# Wiener Filter Using the Conjugate Gradient Method and a Third-Order Tensor Decomposition
## 4 Simulation Results
### Iw-Tot-Cg versus Iwf-Nkp-Cg  
The last set of experiments aims to evaluate the Wiener filters based on the CG method in conjunction with the decomposition-based approaches. The proposed IWF-TOT-CG is compared with a recently designed version of the iterative Wiener filter [33]. This algorithm (namely IWF-NKP-CG) also uses the CG method to solve the associated Wiener-Hopf equations, but it relies on the second-order NKP decomposition, thus following the initial approach from [8]. In this case, the length of the filter is factorized as \(L=L_{1}^{*}L_{2}^{*}\), while the low-rank approach relies on the decomposition parameter \(P^{*}<L_{2}^{*}\). In our scenario that considers an impulse response of length \(L=512\), this decomposition is performed using \(L_{1}^{*}=32\) and \(L_{2}^{*}=16\). The IWF-NKP-CG involves two sets of Wiener-Hopf equations, which correspond to two component Wiener filters of lengths \(P^{*}L_{1}^{*}\) and \(P^{*}L_{2}^{*}\). Their solutions are obtained based on the CG method, using a maximum number of updates (denoted by \(K^{*}\)). Since in the involved setup, we have \(L_{2}^{*}=L_{12}\), it is reasonable to use \(K^{*}=K_{1}\).  
In Figure 10, the required statistics are obtained by averaging across \(N=5L\) available data samples, while \(\text{SNR}=20\,\text{dB}\). These represent reasonably good conditions, which favor reliable estimates. Nevertheless, the IWF-NKP-CG [33] is outperformed by the proposed IWF-TOT-CG for different values of their decomposition parameters. This performance gain results from the TOT decomposition compared to the second-order NKP-based approach, which further supports the initial findings from [18]. For this experiment, the values of \(P\) and \(P^{*}\) were selected according to the low-rank approach. Concerning the values of \(P\) (for IWF-TOT-CG), these considerations were previously indicated related to Figures 6 and 8. Similar aspects can be outlined when choosing the values of \(P^{*}\) (for IWF-NKP-CG). As recently indicated in [33] and previously supported in [8], for network impulse responses, the rank of the corresponding matrix of size \(L_{1}^{*}\times L_{2}^{*}\) (where \(L_{1}^{*}\times L_{2}^{*}=L\)) associated with the reshaped impulse response is much lower than \(L_{2}^{*}\), e.g., usually less than \(L_{2}^{*}/6\). In this context, it is reasonable to consider \(P^{*}<3\) in Figure 10, in order to properly address the trade-off between performance and complexity. Otherwise, using larger values of \(P^{*}\) will not justify the performance gain, while increasing the computational amount.  
Figure 9: Normalized misalignment of the IWF-TOT [18] and IWF-TOT-CG (with \(K_{1}=12\) and \(K_{2}=1\)) using \(P=1\). The required statistics are obtained by averaging across \(N=5L\) data samples (with \(L=512\)), while (**a**) \(\text{SNR}=10\) dB and (**b**) \(\text{SNR}=0\) dB.  
Finally, the IWF-NKP-CG [33] and the proposed IWF-TOT-CG (using \(P^{*}=P=1\)) are compared in more challenging conditions, when using smaller amounts of data for estimating the required statistics, i.e., \(N=2L\) and \(N=L\). As we can notice in Figure 11, the performance gain of the IWF-TOT-CG is more apparent in these cases, which shows the robustness of the TOT-based decomposition in conjunction with the CG method.