Distributed and Collaborative Processing in Wireless Sensor Networks

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LI, WENJUN. Distributed and Collaborative Information Processing in Wireless Sensor Networks. (Under the direction of Professor HUAIYU DAI).

Wireless sensor networks, formed by numerous tiny devices capable of sensing, computing, and wireless communication, are emerging as a revolutionary technology with applications in diverse areas. The unique features of wireless sensor networks, and in par-ticular the power scarcity of sensor nodes, have brought new challenges and problems to the field of distributed and collaborative information processing. In this dissertation, we address some important problems within this broad topic, including data gathering, dis-tributed detection, and disdis-tributed consensus, with the emphasis on efficient use of stringent system resources to achieve certain application-specific objectives.

We start in Chapter 2 with an investigation of schemes for collecting sensor data at a sink node, which are basic building blocks for all sensor network applications in hierarchi-cal networks. A central problem is to explore the inherent tradeoff between two inconsistent performance measures: throughput and energy efficiency. We consider a cross-layer frame-work, where spatial diversity is exploited through multiuser detection techniques at the physical layer to achieve dramatically increased throughput, and deterministic or random-ized medium access methods are employed to avoid excessive interference. Our results on the optimal performance of different medium access control schemes coupled with different linear multiuser detectors provide useful insights into cross-layer design in WSN.

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mal local mapping rules yields better detection performance as measured by error exponents compared with PAC fusion under the same transmission power constraint. Subsequently, in Chapter 4, we investigate distributed detection via multi-hop transmissions, which signif-icantly reduces the energy consumption over direct transmission in networks where nodes have widely different distances towards the fusion center. Several multihop fusion rules, including Multihop Forwarding (MF), Histogram Fusion (HF) and Log-likelihood Ratio Fu-sion (LF) are investigated. We demonstrate how transmisFu-sion structures are designed along with fusion rules to achieve optimal tradeoffs between detection performance and energy efficiency.

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by

Wenjun Li

A dissertation submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the Degree of

Doctor of Philosophy

Electrical Engineering

Raleigh, NC 2007

Approved By:

Dr. Hamid Krim Dr. Hien Tran

Dr. Brian Hughes Dr. Alexandra Duel-Hallen

Dr. Huaiyu Dai

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Dedication

To my parents

To my husband Kai

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Biography

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Acknowledgements

The printed pages of this dissertation hold far more than the culmination of years of study. Thanks are due to many who have afforded this truly learning and enjoyable experience.

My deepest gratitude goes first and foremost to Professor Huaiyu Dai, my advisor, for his constant guidance and support. He has walked me through all stages of my graduate study and the writing of this dissertation. His wisdom, vision and rigorous scholarship have immensely influenced my academic growth. Without his guidance on the overall picture as well as his attention to details, this dissertation could not have reached its present form.

Second, I would like to express my heartfelt gratitude to Professor Brian Hughes, Professor Alexandra Duel-Hallen and Professor Hamid Krim, for their illuminating instruc-tion, encouragement, and valuable feedbacks on my research work. I learned fundamentals of communication and signal processing theory from their courses. I especially enjoy their organized and inspiring way of teaching, from which I learned not only the knowledge, but also a logical and artful fashion in presenting technical ideas. I would like to acknowledge and thank Professor Hien Tran for taking time off his busy schedule to serve as my commit-tee member, and to provide thoughtful comments. I would also like to take the opportunity to thank all professors and teachers who have imparted to me knowledge in my student life. It was my pleasure to have worked closely with my colleagues: Dr. Hongyuan Zhang, Dr. Quan Zhou and Yanbing Zhang. I have benefited significantly from enlightening discussions with them. I am also grateful to many friends I have made at NC State, for helping me out in my difficult times, and for the great memory we shared.

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List of Figures

2.1 Maximum asymptotic throughput of round-robin with different linear detectors 22 2.2 Maximum asymptotic throughput of round-robin and slotted ALOHA with

different linear detectors, β = 1 . . . 23 2.3 Minimum effective energy with throughput constraint for different MAC

schemes with the decorrelator (fixed m or a), ∆ = 5, N = 10, β = 1, σ2_{T /G}_{= 1 . . . .} ₂₄

2.4 Minimum effective energy with throughput constraint for different MAC schemes with the decorrelator (joint optimization),N = 10,β = 1,σ2_{T /G}_{= 1 25}

2.5 Optimalmorafor minimum effective energy with throughput constraint for different MAC schemes with the decorrelator (joint optimization), N = 10, β = 1,σ2_{T /G}_{= 1 . . . .} ₂₅

2.6 Minimum effective energy with throughput constraint for round-robin with different linear detectors (joint optimization),N = 10,β = 1, σ2_{T /G}_{= 1 .} ₂₆

2.7 Throughput comparison: multiuser scheduling v.s. round-robin (with the decorrelator),N = 10,n= 1000,σ_L= 8dB,β = 1 . . . 29 3.1 Distributed detection fusion schemes: (a) PAC fusion; (b) MAC fusion . . . 33 3.2 Error probabilities for detection of a sinusoid signal,m = 1, σ = 1, ρ= 0.5,

ω0 =π/4, Pav = 2,Ptot= 10 . . . 46 3.3 Error exponents for detection of a sinusoid signal, m = 1, σ = 1, ρ = 0.5,

ω0 =π/4, Pav = 2,Ptot= 10 . . . 47 3.4 Error probabilities for detection of a constant signal under average power

constraint, m= 1,σ = 0.5,ρ= 0.5,Pav = 1 . . . 50 3.5 Error exponents for detection of a constant signal under average power

con-straint,m= 1, σ= 0.5,ρ= 0.5,Pav = 1 . . . 51 3.6 Error exponents for detection of an autoregressive signal under average power

constraint, Pav = 1. (a) ρ = 0.5,Π0 = 1, σ = 1 (Γ = 1); (b) ρ = 0.8,Π0 =

1, σ = 3 (Γ = 1/9); (c) ρ = 0.8,Π0 = 1, σ = 1 (Γ = 1); (d) ρ = 0.8,Π0 =

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4.3 Total transmission energy for direct transmission and various multihop fusion schemes,n= 100,κ= 2.5. . . 71 5.1 A realization ofG(100,0.3) . . . 76 5.2 Illustration of the induced graph from distributed clustering of a realization

of G(300, r(300)). Nodes are indicated with small dots, cluster-heads are indicated with small triangles, cluster adjacency are indicated with solid lines, and the transmission range of cluster-heads (not clusters) are indicated with dashed circles. . . 78 5.3 Decay of relative norm-2 error for conventional and cluster-based gossip

al-gorithms onG(100, r(100)) . . . . 85 6.1 Upper bound for the conductance of a Markov chain onG(n, r) . . . . 97 6.2 Nonreversible chain used in the centralized algorithm: incoming probabilities

for the east state of a west boundary cluster (left) and an internal cluster (right) are depicted. . . 101 6.3 Illustration of neighbor classification and virtual neighbors for boundary

nodes. Note that for an east boundary node i, there can only be virtual east neighbors of the first category (i, j, k ∈ Ne0

i ), and virtual north and south neighbors of the second category (l∈Nb3

i ). . . 105 6.4 The Markov chain used in LADA: combined outgoing probabilities (solid

lines) and combined incoming probabilities (dotted line) for the east state of node iare depicted . . . 107 6.5 The Markov chain used in LADA-U: outgoing probabilities (solid lines) and

