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The data transmission process in Wireless Sensor Networks (WSNs) often experiences errors and packet losses due to the environmental interference. In order to address this problem, we propose a Compressive Sensing data gathering algorithm based on Packet Loss Matching (CS-PLM). It is proven that, under tree routing, the packet loss on communication links would severely undermine the data reconstruction accuracy in Compressive Sensing (CS) based data gathering process. It is further pointed out that the packet loss in CS based data gathering exhibits the correlation effect. Meanwhile, we design a sparse observation matrix based on packet loss matching and verify that the designed matrix satisfies the Restricted Isometry Property (RIP) with a probability arbitrarily close to 1. Therefore, reliable transmission of the compressed data can be guaranteed by adopting the multipath backup routing among CS nodes. It is shown in the simulation results that, with a 60% packet loss ratio of the link, the CS-PLM algorithm can still ensure the effective reconstruction of the data gathered by the CS algorithm and the relative reconstruction error is lower than 5%. Therefore, it is verified that the proposed algorithm could effectively alleviate the sensitivity to packet losses for the CS based data gathering algorithm on unreliable links.

The nodes in the wireless sensor networks (WSNs) are usually densely deployed and a lot of redundancy exists in the data gathered, which leads to the waste of the energy of the nodes. The compressive sensing (CS) algorithm is a new technique which could largely reduce the sampling frequency and execute the sampling process in parallel with the compression process. As a result, this technique has drawn much attention by researchers.

In order to balance and reduce the energy consumption of the nodes as well as prolong the network lifetime, researchers have proposed data gathering algorithms based on compressed sensing. At present, most of these algorithms are focused on how to effectively reduce network energy consumption and extend the network lifetime [

For the CS based data gathering problem on unreliable links, we propose a compressive sensing data gathering algorithm based on packet loss matching (CS-PLM). In this algorithm, the nodes in the network are divided into two types, i.e., the traditional forwarding (TF) nodes and the compressive sensing (CS) nodes. The packet loss of TF nodes does not exhibit the correlation while the lost packets of CS nodes are strongly correlated. In the process of the Compressive Sensing data gathering, a packet loss will lead to the loss of the data gathered from multiple nodes; the packet loss correlation effect is caused by the superimposed transmission of the collected data from each node of the multihop link in the CS compression sampling process. The closer the packet loss node

The main contributions of this paper are as follows.

①By analyzing the routing with tree structure, it is pointed out that the packet loss would seriously undermine the reconstruction accuracy of the CS based data gathering process and the packet loss in the CS based data gathering process exhibits the correlation effect.

②We design a SPLM measurement matrix and further prove that this matrix satisfies the Restricted Isometry Property (RIP) with a probability arbitrarily close to 1.

③We propose a multipath backup routing transmission scheme based on Hybrid CS to guarantee the reliable handover of the CS projection data.

Due to its feature of simple encoding and complex decoding, compressed sensing theory has been widely applied to the area of data collection in WSNs. At present, the research of CS based data collection algorithms in wireless sensor networks mainly focuses on how to use CS technology to reduce the network energy consumption of the data gathering process in WSNs. Most of these works assume that the network link is an ideal link where the impact of packet loss on the CS based data gathering process is ignored. The number of transmitted data packets was presented in paper [

The application of CS was investigated in [

It was shown via simulations in [

There are other data gathering methods in conventional WSNs such as ARQ, multipath transmission, network coding, etc. However, there are relatively few works studying the reliable CS based data gathering algorithm in WSNs. Furthermore, the CS based data gathering algorithm is much more sensitive to packet losses than conventional methods. Therefore, the study of CS based data gathering algorithm on unreliable links is quite meaningful to the application of CS theory in practical scenarios.

In wireless sensor network, serious packet loss will undermine the communication performance, service quality, and application effect of sensor network. In recent years, the research premise of the CS based data gathering theory is the ideal link, and, because of the dynamic characteristics of the wireless link, channel interference and asymmetry of conflict, the wrong direction, and height of antenna, the unreliable link issues are commonly encountered in practical applications. There are many methods to ensure the reliable transmission of links in the traditional data gathering methods of wireless sensor networks, but to the best of our knowledge, there is little work for reliable sensor network data gathering method based on compression perception. In addition, the sensitivity of the CS data gathering method to link packet loss is much higher than that of traditional data gathering, so the research on compressed sensing data gathering algorithm under unreliable link is of great significance to the application of compressed sensing technology in real sensor network.

The CS is a new technique which samples the sparse signal with a frequency below the Nyquist sampling frequency and achieves the projective transformation of the target signal from a high-dimension space to a low-dimension one. The accurate reconstruction of the compressed signal is achieved via the optimal reconstruction algorithm which is widely studied and applied in many areas due to its excellent compression performance.

Assume that

where

In the data gathering process of WSNs, each round of CS based data gathering is performed with

Assume that

The CS based data gathering process on unreliable links under the tree-like topology is illustrated in Figure

CS based data gathering on unreliable links.

