Pure order recursive least-squares ladder algorithms

The new class of pure order recursive ladder algorithms (PORLA) is presented in this paper. The new method obtains the true, not approximate, least-squares (LS) ladder solution by performing two steps. First, the covariance matrix of the estimated signal is calculated time recursively, and second, the reflection coefficients of the ladder form are determined by a pure order recursive procedure initialized from the covariance matrix. Since time updates in the ladder recursions have been eliminated by the new approach, error propagation does not occur, and substantial improvements in numerical accuracy compared to conventional mixed time and order recursive LS ladder algorithms are efficiently achieved by the presented algorithms. In contrast to conventional LS ladder algorithms, fast initial convergence is not corrupted by roundoff error in the new method. The true LS pure order recursive ladder algorithm is derived and extended to joint process estimation. Additionally, four computationally efficient approximate ladder algorithms, derived from the new approach, are given. One of them identically represents the well-known Makhoul covariance ladder algorithm (1977), which can now be computed without Levinson recursions. Therefore, this new formulation leads to an improved numerical performance, a much simpler implementation scheme, and drastically reduced computational costs compared to its widely used traditional counterpart. Finally, dynamic-range-increasing power normalized versions of the algorithms are also given in the paper.