## 2 Related Work  
Recently there has been a growing interest in WSNs. One of the major issues in WSN is developing an energy-efficient routing protocol. Since the sensor nodes have limited available power, energy conservation is a critical issue in WSN for nodes and network life. The issue of heterogeneity (in terms of energy) of nodes is addressed in [21]. In [22], the proposed protocol is based on random selection of CHs weighted according to the remaining energy of the node. This approach addresses the problem of varying energy levels and consumption rates but still assumes that the BS can be reached directly by all the nodes. In [23], the authors provided the optimal heterogeneous sensor deployment that minimizes the deployment cost in different communication modes. In their model, the cost of the cluster head device is determined by the amount of initial battery energy, which depends on the number of cluster members and communication mode. They do not consider the sensing coverage and aging process over time. Low-Energy Adaptive Clustering Hierarchy (LEACH) [9] is one of the most popular distributed cluster-based routing protocols in WSNs. The operation of LEACH is generally separated into two phases, the set-up phase and the steady-state phase. In the set-up phase, CHs are selected and clusters are organized, each node decides whether to become a CH for the current round. This decision is based on a predetermined fraction of nodes and the threshold \(T(s)\), which is given by:  
\[T(s)= \begin{cases}\frac{p_{opt}}{(1-p_{opt}\times(r\ mod(1/p_{opt})))}&\text{ if }s\in G,\\ 0&otherwise,\end{cases}  \tag{1}\]  
where \(p_{opt}\) is the predetermined percentage of CHs, \(r\) is the count of current round and \(G\) is the set of sensor nodes that have not been CHs in the last \(1/p_{opt}\) rounds. Using this threshold, each node will be a CH at a round within \(1/p_{opt}\) rounds. After \(1/p_{opt}\) rounds, all nodes are once again eligible to become CHs. In this way, the energy concentration on CHs is distributed. LEACH does not consider the residual energy of each node so the nodes that have relatively smaller energy remaining can be selected as CHs. This makes the network lifetime shortened. In the steady-state phase, the actual data transmissions to the BS take place. After the steady-state phase, the next round begins. In [24], the authors proposed a new routing protocol and data aggregation method in Leach-heterogeneous system where the sensor nodes form the cluster and the CH elected based on the residual energy of the individual node calculation with re-clustering scheme is adopted in each cluster of the WSNs. The proposed protocol works with 3 types of nodes (normal, advanced and super), CHs are selected on the basis of their residual energy and their distances to the BS. CS is a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition, the theory of CS extends traditional sensing and sampling systems to a much broader class of signals. The promise of CS is that a sparse or compressible signal can be recovered from a small salient set of projections. To make this possible, there are two principles [16]: sparsity, which pertains to the signals of interest, and incoherence, which pertains to the sensing modality. Under CS framework, any compressible signal \(X\in\mathbb{R}^{N}\) can be represented in the form of:  
\[X=\Psi\alpha, \tag{2}\]where \(\Psi\in\mathbb{R}^{N\times N}\) is the transform matrix and \(\alpha\) is the sparse representation of \(X\). There are no more than \(K\) nonzero entries in vector \(\alpha\), where \(K\) is much smaller than \(N\), and the signal \(X\) is called \(K\) sparse signal. In contradiction to sparsity, incoherence means that the measurement matrix \(\Phi\) has dense representation in basis \(\Psi\), and \(\Phi\) is independent to \(\Psi\). In practice, most of natural signals are sparse or near sparse, and they can be recovered from their compressible samples.