## 3 Proposed Method
### Analysis of Phase Transition  
We analyzed the empirical phase transition of the proposed measurement matrix and compared it with several state-of-the-art alternatives. It is worth noting that for deterministic sparse measurement matrices with inflexible measurement numbers, analysis of their phase transitions becomes unfeasible. The compared measurement matrices included the Gaussian random measurement matrix [7], the random symmetric measurement matrix [9], the random binary measurement matrix [9], and the bipartite graph measurement matrix [17]. The latter is a deterministic sparse measurement matrix with a flexible measurement number, and the others are random measurement matrices. The Gaussian random measurement matrix is a classic and highly effective measurement matrix. It is one of the most commonly used benchmarks in the study of measurement matrices due to its excellent performance. The random symmetric and random binary measurement matrices are classic matrices that share structural similarities with our proposed measurement matrix. Additionally, we included the bipartite graph measurement matrix in our comparisons due to the fact that it is a classic deterministic sparse matrix, known for its flexibility in dimensions and good sparsity properties. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
103 & 107 & 127 & 1  
In the Gaussian random measurement matrix, the entries are independently sampled from a Gaussian normal distribution with a mean of 0 and a variance of 1. In the random symmetric measurement matrix, the entries are independently sampled from a symmetric signs distribution, where each element takes the value of \(+\frac{1}{\sqrt{N}}\) or \(-\frac{1}{\sqrt{N}}\) with equal probability. In the random binary matrix, the entries are independently sampled from a distribution where each element takes the value of \(+\frac{2}{\sqrt{N}}\) or 0 with equal probability. The bipartite graph measurement matrix, constructed using the Progressive Edge-Growth (PEG) algorithm, is a low-density parity-check (LDPC) matrix that generally contains \(d\) non-zero elements in each column, where \(d<<N\). Phase transition was achieved utilizing the methodology described in references [25; 32; 33], as follows. For a fixed \(N=239\), \(\alpha\) varied from 0.1 to 0.95 with a step size of 0.05, and \(k\) varied from 1 to \(M\) with a step size of 1. At each combination of \((\alpha,\beta)\), 300 sparse signals were measured using different measurement matrices and then recovered using the OMP algorithm [22]. Let \(\mathbf{x}\) be the original signal and \(\mathbf{\hat{x}}\) be the recovered signal. For each trial, if  
\[\frac{\left\|\mathbf{x}-\mathbf{\hat{x}}\right\|_{2}}{\left\|\mathbf{x}\right\| _{2}}\ <\ 0.01 \tag{11}\]  
we declare that the recovery is successful. Then, for each \((\alpha,\beta)\)-tuple and each measurement matrix, we can obtain the exact recovery rate. Since it was observed that the empirical 50% success curve closely aligned with the theoretical bound [27], we employed the logistic regression method as outlined in references [25; 32; 33]. That is, for each \(\alpha\), the phase transition referred to the value of \(\beta\) that yielded a 50% exact recovery rate in this study. For greater universality, empirical analysis of the phase transitions was conducted on various measurement matrices for two types of randomly synthesized sparse signals: Gaussian sparse signals and binary uniform sparse signals. The generation process was as follows: the non-zero elements of the Gaussian sparse signal followed a standard Gaussian distribution, whereas the non-zero elements of the binary uniform signal followed a uniform distribution in the range [0; 1]. The empirical analysis results are depicted in Figure 2. Obviously, the Gaussian random measurement matrix, the random symmetric measurement matrix, and the random binary measurement matrix exhibited similar phase transition curves.