If packet loss occurs on the link between S_{5} and the Sink end, all the packets corresponding to S_{5} as well as the child nodes of S_{5} will be lost, which is shown in the frame in Figure

Therefore, the CS based data gathering on unreliable links exhibits the following features: ① one packet loss on the link will result in the data loss of multiple nodes; i.e., the packet loss exhibits the correlation effect. ② The Sink node has no information on the packet loss situation for the nodes in the network and regards the measurement data of the nodes in the network as the data projection to perform reconstruction. That is, the sensing data for compression does not match the sampling of the measurement matrix.

In order to solve the mismatch problem between the sensing data and the sampling of the measurement matrix, we design a sparse measurement matrix based on Packet-loss Matching (SPLM). In each measurement, the information of the packet loss node is omitted by the measurement matrix. As a result, the packet loss problem for CS based data gathering under tree topology is transformed into the measurement matrix projection problem based on sparse matching. Hence, the large-scale measurement and sampling are accomplished for the data in the network; meanwhile erroneous judgement for the data gathering situation can be avoided at the Sink end. The detailed realization of this process is as follows.

The LSM is defined as the matrix which records the link state information with size

Each row of the DRP matrix contains O(

The sparse measurement matrix based on packet loss matching employs the randomness of the packet losses on realistic links to construct the random sparse measurement matrix. The construction process can be achieved by multiplying the LSM with the DRP element-wise, as shown in

The design of the measurement matrix should guarantee that most of the orthogonal base satisfies the RIP constraint. However, the proof of the RIP condition is a NP-hard problem. It was pointed in [

For a matrix

Assuming that matrix

And not all of the coefficients

Define the random process

Therefore,

Define the case that (

Solving the probability

To evaluate the performance of the SPLI matrix, the classical CS data gathering algorithm, i.e., the CDG algorithm [

Performance comparisons for sparse measurement matrices based on packet loss tags.

For the unreliable link under the tree topology, the CS based data gathering not only exhibits the correlation effect for packet losses but also suffers from the problem of misjudgment on the data packet reception situation at the Sink end. However, the SPLI matrix can only solve the misjudgment at the Sink end. Therefore relevant mechanisms still need to be studied to solve the problem of correlation effect in the process of CS based data gathering. Essentially, the correlation effect of the packet loss is caused by the weighted superposition processing of data packets during the CS based data gathering process, which is also the advantage of CS based data gathering. Therefore, the most effective method for solving the correlation effect is to guarantee the reliability of link transmission and avoid the appearance of correlated loss.

The cost of the guarantee for the reliable transmission in the whole network link is huge. To reduce the maintenance cost of the network, according to the performance analysis of the SPLM measurement matrix, this paper designs a hybrid CS method for the data gathering in the network and divides the nodes of entire network node into traditional forwarding (TF) nodes and CS node, where the TF node only forwards data in a traditional data gathering manner, and the packet loss does not exhibit the correlation effect. The CS node transmits and receives data in a CS based data gathering manner and the packet losses are correlated. Therefore, for the data gathering between TF nodes, simply adopting the SPLM measurement matrix can overcome the impacts of packet losses on CS based data reconstruction. However, for CS nodes, in addition to using the SPLM measurement matrix, a corresponding mechanism must be designed to ensure the reliability of data transmission between CS nodes.

This paper designs a transmission mechanism based on multipath backup routing to ensure the reliability of data transmission between CS nodes. Under normal conditions, the CS node uses MST routing to transmit and receive data packets. If a packet loss occurs on the CS link, the transmitting node

where _{1} is the energy consumption coefficient of the circuit and _{2} is the power amplifying coefficient,

The above optimization problem is solved using the Lagrangian multiplier method. The energy consumption of the network is the minimum if and only if the distance between the destination node and the source node is the same for each hop. We further present the value of the characteristic distance

The CS-PLM algorithm first divides the nodes in the entire network into TF nodes and CS nodes. The MST routing tree is built for nodes in the entire network where the number of child nodes for node

During the data gathering process of the CS-PLM algorithm, the TF nodes gather data along the MST routing in the traditional relay method. As shown by the white nodes in Figure

Distributions of different node types in the CS-PLM algorithm.

In the case of an unreliable link, no processing will be performed if packet losses occur in the link between TF nodes. However, if a packet loss occurs in the link between CS nodes, the data packet is directly sent to the Sink end using the minimum energy consumption backup path. After each round of data gathering, the Sink node builds an SPLM measurement matrix according to the packet loss during each measurement process and further employs the SPLM measurement matrix and the

The data gathering process of the CS-PLM algorithm can be divided into three stages. The first stage is initialization of the sensor network, in which the node networking, the accumulation of the a priori information for the reception state of each link, and the configuration for the node measurement vector are accomplished. At the second stage, the CS based data gathering is performed on the lossy link and the effective CS based sampling and gathering is achieved for all the nodes in network. At the third stage, the CS based reconstruction is achieved for the sampled data and the original data is further acquired for the nodes in the network. The operation of the algorithm is detailed as follows:

_{i1},_{i2},…,

_{ij} according to the routing. Then the messages are added up sequentially and relayed to the Sink end. When a node recognizes its packet loss in this process, the probability density functions

In order to evaluate the performance of the CS-PLM algorithm, we employ the tool MatlabR2014a to perform the simulations. Based on the network model in this paper, we assume that the network is deployed in a

The reconstruction accuracy of the CS-PLM algorithm is illustrated with different network packet loss ratio in Figure

Reconstruction error with different packet loss ratio.