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List of Tables

3.1 Bayesian error exponents for the centralized and distributed detection schemes 45 3.2 Scaling factors and Bayesian error exponents for the centralized and

dis-tributed detection schemes, ρ = 0.5, m = σ = 1, ω₀ = π/4, P_av = 2, and Ptot= 10 . . . 46 4.1 Scaling Laws of the Total Transmission Energy . . . 70 4.2 Best Bhattacharyya distances and error probabilities for detecting

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List of Abbreviations and Acronyms

APC average power constraint

AR(1) first-order autoregressive

CDF cumulative distribution function CDMA code-division multiple access

C-LADA cluster-based location aided distributed averaging CSI channel state information

DFT delayed first transmission

DS-CDMA direct-sequence code-division multiple access

FC fusion center

HF histogram fusion

i.i.d. independent and identically distributed

K-L Kullback-Leibler

LADA location aided distributed averaging

LADA-U location aided distributed averaging - uniform LDP large deviation principle

LF log-likelihood ratio fusion

LLR log-likelihood ratio

LRQ log-likelihood ratio quantizer

MAC medium access control / multi-access channel

MDS minimum dominating set

MF matched filter / multihop forwarding MIMO multiple-input multiple-output

MMSE minimum mean squared error

MPR multi-packet reception

MST minimum spanning tree

N-P Neyman-Pearson

PAC parallel access channel pdf probability density function

PHY physical

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QoS quality of service

SDMA space-division multiple access SENMA sensor network with mobile agents SINR signal-to-interference-and-noise ratio

SNR signal-to-noise ratio

SPT shortest path tree

TBMA type-based multiple access

TBRA type-based random access

TDMA time-division multiple access

TPC total power constraint

WLAN wireless local area network

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Chapter 1

Introduction

1.1 Wireless Sensor Networks

A wireless sensor network (WSN) is a wireless network consisting of spatially distributed autonomous devices using sensors to cooperatively monitor physical or environ-mental conditions, such as temperature, sound, vibration, pressure, motion or pollutants, at different locations [3]. As the advances in hardware technology enables the manufacturing of low-cost sensors, wireless sensor networks have become one of the most active research fields in recent years. Although originally used for military surveillance, wireless sensor net-works have found wide applications in many other fields, such as environmental monitoring, seismic detection, inventory tracking and health monitoring.

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number of sensors improves the reliability of the decision. An immediate consequence of dense deployment is that data observed by co-located sensors are highly correlated, and only the most useful information needs to be extracted from the redundant data. Vari-ous compression techniques such as distributed source coding and multihop fusion can be employed to reduce the required data rate for transporting the desired information to the sink. Thirdly, sensor nodes are tiny, low-cost electronic devices that are prone to fail-ure, and wireless links are highly instable. Therefore, algorithms and protocols in WSN must be robust to frequent topology changes under unattended operation. Last but not the least, sensor nodes are extremely power-limited: they rely on irreplaceable batteries, whose technologies are still lagging behind the advances in processor designs. Consequently, energy resource continues to be an ultimate bottleneck in wireless sensor networking, and energy conservation for network lifetime maximization is a central theme in all WSN studies [26, 28, 29, 43, 92]. All the above unique features bring new challenges to the design and development of WSN.

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tradeoff between energy efficiency and scalability.

The advances in wireless sensor networks have created new problems and opened new research directions in the field of distributed and collaborative information processing. In this dissertation, we address some important issues of this broad topic, including data gathering, distributed detection and distributed consensus. In subsequent sections of this chapter, we give a more detailed introduction of each sub-topic treated this dissertation.

1.2 Optimal Throughput and Energy Efficiency in WSN

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schemes, round-robin and slotted ALOHA, as well as three linear detectors, the matched filter (MF), the decorrelator, and the linear minimum-mean-squared-error (MMSE) detec-tor. The throughput (defined as the number of successfully received packets per slot) and the effective energy (defined as the average energy consumption per successfully received packet) are derived for each MAC scheme and linear detector pair. We optimize the average number of transmissions per slot and the transmission power for two purposes: maximiz-ing the throughput, or minimizmaximiz-ing the effective energy subject to a throughput constraint. By comparing the optimal performance of different MAC schemes equipped with different detectors, we draw important tradeoffs involved in the sensor network design. We also con-sider the impact of shadowing effect on the system performance and show that multiuser scheduling greatly boosts the throughput in low SNR region and hence is of particular significance for sensor network applications.

1.3 Distributed Detection in WSN

Large-scale sensor networks are suitable for distributed detection of events or tar-gets, as the detection error probability decays exponentially with the number of sensors under appropriate processing at sensors and the fusion center. The distributed detection problem can be formulated as follows: distributed sensors take observations and commu-nicate local decisions with the fusion center, which makes a final decision about whether Hypothesis 0 or Hypothesis 1 is true based on the received information. Most existing works on distributed detection have focused on designing local mapping rules and fusion rules to maximize the detection performance under a given bandwidth constraint, and few has framed the problem in network settings and considered power efficiency. In fact, the conventional way of treating the decision fusion independently from communication often leads to energy-inefficient designs. It is our aim to improve such designs by taking realistic communication constraints and costs into account.

