Cross-layer design for decentralized detection in WSNs

4.1 Wireless channel model

We focus on the case where the sensor nodes and the fusion center have minimal movement and the environment changes slowly. Accordingly, only the slow-fading component of the wireless channel is considered. Figure 3 shows the fading channel model, where w(t) is an AWGN with PSD N₀ / 2, and m(d_c) is the mean path attenuation for a sensor node at a distance d_c from the fusion center. Using the Hata path-loss model, the total decibel power loss is given by [26]

$\begin{array}{ll}{P}_L& =\underset{{\mu }_{\mathrm{c}}}{\underbrace{20log\left(4\pi d_0/{\lambda }_p\right)+10{\rho }_{\mathrm{c}}log(d_{\mathrm{c}}/d_0)}}+X_{{\sigma }_{\mathrm{c}}}\text{dB}\end{array}$

(1)

where d₀ is a reference distance corresponding to a point located in the far field of the transmit antenna, λ _p is the wavelength of the propagating signal, ρ_c is the path loss exponent, and $X_{{\sigma }_{\mathrm{c}}}\sim \mathcal{N}(0,{\sigma }_{\mathrm{c}}^2)$.

where P_t is the average transmitted signal power, and Φ(.) is the cumulative distribution function for the standard normal PDF. We note that the CSI relevant to our model is represented by the statistics σ_c, μ_c, and N₀. These statistics are required to be estimated by each sensor, and no instantaneous channel state information is required for the XCP design. Since we assume fixed nodes and a slowly varying channel, the estimation process could be executed less frequently to save sensor node resources. This is particularly important in wireless sensor networks since the estimation of the channel state is both time and power consuming.

It should be highlighted that the large-scale fading model presented here allows us to obtain the closed form solution in (2). More complex fading models, e.g., small-scale fading, can be integrated similarly, but they may allow only numerical solutions.

4.2 Media access control protocol model

The detection cycle initiated by the fusion center is illustrated in Figure 4. We assume a slotted multiaccess communication scheme with number of communication slots L per detection cycle, where each packet requires one time slot for transmission, all time slots have the same length τ / L, τ is the delay for detection, and all transmitters are synchronized. Each local sensor i collects a number of observations n _i and forms an information packet for transmission over the wireless channel, with communication rate:

$\begin{array}{l}{R}_i=\mathit{\text{bL}}{n}_i/\tau ,\end{array}$

(3)

Unequal priority TDMA, where a single sensor may be assigned more than one time slot, could also be used and may lead to a better detection performance. However, the resulting optimization problem is an integer programming problem that is generally hard to solve in real time. Therefore, only equal-priority TDMA is considered in this work. Without loss of generality, we assume that L = m N, where m is a positive integer, i.e., at each detection cycle, all sensors transmit the same number of times. This assumption facilitates the comparison with the slotted ALOHA scheme. The decision takes place at the end of the detection cycle by the FC. The process repeats for every detection request initiated by the fusion center. To simplify the analysis, the MAC protocol does not consider the acknowledgement slots and any protocol specifics required for synchronization or rate negotiation. Also, we ignored the packet overhead, which is a reasonable approximation for practical WSN protocols with large packet payload.

4.3 Energy model

where ${\mathcal{E}}^0=\sum _{i=1}^N{e}_i^0$ is the total initial energy in all sensors at the time of deployment, ${\mathcal{E}}^{\mathrm{w}}=\sum _{i=1}^N{e}_i^{\mathrm{w}}$ is the total wasted energy remaining in sensor nodes when the network dies, f _r is the average sensor reporting rate defined here as the number of detection cycles per unit time, and ${\mathcal{E}}^{\mathrm{r}}=\sum _{i=1}^N{e}_i^{\mathrm{r}}$ is the expected reporting energy consumed by all sensors in one detection cycle.

Our objective is to allocate the reporting energy $e_i^{\mathrm{r}}$ for each sensor in such a way that maximizes the detection performance. In what follows, we derive the energy constraints for the sensor network.

4.4 Sensing model

where ε is the amplitude of the emitted signal at the object, η is a known attenuation coefficient, typically between 2 and 4, and w _i is an additive white Gaussian noise with zero mean and variance ${\sigma }_s^{i^2}$. We note that the above model considers passive sensing [25]. In the active sensing case, the observation model is given by x _i  = ζ ε _tr  / (2d _i )^η/2 + w _i , where ζ is a known reflection coefficient at the object, ε _tr is the amplitude of the signal transmitted by the active sensor (illuminating signal), and 2d _i is the round trip distance traveled by the signal [19]. We note that the two observation models differ only in the scaling factor ζ / 2^η/2. Therefore, without loss of generality, we assume the active sensing model in the following discussion. If passive sensing is assumed, then the detection problem will be slightly different than the problem presented here, since the amplitude of the source signal, ε, is unknown and has to be estimated from sensor observations.

