# Low-Complexity Recursive Least-Squares Adaptive Algorithm Based on Tensorial Forms
## 4 Simulations and Practical Considerations
For each of the tensor-based update processes (i.e., for each simulated channel), this setting leads to arithmetic complexities proportional to \(L_{i}\). Although the tensor-based adaptive filters are initialized using the method recommended in [11], the initial convergence is not shown, as it is not considered relevant. For comparison reasons, the classical RLS method based on [PERSON]'s identity for the direct identification of the _unknown_ global impulse response is also simulated as a reference, as it is an already validated/established solution in scientific literature. The corresponding forgetting factors were chosen in a similar manner as the equivalent parameters of the tensor based methods (i.e., \(\lambda=1-1/(2^{k}L)\)). The first experiments were performed with a configuration of \(N=3\) filters for the identification of a global system with the length \(L=1024\) (\(L_{1}=64\), \(L_{2}=L_{3}=4\)), where the input signal is Gaussian noise. It can be noticed in Figure 1 that the RLS-DCD-T with \(N_{u,i}=1\) is enough to generate practically identical performances in comparison to the RLS-T algorithm. Moreover, the RLS adaptive method with different forgetting factors was simulated. It is obvious from the illustration that decreasing the forgetting factor for the RLS in order to increase the tracking speed deteriorates the system identification performance at a convergence state. It is also noteworthy that the individual adaptive filters are connected, i.e., the solution of one filter at time index \(n\) depends on the estimates of the other \(N-1\) filters at time index \(n-1\), due to Equation (19). Consequently, the estimation errors also cumulate in this manner, which could explain the steady-state variation of the tensor-based algorithms (RLS-T and RLS-DCD-T), as compared to the conventional RLS algorithm. The experiment was repeated for an AR(1) process in Figure 2. Despite the fact that the RLS-DCD-T slightly loses performance in comparison to the RLS-T, similar conclusions can be applied when performing the comparison with the RLS. The tensor based methods outperform the RLS adaptive filter, which cannot simultaneously match them in terms of tracking speed, respectively performance when convergence is achieved. For the third and fourth scenarios, the MISO configuration was changed to a decomposition based on \(N=4\) filters. The global system has \(L=2048\) coefficients and for the tensor-based systems \(L_{1}=16\), \(L_{2}=8\), respectively \(L_{3}=L_{4}=4\). In Figure 3, the input signal is Gaussian noise. The performances corresponding to the tensor based algorithms are again very similar. In comparison to the approach which directly identifies the global impulse response (i.e., the RLS algorithm), the RLS-DCD-T and RLS-T have much better tracking capabilities, when targeting for all the methods similar performance after considering the RLS-T.  
Figure 1: Performance of the RLS-T, RLS-DCD-T, and RLS algorithms, for the identification of the global impulse response \(\mathbf{g}\). The tensor-based filters have \(q_{1}=6\), and \(q_{2}=q_{3}=2\). The input signal is Gaussian noise, \(N=3\), and \(L=1024\). Both RLS versions have forgetting factors with the form \(\lambda=1-1/(2^{k}L)\). The performance of the RLS-T and the RLS-DCD-T are almost identical. Figure 2: Performance of the RLS-T, RLS-DCD-T, and RLS algorithms, for the identification of the global impulse response \(\mathbf{g}\). The tensor-based filters have \(q_{1}=6\), and \(q_{2}=q_{3}=2\). The input signal is an AR(1) sequence with the pole 0.85, \(N=3\), and \(L=1024\). Both RLS versions have forgetting factors with the form \(\lambda=1-1/(2^{k}L)\). vergence is reached. When repeating the experiment for an AR(1) sequence (Figure 4), it can be considered clear that, for both situations, any attempt of improving the tracking capabilities of the RLS adaptive filter by reducing its memory (i.e., by using a smaller value of \(\lambda\)) leads to a compromise with the quality the unknown global system identification at the convergence state.