Data Aggregation and Recovery in Wireless Sensor ...

Data Aggregation and Recovery in Wireless Sensor Networks Using Compressed Sensing Guangming Cao Department of Integrated Electronics, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China 518055; The Chinese University of Hong Kong, Hong Kong; Graduate University of Chinese Academy of Sciences, Beijing, China 100049. E-mail: [email protected]

Peter Jung, Slawomir Sta´ nczak TU Berlin Einsteinufer 25, 10587 Berlin, Germany

E-mail:

Fengqi Yu Department of Integrated Electronics, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China 518055 E-mail: Abstract: QoS support for data aggregation in large–scale multi–hop wireless sensor networks (WSNs) inevitably faces two crucial but hard issues: packet loss and energy dissipation. But fortunately, most sensing data is spatially and temporally correlated and compressible. Therefore, compressed sensing (CS), as an emerging area in the field of sparse recovery, is a promising reconstruction scheme having the potential of packet error correction with low–energy consumption. In this paper we present such a CS– oriented data aggregation technique for the multi–hop topology. Our scheme is balanced in energy consumption among the nodes and recovers lost packets at fusion center without additional transmitting costs. Simulations show that our approach offers accurate recovery with high probability and works well even for 50% data loss rate when environmental data is sparse in a certain domain. Comparing with the existing methods, we achieve with our method a higher recovery accuracy and less energy consumption on TinyOS. Furthermore, the system is demonstrated in the experiment of monitoring grid computer facilities set up at Shenzhen Institutes of Advanced Technology. Keywords: large–scale wireless sensor networks; compressed sensing; packet loss; energy balance. Reference to this paper should be made as follows: Cao, G., Jung, P., Sta´ nczak, S. and Yu, F. (xxxx) ‘Data Aggregation and Recovery in Wireless Sensor Networks Using Compressed Sensing’, International Journal of Sensor Networks, Vol. x, No. x, pp.xxx–xxx. Biographical notes: Guangming Cao received the BEng degree from Huazhong University of Science and Technology, Wuhan, China, in 2007. Then he began his PhD studying in the Department of Integrated Electronics at Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China. He was also a visiting student at Fraunhofer Heinrich Hertz Institute (HHI), Berlin, Germany, from 2011 to 2012. His current research interests are in wireless sensor networks and compressed sensing. Peter Jung, Member IEEE, Member VDE/ITG, received the Dipl.-Phys. in high energy physics in 2000 from Humboldt University, Berlin, Germany, in cooperation with DESY Hamburg. He received the Dr.-rer.nat (Ph.D.) degree in 2007 (on Weyl–Heisenberg representations in communication theory) at the Technical University of Berlin (TUB), Germany. P. Jung is currently working under DFG grants JU 27951&2 at TUB. His current research interests are in the area compressed sensing, time–frequency analysis, dimension reduction and randomized algorithms. Dr.SlawomirStanczak studied control systems engineering at the Wroclaw University of Technology, Poland, and at the Technical University of Berlin (TU Berlin), Germany. He received his Dipl.-Ing.degree and Dr.-Ing. degree with distinction (summa cum laude) in Electrical Engineering from the TUBerlin in 1998 and 2003, respectively. Dr.Stanczak has been involved in research and development activities in wireless communications since 1997. Since 2003 he leads a research group at the Fraunhofer Heinrich Hertz Institute and since 2010 Dr.Stanczak has been the acting director of the Heinrich-Hertz-Lehrstuhl at the TU Berlin. Fengqi Yu was born in China and earned his Ph.D. degree in Integrated Circuits and Systems Lab (ICSL) at the University of California, Los Angles (UCLA). In 2006, he jointed Shenzhen Institutes of Advanced Technology (SIAT) in China as a full professor and director of the Integrated Electronics Department. Before joining SIAT, he worked at Rockwell Science (United States), Intel (United States), Teradyne (United States), Valence Semiconductor (United States), and Suzhou CAS IC Design Center (China). His R&D interests include CMOS RF integrated circuit design, CMOS sensor design, wireless sensor networks, RFID, and wireless communications.

