AbstractâIn this paper, we study the integration between. Compressed Sensing (CS) and clustering methods in Wireless. Sensor Networks (WSNs) that ...
2014 IEEE International Conference on Distributed Computing in Sensor Systems
Compressive Sensing Based Data Gathering in Clustered Wireless Sensor Networks Minh Tuan Nguyen⋆ and Keith A. Teague School of Electrical and Computer Engineering Oklahoma State University Stillwater, OK 74078 Email: [tuanminh.nguyen, keith.teague]@okstate.edu
Abstract—In this paper, we study the integration between Compressed Sensing (CS) and clustering methods in Wireless Sensor Networks (WSNs) that significantly reduce power consumption for the networks. In theory, a base station (BS) needs to collect M measurements from the network with N sensors, then applies CS to obtain precisely all N sensor readings. In clustered networks, a cluster-head (CH) collects data from nonCH sensors in its cluster, adds all received and its own data and then sends the combined measurement to the BS. We further analyze the clustered network with the measurement matrix created by clustering methods, and formulate the total power consumption. Finally, we suggest the optimal number of clusters for the networks to consume the least power in practice.
I. I NTRODUCTION Saving energy in wireless sensor networks (WSNs) is always a critical problem that has been shown in many research areas. The networks have applications in both military and civil applications [1] live on inexpensive and small sensors that are dropped in an area randomly. Since they usually work in harsh conditions without maintenance, the on-board batteries deplete power very quickly if we do not have a balanced energy saving plan. Since Compressive Sensing (CS) [2], [3], [4], [5] provides a novel paradigm for data collection in WSNs, we only need M CS measurements at the BS to recover N sensor readings precisely (M ≪ N ). The sensor readings collected from such networks are highly correlated and compressible that satisfy the requisition to apply CS. They also can be considered as sparse signals in some bases, such as Wavelet or DCT. Our goal is to prolong sensors’ lives and keep the network connected as long as possible by balancing sensor power consumed and proposing an energy-efficient data gathering method for the networks. In our paper, we work on the clustering problems that have been shown to save and balance energy for WSNs. Sensors multiply their reading to Gaussian coefficients and send the products to their own cluster-heads (CH). All data received at CHs are added together and sent to the base-station (BS) to make CS measurements. Based on M measurements collected from sensors, all raw reading data from N sensors will be recovered based on those linear measurements at the BS. We formulate the total power consumption for the networks. We ⋆ Minh T. Nguyen is a lecturer with Thai Nguyen University of Technology (TNUT), Thai Nguyen city, Vietnam.
978-1-4799-4618-1/14 $31.00 © 2014 IEEE DOI 10.1109/DCOSS.2014.11
analyze the total power consumption of the network versus number of clusters. Both common positions of the BS are considered: the BS at the center and outside the sensing area. Based on that, we can obtain the optimal number of clusters that provides the minimum power consumption for our networks. In our simulation, we work on both random sparse signals (sparse in canonical basis) and real sensor readings. These real sensor readings are supposed to be dense signals but sparse in some domains such as DCT or Wavelet. We compare the combination between the measurement matrix and the sparsifying matrix to clarify the simulation results. The paper is organized as follows. We provide a brief overview of CS and related work in Section 2. We formulate the problem in Section 3 and show some simulation results in Section 4. Finally, conclusions and suggestions for future work are presented. II. BACKGROUND AND R ELATED WORK A. Brief Overview of CS 1) Sparse presentation of signals: Compressed sensing (CS) [2], [3] offers novel techniques to recover a compressible signal from its undersampled random projections, also called measurements. A signal x = [x1 x2 . . . xN ]T ∈ RN is defined to be k-sparse if it has a sparse representation in a proper basis ψ = [ψi,j ] ∈ RN ×N , where x = ψθ and θ has only k nonzero elements. Based on the CS paradigm, a k-sparse signal can be under-sampled and be recovered from only M ≪ N random measurements y = [y1 y2 . . . yM ]T ∈ RM . 2) Signal Sampling: : The random measurements are generated by y = ϕx, where ϕ = [φi,j ] ∈ RM ×N is called the measurement matrix and is often a dense Gaussian matrix or a sparse binary matrix [6]. The∑ith element in the measurement N vector y is formed by yi = j=1 φi,j xj . 3) Signal Recovery: : It has been shown that we can reconstruct a k-sparse signal with high probability from only M = O (k log N/k) CS measurements [5] employing the following l1 optimization problem [2] ˆθ = arg min || θ ||1 , subject to y = ϕψθ, (1) ∑n where || θ ||1 = i=1 |θi | and x b = ψ ˆθ. The l1 optimization problem can be solved with linear programming techniques such as Basis Pursuit (BP) [2].
