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Cluster-Based Energy-Efficient Data Collection in. Wireless Sensor Networks utilizing Compressive. Sensing. Minh Tuan Nguyen and Nazanin Rahnavard.
2013 IEEE Military Communications Conference

Cluster-Based Energy-Efficient Data Collection in Wireless Sensor Networks utilizing Compressive Sensing Minh Tuan Nguyen and Nazanin Rahnavard School of Electrical and Computer Engineering Oklahoma State University Stillwater, OK 74078 Email: [tuanminh.nguyen, nazanin.rahnavard]@okstate.edu

Abstract—In this paper, an integration of compressive sensing (CS) and clustering in wireless sensor networks (WSNs) is proposed to significantly reduce the energy consumption related to data collection in such networks. Both compressive sensing (CS) and clustering have been proved to be efficient ways to reduce the energy consumptions in WSNs, however, there is little study about the integration of them for further gains. The idea is to partition a WSN into clusters, in which each cluster head collects the sensor readings within its cluster and forms CS measurements to be forwarded to the base station. The spatial correlation of the readings in a WSN results in an inherent sparsity of data in a proper basis such as discrete cosine transform (DCT) or Wavelet. This sparsity can then facilitate the application of the CS in data collection in WSNs. This way, we only need to forward M  N CS measurements from N sensor nodes. An important issue that needs to be considered for applying CS in the data collection problem is the underlying routing mechanism. Some related studies employ minimum spanning tree, random walk, or gossiping as the routing mechanism. However, we propose applying CS on top of a clustering algorithm to reduce the energy consumption. Under this novel framework, we study different clustering techniques and the properties of the block diagonal measurement matrix that is formed based on the clustering algorithm. We further formulate and analyze the total power consumption, based on that we can obtain the optimal number of clusters for reaching the minimum power consumption.

the BS. All raw reading data from N sensors will be recovered based on those measurements at the BS. The algorithm helps to reduce a significant energy consumption to transmit data from the network to the BS. Furthermore, we formulate the total power consumption for clustered WSNs that apply the algorithm. We 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 and real sensor readings. The 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 works in Section 2. We are going to formulate of our problem in Section 3 and show some simulation results in Section 4. The rest of this paper contains conclusion. II. BACKGROUND

AND

R ELATED WORK

A. Brief Overview of CS I. I NTRODUCTION

Compressed sensing (CS) [2], [3], [4], [5] 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 non-zero elements. Based on the CS paradigm, a k-sparse signal can be under-sampled and be recovered from only M L).

in which ρ(x, y) is the node distribution. To make the analysis tractable, similar to [19],√we assume each cluster area is a circle with radius R = L/ πNc and the density of the nodes is uniform throughout the cluster area, i.e. ρ(r , θ) = 1/(L2 /Nc ). We have [19]:  2π  R 1 L2 r3 dr dθ = . (8) E[r2 ] = 2 (L /Nc ) θ=0 r =0 2πNc and accordingly N L2 − 1) (9) Nc 2π As we see, the total intra-cluster power consumption is a decreasing function of the number of clusters. 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 M required 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 = M E[d2 ], (10) Pintra−cluster = (

D. Power Consumption Analysis As stated in the previous section, non-CH sensors send their readings to their own CH. We refer to the communication cost associated with the communication between the non-CH nodes to CHs as the intra-cluster power consumption and is denoted as Pintra−cluster . Next, the CHs create the CS measurements as the combinations of all reading data within each cluster (yi = φi xi ) 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 ).





where d is the random variable representing the distance between CHs and BS. Assuming that all CHs are randomly 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 by  L L L E[d2 ] = [(x − Li )2 + (y − )2 ]f (x, y)dxdy (11) 2 0 0 3 3 2 1 (L − Li ) L L = [ + i]+ , (12) L 3 3 12 in which f (x, y) = L12 (uniform distribution of CHs). From (10) and (12) we conclude that the average power consumption related to the communication between the CHs and the BS is independent of the number of the clusters. Using (4), (9), (10), and (12), the total power consumption can be formulated as