Int. J. Sensor Networks, Vol. x, No. x, 2013

1 Introduction Energy–efficient QoS (quality of service) support for data aggregation is an important issue for large–scale wireless sensor networks (WSNs). This kind of QoS covers several aspects, such as secure data aggregation, which preserves data confidentiality and integrity in security sensitive cases [WDSX07, OX11, OX09]. However, under resource constrains, packet loss rate is one of the most fundamental parameters [Xia08] to satisfy the service requirements of different applications. Generally, packet loss and energy dissipation are two crucial and inevitable problems for large–scale wireless sensor networks. Sensor nodes, with salient feature of multi-hop communication links, are deployed to surveil environmental properties of some interesting area, such as forest fire monitoring. Unlike communication networks intending to connect mutual nodes, large scale WSNs aim to gather sensing data from some interesting field which make the data stream mainly unidirectionally pour to the fusion center (the sink). A conventional scheme here is the tree routing topology, and data is streamed in “sense–and–send” and “receive– and–forward” scenario. Since this gathering approach is simple, easy to implement by network protocols and without special hardware requirements, it is therefore universally used in several kinds of applications. However, the weakness is obvious and natural. Every node not only generates self–data but also forwards network packets. If we imagine each node be elevated to a height proportional to its load, the whole traffic load field would look like a pyramid: nodes closer to the sink have heavier load while those nodes farther away in the tree topology have smaller load. This directly causes two problems: (i) inefficient communications near sink in sense of congestion and (ii) more importantly, the nodes closer to sink unbalancedly consuming more energy and failing down earlier. At the same time, besides single–hop packet loss rate due to external or internal causes like channel interference or exhausted energy in large–scale WSNs is serious enough to not be ignored, i.e. 3% in our experiments, end–to–end packet loss rate, from the sampling node to the sink, is considerable, even serious enough to the order of 50% in long term, and difficult to improve without cost. Paradoxically, if saving energy is not a problem anymore, for example, sensor nodes can obtain external energy, packet loss rate becomes much better, as shown in 3.2. To improve this situation, the sparse structure of environmental parameters should be used already in the design of the network protocol and for the data recovery at the fusion center. Therefore, compressed sensing (CS), as an emerging area in the field of sparse recovery, is well suited for such large–scale WSNs. The source compression at the sensor node performs much more inexpensively than conventional encoding and this is suitable for WSN hardware configurations. Based on c 2008 Inderscience Enterprises Ltd. Copyright ⃝ c 2009 Inderscience Enterprises Ltd. Copyright ⃝

2 the theory, as long as the sensor readings are sparse in some basis, such as frequency or wavelet domain, it can be reconstructed through a small number of measurements with high precision. More importantly, due to the linearity in such type of compressed encoding the sensor nodes can always transmit a globally fixed number of packets hop by hop regardless of the number of nodes which naturally avoids the “pyramid–shape”. Meanwhile, CS erasure coding (CSEC) [CCZ+ 10] firstly addresses the problem of packet loss in WSNs through compressed sensing, which offers accurate recovery of sensing data in the case of packet loss rates up to 20% by increasing the compressed dimension by the same percentage. Due to their mechanism, processing more serious loss rate inevitably sacrifices the precious capacity and energy of sensor networks. However, from applications, maximum end–to–end packet loss rate of 20% can’t be of sufficiency, and multi–hop scheme is not covered explicitly. Although combining these areas of sensing networks and CS is among the first set of issues in compressed sensing applications [BHSN06], it is still a challenge to implement this theoretical methods in a network– oriented fashion and to improve packet loss tolerance in multi–hop topology. In this paper, we present a direct CS–oriented multi–hop node protocol for the tree topology which also operates at a fixed compression dimension, but with a new methodology of handling packet errors. Here, lost packets are recovered at the fusion center without additional transmitting costs which also comes with a better balancing in energy consumption. We consider peer–to–peer and end–to–end packet loss rates and compare greedy–based and convex CS–oriented recovery methods with conventional data aggregation and with CSEC reception.

2 Related work Confronted by the inevitable packet loss during data aggregation in WSNs, several solutions have been addressed recently. Reference [OX09] gives a comprehensive overview of data aggregation, where the routing protocols are widely used to improve aggregation efficiency, like preventing data loss. Joint source-channel coding has also been used for data aggregation. For example, in [Cam07], Multiple-Input Turbo (MIT) code was introduced to improve turbo code in bit error rate by exploiting the sampling data correlation in dense wireless sensor networks. Protocol and coding are the main methods to mitigate data loss over wireless communications. In network layer protocols, retransmission schemes, such as Automatic Repeat reQuest (ARQ), are popularly applied due to its simplicity [ZG03]. In reference [NK12], combined ARQ with optimal transmission power and varied packet size achieved a guaranteed connectivity and efficient energy consumption.

Data Aggregation and Recovery in Wireless Sensor Networks Using Compressed Sensing However, retransmissions are not energy efficient in multi–hop applications, especially in mutable wireless channels. Moreover, retransmissions do not work well in delay sensitive applications too, like emergency detection. Alternatively, forward error correction (FEC) is better suitable to reduce the additional retransmission cost and overhead. Reference [SO06] uses FEC, together with a coarse-grained synchronization, called cascading timers, to handle packet loss and improve energy efficiency. However, the usefulness of coding in sensor network applications is limited by greater decoding complexity or bandwidth overhead [RO12]. Reference [SBW09] used a modified turbo code on a MicaZ platform and showed a practical computational complexity, while they only considered the single–hop case and the performance varied strongly depend on channel quality. The improvement of error resiliency in the multi–hop scenario was exploited in [VA09] by sending redundant bits with lightweighted and simple FEC, and energy consumption and end–to–end latency performed better comparing with ARQ. But it based on channel–aware and geographical routing protocols which made the networks much too complicated. Fortunately, recent developments in compressed sensing theory have provided a wide range of interest in in–network data compression. Reference [BHSN06] discusses the concept of CS in wireless network scenarios, which is called Compressive Wireless Sensing (CWS). CWS reduces latency of information retrieval and improve power–distortion trade–off with little prior knowledge about the sensing data, assuming that the wireless sensor networks are one–hop star–topology network with analog transceivers within zero–mean additive Gaussian communication noise. While the setup in [BHSN06] allows direct application of the compressive sensing theory, it does not reflect the structure of current self–organized multi–hop energy constrained sensor networks. Compressive data gathering has been introduced in [LWSC09] which uses compressed sensing in multi–hop large–scale WSNs. An important advantage of this approach is that the sensor nodes always transmit a fixed number of packets regardless of the number of nodes, which means that it operates at a fixed compressed dimension. It achieves the network capacity gain in order of compression rate, while energy influence and data loss tolerance are not covered. Reference [CCZ+ 10] addresses the problem of packet loss in WSNs. The proposed approach called CS erasure coding (CSEC) offers accurate recovery of sensing data through compressive sensing. Numerical results show that the proposed method recovers the data with high probability even in the case of packet loss rates up to 20%. However, accurate recovery is achieved here by increasing the compressed dimension, just increasing by 20% for an expected packet loss rate of 20%, which does not utilize the full potential of CS based recovery. Meanwhile, along with large scale and multi–hop characters, end– to–end packet loss rate could easily exceed 20% due to

