# On Fast Converging Data-Selective Adaptive Filtering
## 3 Data-Selective Adaptive Filtering Algorithms
### Simulation 3: System Identification  
In this simulation, our problem is to identify an unknown channel impulse response, described as:  
\[\mathbf{h}=[0.1010\ 0.3030\ 0\ -0.2020\ -0.4040\ -0.7071\ -0.4040\ -0.2020]^{T}. \tag{51}\]  
Figure 4: Simulation 2: (**a**) Learning curves for the data selection and (**b**) Comparison between the desired \(P_{\mathrm{up}}\) and achieved \(\hat{P}_{\mathrm{up}}^{LMSN}\), \(\hat{P}_{\mathrm{up}}^{LMSQN}\) and \(\hat{P}_{\mathrm{up}}^{CG}\) by the data-selection algorithms.  
Figure 5: Simulation 2: Comparison between the desired signal and the predicted by the DS-CG, DS-LMSQN and DS-LMSN algorithms for (**a**) \(P_{\mathrm{up}}=0.4\) and (**b**) \(P_{\mathrm{up}}=0.7\).  
The unknown system output is written as \(d(k)=\mathbf{h}^{T}\mathbf{x}(k)+n(k)\), where \(n(k)\) is a Gaussian noise with zero mean and variance \(\sigma_{n}^{2}=10^{-3}\). We consider two cases of input signals: a first-order and a fourth-order AR process, given by  
\[x(k)= 0.88x(k-1)+n_{1}(k),\] \[x(k)= -0.55x(k-1)-1.221x(k-2)-0.49955x(k-3)\] \[-0.4536x(k-1)+n_{2}(k),\]  
where \(n_{1}(k)\) and \(n_{2}(k)\) are samples from a Gaussian noise uncorrelated with the additional noise \(n(k)\). The variances \(\sigma_{n_{1}}^{2}\) and \(\sigma_{n_{2}}^{2}\) are set such as the input signal is of unit variance. The parameters employed in the system identification problem for conjugate gradient are \(\lambda=0.98\) and \(\eta=0.48\), for both DS-LMSN and DS-LMSQN we set \(\
u=0.1\) and for DS-LMSN \(\theta=0.1\). The filter order is \(N=7\) with the purpose of ensuring the convergence of the filter coefficients to the optimal coefficients due to channel size.  
The learning curves of the algorithms are compared in Figure 6 for a prescribed probability of update \(P_{\text{up}}=0.4\) and first-order AR input signal. It can be noted that all algorithms achieve good performance even with a smaller amount of update. The DS-CG algorithm attains better performance than the DS-LMSN and DS-LMSQN since it converges faster to the steady-state. The observed \(\hat{P}_{\text{up}}\) and the prescribed \(P_{\text{up}}\) probabilities of update are depicted in Figure 6. In all cases, these values are close, except at low values of \(P_{\text{up}}\) where we obtained a bit more updates than the ones prescribed. By using a fourth-order AR process as input signal in Figure 7, the results are similar to the first-order AR, confirming the expected robustness with respect to the statistical properties of the input signal, as long as the rank of its autocorrelation matrix does not become too small.  
In another example utilizing a fourth-order AR input signal, we included an outlier signal affecting the reference signal \(1\%\) of the time with an amplitude equal to five. The desired \(P_{\text{up}}=0.3\) was set, and we measured the misalignment in the adaptive filter coefficients, defined as \(\frac{\|\mathbf{w}(k)-\mathbf{w}_{o}\|}{\|\mathbf{w}_{o}\|}\) where \(\mathbf{w}_{o}\) represents the optimal vector of coefficients, for the algorithms discussed in the paper. As observed in Table 1, the misalignment is higher when ignoring the outliers and that the level of misalignment achieved by considering outliers for \(P_{\text{up}}=0.1\) matches the one for \(P_{\text{up}}=0.3\) addressing the outliers. It is also possible to verify that the proposed solutions approach the solution when the algorithms are updated all the time.  
Figure 6: Simulation 3 for first-order AR input signal: (**a**) Learning curves for the data selection and (**b**) Comparison between the desired \(P_{\text{up}}\) and achieved \(\hat{P}_{\text{up}}^{LMSN}\), \(\hat{P}_{\text{up}}^{LMSQN}\) and \(\hat{P}_{\text{up}}^{CG}\) by the data-selection algorithms.