# A Connection Between the Kalman Filter and an Optimized LMS Algorithm for Bilinear Forms
## 1 Introduction  
Bilinear systems were previously treated in various contexts [1], such as the nonlinear systems approximation. The applications derived from there are numerous, among which we can mention system identification [2; 3; 4; 5], design of digital filters [6], echo cancellation [7], chaotic communications [8], active noise control [9], neural networks [10], etc. In all these papers, the bilinear term is understood in terms of an input-output relation (with respect to the data).  
More recently, a new approach was studied in [11], where the bilinear term was defined in the context of a multiple-input/single-output (MISO) system, with respect to the impulse responses of a spatiotemporal model. In [11], the Wiener filter solution for the identification of such bilinear forms was proposed, and then, in [12; 13; 14], some adaptive solutions based on different basic algorithms were provided. Similar frameworks can be found in [15; 16; 17; 18], in conjunction with specific applications, such as channel equalization and nonlinear acoustic echo cancellation; however, these works were not associated with the identification of bilinear forms or analyzed in this context.  
The Kalman filter was first introduced in [19] and it has a wide range of applications in technology, as well as in signal processing [20]. Some of its applications in the automotive field are mentioned in [21; 22], and the references therein, while [23] also provides a computational complexity analysis.  
In this paper, we focus on a comparative study of two different types of algorithms tailored for the identification of bilinear forms. The first one is based on the Kalman filter [19], whose bilinear form was introduced in our previous work [13], together with a simplified (i.e., low complexity) version, which could be more suitable in real-world applications. The second algorithm is a version of the least-mean-square (LMS) adaptive filter, but where the step-size parameter has been optimized inorder to meet a proper compromise between the convergence rate and misadjustment. In addition, as it will be shown, there is a strong similarity between the simplified Kalman filter and the optimized LMS algorithm. Simulations show the good performance of the proposed solutions, in comparison with other related works based on the conventional algorithms. Moreover, the gain could be twofold, in terms of both convergence performance and computational complexity.  
The paper structure is as follows. Section 2 introduces the system model in the context of bilinear forms. In this framework, we present the Kalman filter together with its simplified version in Section 3. The optimized LMS algorithm is derived in Section 4. Next, a comparison between the simplified Kalman filter and the optimized LMS algorithm for bilinear forms is presented in Section 5. Section 6 provides some practical considerations. Simulations are performed in the framework of system identification and the results are given in Section 7. Finally, some conclusions are drawn in Section 8. Also, Appendix A provides detailed computations of some terms required within the optimized LMS algorithm, while Appendix B summarizes and briefly explains the main parameters used in the paper, in order to facilitate the reading. For the sake of clarity, the main notation used in this work is provided in Table 1.