# An Insightful Overview of the Wiener Filter for System Identification
## 7 Simulations
### Conventional Wiener Filter with \(\ell_{2}\)-Norm Regularization  
First, it is important to assess the performance of the conventional Wiener filter, which depends on several important factors, such as the accuracy of the statistical estimates, the SNR, and the regularization parameter. The performance measure is the normalized misalignment (in dB), which is evaluated based on (16). The input signal, \(x(k)\), is an AR(1) process with a pole at 0.9, while the additive noise, \(w(k)\), is white and Gaussian, and different values of the SNR are used. The unknown system, \(\mathbf{h}\), is the first impulse response from G168 Recommendation [33], which is padded with zeros up to the length \(L=500\). It represents a network echo path, which is sparse in nature. Its sparsity measure can be evaluated based on (15) and results in \(\xi_{12}(\mathbf{h})=0.8957\). The first critical factor that influences the overall performance of the Wiener filter is the accuracy of the statistical estimates--namely, the covariance matrix, \(\mathbf{R}\), and the cross-correlation vector, \(\mathbf{p}\). Let us consider that \(N\) data samples are used to estimate these quantities, which result in  
\[\widehat{\mathbf{R}} =\frac{1}{N}\sum_{k=1}^{N}\mathbf{x}(k)\mathbf{x}^{T}(k), \tag{63}\] \[\widehat{\mathbf{p}} =\frac{1}{N}\sum_{k=1}^{N}\mathbf{x}(k)d(k). \tag{64}\]  
A higher value of \(N\) (as compared to the filter length \(L\)) leads to a reliable set of estimates. On the other hand, when a low quantity of data samples is used for the estimation of the statistics (i.e., \(N\approx L\)), the conventional Wiener filter is not able to provide an accurate solution. Such a scenario is very challenging in different practical applications, where only an incomplete set of data/information is available. Moreover, in the framework of a low SNR environment, the challenges become even more apparent. In order to highlight these aspects, the performance of the conventional Wiener filter depending on \(N\) and for different SNRs is illustrated. Having available the statistical estimates from (63) and (64), the conventional Wiener solution is obtained based on (19). In this simulation, a very small value of the regularization parameter is used, i.e., \(\delta=10^{-8}\). The results are presented in Figure 2. As expected, a larger value of \(N\) improves the Wiener solution, while the accuracy is also influenced by a lower SNR. The critical scenario appears for \(N\approx L\), where the conventional Wiener filter is not able to obtain a reliable solution even for a high SNR. Next, the influence of the regularization parameter, \(\delta\), on the performance of the conventional Wiener filter is evaluated in different SNR conditions. First, in Figure 3a,b, good SNR conditions are considered, by setting \(\text{SNR}=30\) dB and \(\text{SNR}=20\) dB, respectively, while choosing different values for \(\delta\). As it can be noticed, the influence of the regulariza  
Figure 2: Misalignment of the conventional Wiener filter with \(\ell_{2}\)-norm regularization for different values of \(N\) and SNRs, using \(\delta=10^{-8}\). tion parameter becomes more apparent when using a low quantity of data (e.g., \(N\approx L\)) to estimate the statistics. In this case, a larger value of \(\delta\) can improve the accuracy of the solution. The influence of the regularization parameter becomes less important for \(N\gg L\); in these conditions, a larger value of \(\delta\) can even bias the Wiener solution. Nevertheless, in low SNR environments, a larger value of the regularization parameter is required. This is supported in Figure 3c,d, in which the SNR is set to 10 dB and 0 dB, respectively. As it can be observed, the largest value of \(\delta\) improves the performance for any value of \(N\). However, the \(\ell_{2}\)-norm regularization used in (19) does not exploit the sparse character of the system, which is taken into account within the \(\ell_{1}\)-norm framework presented in Section 4.