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networks. Alternatively, we can employ a multiple access channel (MAC), whose bandwidth requirement does not depend on the number of sensors. However, due to the additive nature of the channel, the received signal at the fusion center is generally not sufficient for reliable detection. It is known that when the sensor observations are independent, MAC fusion of local log-likelihood-ratio (LLR) or local histograms yields desirable performance [75, 76, 81, 82, 83, 84]. We explore the possibility of performing distributed detection over a MAC with correlated sensor observations, which arise naturally when sensors are packed densely together. We consider a one-dimensional sensor network and two exemplary problems–detection of a deterministic signal in correlated Gaussian noise and detection of a first-order autoregressive signal in independent Gaussian noise, under the Neyman-Pearson (N-P) and Bayesian criteria. We assume that local observations are mapped according to a certain function subject to a power constraint. Using the large deviation approach, we demonstrate that for the deterministic signal in correlated noise problem, with a specially-chosen mapping rule, MAC fusion achieves the same asymptotic performance as centralized detection under the average power constraint (APC), while there is always a loss in error exponents associated with PAC fusion. Under the total power constraint (TPC), MAC fusion still results in exponential decay in error exponents with the number of sensors, while PAC fusion does not. For the autoregressive signal problem, we propose a suboptimal MAC mapping rule which performs closely to centralized detection for weakly-correlated signals at almost all SNR values, and for heavily-correlated signals when SNR is either high or low. Finally, we show that although the lack of MAC synchronization always causes a degradation in error exponents, such degradation is negligible when the phase mismatch among sensors is sufficiently small.

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scheme, the quantized observations are sent to the fusion center along the shortest path tree without further processing at relay nodes. For HF, each sensor transmits the histogram of the observations of its descendants and itself, which achieves further energy reduction relative to MF when the number of quantization bits is small. For LF, the normalized log-likelihood ratio (LLR) values for subsets of nodes are computed and propagated along a minimum spanning tree, such that the fusion center acquires an estimate of the normalized LLR of all sensors’ observations, which is used to decide the hypothesis. It is demonstrated that the LF scheme outperform MF and HF in terms of the required energy to achieve a certain probability of error, as well as the energy scaling law with the network size.

1.4 Distributed Consensus in WSN

In most sensor network applications, it is only an aggregate function of sensor values that is needed. For example, in the distributed detection problem mentioned above, a sufficient statistic for N-P and Bayesian detection is the normalized LLR value, which is equal to the average of individual LLR values when sensor observations are conditionally independent. Thus the problem boils down to computing the average of node values. De-pending on the network structure, this can be done with a spanning-tree based centralized algorithm in hierarchical networks, or a purely distributed algorithm in flat networks. Lines of research on the first approach has been surveyed in [41, 42]. In this dissertation, we will focus on purely distributed algorithms due to their simplicity, salability and robustness. We are interested in particular the distributed consensus problem, where the aim is to compute the average of node values (which can be easily extended to general weighted sums) through iterative local information exchange. For this problem, two performance measures, the time complexity and the message complexity (relating to energy efficiency) are of interest. We investigate methods to mitigate the slow convergence and high communication complexity of known algorithms in literature.

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be-having as a single entity. In this scenario, after initiation, only inter-cluster communication and intra-cluster broadcast are needed to update the values of all nodes. A distributed clustering algorithm is provided with which nodes are clustered in a distributed manner without the requirement of global knowledge. We then propose cluster-based distributed averaging algorithms in forms of fixed iteration and random gossiping following known algo-rithm in literature [17, 18, 107]. Clustering essentially allows nodes in neighboring clusters to be joined, hence the resultant graph is better-connected and the algorithm convergence is faster. Moreover, since the number of clusters is much smaller than the number of nodes, the communication and computation burden is significantly reduced.

While the message complexity is improved in the order sense through clustering, the order of time complexity remains the same, mainly because the proposed algorithms are still based on symmetric weight matrices, or reversible Markov chains as those in literature [17, 18, 107]. Our second approach to improve the performance of distributed consensus is to explore algorithms based on nonreversible chains, which have been shown to mix substantially faster than related reversible ones by overcoming the diffusive behavior [23, 32]. We propose a class of Location-Aided Distributed Averaging (LADA) algorithms for wireless networks, where nodes’ location information is used to construct nonreversible chains that facilitates distributed computing and cooperative processing. We first show that it is possible to achieve an ²-averaging time of O(r−1log(²−1))1 in a wireless network with transmission radius r with a centralized algorithm, which is close to a performance lower bound derived based on resistance, an invariant of Markov chains. We then present a distributed LADA algorithm, which utilizes only the direction information of neighbors to construct nonreversible chains. The constructed chain does not naturally possess a uniform stationary distribution, which is in turn compensated by a weight estimation procedure to yield the average estimate. It is shown that LADA achieves the same scaling law in averaging time as the centralized scheme in wireless networks for all r satisfying the connectivity requirement. Finally, a cluster-based LADA (C-LADA) varaint is also proposed to further

1_{We use the following order notations in this dissertation: Let}_f₍_n_{) and}_g₍_n_{) be nonnegative functions for} n≥0. We sayf(n) =O(g(n)) andg(n) = Ω(f(n)) if there exists somekandc >0, such thatf(n)≤cg(n) forn≥k;f(n) = Θ(g(n)) iff(n) =O(g(n)) as well as f(n) = Ω(g(n)). We also sayf(n) =o(g(n)) and

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improve on the message complexity.

1.5 Dissertation Organization

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Chapter 2

Optimal Throughput and Energy

Efficiency of WSN

2.1 Introduction

In this chapter, we study the problem of transporting data from multiple sensor nodes to a common receiver. As mentioned in Chapter 1, the need for transporting observed data or functions of them to a central controller arises in many sensor network applications, such as distributed detection and estimation, and distributed tracking. Power, the scarcest resource in sensor networks, must be conserved. Meanwhile, the sensor network should be able to maintain a certain throughput (which is equivalent to a certain delay constraint), in order to fulfill the QoS requirement of the end user, and to ensure the stability of the network. Typically, the throughput and the energy efficiency are inconsistent, and there exists a tradeoff between the two measures. The objective of this study is to explore the maximum achievable throughput under certain network configurations and receiver structures, as well as optimal network designs that achieve the desired throughput with minimal energy consumption.

