## 2 Background  
### Theory of CS  
The underlying assumption in this paper is that the signal \(\mathbf{x}\in\mathbf{R}^{N}\) itself is naturally sparse. That means than \(\mathbf{x}\in\mathbf{R}^{N}\) has \(k\ll N\) non-zero elements naturally.  
There are mainly two steps in CS: the measurement process and the recovery process. In the measurement process, \(\mathbf{x}\) is measured by projecting it onto the measurement matrix \(\mathbf{\Phi}\), as shown in Equation (1):  
\[\mathbf{y}=\mathbf{\Phi}\mathbf{x} \tag{1}\]  
Since \(\mathbf{\Phi}\) is a matrix with a size of \(M\times N\), where \(M<N\), \(\mathbf{x}\) is projected into a lower-dimensional space. Clearly, the measurement ratio is \(M/N\).  
In the recovery process, \(\mathbf{x}\) needs to be reconstructed from \(\mathbf{y}\) by solving Equation (1). However, because \(M<N\), this process becomes ill-posed. According to compressed sensing theory, if \(\mathbf{x}\) is sparse, it can be approximately reconstructed by solving the following \(l_{0}-\) optimization problem (2):  
\[\min\lVert\mathbf{x}\rVert_{0}\;\;s.t.\;\mathbf{y}=\mathbf{\Phi}\mathbf{x} \tag{2}\]  
However, this problem is NP-hard, thus a greedy algorithm, such as orthogonal matching pursuit (OMP) [22] can be used to find a good-enough solution. [PERSON] et al. [4] demonstrated that if the measurement matrix satisfies the RIP under a specific condition, problem (2) can be replaced with the \(l_{1}-\) optimization problem (3), which can be solved efficiently using the basis pursuit (BP) algorithm [23].  
\[\min\lVert\mathbf{x}\rVert_{1}\;\;s.t.\;\mathbf{y}=\mathbf{\Phi}\mathbf{x} \tag{3}\]  
The RIP is an important criterion to verify whether \(\mathbf{\Phi}\) is a proper measurement matrix.  
Definition 1: For a measurement matrix, \(\mathbf{\Phi}\in\mathbf{R}^{M\times N}\), and any \(k\)-sparse signal, if there exists a constant \(\delta_{s}\in(0,1)\), such that  
\[(1-\delta_{s})\lVert\mathbf{x}\rVert_{2}^{2}\leq\lVert\mathbf{\Phi}\mathbf{ x}\rVert_{2}^{2}\leq(1+\delta_{s})\lVert\mathbf{x}\rVert_{2}^{2} \tag{4}\]  
then, \(\mathbf{\Phi}\) is said to satisfy the RIP with order \(k\). And the smallest number \(\delta_{k}\) is called the restricted isometry constant (RIC) of order \(k\).  
We can evaluate whether a matrix is a good measurement matrix by checking whether it satisfies the condition (4) for a large \(k\). However, RIP merely provides a sufficient condition to guarantee the accurate recovery of signals, which is very conservative in practical applications.  
Phase transition is more precise, serving as both a necessary and sufficiency condition. It is widely used to evaluate the effectiveness of measurement matrices [24] and recovery algorithms [25; 26]. Let \(\alpha=M/N\) represent the sampling ratio and \(\beta=k/N\) denote the sparsity ratio. It was observed that the plane of \((\alpha,\beta)\) can be partitioned into two distinct phases, as shown by the significant phase transition curve during signal recovery, as shown in Figure 1. In one phase, the original signal can be perfectly recovered, and in the other phase, recovery fails, with equally high probability. Phase transition is an important criterion and has garnered extensive research attention. The phase transition threshold for the Gaussian random measurement matrix was investigated in [27]. Numerical experiments demonstrated a nearly perfect alignment between the empirical 50% success curve and the theoretical bound [27].