On optimal signal sets for digital communications with finite precision and amplitude constraints

Given a linear, time-invariant, dispersive channel, a receiver that samples the channel output to within an accuracy of ±d where d > 0, and a transmitter with an output amplitude constraint, what is the maximum data rate that can be reliably communicated. For any dispersive channel the maximum rate depends on d, and is finite. The transmitted waveforms must be designed so that two channel outputs associated with two distinct transmitted signals are separated in amplitude at a particular time by d. It is shown that given any channel impulse response with rational Laplace transform, there exists an optimal set of inputs that are ±A everywhere where A is the maximum allowable amplitude. Furthermore, in any finite time interval each input changes sign a finite number of times. If the channel impulse response is a single decaying exponential, it is shown that simple binary signaling, in which A or -A, depending on the current message bit, is transmitted during each symbol interval, maximizes the data rate.