Joint Spatiotemporal Correlation in Wireless Sensor Networks In the previous sections, the spatial and temporal correlations were separately

cap-tured. In order to understand the joint effects of spatial and temporal correlation, spatiotemporal correlation characteristics of point and field sources are investigated next.

Spatiotemporal Characteristics of Point Sources. In many WSN applica-tions such as target detection and fire detection, the goal is to estimate the properties of an event generated by a single point source, through collective observations of sensor nodes. In this section, we first introduce our model for the point source and formulate its spatiotemporal characteristics. Next, we derive the distortion function for the estimation of the point source.

Here, we are interested in observing the joint behavior of spatial and temporal cor-relation. Therefore, in order to capture the spatiotemporal correlation characteristics, we follow a different approach than in Section 5.3. Here, the point source is assumed to generate a continuous signal that is modeled by a random processf_S(s, t), wheres denotes the outcome andtdenotes time. For ease of illustration, we usef_S(t) in the remainder of the chapter. We model the point source,f_S(t), as a Gaussian random process such thatf_S(t) is first-order stationary; that is,µ_S(t)=µ_Sand has a variance σ_S². Without loss of generality, we assumeµS=0.

For ease of illustration, we assume that the coordinate axis is centered at the point source. As a result, the received signal,f(x, y, t), at timetat a location (x, y) can be

which is the delayed and attenuated version of the signalf_S(t). In this model, we assume that the event signal travels with the speed,v, and is attenuated based on an exponential law, whereθ_sis the attenuation constant. Note that the functionf(x, y, t) is also a Gaussian random process and that the samples taken by the sensors are jointly Gaussian random variables (JGRVs). Sinceµ_S=0, the mean of the received signal is given byµ_E=0.³The variance of the received signal is also given as follows:

σ_E²(x, y)=E An interesting result from (5.25) is that the variance of the signal observed at location (x, y) depends on the distance between the observation location and the point source. As in Figure 5.1, the received signal at timet_kby a sensorn_iat location (xi, y_i) is given by

Si[k]=f(xi, yi, tk) (5.26)

3The subscriptsSandEthat are used here represent thesourceandevent, respectively.

Assuming wide-sense stationarity, thespatiotemporal correlation functionfor two samples of a point source taken at locations (xi, y_i) and (xj, y_j) and at timest_k and t_l, respectively, is given by

ρp(i, j, k, l)= E

S_i[k]S_j[l]

σ_E(xi, y_i)σ_E(xj, y_j)

=ρS( t) (5.27)

where t = |t_k−t_l−(di−d_j)/v|,d_i =

x²_i +y²_i is the distance of the sensorn_ito the point source, andρ_S( t)=E[fS(t)f_S(t+ t)]/σ_S²is the correlation function of the point source which is given byρ_S( t)=e^{− t/θt}, whereθ_tis a constant govern-ing the degree of correlation. Note that the spatiotemporal correlation between two samples,ρ_p(i, j, k, l), depends mainly on the difference between sample timest_kand t_lsince generallyv(di−d_j).

In WSNs, we are interested in estimating the signal generated by the point source using the samples collected by the sensor nodes. The expectation of the generated signal,fS(t), over an intervalτis given by

S(τ)= 1 τ

_τ

0

f_S(t)dt (5.28)

Each sensor node,n_i, receives the attenuated and delayed version of the generated signalf_S(t), that is,S_i[k]. Due to the impurities in the sensor circuitries, the sampled signal is the noisy version of this received signal which is given by

X_i[k]=S_i[k]+N_i[k] (5.29) where the subscriptidenotes the location of the nodeni, that is, (xi,yi),kdenotes the sample index which corresponds to timet=t_k,X_i[k] is the noisy version of the actual sample S_i[k], and N_i[k] is the observation noise, that is, N_i[k]∼N(0, σ_N²).

S_i[k] is given by (5.24) and (5.26).