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constrained energy strategy, which will be detailed in the following section. Moreover, compensation for packet loss by increasing the compressed dimension by the same amount causes at least the same additional scale in energy consumption straightforwardly. On the other hand, energy factor is an important reason to adopt CS in WSNs. Our approach is to recover the packets which are lost during transmission at sink by compressed sensing due to its inherent environmental sparsity and without conveying much additional redundant data. The paper is organized as follows: Section 3 establishes the transmission pattern in wireless sensor networks including a model to capture packet losses. Then we illustrate and discuss the exemplary packet loss situation in a particular running application. After that, in Section 4 we introduce the compressed sensing background and verify in our example the sparsity assumption of the sensor data. Then we present our proposal for reconstructing missing data based on compressed sensing. In Section 5 we discuss our implementation setup and evaluate the method through simulations by Matlab and on the practical level by TinyOS. Finally, we complete the paper by some concluding remarks.

3 System Model The demand for compensating packet loss is obvious because such large scale sensor networks are typically composed of a number of sensors and transmitting mass amount of measurement data through multiple hops to the sink, where packet loss could happen in any hop. Taking forest fire monitoring for example, sensor nodes measure parameters of the environment such as temperature, smoke and illumination and send or relay this data to the central controller. The main data stream is unidirectional aggregation having tree topology. Fig.1a shows a simple schematic diagram of forest fire monitoring by wireless sensor networks. Peer–to–peer communication may break due to several reasons, such as for example fragile and volatile channels. Moreover, end–to–end fault would accumulate through routing links. To construct our system model, we first briefly establish the data aggregation pattern in WSNs. We then discuss the packet loss model and illustrate the packet loss situation in a running application, which is followed by our approach in the next section.

3.1 Data Aggregation Under the condition that the main data stream of large– scale WSNs is unidirectional aggregation having tree topology, let’s illustrate the data aggregation process. Obviously, only a fraction of whole routing topology shown in Fig. 1b can be discussed here. The network self–organizes data gathering, tree topology routing and collects data at the sink node.

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Figure 2: Linear topology of wireless sensor networks.

(a)

(b)

Figure 1: Large scale wireless sensor network architecture: (a) application diagram for data aggregation; (b) a fraction of the tree topology. To establish a simplified system model and present our approach, we first briefly discuss the conventional data aggregation in wireless sensor networks. There are N sensor nodes in a network. The network samples data periodically and stores the values in the N -dimensional data vector d ∈ RN . The data vector d is sent through the network to the sink node with or without encoding, and finally the sink receives the vector y ∈ RM where the dimension M depends on the aggregation strategy. Let us abbreviate this aggregation process by y = Enc(d). Generic data aggregation: In generic data aggregation, we look locally at an arbitrary sensor node k and its one–hop child nodes set Lk ⊆ [1, 2...N ]\{k}. Node k now generates the Mk –dimensional vector Sk ∈ RMk Sk = Enck (dk , Rk )

(1)

based on its own data dk ∈ R (scalar for simplicity) and the vector Rk ∈ RKk received from its child nodes. The individual encoding process at node k is again denoted with Enck . After N hops the sink is reached, i.e. for k = N we write y = SN = EncN (dN , RN ). Linear encoding in WSNs: In this case the function Enck is represented by an encoding matrix Ek ∈ RMk ×(1+Kk ) and (1) gives: [ ] dk Sk = E k . (2) Rk Assume that the N sensor nodes are arranged in B branches with each branch consists of H hops, i.e. N = BH. For simplicity we assume also that indices of nodes are arranged such that Bb = [bH + 1, . . . , (b + 1)H]. In the conventional way of data aggregation by tree routing, data is streamed in “sense–and–send” and “receive–and–forward” scenario. Thus, in the case of error–free communication the gathered data y ∈ RM at the sink is then simply equal to the sensing data d ∈ RN by the N nodes, y = [d1 , ...dH , dH+1 , ...dBH ]T = d. Correspondingly, the encoding matrix Ek equals the identity and Sk ∈ RMk reads as: [ ] dk Sk = . (3) Rk

Since the output dimension Mk increases linearly hop by hop and finally the sink receives M = N packets containing the primitive data y = d from the N nodes. Because this gathering approach is simple, easy to implement by network protocols and without special hardware requirements, it is therefore universally used in several kinds of applications. Packet–loss model in WSNs: In order to describe the transmission chain with packet loss more concisely, without losing much generality, we consider at first a more simple network structure having a linear topology which could be a certain branch from a tree topology, as shown in Fig. 2. Network structures like tree topology can be expressed in a similar form, shown in the following section. In this topology, node k has only one child node Lk = {k − 1}. Therefore Rk = Sk−1 , Mk = k and for linear topology data aggregation (3) would be like this: [ ] dk Sk = . (4) Sk−1 Let us now model the effect of packet loss. We assume that the probability of packet loss in every hop is independent and identically distributed (i.i.d.). Hence, let qk be a Mk –dimensional random vector representing the error pattern at node k. Its jth component qjk for j ∈ [1 . . . Mk ] equals 1 means successful reception of the jth packet of node k while 0 means that the packet is lost. We assume a success probability p, i.e. Pr{qjk = 1} = p.

