Optimum pulse shapes for pulse-amplitude modulation data transmission using vestigial sideband modulation

Vestigial sideband (VSB) pulses are derived that have no inherent intersymbol interference (isi), and in addition have maximum immunity in a mean-squared-error (MSE) sense to timing and carrier phase errors. The optimum VSB pulses are shown to be identical to those of Franks (1968) at the band edges, and to have vestigial roll-off characteristics of a similar discontinuous nature. It is also shown that one may allocate available bandwidth to the vestigial roll-off band band and Nyquist roll-off band in any way desired. The final performance for the optimum pulse depends only on the total bandwidth used. These results are shown to apply as well to a system employing a matched filter.