Random Field Estimation with Delay-Constrained and Delay-Tolerant Wireless Sensor Networks

In recent years, research Wireless Sensor Networks (WSNs) has attracted considerable attention. This is in part motivated by the large number of applications in which WSNs are called to play a pivotal role, such as parameter estimation (i.e., moisture, temperature), event detection (leakage of pollutants, earthquakes, fires), or localization and tracking (e.g., border control, inventory tracking), to name a few [1].

Typically, a WSN consists of one Fusion Center (FC) and a potentially large number of sensor nodes capable of collecting and transmitting data to the FC over wireless links. In many cases, the underlying phenomenon being monitored can be modeled as a spatial random field. In these circumstances, the set of sensor observations are correlated, with such correlation being typically a function of their spatial locations (see, e.g., [2]). By effectively handling correlation in the data encoding process, substantial energy savings can be achieved.

In a source coding context, the work in [3] constitutes a generalization to sensor trees of Wyner-Ziv's pioneering studies [4]. The authors propose two coding strategies, namely Quantize-and-Estimate (Q&E) and Compress-and-Estimate (C&E), and analyze their performance for various networks topologies. The Q&E encoding scheme is a particularization of Wyner-Ziv's to scenarios with no side information at the decoder. Consequently, each sensor observation is encoded (and decoded) independently. Conversely, C&E turns out to be a successive Wyner-Ziv-based coding scheme and, for this reason, it is capable of exploiting spatial correlation.

In a context of random field estimation with WSNs, the pioneering work of [5] introduced the so-called "bit-conservation principle". The authors prove that, for spatially bandlimited processes, the bit budget per Nyquist-period can be arbitrarily reallocated along the quantization precision and/or the space (by adding more sensor nodes) axes, while retaining the same decay profile of the reconstruction error. In [6] and, again, for bandlimited processes with arbitrary statistical distributions, the authors propose a mathematical framework to study the impact of the random sampling effect (arising from the adoption of contention-based multiple-access schemes) on the resulting estimation accuracy. For Gaussian observations, [7] presents a feedback-assisted Bayesian framework for adaptive quantization at the sensor nodes.

From a different perspective but still in a context of random field estimation, [2] proposes a novel MAC protocol which minimizes the attempts to transmit correlated data. By doing so, not only energy but also bandwidth is preserved. Besides, in [8], the authors investigate the impact of random sampling, as opposed to deterministic sampling (i.e., equally-spaced sensors) which is difficult to achieve in practice, in the reconstruction of the field. The main conclusion is that, whereas deterministic sampling pays off in the high-SNR regime, both schemes exhibit comparable performances in the low-SNR regime.

Contribution

In this paper, we address the problem of (nonnecessarily bandlimited) random field estimation with wireless sensor networks. To that aim, we adopt the Q&E and C&E encoding schemes of [3] and analyze their performance in two scenarios of interest: delay-constrained (DC) and delay-tolerant (DT) sensor networks. In DC scenarios, the observations collected in a particular timeslot must be immediately encoded and conveyed to the FC. In DT networks, on the contrary, the time horizon is enlarged to consecutive timeslots. Clearly, this entails the use of local buffers but, in exchange, the distortion in the reconstructed random field is lower. To capitalize on this, we derive closed-form expressions of the distortion attainable in DT scenarios (unlike in [2, 6, 8], we explicitly take into account quantization effects). From this, we determine the optimal number of samples to be encoded in each of the timeslots as a function of the channel conditions of that particular timeslot. This constitutes the first original contribution of the paper. Along with that, we identify under which circumstances buffers are stable (i.e., buffer occupancy does not grow without bound) and, besides, we study a number of distortion versus buffer occupancy tradeoffs. To the best of our knowledge, such analysis has not been conducted before in a context of random field estimation. Complementarily, we analyze the latency in the reconstruction of consecutive realizations (i.e., those collected in one timeslot) of the random field, this being an original contribution, as well.

The paper is organized as follows. First, in Section 2, we present the signal and communication models, and provide a general framework for distortion analysis. Next, in Section 3, we focus on delay-constrained scenarios and particularize the aforementioned distortion analysis. In Sections 4 and 5 instead, we address delay-tolerant scenarios and analyze the behavior of the Q&E and C&E encoding schemes, respectively. Next, Section 6 investigates latency issues associated with DT networks. In Section 7, we present some computer simulations and numerical results and, finally, we close the paper by summarizing the main findings in Section 8.

Let be a one-dimensional random field defined in the range , with denoting the spatial variable. As in [2, 8, 9], we adopt a stationary homogeneous Gaussian Markov Ornstein-Uhlenbeck (GMOU) model [10] to characterize the dynamics and spatial correlation of . GMOU random fields obey the following linear stochastic differential equation

where, by definition, with , denotes Brownian Motion with unit variance parameter, and , are constants reflecting the (spatial) variability of the field and its noisy behaviour, respectively. According to this model, the autocorrelation function is given by and, hence, the process is not (spatially) bandlimited.

