## 4 Simulation Example  
**Example 1**. : _Consider the following MISO-OE system with time-delays,_  
\[y(t) =\sum\limits_{i=1}^{3}z^{-d_{i}}\frac{B_{i}(z)}{A_{i}(z)}u_{i}(t) +v(t),\] \[A_{1}(z) =1-0.1z^{-1}+0.7z^{-2},\ \ B_{1}(z)=1.5z^{-1}+0.9z^{-2},\] \[A_{2}(z) =1+0.3z^{-1}+0.5z^{-2},\ \ B_{2}(z)=0.2z^{-1}+1.8z^{-2}, \tag{34}\] \[A_{3}(z) =1-0.2z^{-1}-0.4z^{-2},\ \ B_{3}(z)=-0.1z^{-1}+2z^{-2},\] \[d_{1} =20,\ \ \ d_{2}=10,\ \ \ d_{3}=30.\]  
The system in Equation (34) is a second order system with three inputs and one output. The inputs \(\{u_{i}(t)\}\), \(i=1,2,3\) are taken as random uncorrelated signal sequences with zero mean and unit variances, and \(\{v(t)\}\) as a white noise sequence with zero mean and variances \(\sigma^{2}\). Let the maximum input regression length be \(l=50\). Then, the parameter vector to be identified is  
\[\mathbf{\theta}_{1}=[-0.1,0.7,\mathbf{0}_{20},1.5,0.9,\mathbf{0}_{28},0.3,0.5,\mathbf{0}_{10},0.2,1.8,\mathbf{0}_{38},-0.2,-0.4,\mathbf{0}_{30},-0.1,2,\mathbf{0}_{18}]^{T}\in\mathbb{R} ^{156},\]  
where \(\mathbf{0}_{i}\) denotes the zero-block with \(i\) zeros. Note that the number of non-zero elements is \(K=\|\mathbf{\theta}\|_{0}=\sum\limits_{i=1}^{3}(n_{ai}+n_{bi})=12\). Taking \(m=130\) and \(\mathrm{TH}=0.001\), apply the AM-LSI algorithm and the AM-BPDNI algorithm to perform the identification, respectively. The parameter estimation errors \(\delta:=\|\mathbf{\theta}-\hat{\mathbf{\theta}}\|/\|\mathbf{\theta}\|\) versus different noise levels are shown in Table 1. When \(\sigma^{2}=0.10^{2}\), the estimation errors \(\delta\) versus the iterative number \(k\) are shown in Figure 1. It can be seen that the AM-BPDNI algorithm performs better than the AM-LSI algorithm and is insensitive to noise. Let the variance of \(\{v(t)\}\) be \(\sigma^{2}=0.10^{2}\). Apply the AM-BPDNI algorithm to obtain the estimated model of the system in Equation (34) with the first \(m=130\) data. Then, validate the estimated model by using \(m_{e}=200\) samples from \(t=131\) to \(330\). The predicted output of the estimated model, the true output of the system and their errors are plotted in Figure 2. It is shown that the predicted outputs \(\hat{y}(t)\) are close to the true outputs \(y(t)\). Moreover, the average predicted output error  
\[\delta_{v}:=\sqrt{\frac{1}{m_{e}}\sum_{t=131}^{330}[\hat{y}(t)-y(t)]^{2}}=0.1422\]  
is small and close to the standard deviation of the noise \(\sigma=0.10\). It follows that the estimated model can well capture the system dynamics. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\sigma\) & **0.10** & **0.15** & **0.20** & **0.25** & **0.30** & **0.40** \\ \hline AM-LSI & 53.4284 & 53.7175 & 54.2753 & 55.3337 & 55.8435 & 57.2966 \\ AM-BPDNI & 1.9974 & 2.9825 & 4.7985 & 8.6665 & 8.9512 & 9.8808 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameter estimation errors \(\delta\) (%) versus the noise variances \(\sigma^{2}\) (\(m=130\)). Figure 1: The parameter estimation error \(\delta\) versus \(k\) (\(m=130,\sigma^{2}=0.10^{2}\)). With 10 iterations, the estimated parameter vector is  
\[\hat{\theta}_{1}=[-0.1001,0.6938,\mathbf{0}_{20},1.4737,0.8882,\mathbf{0}_{28}, 0.2902,0.4818,\mathbf{0}_{10},0.1699,\] \[1.7783,\mathbf{0}_{38},-0.1917,-0.3961,\mathbf{0}_{30},-0.08002,1.96 28,\mathbf{0}_{18}]^{T}. \tag{35}\]  
It can be seen from Equation (35) that there are six zero-blocks in \(\hat{\theta}\) and the number of zeros in each zero-block are \(z_{1}=20\), \(z_{2}=28\), \(z_{3}=10\), \(z_{4}=38\), \(z_{5}=30\) and \(z_{6}=18\). Then, the time-delay of each input channel can be estimated according to Equation (33),  
\[\hat{d}_{1}=z_{1}=20,\ \hat{d}_{2}=z_{3}=10,\ \hat{d}_{3}=z_{5}=30.\]  
Obviously, the time-delay estimates are agreement with the true time-delays. **Example 2**.