(5)

Furthermore, we denote with Qk the Mk × Mk – dimensional random error matrix with elements ∏k−1 (Qk )ij := l=j qil such that linear encoding and packet loss can now be written as Qk ⊙ Ek where ⊙ denotes pointwise (Hadamard) product. Under consideration of packets loss, received data Rk at node k is changed to Rk = qk−1 ⊙ Sk−1 , so (3) becomes: [ ] dk (Sk ) = , (6) qk−1 ⊙ Sk−1 which is equivalent to: { dk (Sk )n = ∏k−1 ( l=n qln )dn

n=k n = 1, 2, ...k − 1.

(7)

Finally, the sink will observe y = SN .

3.2 Packet Loss in Wireless Sensor Networks Before we present our approach in Section 4, we first show how serious the packet loss is in real applications here. Generally, packet loss can be caused

Data Aggregation and Recovery in Wireless Sensor Networks Using Compressed Sensing sensor node id

transmission Ntx

received Nrx

success rate (%) rmulti

3 4 5 6 7

17625 12972 10771 13013 13397

22578 22578 22578 22578 22578

78.06 57.45 47.71 57.64 59.33

Table 2

Figure 3: High Performance Computer (HPC) building monitoring system, detecting temperature, humidity, power grid status, cooling device status etc. group id indoor 1 2 3 outdoor 1 2 3 Table 1

transmission packets number Ntx

received Nrx

success rate(%) rsingle

1924 8161 22370

1879 7930 21687

97.66 97.17 96.95

2937 8849 11841

2858 8646 11574

97.31 97.71 97.75

by a number of factors, including channel congestion, signal degradation over wireless media due to multi– path fading, broken routing chains, hardware fault resulting from out of battery. The application we consider here has been developed to monitor High Performance Computer (HPC) room, i.e. monitoring various operating conditions to ensure HPC working is safe. Fig.3 shows the basic outline of this system. It’s a typical in-door environment with low power consumption. We discuss packet loss rate in two aspects, point-to-point and end-to-end respectively. Single hop packet loss rate: We have tested the single hop packet loss rate in three groups. Each group consists of two sensor nodes, which are self-designed and compatible with TinyOS, one sensor transmits and the other receives. The transmitting power is 10dBm. Test fields are in-door and out-door separately, in distance of 20m and line-of-sight propagation. Results in Table 1 show that the single hop success rate is on average 97%, where the out-door case is a little better than indoor. In this scenario, there are no collisions on the MAC layer since each node has an exclusive channel without interfering with others nodes. Furthermore, switching between different energy saving modes is disabled. Multi-hop packet loss rate: Packet loss in end-toend multi-hop network is more fragile than in pointto-point transmission. For the system of HPC building monitoring, our statistics of received packets is 189 days, including a 32 days pause for maintenance. We take the temperature sensor nodes for example. Every node sends

End-to-end packet loss rate in energy constrained network.

sensor node id

transmission Ntx

received Nrx

success rate (%) rmulti

1 2 4 6 8 9 11

7721 7275 7715 7718 6737 7695 7729

9112 9094 9108 9099 8111 9098 9098

84.73 80.00 84.71 84.82 83.06 84.58 84.95

Table 3

Point-to-point packet loss rate (statistic in packets numbers).

5

End-to-end packet loss rate in energy non-constrained network.

one packet per 10 minutes, so Ntotal = 16578 packets should be sent in total. We count the number of received packets Nrx and compute the success rate by Nrx /Ntotal . Table 2 shows the packet success rates rmulti which are on the average at 50%. Several reasons cause that the result is worse than we expect theoretically. For example, our software based on TinyOS didn’t tackle the mode switching smoothly between deep sleep and wake up which is for energy saving. Another reason is that the routing protocol wasn’t suitable enough to cope with the volatile channels presented in this particular HPC scenario. Summarizing, in the multi–hop case it is much more serious and difficult to ensure a sufficient performance than in single–hop transmission. Now, if we don’t turn down radio power and do not switch nodes such frequently into sleep mode, the packet loss rate is better as shown in Table 3. The results are from a similar application monitoring power consumption at the same building. The main difference is that the current network has access to energy from the environment, i.e. the sensor nodes are directly powered by electric grid rather than batteries. The average success rate is now around 83%. Taking into account that the average number of hops of this network is 5, the end–to–end success rate is consistent with the single-hop rate rsingle , as (rsingle )5 = (0.97)5 = 0.858 ≈ rmulti . The motivation for our new approach is to achieve such rates also in the case energy–constraint networks.

