## 4 Testing the Performance of Measurement Matrix Using Simulated Underwater Echo
### Simulation of Underwater Echo and Its Compression and Reconstruction  
To verify the performance of the measurement matrix constructed with the proposed method, the highlight model [32] is used to simulate underwater echo, which is compressed and reconstructed by CS to evaluate the performance of the measurement matrix. For an acoustic emission signal \(x(t)=x_{0}(t)e^{-i\omega_{c}t}\), in which \(\omega_{c}\) is the carrier frequency at 30 kHz, and \(x_{0}(t)\) is the envelope assuming a trapezoidal shape. Assuming a sound velocity in fresh water of 1500 m/s and the pulse width is 10 ms. The number of points sampled is 4096. The underwater echo is simulated by the three-highlight model. Figure 3 shows the simulated echo with different signal-to-noise ratios (SNRs). The simulated underwater echo in Figure 3 is then compressed and reconstructed with CS. In this study, the discrete cosine transform [33; 34] is used as the sparse matrix, and the reconstruction algorithm adopted is the orthogonal matching pursuit (OMP) [35; 36; 37]. The measurement matrix based on the cyclic direct product and QR decomposition is compared respectively with Gaussian matrix, Bernoulli matrix, partial Hadamard matrix and Toeplitz matrix. The results are shown in Figures 4-6. As seen in the performance comparisons above, the signal compressed by the measurement matrix developed in this study provides better reconstruction results than those by other measurement matrices. Firstly, the SNR of the reconstructed signal is improved significantly. When the compression ratio is small, the performance of this matrix is nearly the same as other measurement matrices. As the compression ratio increases, our method shows greater improvement in the SNR. At a compression ratio of above 0.7, the SNR of this matrix is approximately 2 dB higher than those provided by other measurement matrices. In terms of the matching rate, the measurement matrix based on the cyclic direct product and QR decomposition yields higher reconstruction accuracy than the others, especially as the  
Figure 3: Simulated echo signals with different signal-to-noise ratios. Figure 4: Reconstruction results by different types of measurement matrices (SNR = \(-3\) dB). Figure 5: The matching rate comparison using different types of measurement matrices. (**a**) SNR = 4 dB; (**b**) SNR = 0 dB; (**c**) SNR = \(-3\) dB. Figure 6: _Cont._  
Table 1 shows the construction time for various types of measurement matrices at different compression ratios. As shown in the table, the measurement matrix based on the cyclic direct product and QR decomposition requires significantly less construction time than the others, as it only needs one two-dimensional matrix to be stored during construction, i.e., four elements. The storage space required is also less, as it is independent of the number of measurements made and only depends on the signal length N. The proposed measurement matrix thus saves storage space and computation time. Testing the Performance of the Measurement Matrix Using Experimentally Measured Underwater Echo Data  
The test was carried out at the experimental station of Qiandao Lake (Chunan, China). The test target was a scaled model of a submarine. The model is the Benchmark Target Strength Simulation Submarine (BeTSSi-Sub) manufactured by Dalian University of Technology. The deployment method is shown in Figure 7[28]. The signal is emitted by the signal generator in the form of a Linear Frequency Modulation (LFM) signal with a frequency of 40 kHz-80 kHz, a pulse width of 1 ms, and a trigger interval of 200 ms. Through the power amplifier, an L6 linear power amplifier is used in the test, and its parameters adjusted according to the transducer's working mode and impedance matching, with different gains; the underwater transducer uses a transmitting and receiving transducer through its internal transceiver conversion module; the transmission voltage response (TVR) is 162 dB and the receiving sensitivity is \(-172\) dB.