# Wiener Filter Using the Conjugate Gradient Method and a Third-Order Tensor Decomposition
## 4 Simulation Results
### IWF-TOT-CG versus WF-CG  
In the first set of simulations, the performance of the proposed IWF-TOT-CG is analyzed according to its main parameters, i.e., \(K_{1}\), \(K_{2}\), and \(P\). The benchmark algorithm involved in comparisons is the WF-CG, using \(K=50\) CG updates. As shown in Figures 2 and 3, this value of \(K\) is sufficient for the WF-CG to reach the conventional WF solution. Since \(L=512\), the decomposition of the IWF-TOT-CG is performed using \(L_{11}=L_{12}=16\) and \(L_{2}=2\). In Figure 4, the influence of \(K_{1}\) is assessed, i.e., the maximum number of CG updates used for the two longer filters within the IWF-TOT-CG. Since the third filter (of length \(L_{2}^{2}=4\)) is very short and the corresponding CG parameter (\(K_{2}\)) should be much smaller than its length, it is natural to use the smallest value \(K_{2}=1\). In this simulation, the decomposition parameter of the IWF-TOT-CG is set to \(P=2\); the influence of this parameter will be analyzed in an upcoming experiment. Also, the amount of data available for estimating the statistics is \(N=5L\), while \(\text{SNR}=20\) dB. These represent good conditions for the WF-CG (i.e., the comparing algorithm) to obtain a reliable estimate in terms of accuracy. Nevertheless, as we can notice from Figure 4, the proposed IWF-TOT-CG reaches a significantly lower misalignment level (i.e., a better accuracy) for all the values of \(K_{1}\). A larger value of this parameter leads to a faster convergence rate, but up to a certain limit. On the other hand, a larger number of iterations also increases the computational complexity in terms of the number of operations. Consequently, a compromise should be made. As we can notice from Figure 4, increasing the value of \(K_{1}\) to more than 12 (e.g., 14) does not lead to performance improvements. Besides, the difference between the cases \(K_{1}=10\) and \(K_{1}=12\) is not so apparent (as compared to the difference between \(K_{1}=8\) and \(K_{1}=10\)), slightly improving the convergence rate, while reaching the same misalignment level (i.e., accuracy). In this context, using \(K_{1}=10\) or 12 are reasonable choices. Figure 4: Normalized misalignment of the WF-CG (after \(K=50\) CG updates) and IWF-TOT-CG with \(P=2\), \(K_{2}=1\), and using different values of \(K_{1}\). The required statistics are obtained by averaging across \(N=5L\) data samples, \(L=512\), and \(\text{SNR}=20\) dB. A similar experiment is considered in Figure 5, but using a fixed value for the first CG parameter (i.e., \(K_{1}=10\)) and two different values for \(K_{2}\). These represent the minimum and maximum values for this parameter, i.e., \(K_{2}=1\) and \(K_{2}=4\), respectively. The maximum value is associated with the length of the corresponding filter (\(L_{2}^{2}\)). The other conditions remain the same as in the previous simulation. Under these circumstances, it can be noticed in Figure 5 that a higher value of \(K_{2}\) does not significantly influence the overall performance of the IWF-TOT-CG so it is natural to use \(K_{2}=1\) in the following experiments. The experiment from Figure 5 was performed to assess the influence of \(K_{2}\), i.e., the number of CG iterations for the shorter filter. We should note that similar conclusions can be obtained when using other values of \(K_{1}\) (for the two longer filters). Next, the influence of the decomposition parameter \(P\) is analyzed in Figure 6. Based on the previous experiments, the setup used for the IWF-TOT-CG is \(K_{1}=12\) and \(K_{2}=1\), while \(N=5L\) and \(\text{SNR}=20\) dB. As shown in Section 2, the decomposition parameter is chosen such that \(P\ll L_{12}\), relying on the low-rank feature of the impulse response. For all the values of \(P\) considered in Figure 6, the IWF-TOT-CG outperforms the WF-CG. Even the minimum value \(P=1\) produces a reasonable attenuation of the normalized misalignment, showing improved accuracy as compared to the benchmark algorithm.