# Kalman Filter Using a Third-Order Tensorial Decomposition of the Impulse Response
## 5 Conclusions and Future Work  
In the current paper, following the recently proposed TOT-based decomposition of the impulse response [36], we have developed a Kalman filter that exploits this technique, in the framework of system identification. The proposed KF-TOT combines (with the Kronecker product) the estimates provided by three filters, which can be much shorter as compared to the full length of the global impulse response. Since the length of an adaptive filter is related to its main performance criteria, this decomposition-based solution leads to important gains. As a result, following the announced goals from Section 1, the KF-TOT first outperforms the conventional Kalman filter in terms of both convergence rate and tracking, but also in terms of accuracy, reaching a significantly lower misalignment level. In addition, as supported in Table 1, these performance gains are achieved owning a lower computational complexity. Second, the KF-TOT leads to improved performances as compared to its KF-NKP counterpart [27], which exploits a second-order decomposition level by combining the estimates provided by two filters. Third, based on the evaluation of the KF-TOT as compared to the recently developed RLS-TOT [37], we reach the conclusion that the proposed algorithm is more reliable in nonstationary conditions, owning to a better tracking behavior and an improved accuracy of the solution.  
Due to its performance features, the proposed KF-TOT fits very well for applications that involve the identification of long-length low-rank impulse responses, especially for time-varying systems. In this context, echo cancellation would represent an excellent application framework, since the echo paths (for both network and acoustic scenarios) are characterized by impulse responses that could easily reach hundreds/thousands of coefficients, while also being highly variable in time. Since this is a real-time application, a fast identification (and tracking) of the echo path impulse response, with low computational demands, would be a very appealing asset. In addition, different long-length low-rank systems can be involved in various applications, which include digital TV transmission [56], satellite-linked communications [57], underwater communications [58], and adaptive beamformers [59]. Hence, the developed KF-TOT could also represent a practical solution for such real-world applications.  
Future work will focus on analyzing a higher-order decomposition level in conjunction with the Kalman filter. In this context, the main challenge is related to the way we handle the rank of a higher-order tensor that results from the development. Usually, approximation techniques are used to find such a tensor rank, which could represent a practical limitation. In the TOT framework, i.e., third-order decomposition level [36], the corresponding tensor associated with the reshaped impulse response results as a sum of \(P\) third-order tensors of rank \(L_{2}\), so this parameter is controlled and limited to small values. Since there is not a straightforward extension from one decomposition level to another, applying the decomposition technique to a higher order while still controlling the tensor rank represents a challenging task. Several preliminary ideas can be found in [60]. Also, we aim to involve the KF-TOT and the related decomposition-based algorithms in the framework of other applications that require the identification of long-length impulse responses, especially related to the acoustic domain.  
Conceptualization, L.-M.D.; methodology, C.P.; validation, J.B.; formal analysis, F.A. All authors have read and agreed to the published version of the manuscript.  
The work of [PERSON] and [PERSON] was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS-UEFISCDI, project number PN-III-P4-PCE-2021-0780, within PNCDI III.  
Not applicable.  
Data Availability Statement:Data is contained within the article.  
The authors declare no conflicts of interest.