(4)

1) Analysis of Pintra−cluster : We assume to have a uniformly distributed WSN divided into Nc clusters with the same number of sensors as N/Nc , consisting of one CH and ( NNc −1) non-CH nodes. We have N Pintra−cluster = NC ( − 1) E[rα ], (5) Nc where r is a random variable representing the distance of a non-CH sensor to its corresponding CH and α is path loss exponent that we assume to be 2 throughout the paper. We can calculate the expected value of r2 (E[r2 ]) as following:   2 E[r ] = (x2 + y 2 ) ρ(x, y) dx dy (6)   = r2 ρ(r , θ) r dr dθ. (7)

N L2 M (L − Li )3 + L3i M L2 − 1) + [ ]+ (13) Nc 2π L 3 12 We usually have two common positions for the BS, at the center of the sensing area (Li = L/2) and outside the sensing area (Li ≥ L). For the former case, (13) is simplified as Ptotal = (

Ptotal = (

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N L2 M L2 − 1) + . Nc 2π 6

(14)



In this section, we work with both random k-sparse signals (sparse in canonical basis, i.e, ψ is identity matrix) and real sensor readings (which are sparse in DCT or wavelet bases). We create a random network with N = 2000 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 and calculate the total power consumption of the network for collecting M CS measurements required for reaching a target error rate 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 K-means and  LEACH clustering as well as the analytical results derived in Section III-D. Fig. 2 shows the histogram for number of sensors in each cluster for both K-means and LEACH when Nc = 10. As we see, K-means generates clusters more uniform in size, resulting in a lower expected intra-cluster power consumption. Next, we find the number of CS measurements required to



  

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Fig. 3. Number of measurements required to satisfy target error = 0.1 for a 100-sparse signal of length 2000 (sparse in canonical basis). 



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IV. S IMULATION R ESULTS











 

 

 













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Fig. 2. Histogram of number of sensors in each cluster for K-means and LEACH.

satisfy a target error for our network when it is clustered into different number of clusters (Nc = [1 2 ... 50]). Each method provides a different BDM as the measurement matrix. K-means and LEACH result in matrices with different size blocks equal to the number of sensors per each cluster. For the sake of comparison, we also consider the ideal case of clusters with equal size. We generate a BDM with equal size blocks and referred to it as CS-based uniform clustering. We choose a fixed target error in all our simulations as target error = 0.1. After finding the number of required measurements, we will find the power consumption for different choices for the location of the BS as mentioned before. We present our first simulation with random k-sparse signals. Then real sensor readings as actual temperatures from a WSN are considered.



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Fig. 4. Total power consumption (A) BS is at the center of the sensing area. Here, Nc∗ = 14. (B) BS is outside the sensing area with Li = L. Here, Nc∗ = 9.

the other hand, Pintra−cluster is a decreasing function of Nc . Therefore, there is an optimal Nc∗ , for which the total power consumption is minimized. Figure 4A depicts Ptotal when the BS is at the center. In this case, we have Nc∗ = 14. Figures 4B depicts Ptotal when BS outside the sensing area at a different location. The minimum power consumption happens in this case at Nc∗ = 9. The optimal number of clusters reduces when the BS is far from the sensing area. In such cases, Pto BS will be a dominating factor in Ptotal and Nc∗ becomes smaller.

A. x as a random k-sparse vector In this example, we consider x to be sparse in the canonical basis. We create a 100-sparse vector X with length N = 2000. The measurement matrix is a M × N BDM, where M is the number of measurements required to satisfy the target error = 0.1. We obtain the number of required measurements for three clustering algorithms as shown in Fig. 3. In Fig. 3, increasing Nc leads to a degradation in the CS performance and a linear increase in the number of required measurements (as also discussed in SectionIII-B). This increases Pto BS . On

B. x as real sensor readings We use real sensor readings from Sensorscope: Sensor Networks for Environmental Monitoring [26]. x is dense in