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conserves sensors’ energy by freeing them from packet relaying, routing and data processing routines, and good performance can be guaranteed even with minimal transmission power. We assume that each node constantly has packets to transmit; the transmission is slotted and the slot length T equals the transmission time of one packet. The sensors and the receiver constitute a multiple access network. Under the traditional collision channel model (i.e., single transmission means success and simultaneous transmissions results in failure), the throughput of the multiple access network is limited: the maximum throughput per slot is 1 for time-division-multiple-access (TDMA), and is only 1/e for slotted ALOHA with optimal decentralized control [90]. Such a throughput may not be sufficient for sensor network applications. Nevertheless, advanced signal processing techniques such as multiuser detection [103] enable correct reception of simultaneous transmitted packets at the physical layer, and consequently, Ghez et. al. proposed the multi-packet reception (MPR) model [39], which revolutionized the underlying assumption of MAC layer design. We assume that the receiver is equipped with N antennas and a linear multiuser detector followed by single-user decoders. The packet transmission is considered successful as long as the output signal-to-interference-ratio (SIR) of the linear detector is above a certain threshold β [48]. The transmitting sensors and the receive antenna array thus form a virtual multiple-input-multiple-output (MIMO) system, which can also be viewed as a space-division multiple access (SDMA) system. Note that due to the analogy between the direct-sequence code-division multiple access (DS-CDMA) system and the MIMO system, the analysis in this chapter can also be adapted to the DS-CDMA system with a single receive antenna and spreading gain N. But since the received power adds up across the antennas, the MIMO system requires only 1/N of the transmission power of the corresponding DS-CDMA sys-tem. A hybrid of CDMA and multiple receive antenna system is also possible, in which case the performance is further enhanced by the effect of “resource-pooling” [49].

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purpose can be either coordinated or random. For coordinated access, we consider round-robin, which is TDMA in essence: the adjacent sensors form a transmission group and the groups are scheduled for access one by one. For random access, we consider the simplest form of slotted ALOHA, known as Delayed First Transmission (DFT) [40]: in each slot every sensor node transmits a packet (new or retransmission) with the same probability p independently. We assume that the receiver transmits a beacon at the beginning of each slot for synchronization [1, 98]. It might require some overhead for the sensor nodes to get some delay estimates for synchronization purpose, and then they can adjust their timing when simultaneously transmitting. It is known that slotted ALOHA is simple and is preferred when the traffic is bursty, but it suffers from certain performance degradation from centrally-controlled networks, and we will investigate the exact performance loss in our system. In addition to different MAC schemes, the linear multiuser detector at the receiver can be the single-user matched filter, the decorrelating detector, or the linear MMSE detector. As we will see, both the MAC scheme and receiver structure employed have significant impact on the system performance. For a given MAC scheme with a given linear detector, we optimize the transmit power, as well as the transmission group size (for round-robin) or the transmission probability (for slotted-ALOHA). We study two optimization problems: one is to maximize the throughput, and the other is to minimize the energy consumption subject to a throughput constraint.

We then modify our assumption of pure Rayleigh-fading by admitting shadow fad-ing into our system model. Multiuser diversity can be realized in such a system by allowfad-ing the sensor group with the best shadowing coefficient to transmit during each slot, and is shown to have great significance in energy-conservation for sensor networks. Fairness con-cerns of multiuser scheduling can be remedied by enabling the movement of the receiver to induce a dynamic shadowing environment, or other known algorithms with little throughput sacrifice (see Section 2.6).

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reception model [1, 102]. The design of transmission probability of slotted ALOHA by ex-ploiting uplink CSI in a distributed fashion is studied in [1]. In [102], the authors analyze slotted-ALOHA sensor networks with multiple mobile agents, whose covering areas can be optimally designed to maximize the throughput or to maximize the energy efficiency. The performance analysis of sensor networks using both CDMA and multiple receive antennas is presented in [67] based on the results on large random networks in [49]. The analysis in this study does not rely on the large network approximation. Meanwhile, most studies on multiuser scheduling for uplink or downlink wireless networks have focused on maximizing the information-theoretic capacity [2, 50, 51, 59]. In [77], the authors present a scheduling algorithm which maximizes a certain performance value estimated by the user or calculated by the base station, such as a linear function of the signal-to-interference-and-noise ratio (SINR). On the other hand, we study multiuser scheduling to maximize the throughput in terms of the average number of successful packets per slot.

The main results of this chapter are summarized as follows:

1. We derive the throughput and the effective energy (average energy consumption for each successful packet) for a multiple access network employing round-robin and slot-ted ALOHA schemes and multiuser receivers in a Rayleigh flat fading environment. 2. We optimize the transmission power and the average number of transmissions per slot

to

(a) maximize the throughput; for each MAC scheme with a linear detector, we also derive the maximum asymptotic throughput when the signal-to-noise ratio goes to infinity.

(b) minimize the effective energy subject to a throughput constraint; it is shown that the minimum effective energy grows rapidly as the throughput constraint approaches the maximum asymptotic throughput.

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4. We show that multiuser scheduling can significantly enhance the system performance in a shadow fading environment.

The organization of the chapter is as follows. In Section 2.2, we introduce the system model and two measures: throughput and energy efficiency. In Section 2.3, we briefly describe the three linear detectors of interest and derive the analytical results to be used later. In Section 2.4, we first derive the energy efficiency and the throughput of the round-robin and slotted ALOHA schemes, and then study the two optimization problems, throughput maximization and throughput-constrained energy minimization respectively. Numerical results and discussions are presented in Section 2.5. Section 2.6 studies multiuser scheduling in the shadow fading environment. Section 2.7 summarizes the whole chapter.

2.2 System Description

We assume that there are totally n sensors in the sensor field, the receiver is equipped withN antennas, and the SIR threshold isβ. The diameter of the sensor field is much smaller than the distance between the sensor field and the receiver, and there exists a rich-scattering environment between the sensor field and the receiver, e.g., the sensors are deployed in a building or a forest. Therefore the channel states between each sensor and each receive antenna can be modeled as independent, identical Rayleigh variables. We assume that sensors have no knowledge of uplink CSI, and transmit with equal power P. Ifm sensors simultaneously transmit, them sensors andN receive antennas form a virtual MIMO system, and the discrete-time model is given by

y=√G m

X

i=1

hixi+n, (2.1)

where xi is the transmitted signal of the i-th sensor and E[kxik2] = P, hi is the N ×1 spatial signature of the i-th sensor, whose entries are independent circularly-symmetric complex Gaussian variables with zero mean and unit variance, Gis the common pathloss,

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packet at one receive antenna is given by ρ = P G_σ2 . In the following we denote the matrix

H= [h₁,h₂,· · ·,h_m].