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B. Related work Many different clustering algorithms have been studied so far [7], [8], [9], [10], [11]. Each cluster has a cluster head (CH) and CHs can be pre-determined [7] or be selected while doing clustering as in the following algorithms. K-means [12], [13], [14] is a very well-known and simple clustering algorithm that chooses CHs for K clusters at the central point of each cluster. This helps to minimize the intra-cluster power consumption. In general, CHs drain power much more than other sensors as they transmit an entire cluster’s data to the BS. In LEACH [11], sensor nodes randomly elect themselves to be CHs. This way, the high-energy dissipation in communicating with the BS will spread among the nodes in the network. The HEED algorithm [15] chooses CHs based on the highest residual energy of sensors to balance network energy. Faulttolerant clustering is mentioned in [16] in order to recover sensors in a failed cluster. The load-balancing clustering [17] makes the whole network consume energy equally and [18] discovered the optimal number of clusters to get the lowest energy consumption for WSNs. CS based routing methods have been shown to save consumed power significantly. Due to the correlation between the sensor readings in a WSN, the monitored signal can have a sparse representation in a proper domain such as DCT or wavelet. In [19], [20], [21], [22], CS is integrated into treebased routing. [23] and [24] exploited random walk routing applied CS with sparse binary measurement matrices. [25] worked on clustered networks and proposed a novel idea in which, all CHs receive raw data from their non-CH sensors and generate CS measurements and finally directly send them to the BS for recovery processes. The approach in [26] is to optimize the transportation cost for multi-hop WSNs using CS. It is mentioned in the paper that the measurements are randomly picked up from each cluster that make a block diagonal matrix (BDM) that we will analyze later. Our work is the first to investigate both the integration of the CS and clustering in WSNs and the combination between clusters in recovery processes. We analyze the sampling processes that create measurement matrices. The total power consumption is formulated. We also compare results between analysis and simulation, and eventually suggest the optimal number of clusters in WSNs. Based on that, the network consumes the least energy and lives longer. III. P ROBLEM F ORMULATION A. System Model We assume that N sensors have been uniformly randomly distributed in a square sensing area (L × L distance unit2 ). The network is divided into Nc non-overlapping clusters. In our simulations, we considered two well-known clustering mechanisms, LEACH [11] and K-means [13]. In our analysis, we assume a random clustering similar to LEACH, in which Nc out of N nodes in the network are selected uniformly at random as CHs and the other nodes find the closest CH to connect to. So, every node has same probability ( M N ) to be
CH that help balance the energy for the network. We assume that the clustering is uniform in analysis, then the number of sensors in each cluster is N/Nc . We also consider two cases for the location of the BS: the center and outside of sensing area. We also assume that sensors can adjust power level based on real transmitting distances. So, the consumed power for reaching a destination node j with distance dij from the node i is Pij = dα ij . Parameter α is usually between 2 and 4, depending on the characteristics of the channel [27]. For simplicity, we chose α = 2. For the reconstruction error related to CS signal recovery we considered the normalized x||2 reconstruction error ||x−b ||x||2 . B. Data Sampling and Measurement Matrices After clustering the networks as we have chosen LEACH [11] and K-means [13] as two clustering methods for our simulations, we deploy CS-based data collection following these steps: 1) Non-CH sensors multiply their data to random Gaussian coefficients and send the products to the CHs they belong to. 2) Each CH adds its own data to the received data and then sends the CS measurements directly to the BS. 3) The BS implements a CS reconstruction algorithm to find sensor readings x, given the measurement matrix ϕ and the collected measurements y = [y 1 , y 2 , . . . , y N ]. c The measurement matrix ϕ is created based on the way we collect measurements. In other words, all non-zero entries of ϕ represent the random Gaussian coefficients generated by the sensors which are sampled. Therefore, if all clusters send data to form every measurement, a dense full Gaussian sensing matrix is created. If the BS only collects each measurement from one cluster at a time, we are going to have a block diagonal matrix (BDM) as the sensing matrix as follows y1 ϕ1 x1 y x2 ϕ2 2 .. = .. (2) . .. . . yN xN ϕNc | {z } | {zc } | {zc } y: M ×1
ϕ: M ×N
x: N ×1
The restricted isometry property (RIP) of BDMs has been studied in [28], [29] and it has been shown that BDMs can satisfy RIP and therefore can be used as efficient measurement matrices. Some explanations and results of using these matrices are also addressed clearly in [25]. One more sampling case we propose is that we can uniformly randomly collect samples from W clusters (2 ≤ W < Nc ). The measurement matrix is neither a full Gaussian matrix (W = Nc ) nor sparse one as BDMs (W = 1). If a sleeping schedule is applied for sensors, only random W clusters active to generate and send measurements while the rest could be in standby. In order to compare which sampling method is the most energy-effective, we compare these matrices with
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reconstruction errors and then the total power consumption in the simulation section to decide the lowest consumed power case to apply. In a real WSN, each cluster may have a different number of sensors and corresponding different number of measurements required. It is mentioned in [25] that the number of measurements taken from a cluster should be linearly proportional to the number of sensors in the corresponding cluster to achieve the best CS recovery performance.
