Optimal Network Size and Encoding Rate for Wireless Sensor Network-Based Decentralized Estimation under Power and Bandwidth Constraints

Abstract

In this paper, we address the problem of decentralized parameter estimation via Wireless Sensor Networks (WSNs). We consider two different encoding strategies, namely, Quantize-and-Estimate (Q&E) and Compress-and-Estimate (C&E) and assume that sensor observations are conveyed to the Fusion Center (FC) over a number of orthogonal Gaussian or Rayleigh-fading channels. We constrain both power and bandwidth to be constant irrespectively of the network size and find approximate closed-form expressions of the optimal number of sensors for a number of cases of interest. Besides, we derive the optimal encoding rate for the Q&E scheme when, in the absence of Transmit Channel State Information (CSIT), sensors must encode their observations at a common and constant rate. For the (successive) C&E strategy, we also determine the encoding order that minimizes the resulting distortion in the FC estimates. We complement the analysis by deriving an expression of the asymptotic distortion when the number of sensors grows without bound, and the rate at which distortion decreases in the high-SNR regime. Finally, we close the paper by presenting some computer simulation results.

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Optimal Network Size and Encoding Rate for
Wireless Sensor Network-based Decentralized
Estimation under Power and Bandwidth
Constraints
Javier Matamoros Student Member, IEEE and Carles Ant´ on-Haro Senior
Member, IEEE
Abstract
In this paper, we address the problem of decentralized parameter estimation via Wireless Sensor
Networks (WSN). We consider two different encoding strategies, namely, Quantize-and-Estimate (Q&E)
and Compress-and-Estimate (C&E) and assume that sensor observations are conveyed to the Fusion
Center (FC) over a number of orthogonal Gaussian or Rayleigh-fading channels. We constrain both
power and bandwidth to be constant irrespectively of the network size and find approximate closed-
form expressions of the optimal number of sensors for a number of cases of interest. Besides, we
derive the optimal encoding rate for the Q&E scheme when, in the absence of Transmit Channel State
Information (CSIT), sensors must encode their observations at a common and constant rate. For the
(successive) C&E strategy, we also determine the encoding order that minimizes the resulting distortion
in the FC estimates. We complement the analysis by deriving an expression of the asymptotic distortion
when the number of sensors grows without bound, and the rate at which distortion decreases in the
high-SNR regime. Finally, we close the paper by presenting some computer simulation results.
This work is partly supported by the European Commission through the NEWCOM++ project (216715), the Generalitat de
Catalunya (2009 SGR 1046) and the Spanish Ministry of Science and Innovation (FPU AP2007-01654).
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I. INTRODUCTION
In recent years, research on Wireless Sensor Networks (WSN) has attracted considerable
attention. Tipically, a WSN consists of one Fusion Center (FC) and a potentially large number
of sensing devices capable of transmitting their measurements over a wireless link. Typical
applications for WSNs encompass parameter estimation (for e.g. environmental monitoring ap-
plications), event detection (for example, accidental release of substances) or positioning and
tracking (see [1] and references therein).
In our setting, the sensors observe a common source of interest, of a random nature, in order
to produce an estimate of some parameter of interest (e.g. humidity, temperature, etc). For a
number of reasons, such as hardware limitations, the observations in the sensors are affected
by some measurement noise. Such noisy observations are then transmitted to the FC where the
parameter of interest associated to the underlying source is estimated. This is the so-called CEO
(Chief Executive/Estimation Officer) which was introduced by Berger [2], and for which the
rate-distortion function for the quadratic Gaussian case was recently derived in [3].
The source-channel coding separation theorem, by which the source and channel coding
problems can be optimally and independently solved [4, Ch. 8], does not hold for Multiple
Access (MAC) channels with correlated sources [5]. A (much simpler) amplify-and-forward
(A&F) strategy is known to scale optimally, in terms of estimation distortion, when the number
of users grows without bound. However, such asymptotic optimality is achieved if distributed
synchronization of the sensor signals can be orchestrated at the physical layer to achieve beam-
forming gains. Conversely, for orthogonal channels, solutions based on the source-channel coding
separation theorem outperform A&F-based ones. Indeed, in [6] the source-coding theorem is
shown to be optimal for the quadratic Gaussian CEO problem.
