# A Low-Complexity Volterra Filtered-Error LMS Algorithm with a Kronecker Product Decomposition
## 1 Introduction  
Active noise control has developed rapidly in the last few decades [1; 2], and it has been successfully applied in many fields, such as heating, ventilating and air-conditioning systems [3], motor systems [4], and active headsets [5; 6]. An active noise control system can generate a sound wave with the same amplitude but inverse phase at the canceling point by utilizing the secondary loudspeaker. Compared with passive methods, active noise control has shown its potential in canceling low-frequency environmental noises. A huge amount of research has been conducted in this field that focuses on the performance limitation of the active noise control system [7; 8], algorithms [9], and stability [10]. The most commonly used algorithm for a single-channel feedforward active noise control system is the filtered-reference least mean square (FXLMS) algorithm modeled by the finite impulse response (FIR) filter due to its stability and low computational complexity. Although the FXLMS algorithm has already been successfully applied in active noise control systems, its performance may degrade in controlling nonlinear noise processes [11]. It is well-known that nonlinear phenomena may occur in many circumstances. First, when the primary noise has a high sound pressure level, the primary path may suffer nonlinear processes. Second, when the secondary actuator works at a high sound pressure level or the size of the secondary loudspeaker is small but works at low frequencies, nonlinear distortion may occur at the secondary actuator [12]. Third, the primary noise may come from a dynamic system, such as a chaotic noise process. The transfer function of the secondary path may have a non-minimum phase, so that the causality constraint willbe violated [11]. In all of these situations, the traditional linear structure cannot model the nonlinear part of the system accurately enough. To overcome these challenges, many researchers have proposed numerous ways when taking the nonlinearity of the system into account, which can be roughly divided into two categories [13; 14]. The former one uses artificial neural networks to model the nonlinearity of the system [15; 16]. However, most artificial neural network-based algorithms are too computationally consuming for real applications of active noise control systems, which is out of the scope of this paper. The latter one employs the modeling capability of adaptive nonlinear filters including adaptive Volterra filters [11; 17], the functional link network (FLNN) [18] and its generalized form (GFLNN) [19], the exponential functional link network (EFLNN) [20], the even mirror Fourier nonlinear (EMFN) filter [21], the Legendre neural network (LeNN) [22], Chebyshev filters (CF) [23; 24], and so on. Furthermore, due to the different properties of different algorithms, convex combinations of the above-mentioned nonlinear filters have also been studied to achieve better performance [25; 26]. For active noise control systems, the influence of the secondary path must be considered for most algorithms. The NANC algorithm with a linear secondary path (LSP) and with a nonlinear secondary path (NSP) was unified in [27] by introducing a virtual secondary path into NANC with the NSP case. For most nonlinear filters, NANC with NSP case is more computationally consuming compared to NANC with the LSP case. The NANC problem with a nonlinear secondary path (NSP), however, is more important in real applications [12]. When the memory length becomes large, the algorithms mentioned above often require high computational complexity that cannot be neglected. There are two ways to reduce the computational complexity of the NANC algorithm. One way is to use the recursive structure, which can be seen as an expansion of the infinite impulse response (IIR) filter. This kind of NANC algorithm includes the bilinear filter algorithm [28; 29], recursive second-order Volterra filter [30], recursive even mirror Fourier nonlinear filter [31; 32], adaptive function expansion bilinear filter [33; 34], and so on.