# Kalman Filter Using a Third-Order Tensorial Decomposition of the Impulse Response
## 1 Introduction  
The Kalman filter is recognized as a popular signal processing tool in the framework of many real-world applications [1; 2; 3]. Among them, it has successfully been used in echo cancellation scenarios [4; 5; 6; 7; 8; 9], aiming to model an unknown impulse response corresponding to the echo path. Nevertheless, the finite impulse response filter associated with this impulse response is usually characterized by a very large number of coefficients, which reaches the order of hundreds or even thousands. Such a large parameter space significantly impacts the overall performance of the Kalman filter, especially in terms of the convergence behavior, solution accuracy, and computational complexity. In system identification problems like echo cancellation it is always helpful to exploit the particular characteristics of the impulse responses to be estimated. For example, echo paths are sparse in nature, i.e., only a small number of coefficients (as compared to the full length of the impulse response) have a significant magnitude, while the rest of them are close to zero or even null. The adaptive algorithms that take advantage of this feature are designed based on a \"proportionate\" approach [10], in which the significant coefficients are emphasized during the updating process. Due to this technique, the resulting proportionate-type adaptive filters achieve higher converge rates as compared to their conventional counterparts. Following the proportionate normalized least-mean-squares (PNLMS) algorithm from [10], different related versions were developed, like the mu-law PNLMS [11], the sparseness-controlled PNLMS [12], the PNLMS with individual activation factors [13], or the block-sparse PNLMS [14]. Also, the sparseness featureswere exploited in the framework of other categories of adaptive filtering algorithms, like the sign affine projection algorithm [15], recursive least-squares (RLS) algorithms with penalties [16; 17], nonlinear filtering architectures [18], and subband adaptive filters [19; 20]. In addition to echo cancellation, these sparsity-aware filters have been applied in the context of a wide range of applications, including underwater acoustics [21; 22], channel estimation [23], and system identification [24]. A connection between the Kalman filter and several proportionate-type algorithms was established in [7]. Nevertheless, all these proportionate adaptive filters still involve the full length of the impulse response, so they face the related difficulties associated with a large number of coefficients. In this context, a useful technique to reduce the parameter space takes advantage of the low-rank characteristic of the impulse responses, which fits very well for the estimation of echo path models [25]. In other words, the corresponding matrices associated with the reshaped impulse responses of the echo paths are never full-rank. This can be further exploited based on the nearest Kronecker product (NKP) decomposition, so that the adaptive filter estimate results based on a combination of shorter impulse responses, thus involving a smaller number of coefficients. In this decomposition framework, the NKP-based technique was applied to the RLS algorithm [26], the Kalman filter [27], the normalized least-mean-squares (NLMS) algorithm [28], correntropy-based methods [29], and nonlinear adaptive filters [30; 31; 32]. Related applications include acoustic feedback cancellation [33], adaptive beamforming [34], and acoustic echo cancellation [35], among others. Following the NKP-based approach, the decomposition technique has been recently applied in conjunction with higher-order tensors [36; 37], thus leading to higher-dimensionality reduction. It is known that tensor-based signal processing [38; 39; 40] represents an important asset in the framework of many applications, like big data analysis [41], biomedical engineering [42], blind source separation [43], and machine learning [44].