4 Proposed solution Recently, compressed sensing [CRT06a, Don06], has attracted more and more attention in a number of areas,

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such as applied mathematics and computer science. As a novel sampling paradigm, CS based algorithms can recover signals from far fewer measurements than what the sensor nodes have acquired, if those signals can be sparsely represented in a proper basis. In this case compressed sensing is therefore more efficient than traditional sampling. It has also the advantage that encoding is simplified and complexity is transferred to decoding end. This is beneficial for several kinds of emerging applications because generally the terminals are distributed and cost-sense while decoders are centrally equipped. Particularly, in WSN scenarios, the sampling process in CS is at the side of sensor nodes and the reconstruction is done at sink.

4.1 Compressed Sensing Background N Let the vector of sensing data d = (di )N i=1 = Ψx ∈ R being sparse in a certain orthonormal basis Ψ (a unitary matrix). Thus, x has very few non-zero elements, i.e. ∥x∥ℓ0 := |{i : xi ̸= 0}| ≪ N (x is sparse in the canonical basis). The vector of observations y is obtained through a measurement represented by the measurement matrix Φ ∈ RM ×N :

y = Φd.

(8)

Without sparsity, M ≥ N observations are necessary for the reconstruction of arbitrary d or x. But in the case of s–sparse x, meaning that ∥x∥ℓ0 ≤ s, it is known that M = O(s log N ) measurements are sufficient. Then, solving the under–determined problem for the sparsest solution can be written as follows: min ∥z∥ℓ0

z∈RN

s.t. y = Az.

(9)

with A = ΦΨ and y = Ax. If any 2s columns of Φ are linear independent x is uniquely determined by y, i.e. M = 2s measurements are sufficient. However, proving this condition is as hard as solving (9) which is known to be NP complete. ∑N Let us define the ℓq -norms ∥x∥ℓq := ( i=1 |xi |q )1/q for 1 ≤ q < ∞. One of the main insights of sparse optimization and compressed sensing is that under certain conditions on A (on Φ) the solution to (9) can be obtained by solving instead the convex program where the ℓ0 –term is replaced by ℓ1 –norm. In the noiseless case, which is considered here, this can formulated as a linear program. The feasibility of this relaxation relies on several characterizations of the null space of A (see here for example also [DDEK11]). More well–known is the stronger condition of the restricted isometry property (RIP) introduced in [CT05]. The matrix A satisfies the RIP of order s if there exists δs ∈ (0, 1) such that: (1 − δs )||x||2ℓ2 ≤ ||Ax||2ℓ2 ≤ (1 − δs )||x||2ℓ2

(10)

holds for all s-sparse vectors x. For example, it is known [CRT06b, Can08] that the basis pursuit: min ∥z∥ℓ1

z∈RN

s.t. y = Az

(11)

exactly recovers any s–sparse x from its observation y = Ax √ if the matrix A satisfies RIP of order 2s with δ2s ≤ 2 − 1. Even more, in the non–sparse setting the solution x ˆ to (11) is in the order of sparsest approximation xs (a vector which contains the largest s values of x) to x [Can08], which means that ∥ˆ x − x∥ℓ1 ≤ C0 ∥x − xs ∥ℓ1 and ∥ˆ x − x∥ℓ2 ≤ C0 s−1/2 ∥x − xs ∥ℓ1 for some constant C0 . However, it is of general interest how many measurements are needed to reconstruct x (and therefore d) with small probability of error. Small δs can be achieved with random matrices with certain sub–Gaussian rows or columns, details see [Ver10]. For example, matrices drawn from the Gaussian distribution Φij = N (0, 1/M ) or an equal probable Bernoulli distribution Φij = ± √1M have for M = O(s log(N/s)) the RIP property with overwhelming probability [BDDW08]. This scaling is almost optimal in this uniform setting, since from [FPRU10] it is known that M ≥ c1 s ln cN2 s measurements are also necessary to recover any 2s-sparse x with (11) without error, where c1 = 1/ ln 9 ≈ 0.455 and c2 = 4.

4.2 Data Aggregation using Compressed Sensing In the previous section we have mentioned that if the sensor data d is sparse in a basis Ψ much less samples are necessary for its reconstruction when using methods from compressed sensing. Motivated by this fact we will now establish an aggregation method based on these ideas. The sparsity assumption for our particular example will be verified in Section 4.3 below. New aggregation scheme: In the wireless sensor network application, the vector y = Φd ∈ RM is observed at the sink where Φ is the M × N –dimensional measurement matrix and d ∈ RN is the data vector of sensor readings of the N nodes. Comparing with conventional gathering, aggregating by compressed sensing is a little different. The data transmitted by node k is now always a M –dimensional vector: [ ] dk Sk = [ϕk I] , (12) Rk where ϕk is kth column of Φ and I ∈ RM ×M is a corresponding identity. Hence, the generic encoding matrix Ek in (2) is here [ϕk I], i.e. meaning that Kk = M . Up to this point such an aggregation strategy covers linear topology and tree topology with several branches due to the linearity in processing. Packet loss in compressed sensing: Facing inevitable packet loss, we will now specify what will be done in this case at the sink, whereas we don’t need to do anything at the sensors. Lost transmissions are counted as zeros at the corresponding positions in measurement matrix, i.e. the matrix Φ is replaced by an effective measurement ˆ = Q ⊙ Φ, where Q = QN is the error matrix at matrix Φ ∏N sink having elements Qij = l=j qil . We can see, that the sampling process is not interrupted by lost transmissions and continues its aggregation.