The random field is uniformly sampled by sensor nodes, with intersensor distance given by (see Figure 1). The spatial samples can thus be readily expressed as follows [11]:

where .

In delay-constrained (DC) networks, the samples collected in the sensing phase of a given timeslot must be necessarily encoded and transmitted to the FC in the corresponding transmission phase. Bearing this in mind, we particularize the analysis of Section 2.2 and compute the average distortion for the cases of Delay-Constrained Quantize-and-Estimate (QEDC) and Compress-and-Estimate (CEDC) encoding strategies.

Here, we impose a long-term delay constraint: the samples collected in consecutive timeslots must be conveyed to the FC in such timeslots. In other words, sensors have now the flexibility to encode and transmit a variable number of samples in each time slot (according to channel conditions) and, by doing so, attain a lower distortion.

Let be the number of samples encoded in channel uses by sensor in time-slot . As in the previous section, we have that

By replacing from (25) into (7), the distortion per timeslot yields

In order to minimize the average distortion over the timeslots at an arbitrary spatial point , we need to solve the following optimization problem, implicitly, we are assuming that sensor ( )th encodes at a constant rate over timeslots. This extent will be verified later on in this section:

where the constraint in (27) is introduced to ensure the stability of the system. Unfortunately, a closed-form solution for cannot be obtained for this problem. Instead, we attempt to solve an approximate problem in which we assume that only codeword will be used by the FC to reconstruct the random field in . Yet, suboptimal (the FC will actually use both codewords, namely and ), this solution outperforms those obtained in delay-constrained scenarios (see computer simulations section). Bearing all this in mind, the new cost function which follows from (26) can be readily expressed as

Clearly, only the second term in the summation of the cost function is relevant to the optimization problem, which can be rewritten as

It is straightforward to show that this problem is convex. Hence, one can construct the lagrangian as follows:

where is the Lagrange multiplier. By setting the first derivative of (30) w.r.t. to zero we obtain

with denoting the negative real branch of the Lambert function [15]. As for the computation of , the future channel gains ( ) would be needed, in principle. However, as this noncasuality requirement vanishes: by the law of large numbers, we have that

and, hence, can be readily obtained by replacing this last expression into the constraint of (29), namely

where we have defined

Finally, replacing into (31) yields

and, by using into (40), the quantization noise for the th sensor node reads:

which evidences that the encoding rate is constant over timeslots (as initially assumed) and over sensors too.

As in previous section, we let be the number of samples encoded in channel uses (i.e., one timeslot). Again, the rate at the output of the C&E encoder must satisfy

To stress that expression (40) differs from (25) in that the C&E encoder assumes that the FC will use to decode and, hence, has been replaced by . Therefore, from (7) and the definition of in (20), we have that for the current block of samples the distortion reads

By averaging over timeslots, the following optimization problem results:

Solving this problem leads to a closed-form solution that is identical to that of the QEDT case, namely,

Finally, replacing into (40) yields

that is, the encoding rate in CEDT networks is constant over sensors and timeslots, as implicitly assumed in the score function (43). To remark, the stability analysis of Section 4.2 also applies here.

In delay-tolerant networks, each sensor encodes and transmits a variable number of samples per timeslot. As a result, the time elapsed until the FC receives the first samples from all the sensors in the network (which allows for the reconstruction of the first realizations of the random field) is unavoidably larger than in delay-constrained networks. In this section, we attempt to characterize such latency. To that aim, we start by analyzing the time needed for one sensor to transmit consecutive samples of the random field. Next, we derive the latency of the QEDT and CEDT encoding strategies, respectively.

Figure 5 depicts the (pertimeslot) distortion in the reconstructed random field for both the QEDC and QEDT encoding strategies and different SNR values. For the QEDC strategy, we show the average value along with the confidence interval (to recall that, unlike in the QEDT case, the distortion in QEDC encoding varies from timeslot to timeslot). Several conclusions can be drawn. First, for each curve there exists an optimal operating point; that is, a network size for which distortion can be minimized. The intuition behind this fact is that, despite that spatial variations of the random field are better captured by a denser grid of sensors, for a total bandwidth constraint the available rate per sensor progressively diminishes, this resulting into a more rough quantization of the observations. Thus, the optimal trade-off between these two effects needs to be identified. Second, the distortion associated to delay-tolerant strategies is, as expected, lower than for delay-constrained ones. Moreover, the lower the average SNR in the sensor-to-FC channels (namely, sensors with lower transmit power), the higher the gain (up to 3 dB for SNR = 0 dB). Third, guaranteing buffer stability in the QEDT scheme only results into a marginal penalty in distortion, as shown in the curves labeled with and . Complementarily, in Figure 6, we depict buffer occupancy for several values of . For , the system is clearly unstable. Conversely, by letting take positive values, for example, for as in Figure 5, the average buffer occupancy can be kept under control (with a relatively small average buffer occupancy of samples, in this case). Clearly, increasing has a two-fold effect: the average buffer occupancy diminishes but, simultaneously, the resulting distortion increases.

