On the duality between fast QR methods and lattice methods in least squares adaptive filtering

The authors show that fast QR methods and lattice methods in least squares adaptive filtering are duals and follow from identical geometric principles. Whereas the lattice methods compute the residuals of a projection operation via the forward and backward prediction errors, the QR methods compute instead the weights used in the projections. Within this framework, the parameter identification problem is solved using fast QR methods by showing that the reflection coefficients and tap parameters of a least squares lattice filter operating in the joint process mode are immediately available as internal variables in the fast QR algorithms. This parameter set can be readily exploited in system identification, signal analysis, and linear predictive coding, for example. The relations derived also lead to a fast least squares algorithm of minimal complexity that is a hybrid between a QR and a lattice algorithm. The algorithm combines the order recursive properties of the lattice approach with the robust numerical behavior of the QR approach