Data Aggregation and Recovery in Wireless Sensor Networks Using Compressed Sensing

(Sk )n = ϕnk dk +

k−1 ∑ k−1 ∏

(

i=1 l=i

|

qnl ) ϕni di . {z

(13)

}

Temperature (Celsius)

Original signal and recovered signal original

25.5

recovered

25 24.5 24 100

200

300

400

500

600

700

800

900

1000

Wavelet coefficients 300 Wavelet coefficient

This approach is easy to implement and doesn’t consume extra energy for packets once they are lost, contrary to the existing solutions, e.g. CSEC. Although this approach seems quite simple whereby from WSN experience it is more intuitive to use some similar data instead of zeros. Later on, in the simulation section, we will compare this also to the strategy where last time measurements are used due to the temporal correlations of sensing data. However, for our system setup we observed a much worse performance. Taking packet loss into account with Rk = qk−1 ⊙ Sk−1 , we get:

7

200 100 0 0 10

1

2

10

10

3

10

signal index

(Rk )n

In the sense of whole network with B branches, the data aggregated at the sink is then: y=

B−1 ∑ (b+1)H ∏ ∑ (b+1)H

(

b=0 i=bH+1

qnl ) ϕni di ,

(14)

l=i

where, the branch number b = 0, 1, . . . B − 1, and every branch equally has H nodes. The elements of the ˆ = Q ⊙ Φ are therefore effective measurement matrix Φ ∏(b+1)H ˆ ϕni := Qni · ϕni = ( l=i qnl ) ϕnl in branch b. If the tree network∏has only one branch as a linear topology, N then ϕˆni = ( l=i qnl ) ϕnl . The concept of end–to–end packet loss rates as discussed at the end of Section 3.2 for the conventional data aggregation scheme can not be used anymore in this CS based approach. Here, the sink receives M linear ˆ of B · H data symbols collected combinations y = Φd ˆ =Q⊙Φ in the vector d. The number of zeros in Φ then determines the number of lost data contributions (multiplied with zero) in these combinations and is a random variable. We then define the ratio: ( ) # zeros in Q P =1−E MN ( ) # zeros in a branch of Q =1−E MH (a)

= 1−

H 1 ∑ h Pr{qjh = 0|(qj,h+1 . . . qj,H ) = 1} H h=1

∑ 1 (1 − p)pH h p−h H H

=1−

(15)

h=1

where the expectation is taken over the statistics of Q. Step (a) follows from the properties of Q since its zeros occur as a series which always starts independently for each compressed dimension j ∈ [1 . . . M ] and for each branch [1 . . . B] in its first hop. We call P in (15) as the end–to–end data success rate since it indicates the averaged proportion of d contained in the observation y and depends on the peer–to–peer packet success rate p. For example, taking H = 5 and p = 77.7% causes a end–to–end data success rate P = 50%.

Figure 4: Temperature samples in wavelet domain. They are sparse and can be well recovered by 40 biggest coefficients out of 1024. (a) comparison between the original and recovered signals; (b) the wavelet coefficients from 1024 temperature data.

4.3 The Sparsity in the Sensor Readings From CS theory, data can be compressed if it is sparse, which is the assumption of our approach. In this section, we illustrate that in our example this is indeed the case. We have observed the sensor data about the temperature distribution in the office building, which are sampled by sensor of type SHT10 with resolution 0.01◦ C. Usually, environment sampled data is piece-wise smooth, spatially and temporally correlated and often sparse in the wavelet domain. The spatial correlation of temperatures has already been observed in [LWSC09], where temperature samples at the same time in different depths of ocean water are sparse in wavelet domain. This is also confirmed by our system, where we used Daubechies wavelets in Ψ to process data. Furthermore, we collect also 1024 temperature values from each sensor node in different time instants. We will use this data later on for comparisons, i.e. to exploit temporal correlations in replacing lost data by previous ones during aggregation, To visualize this, we keep the largest 40 coefficients which takes 4% of total, to reconstruct the original data. From Fig. 4, we can see that the recovered values sufficiently agree with the sensing data. The relative error to the original data is merely 0.2% which is precise enough for room temperature.

4.4 Discussion of the Proposal A key assumption is that we recover the data at sink ˆ i.e. the corresponding positions by the same matrix Φ, have to be known at the sink. Fortunately, this can be achieved without much cost. In most data collection scenarios, every packet has a certain sequence number

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based on its node id such that the sink can identify it. Therefore, the sink knows the status of all packets. In CS, the following should be handled appropriately. Sensor reading are always encoded into M packets, i.e. the packets need be numbered to be identified. Lost positions have to be known at the sink which means that one sensor node should record a lost event once it detects a packet loss. It also should be noted that our application motivates the investigation of more complex random models for measurement matrices in compressed sensing since here the columns have specific dependencies caused by the multi-hop transmission model. Several results are known for the independent case and then much more simple distribution can be used for the elements Φij of Φ. For ˆ ij is example, if Φij follows the Bernoulli distribution Φ {−1, 0, +1}–valued. Combining the results of Achlioptas [Ach03] with proof strategy of [BDDW08] shows that if the probability of zeros is no more than 2/3 small δs can be ensured. Due to the limited space we will not discuss this here since such assumptions are not really feasible in multi–hop WSNs.