The rate at which distortion decreases for the QEDC and QEDT schemes (evaluated at their respective optimal operating points) for an increasing SNR is shown in Figure 7. For intermediate distortion values, the gap is approximately 4 dB. That is, for a prescribed distortion level, the energy consumption in delay-constrained networks is 2.5 times higher.

Figure 8 illustrates the average distortion in the reconstructed random field for the CEDC and CEDT encoding strategies. As in quantize-and-estimate encoding, there exists an optimal number of sensors nodes. Finding such reveals particularly useful for random fields with low per sensor, since the curve is sharper in this case. The gap between the minimum distortion attainable by the CEDC and CEDT schemes (which results from an adequate exploitation of channel fluctuation in the delay-tolerant approach) is approximately 2-3 dB. Concerning buffer occupancy-distortion tradeoffs, the same comments as in the quantize-and-estimate case apply.

Next, in Figure 9, we compare the distortion attained by QEDT/CEDT encoding strategies for random fields with low and high spatial variabilities ( , , resp.). Due to the fact that CEDT is capable of exploiting spatial correlation, it always outperforms QEDT. Moreover, the higher the spatial correlation ( ), the larger the gap between the curves.

Finally, in Figures 10 and 11 we depict the average latency for the QEDT and CEDT strategies, respectively. Interestingly, there exists a trade-off in terms of attainable distortion versus latency. Whereas in CEDT encoding latency exhibits a linear increase in the number of sensors, in QEDT encoding latency grows logarithmically (i.e., more slowly). However, CEDT schemes attain a lower distortion than QEDT ones. Besides, in Figure 10 it is also worth noting the perfect match between simulations and numerical results and, unsurprisingly, that the higher the average , the lower the latency. Also, Figure 11 reveals that by using an appropriate value of (i.e., ), the latency associated to the approximate model described in Section 6.3 matches the actual one.

In this paper, we have extensively analyzed the problem of random field estimation with wireless sensor networks. In order to characterize the dynamics and spatial correlation of the random field, we have adopted a stationary homogeneous Gaussian Markov Ornstein-Uhlenbeck model. We have considered two scenarios of interest: delay-constrained (DC) and delay-tolerant (DT) networks. For each scenario, we have analyzed two encoding schemes, namely, quantize-and-estimate (QE) and compress-and-estimate (CE). In all cases (QEDC, QEDT, CEDC and CEDT), we have carried out an extensive analysis of the average distortion experienced in the reconstructed random field. Moreover, for the QEDT and CEDT strategies we have derived closed-form expressions of (i) the average distortion in the estimates, and (ii) the optimal number of samples of the random field to be encoded in each timeslot (under some simplifying assumptions). Interestingly, the resulting pertimeslot distortion in DT scenarios is deterministic and constant whereas, in DC scenarios, it ultimately depends on the fading conditions experienced in each timeslot. Next, we have focused on the latency associated to the QEDT and CEDT strategies. We have modeled our system as an absorbing Markov chain and, on that basis, we have fully characterized the pdf, CDF, and the average latency for the QEDT case. For CEDT encoding, we have identified an approximate system model suitable for the computation of the average latency. Simulation results reveal that, under a total bandwidth constraint, there exists an optimal number of sensors for which the distortion in the reconstructed random field can be minimized (QEDC, QEDT, CEDC and CEDT cases). This constitutes the best trade-off in terms of, on the one hand, the ability to capture the spatial variations of the random field and, on the other, the persensor channel bandwidth available to encode observations. Besides, the distortion associated to delay-tolerant strategies is, as expected, lower than for delay-constrained ones: some 2-3 dB for both the QE and CE encoding schemes. Moreover, buffer occupancy can be kept at very moderate levels (3 timeslots) with a marginal penalty in terms of distortion (less than 0.3 dB). We also observe that CE schemes effectively exploit the spatial correlation and, by doing so, attain a lower distortion than their QE counterparts (DC and DT scenarios). As far as latency is concerned, we have empirically shown that CEDT exhibits a linear increase in the number of sensors whereas in QEDT encoding latency grows logarithmically (i.e., more slowly). However, CEDT schemes attain a lower distortion than QEDT ones. Besides, for the QEDT case, there is a perfect match between simulations and the theoretical model and, for the CEDT case, latency can be accurately represented by adequately parameterizing the aforementioned approximate system model.