Compressive Sensing Based Data Collection in Wireless Sensor Networks

In order to improve the energy efficiency by reducing the amount of the data delivered in Wireless Sensor Networks(WSNs), a Compressive Sensing(CS) based data collection scheme considering the correlation in temporal-spatial domain is studied in this paper. Kronecker product is applied to construct the sparse basis in the joint domain. The simulation results show that due to the huge amount of sensor nodes, by exploiting the dependency in spatial domain, the data number can be reduced distinctly. The high recovery accurancy can still be achieved.


Introduction
In most cases, communication cost is significant in WSNs. In order to improve the energy efficiency, optimal algorithms [1,3] for the compression of sensed data, communication and sensing have been investigated. In this paper, a CS-based data collection scheme using WSN is designed and simulated to show the improvement of the engergy efficiency.

Compressive Sensing
To describe CS theory [4,6] concisely, we describe a general linear system with where x is the target signal in vector form of length n , the vector y is the observations with length m , and A is a matrix of size m by n , m n  . If the signal is sparse, Eq. (1) can be rewritten as where A    of size m n  and x c   , c is a sparse representation when the transform matrix is adopted. In general,  is n l  with l greater or equal to n . CS uses a basis pursuit (BP) approach to find the solution to Eq.(2). For measurements contaminated by noise, BP can be replaced by Basis Pursuit De-Noise (BPDN). The solution is given by 1 2 min . .
Because the 1 L norm is convex, it can then be calculated by modern linear programming optimization algorithms.

Model of the Signal in CS-based WSN
Suppose a WSN consists of N sensor nodes indexed by vector and a sink node.
The samples collected over T sensing periods are represented by a 3-D signal ensemble 1 2  , , , , , ,

CS-Based Signal Model in Temporal Domain
As assumed above, , 1,2, , where , , , , signal ensemble X with temporal tranformation can be compactly written where i m i y   are the measurements. Then, each sensed data can be recovered according to (3)  by solving  1   2   , , , arg min . .
then, we can obtain the estimate as , , 1,2, Assume that at instant t only t n sensor nodes are in the active set A n  , and accordingly the entries are the available measurements while all the others are zero. Thus, the measurements at instant t can be expressed as The total number of the measurements is

CS-Based Signal Model in Spatial Domain
With the assumption of the compressibility of the data sensed in WSN, there are also such basises as 1, 2, where matrix , S t C consists of the associated coefficients in spatial domain. According to rules of linear algebra, we can express the vector t x as Assume we acquire t n N  measurements of the sensor samples at each time instant 1, 2, , t T   , then also by (2) in CS theory above, we have

CS-Based Signal Model in Spatial-Temporal Domain
In order to derive the sparsifying basis matrix of the joint correlation structure of  , we combine the basis matrixs above applying the rules of Kronecker product. Assume x is the vector-reshaped expression of  which can be expressed as where     1 , , where N I is the identity matrix of size N N  .
Continue with the recovery of the signal ensemble for each sensor in WSN, the decoder in sink node can carry it out by solving the following joint optimization based on CS theory as 1 2 arg min . .
resulting in 1 , , . Then x  can be further reshaped into the estimate of 1 , ,

Conclusions
By expoiting the correlatlion in temporal and spatial domains, the number of the measurements is obviously decreased which means the energy consumed by data communication is consequently reduced. Due to the huge amount of the sensors result in high spatial correlation, the energy saving can be efficiently performed with good recovery quality.