Fast projection algorithm and its step size control

Abstract

This paper provides a fast Projection algorithm and a step size conl.rol to obtain the same steady-state excess mean squared error (MSE) for various projection orders. Computer simulations for colored noise and speech input signal confirm the effectiveness of the Projection algorithm and the step size control. lntroductiionOf the many adaptive filtering algorithms, the Normalized LMS (NLMS) algorithm is generally used in practice because of its simplicity. The computational complexity of the NLMS algcirithm is low, however, convergence is very slow and tracking is poor for a colored input signal such as speech.Thc RLS (Recursive Least Squares) algorithm, on the other hand, has the same convergence speed for both a colored input signal and a white signal, but its large computational complexity is a drawback.In recent years, an algorithm called Projection (or Affine Projection) [ 11 has been drawing attention. This algorithm has properties that lie between those of the NLMS and RLS, i.e. less computational complexity than RLS but much faster convergence than NLMS for an input signal such as speech which can be modeled as a low-order AR process. The Projection algorithm, however, still needed more computation than NLMS, which was a problem.Recently, efforts to reduce the amount of computation have been made [2] [3] [4], and computational complexity has been significantly reduced. This paper describes the fast Projection algorithm and proposes a control method for so-called step size parameter to obtain the same steady-state excess MISE for various projection orders.

Underdetermined-order recursive least-squares adaptive filtering: the concept and algorithms

“…Rearrangement of (6) yields (8) where is the identity matrix, is the projection operator onto the column space of the matrix , and is the projection operator onto the orthogonal complement of the column space of [20]. When the algorithm is optimal , the first term on the right-hand side is the result of minimization of the following sum with respect to (9) Hence, it is clear that the algorithm updates the minimumnorm solution to an RLS-like least-squares error criterion that is underdetermined: The number of equations is less than the number of unknowns. This is not a well-defined problem, and it is necessary to resort to finding the minimum-norm solution among several candidates and update it by projecting the last solution onto the orthogonal complement of the subspace spanned by the input vectors.…”

“…Their approach is based on and, hence, is dependent on the relative sizes of prediction order and adaptive filter size. The URLS-type algorithms have gained practical importance recently, particularly in acoustic echo cancellation [6], [8], [9], which have reported encouraging results. In this paper, we also demonstrate the feasibility of the URLS algorithms in practical applications.…”

Double-talk robust fast converging algorithms for network echo cancellation

“…A robust version of PAPA (and hence of APA) is obtained straightforwardly, by applying the principles presented previously Additionally, most of the computational procedures of the fast affine projection (FAP) algorithm [9], [10] can be incorporated in order to reduce the computational complexity of PAPA. Unfortunately, introducing an alternative coefficient vector, as in [9], cannot be done in PAPA, because invariance of the product of the step-size matrix and the excitation vector is destroyed, since varies from one iteration to the next.…”

“…The robust PNLMS algorithm is also generalized to a robust proportionate affine projection algorithm (PAPA)-an algorithm which is a combination of the affine projection algorithm, [8]- [10] and the proportionate step-size technique.…”

Filtrado adaptativo multicanal para control local de campo sonoro basado en algoritmos de proyección afín.

“…Aunque los algoritmos de proyección afín llevan estudiándose y aplicándose a problemas similares a los planteados en esta tesis doctoral desde los años 90 (véase [63], [62] o [89]), en este trabajo hemos querido hacer una reflexión sobre las estrategias que se han venido usando para la optimización del coste computacional de estos algoritmos, particularizados en una aplicación concreta: la de control activo de ruido. Esta aplicación particular, no quita generalidad a los resultados alcanzados, puesto que en todo caso es una aplicación que añade Por tanto, el algoritmo de proyección afín basado en la estructura convencional de filtrado-x presenta un coste computacional menor que su homónimo basado en la estructura modificada aunque claro está que su velocidad de convergencia será menor, sobre todo si tratamos de controlar sistemas cuyos caminos acústicos introducen retardos largos.…”