Faster, Stabler, and Simpler—A Recursive-Least-Squares Algorithm Exploiting the Frisch–Waugh–Lovell Theorem

Abstract

We propose a novel recursive least squares (RLS) algorithm that exploits the Frisch–Waugh–Lovell theorem to reduce digital complexity and improve convergence speed and algorithmic stability in fixed-point arithmetic. We tested the new algorithm in the digital background calibration section of a four-channel time-interleaved analog-to-digital converter, obtaining better stability and faster convergence. The digital complexity of the new algorithm in terms of multiplications and divisions is 33% lower asymptotically than that of the conventional Bierman algorithm if the model parameters need not be computed at each update; otherwise, it is the same. Memory requirements are also the same. Because, in calibration, the distance between the ideal and calibrated outputs of the system is to be minimized, the actual value of the model parameters is usually not of interest. Convergence time can be up to 10 or 20 times better in fixed-point arithmetic, and stability for large models is also better in our simulations. In our simulations, when the conventional Bierman RLS algorithm is stable, the steady-state accuracy of the new algorithm is either comparable or better, depending on the simulation setup.