We assume that a feedback channel exists from the receiver to the sensor nodes, which is used for synchronization, acknowledgements, group selection and other signaling on the MAC layer. The bandwidth of the feedback channel is typically small and thus the energy consumption for receiving the signaling is assumed to be negligible throughout the chapter. For simplicity, we also ignore the circuit energy consumption, which can be incor-porated and the optimizations described here can be performed with minor modifications. Some measures of sensor network’s energy efficiency have been explored in the literature: In [28], the energy consumption per bit to achieve a desired bit-error-rate is evaluated, and in [102], the metric efficiency, defined as the average number of successes over the total number of transmissions, is studied for SENMA networks. The former metric does not as-sume a multi-packet reception model, and the latter does not characterize the exact energy expenditure, as a transmission scheme with high efficiency is not necessarily energy efficient if the transmit power is not constrained. We combine the ideas in these two papers and measure the energy efficiency by the effective energy [67], defined as the average energy consumption per successfully transmitted packet:

Ee=

P T

Pr[succ], (2.2)

where Pr[succ] is the average probability of success for a transmitted packet. Note that the effective energy directly determines the number of packets a sensor can successfully transmit during its life time. The throughput, denoted by λ, is defined as the average number of successful transmissions per slot. Denoteaas the average number of transmissions per slot, we have

Pr[succ] = λ

a. (2.3)

Throughout the chapter we assume that the number of receive antennas N, the total number of sensorsn, the SIR thresholdβ, the common pathlossG, as well as the noise varianceσ2 _{are fixed. When}_G_and_σ2_{are fixed, the optimization of the transmission power}

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2.3 Linear Multiuser Detectors in Rayleigh Fading Channels

Assume that m sensors simultaneously transmit and the SNR is ρ, then the out-come of the ith transmitted packet (success is denoted by 1 and failure is denoted by 0) is a random variable determined by the channel realization:

o_i(H) =I(SIR_i≥β|m, ρ,H), (2.4) whereI(·) denotes the indicator function. The expected value of the outcome averaged over all channel realizations is denoted by q(m, ρ), which is the same for all i:

q(m, ρ) =EH[oi(H)] = Pr [SIRi≥β|m, ρ]. (2.5)

In an ergodic channel, the average number of successes when there aremtransmissions per slot and SNR isρ is given by

EH

" _m X

i=1

oi(H)

#

= m

X

i=1

EH[oi(H)] =mq(m, ρ). (2.6)

As we will see, the throughput and the effective energy for round-robin and slotted-ALOHA are functions of q(m, ρ), which is determined by the physical channel and the linear detec-tor used. In general q(m, ρ) decreases with m and increases with ρ. In this section, we briefly describe the three linear detectors of interest, and derive the expression of q(m, ρ) in Rayleigh fading channels for each detector. Moreover, as we will use the asymptotic value of q(m, ρ) as ρ → ∞ frequently in later analysis, we also derive the expression of q(m,∞) = lim. ρ→∞q(m, ρ). The readers are referred to [103] for more details of these

multiuser detectors.

2.3.1 Matched Filter

The matched filter only requires the knowledge of the spatial signature of the desired user, which is suitable for the downlink but not much of an advantage for the uplink where the knowledge of spatial signatures of all users are known. The SIR of the i-th user after matched-filtering is given by

SIRi = P Gkhik

4

σ2_k_h_i_k2₊_{P G}Pm

j=1,j6=ikh†ihjk2

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where† _{denotes conjugate transpose.}

Lemma 2.1. The q(m, ρ) of the matched filter in the Rayleigh fading channel is given by

qmf(m, ρ) =

  

1−Γ(β/ρ , N), m= 1,

1 (m−2)!

R_∞

0 [1−Γ(βy+β/ρ) , N)]ym−2e−ydy, m >1.

(2.8)

where Γ(a, x) is the regularized gamma function given by Γ(a, x) =

Rx

0 ta−1e−tdt

R∞

0 ta−1e−tdt

.

In the case ρ→ ∞,

qmf(m,∞) =

  

1, m= 1, 1−I

³

β

1+β;N, m−1

´

, m >1, (2.9)

where I(x;a, b) is the regularized beta function, given by I(x;a, b) =

Rx

0 ta−1(1−t)b−1dt

R₁

0 ta−1(1−t)b−1dt

.

Proof: See Appendix A.

2.3.2 Decorrelating Detector

The decorrelating detector is optimal according to three different criteria: least squares, near-far resistance, and maximum-likelihood when the received amplitudes are unknown [103]. When the spatial signatures are independent, the decorrelator exhibits improved performance than the matched filter except at low signal-to-noise ratios, and it converges to the linear MMSE detector at high signal-to-noise ratios. Generally, the decorrelator allows simpler expressions as it decomposes a multiuser channel into paral-lel single-user Gaussian channels. If H†_H _{is invertible, the SIR of the} _{i-th user using a}

decorrelating detector is given by

SIRi= h ρ (H†_H₎−1

i

ii

, (2.10)

and when H†_H_{is singular, SIR}

i is zero.

Lemma 2.2. The q(m, ρ) of the decorrelator in the Rayleigh fading channel is given by (c.f. (2.8) for the definition of the Γ(a, x) function),

qdec(m, ρ) =

  

1−Γ(β/ρ , N −m+ 1), m≤N, 0, m > N.

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When ρ→ ∞,

qdec(m,∞) =

  

1, m≤N,

0, m > N. (2.12)

Proof: See Appendix B.

2.3.3 Linear MMSE Detector

The linear MMSE detector cancels the interference and noise in an optimal way, such that the mean squared error is minimized among linear detectors. It can be shown that the linear MMSE detector also maximizes the SIR [103], hence it is optimal among linear detectors under the multiple packet reception model where the success probability only depends on the SIR. For the linear MMSE receiver, it can be shown that the SIR of thei-th user is given by

SIRi =h†_i

µ

HiH†_i + 1_ρI

¶₋₁

hi, (2.13)

where Hi denotes the matrix obtained by striking out the i-th column of H. There is no straightforward closed-form expression of q(m, ρ) for the linear MMSE detector in the Rayleigh fading channel. An approximation ofqmmse_{(m, ρ) can be obtained by using recent}

results on linear multiuser detectors in large random networks [99], where the SIR is shown to approach a Gaussian distribution as N approaches infinity, withα=m/N fixed. However, simulations show that such approximations are not accurate enough when N is relatively small, so we use exact success probabilities obtained through simulations for the linear MMSE detector. Nevertheless, when ρ → ∞, the success probability of the linear MMSE detector has a simple form, given by the following lemma.

Lemma 2.3. For Rayleigh fading channels (c.f. (2.9) for the definition of the I(x;a, b)

function),

qmmse(m,∞) =

  

1, m≤N, 1−I

³

β

1+β;N, m−N

´

, m > N. (2.14)

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2.4 Throughput and Energy Optimizations

In this section, we first derive the general expressions of the throughput and the effective energy for the round-robin and slotted ALOHA schemes, and then study the two optimization problems, throughput maximization and throughput-constrained energy-minimization for both MAC schemes.

