## 2 Direct Under-Sampling Compressive Sensing Method
### Direct Under-Sampling Compressive Sensing Method (DUS-CS)
#### 2.2.2 Direct Under-Sampling Compressive Sensing (DUS-CS) Method
The measurement matrix now becomes  
\[\Phi(m,n)=\frac{1}{N}\sum_{k=-N/2+1}^{N/2}\exp\left(2\pi j\frac{k}{N}\Big{(} \frac{t_{m}}{T_{s}}-n\Big{)}\right) \tag{13}\]  
This approach improves the accuracy of observation of the signals and increases the calculation speed. Sparse representation method incorporating prior information  
The key to signal sparse representation is the construction of a sparse dictionary and selection of the appropriate atom from the dictionary to represent the signal. The more similar the characteristics of the atom are to the original signal, the better the effect of signal observation and processing. Based on the prior information of the incident signal combined with the principle of signal sparse decomposition, it is possible to construct an a priori complete dictionary. The specific construction method is as follows: the number of sample points, \(n\) generated by one pulse width forms a vector quantity, \(s_{p}\), of size \(N\times 1\) and normalization identity \(\left\|s_{p}\right\|_{2}=1\). This defines the prototype atom \(s_{p}=\left[s_{p}(T_{s}),s_{p}(2T_{s}),\ldots,s_{p}(NT_{s})\right]\). Each atom \(s_{p}\) is treated as a block and recorded as \(\Psi_{1}\). Then, by blocking and shifting, the sparse complete dictionary \(\Psi^{n\times n}=\left[\psi_{1}^{T},\psi_{2}^{T},\ldots,\psi_{n}^{T}\right]^{T}\) is achieved. The block sparse decomposition is expressed as  
\[x_{N}=\left[ \begin{array}{cccc}\psi_{11}&\psi_{12}&\cdots&\psi_{1N}\\ \psi_{21}&\psi_{22}&\cdots&\psi_{2N}\\ \vdots&\vdots&&\cdots\\ \psi_{N1}&\psi_{N2}&\cdots&\psi_{NN}\\ \end{array} \right]=\left[ \begin{array}{c}\psi_{1}\\ \vdots\\ \psi_{m}\\ \end{array} \right]\left[\alpha_{1},\cdots,\alpha_{n}\right]^{T} \tag{14}\]  
where \(\alpha\) is the block sparse coefficient vector in sparse decomposition theory. 2 Reconstruction method of block sparse orthogonal matching pursuit  
This is first performed by integrating the block sparseness of prior information into an orthogonal matching pursuit (OMP) algorithm, to form a block-sparse orthogonal matching pursuit (BOMP) reconstruction. Then, the most relevant sub-block of the signal is chosen from the sparse complete dictionary and its projection on the sub-block from the signal is subtracted to obtain a residual signal. This process is iterated until the energy of the residual signal is less than a given threshold, or the algorithm reaches its termination condition. The BOMP algorithm flow is as follows:  
1. Initialize the residual \(r_{0}=x\). The block structure of the dictionary is \(\Psi=\left[ \begin{array}{cccc}\psi_{1}&\psi_{2}&\cdots&\psi_{n}\\ \end{array} \right]^{T}\). 2. Selection of atomic blocks. On the \(i\)-th iteration, select the index to be the largest inner product value of the residual, such that \[index=\underset{i}{\text{argmax}}\left(mean\left(\left|\Theta^{T}[i]r_{i-1} \right|\right)\right)\ \ \ \ i\in[1,K]\] (15)
3. Update the set of support blocks (choose sub-blocks that best match the signal) and the residuals, such that \[I=l_{i-1}\cup\{i_{i}\}\ \ \ \ \ r_{i}=r_{i-1}-\psi_{i}x\] (16)
4. Convergence conditions. When \(i<k,i=i+1\), return to the step 2 or stop iterating, then output the reconstructed signal. The biggest difference between the BOMP algorithm and the traditional method is that the inner product of each atom block with the current residual is used to select the most correlated atomic block. Now there is no longer an atom, but an atomic sub-block. This reduces the number of calculations and improves the reconstruction accuracy. 3. Application of Compressive Sensing Method with Direct Under-Sampling in Underwater Echo Signals  
According to the highlight model [32; 33], an echo signal can be represented as the linear superposition of \(K\) sub-echoes  
\[x(t)=\sum_{i=1}^{K}A_{i}*s(t-\tau_{i}) \tag{17}\]  
where \(s(t)=e^{-i\omega_{i}t}\) is the emitted pulse signal, with a pulse time of 0.01 s and a frequency of 30 kHz, \(x(t)\) is the echo signal, \(A_{i}\) is the amplitude of a sub-echo, \(\tau_{i}\) is the time-delay of a sub-echo.