## 1 Introduction  
Extracting useful information from noise has always been a problem of extensive research in various fields. For example, in the field of image, image denoising and super-resolution analysis can enrich the information of the images [1]; in the field of condition monitoring, fault signal extraction can identify the existence of early fault [2, 3]; in the field of electronics, the extraction of small changes in current can identify kinds of weak measurement changes [4]. Therefore, it is crucial and necessary to study effective signal denoising methods. The evaluation criteria of a \"weak signal\" are not fixed, and there is no clear and widely agreed definition and interpretation of the weak signal. As weak signal detection is widely used in many fields, the detection objects in each field are completely different, and the detection methods vary greatly. Moreover, the degree of signal weakness in different detection objects is inconsistent, so it is difficult to define \"weak\". Therefore, in order to distinguish it from weak signals in other fields, it should be emphasized that what is studied in this paper is weak sensor electrical signals, and weakness means that the energy of electrical signals is relatively weak compared with other electrical noises. Alternatively, weak signals can be interpreted from the perspective of signal detection; that is, weak signals refer to signals that are difficult to detect with conventional and traditional signal analysis methods. The commonly used signal processing methods include wavelet transform (WT), short-time Fourier transform (STFT) [5], empirical mode decomposition (EMD) [6], variational mode decomposition (VMD) [7], fast kurtogram [8], and singular value decomposition (SVD) [9], etc. The sparse feature is a universal property in the signal [10]. In recent years, scholars have carried out extensive discussions and studied various methods by using the sparsity priori. In order to extract the sparse features of signals, the most representative method is the sparse representation (SR) algorithm [11]. The core idea of SR is that any signal can be represented as the superposition of some optimal basic functions from the basic waveform library, which matches the main structure of the signal best. The functions of the basic waveform library are called dictionaries, and dictionaries can be constructed based on prior knowledge of the signal, or based on paradigms of nonparametric signals. Since the dictionary matches some major features of the original signal, the coefficients of the signal are sparse under the decomposition of the dictionary. The theoretical system of sparse representation algorithm was first established on the matching tracking algorithm proposed by [PERSON] in 1993. In 1998, [PERSON] proposed the sparse decomposition theory. In essence, it reasonably loosed the regularization term of \(l_{0}\) in the mathematical model of matching tracking algorithm to \(l_{1}\) regularization, which also marks the formal establishment of the two core algorithms of sparse representation, and the subsequent sparse representation algorithms are improved or extended on this basis. Greedy tracking algorithms based on \(l_{0}\) regularization mainly include orthogonal matching pursuit (OMP) [12] in 1993, tree based tracking algorithm proposed by [PERSON] et al. [13] in 2006, and phased orthogonal matching pursuit (StOMP) proposed by [PERSON] [14] in 2012. Algorithms based on \(l_{1}\) or \(l_{p}\) regularization mainly include the least absolute shrinkage selection operator (LASSO) [15] in 1996, focal underdetermined system solver (FOCUSS) [16] in 1997 and gradient projection-based sparse representation (GPSR) [17] in 2008. In the same year, [PERSON] [18] proposed greedy basis tracking, and [PERSON] [19] proposed morphological component analysis (MCA), also belonging to this category. In 2004, [PERSON] et al.