Wireless Sensor Network

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NN X N [n] S N [n] E YN [n]

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Figure 5.1. Correlation model and architecture.

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SPATIOTEMPORAL CORRELATION THEORY FOR WIRELESS SENSOR NETWORKS

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Xi [n], the noisy version of the event information, Si [n], which is spatially correlated to the event source, S. In order to communicate this observation to the sink, each node has to encode its observation. The encoded information, Yi [n], is then sent to the sink through the sensor network. The sink, at the other end, decodes this information to get the estimate, S, of the event source S. The encoders and the decoders are labeled as E and D, respectively, in Figure 5.1. Using this model, we will exploit various aspects of correlation among sensor readings in terms of both time and space. Each observed sample, Xi [n], of sensor ni at time n is represented as Xi [n] = Si [n] + Ni [n] (5.1)

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where the subscript i denotes the spatial location of node ni [i.e., (xi , yi )], Si [n] is the realization of the space time process s(t, x, y) at time t = tn 1 and (x, y) = (xi , yi ), and Ni [n] is the observation noise. {Ni [n]}n is a sequence of i.i.d Gaussian random 2 variables of zero mean and variance N . We further assume that the noise each sensor node encounters is independent of each other; that is, Ni [n] and Nj [n] are independent for i = j and n. / As shown in Figure 5.1, each observation Xi [n] is then encoded into Yi [n] by the source-coding at the sensor node as Yi [n] = fi (Xi [n]) (5.2)

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and then sent through the network to the sink. The sink decodes the received data to reconstruct an estimation S of the source S S = g(Y1 [n1 ], ..., Y1 [n ]; ...; YN [n1 ], ..., YN [n ]) (5.3)

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based on the data received from N nodes in the event area over a time period = tn tn1 . The sink is interested in reconstructing the source S according to a distortion constraint D = E d(S, S) (5.4)

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In the next sections, the general distortion function in (5.4) will be used to independently obtain the distortion functions for spatial and temporal correlation in the WSN, which will be further extended to capture the joint spatiotemporal correlation analysis of point and eld sources as well.

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1 Note

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that we use a discrete-time model since each node is assumed to sample the physical phenomenon synchronously after the initial wake-up.

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SPATIOTEMPORAL CORRELATION IN WIRELESS SENSOR NETWORKS

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5.3 SPATIOTEMPORAL CORRELATION IN WIRELESS SENSOR NETWORKS 5.3.1 Spatial Correlation in Wireless Sensor Networks In this section, based on the communication architecture and the theoretical correlation model presented in Section 5.2, the spatial correlation between observations of each sensor node is modeled. The information gathered by N sensors in an event area can be modeled as shown in Figure 5.1. The sink is assumed to be interested in a point source S. Since we only consider the spatial correlation between nodes, in this analysis we assume that the samples are temporally independent. Hence, by dropping the time index n, (5.1) can be restated as Xi = Si + Ni , i = 1, ..., N (5.5)

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The sink is interested in reconstructing the source S according to observations of nodes ni which observe the spatially correlated version of S at (xi ,yi ), that is, Si . The physical phenomenon is modeled as joint Gaussian random variables (JGRVs) at each observation point as E{Si } = 0, var{Si } =

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2 S ,

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i = 1, ..., N i = 1, ..., N

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2 cov{Si , Sj } = S corr{Si , Sj }

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corr{Si , Sj } = i,j = K (di,j ) =

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