Sensor node
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C. Power Consumption Analysis As mentioned in section III-B, we first have non-CH sensors multiplying their own readings to Gaussian coefficients and then sending the products to the CHs they belong to. We refer to the communication cost associated with the communication between the non-CH nodes to CHs as the intracluster power consumption and is denoted as Pintra−cluster . Next, the CHs add all received data included their own data as the combinations of all reading data within each cluster, and send the measurements to the BS. The corresponding power consumption is referred to as Pto BS . The total power consumption is formed as Ptotal = (Pintra−cluster + Pto BS ).
(3)
1) Analysis of Pintra−cluster : We assume to have a uniformly distributed WSN divided into Nc non-overlapped clusters with the same number of sensors as N/Nc , consisting of one CH and ( NNc − 1) non-CH nodes. We have Pintra−cluster = Nc (
N M − 1) E[rα ] W, Nc Nc
(4)
where r is a random variable representing the distance of a non-CH sensor to its corresponding CH and α is the path loss exponent that we assume to be 2 throughout the paper. The expected value of r2 (E[r2 ]) can be calculated as follows [18]: E[r2 ] =
L2 , 2πNc
Pintra−cluster = (
N L2 M − 1) W. Nc 2πNc
(6)
As we see, the total intra-cluster power consumption is a decreasing function of the number of clusters Nc . 2) Analysis of Pto BS : Next, we need to find Pto BS , which is based on the distances between CHs and the BS and the total number of measurements to be transmitted from each CHs to the BS. We assume the BS is located at the location (Li , L2 ) with respect to our reference point (see figure 1). The average consumed power by all CHs is given by Pto BS = W M E[d2 ],
Fig. 1.
(7)
where d is the random variable representing the distance between CHs and BS. Assuming that all CHs are randomly
Li
A clustered WSN with BS outside the sensing area (Li > L).
distributed in the whole area to balance the energy consumption for the whole network, the expected squared distance between CHs and the BS is given [24] by (L − Li )3 + L3i L2 + . (8) 3L 12 The average power consumption related to the communication between the CHs and the BS is independent of the number of the clusters. Using (3), (6), (7), and (8), the total power consumption can be formulated as { } N L2 (L − Li )3 + L3i L2 Ptotal = ( − 1) + + WM Nc 2πNc 3L 12 (9) We usually have two common positions for the BS, at the center of the sensing area (Li = L/2) and outside the sensing 2 area (Li ≥ L). When BS is at the center, E[d2 ] = L6 and equation (9) is simplified as { } N L2 M M L2 Ptotal = ( − 1) + W. (10) Nc 2πNc 6 E[d2 ] =
(5)
and accordingly,
L
0
IV. S IMULATION R ESULTS In this section, we work with both random k-sparse signals (sparse in canonical basis, i.e, ψ is the identity matrix) and real sensor readings (which are sparse in DCT or wavelet bases). We create a random network with N = 500 and L = 100 according to the network model mentioned in Section III-A. We use K-means and LEACH clustering algorithms to arrange sensors into Nc clusters. Then, we apply our CS-based data collection method and calculate the total power consumption of the network for collecting M CS measurements required for reaching a target error rate of 0.1. The number of measurements from each cluster is proportional to the size of the cluster. We will provide our simulation results based on Kmeans and LEACH clustering as well as the analytical results derived in Section III-C. Figure 2 shows the histogram for the number of sensors in each cluster for both K-means and LEACH when Nc = 10. We can see that K-means algorithm generates clusters more
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Compare 3 projection matrices
uniform in size than LEACH, resulting in a lower expected intra-cluster power consumption. 4
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Fig. 4. Reconstruction errors obtained from 3 measurement matrices with 50-sparse signals divided into 10 clusters
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Fig. 2. Histogram of number of sensors in each cluster for K-means and LEACH.
As shown in figure 3, the uniform clustering method gains the smallest intra-cluster power consumption between three methods. 8
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Fig. 5. Compare the total power consumptions between different values of W = 1; 2; 10 at error-target = 0.1
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to reach the error-target: M = 162, M = 169 and M = 248 corresponding to W = 1, W = 2 and W = N c, respectively. So, figure 5 shows the total power consumption in the same network with three different ways of sampling while BS is at the center of the sensing area. The case (W = 1) where we collected measurements from each cluster consumes the least power. We apply this method in our next simulations.
