## 2 Linearization System for Loudspeaker
Figure 2 shows a schematic diagram to understand the proposed realization method. Figure 2a shows all elements of the third-order Volterra kernel when the memory length is \(N=4\). It can be seen that the third-order Volterra kernel is a three-dimensional digital filter. Next, since the third-order Volterra kernel has symmetry given by (2), the elements required to compute the output signal of the third-order Volterra filter, denoted by the third term in (1), are shown in Figure 2b. Then, Figure 2c shows the four 2D elements obtained by slicing the structure shown in (b) in the \(l\) direction. This operation results in the decomposition of a three-dimensional filter structure into multiple two-dimensional filter structures. Thus, the \(l\)-layer 2-D filter can be represented by the following matrix:  
\[\quad+\sum_{k_{1}=0}^{N-1}\sum_{k_{2}=0}^{N-1}h_{2}(k_{1},k_{2})x (n-k_{1})x(n-k_{2})\]0 Block diagram of the 3 rd-order linearization system for loudspeakers with higher-order nonlinearity. \[\quad+\sum_{k_{1}=0}^{N-1}\sum_{k_{2}=0}^{N-1}h_{2}(k_{1},k_{2})x (n-k_{1})x(n-k_{2})\]1 Third-order Volterra kernel considering its symmetry characteristic and its separated layers. (**a**) All elements (\(N=4\)). (**b**) Elements required for convolution. (**c**) Elements of each layer. \[\mathbf{H}_{3,l}=\left( \begin{array}{cccc}h_{3}(0,l-1,l-1)&h_{3}(0,l,l-1)&\cdots& h_{3}(0,N-1,l-1)\\ h_{3}(1,l-1,l-1)&h_{3}(1,l,l-1)&\cdots&h_{3}(1,N-1,l-1)\\ \vdots&\vdots&\ddots&\vdots\\ h_{3}(l-1,l-1,l-1)&h_{3}(l-1,l,l-1)&\cdots&h_{3}(l-1,N-1,l-1)\end{array} \right). \tag{4}\]  
Then, the input-output relation equation of the \(l\)-layer 2D filter is:  
\[y_{3,l}(n)=\sum_{k_{1}=0}^{l-1}\sum_{k_{2}=l-1}^{N-1}h_{3}(k_{1},k_{2},l-1)x(n- k_{1})x(n-k_{2}), \tag{5}\]  
or,  
\[y_{3,l}(n)=\mathbf{x}^{T}(n)\mathbf{H}_{3,l}\mathbf{x}^{\prime}(n), \tag{6}\]  
where \(\mathbf{x}(n)\) and \(\mathbf{x}^{\prime}(n)\) are the input signal vectors represented by:  
\[\mathbf{x}(n) =[x(n)\ x(n-1)\ \cdots\ x(n-l+1)]^{T}, \tag{7}\] \[\mathbf{x}^{\prime}(n) =[x(n-l+1)\ x(n-l)\ \cdots\ x(n-N+1)]^{T}. \tag{8}\]  
Thus, the output of the third-order Volterra filter is:  
\[y_{3}(n)=\sum_{l=1}^{N}y_{3,l}(n)x(n-l+1). \tag{9}\]  
In the proposed realization, these matrices \(\mathbf{H}_{3,l}\) are decomposed by singular value decomposition as follows:  
\[\mathbf{H}_{3,l}=\sum_{m=1}^{M}\sigma_{m,l}\mathbf{u}_{m,l}\mathbf{v}_{m,l}^{T}, \tag{10}\]  
where \(M\) is the rank of \(\mathbf{H}_{3,l}\), \(\sigma_{m,l}\) is the \(l\)-layer \(m\)th singular value, \(\mathbf{u}_{m,l}\) is a left singular vector, and \(\mathbf{v}_{m,l}\) is a right singular vector, respectively. Hence, the \(l\)-layer output is given by:  
\[y_{3,l}(n)=\sum_{m=1}^{M}\sigma_{m,l}\left[x^{T}(n)\mathbf{u}_{m,l}\mathbf{v}_ {m,l}^{T}\mathbf{x}^{\prime}(n)\right]. \tag{11}\]  
This input-output relationship is represented by a block diagram shown in Figure 3. Figure 3 shows that this filter structure has independent paths for each singular value. Thus, even if we remove paths with small singular values, we expect to be able to model the original \(l\)-layer 2D filter with high accuracy. If only the paths involving the singular values up to the \(a_{l}\)th are kept and the other paths are removed, the input-output relation is:  
\[y_{3}(n) =\sum_{l=1}^{N}y_{3,l}(n)x(n-l+1), \tag{12}\] \[y_{3,l}(n) =\sum_{m=1}^{a_{l}}\sigma_{m,l}\left[x^{T}(n)\mathbf{u}_{m,l} \mathbf{v}_{m,l}^{T}\mathbf{x}^{\prime}(n)\right]. \tag{13}\]  
The concrete procedure of the proposed implementation method is as follows. 1. Obtain the third-order Volterra kernel of the target loudspeaker using an identification method such as FRM [16]. 2. Reduce the obtained third-order Volterra kernel to only the necessary components as shown in Figure 2b based on the symmetry of (2). 3. Slice the third-order Volterra kernel with only the necessary components as shown in Figure 2b into multiple two-dimensional components \(H_{3,l}\), as shown in Figure 2c. 4.