## 4 RLS Adaptive Algorithm
Consequently, the filters' updates become  
\[\widehat{\mathbf{g}}(t) =\widehat{\mathbf{g}}(t-1)+\mathbf{k}(t)\varepsilon(t),\]5 \[\widehat{\mathbf{g}}(t) =\widehat{\mathbf{g}}(t-1)+\mathbf{k}(t)\varepsilon(t),\]6  
Finally, the components of the vectors \(\widehat{\widehat{\mathbf{g}}}_{1}(t)\) and \(\widehat{\mathbf{g}}_{2}(t)\) [see (31) and (33)] are used in (29) to obtain the symmetric filter \(\widehat{\mathbf{g}}^{+}(t)\), then the estimate \(\widehat{\mathbf{g}}(t)\) results immediately. Nevertheless, we should note that in some applications (like echo cancellation), the coefficients of the adaptive filter are not really required, because only the error signal is needed. In such cases, the evaluation of \(\widehat{\mathbf{g}}(t)\) based on (29) is not necessary, and the adaptive filter provides only the a priori error signal \(\varepsilon(t)=\varepsilon_{1}(t)=\varepsilon_{2}(t)\). The proposed RLS algorithm with symmetric filter (RLS-SF) is summarized in \[\widehat{\mathbf{g}}(t) =\widehat{\mathbf{g}}(t-1)+\mathbf{k}(t)\varepsilon(t),\]7 As we can see, the RLS-SF combines the solutions of two adaptive filters of much smaller lengths (i.e., \(PL_{1}\) and \(PL_{2}\)), as compared to the conventional RLS algorithm that updates a single adaptive filter of length \(L=L_{1}L_{2}\). Because usually \(P\ll L_{2}\), the expected gain of the RLS-SF is twofold, in terms of both performance (i.e., convergence rate/tracking) and computational complexity, which is proportional to \(\mathcal{O}\big{[}(PL_{1})^{2}+(PL_{2})^{2}\big{]}\). Simulation results provided in Section 5.2 support these performance features. The computational complexities of the proposed algorithms developed in Sections 3 and 4 are similar to those proposed in [16; 17], respectively. Basically, in both approaches, the original system identification problem of length \(L=L_{1}L_{2}\) is reformulated as a combination of two shorter filters of lengths \(PL_{1}\) and \(PL_{2}\), where usually \(P\ll L_{2}\). However, although the solutions developed in [16; 17] follow the direct approach based on (5), the current proposed algorithms exploit (11), where the motivation is mainly twofold. First, the intrinsic characteristics of the resulting impulse response can be exploited (i.e., the particular symmetry feature) within the identification process. Second, using the matrix \(\mathbf{Q}\) could lead to potential benefits in terms of further exploiting the low-rank features. This is different as compared to the approach used in [16; 17], and also represents a step toward our future works on this topic. The final goal would be to further reduce the value of \(P\), which represents an important gain, especially for the identification of acoustic impulse responses. autoregressive model with a pole at 0.9) or a speech sequence. The output of the echo path (i.e., the echo signal) is corrupted by a background noise, \(w(t)\), which is considered white and Gaussian (with different SNRs). All the experiments were performed by using MATLAB R2018b, running on an Asus GL552 VX device (Windows 10 OS), which has an Intel Core i7-6700 HQ CPU @ 2.60 GHz, with 4 Cores, 8 Logical Processors, and 16 GB of RAM. Four echo paths are used in simulations and their corresponding impulse responses are depicted in \[\widehat{\mathbf{g}}(t) =\widehat{\mathbf{g}}(t-1)+\mathbf{k}(t)\varepsilon(t),\]8 The impulse response from Figure 1a is the first network echo path from the ITU-T G168 Recommendation [42], which is a cluster of 64 coefficients padded with zeros up to the length \(L=500\). The second impulse response from Figure 1b is obtained in a similar manner, by using the fifth network echo path from the same recommendation, i.e., a cluster of 96 coefficients padded with zeros. The impulse responses from Figure 1c,d are two measured acoustic echo paths, with different sparseness degrees.