# A Connection Between the Kalman Filter and an Optimized LMS Algorithm for Bilinear Forms
## 7 Results
The regular SKF is considered for comparison, using a similar way to estimate its specific parameter, i.e., \(\widehat{\sigma}^{2}_{w}(n)=1/(ML)\left\|\widehat{\mathbf{f}}(n)-\widehat{ \mathbf{f}}(n-1)\right\|^{2}\)[26]. Because of the nature of the estimators (as it was explained in Section 6), the algorithms behave like variable step-size adaptive filters, achieving both low misalignment and fast convergence/tracking. Moreover, as we can notice from these two figures, the proposed SKF-BF still outperforms the regular SKF in terms of both performance criteria. Figure 5: Normalized misalignment of the SKF-BF and regular SKF using WGNs as input signals. The length of the global impulse response is \(ML=512\). The specific parameters are set to \(\sigma^{2}_{\overline{\mathbb{H}}_{\mathbb{B}}}=\sigma^{2}_{\overline{\mathbb{ H}}_{\mathbb{B}}}=\sigma^{2}_{\overline{\mathbb{H}}_{\mathbb{B}}}=0^{-9}\). Figure 6: Normalized misalignment of the SKF-BF and regular SKF using AR(1) processes as input signals. The length of the global impulse response is \(ML=512\). The specific parameters are set to \(\sigma^{2}_{\overline{\mathbb{H}}_{\mathbb{B}}}=\sigma^{2}_{\overline{\mathbb{ H}}_{\mathbb{B}}}=0^{-9}\). As outlined in Section 5, there are strong similarities between the SKF-BF and OLMS-BF algorithms. In Figures 9 and 10, we compare the performances of these algorithms using two types of input signals, i.e., WGNs and AR(1) processes, respectively. Both algorithms use the recursive estimate \(\hat{\sigma}^{2}_{\overline{\mathbf{n}}_{\mathbf{k}}}(n)\) (from (76)) and \(\sigma^{2}_{\overline{\mathbf{n}}_{\mathbf{g}}}=0\). As we can notice, the SKF-BF and OLMS-BF algorithms behave quite similar, especially when the input signals are WGNs (Figure 9). When the input signals are AR(1) processes (Figure 10), the SKF-BF outperforms the OLMS-BF in terms of the initial convergence rate; however, it pays with a slower tracking reaction. Nevertheless, the overall performances of these algorithms are very similar, as supported by the comparison provided in Section 5. In the second set of experiments, the behavior of the OLMS-BF algorithm is analyzed, as compared to the NLMS-BF algorithm [12]. The NLMS-BF algorithm uses different values of its step-size parameters, \(\alpha_{\tilde{\mathbf{h}}}\) and \(\alpha_{\tilde{\mathbf{g}}}\). The performances are now evaluated in terms of both NPMs and NM, using both types of input signals as before (WGNs and AR(1) processes). The results are presented in Figures 11 and 12, using WGNs as inputs, and in Figures 13 and 14, where the input signals are AR(1) processes. It can be noticed that the proposed solution achieves similar convergence rate but a much lower misalignment level than the NLMS-BF algorithm with \(\alpha_{\tilde{\mathbf{h}}}=\alpha_{\tilde{\mathbf{g}}}=0.5\) (which provides the fastest convergence rate [12]). On the other hand, if we target a lower misalignment and set the step-sizes of the NLMS-BF to smaller values (i.e., \(\alpha_{\tilde{\mathbf{h}}}=\alpha_{\tilde{\mathbf{g}}}=0.1\) and \(\alpha_{\tilde{\mathbf{h}}}=\alpha_{\tilde{\mathbf{g}}}=0.01\)), the convergence rate also decreases. The results are presented in Figures 11 and 12, using WGNs as inputs, and in Figures 1  
decreases. However, the OLMS-BF algorithm leads to a misalignment level similar to the NLMS-BF algorithm using the smallest step-sizes. In addition, when the input signals are AR(1) processes, the improvement offered by the OLMS-BF algorithm is even more apparent. Figure 11: Normalized projection misalignment of the OLMS-BF and NLMS-BF (using different step-size parameters): (**Top**) identification of the temporal impulse response \(\mathbf{h}(n)\), (**Bottom**) identification of the spatial impulse response \(\mathbf{g}(n)\). The input signals are WGNs, \(L=64\), and \(M=8\). Figure 12: Normalized misalignment of the OLMS-BF and NLMS-BF (using different step-size parameters). The input signals are WGNs and \(ML=512\).