The application of the orthogonal decomposition method for the algebraic solver separator

Abstract

The method which is valid for improving the transmission of information and a clean separation of signal and noise is suggested in [1]. The basis of the proposed new way of developing the theory and methods of communication is the rejection of the probabilistic method for evaluating noisy signals according to the maximum likelihood rule. This method contains the mathematical procedure for absolutely accurate separation, as well as proving the absence of any fundamental theoretical limits [2,3] on the effectiveness of communications, including the absence of channel capacity limitations [4]. This approach includes the fundamentally new concept and the technical aspects of implementing telecommunication systems and uses systems of linear algebra equations (SLAE) to filter signals from noise. The SLAE matrix is the linear algebraic matrix (LSM) that separates and extracts the true values of informative signal parameters. Such a SLAE always has the solution, but obtaining this solution sometimes requires a particular method to be used, because its matrix is not always square, but can be rectangular as well. Therefore, the method of the orthogonal decomposition is proposed in this paper. For obtaining the matrices of orthogonal decomposition the Gram–Schmidt process, which is suitable for matrices of any size and composition, can be used. The method of solving a SLAE includes full description of solution and acceptable for matrices of any size. The example of solving the SLAE with a small matrix is presented in the paper. The MathCad Prime has been implemented for a bigger matrix. The implementation includes the functions that can be used in any other programming language. The solution has minimal norm and acceptable for linear algebraic matrices that separate signal and noise.