## 2 Related Work  
Finding an optimal CS matrix for compression and reconstruction is one of the most difficult problems in CS. Randomly constructing CS matrix elements that meet the RIP requirement does not produce the optimal solution. Because of this, random sensing matrices have been abandoned in favor of deterministic matrices in a number of research studies. The Ternary matrix [11], Walsh-Hadamard [12], and low-density parity checks [13] are examples of deterministic matrices that can be used in computations. Another challenge is finding suitable algorithms for compressed speech enhancement and a domain of transformation that is suitable for sparse signals. Many basis functions are available for speech signals [14; 15], like the Fourier Transform, Cosine Transform, and Wavelet Transforms [16]. There are several different types of sensing matrices; for example, the Gaussian random matrix, Bernoulli, Fourier, wavelets, etc. [14]. Several optimization techniques, such as \(l_{1}\) minimization [17], orthogonal matching pursuit (OMP) [18], and compressive sampling matching pursuit (CoSaMP) [19], were also utilized to recover the compressively sampled speech signals. The study presented by [PERSON] et al. [20] analyzed the performance of several CS reconstruction algorithms applied to image reconstruction and suggested that the \(l_{1}\) minimization-based basis pursuit algorithm outperforms the other OMP methods. As presented in [21], the basis pursuit algorithm also provides a slight edge over the CoSaMP for speech enhancement. Several CS-based methods for speech enhancement have also been proposed in [22; 23; 24]. [PERSON] et al. [22; 24] employed a random sampling matrix in their sensing process. At present, the most accurate approach for determining the quality of speech is to conduct subjective listening tests. However, subjective evaluations of speech enhancement algorithms are costly and time-consuming, despite the fact that they are accurate, reliable, and performed under rigorous conditions [25; 26; 27; 28]. Therefore, many efforts have been made to develop objective measures that are highly correlated with speech quality. Several attempts have been made over the years to predict subjective speech quality using objective speech quality metrics [25]. Objective measures based on LPC, such as cepstrum distance measures (CEP) [29] and frequency-weighted segmental SNR (fwsegSNR) [25], are among the objective speech quality measures examined in [28], along with others, such as segmental SNR (segSNR) [30], weighted-slope spectral distance (WSS) [31], and PESQ [32; 33]. The preceding discussion has highlighted that the basic compressive sensing idea has been modified in different ways to achieve better speech enhancement performance. Many ideas have been proposed for efficient reconstruction of the compressed signal [34], as shown in Figure 1. However, this work is primarily concerned with the efficient reconstruction of compressed signals while achieving better enhancement using greedy algorithms. Greedy algorithms recover the signal iteratively by choosing a local optimal solution at each iteration with the intention of eventually discovering the global optimal solution. Ingeneral, greedy algorithms take the steps depicted in Figure 2. In each repetition, the quantity of the column selected varies slightly. Only one column is selected for each iteration of the matching pursuit (MP) [35], orthogonal matching pursuit (OMP), and matching pursuit based on least squares (MPLS) [36] algorithms.