5 Implementation and simulation 5.1 Implementation The implementation mainly includes three parts: generation of the measurement matrix, routing establishment and the reconstruction of the sensor data. Generation of measurement matrix: According to the theory of compressed sensing, it is necessary that the measurement matrix Φ satisfies RIP with a sufficiently small constant δs . For example, as discussed in Section 4.1, Gaussian matrices are advised. However, for Atmega128 with 8-bit microprocessor, it takes many hardware resources to generate the numbers because of floating point operations. We therefore generated the components of the vector ϕk (columns of Φ) as uniformly distributed pseudo–random integers by “Park-Miller Minimal Standard Generator” which is a multiplicative linear congruential generator [PM88] between [1 . . . 216 ] seeded by the id of the node. With proper normalization at the sink this yields a sub–Gaussian distribution as well (it is bounded). Another approach is to generate the random matrix Φ offline and broadcast it to the nodes. Establish routing: Compressed sensing compresses data through transferring packets hop by hop. It decreases the load of the parent nodes, while increases packet latency. During the routing establishment, all nodes broadcast beacons to estimate link qualities and maintain the expected transmission rate (the value of etx in TinyOS). Based on the updated list of link estimation (periodically or event-triggered), each node maintains its parent address to form the routing topology. In compressed sensing, each node not only has its parent list, but also has a list of descendants. Broadcasting beacons periodically to check all descendants is more

feasible as compared to notifying. Although it seems more energy efficient for the child node to tell its parent node about connectivity, it is not possible to notify its parent in time on a link failure. Reconstruction of the data: After the M –dimensional compressed data y has been collected at sink, it is reconstructed on a PC by solving the ℓ1 –minimization problem in (11) as mentioned above. In this noiseless setting the basis pursuit (BP) can be casted as a linear program with complexity O(N 3 ) [SBB06]. Greedy algorithms like Orthogonal Matching Pursuit (OMP) are more advisable due its reduced complexity as compared to BP (OMP has reconstruction complexity O(N s2 ) [TAG07]) whereby BP usually needs fewer measurements and provides better reconstruction precision.

5.2 Simulation In order to verify the performance of our approach, we first illustrate the influence of packet loss. The results of our approach are then shown for different reconstruction algorithms. After that, we compare possible packet loss patterns. Finally, the energy efficiency is analyzed. Simulation setup: Tossim [LLWC03] and Matlab are adopted in our simulation. Tossim is a discrete event-based simulator in TinyOS, which models hardware functions of Atmega128 and chip-related protocols (such as physical layer and MAC layer) by its own software modules. The MAC protocol in Tossim is CSMA1 as also adopted for the practical application. The distance between any two adjacent nodes in the network is 100 meters. The ”log distance path loss” model is used to calculate radio attenuation ratio. For the given transmission, interference range and noise floor, each node can only communicate with its adjacent nodes. The nodes are organized in tree topology. As mentioned in Section 3.1, it is a general scheme for applications of data aggregation in wireless sensor networks. In our experiment, the sensor data d are the temperature values from field environment, which have a strong spatial correlation. We assume that the data is K– sparse in some fixed basis Ψ. If Φ satisfies the RIP with sufficiently small δs for s = 2K, CS based algorithms like BP in (11) can reconstruct the sensing data. For random matrices as mentioned above the probability for small δs increases with increasing M . To illustrate this we have repeated the experiment 500 times for each fixed M . We observed that for N = 100 and K = 5 in the tree topology with B = 10 branches, the sink can recover the original data for M = 24 with sufficient precision. By this we mean that the reconstruction error ∥dˆ − d∥ℓ2 /∥d∥ℓ2 is less than 1%. Fig. 5 shows the simulation result for M = [1 . . . 64]. Summarizing, a reliable reconstruction even in the presence of packet loss can be achieved in our setting for M greater than 4 ∼ 5K. Recovery of lost data using CS: At the beginning, we compare the case of no loss and 5% data loss. Fig. 5 shows that our solution recovers the sensor data under

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Figure 5: Data recovery with OMP for data loss rates ˆ (1 − P ) of 0% and 5% when using the modified matrix Φ at the sink (N = 100, K = 5). It can be seen that there is no significant difference. nearly the same quality as without loss. This indicates that the solution could recover sensed data perfectly even though some data were lost before. In [TAG07], it is suggested that OMP gives accurate reconstruction with probability greater than 1 − 2ϵ when M ≥ CK ln(N/ϵ) for some ϵ ∈ (0, 0.36) where C ≤ 20 (C can be improved for large K). From the results above we know that M ≈ 5K is sufficient for reconstruction without packet loss. From Fig.5 we see that M = 5K is also enough for a data ˆ is used loss rate 1 − P = 5% if the modified matrix Φ at sink recovery. We also simulated the case where original measurement matrix Φ is used for reconstruction which is of course the simplest solution. However, the results in Fig.6 indicate that this approach will not work satisfactory. The case that the elements of the measurement matrix Φ are known only with a certain probability has not been analyzed much in compressed sensing. In [CC12] a modified OMP algorithm has been presented to cope with missing elements in Φ. The authors propose to use OMP only for support recovery (the non–zero positions in x) and then apply a statistical least squares stage on this support which accounts for the distribution of the missing elements. However, in application we found no further improvement in the reconstruction performance. But, in real application, the data loss rate is more serious than 5%. As previously mentioned, 50% data may be lost in energy constrained scenario. Fig. 7 shows that M has to be increased slightly to achieve the same probability of precise reconstruction. Discussion of recovery via the RIP properties: To motivate why our approach recovers lost data we have plotted in Fig.8 an indication for the RIP constant ˆ for data δs of the effective measurement matrix Φ loss rates 1 − P = 0%, 5% and 50% for one particular fixed matrix Φ. Obviously, computing the true constant numerically is not feasible. Instead we selected for each sparsity level a fixed number of random subsets of

