Nonlinear System Identification with LAR Catia Valdman, Marcello L. R. de Campos and Jos´ e A. Apolin´ ario Jr. Abstract—In this paper, the use of the Least Angle Regression (LAR) algorithm in combination with a Volterra filter is proposed for nonlinear system identification. The LAR algorithm has been used successfully in applications with sparse systems. Volterra filters are known as a good model of nonlinear systems and have been useful in a number of applications. However, only low order Volterra models are usually employed due to the large number of coefficients. Since the LAR algorithm indicates the most correlated coefficients in an increasing way, one by one, we propose to use this information to stop the LAR algorithm when a number of desired coefficients are already calculated. Hence, for a large order Volterra filter, the most important coefficients will be evaluated independently of its kernel position. To validate the proposition, we use third-order and fifth-order Volterra filters with the LAR algorithm to identify two nonlinear systems. Results of the LAR algorithm are compared to results of the Least Squares and the Subset Selection algorithms. Index Terms— LAR algorithm, nonlinear system, Volterra filter. Abstract— Neste artigo o uso do algoritmo Least Angle Re- gression (LAR) em conjunto com um filtro Volterra ´ e proposto para a identificac ¸˜ ao de sistemas n˜ ao lineares. O algoritmo LAR vem sendo usado em sistemas esparsos com resultados satisfat´ orios. Filtros Volterra s˜ ao conhecidos por serem eficientes para modelagem de sistemas n˜ ao lineares e j´ a foram utilizados em diversas aplicac ¸˜ oes. Entretanto, na maioria das vezes, apenas modelos Volterra de baixa ordem s˜ ao utilizados devido ao seu elevado n ´ umero de coeficientes. Uma vez que o algoritmo LAR indica os coeficientes mais correlatos acrescentando-os ao modelo sempre de um em um, nossa proposta ´ e utilizar esta informac ¸˜ ao para interromper o algoritmo LAR quando um n´ umero desejado de coeficientes j´ a tiver sido calculado. Desta forma, para um filtro Volterra de alta ordem, os coeficientes mais importantes ser˜ ao estimados, independente de sua posic ¸˜ ao no kernel. Para validar esta proposta, utilizamos um filtro Volterra de terceira ordem e um filtro Volterra de quinta ordem para identificar dois sistemas n˜ ao lineares. Os resultados obtidos pelo algoritmo LAR s˜ ao comparados com os resultados dos algoritmos Least Squares e Subset Selection. Index Terms— Algoritmo LAR, sistema n˜ ao linear, filtro Vol- terra. I. I NTRODUCTION Nonlinear system models are used in many areas, such as communication systems, power amplifiers, loudspeakers with harmonic distortion and others [1]. The Volterra filter is commonly used to identify nonlinear systems, however, standard approaches tend to limit the order of the filter to avoid a large number of coefficients. For example, in [2] and Manuscrito recebido em 30 de novembro de 2010; revisado em 1 de abril de 2011. C. Valdman (catia@valdman.com) and M. L. R. de Campos (mcampos@ieee.org) are with the Federal University of Rio de Janeiro - UFRJ - P.O. Box 68504 - 21941–972 - Rio de Janeiro - RJ - Brazil. J. A. Apolin´ ario Jr. (apolin@ime.eb.br) is with the Military Institure of Engineering - IME - Prac ¸a General Tib´ urcio, 80 - 22290–270 - Rio de Janeiro - RJ - Brazil. [3], adaptive second-order Volterra filters were used to model nonlinear acoustic echo paths using the NLMS algorithm. In [4], stationary and non-stationary signals which arise from Volterra models were estimated using neural networks; again, the second-order Volterra model was considered sufficient for the purpose. In [5], a study was carried out for several algorithms combined with Volterra filter to identify nonlinear systems. The algorithms studied were: LMS, NLMS, RLS, affine projection and summation affine projection; once again, only second-order nonlinear components were treated [5]. The LAR algorithm was first developed and based on diabetes studies [6]. Since then, it has been used in several applications. In [7] and [8], models to text classification were developed and up to 2,000 coefficients were chosen. In [9], the LAR algorithm was used to estimate performance variability of integrated circuits with a larger number of coefficients, in the order of 10 4 to 10 6 . By comparing the response of the LS and the LAR algorithms, the authors concluded that the LAR algorithm achieves up to 25x runtime speedup without compromising any accuracy [9]. Recently, it has been employed in image processing, for face representation and recognition [10], and for face age estimation [11]. Based on the fact that the LAR algorithm is very useful when dealing with sparse systems, indicating the most im- portant coefficients to be used, and on its proven success in several applications, we propose its use in combination with the Volterra filter to identify nonlinear systems. In [12], one nonlinear system using a third-order Volterra filter and one using a fifth-order Volterra filter were identified with the LAR algorithm. In this paper we provide a more in depth description of this algorithm and add a new simulation where the coefficients have higher magnitudes. This paper is organized as follows. Section II describes how a nonlinear system can be modeled and the importance of the Volterra filter for this task. The LAR algorithm is addressed in Section III. The proposed configuration is tested in simulated scenarios and the results are shown in Section IV for Volterra filters of third and fifth orders. Conclusions are drawn in Section V. II. NONLINEAR SYSTEMS AND THE VOLTERRA SERIES Certain classes of nonlinear systems can be represented by one of the three following cascade models [13]: • LN – a linear filter followed by a memoryless nonlinea- rity, known as the Wiener model; • NL – a memoryless nonlinearity followed by a linear filter, known as the Hammerstein model; and • LNL – a linear filter, a memoryless nonlinearity and a second linear filter. It may be desirable to model them using a single Volterra based filter [13], i.e., to use a Volterra series for describing the REVISTA TELECOMUNICAÇÕES, VOL. 13, Nº. 02, DEZEMBRO DE 2011 12