The observed information, X_i[k], is then encoded and sent to the sink through the WSN. It has been shown that joint source-channel coding outperforms separate coding. Moreover, as discussed in Section 5.3, for WSNs with a finite number of nodes, uncoded transmission outperforms any approach based on the separation paradigm leading to the optimal solution for infinite number of nodes [3]. In the light of these results, we assume that uncoded transmission is deployed in each node. Hence, the transmitted observation,Y_i[k], is given by

Yi[k]=

P_E

σ_S²+σ_N²Xi[k], i=1, ..., N (5.30)

SPATIOTEMPORAL CORRELATION IN WIRELESS SENSOR NETWORKS 117

whereσ_S²andσ_N² are the variances of the event informationS_i[k] and the observation noiseN_i[k], respectively.

The transmitted information is decoded at the sink. Since uncoded transmission is used, it is well known that minimum mean square error (MMSE) estimation is the op-timum decoding technique [3]. Hence the estimation,Z_i[k], of the event information S_i[k] is simply the MMSE estimation ofY_i[k], which is given by The sink is interested in estimating the expected value of the event during a decision intervalτthat is given by (5.28). Assuming each sensor node sends information at a rate off samples/sec, this estimation can simply be found by

S(τ, f, M)ˆ = 1

whereMis the number of sensor nodes that send samples of the observed point source.

Mnodes are chosen among the nodes in the network to represent the point source and, hence, are referred to asrepresentative nodes. Consequently, the distortion achieved by this estimation is given by [2]

Dp(τ, f, M)=E

(S(τ)−S(τ, f, M))ˆ ²

(5.33) where the subscriptpdenotes the point source. Using (5.24), (5.25), (5.28), (5.31), and (5.32), (5.33) can be expressed as

D_p(τ, f, M)=σ_S²− 2

anddi =√

(xi+yi), andρ(i, j, k, l) is the spatiotemporal correlation function given in (5.27).

Spatiotemporal Characteristics of Field Sources. In some WSN applications such as temperature monitoring and seismic monitoring, the physical phenomenon is dispersed over the sensor field and, hence, can be modeled as a field source. Thus, here we explore the spatiotemporal characteristics of observing such a phenomenon in WSNs.

As in Section 5.3.3, the event signalf(x, y, t) is assumed to be a Gaussian random process withN(0, σ_s²). The sink is interested in estimating the signalf(x0, y0, t) over the decision intervalτat location (x0, y0). Assuming that the observed signalf(x, y, t) is wide-sense stationary (WSS), the expectation of the signal over the decision interval τ[i.e.,S(τ)] can be calculated by the time average of the observed signal as

S(τ)= 1 τ

_τ

0

f(x0, y₀, t)dt (5.35) where (x0, y₀) is the event location. The signalS_i[k] received at timet_kby a sensor node at location (xi, y_i) is defined as in (5.26), and theS_i[k]’s are JGRV withN(0, σ_s²).

The covariance of two samples,S_i[k] andS_j[l], is given by

cov{S_i[k], Sj[l]} =σ²_Sρ_s(i, j)ρ_t(δ) (5.36) where

ρs(i, j)=e^−di,j/θs and ρt(δ)=e^−|δ|/θt (5.37) are spatial and temporal correlation functions, respectively,δ=(k−l)/f,f is the sampling rate,di,j=

(xi−xj)²+(yi−yj)²is the distance between two nodesni

andn_j, andθ_sandθ_tare spatial and temporal correlation coefficients, respectively.

Following the discussion and derivations in Section 5.3.3, the noisy version of the signal,X_i[k], and the transmitted signal,Y_i[k], are given by (5.29) and (5.30), respectively. The estimationZ_i[k] can be found as

Z_i[k]= σ_S² σ_S²+σ_N²

S_i[k]+N_i[k]

(5.38) After collecting the samples of the signal in the decision intervalτfromMnodes, the sink estimates the expectation of the signal over the last decision interval as given in (5.32). As a result, the distortion achieved by this estimation is given as in (5.33).

Using the definitions above and substituting (5.35), (5.38), and (5.32) into (5.33), the distortion function can be derived as [2]

COROLLARIES AND EXPLOITING CORRELATION IN WIRELESS SENSOR NETWORKS 119 In order to provide further insight into the spatiotemporal correlation charac-teristics and distortion analysis derived in this section, next we discuss possible approaches that can be used in the design of efficient communication techniques exploiting the spatiotemporal correlation observed in the wireless sensor networks.

5.4 COROLLARIES AND EXPLOITING CORRELATION IN WIRELESS