Accurate relations and probability bounds of errors in the systems with multipositional multidimensional quadrature amplitude shift keying

One of the trends of increasing information transmission system frequency-energetic efficiency i.e. increasing the exchange relations between noise immunity and transmission speed, is connected with the development and application of multidimensional signals. Signal quadrature amplitude shift keying (QASK) has been widely used. The report discusses the noise immunity of systems with N-dimensional M-positional QASK: QASK-M/N. Formulas for the calculation of a minimal Euclidean distance under a given maximal and medium signal energies are obtained. A formula for symbol error probability is shown under coherent reception in a constant channel with additive white Gaussian noise, supposing that transmitted messages are probabilistic. Relations equivalent to upper and lower boundaries for P_e are obtained. On the base of the obtained relations we made a numerical analysis of frequency-energetic efficiency and estimated gain with increasing QASK-M/N signal dimensionality