where μ ⁱ  = ζ ε _tr  / (2d _i )^η/2, and n _i is the number of independent and identically distributed (IID) observations obtained by sensor i at each time slot. We note that noise samples are independent across sensors, i.e., the observations at local sensors are independent across time and space, but not necessarily identically distributed since some sensors may be closer to the resolution cell, and noise variances are assumed unequal. In the following, we designate the vector of sensor observations at time slot k by x _i [k] = [x _i [1, k] x _i [2, k] … x _i [n _i , k]]. We note that x _i [k] has the multivariate Gaussian distribution $\mathcal{N}(0,\mathbf{C})$ under hypothesis ${\mathcal{H}}_0$ and $\mathcal{N}(\mu ,\mathbf{C})$ under hypothesis ${\mathcal{H}}_1$, where μ = [μ¹μ² … μ ^N ] and $\mathbf{C}={\sigma }_s^{i^2}I$.

Proof.

At the fusion center, the log likelihood ratio (LLR) is the sum of the individual observations received at each time slot. Therefore, the test could be expressed as

$\begin{array}{l}\sum _{k=1}^{n_s}\sum _{i=1}^N{r}_i\left[k\right]l\left({\mathbf{x}}_i\right[k\left]\right)\begin{array}{c}{\mathcal{H}}_1\\ \gtrless \\ {\mathcal{H}}_0\end{array}ln\gamma ,\end{array}$

(16)

where l(x _i [k]) is given by (17). The LLR test then reduces to (18), where $z_i=\sum _{k=1}^{n_s}{r}_i\left[k\right]$ indicates the number of times sensor i successfully transmitted to the fusion center in n _s time slots. The random vector z = [z₁z₂ … z _N z _e ], where $z_e=n_s-\sum _{i=1}^N{z}_i$, is multinomially distributed with probability vector p = [λ₁λ₂ … λ _N λ _e ], where the probability of collision or idle slot ${\lambda }_e=1-\sum _{i=1}^N{\lambda }_i$, and the sample space represents all possible combinations of z _i such that $z_e+\sum _{i=1}^N{z}_i=n_s$.

$\begin{array}{ll}l\left({\mathbf{x}}_i\right[k\left]\right)& =ln\frac{p_{{\mathbf{x}}_i}\left({\mathbf{x}}_i\left[k\right];{\mathcal{H}}_1\right)}{p_{{\mathbf{x}}_i}\left({\mathbf{x}}_i\left[k\right];{\mathcal{H}}_0\right)}\\ =ln\frac{exp\left[-\frac{1}{2}{\left({\mathbf{x}}_i\right[k]-{\mu }^{i}1)}^T{\mathbf{C}}^{-1}\left({\mathbf{x}}_i\right[k]-{\mu }^{i}1)\right]}{exp\left[-\frac{1}{2}{\mathbf{x}}_i^T\left[k\right]{\mathbf{C}}^{-1}{\mathbf{x}}_i\left[k\right]\right]}\\ =\frac{{\mu }^i}{{\sigma }_s^{i^2}}\sum _{j=1}^{n_i}{x}_i[j,k]-\frac{1}{2}{n}_i{\left(\frac{{\mu }^i}{{\sigma }_s^i}\right)}^2\end{array}$

(17)

$\begin{array}{ll}V=& \sum _{k=1}^{n_s}\sum _{i=1}^N\sum _{j=1}^{n_i}\left(\frac{{\mu }^i}{{\sigma }_s^{i^2}}\right)r_i\left[k\right]x_i[j,k]\begin{array}{c}{\mathcal{H}}_1\\ \gtrless \\ {\mathcal{H}}_0\end{array}\frac{1}{2}\sum _{k=1}^{n_s}\sum _{i=1}^N{n}_i{r}_i\left[k\right]\\ \times {\left(\frac{{\mu }^i}{{\sigma }_s^i}\right)}^2+ln\gamma =\frac{1}{2}\sum _{i=1}^N{z}_i{n}_i{\left(\frac{{\mu }^i}{{\sigma }_s^i}\right)}^2+ln\gamma =\gamma \prime .\end{array}$

(18)

4.5 Measurement of detection performance

Under non-Gaussian assumptions, there is no general result that enhancement of the deflection coefficient will lead to a better performance in terms of the ROC curve. However, it is likely that more separation between the two density functions will lead to a better detection performance.

4.6 Handshaking overhead

Example 1.

