# A Connection Between the Kalman Filter and an Optimized LMS Algorithm for Bilinear Forms
## 6 Practical Considerations  
The previously developed algorithms are designed to identify the individual impulse responses of the bilinear form. The global (spatiotemporal) impulse response can be computed based on the Kronecker product between them. An alternative solution is to use the regular Kalman filter to identify the spatiotemporal impulse response directly, relying on the observation Equation (4) and identifying the state equation:  
\[\mathbf{f}(n)=\mathbf{f}(n-1)+\mathbf{w}(n), \tag{75}\]  
where \(\mathbf{w}(n)\) is a zero-mean white Gaussian noise signal vector. The covariance matrix of \(\mathbf{w}(n)\) is \(\mathbf{R}_{\mathbf{w}}(n)=\sigma_{w}^{2}\mathbf{I}_{ML}\), where \(\mathbf{I}_{ML}\) is the identity matrix of size \(ML\times ML\) and the variance \(\sigma_{w}^{2}\) captures the uncertainties in \(\mathbf{f}(n)\). In this way, following the approach from [26], we can easily derive the regular Kalman filter (KF) and its simplified version (namely SKF), which can identify the global impulse response using a single adaptive filter \(\widehat{\mathbf{f}}(n)\); for further details, please see Sections VI and VII in [26]. However, we need to mention that the solution found using the regular KF and SKF involves an adaptive filter of length \(ML\), whereas their counterparts tailored for bilinear forms (i.e., KF-BF and SKF-BF) use two shorter filters of lengths \(L\) and \(M\), respectively. As a consequence, besides a lower computational complexity, a much faster converge rate and tracking are expected for the bilinear algorithms with respect to the conventional ones. The same ideas apply for the OLMS-BF algorithm, as compared to its regular counterpart, i.e., the joint-optimized normalized LMS (JO-NLMS) algorithm [29], which could be used to identify the global impulse response \(\widehat{\mathbf{f}}(n)\). The computational complexity of the previously discussed algorithms is summarized in Table 4. It can be easily seen that the SKF-BF offer a great reduction in terms of complexity with respect to KF-BF. Also, the SKF-BF and OLMS-BF differ only by a small number of operations, thus confirming the similarity that was highlighted in Section 5. Finally, when \(ML\gg M+L\) (which is usually the case in practice), we can notice that the algorithms tailored for bilinear forms (namely KF-BF, SKF-BF, and OLMS-BF) offer lower computational complexities as compared to their regular counterparts (i.e., KF, SKF, and JO-NLMS, respectively). Next, a few important observations ought to be made regarding the specific parameters that must be set within the algorithms. Here, the noise power \(\sigma_{v}^{2}\) is required in order to compute the Kalman gain vectors (for KF-BF and SKF-BF) or the optimal step-sizes (for OLMS-BF). In practice, we can estimate this parameter in different ways; some simple and efficient methods for this purpose are presented in [30; 31]. Although there are different other methods that can be used to estimate the noise power, the analysis of their influence on the performance of the algorithms lies beyond the scope of this paper. The parameters related to the uncertainties in the unknown systems also need to be set or estimated, i.e., \(\sigma_{\overline{m}_{h}}^{2}\) and \(\sigma_{\overline{m}_{g}}^{2}\). Choosing small values for these parameters yields a small misalignment, but at the same time a poor tracking. On the other hand, large values (meaning that there are high uncertainties in the unknown systems) lead to a good tracking but also a high misalignment. This means that we always need to have a good compromise between fast tracking and low misalignment. In practice, if we have some a priori information about the systems which we need to