Efficient Algorithms for Linear System Identification with Particular Symmetric Filters

In linear system identification problems, it is important to reveal and exploit any specific intrinsic characteristic of the impulse responses, in order to improve the overall performance, especially in terms of the accuracy and complexity of the solution. In this paper, we focus on the nearest Kronecker product decomposition of the impulse responses, together with low-rank approximations. Such an approach is suitable for the identification of a wide range of real-world systems. Most importantly, we reformulate the system identification problem by using a particular symmetric filter within the development, which allows us to efficiently design two (iterative/recursive) algorithms. First, an iterative Wiener filter is proposed, with improved performance as compared to the conventional Wiener filter, especially in challenging conditions (e.g., small amount of available data and/or noisy environments). Second, an even more practical solution is developed, in the form of a recursive least-squares adaptive algorithm, which could represent an appealing choice in real-time applications. Overall, based on the proposed approach, a system identification problem that can be conventionally solved by using a system of L=L1L2 equations (with L unknown parameters) is reformulated as a combination of two systems of PL1 and PL2 equations, respectively, where usually P≪L2 (i.e., a total of PL1+PL2 parameters). This could lead to important advantages, in terms of both performance and complexity. Simulation results are provided in the framework of network and acoustic echo cancellation, supporting the performance gain and the practical features of the proposed algorithms.