Adaptive filtering for the identification of bilinear forms

Bilinear systems are involved in many interesting applications, especially related to the approximation of nonlinear systems. In this context, the bilinear term is usually defined in terms of an input-output relation (i.e., with respect to the data). Recently, a different approach has been introduced, by defining the bilinear term with respect to the impulse responses of a spatiotemporal model, which resembles a multiple-input/single-output (MISO) system. Also, in this framework, the Wiener filter has been studied to address the identification problem of these bilinear forms. Since the Wiener filter may not be always very efficient or convenient to use in practice, we propose in this paper an adaptive filtering approach. Consequently, we develop and analyze some basic algorithms tailored for the identification of bilinear forms, i.e., least-mean-square (LMS), normalized LMS (NLMS), and recursive-least-squares (RLS). Simulations performed in the context of system identification (based on the MISO system approach) support the theoretical findings and indicate the appealing performance of these algorithms.