Search for ways to reduse optimal algorithms for separation demodulation of mutially nonortogonal digital signals

In the statistical theory of separation (STS) of digital signals (DS), as in an independent branch in the theory of multi-user detection, the problems of separation-demodulation of signals that are mutually non-orthogonal in the length of the information clock interval are considered. In this case, the minimum probability of error in the estimation of group or individual information discrete parameters is chosen as the optimality criterion. Analysis of the noise immunity of algorithms for processing an additive mixture of mutually non-orthogonal DS synthesized by STS methods according to an individual criterion allows us to obtain answers to questions about the limits of their potential noise immunity. To date, the results of such an analysis have already been obtained for the cases of Binary Phase Shift Keying (BPSK), when there are no more than three mutually non-orthogonal signals. However, these results are rather of theoretical significance. The fact is that the complexity of the separation-demodulation algorithms, optimal according to the individual criterion, is characterized by an exponential dependence of their complexity on the number and position of the DS to be separated. A linear dependence of the complexity is observed only in some cases – for example, when the observation contains a certain, predetermined number of mutually orthogonal signals and one, non-orthogonal to all. The possibilities of simplifying optimal division-demodulation algorithms are evaluated and the case is considered in detail when three mutually non-orthogonal BPSK signals are observed, in the general case asynchronous in clock points. A suboptimal reduced algorithm is proposed, significantly simplified in comparison with the one that is optimal in terms of the individual optimality criterion. The condition for its operability is given for the worst case of mutual linear dependence between the signals. Under more general conditions - arbitrary degrees of non-orthogonality between the signals – it is demonstrated that a significant simplification of the division-demodulation algorithm for three DS can be achieved by providing certain cancellations in the instantaneous powers between the signals. Examples of reduced structural schemes of division-demodulation algorithms for three mutually non-orthogonal BPSK signals, synchronous and asynchronous in clock points, are given. Estimates of noise immunity losses due to the proposed simplifications were made.