As far as source coding is concerned, the work in [7] constitutes a generalization to sensor trees
of Wyner-Ziv’s pioneering studies [8]. More precisely, the authors compare two different coding
schemes, namely Quantize-and-Estimate (Q&E) and Compress-and-Estimate (C&E). The former
is a particularization of Wyner-Ziv’s problem to the case where no side information is available
at the decoder; whereas the latter is a successive Wyner-Ziv-based coding strategy capable of
exploiting the correlation among sensor observations. Yet sub-optimal, the performance of this
successive encoding scheme is not far from that of other more complex joint (over sensors) coding
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strategies. Besides, in [9] it is shown to achieve any point of the rate-distortion region of the
quadratic Gaussian CEO problem. In both cases, it is assumed that sensors experience Gaussian
(or, more generally, deterministic) channels, this meaning that the transmissions rates are known
and, consequently, all sensors convey their observations reliably. Going one step beyond, the
authors in [10] derive the rate-distortion region for the case of erasure channels where a fixed and
known number of observations k are reliably received. Nevertheless, in the case that the erasure
pattern is unknown and/or random, joint coding techniques or successive encoding techniques
such as [7], exhibit poor performance. This problem is addressed in [11], where a number of
tradeoffs between reliability and efficiency are examined for a lossless compression setting. [9].
In scenarios with non-reciprocal (e.g. FDD) fading channels, it is often assumed that only
statistical CSI is available at the transmitter. Consequently, the encoding rate at the sensor
nodes cannot be dynamically adjusted to match instantaneous channel conditions. In this context,
the point-to-point communication of Gaussian sources over fading channels has been recently
investigated in [12] and [13] with the ultimate purpose of minimizing the expected distortion
at the FC. These works stand on the seminal work of [14], in which the source is encoded
in multiple layers (each one representing a different channel state) by exploiting the so-called
successive refinement property [15]. Accordingly, the receiver adaptively decodes the information
according to the realization of the channel state.
Contribution: In this paper, we extend and carry out an in-depth analysis of the Q&E and
C&E encoding strategies presented in [7]. First, we focus on a scenario with orthogonal Gaussian
channels where, as commented above, the source-channel coding separation theorem holds. We
go one step beyond [7] and further constrain not only power but also bandwidth to be constant
irrespectively of the number of sensors in the network. With these assumptions, we ask ourselves
whether increasing the population of sensors is always worth doing (in terms of estimation
accuracy) or if, alternatively, an optimal operating point exists. To answer this question, we derive
an approximate closed-form expression of the optimal number of sensors in the network. Next,
we move on the case of orthogonal Rayleigh-fading channels. For benchmarking purposes, we
initially assume that sensors are capable of acquiring instantaneous transmit CSI (CSIT). Being
C&E a successive encoding/decoding scheme, we analytically determine the optimal encoding
order (over sensors) which minimizes the resulting distortion. We complement the analysis by
deriving the asymptotic distortion attainable by the Q&E and C&E strategies for both Gaussian
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and Rayleigh-fading channels. Next, we realistically constrain the sensor nodes in the network
(and, by doing so, we go beyond the point-to-point analysis of [12] and [13]) to operate without
instantaneous transmit CSI. Accordingly, the observations must be encoded at a constant and
common (i.e. identical for all sensors) rate that we determine on the basis of statistical CSI
only. Such encoding rate has an impact on (i) the number of quantization bits allocated to the
encoders; and (ii) the actual number of observations reliably received at the FC due to the
outage probability experienced in the sensor-to-FC fading channels. This is in stark contrast
with [10] and [11] where the quantization process and the transmission problems are regarded
as decoupled. In other words, we analytically find the optimal trade-off in terms of quantization
bits (more finely-quantized observations are beneficial in terms of the overall distortion) vs. the
number of observations actually received at the FC (the higher the number observations, the
better the smoothing of the observation noise). We solve the problem for two cases of interest,
namely, sensors with high and low observation noise. Finally, we determine the asymptotic rate
at which distortion decreases in the high-SNR regime of a large sensor network.