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Figure 8: Indication for the RIP constant δs of a ˆ (N = 1000 and M = 250) for data loss particular fixed Φ rates (1 − P ) of 0%, 5% and 50%. columns and computed the corresponding minimum and maximum singular values of all these submatrices. From this ensemble it is then straightforward to compute a restricted value of δs . From √the discussion of (11) we know that a value below 2 − 1 is desirable. In Fig.8 we see that the selected Φ supports a sparsity of K . 5 for a data loss rate 1 − P = 5% without further degradation. However, in the case of 50% we sparsity is reduced to K . 2, approximately. Finally, we would like to mention that performance guarantees based on RIP are usually much stronger than necessary for practical implementations since these results have uniform character. This means that once such a matrix has been found it can be used to recover any K–sparse d (or x), i.e. there is no need to regenerate Φ. Based for example on the RIP’less theory in [CP11], much more direct estimates are possible with regeneration. Different algorithms for recovery: We have compared the greedy–based (OMP) algorithm under the data loss model also with BP (ℓ1 –optimization) by using the Matlab toolbox CVX (recall that BP can also be efficiently implemented as a linear program). As already mentioned, as long as the RIP constant is small enough, exact reconstruction in (11) can be guaranteed which is not the case for OMP. As already mentioned again, the results in [TAG07] ensure reconstruction by OMP with a certain probability for admissible measurement matrices. In this simulation, the two different recovery probabilities are presented in Fig. 9, where BP provides a more stable performance, especially in the high packet loss rate. These results illustrate the different precision of our approach under two typical reconstruction algorithms. It can be seen that, although OMP is not as robust as BP, it is still good enough for recovering lost data. Comparisons with CSEC: Fig. 10 compares our greedy–based (OMP) with CSEC (using OMP) and with the conventional data aggregation. It shows that for all methods the recovery precision gets worse with

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Figure 10: Compare with 3 different data recovery approaches. (1) CS with OMP (M = 25, K = 5), (2) CSEC [CCZ+ 10] with OMP (M = 30, K = 5) (3) conventional aggregation. increasing packet loss rate, where our OMP–based method offers the strongest robustness. On the contrary, CSEC discards the whole row of measurement matrix when one packet loss happened during this sampling. Here, it precisely reconstructs the data for packet loss rates up to 20% after we increased the compression dimension by 20%, i.e. M = 30 instead of M = 25. The conventional aggregation shows a straight line in the figure since every lost packet causes a corresponding data vacancy at sink. Energy dissipation comparisons: Energy dissipation is one of the most important issue for energy constrained WSNs. Our approach also keeps the advantages of balanced energy consumption that CS promises because in every hop the same number of M packets are sent in the whole network. In order to make clear the energy efficiency gain with different methods, we decrease the energy consumption for listening to minimum, since this part is decided by lower-level of MAC protocol and is the same whether using CS or not. In the simulation,

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Status

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8.93 mA 4.93 mA 8 uA 0.3 uA 10.4 mA 9.3 mA 0.2 uA 3 mA 4 mA

RADIO(CC1000) LED SENSOR (SHT10)

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The energy model of sensor node.

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TinyOS and showed that it is also efficient in energy balance.

Acknowledgment This work is supported in part by Deutsche Forschungsgemeinschaft (DFG) grants JU 2795/2-1 and STA 864/3-2, and Shenzhen Key Laboratory for RF Integrated Circuits.

References [Ach03] D. Achlioptas. Database-friendly random projections: Johnson-lindenstrauss with binary coins. Journal of computer and System Sciences, 66(4):671–687, 2003. [BDDW08] R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin. A simple proof of the restricted isometry property for random matrices. Constructive Approximation, 28(3):253–263, 2008. Figure 11: Energy consumption of every sensor node in the network under packet loss. CS data collection is more balanced than linear routing protocol, and CSEC consumes more energy than CS. we compare CS data collection with traditional tree collection protocol (CTP) which is widely used in TinyOS. The two kinds of networks both sample data every 10 seconds during 1000 seconds in a linear topology as B = 1. The linear topology is composed of 100 nodes and the sink node 0 is at one end of the chain, where BMAC (low-power listen CSMA) is adopted for wireless channel access and routing. We use the energy model from TABLE 4 to calculate the consumption of every node. For packet loss rate 1 − P = 20% we can see in Fig.11 that CTP is unbalanced which causes the nodes near sink expire earlier and disable the network. But, both two CS scenarios are more balanced whereby CSEC consumes more energy due to its overhead as explained above.

6 Conclusions We have described a novel data aggregation scheme that is able to reconstruct lost packets in large–scale wireless sensor networks using compressed sensing. This approach addresses one of the major technical challenges of QoS in WSNs. We have shown that this multi–hop and cascade encoding scheme has a low–cost implementation but accurate data recovery due to sparse nature of sensor data and balanced energy consumptions. Our method is stable even if a considerable part of packets are lost. Furthermore, we have implemented this scheme in

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