## 6 Discussion  
The motivation behind the approach discussed in this paper was mainly twofold. First, we have shown how the intrinsic characteristics of the resulting impulse response can be exploited (i.e., the particular symmetry feature) within the identification process. As mentioned before, the identification of linearly separable systems that own particular intrinsic symmetric/antisymmetric properties has been addressed in our recent works [29; 30]. In this paper, as compared to these previous works, we did not limit the current solution only to the identification of linearly separable systems, but for more general forms of impulse responses. Second, using the matrix \(\mathbf{Q}\) within the identification process could represent a potential benefit in terms of further revealing and exploiting the low-rank features. This is different as compared to the approach used in [16; 17], and also represents a step toward other potential future works on this topic. The final goal is to further reduce the value of \(P\), which represents an important gain, especially for the identification of acoustic impulse responses.  
The preliminary findings in this direction indicate promising results. For example, let us consider that we are able to find a matrix \(\widetilde{\mathbf{Q}}\), such that the coefficients of the impulse response \(\widetilde{\mathbf{Q}}\mathbf{h}\) result in the descending order of their absolute values. Also, let us focus on the singular values of the corresponding matrices, i.e., \(\mathbf{H}=\text{ivec}(\mathbf{h})\) and \(\widetilde{\mathbf{H}}=\text{ivec}\left(\widetilde{\mathbf{Q}}\mathbf{h}\right)\), where \(\text{ivec}(\cdot)\) is the inverse vectorization operation, which converts a vector of length \(L=L_{1}L_{2}\) into a matrix of size \(L_{1}\times L_{2}\). In case of acoustic impulse responses, these matrices are closer to full-rank. However, the faster their singular values decrease to zero, the better is the approximation that justifies (5). In order to provide a supporting example, the singular values of \(\mathbf{H}\) and \(\widetilde{\mathbf{H}}\) (normalized with respect to the maximum one), together with the corresponding impulse responses, are represented in Figure 17, considering the acoustic impulse response \(\mathbf{h}\) from Figure 1c. Clearly, a smaller value of \(P\) can be used for the approximation of the permuted impulse response from Figure 17b, because the singular values of \(\widetilde{\mathbf{H}}\) decrease faster to zero, as compared to the singular values of \(\mathbf{H}\).  
Figure 16: Performance of the conventional RLS filter [with \(\lambda=1-1/(KL/2)\)] and the RLS-SF [with \(\lambda_{1}=1-1/(2 KPL_{1})\), \(\lambda_{2}=1-1/(2 KPL_{2})\), and \(P=8\)], using \(K=10\), for the identification of the acoustic echo path from Figure 1c. (**a**) The true acoustic impulse response, (**b**) estimated impulse response (after 5 s) with the conventional RLS filter, and (**c**) estimated impulse response (after 5 s) with the RLS-SF. Other conditions are the same as in Figure 15.  
The previous discussion outlined the main features of the proposed solutions, but it also represents a motivation for future investigations on this approach. This could further lead to potential performance improvements in the framework of low-rank system identification problems, by fully exploiting the intrinsic characteristics and the specific nature of the acoustic impulse responses.