## 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 basic idea of DUS-CS is that, in combination with the inherent characteristics of the signal, one can use the prior information sparse matrix [28; 29] to represent the original signal. The signal can be represented by a complete prior atomic library and K-sparse coefficient vector. Then, according to the implementation method of direct under-sampling and the RIP (restricted isometry property) [30] condition of the measurement matrix to constructed measurement matrix, signal observation sampling can be performed to obtain under-sampling observation signals. Finally, based on the orthogonal matching pursuit method, combined with the block sparse property of the signal, it becomes possible to reconstruct the original signal (see flow diagram in Figure 1). As shown in Figure 1, the core of the direct under-sampling compression sensing method lies in the construction of the measurement matrix and the sparse representation and reconstruction method, leveraging the prior information. Therefore, the following two parts are described in detail. (1) Construction of measurement matrix  
The direct under-sampling method is used on an input signal \(x(t)\), with signal length \(N\), sampling rate \(f_{s}\), to yield \(M\) observation signals  
\[y=[y_{1},y_{2},\ldots,y_{M}]^{T} \tag{5}\]  
where  
\[y_{m}=x(t_{m}) \tag{6}\]  
The sampling moment satisfies the following conditions  
\[\min(|t_{m}-t_{m-1}|)=kT_{s} \tag{7}\]  
Figure 1: Direct under-sampling compressive sensing method. where \(T_{s}=1/f_{s}\) is the sampling period of the signal and \(k\geq 2\) is an integer to ensure that the under-sampled measurements are low speed and random. We use the implementation of direct under-sampling and combined with the theory of compressive sensing to construct the observation matrix. The observable structure of the original signal, \(x\), is expressed as  
\[y=\Phi x \tag{8}\]  
where \(y\in R_{M\times 1}\) is the observed signal and \(\Phi\in R_{M\times N}\) is observation matrix (\(M\ll N\)). According to the Whittaker interpolation theorem [31], continuous signals can be reconstructed by interpolation of their uniform discrete samples by  
\[x(t)=\sum_{n=-\infty}^{\infty}x_{n}\sin c\Big{(}\frac{t}{T_{s}}-n\Big{)} \tag{9}\]  
So, the observation matrix can be expressed as  
\[\Phi(m,n)=\sin c\Big{(}\frac{t_{m}}{T_{s}}-n\Big{)},1\leq m\leq M,1\leq n\leq N \tag{10}\]  
According to the Nyquist sampling theorem, signal reconstruction is obtained by filtering the periodic spectrum of the signal. Assuming a periodic signal such as Equation (10), each element in the observation matrix is calculated considering a certain signal period. According to the slow decay and non-compact support characteristics of the \(\sin c\) function, the role of other cycles, while not dominant, cannot be simply ignored directly. Therefore, the R cycles are accumulated to obtain the observation matrix  
\[\Phi(m,n)=\sum_{r=-\infty}^{\infty}\sin c\Big{(}\frac{t_{m}}{T_{s}}-n+rN\Big{)} =\sum_{r=-R/2+1}^{R/2}\sin c\Big{(}\frac{t_{m}}{T_{s}}-n+rN\Big{)} \tag{11}\]  
In this formula, the larger the period R, the higher the calculation accuracy. However, as the number of cycles increases to obtain high precision, it takes longer to reach convergence of the Equation (11). In order to guarantee a certain precision, the Poisson superposition summation formula is used to speed up the calculation summation  
\[\sum_{r=-\infty}^{\infty}f(t+rN)=\frac{1}{N}\sum_{k=-\infty}^{\infty}F\Big{(} \frac{k}{N}\Big{)}\exp\Big{(}2\pi j\frac{k}{N}t\Big{)} \tag{12}\]  
In the formula, the function, \(F\), is the Fourier transform of the signal \(f\). Owing to the nature of the sinc function, the continuous Fourier transform is a finitely supported rectangular pulse.