II. SIGNAL MODEL
Consider a WSN composed of one Fusion Center (FC) and N energy-constrained sensor
nodes. The sensors observe a common source of interest which can be modeled as a length-n
(with n sufficiently large) vector x = [x(1),...,x(n)]Tof independent and identically-distributed
(i.i.d) complex circular Gaussian random variables, namely, p?x(1),...,x(n)?=?n
of variance σ2
accuracy. According to Fig. 1, the observation at sensor k can be expressed as
i=1p?x(i)?,
x. Our goal is to estimate x from the sensor observations with the highest possible
yk= x + vk
(1)
where vk denotes memoryless AWGN noise (measurement noise) of variance σ2
over sensors. From this model, it follows that the observations are correlated with correlation
coefficient given by ρ = cov(yk,yl)/(σykσyl) = σ2
surements to the Fusion Center, each sensor encodes a block of n consecutive observations
yk = [y(1)
uk(s) ∈ C. Next, the corresponding indexes1sk∈ {1,...,2nRk};k = 1...N are sent to the FC
vand i.i.d.
x/(σ2
x+ σ2
v). In order to convey the mea-
k,...,y(n)
k]T, where n denotes the temporal variable, into a (length-n) codeword
1As it will become apparent later, the codebook C consists of, at most, 2nRkcodewords.
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over the set of N orthogonal channels. At this point, we introduce Wsas temporal sampling
rate of the source and, hence, for a reliable transmission to occur, the encoding rate Rkmust
satisfy:
?
where wkand pkstand for the transmit power and bandwidth allocated to the k-th sensor/channel,
and Nodenotes the noise spectral density. For the ease of notation and without loss of generality,
in the sequel we consider Ws= 1. The channel (squared) gains, γk;k = 1,...,N, are subse-
quently modeled as Gaussian channels (namely, γk= 1 ∀k), or as independent exponentially-
distributed unit-mean random variables (i.e. Rayleigh-fading channels). Besides, we further
impose a total bandwidth constraint, W, and a sum-power constraint, P. For simplicity, we
evenly allocate power and bandwidth to the set of N sensors, this yielding pk = P/N and
wk= W/N. Hence, the transmission rate at the output of the k-th encoder reads
Rk≤W
with SNR ?
the source. Due to the measurement noise and the resource constraints, some distortion results
which, in the sequel, will be characterized by the following metric:
n
?
III. QUANTIZE-AND-ESTIMATE (Q&E): DISTORTION ANALYSIS
WsRk≤ wklog2
1 +pkγk
wkNo
?
[b/s]
(2)
Nlog2(1 + SNR · γk) [b/s]
(3)
P
WNo. Finally, the FC decodes the received signals and produces an estimate ˆ x of
D =1
n
i=1
E?|ˆ xi− xi|2?.
(4)
In the Q&E scheme, each observation is encoded regardless of any side information that could
be made available by the FC. From [4], the following inequality holds for the rate at the output
of the k-th encoder (quantizer):
Rk≥ I(yk;uk)[b/sample]
(5)
with I(·;·) standing for the mutual information. The encoding process is modeled through the
auxiliary variable uk = yk+ zk, with zk ∼ N(0,σ2
From this, one concludes that uk←→ yk←→ x form a Markov chain and, hence, the mutual
information reads:
zkI) and statistically independent of yk.
I(yk;uk) = H(uk) − H(uk|yk)
= log
?
1 +σ2
x+ σ2
σ2
zk
v
?
[b/sample]
(6)
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