# Adaptive Bat Algorithm Optimization Strategy for Observation Matrix
## 4 Experimental Simulation
### Signal Reconstruction Experiment
#### 4.2.2 Signal Reconstruction Experiment and Analysis  
Figure 3 shows the performance comparison of four signals reconstructed by the adaptive bat algorithm with a compression ratio of 0.3. And the randomly generated 30 observation matrices corresponding to the various observation matrices are used for the test experiments. The trend of the original signal and the recovered signal are compared by calculating the mean and variance. As can be seen from Figure 3, the ABA has a better reconstruction effect on the original signal. In addition, the proposed algorithm can effectively reconstruct the original signal in a relatively uniform area and achieve stable characteristics. Therefore, the ABA optimized observation matrix below is introduced. However, since the test process of the signal reconstruction has not selected a good observation matrix, there will be a case where the algorithm fluctuates in the signal reconstruction effect. This also reflects the innovation of the paper using the proposed algorithm to optimize the observation matrix. The difference in reconstruction accuracy for each algorithm and different observation matrices is shown in Figures 4-7. \begin{table}
\begin{tabular}{l c c} \hline \hline
**The Test of Sparse Signal:** \\ \hline \(S_{1}=0.3\cos(2\pi f_{1}T_{s}t_{s})+0.6\cos(2\pi f_{2}T_{s}t_{s})+0.1\cos(2 \pi f_{3}T_{s}t_{s})+0.9\cos(2\pi f_{4}T_{s}t_{s})\) \\ \hline \(S_{2}=6\cos(80\pi f_{1}T_{s}t_{s})-7\cos(10\sin(\pi f_{2})T_{s}t_{s})-3\cos(2 0\sin(\pi f_{4}T_{s}t_{s}))\) \\ \hline \(S_{3}=\sin(11\pi f_{1}T_{s}t_{s})+0s(17\pi f_{2}T_{s}t_{s})-0.4\cos(6\pi f_{3} T_{s}t_{s})\) \\ \hline \(S_{4}=32\sin(4\pi f_{1}T_{s}t_{s})+60\cos(\pi f_{2}T_{s}t_{s})-2.2\sin(60\pi f_{ 3}T_{s}t_{s})+0.2\sin(2\sin(16\pi f_{4})T_{s}t_{s})\) \\ \hline \end{tabular} \(F_{s}\) represents sampling frequency, \(t_{s}=\frac{1}{F_{s}}\) represents sampling interval, \(T_{s}=1:N\) represents sampling sequence, \(f_{1}\), \(f_{2}\), \(f_{3}\), and \(f_{4}\) represent signal frequency, respectively, \(rate=\frac{M}{N}\) represents the compression ratio, \(K\) represents the sparse degree and \(K=\left\lceil\frac{M}{\log N}\right\rceil\).  
\end{table} 
Table 1: Sparse signals. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline signal length \(N\) & 256 & signal frequency \(f_{1}\) & 50 Hz \\ compression rate & 0.3, 0.35, 0.4, 0.45, 0.5 & signal frequency \(f_{2}\) & 100 Hz \\ sampling frequency \(F_{s}\) & 800 & signal frequency \(f_{3}\) & 200 Hz \\ population size \(P\) & 100 & signal frequency \(f_{4}\) & 400 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter setting. We can see from Figures 4-7 that it is feasible for the ABA to optimize the observation matrix in terms of improving the stability of the observation matrix or enhancing the reconstruction accuracy of the signal. Therefore, the observation matrix optimized by ABA is more reliable. Figure 4: _Cont._  
Figure 3: Reconstruction effect diagram for different signals: (**a**) Signal 1; (**b**) Signal 2; (**c**) Signal 3; (**d**) Signal 4. Figure 4 shows the reconstructed error values obtained for the \(S_{1}\) with different compression ratios. It can be seen from Figure 4 that the reconstructed error curves of Gaussian, Bernoulli, and SVD-M have large amplitude fluctuations and poor error values. However, the BA and ABA have smaller error values and a minor fluctuation. This also proves that the reconstruction effects of BA and ABA are better than observation matrixes. Figure 4: Reconstruction error of each algorithm to signal-1 with different compression ratio: (**a**) Rate = 0.3; (**b**) Rate = 0.35; (**c**) Rate = 0.4; (**d**) Rate = 0.45; (**e**) Rate = 0.5. Figure 5: _Cont._  
For the \(S_{2}\) signals, Figure 5 depicts a reconstruction error plot of the observation matrix and algorithms with different compression ratios.