Error propagation and methods of error correction in LS FIR filtering and l-step ahead linear prediction

In the paper, a general method is presented, which leads to the exact prediction and tracing of the finite precision error generated in the solution of the l-step ahead linear prediction and the optimal FIR filtering problems. It is shown that two sources of numerical error exist, in the algorithms used for the solution of these problems. The first source lies in the formulas used for the forward linear prediction, while the second source is intimately connected with the formulae used specifically for the solution of the l-step ahead and LS FIR problems. The propagation of this numerical error is determined precisely, and it is shown that there exist specific intermediate quantities, calculated in the evolution of the algorithms, which are indicators of the exact magnitude of the overall finite precision error. Clearly, the way for the error correction is open and, in fact, in the paper, some methods are presented for improving the numerical accuracy of the solution of these problems