Pulse templates for transmission over an ensemble of channels

A method is given for finding the envelope of a family of pulses whose shapes are determined by a finite number of parameters. In addition, it is assumed that the parameters are analytically constrained collectively. In the second part of the paper, a family of pulse shapes is determined that can be efficiently transmitted through a channel that is selected at random from an ensemble of channels. The pulse envelope theory of the first part of the paper is then applied using this family of pulses to determine the envelope of mean-square channel outputs of the family of pulses that provide relatively low loss through an ensemble of channels