QRD Least-Squares Lattice Algorithms

Abstract

This chapter presents a full derivation of the square-root-free square-root-free (SRF) QR-decomposition-based least-squares lattice (QRD-LSL) algorithms in complex form, based on linear interpolation (or two-sided prediction) theory as a generalization of linear prediction theory. The conventionally adopted QRD-LSL prediction algorithm can be derived directly from the QRD-LSL interpolation algorithm and then extended to solve the joint process estimation problem. The QRD-LSL interpolation algorithm that produces interpolation errors (residuals) of various orders may have potential implications for some signal processing and communication problems. Interestingly, the QRD-LSL interpolation algorithm can also be used to calculate the Kalman gain vector to implement the widely known recursive least-squares (RLS) algorithm in a transversal structure to generate the least-squares filter weights at each time step. Therefore, linear interpolation theory may provide a bridge between lattice filters and transversal filters. The chapter is organized as follows. Section 5.1 presents the fundamentals of QRD-LSL algorithms. The LSL interpolator and the LSL predictor are briefly presented in Section 5.2. Section 5.3 presents the SRF Givens rotation with feedback mechanism that is employed to develop the SRF QRD-LSL algorithms. In Section 5.4, the SRF QRD-LSL interpolation algorithm is derived, and then reduced to the SRF QRD-LSL prediction algorithm, which is then extended to develop the SRF joint process estimation. The RLS algorithm in the transversal structure based on the SRF QRD-LSL interpolation algorithm is presented in Section 5.5 followed by some simulation results in Section 5.6. Section 5.7 draws conclusions.