## 2 Related Work
For a signal \(X\in\mathbb{R}^{N}\), we can obtain its \(M\) linear observations, as in Figure 1.  
If \(\Theta=\Phi\)\(\Psi\) satisfies the restricted isometry property (RIP) [16], condition \(M\leq cK\log(N/K)\) such that \(c\) is a small constant with \(c>0\), the vector \(\alpha\) can be accurately recovered from \(Y\) as the unique solution of:  
\[\hat{\alpha}=argmin_{\alpha}\left\|\alpha\right\|_{1}\quad s.t.\quad Y=\Phi \Psi\alpha, \tag{4}\]  
The original networked data \(X\) may be sparse itself or can be sparsified with a suitable transform such as Discrete Cosine Transform or Discrete Wavelet transform [25, 26, 27]. One example of the self-sparse \(X\) is a linear combination of just \(K\) basis vectors, with \(K\ll N\), that is; only \(K\) are nonzero and \((N-K)\) are zero [28]. Usually, the networked data vector \(X\) is sparse with a proper \(\Psi\) in Equation (2). Most of the previous works [25, 26, 27] consider a regular sensor network. However, sensors are usually deployed on an irregular grid. So it is expected to find some sparse representations to sparsify irregular grid sensor networked data. In WSNs, sampling matrix \(\Phi\) is usually pre-designed, _i.e._, each sensor locally draws \(M\) elements of the random projection vectors by using its network address as the seed of a pseudorandom number generator. Based on CS theory, [PERSON] _et al_. [28] considers a sparse event detection scenario where the channel impulse response (CIR) matrix is used as a natural sampling matrix. In [29], a basic global superposition model to obtain the measurements of sensor data is proposed, where a sampling matrix is modeled as the channel impulse response (CIR) matrix while the sparsifying matrix is expressed as the distributed wavelet transform. In [30], compressive distributed sensing using random walk CDS(RW) algorithm is proposed that uses rateless coding. In our proposed protocol we use CS to fuse data efficiently in EDACP-CS and consider three types of nodes and different transmission ranges to achieve a robust self configured WSN that maximizes lifetime. In [31], energy efficient clustering and data aggregation protocol for heterogeneous WSNs (EECDA) has been proposed. EECDA combines energy efficient cluster based routing and data aggregation for improving the performance in terms of lifetime and stability. However, in the proposed protocol we  
Figure 1: The scheme of compressive sampling. discuss effectively the aggregation using CS and consider the heterogeneity of nodes due to energy as well as transmission ranges. We assume that CHs are randomly selected based on their residual energy and the distance form the BS. In [32], the authors have proposed a novel data collection algorithm using compressive sensing (CS data collection). In CS data collection, each sensor node only communicates with its neighbor node in one hop. However, in the proposed protocol all sensor nodes organize themselves into clusters with one node elected as a CH based on their weighted election probabilities and their distance to the BS. All the above protocols do not consider efficient data compression and link heterogeneity. In the proposed protocol EDACP-CS, each sensor node independently elects itself as a CH based on its residual energy and its distance to the BS. Therefore, nodes that are closer to the BS and contain more energy than the other nodes have more chance to be selected as a CH for current round. EDACP-CS introduces a new scheme to combine clustering strategy with CS theory for increasing both energy and stable period constrains under heterogeneous environment in terms of node energy and transmission range. Simulation shows that the network lifetime and the stability period are much better in the proposed protocol than CS data collection, EECDA and LEACH-Hetero protocols.