Adaptive sparse random projections for wireless sensor networks with energy harvesting constraints
 Rong Ran^{1} and
 Hayong Oh^{2}Email author
https://doi.org/10.1186/s1363801503243
© Ran and Oh; licensee Springer. 2015
Received: 15 November 2014
Accepted: 10 March 2015
Published: 21 April 2015
Abstract
Considering a largescale energyharvesting wireless sensor network (EHWSN) measuring compressible data, sparse random projections are feasible for data wellapproximation, and the sparsity of random projections impacts the mean square error (MSE) as well as the system delay. In this paper, we propose an adaptive algorithm for sparse random projections in order to achieve a better tradeoff between the MSE and the system delay. With the energyharvesting constraints, the sparsity is adapted to channel conditions via an optimal power allocation algorithm, and the structure of the optimal power allocation solution is analyzed for some special case. The performance is illustrated by numerical simulations.
Keywords
1 Introduction
Energy supply is a major design constraint for conventional wireless sensor networks (WSNs), and the lifetime is limited by the total energy available in the batteries. Some specific sensors in WSNs may consume more energy than the radio during a long acquisition time [1]. Replacing the batteries periodically may prolong the lifetime but not be a viable option when the replacement is considered to be too inconvenient, too dangerous, or even impossible when sensors are deployed in harsh conditions, e.g., in toxic environments or inside human bodies. Therefore, harvesting energy from the environment is a promising approach to cope with battery supplies and the increasing energy demand [2]. The energy that can be harvested includes solar energy, piezoelectric energy, or thermal energy, etc. and is theoretically unlimited. Besides, background radiofrequency (RF) signals radiated by ambient transmitters can also be a viable new source for wireless power transfer (WPT) [3,4] and (Ng et al.:Secure and Green SWIPT in Distributed Antenna Networks with Limited Backhaul Capacity, submitted). Unlike the conventional WSNs that are subject to a power constraint or sum energy constraint, each sensor with energy harvesting capabilities is, in every time slot, constrained to use the most amount of stored energy currently available, although more energy may be available in the future slot. Therefore, a causality constraint is imposed on the use of the harvested energy. Current researches on the energy harvesting issues mostly have focused on wireless communication systems. Gatzianas et al. [5] considered a crosslayer resource allocation problem to maximize the total system utility, and Ho and Zhang [6] studied the throughput maximization with causal side information and full side information for wireless communication systems. Ng et al. [3] studied the design of a resource allocation algorithm minimizing the total transmit power for the case when the legitimate receivers are able to harvest energy form RF signals for a multiuser multipleinput singleoutput downlink system. Energy management policies were studied for energyharvesting wireless sensor networks (EHWSNs) in [7], where sensor nodes have energyharvesting capabilities, aiming at maximizing the system throughput and reducing the system delay.

Considering the wireless fading channels, we verify that the random projection matrix satisfies the property that the inner product between two projected vectors are preserved in expectation, and then provide a lower bound of the system delay for achieving an acceptable data approximation error probability.

We give a new definition of the sparsity of random projections and formulate the optimal sparsity problem which is converted into an optimal power allocation problem for maximizing the system throughput. Unlike the conventional energy allocation problem, due to battery dynamics and channel dynamics, the closedform solution may not be available. Therefore, we study a special case that the battery capacity is not bounded to find the structure of the optimal solution. Specifically, in the case the problem is converted into a convex optimization problem, then the closedform solution is obtained in terms of Lagrangian multipliers.
The rest of paper is organized as follows. Section 2 gives the system model and overview previously known results on sparse random projections. Section 3 redefines the sparsity and formulate the optimal sparsity problem for EHWSNs. Section 4 considers a specific case and address the structure of the optimal solution. Section 5 provides the simulation results. Finally, Section 6 concludes the paper.
2 System model
2.1 Compressible data and sparse random projections
Therefore, each random projection vector is pseudorandomly generated and stored in a small space.
Corollary 1.
□
3 Adaptive sparse random projections
3.1 Sparse random projections with channel fading
where g _{ ij } gives the probability of a projection from sensor node j at time slot i. The details of g _{ ij } will be illustrated in the next section.
Proposition 1.
□
Proposition 1 states that an estimation of the inner product between two vectors, using the matrix of sparse random projections (4), are correct in expectation and have bounded variance. If there is a signal and a matrix of sparse random projections satisfy the conditions (8) and (16), respectively, we can achieve the following proposition:
Proposition 2.
with probability at least 1−n ^{−γ }, if the k largest transform coefficients in magnitude give an approximation with error \(\left \\textbf {u}\hat {\textbf {u}}_{\textit {opt}}\right \_{2}^{2} \leq \eta \left \\textbf {u}\right \_{2}^{2}\).