2.4.1 Throughput and Effective Energy of Round-Robin and Slotted ALOHA

Round-robin

Round-robin is a fair scheduling scheme and is relatively easy to implement: m sensors in close proximity form a group. For simplicity we assume that n is a multiple of m, so there are totallyK =n/m groups. Groups are scheduled for access one by one, and when a group is scheduled in a slot, all the sensors in that group transmit simultaneously. It is easily seen that in an ergodic fading channel (shown at the beginning of Section 2.3), the throughput of round-robin is

λrr(m, ρ) =mq(m, ρ). (2.15)

WithP =ρσ2/G, the effective energy of round-robin is given by E_e_,_rr(m, ρ) = ρσ2T /G

q(m, ρ) . (2.16)

Slotted ALOHA

To employ the decorrelating detector or the linear MMSE detector in a slotted ALOHA system requires that the receiver knows the number and the channels of the trans-mitting nodes. For example, the sensors can signal their intention of transmission in a short reservation period at the beginning of each slot. We consider the type of slotted ALOHA where the transmission probability for all packets (new or retransmissions) is the same. Denoting the transmission probability of each user byp, the throughput of slotted ALOHA is given by

λ_sa = n

X

k=1

  n

k



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The average number of transmissions per slot is a = np. In the case n is large and p is small, we can approximate the binomial probabilities with Poisson probabilities and obtain

λsa(a, ρ) =e−a

n

X

k=1

ak

k!kq(k, ρ) =e

−aXn k=1

ak

(k−1)!q(k, ρ). (2.18) The average success probability is Pr[succ] =λsa(a, ρ)/a, thus the effective energy is given

by

Ee,sa(a, ρ) = ρσ 2_{T /G}

λsa(a, ρ)/a. (2.19)

The receiver can simply inform the sensors of the transmission probability, or the sensors can compute the optimum transmission probability if they have the knowledge ofn. Slotted ALOHA also has built-in fairness, since the transmission probability is independent of the channel states of individual sensors.

2.4.2 Throughput Maximization

As we have shown, the throughput depends on both the MAC scheme as well as the type of the linear detector used. For a given MAC scheme with a given linear detector, the throughput is a function of the SNR ρ and the average number of transmissions per slot a (for round-robin, a = m, and for slotted ALOHA, a = np). These parameters can be chosen judiciously such that the throughput is maximized. The performance of various MAC schemes with different linear detectors can then be compared, in terms of the maximum throughput. In the following we focus on the joint optimization of aand ρ; the optimization of a single parameter is straightforward and is therefore omitted.

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throughput that can be achieved by a MAC scheme with a given type of linear detector. In the following we discuss this case in detail.

For a given MAC scheme with a given linear detector, we define the maximum asymptotic throughput as the maximum throughput achievable with a given number of receive antennas as SNR ρ approaches infinity, and denote it by Λ(∞) = max. aλ(a,∞). The maximum asymptotic throughput plays an important role in throughput-constrained energy minimization to be discussed in Section 2.4.3, in the sense that any throughput constraint larger than Λ(∞) can not be attained. With a general linear detector, we have the following theorem:

Theorem 2.1. The maximum asymptotic throughput of round-robin and slotted ALOHA are respectively given by

Λrr(∞) = max

m mq(m,∞); (2.20)

Λsa(∞) = max_a e−a

n

X

k=1

ak

(k−1)!q(k,∞), (2.21) The above expressions can be evaluated for different detectors using (2.9), (2.12) and (2.14).

Remark 1: With the decorrelating detector, the maximum asymptotic throughput of the two MAC schemes are respectively given by

Λdec_rr (∞) =N, withm=N; (2.22) Λdec_sa (∞) = max

a e

−aXN k=1

ak

(k−1)!. (2.23)

The above are direct consequences of applying (2.12). Note that with the decorrelator, the maximum asymptotic throughput of slotted ALOHA can be much smaller than that of round-robin. For example, whenN = 10, the maximum asymptotic throughput of slotted ALOHA with the decorrelator is 5.831, which is achieved at a= 7.297.

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1. Λmmse

rr (∞)≥Λmfrr (∞), with the equality held whenN = 1.

2. Λmmse

rr (∞) ≥Λdecrr (∞); the equality holds if and only if the throughput of the linear

MMSE withm=N + 1 is smaller than withm=N, i.e.,

(N+ 1)

·

1−I

µ

β

1 +β;N,1

¶¸

≤N,

which yields

β ≥ 1

(N + 1)1/N ₋₁.

In other words, the linear MMSE detector can support a throughput larger than the number of receive antennas N (and surpass the decorrelator) if and only if β < ₍_N₊₁₎11/N₋₁. Note that the right-hand-side of the above inequality is a strictly increasing function ofN, going from 1 to +∞.

3. The relative performance of the decorrelator and the matched filter depends onβ. It can be shown that whenβ ≥1, Λdec_rr (∞)≥Λmf_rr (∞).

Remark 3: As for slotted ALOHA, since we have for all m, qmmse(m,∞) ≥ max{qmf_(m,_∞_{), q}dec_(m,_∞₎_}_{, the maximum asymptotic throughput with the linear MMSE}

is always the best, while it is not immediate whether the matched filter or the decorelator is the worst.

2.4.3 Throughput-Constrained Energy Minimization

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throughput constraint must not exceed this limit, otherwise it cannot be met. Comparing (2.16) and (2.19), we observe that σ2_{T /G} _{is a common factor and is fixed. Therefore to}

minimizeEe it suffices to find

min a,ρ

aρ

λ(a, ρ), (2.24)

subject to

λ(a, ρ)≥∆. (2.25)

In the following we briefly describe both single-parameter optimization as well as joint optimization.

Fixed ρ

For a fixed ρ, the throughput constraint ∆ can be met if and only if Λ(ρ) =. maxaλ(a, ρ), the maximum throughput given ρ, satisfies Λ(ρ) ≥ ∆. When ρ is fixed, for each MAC scheme, the values ofathat satisfyλ(a, ρ)≥∆ form a closed interval (of reals or integers). Since Pr[succ] decreases witha, the effective energy is minimized by the minimum awith which the throughput constraint is satisfied, i.e.,

aopt(ρ) = min{a|λ(a, ρ)≥∆}. (2.26)

Fixed a

When ais fixed, the throughput constraint ∆ can be met if and only ifλ(a,∞)=. limρ→∞λ(a, ρ), the maximum throughput givena, satisfiesλ(a,∞)≥∆. Since the

through-put is a monotone increasing function of ρ, we can find the smallest ρ that meets the throughput constraint, which is denoted by ρmin(a) = min{ρ | λ(a, ρ) ≥ ∆}. Thus the

minimum effective energy for fixedais given by

Ee,min(a) = min

ρ≥ρmin(a)

aρ

λ(a, ρ). (2.27)

Joint Optimization

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joint optimization can be proceeded in two steps: first, find the minimum effective energy when a is fixed, as described above; then find the global minimum across all a. This is characterized by the following theorem.

