Reception noise immunity of multiposition signals

The noise immunity of coherent reception of signals employing multiple phase modulation, and of optimum noncoherent reception of signals employing multiple phase-difference modulation, is analyzed for channels with fixed parameters and white noise for a Gray keying code. Formulas are derived for the error probability in k-tuple phase modulation and for k-tuple phase-difference modulation. The findings are particularly valuable when the signal energy to spectral noise power density ratio is relatively small, in which case the existing approximate relationships are too rough.