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Nc∗ = 12, respectively. As we make the BS farther from the sensing area, Pto BS become a more dominating factor in Ptotal and this leads to a decrease in Nc∗ . 2) DCT as the sparsifying basis: In this case we employ DCT as the sparsifying basis. As discussed in Section III-B, DCT is incoherent with the canonical basis and the CS performance does not degrade with increasing Nc . This can be seen in our simulation results shown in Figure 7. The number of required measurements to reach a target reconstruction error is almost constant versus changing Nc . Lets call this constant value as M0 . Given that M does not change with Nc , we can

the canonical domain. In order to apply CS, as mentioned in the background section, we need a sparsifying basis. Next, we will study the utilization of both DCT and Wavelet bases. 1) Wavelet as the sparsifying basis: This case is similar to the case discussed in section IV-A in a sense that the Wavelet basis also has a large coherence μ. As discussed in Section III-B, this causes a linear increase in the number of required measurements versus Nc . Our simulation results in Fig. 5 depicts this fact. Similarly, there will be an optimal Nc∗ , for which the total power consumption is minimized. Figures 6A, 4    , *) %* 567)  6

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see from (13) that Ptotal is a decreasing function of Nc . This is also shown in Figure 8A and 8B for the BS being at the center and Li = 3L, respectively. On the other hand, since we are collecting M measurements from the networks, we have Nc ≤ M . Therefore, Nc∗ = M and the smallest size of each cluster in average is N/M sensors. 3) Remarks on the effect of the sparsifying basis on the performance: Based on our discussions, 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 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.



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Fig. 7. Number of measurements required when DCT is considered as the sparsifying basis.



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Fig. 5. Number of measurements required when Wavelet is considered as the sparsifying basis.



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V. C ONCLUSION 

In this paper we proposed an energy-efficient data collection in wireless sensor networks (WSNs) that is based on an integration of the clustering and compressive sensing (CS). It is well known that natural signals have spatial correlation and therefore the sensor readings in a WSN are sparse in a proper basis such as DCT or wavelet. This sparsity facilitates the utilization of CS for energy-efficient data collection in WSNs. As opposed to other contributions that mostly consider spanning trees or random walks as the underlying routing method, in this paper we introduced the integration of the CS with clustering to benefit from the power saving offered by the





 













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Fig. 6. Total power consumption (A) BS is at the center of the sensing area. Here, Nc∗ = 18. (B) BS is outside the sensing area with Li = L. Here, Nc∗ = 12.

B depict Ptotal when the BS is at the center and Li = L, respectively. The optimal number of clusters are Nc∗ = 18,