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Fig. 3. Compare the total intra-cluster power consumptions between three clustering methods: Uniform, K-means and LEACH
We first want to decide based on simulation the value W mentioned in Section III-C. If W = Nc , ϕ is the full Gaussian matrix. If W = 1, ϕ is a BDM. Figure 4 shows that the combined measurement matrix with W = 2 works as good as W = Nc (full Gaussian). So we need to compare the total power consumption between three types of data collection methods. Based on figure 4, we chose the error-target = 0.1, the total number of measurements required
We created random signals with 50 non-zero elements (k = 50). Figure 6 shows the number of measurements needed to satisfy an error target set as 0.1. As mentioned in [25], the number of measurements required is a linear function of Nc . The projection matrix ϕ becomes sparser as the number of clusters is increased. This affects CS performance in that we need more measurements to compensate for the reconstruction error as shown in figure 6. With the number of measurements required, the total consumed power is calculated following each clustering method. Figures 7 and 8 show total power when the BS is at two positions outside the sensing area. The optimal numbers of
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50-sparse random signals; N = 500 300 8
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clusters for the network to consume the least power at Li = 2L and Li = 5L, respectively, are 7 and 4. This means that when BS is close to the sensing area, we need a larger number of clusters to reduce intra-power consumption. When the BS is far from the sensing area, the power to transmit measurements from CHs to the BS dominates the total power consumption.
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The real sensor readings we chose to use are collected from Sensorscope: Sensor Networks for Environmental Monitoring [30]. We use DCT as the sparsifying matrix. As previously mentioned, sparse BDMs do not affect CS performance if the DCT basis is used. So, the measurements shown in figure 9 can be considered as constant. The intra-power consumption is reduced as we deploy more clusters based on the smaller sizes of clusters as shown in figure 3. Therefore, in this case, we can increase the number of Nc to reduce total power consumption as shown in figure 10. Based on our analysis and discussion, we can conclude that under the given clustered scenario and assuming that the signal of interest is sparse in both Wavelet and DCT bases, employing
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the DCT will be more energy efficient. This is because when ψ is a DCT matrix, ϕ can get very sparse (by increasing Nc ) without a considerable loss in the CS performance. Our analytical and simulation results showed that in this case the consumed power is a decreasing function of Nc , and more clusters results in more power savings. V. C ONCLUSION AND FUTURE WORKS In this paper, we proposed and analyzed an energy-efficient data gathering method in clustered wireless sensor networks (WSNs) that is based on an integration of the clustering and compressive sensing (CS). We formulated the total power consumption for the networks and applied both common types of signals: sparse random signals and real sensor readings. We exploited the spatial correlation in natural signals by using DCT sparsifying basis. Based on that, whether we choose the sparsest BDMs as measurement matrices does not affect CS performance. With the basis, we do not have to increase the number of measurements required to satisfy the error-target if the number of clusters is increased. We also suggest the optimal number of clusters that leads to the lowest consumed power for WSNs deploying CS. Experiments with K-means and LEACH verify that the method works in both simulation and practice. In future works, we will exploit further the measurement matrices, BDMs and the combined clusters (W ≥ 2). We will also consider the power consumption of multi-hop inter-cluster communication in the networks for more energy saving. VI. ACKNOWLEDGEMENTS This work is supported by Vietnamese Ministry of Education and Training (MOET) and School of Electrical and Computer Engineering (ECE), Oklahoma State University. R EFERENCES [1] I. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “Wireless sensor networks: a survey,” Computer Networks, vol. 38, no. 4, pp. 393 – 422, 2002. [2] D.L.Donoho, “Compressed sensing,” Information Theory, IEEE Transactions on, vol. 52, pp. 1289 – 1306, 2006. [3] E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” Information Theory, IEEE Transactions on, vol. 52, pp. 489 – 509, Feb. 2006. [4] E. Candes and M. Wakin, “An introduction to compressive sampling,” Signal Processing Magazine, IEEE, vol. 25, no. 2, pp. 21–30, 2008. [5] J. Haupt, W. Bajwa, M. Rabbat, and R. Nowak, “Compressed sensing for networked data,” Signal Processing Magazine, IEEE, vol. 25, no. 2, pp. 92–101, 2008. [6] R. Berinde and P. Indyk, “Sparse recovery using sparse random matrices,” preprint, 2008. [7] A. A. Abbasi and M. Younis, “A survey on clustering algorithms for wireless sensor networks,” Computer Communications, vol. 30, no. 1415, pp. 2826 – 2841, 2007. [8] R. Xu and I. Wunsch, D., “Survey of clustering algorithms,” Neural Networks, IEEE Transactions on, vol. 16, pp. 645 –678, May 2005. [9] S. Bandyopadhyay and E. Coyle, “An energy efficient hierarchical clustering algorithm for wireless sensor networks,” in INFOCOM 2003. Twenty-Second Annual Joint Conference of the IEEE Computer and Communications. IEEE Societies, vol. 3, pp. 1713 – 1723 vol.3, March-3 April 2003.
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