Data Gathering with Compressive Sensing in Wireless Sensor Networks a Random Walk Based Approach

In this paper, we study the problem of data gathering with compressive sensing (CS) in wireless sensor networks (WSNs). Unlike the conventional approaches, which require uniform sampling in the traditional CS theory, we propose a random walk algorithm for data gathering in WSNs. However, such an approach will conform to path constraints in networks and result in the non-uniform selection of measurements. It is still unknown whether such a non-uniform method can be used for CS to recover sparse signals in WSNs. In this paper, from the perspectives of CS theory and graph theory, we provide mathematical foundations to allow random measurements to be collected in a random walk based manner. We find that the random matrix constructed from our random walk algorithm can satisfy the expansion property of expander graphs. The theoretical analysis shows that a k-sparse signal can be recovered using 1 minimization decoding algorithm when it takes m = O(k log(nk)) independent random walks with the length of each walk t = O(nk) in a random geometric network with n nodes. We also carry out simulations to demonstrate the effectiveness of the proposed scheme. Simulation results show that our proposed scheme can significantly reduce communication cost compared to the conventional schemes using dense random projections and sparse random projections, indicating that our scheme can be a more practical alternative for data gathering applications in WSNs