Exact analysis of the finite precision error generation and propagation in the FAEST and the fast transversal algorithms: A general methodology for developing robust RLS schemes

In this paper, an analysis for the actual and deeper cause of the finite precision error generation and accumulation in the FAEST-5p and the fast transversal filtering (FTF) algorithm is undertaken, on the basis of a new methodology and practice. In particular, it is proved that, in case where the input data in these algorithms is a white noise or a periodic sequence, then, out of all the formulas that constitute these two schemes, only four specific formulas generate an amount of finite precision error that consistently makes the algorithms fail after a certain number of iterations. If these formulas are calculated free of finite precision error, then all the results of the two algorithms are also computed error-free. In addition, it is shown that there is a very limited number of specific formulas that transmit the finite precision error generated by these four formulas. Moreover, a number of very general propositions is presented that allow for the calculation of the exact number of erroneous digits with which all the quantities of the FAEST and FTF schemes are computed, including the filter coefficients. Finally, a general methodology is introduced, based on the previous results, that allows for the development of new RLS algorithms that, intrinsically, suffer less of finite precision numerical problems and that therefore are, in practice, suitable for high quality fast Kalman filtering implementations.