Theorem 2.2. For a given throughput constraint ∆, if∆≤Λ(∞), the minimum effective energy jointly optimized over a andρ is given by

Ee,min= min

a Ee,min(a) = mina ρ≥minρmin(a)

aρ

λ(a, ρ), (2.28)

while if ∆>Λ(∞), the throughput constraint cannot be met.

2.5 Numerical Results and Discussions

In this section we present the numerical results and draw some observations on the comparative performance of different MAC schemes, as well as on the comparative performance of different linear detectors.

2.5.1 Maximum Throughput

Example 1 [Comparison of detectors; Joint Optimization] In Fig. 2.1 we plot the maximum asymptotic throughput (result of joint optimization) of round-robin with three linear detectors when β = 1 and β = 3. Note that the two curves for the decorrelator coincide. When β = 1, the maximum asymptotic throughput of the linear MMSE de-tector exceeds that of the decorrelator (which is N) for all values of N except N = 1,

since 1

(N+1)1/N₋₁ > 1 for all N > 1. When β = 3, the maximum asymptotic throughput of the linear MMSE detector exceeds that of the decorrelator when N ≥ 8, with which

1

(N+1)1/N₋₁ >3. As β gets larger, it requires a largerN for the linear MMSE detector to surpass the decorrelator in terms of the maximum asymptotic throughput.

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0 2 4 6 8 10 12 14 16 18 20 0

5 10 15 20 25 30

Number of receive antennas N

Λ

(

∞

)

Maximum Asymptotic Throughput of Linear Detectors

MF, β=1 Deccorelator, β=1 LMMSE, β=1 MF, β=3 Decorrelator, β=3 LMMSE, β=3

Figure 2.1: Maximum asymptotic throughput of round-robin with different linear detectors

matched filter and the linear MMSE detector. WhenN is small, the matched filter outper-forms the decorrelator for slotted ALOHA, and when N is large, the opposite is true. For both MAC schemes the linear MMSE detector assumes great superiority, and can achieve a maximum asymptotic throughput greater than N with the linear MMSE detector when β = 1.

2.5.2 Minimum Effective Energy with Throughput Constraint

In the following we present the results of throughput-constrained energy minimiza-tion described in Secminimiza-tion 2.4.3. We show the results of optimizaminimiza-tion with fixedaand joint optimization. For all simulations in this section we use the following values: N = 10,β = 1 and σ2_{T /G}_{= 1 (scaling factor of}_E

e).

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0 2 4 6 8 10 12 14 16 18 20 0

5 10 15 20 25 30

number of receive antennas N

maximum asymtptotic throughput

Λ

(

∞

)

round−robin, MF round−robin, DEC round−robin, LMMSE slotted ALOHA, MF slotted ALOHA, DEC slotted ALOHA, LMMSE

Maximum Asymptotic Throughput, β=1

Figure 2.2: Maximum asymptotic throughput of round-robin and slotted ALOHA with different linear detectors, β= 1

except form= 5, (where the minimum effective energy of round-robin goes to infinity and is not shown in the figure), round-robin is much more energy-efficient than slotted ALOHA for the same value ofa.

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5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Average number of transmissions in each slot (m or a)

Minimum Effective Energy

Minimum effective energy with throughput constraint, ∆=5, N=10, β=1, σ2T/G=1

round−robin slotted ALOHA

Figure 2.3: Minimum effective energy with throughput constraint for different MAC schemes with the decorrelator (fixedm ora), ∆ = 5,N = 10, β = 1,σ2_{T /G}_{= 1}

approaches infinity for slotted ALOHA and round-robin respectively as ∆→5.831 and as ∆→ 10. When ∆ is relatively small (e.g., ∆≤ 3), slotted ALOHA does not incur much extra energy expenditure than robin. As ∆ increases, the energy saving by round-robin relative to slotted ALOHA becomes increasingly larger, and round-round-robin can support a throughput that cannot be achieved by slotted-ALOHA.

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0 1 2 3 4 5 6 7 8 9 10 0

1 2 3 4 5 6 7 8 9 10

Throughput Constraint ∆

Minimum effective energy with throughput constraint, N=10, β=1, σ2T/G=1

Figure 2.4: Minimum effective energy with throughput constraint for different MAC schemes with the decorrelator (joint optimization),N = 10, β= 1, σ2_{T /G}_{= 1}

0 1 2 3 4 5 6 7 8 9 10

Optimum m or a

Optimum m / a for minimum effective energy with throughput constraint, N=10, β=1, σ2 T/G=1

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0 2 4 6 8 10 12 14 0

1 2 3 4 5 6 7 8 9 10

Minimum effective energy with throughput constraint, N=10, β=1, σ2T/G=1

MF Decorrelator LMMSE

Figure 2.6: Minimum effective energy with throughput constraint for round-robin with different linear detectors (joint optimization),N = 10,β = 1,σ2_{T /G}_{= 1}

2.6 Multiuser Scheduling Under Shadow Fading

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Another verified problem with multiuser scheduling for a system described in Sec-tion 2.2 is that, under pure Rayleigh fading, multiuser scheduling has a vanishing relative scheduling gain as m and N increases (indicating a tradeoff between multiple antennas and multiuser diversity) [50]. While shadow fading generally increases the dynamism of individual link quantity, which leads to larger outage probability and is unfavorable to real-time applications, it can actually enhance the scheduling gain in a multiuser environment for delay-tolerant applications [50]. By slightly modifying our system model, we can investigate the multiuser scheduling gain that is realizable under the shadow fading.

We assume that the sensors in each group are adjacent to each other such that they experience the same shadow fading while sensors in different groups experience inde-pendent identically-distributed shadow fading. In each slot, the scheduler selects the group with the highest shadowing coefficient. Although this scheduler is not optimal in terms of throughput, it only requires about 1/N m amount of channel knowledge compared to the optimal scheduler. Ideally, the receiver is a mobile agent which moves at the end of each slot to induce a dynamic environment such that all groups have similar chances to enjoy the best channel in the long run. Fairness can be further guaranteed by employing other methods, such as those in [104].