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[5] R. Baraniuk, “Compressive sensing [lecture notes],” Signal Processing Magazine, IEEE, vol. 24, pp. 118 –121, July 2007. [6] R. Berinde and P. Indyk, “Sparse recovery using sparse random matrices,” 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. [10] S. Heikalabad, N. Firouz, A. Navin, and M. Mirnia, “Heech: Hybrid energy effective clustering hierarchical protocol for lifetime prolonging in wireless sensor networks,” in Computational Intelligence and Communication Networks (CICN), 2010 International Conference on, pp. 325 –328, Nov. 2010. [11] M. Handy, M. Haase, and D. Timmermann, “Low energy adaptive clustering hierarchy with deterministic cluster-head selection,” in Mobile and Wireless Communications Network, 2002. 4th International Workshop on, pp. 368 – 372, 2002. [12] H. Steinhaus, “Sur la division des corp materiels en parties,” Bull. Acad. Polon. Sci, vol. 1, pp. 801–804, 1956. [13] J. B. MacQueen, “Some methods for classification and analysis of multivariate observations,” in Proc. of the fifth Berkeley Symposium on Mathematical Statistics and Probability (L. M. L. Cam and J. Neyman, eds.), vol. 1, pp. 281–297, University of California Press, 1967. [14] S. P. Lloyd, “Least squares quantization in pcm,” IEEE Transactions on Information Theory, vol. 28, pp. 129–137, 1982. [15] O. Younis and S. Fahmy, “Distributed clustering in ad-hoc sensor networks: a hybrid, energy-efficient approach,” in INFOCOM 2004. Twenty-third AnnualJoint Conference of the IEEE Computer and Communications Societies, vol. 1, pp. 4 vol. (xxxv+2866), March 2004. [16] C. Li, M. Ye, G. Chen, and J. Wu, “An energy-efficient unequal clustering mechanism for wireless sensor networks,” in Mobile Adhoc and Sensor Systems Conference, 2005. IEEE International Conference on, pp. 8 pp. –604, Nov. 2005. [17] G. Gupta and M. Younis, “Fault-tolerant clustering of wireless sensor networks,” in Wireless Communications and Networking, 2003. WCNC 2003. 2003 IEEE, vol. 3, pp. 1579 –1584 vol.3, March 2003. [18] A. Amis and R. Prakash, “Load-balancing clusters in wireless ad hoc networks,” in Application-Specific Systems and Software Engineering Technology, 2000. Proceedings. 3rd IEEE Symposium on, pp. 25 –32, 2000. [19] W. Heinzelman, A. Chandrakasan, and H. Balakrishnan, “An application-specific protocol architecture for wireless microsensor networks,” Wireless Communications, IEEE Transactions on, vol. 1, pp. 660 – 670, Oct 2002. [20] C. Luo, F. Wu, J. Sun, and C. W. Chen, “Efficient measurement generation and pervasive sparsity for compressive data gathering,” Wireless Communications, IEEE Transactions on, vol. 9, no. 12, pp. 3728–3738, December. [21] J. Luo, L. Xiang, and C. Rosenberg, “Does compressed sensing improve the throughput of wireless sensor networks?,” in Communications (ICC), 2010 IEEE International Conference on, pp. 1–6, May. [22] R. Xie and X. Jia, “Minimum transmission data gathering trees for compressive sensing in wireless sensor networks,” in Global Telecommunications Conference (GLOBECOM 2011), 2011 IEEE, pp. 1–5, Dec. [23] S. Lee and A. Ortega, “Joint optimization of transport cost and reconstruction for spatially-localized compressed sensing in multi-hop sensor networks,” Asia Pacific Signal and Info. Proc. Assoc. Summit (APSIPA). Singapore, 2010. [24] H. L. Yap, A. Eftekhari, M. Wakin, and C. Rozell, “The restricted isometry property for block diagonal matrices,” in Information Sciences and Systems (CISS), 2011 45th Annual Conference on, pp. 1 –6, March 2011. [25] M. Wakin, J. Y. Park, H. L. Yap, and C. Rozell, “Concentration of measure for block diagonal measurement matrices,” in Acoustics Speech and Signal Processing (ICASSP), 2010 IEEE International Conference on, pp. 3614 –3617, March 2010. [26] http://lcav.epfl.ch/op/edit/sensorscope en.



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Fig. 8. (A): Total power consumption when the BS at the center of the sensing area. (B): Total power consumption when Li = 3L.

two techniques. We refer to our scheme as clustered-base CS (CCS). The resulting CS measurement matrices in CCS is in the form of block diagonal matrices (BDMs). We formulated the total power consumption and discussed the effect of different sparsifying bases on the CS performance as well as the optimal number of clusters for reaching the minimum power consumption. We employed K-means, LEACH, and uniform clustering techniques in our simulations and found the optimal cluster sized when the signal of interest in sparse in canonical, wavelet, and DCT bases. We showed that the integration of the CS and clustering provides a significant energy savings for data collection purposes in WSNs. VI. ACKNOWLEDGEMENTS This work is supported by National Science Foundation under Grants ECCS-1056065 and CCF-0915994, MOETVietnam and 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] N. Rahnavard, A. Talari, and B. Shahrasbi, “Non-uniform compressive sensing,” in Communication, Control, and Computing (Allerton), 2011 49th Annual Allerton Conference on, pp. 212 –219, Sept. 2011.

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