# A Recursive Least-Squares Algorithm for the Identification of Trilinear Forms
## 3 RLS Algorithm for Trilinear Forms
\right], \tag{52}\] \[\widehat{\mathbf{h}}_{2}(0) =\frac{1}{L_{2}}\mathbf{1}_{L_{2}},\] (53) \[\widehat{\mathbf{h}}_{3}(0) =\frac{1}{L_{3}}\mathbf{1}_{L_{3}}, \tag{54}\]  
where \(\mathbf{0}_{N}\) and \(\mathbf{1}_{N}\) are two vectors of length \(N\), all elements of which are equal to zero and one, respectively. The proposed RLS algorithm for trilinear forms, namely RLS-TF, is summarized in Table 1. It could also be interpreted as an extension of the RLS-based algorithm tailored for bilinear forms, which was presented in [12]. However, if the MISO system identification problem results based on (12), it is natural to use the RLS-TF algorithm, which is designed for the identification of third-order (rank-1) tensors. In terms of computational complexity, it can be noticed that the proposed RLS-TF algorithm combines the solutions provided by three RLS-based filters, i.e., \(\widehat{\mathbf{h}}_{1}(n)\) (of length \(L_{1}\)), \(\widehat{\mathbf{h}}_{2}(n)\) (of length \(L_{2}\)), and \(\widehat{\mathbf{h}}_{3}(n)\) (of length \(L_{3}\)). Since the complexity of a regular RLS-based algorithm is proportional to the square of the filter length, the overall complexity of the RLS-TF algorithm roughly results in \(\mathcal{O}(L_{1}^{2}+L_{2}^{2}+L_{3}^{2})\). On the other hand, the system identification problem can also be handled based on the conventional RLS algorithm, which results following (20) and (21), together with the cost function from (36). However, in this case, there is a single adaptive filter \(\widehat{\mathbf{h}}(n)\), of length \(L=L_{1}L_{2}L_{3}\), so that the computational complexity is of the order of \(\mathcal{O}(L^{2})\). This could be much more computationally expensive as compared to the proposed RLS-TF algorithm. Basically, the RLS-TF algorithm \"transforms\" a large system identification problem of length \(L=L_{1}L_{2}L_{3}\) into three \"smaller\" problems of lengths \(L_{1}\), \(L_{2}\), and \(L_{3}\), respectively, with advantages in terms of both performance (as will be shown in simulations) and complexity. As outlined before, the proposed RLS-TF algorithm combines the solutions provided by three adaptive filters of lengths \(L_{1}\), \(L_{2}\), and \(L_{3}\), respectively, while the conventional RLS algorithm deals with a single filter of length \(L=L_{1}L_{2}L_{3}\), which is usually much longer. Since the length of the filter highly influences the main performance criteria, i.e., convergence rate and misadjustment [18], the proposed algorithm is able to outperform the conventional one in terms of both criteria. In other words, the shorter the length, the faster the convergence and the lower the misadjustment. This expected behaviour will be supported by the simulation results provided in the next section. Finally, we should observe that there are some extra operations specific to the RLS-TF algorithm. For example, the \"input\" signals \(\mathbf{x}_{32}(n)\), \(\mathbf{x}_{31}(n)\), and \(\mathbf{x}_{21}(n)\) result based on (30)-(32), which rely on (27)-(29). Furthermore, the global impulse response (if required within the application) can be evaluated based on (22). These operations require Kronecker products, but the related computational complexity is moderate, i.e., of the order of \(\mathcal{O}(L_{1}L_{2}L_{3})=\mathcal{O}(L)\). The detailed computational complexities of the proposed RLS-TF algorithm and other benchmark algorithms (i.e., the conventional RLS and NLMS algorithms) are summarized in Table 2. For a better visualization, the computational complexities are also illustrated in Figure 1, in terms of the number of multiplications and additions (per iteration), for different values of \(L_{1}\); the other lengths are fixed to \(L_{2}=8\) and \(L_{3}=4\) (similar to the experimental setup from Section 4). As we can notice, the computational complexity of the conventional RLS algorithm was significantly greater, while the computational amount of the proposed RLS-TF algorithm was closer to the conventional NLMS algorithm, especially for higher lengths.