# Low-Complexity Recursive Least-Squares Adaptive Algorithm Based on Tensorial Forms
## 2 System Model
Then,  
\[c_{N}\mathbf{h}_{N}\otimes c_{N-1}\mathbf{h}_{N-1}\otimes\cdots\otimes c_{1} \mathbf{h}_{1}=\mathbf{h}_{N}\otimes\mathbf{h}_{N-1}\otimes\cdots\otimes \mathbf{h}_{1}. \tag{9}\]  
Nevertheless, the global impulse response \(\mathbf{g}\) can be identified with no scaling ambiguity. Furthermore, considering (6)-(8), \(y(n)\) can be rewritten as  
\[y(n)=\mathbf{g}^{T}\mathbf{x}(n). \tag{10}\]  
The target is to identify the unknown global system \(\mathbf{g}\) given by its individual components \(\mathbf{h}_{i}\) by using the reference signal:  
\[d(n)=\mathbf{g}^{T}\mathbf{x}(n)+w(n), \tag{11}\]  
where the measurement noise, \(w(n)\), is uncorrelated with the incoming signal. The reference signal from (11) is described by the variance:  
\[\sigma_{d}^{2}=\mathbf{g}^{T}E\Big{[}\mathbf{x}(n)\mathbf{x}^{T}(n)\Big{]} \mathbf{g}+\sigma_{w}^{2}=\mathbf{g}^{T}\mathbf{R}\mathbf{g}+\sigma_{w}^{2}. \tag{12}\]  
\(E[\cdot]\) denotes mathematical expectation, \(\mathbf{R}=E\big{[}\mathbf{x}(n)\mathbf{x}^{T}(n)\big{]}\), and \(\sigma_{w}^{2}=E[w^{2}(n)]\). The error signal has the expression:  
\[e(n)=d(n)-\widehat{\mathbf{g}}^{T}\mathbf{x}(n), \tag{13}\]  
where \(\widehat{\mathbf{g}}\) describes an estimator of the global impulse response. The Wiener approach for the estimate \(\widehat{\mathbf{g}}\)[14] is not suitable in nonstationary conditions. Moreover, the requirement of real-time processing makes it only a starting point in providing more adequate solutions based on adaptive algorithms. The majority of adaptive systems rely on the LMS-based methods, which offer a satisfactory balance between performance and arithmetic workload. However, state-of-the-art literature provides several versions of the RLS adaptive filter as appealing alternatives with better performance indicators (i.e., tracking speed, convergence rate). Despite of the advantages which the RLS-based versions have over their LMS-based counterparts, most of the RLS algorithms cannot provide stable functionality combined with proper arithmetic complexity. In recent years, the integration between the dichotomous coordinate descend (DCD) iterative methods [20] with the RLS adaptive filters [14] successfully contested the _prohibitive_ tag associated with the RLS family implementations. The RLS-DCD delivered promising results with acceptable processing requirements and performance similar to the classical RLS versions. Now, having (13), we can define the mean-squared error (MSE) criterion:  
\[J(\widehat{\mathbf{g}})=E\Big{[}e^{2}(n)\Big{]}=\sigma_{d}^{2}-2\widehat{ \mathbf{g}}^{T}\mathbf{p}+\widehat{\mathbf{g}}^{T}\mathbf{R}\widehat{\mathbf{ g}}, \tag{14}\]  
with \(\mathbf{p}=E[\mathbf{x}(n)d(n)]\) representing the cross-correlation vector between \(\mathbf{x}(n)\) and \(d(n)\). Our focus is the minimization of \(J(\widehat{\mathbf{g}})\), and it represents the optimization criterion. In order to do this, we may use the well-known solution [14] given by the Wiener filter:  
\[\widehat{\mathbf{g}}_{\text{W}}=\mathbf{R}^{-1}\mathbf{p}. \tag{15}\]  
With the iterative Wiener filter [8], we are able to speculate the decomposition of the global impulse response, considering the topic of the multilinear forms. The optimization criterion consists on a block coordinate descent method applied on the individual components [25]. Comparing to the standard Wiener filter given by Equation (15), the iterative one boosts the performance. It is suitable, especially when a small amount of data are used in order to estimate the statistics of the signal (i.e., \(\mathbf{R}\) and \(\mathbf{p}\)).