# Wiener Filter Using the Conjugate Gradient Method and a Third-Order Tensor Decomposition
## 5 Conclusions and Future Works  
In this paper, we have developed an iterative version of the Wiener filter using the CG method and exploiting a tensorial decomposition of the impulse response. The resulting  
Figure 11: Normalized misalignment of the IWF-NKP-CG [33] (with \(K^{*}=12\)) and IWF-TOT-CG (with \(K_{1}=12\) and \(K_{2}=1\)) using \(P^{*}=P=1\), while \(\text{SNR}=20\) dB. The required statistics are obtained by averaging across \(N=ML\) data samples (with \(L=512\)), where (**a**) \(M=2\) and (**b**) \(M=1\).  
Figure 10: Normalized misalignment of the IWF-NKP-CG [33] (with \(K^{*}=12\)) and IWF-TOT-CG (with \(K_{1}=12\) and \(K_{2}=1\)) using different values of \(P^{*}\) and \(P\), respectively. The required statistics are obtained by averaging across \(N=5L\) data samples, \(L=512\), and \(\text{SNR}=20\) dB.  
IWF-TOT-CG combines the solutions of three sets of Wiener-Hopf equations which are solved with CG updates, thus avoiding matrix inversion operations. An important gain is related to the dimensionality reduction in a long-length system identification problem, which can be reformulated using a reduced set of coefficients, corresponding to three (much) shorter filters. This approach fits very well with the identification of low-rank impulse responses, like in echo cancellation.  
In terms of its performance, the proposed IWF-TOT-CG outperforms the conventional Wiener filter, especially in some challenging scenarios. Here, we can mention the cases when dealing with a small amount of available data (to estimate the statistics) or working in low SNR environments. While the accuracy of the conventional Wiener filter is highly affected in such conditions, the proposed version (which operates with smaller data structures) is still robust and provides reliable solutions. Moreover, the IWF-TOT-CG performs better as compared to the previously developed IWF-TOT [18], which involves matrix inversion operations to solve the Wiener-Hopf equations. Also, the proposed algorithm provides improved performance as compared to its counterpart based on the second-order NKP decomposition, i.e., the IWF-NKP-CG [33].  
Future works will focus on three main directions. First, we can exploit other line search methods to solve the Wiener-Hopf equations, like those based on the coordinate descent technique [34; 35]. A comparison between these methods is beyond the scope of this paper; however, this could open the path toward using inexact line search methods in conjunction with decomposition-based algorithms. Among them, we can mention the dichotomous coordinate descent technique [40; 41; 42; 43; 44], which is very appealing in terms of computational efficiency. Second, another direction for future works targets the extension to higher-order tensors, which could lead to improved decomposition-based solutions and higher dimensionality reduction. Third, it would be highly useful to extend the decomposition-based approach and the tensorial framework to other potential solutions used for system identification problems, like the Kalman filter and different adaptive filtering algorithms. These developments could be further used in real-world applications, like echo cancellation, active noise control, and interference reduction.  
Conceptualization, J.B.; methodology, C.P.; software, C.-L.S.; validation, R.-L.C.; formal analysis, L.-M.D.; investigation, S.C. All authors have read and agreed to the published version of the manuscript.  
This work was supported by a grant of the Ministry of Research, Innovation and Diglization, CNCS-UEFISCDI, project number PN-III-P4-PCE-2021-0438, within PNCDI III, and by a grant from the National Program for Research of the National Association of Technical Universities - GNAC ARUT 2023 (no. 36/09.10.2023, code 133).