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

- Javier Matamoros
^{1}Email author and - Carles Antón-Haro
^{1}

**2010**:102460

https://doi.org/10.1155/2010/102460

© J. Matamoros and C. Antón-Haro. 2010

**Received: **23 February 2010

**Accepted: **3 May 2010

**Published: **1 June 2010

## Abstract

In this paper, we study the problem of random field estimation with wireless sensor networks. We consider two encoding strategies, namely, Compress-and-Estimate (C&E) and Quantize-and-Estimate (Q&E), which operate with and without side information at the decoder, respectively. We focus our attention on two scenarios of interest: *delay-constrained* networks, in which the observations collected in a particular timeslot must be immediately encoded and conveyed to the Fusion Center (FC); *delay-tolerant* (DT) networks, where the time horizon is enlarged to a number of consecutive timeslots. For both scenarios and encoding strategies, we extensively analyze the distortion in the reconstructed random field. In DT scenarios, we find closed-form expressions of the optimal number of samples to be encoded in each timeslot (Q&E and C&E cases). Besides, we identify buffer stability conditions and a number of interesting distortion versus buffer occupancy tradeoffs. Latency issues in the reconstruction of the random field are addressed, as well. Computer simulation and numerical results are given in terms of distortion versus number of sensor nodes or SNR, latency versus network size, or buffer occupancy.

## 1. Introduction

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.

## 2. Signal Model

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.

### 2.1. Communication Model

*sensing*phase and the

*transmission*phase. In the former, each sensor collects and stores in a local buffer a large block of independent and consecutive observations . Next, in the transmission phase, is block-encoded into a length- codeword in codebook at a rate of bits per sample. The encoding (quantization) process is modeled through the auxiliary random variable with standing for memoryless Gaussian noise with variance and statistically independent of (for the ease of notation, we drop the sample index.). The corresponding codeword index is then conveyed to the FC, in a total of channel uses, over one of the

*orthogonal*channels (for other encoding schemes, such as Compress-and-Estimate in Section 3.2, denotes the index of the

*bin*to which the codeword belongs to. For further details, see [3]). The codeword can only be reliably decoded at the FC if the encoding rate satisfies

### 2.2. Distortion Analysis: A General Framework

*two closest*decoded codewords, namely and , will be used to reconstruct for

*all*the corresponding intermediate spatial points (in noiseless scenarios, that is, for all , this approach turns out to be optimal due to the Markovian property of GMOU processes. For the general case, yet suboptimal, it capitalizes on the codewords which retain more information on the random field at the spatial point ) (see Figure 1), that is

It is worth noting that the variance of the quantization noise and are determined by the encoding strategy in use at the sensor nodes.

## 3. Delay-Constrained WSNs

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.

### 3.1. Quantize-and-Estimate: Average Distortion

### 3.2. Compress-and-Estimate: Average Distortion

*all*the sensor observations but due to the spatial Markov property of the random field model, this is not expected to substantially decrease the encoding rate). Accordingly, the minimum rate per sample can be expressed as follows:

## 4. Delay-Tolerant WSNs with Quantize-and-Estimate Encoding

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.

*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:

*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

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

### 4.1. Average Distortion in the Reconstructed Random Field

*original*cost function of (26), the distortion for an arbitrary point in the th network segment reads

Interestingly, distortion is not a function of the channel gain experienced by the th sensor in timeslot (i.e., distortion does not depend on ). As a result and unlike in QEDC encoding, the distortion experienced in every timeslot is identical. This can be useful in applications where a constant distortion level is needed.

### 4.2. Buffer Stability Considerations

with . By doing so, one can prove (see the appendix) that buffers are stable. Unsurprisingly, this come at the expense of an increased distortion in the estimates (see computer simulation results in Section 7).

## 5. Delay-Tolerant WSNs with Compress-and-Estimate Encoding

*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

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.

### 5.1. Average Distortion in the Reconstructed Random Field

## 6. Latency Analysis

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.

### 6.1. Latency Analysis for a Single Sensor Node

*transient*states ( ) and one

*absorbing*state ( ) defined as follows (see, Figure 3):

for and . For Rayleigh-fading channels, the CDF of the channel gain is given by .

### 6.2. Latency Analysis for QEDT Encoding

Intuitively, latency is a monotonically increasing function in the number of sensors (the more sensors, the larger the time needed to collect all samples). This extent will be verified in Section 7 (Simulation and numerical results).

### 6.3. Latency Analysis for CEDT Encoding

The latency analysis for CEDT strategies if far more involved due to the successive encoding of data that C&E schemes entail. In general, this does not allow for the derivation of closed-form expressions and, thus, we will resort to an approximate (yet accurate) model.

*entire*random field can be reconstructed if, equivalently, samples sent by the first sensor have already been decoded by the FC. The encoding/decoding process for the first sensor is identical in C&E and Q&E schemes and, hence, in order to compute the latency for the reconstruction of the

*random field*, it suffices to compute the time to absorbtion for an

*individual*sensor (sensor #1) as we did in Section 6.1. The only change with respect to the model given in (54) is that the Markov chain has now a total of states (instead of ) and, hence, the size and elements of matrix and vectors and in (57) and (58) must be adjusted accordingly.

As for parameter , which exclusively depends on the pdf of the sensor-to-FC channel gains, it can only be determined empirically (see next section).

## 7. Simulations and Numerical Results

*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.

## 8. Conclusions

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.

## Declarations

### Acknowledgment

This work is partly supported by the Catalan Government (2009 SGR 1046), the EC-funded project NEWCOM++ (216715), and the Spanish Ministry of Science and Innovation (FPU grant AP2007-01654).

## Authors’ Affiliations

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