# An Insightful Overview of the Wiener Filter for System Identification
## 7 Simulations
### Wiener Filter Based on the Kronecker Product Decomposition  
The Wiener filter based on the Kronecker product decomposition (KPD) presented in Section 5 involves smaller data structures with respect to the conventional Wiener filter. While the conventional solution is obtained by directly solving (19), the KPD-based Wiener filter iteratively combines the solutions from (41) and (42). In other words, instead of dealing with a single filter of length \(L=L_{1}L_{2}\) (as in the conventional case), the KPD approach handles two shorter filters of lengths \(PL_{1}\) and \(PL_{2}\), with \(P\ll L_{2}\). The advantages become more apparent for the identification of long-length impulse responses, such as in echo cancellation [10]. In Figure 9, the conventional Wiener filter is compared with the KPD-based solution using different values of the decomposition parameter \(P\). The regularization parameter is set to \(\delta=10^{-8}\) for both algorithms. In this simulation, \(N\) is set to \(5000\) and different SNRs are used. As shown in Section 5, the value of \(P\) is related to the rank of the matrix \(\mathbf{H}\), which reshapes the impulse response \(\mathbf{h}\). In this scenario, the length of \(\mathbf{h}\) is \(L=500\), so that the values \(L_{1}=25\) and \(L_{2}=20\) can be used. In this case, \(\text{rank}(\mathbf{H})=3\). As it can be observed in Figure 9a-c, the performance of the KPD-based Wiener filter is improved as the value of \(P\) increases up to the rank of \(\mathbf{H}\). The advantages become more apparent for lower SNRs, where the KPD solution outperforms the conventional Wiener filter for most of the values of \(P\). It is interesting to notice that in heavy noise conditions (e.g., \(\text{SNR}=0\) dB), a lower value of \(P\) could be used, as indicated in Figure 9d. In terms of the quantity of data available for the estimation of the statistics, the value \(N=5000\) corresponds to a favorable scenario, since \(L=500\), so that \(N=10L\) in this case. Therefore, a reliable set of statistical estimates [see (63) and (64)] are available. However, even in this favorable scenario, the performance of the conventional Wiener filter is still influenced by the SNR level, as already supported in Figure 2. In other words, the accuracy  
Figure 8: Misalignment of the conventional Wiener filter with \(\ell_{2}\)-norm regularization and the Wiener filter with iterative \(\ell_{1}\)-norm regularization, using \(\delta=\delta_{1}\) [based on (65) and (66)] and different values of \(N\), for (**a**) \(\text{SNR}=30\) dB, (**b**) \(\text{SNR}=20\) dB, (**c**) \(\text{SNR}=10\) dB, and (**d**) \(\text{SNR}=0\) dB. of its solution is worse when the SNR decreases, as can also be inferred from Figure 9. On the other hand, the KPD-based Wiener filter operates with smaller data structures, and it is less influenced by the accuracy of the statistical estimates, as compared to the conventional Wiener filter. Consequently, it is also more robust to the SNR level. Especially in noisy conditions, i.e., with moderate and low SNRs (e.g., 10 dB and 0 dB, respectively), the KPD-based Wiener filter (for any value of \(P\)) outperforms the conventional Wiener solution. Next, the performance of the KPD-based Wiener filter (using \(P=3\)) is evaluated for different values of \(N\) and in different SNR conditions, using the conventional Wiener filter as a benchmark for comparison. As it can be observed in Figure 10, the KPD approach outperforms the conventional solution for all the values of \(N\) and in all SNR conditions. The advantages become more apparent for low values of \(N\), i.e., \(N\approx L\). Figure 10: Misalignment of the conventional Wiener filter and the KPD-based Wiener filter using \(P=3\), for different values of \(N\) and (**a**) \(\text{SNR}=30\) dB, (**b**) \(\text{SNR}=20\) dB, (**c**) \(\text{SNR}=10\) dB, and (**d**) \(\text{SNR}=0\) dB.