We evaluate the upper bound for the example network simulated in Section 6, where the parameters are shown in Table 1. We use σ_c = 6 d B and an average value of μ_c = 50 d B for all sensors. Further, we need λ = 0.9 to ensure reliable communication during the handshaking cycle and a percentage of resources consumed in the handshaking α = 0.05. For delay for detection τ = 25 s ec, and network lifetime l = 250 days, we get:

$\begin{array}{l}\begin{array}{ll}\text{Design parameters}:& R_{\mathrm{h}}=965\mathit{\text{bps}}\\ P_{\mathrm{h}}=0.56W\\ \text{Upper bound}:& f_{\mathrm{h}}<22.69\mathit{\text{cycles}}/\mathit{\text{day}}\\ \text{Resources consumed}:& {\tau }_{\mathrm{h}}{f}_{\mathrm{h}}=242\mathit{\text{sec}}/\mathit{\text{day}}\\ e_{\mathrm{h}}{f}_{\mathrm{h}}=1.94J/\mathrm{day.}\end{array}\end{array}$

(42)

Figure 5 plots the operating point for the two conditions (33) and (38). Accordingly, for the given sensor network, the analysis and the developed model could be used with sufficient accuracy as long as the environment dynamics do not require more than 22 cycles/day to update the fusion center. If the environment dynamics are much faster, then the handshaking overhead has to be included in the model development.

Theorem 1.

Proof.

When q _i  = 1, the objective function reduces to $D^2=(\tau /b)c_i{R}_i\Phi \left({\rho }_i\right)\prod _{j\ne i}(1-q_j)$. Any value of q _j  ≠ 0 will cause the objective function value to decrease. Physically, q _j  ≠ 0 corresponds to a guaranteed collision, i.e., loss of information. Therefore, q _j  = 0. Since R _j and $e_j^{\mathrm{r}}$ do not affect the objective function, we arbitrarily set R _j  = 0. $e_j^{\mathrm{r}}$ should be set to 0 to save the energy budget for the non-contributing sensor. The solution q _j  = R _j  = e j r = 0 for j ≠ i could be shown to satisfy the KKT conditions by direct substitution. The objective function monotonically increases with $e_i^{\mathrm{r}}$. Therefore, $e_i^{\mathrm{r}}$ should be set to its maximum value, i.e., $e_i^{\mathrm{r}}=min(\epsilon ,{\epsilon }_t)$. Finally, optimal R _i is set to maximize the objective function and, hence, given by (63).

We conclude that we have a set of N candidate points, (q _i  = 1, q _j  = 0, j ≠ i), for a local maximum, which could be checked easily in N time steps, in addition to the computations required to find the optimal communication rate, which could be implemented efficiently for a single-variable function.

Theorem 2.

A candidate point for a local maximizer of the objective function in (43) is when a subset of the sensors, defined by the index set${\mathcal{S}}_{\epsilon }$, transmit with their maximum energy, while all other sensors remain silent. Optimal design variables for the active sensors are at x^∗, where ∇J(x^∗) = 0. The unallocated energy is equal to${\epsilon }_{\mathrm{t}}-\sum _{i\in {\mathcal{S}}_{\epsilon }}{\epsilon }_i$.

Proof.

For the active sensors, 0 < q _i  < 1. The total energy constraint is inactive, i.e., ${\nu }_{e_{\mathrm{T}}}=0$. If $e_i^{\mathrm{r}}<{\epsilon }_i$, then from the complementary slackness condition ${\nu }_{e_i^0}={\nu }_{e_i^1}=0$. From the stationarity condition, we get ${\nabla }_{e_i^{\mathrm{r}}}J=0$, but ${\nabla }_{e_i^{\mathrm{r}}}J=0$ if and only if $e_i^{\mathrm{r}}=0$, a contradiction. Therefore, the only option left is $e_i^{\mathrm{r}}={\epsilon }_i$. In this case, the constraint $e_i^{\mathrm{r}}\le {\epsilon }_i$ is active; hence, ${\nu }_{e_i^0}=0$. From the stationarity condition, we get ${\nabla }_{e_i^{\mathrm{r}}}J{|}_{e_i^{\mathrm{r}}={\epsilon }_i}={\nu }_{e_i^1}$, which satisfies the dual feasibility condition, since the left-hand side is ≥0. Therefore, this point is a candidate for a local maximizer.

We conclude that all active sensors in this case should transmit with maximum energy. Since all other constraints are inactive, all Lagrange multipliers are equal to 0, and therefore, from the stationarity condition, the optimal values for q and R are equal to the stationary point x^∗, where ∇J(x^∗) = 0.

Theorem 2 results in few candidate points if the individual energy constraint for each sensor is a large fraction of the total energy constraint, such that only few sensors consume the total energy budget. Otherwise, the number of candidate points will be prohibitively large. The solution of the optimization problem is summarized in Algorithm 1, where Theorem 1 is used, and the called optimization algorithm is any optimization method of choice, e.g., interior point method.

Algorithm 1 Optimization Problem Solution