## 2 Linearization System for Loudspeaker  
Loudspeakers increase volume as well as nonlinear distortion. This is due to the fact that as the diaphragm amplitude increases, the voice coil is displaced from the magnetic flux formed by the permanent magnet, causing the magnetic flux density of the voice coil to change depending on the diaphragm amplitude, and the stiffness of the edges and dampers that support the diaphragm to change depending on the diaphragm amplitude. Therefore, the loudspeaker must be modeled as a nonlinear system. Since the nonlinearity in a loudspeaker has a memory, like the impulse response in a linear system, it can be approximated by a Volterra series expansion [9] that can accurately model a nonlinear system with a memory. Then, the Volterra kernels representing each term in the Volterra series expansion are represented as multidimensional digital filters, which are called Volterra filters [13], and are one of the representative examples of nonlinear digital filters. Now, if the nonlinearity of the nonlinear system under consideration can be terminated up to the third order, in other words, if the nonlinearity above the fourth order is relatively small and negligible, the nonlinear input-output relation of the Volterra filter can be described as:  
\[y(n)=\sum_{k_{1}=0}^{N-1}h_{1}(k_{1})x(n-k_{1})\] \[\quad+\sum_{k_{1}=0}^{N-1}\sum_{k_{2}=0}^{N-1}h_{2}(k_{1},k_{2})x (n-k_{1})x(n-k_{2})\] \[\quad+\sum_{k_{1}=0}^{N-1}\sum_{k_{2}=0}^{N-1}\sum_{k_{3}=0}^{N-1 }h_{3}(k_{1},k_{2},k_{3})x(n-k_{1})x(n-k_{2})x(n-k_{3}), \tag{1}\]  
where \(x(n)\) and \(y(n)\) are input and output signals with discrete time \(n\), \(N\) is the memory length of the Volterra filter, and \(h_{1}(k_{1})\), \(h_{2}(k_{1},k_{2})\), and \(h_{3}(k_{1},k_{2},k_{3})\) are the filter coefficients of the first-, second-, and third-order Volterra filters, respectively. From (2.1), the second term is represented by a 2D digital filter and the third term is represented by a 3D digital filter. Hence, as the computation of the third term requires huge computational cost, the efficiency realization is required. In general, Volterra filters (kernels) have symmetry, for example, the symmetric property for the third-order Volterra filter is given by:  
\[h_{3}(k_{1},k_{2},k_{3}) =h_{3}(k_{1},k_{3},k_{2})\] \[=h_{3}(k_{2},k_{1},k_{3})\] \[=h_{3}(k_{2},k_{3},k_{1})\] \[=h_{3}(k_{3},k_{1},k_{2})\] \[=h_{3}(k_{3},k_{2},k_{1}). \tag{2}\]  
A block diagram of the linearization system that compensates for nonlinear distortion in a loudspeaker is shown in Figure 1. \(H_{1}(z)\), \(H_{2}(z_{1},z_{2})\), and \(H_{3}(z_{1},z_{2},z_{3})\) shown in the loudspeaker section in this figure are the first, second, and third order responses of the loudspeaker, respectively. Note that these are expressed in the form of multidimensional \(z\)-transformed Volterra filters for each order in (2.1). Furthermore, \(\hat{H}_{2}(z_{1},z_{2})\) and \(\hat{H}_{3}(z_{1},z_{2},z_{3})\) in the linearization system section represent the multidimensional z transform of the pre-identified second- and third-order Volterra kernels of the loudspeaker, respectively. Furthermore, \(\hat{H}_{1}^{-1}(z)\) is the transfer function of the approximate linear inverse filter of the identified first-order Volterra kernel (linear response) of the loudspeaker, which is designed to satisfy the following condition:  
\[H_{1}(z)\cdot H_{1}^{-1}(z)=z^{-\Delta},\]where \(\Delta\) is called the inverse delay and generally set to half the filter length of \(\hat{H}_{1}^{-1}(z)\). Second- and third-order nonlinear distortions in loudspeakers can be fully compensated for by the linearization system if the second- and third-order Volterra kernels are identified with high accuracy and the approximate linear inverse filter is properly designed. **3. Efficient Realization for Third-Order Volterra Filter Using Singular Value Decomposition**  
In this section, we describe the proposed efficient realization of the third-order Volterra filter based on singular value decomposition.