, and are constrained in the ranges of their values as well as in energy. Corresponding to each of the following cases, we establish sharp bounds for the minimum distance and for the average distance between elements of a fixed size signal set, and construct sets of signals that attain both bounds simultaneously. \begin{list} \item {\em Case A (Energy Constraint Only):} The average energy of the waveforms in the signal set is at most
\sigma
, where
0 \leq \sigma < \infty
. \item {\em Case B (Energy and Peak Amplitude Constraints):} The average energy of the waveforms in the signal set is
\leq \sigma (0 \leq \sigma < 1)
, and the absolute value of each waveform is at most
1
. \item {\em Case C (Energy and Value Constraints):} The average energy of the waveforms in the signal set is at most
b^{2}\sigma + a^{2}(1 - \sigma)
, and each waveform takes values in the set
[a, b]
, where
0 \leq a < b < \infty
, and
0 \leq \sigma \leq 1
. \end{list} Cases A and B are applicable to signal design for communication in channels with additive noise (say Gaussian), and Case C is applicable to signal design for optical channels, where the signal represents the intensity of a photon stream. The general character of the results is that the minimum distance behaves like
\gamma \sigma
in Cases A and B, and like
\gamma \sigma (1 - \sigma)
in Case C, with
\gamma
a suitable constant.