# An Under-Sampling Array Signal Processing Method Based on Improved Hadamard Matrix
## 4 Simulation and Experiments Analysis  
### Validation and Analysis Based on Simulation Signals  
Before using the measured data, we compared the performance of several deterministic measurement matrices through simulation signal verification. The results show that under a high compression ratio, these deterministic measurement matrices have better reconstruction effects on known targets, but in the environment of a low compression ratio, the reconstruction effect of several measurement matrices produces the obvious difference. The relevant parameters of simulation experiments in this paper are listed in Table 1.  
Figure 3: The process of under-sampling array.  
Figure 4 shows four different deterministic measurement matrices--Toeplitz and circulant. Based on m-sequence, Sequential Partial Hadamard and the Improved Hadamard measurement matrix proposed in this paper, when the compression ratio is 44%, the reconstruction effect of known target detection in simulation. It can be found that compared with other measurement matrices, the improved Hadamard matrix reconstruction effect is still significant.  
Furthermore, in order to analyze the effect in different number of sensors and sources on the performance of the improved Hadamard measurement matrix, the following simulations analysis are performed, as shown in Figure 5. To facilitate performance comparison and unified compression ratio and other variables, some simulation parameters are changed on the basis of Table 1, and the change content is shown in Table 2.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Array type & Linear array \\ Sensors number & 27 \\ Underwater speed & 1500 m/s \\ Central frequency & 150 kHz \\ Sampling frequency & 1000 kHz \\ Element spacing & 0.005 m \\ Target angle & [60\({}^{\circ}\), 90\({}^{\circ}\)] \\ Target amplitude & [1, 0.75] \\ gridding & [0:0.5:180] \\ Number of sensors after compression & 11 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters.  
Figure 4: Simulation target reconstruction effect with a compression ratio of 44%.  
Figure 5 shows the reconstruction effect of different number of sensors and sources in known target detection simulation--2014 sensors, when the compression ratio is 50%, the target angle is [60\({}^{\circ}\), 90\({}^{\circ}\), 110\({}^{\circ}\)] or [60\({}^{\circ}\), 90\({}^{\circ}\), 110\({}^{\circ}\), 140\({}^{\circ}\), 160\({}^{\circ}\)], and the target amplitude is or [1, 0.75, 0.5, 0.4, 0.4]. It can be found that compared to other measurement matrices, as the number of sensors decreases or the number of sources increases, the estimation accuracy and reconstruction amplitude will decrease, but it still has good reconstruction estimation performance, and it can reconstruct the relative correct trend for different actual amplitude sources after using the improved Hadamard matrix compression. The simulation results in 14 sensors with 5 sources were not displayed, because the resolution ability of 14 sensors for 5 sources after compression was greatly reduced, resulting in poor results.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Sensors number & 2014 \\ Target angle & [60\({}^{\circ}\), 90\({}^{\circ}\), 110\({}^{\circ}\), 140\({}^{\circ}\), 160\({}^{\circ}\)] \\ Target amplitude & [1, 0.75, 0.5, 0.4, 0.4] \\ compression ratio & 50\% \\ Number of sensors after compression & 10, 7 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation parameter changes.  
Figure 5: Simulation target reconstruction effect in different number of sensors and sources: (**a**) simulation results in 20 sensors with 3 sources; (**b**) simulation results in 14 sensors with 3 sources; and (**c**) simulation results in 20 sensors with 5 sources.