  Citation: Rusu, A.-G.; Ciochin ˘ a, S.; Paleologu, C.; Benesty, J. Cascaded RLS Adaptive Filters Based on a Kronecker Product Decomposition. Electronics 2022, 11, 409. https:// doi.org/10.3390/electronics11030409 Academic Editor: Manohar Das Received: 10 December 2021 Accepted: 27 January 2022 Published: 29 January 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). electronics Article Cascaded RLS Adaptive Filters Based on a Kronecker Product Decomposition Alexandru-George Rusu 1,2 , Silviu Ciochin ˘ a 1 , Constantin Paleologu 2, * and Jacob Benesty 3 1 Department of Telecommunications, University Politehnica of Bucharest, 061071 Bucharest, Romania; alexandru.rusu@rohde-schwarz.com (A.-G.R.); silviu@comm.pub.ro (S.C.) 2 Department of Research and Development, Rohde & Schwarz Topex, 020335 Bucharest, Romania 3 INRS-EMT, University of Quebec, Montreal, QC H5A 1K6, Canada; Jacob.Benesty@inrs.ca * Correspondence: pale@comm.pub.ro Abstract: The multilinear system framework allows for the exploitation of the system identification problem from different perspectives in the context of various applications, such as nonlinear acoustic echo cancellation, multi-party audio conferencing, and video conferencing, in which the system could be modeled through parallel or cascaded filters. In this paper, we introduce different memoryless and memory structures that are described from a bilinear perspective. Following the memory structures, we develop the multilinear recursive least-squares algorithm by considering the Kronecker product decomposition concept. We have performed a set of simulations in the context of echo cancellation, aiming both long length impulse responses and the reverberation effect. Keywords: recursive least-squares (RLS) algorithm; adaptive filters; Kronecker product decomposition; system identification; echo cancellation 1. Introduction In the field of system identification, many applications involve adaptive filtering algorithms [1,2]. One of them is the echo cancellation problem, which has raised many challenges over the years [3,4]. Based on the input-output relation, a dynamic system should be determined (i.e., the echo path), considering various parameters and external factors that must be estimated. These dynamic systems are modeled linearly through an adaptive filter with a finite-impulse-response (FIR) structure [5,6]. The main performance bottlenecks, in terms of computational complexity, tracking, and convergence rate, arise when the length of the impulse response reaches hundreds/thousands of coefficients. The literature presents many approaches to improve the overall performance, also taking into account the fact that the echo paths are sparse in nature [7–13]. Recently, in our previous work [14], we introduced a new approach of splitting a long length impulse response into several impulse responses of shorter lengths, aiming to reduce the computational complexity by maintaining the overall performance. Another challenge arises when the echo path produces multiple reflections, and this effect is called reverberation. From a mathematical point of view, this effect could be described (to some extent) by using the Kronecker product decomposition of the impulse response [15,16]. In this paper, we extend our study on cascaded adaptive filters, aiming to reduce the computational complexity considering both long length impulse responses and the reverberation effect. Our approach is based on multilinear structures and the Kronecker product decomposition. The main goal is to outline the features of this development and its potential. The rest of the paper is organized as follows. Section 2 presents the background for different bilinear structures without memory, while Section 3 introduces bilinear structures with memory. In Section 4, the new development is combined with the recursive least- Electronics 2022, 11, 409. https://doi.org/10.3390/electronics11030409 https://www.mdpi.com/journal/electronics