# Low-Complexity Recursive Least-Squares Adaptive Algorithm Based on Tensorial Forms
## 4 Simulations and Practical Considerations  
Several sets of experiments were performed in the context of the system identification problem with the MISO configuration. The input signals are white Gaussian noises with sub-unitary values. For some scenarios, before being fed to the input of the system, the signals are also filtered through an auto-regressive AR(1) system with the pole 0.85, in order to create highly-correlated input sequences [11]. For \(d(n)\), the signal-to-noise (SNR) ratio is experimentally set to 15 dB. In the context of adaptive filtering, the convergence behavior and performance analysis of any adaptive algorithm consider the Gaussian input as a benchmark. When the input is white Gaussian noise, the conditional number of the correlation matrix of the input signal is equal to one, since this matrix is a diagonal one and all its eigenvalues are equal. On the other hand, in case of more correlated inputs, the conditional number of this matrix increases, resulting in the challenges in terms of convergence rate. However, the recursive least-squares (RLS) algorithm is less influenced by the character of the input signal (in terms of the correlation degree), as compared to the family of least-mean-square (LMS) algorithms [14]. Consequently, it is important to show that the proposed RLS-DCD-T algorithm also inherits this feature. This is the reason for including both cases in our experiments, i.e., white Gaussian noise and AR(1) process as inputs. The criterion employed to analyze the performance is the normalized misalignment, corresponding to the global system identification process. It is defined for each time index, using the global filter estimate \(\widehat{\mathbf{g}}(n)\) and the true impulse response \(\mathbf{g}\), as \(20\text{log}_{10}[\|\mathbf{g}-\widehat{\mathbf{g}}(n)\|_{2}/\|\mathbf{g}\|_ {2}]\), where \(\|\cdot\|_{2}\) denotes the \(\ell_{2}\) norm and \(\mathbf{g}\) is computed based on (8). The experiments use global impulse responses which can be decomposed using (8) into \(N=3\), respectively \(N=4\), smaller systems of lengths \(L_{i},\ i=1,2,\ldots,N\) (depending on the case). The first and the fourth small impulse responses are taken as the first \(L_{1}\), respectively \(L_{4}\), samples of the the B4 and B1 impulse responses from the G.168 ITU-T Recommendation [28]. The second and third systems are generated using \((-0.5^{h})/2\), re spectively \((0.8^{j_{2}})/2\), where \(j_{1}=0...L_{2}-1\) and \(j_{2}=0...L_{3}-1\). In order to evaluate the tracking capabilities of the presented algorithms, the global system is suddenly modified after the initial convergence is achieved by changing the sign of the coefficients corresponding to one of the unknown channels (the longest one, when the case). The experimental results will mainly compare the performances of the RLS-DCD-T, respectively the RLS-T, adaptive algorithms. For both tensor-based methods, the adaptive filter lengths are chosen as \(L_{i}=2^{q_{i}}\) and the forgetting factors are set as \(\lambda_{i}=1-1/(2^{k_{i}}L_{i})=1-1/(2^{k_{i}+q_{i}})\). From a hardware implementation point of view, the multiplications with lambda can be performed more efficiently with only bit-shifts and subtractions [18; 29]. Furthermore, for the RLS-DCD-T, its DCD specific parameters are chosen considering several aspects. Firstly, the tap values for the estimated systems are sub-unitary (\(H=1\)), and they are represented using \(M_{b}=16\) bits--one for the sign and the rest, up to 16, hold the magnitude information. Secondly, the maximum number of allowed updates was successfully used in the past with values lower than 10 for adaptive filters with hundreds and thousands of coefficients [18; 27]. Taking into account that the selected scenarios have filters with the corresponding lengths \(L_{i}\) of maximum tens of coefficients, we chose not to display simulations with \(N_{u,i}>1\) because they showed no improvement in performance, even for the longer adaptive filters in the tensorial decompositions.