Signal Design for the Amplitude-Limited Gaussian Channel by Error Bound Optimization

A necessary and sufficient condition is presented for an input signal alphabet to optimize the random coding exponent for a time-discrete channel with signals restricted in amplitude. It is applied to the white noise Gaussian channel for signals in one and two dimensions. By use of this theorem the best input signal quantizafion is determined by numerical optimization. Results are presented on the number of amplitude or envelope levels needed to maximize the error bound parameter R
0
at different signal-to-noise ratios. For one-dimensional (baseband) signals a binary antipodal configuration is optimum, in the sense of maximizing R
0
, for signal-to-noise ratios below 6.9 dB. For two-dimensional (passband) signals with limited envelope, phase modulation is shown to be optimum for signal-to-noise ratios below 7.35 dB.