where (21) comes from the fact that \(\frac {\textbf {u}_{\infty }}{\textbf {u}_{2}}\leq C\). Thus we can obtain a constant probability p by setting \(m_{1}=\mathcal {O}\left (\frac {2+\sum _{j=1}^{n}\frac {3}{\min _{i} g_{\textit {ij}}}C^{2}}{\epsilon ^{2}}\right)\). For any pair of vectors u and v _{ i }, the random projections \(\frac {1}{L}\Phi \textbf {u}\) and \(\frac {1}{L}\Phi \textbf {v}_{i}\) produce an estimate \(\hat {w}_{i}\) that lies outside the tolerable approximation interval with probability at most \(\phantom {\dot {i}\!}e^{c^{2}m_{2}/12}\) where 0<c<1 is the some constant and L _{2} is the number of independent random variables w _{ i } which lie outside of the tolerable approximation interval with probability p. Setting \(m_{1}=\mathcal {O}\left (\frac {2+\sum _{j=1}^{n}\frac {3}{\min _{i}g_{\textit {ij}}}C^{2}}{\epsilon ^{2}}\right)\) and \(m_{2}=\mathcal {O}((1+\gamma)\log _{n})\) obtain p=1/4, and p _{ e }=n ^{−γ } for some a constant γ>0. Finally, for \(m=m_{1}m_{2} = \mathcal {O}\left (\frac {\left (1+\gamma \right)}{\epsilon ^{2}}\left (2+\sum _{j=1}^{n}\frac {3}{\min _{i}g_{\textit {ij}}}C^{2}\right)\log n\right)\), the random projections Φ can preserve all pairwise inner products within an approximation error ε with probability at least 1−n ^{−γ }.
□
Proposition 2 states that sparse random projections can produce a data approximation with error comparable to the best kterm approximation with high probability.
3.2 Optimal power allocation based sparsity adaption
where \(p_{\textit {ij}}^{*}\) is the allocated energy for node j during the ith time slot. \(p_{\textit {ij}}^{*}\) is determined in term of full information consisting of past and present and future channel conditions and amount of energy harvested. The case of full information may be justified if the environment is highly predictable, e.g., the energy is harvested from the vibration of motors that turned on only during fixed operating hours and lineofsight is available for communications.
where \(\alpha _{i}=\left [\ln 2\sum _{k=i}^{m}(\lambda _{k}\mu _{k})\beta _{i}\right ]^{1}\), μ _{ m }=0 and \(\gamma _{i}=\frac {\left h_{\textit {ij}}\right ^{2}}{\sum _{l=1,l\neq j}^{n}\left h_{\textit {il}}\right ^{2}E_{\textit {il}}}\).
3.3 Structural solution
If the battery capacity is finite, the optimal waterlevel is not monotonic. Therefore, the structure of the optimal energy allocation cannot be described in a simple and clear way, and an online programming may be required. Since we are more interested in an offline power allocation structure, we study the following special case.
Proposition 3.
if E _{ max }=∞, the optimal water levels are nondecreasing as α _{ i }≤α _{ i+1}. In addition, the water level changes when all the energy harvested before the current transmission are used up.
Proof: Without the battery capacity constraint, the water level is given as \(\alpha _{i}=\left (\ln 2\sum _{k=i}^{m}\lambda _{k}\right)^{1}\). Since λ _{ k } ≥0,∀k, we have α _{ i }≤α _{ i+1}. If α _{ i }≤α _{ i+1}, by definition \(\alpha _{i}=\left (\ln 2\sum _{k=i}^{m}\lambda _{k}\right)^{1}\), we get λ _{ i }≠0 and λ _{ i }>0. So the complementary slackness condition (32) only holds when \(\left (\sum _{k=1}^{i}p_{\textit {kj}}\sum _{k=1}^{i1}E_{\textit {kj}}\right) = 0\), which means all stored energy should be used up before the current transmission. □
The case of E _{max}=∞ represents an ideal energy buffer which refers to a device that can store any amount of energy, does not have any inefficiency in charging, and does not leak any energy over time. As an example, consider a sensor node installed to monitor the health of heavy duty industrial motors. Suppose the node operates using energy harvested from the machine’s vibrations, the harvested energy is greater than the consumed power and the health monitoring function is desired only when the motor is powered on. Proposition 3 presents an analytically tractable structure of the optimal sparsity. Intuitively, the harvested energy is reserved in the battery for the use in the later transmission, in order to reduce the effect of causality constraint and improve the flexibility of harvested energy allocation. The optimal water level can be obtained by the power allocation policy and it is structured as follows: the water level is nondecreasing and the harvested energy is used in a conservative way. Based on the structural properties, we can use the following reserve multistage waterfilling algorithm modified based on [18], to achieve the solution:
4 Simulation results
5 Conclusions
In this paper, we proposed to adapt sparsity of random projections according to full channel information for EHWSNs. Compared to the conventional sparse random projections which keep the sparsity constant for the whole transmission slots, the proposed one achieves a better tradeoff between the MSE and the system delay. The optimal sparsity problem is turned into an optimal power allocation maximizing throughput with the energyharvesting constraints. An offline power allocation structure is available for a special case that the battery capacity is infinite. Simulation results have shown that the proposed scheme achieves smaller MSEs than the conventional scheme. Meanwhile, the proposed scheme can also reduce the system delay given an accepted error rate. However, full channel information may not be always available. Therefore, for future work, we will study adaptive sparse random projections with partial channel information.
Declarations
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (Grant. NRF2012R1A1A1014392 and NRF2014R1A1A1003562).
Authors’ Affiliations
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