In order to evaluate the performance of the algorithm under different degree of correlation, we choose two data sets with different degree of sparsity; i.e., the results are illustrated in Figure

Network packet loss ratio with different sparsity Degree.

The network lifetime performances for different algorithms are compared in Figure

Network lifetime for the CS based algorithms.

According to the theory of CS, the correlation between the measurement matrix and the sparse base of the data will affect the reconstruction performance of the algorithm. To verify the performance of the CS-PLM algorithm under different sparse basis, we choose two sparse bases, i.e., the Discrete Cosine Transform (DCT) and Discrete Fourier Transform (DFT) as a comparison. Related results are shown in Figure

Relative reconstruction errors under different sparse bases.

The impacts of packet loss ratio on relative reconstruction error are illustrated in Figure

Relative reconstruction errors with different number of measurements.

When the decay coefficient of the event source is^{−5}, the performances of these three algorithms are almost the same. However, when the bit error rate is as high as^{−3}, the CS-PLM algorithm outperforms the others. It is shown in Figure ^{−3}, the SNR for data reconstruction of the CDG algorithm is 27.33dB and the high bit error rate obviously affects the performance of the CDG algorithm. The SNR for data reconstruction of the SRS-DG algorithm is 29.72dB. Since the SRS-DG algorithm is designed for recovering block loss, the data block is abandoned once transmission error occurs and the error-free nodes are measured with the sparse measurement matrix. Therefore, the measured information is reduced every time. The lost data block is then compensated by increasing the number of measurements through the next round of data gathering and the reconstruction SNR is therefore increased. As a result, the SNR is not high for this round of data reconstruction. The data reconstruction SNR obtained by the CS-PLM algorithm is 35.91dB. The transmission error is predicted by the spatial correlation of the data under certain conditions and the amount of abandoned information is therefore reduced. Henceforth, in the wireless scenarios with high bit error rate, the CS-PLM algorithm does not cause additional communication energy consumption and it can overcome the impacts of erroneous data blocks on data reconstruction. The efficiency of the CS-PLM algorithm is therefore verified.

Performance analyses of the CS-PLM algorithm.

For the CS-PLM algorithm, under the premise of ensuring the constant data compression rate, different network sizes require different measurement numbers in the data gathering process, which in turn causes the changing threshold for the judgment of node type. Therefore, different network size leads to different proportions of node types, which not only affects the total packet throughput of the network, but also affects the complexity of constructing the backup paths between CS nodes as well as the reliability of their transmission. Therefore, the performance of the algorithm is affected by the network size and related results are illustrated in Figure

Relative reconstruction error with different network scale.

Relative reconstruction error under different sparse bases.

In order to address the CS based data gathering problem on unreliable links, we have proposed a CS-PLM algorithm. We designed the SPLM measurement matrix by analyzing the influence of the packet loss on CS based data gathering and further verified through simulations that the correlation of packet losses would undermine the reconstruction performance of the SPLM measurement matrix. Therefore, the nodes in the network are divided into TF nodes and CS nodes. Packet losses between TF nodes do not exhibit correlation and we simply adopt the SPLM measurement matrix to perform measurement projection. However, besides adopting the SPLM measurement matrix for measurement projection, the CS nodes also guarantee the transmission reliability by the minimum energy consumption backup paths and avoid the occurrence of correlated lost packets. It was shown in the simulation results that when the packet loss ratio on the link is 60%, the CS-PLM algorithm could still guarantee the effective reconstruction of the compressed data. Compared with other algorithms, the proposed algorithm showed great improvements in terms of reconstruction accuracy and the sparsity degree of the data set. It can accurately reflect the influences of the network size, sparse base, and the number of measurements for data gathering on the performance of the CS-PLM algorithm. Future work may focus on the impacts of packet loss ratio for mobile nodes on the reconstruction accuracy when the network flow is sufficiently large or small.

The data used to support the findings of this study are included within the article.

The authors declare no conflicts of interest.

This work was supported by Henan Province Education Department Cultivation Young Key Teachers in University under Grant no. 2016GGJS-158; Henan Province Education Department Natural Science Foundation under Grants nos. 19A520006 and 18B520026; Luoyang Institute of Science and Technology High-level Research Start Foundation under Grant no. 2017BZ07; Natural Science and Technology Research of Henan Province Department of Science Foundation under Grants nos. 182102210428 and 162102310474.