# A Connection Between the Kalman Filter and an Optimized LMS Algorithm for Bilinear Forms
## 7 Results  
Experiments are performed in the context of system identification, in order to highlight the performance of the Kalman-based algorithms for bilinear forms (referred to as KF-BF and SKF-BF), in comparison with their regular counterparts (KF and SKF, as mentioned in Section 6). Also, we aim to evaluate the features of the OLMS-BF algorithm, as compared to other existing solutions, e.g., the normalized LMS-BF (NLMS-BF) and the JO-NLMS algorithms, which were introduced in [12] and [29], respectively. In most of the experiments, both the temporal and the spatial impulse responses are randomly generated from a Gaussian distribution, having the lengths equal to \(L=64\) and \(M=8\), respectively. This leads to a length of the spatiotemporal impulse response equal to \(ML=8\times 64=512\). It is also useful to evaluate the tracking capabilities of the algorithms; to this purpose, a sudden change in the temporal impulse response is applied in the middle of simulations, by generating a new random vector of length \(L=64\), also from a Gaussian distribution. Only in the last experiment, the impulse response \(\mathbf{h}(n)\) is an acoustic echo path of length \(L=512\), while the coefficients of \(\mathbf{g}(n)\) are computed as \(g_{m}(n)=0.5^{m}\), with \(m=1,\ldots,M\) and \(M=4\); in this case, the length of the global system is \(ML=4\times 512=2048\). The input signals \(x_{m}(n),\ m=1,2,\ldots,M\) are either white Gaussian noises (WGNs) or AR(1) processes (which were obtained after passing a white Gaussian noise through a first-order system with the transfer function \(1/\left(1-0.8z^{-1}\right)\)). The additive noise \(v(n)\) is white and Gaussian, having the variance \(\sigma^{2}_{v}=0.01\); we assume that this parameter is available in the experiments. In most of simulations, the measure of the performance is the NM (in dB) (see (10)), to evaluate the identification of the global impulse response. In addition, in the second set of experiments (focusing on the OLMS-BF algorithm), we also involve the NPMs (based on (8) and (9)), related to the individual impulse responses. In Figures 3 and 4, the KF-BF is compared to the regular KF for WGN and AR(1) input signals, respectively. The specific parameters of the algorithms are set to \(\sigma^{2}_{\overline{w}_{\mathbf{h}}}=\sigma^{2}_{\overline{w}_{\mathbf{g}} }=\sigma^{2}_{w}=10^{-9}\). It can be noticed from both figures that the KF-BF achieves a faster convergence rate as compared to the regular KF, for both types of input signals, providing also a better tracking capability. The gain is even more apparent in case of AR(1) inputs. The previous experiment is repeated (for the same two types of inputs) in Figures 5 and 6, this time comparing the SKF-BF with the regular SKF [26]. As it can be observed, the simplified versions (SKF-BF and SKF) yield a slower convergence rate (especially in case of AR(1) inputs) as compared to the full versions (KF-BF and KF, respectively); however, the computational complexities for these simplified versions are much lower. As it was expected, the SKF-BF outperforms the regular SKF in terms of the convergence rate; the improvement is much more visible in the case of AR(1) inputs. Figure 4: Normalized misalignment of the KF-BF and regular KF using AR(1) processes as input signals. The length of the global impulse response is \(ML=512\). The specific parameters are set to \(\sigma_{\mathbb{H}_{\mathbf{h}}}^{2}=\sigma_{\mathbb{H}_{\mathbf{g}}}^{2}= \sigma_{w}^{2}=10^{-9}\). Next, the performance of the SKF-BF is evaluated in Figures 7 and 8, but using the recursive estimation \(\widehat{\sigma}^{2}_{\overline{\mathbb{H}}_{\mathbb{B}}}(n)\) from (76) (not a constant value, as in the previous experiments). The spatial impulse response is assumed to be time invariant, so that we can set \(\sigma^{2}_{\overline{\mathbb{H}}_{\mathbb{B}}}=0\).