Denote the channel gain of the kth (k= 1,· · · , K) group byGk, then for the kth group the system model in (2.1) is modified as

y=pGk m

X

i=1

hixi+n. (2.29)

G_k is modeled as log-normal distributed, which has area mean E[G_k] =G=G_L(dB), and decibel spread σL(dB). The average SNR is given by ρ = P G_σ2 . Denote Gk = ezk, then

z_k∼ N(κG_L,(κσ_L)2_{) is a Gaussian variable, where} _κ_{= ln 10/10.}

Lemma 2.4. [30]: If Z1,· · ·, ZK are i.i.d. Gaussian with mean µ and variance σ2, as K → ∞,

max

1≤k≤KZk→µ+σ √

2 lnK. (2.30)

Applying the lemma toz_k, we have maxz_k→κGL+κσL √

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eκσL√2 lnK =. ξρ, whereξ =eκσL√2 lnK roughly characterizes the scheduling gain in terms of the improvement of SNR. The throughput and the effective energy of the scheduling algorithm respectively converge to

λ_sch(m, ρ) =mq(m, ξρ), (2.31)

and

Ee,sch(m, ρ) = ρσ 2_{T /G}

q(m, ξρ). (2.32)

In comparison, the throughput and effective energy of the same system via using the round-robin approach are given by

λrr(m, ρ) =Eρk[mq(m, ρk)] =m

Z ₊_∞

−∞

q(m, ez)e

−(lnz−lnρ)2_/₂₍_κσL₎2

z√2πκσL

dz=. mq(m, ρ). (2.33)

and

Ee,rr(m, ρ) = ρσ 2_{T /G}

q(m, ρ) . (2.34)

The throughput of multiuser scheduling and round-robin in shadow fading, both with the decorrelator, are depicted in Fig. 2.7, where N = 10, n= 1000, σL= 8dB, β = 1, and two SNR values, -10dB and 0dB are shown. The throughput of round-robin without shadowing (i.e., pure Rayleigh fading) are also plotted for comparison. We observe that even for round-robin, shadowing is beneficial when the SNR is low, while the opposite is true when SNR is high: shadowing degrades the throughput. This can be readily explained by Jensen’s inequality by observing the property of theq(m, ρ) function: for all three detectors, it can be shown that the q(m, ρ) function is convex in the low SNR range and is concave in the high SNR range, and approaches q(m,∞) as ρ → ∞ (see (2.9), (2.12) and (2.14)). Meanwhile, the throughput of multiuser scheduling is almost invariant of the SNR, and is roughly equal to the number of transmissions. This means that despite of the average SNR, the group of the best channel has an effective SNR with which the success probability is 1. This demonstrates that multiuser scheduling is most useful when the SNR is low, which is of particular significance for sensor networks.

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1 2 3 4 5 6 7 8 9 10 0

1 2 3 4 5 6 7 8 9 10

Number of transmitting sensors m

throughput

Throughput comparison: multiuser scheduling v.s. round−robin, N=10, σ L=8dB, β=1

scheduling, shadowing, ρ=0dB round−robin, shadowing, ρ=0dB round−robin, no shadowing, ρ=0dB scheduling, shadowing, ρ=−10dB round−robin, shadowing, ρ=−10dB round−robin, no shadowing, ρ=−10dB

Figure 2.7: Throughput comparison: multiuser scheduling v.s. round-robin (with the decor-relator), N = 10, n= 1000,σL= 8dB,β= 1

can be made virtually 1 for a modest ρ when the number of sensors is large implies that there is no loss in the energy consumption, and that the minimum effective energy remains low for all throughput constraints ∆<Λ(∞).

2.7 Summary

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per slot a and the transmission power per sensor node, to meet two objectives: through-put maximization, and throughthrough-put-constrained effective energy minimization. There are interesting connections between these two optimization problems. In particular, the max-imum asymptotic throughput as the SNR goes to infinity defines the upper limit on the throughput constraint that can be achieved.

Under the assumption of Rayleigh flat-fading, we show that slotted ALOHA suf-fers from the greatest performance loss when paired with the decorrelator. While slot-ted ALOHA has similar minimum effective energy as round-robin for small throughput-constraint, it soon turns energy-inefficient as throughput-constraint increases. For both MAC schemes, the linear MMSE detector significantly outperforms the decorrelator and the matched filter in both the throughput and the energy efficiency. Finally we consider the shadowing effect on the system performance and show that multiuser scheduling greatly boosts the throughput in the low SNR region and hence is of particular significance for sensor network applications.

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Chapter 3

Distributed Detection in WSN

over A Multi-Access Channel

3.1 Introduction

In Chapter 2, we have studied the problem of transporting data from multiple sensor nodes to a central controller, with application-specific signal processing details ig-nored. Starting from this chapter, we investigate two specific applications of wireless sensor networks, where our main goal is to maximize some application-oriented performance met-ric. Meanwhile, it is very important to keep the communication complexity and energy consumption low to prolong the sensor network lifetime. As will be demonstrated later, this leads to joint designs of local processing rules and communication strategies.

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of the detection performance as the number of sensors goes to infinity, measured by the error exponent, has gained much research interest [20, 21, 75, 76, 95, 96]. The error exponent gives an estimate of the number of sensors required to reach a certain error probability, and is therefore a useful performance index in the large sample regime.

The traditional approach of studying the distributed detection problem is to as-sume that sensors transmit their observations (possibly quantized versions of them) through a set of independent parallel access channels (PAC) [20, 21, 105], as shown in Fig. 3.1.(a). For large-scale sensor networks, this assumption implies a large bandwidth requirement for simultaneous transmissions or a large detection delay. Alternatively, we can employ a mul-tiple access channel (MAC), as shown in Fig. 3.1.(b). The bandwidth requirement of MAC fusion does not depend on the number of sensors, but due to the additive nature of the chan-nel, the received signal at the fusion center is generally not sufficient for reliable detection. However, it is known that when the sensor observations are conditionally independent, soft decision fusion where each sensor transmits its local log-likelihood-ratio (LLR) value over the MAC is asymptotically optimal [76]. Recently, Type-Based Multiple Access (TBMA) has been proposed by Mergen and Tong [81, 82, 83, 84] as well as by Liu and Sayeed [75, 76], where each sensor transmits the waveform corresponding to its quantized observation over a MAC. For i.i.d. sensor observations and identical channel gains, the fusion center receives a noisy version of the type of sensor observations, which is a sufficient statistic for detection, and TBMA achieves the same error exponent as the centralized detection [76, 83]. The per-formance of TBMA degrades for fading channels, especially when the channel gain has zero mean [81]. Anandkumar and Tong subsequently proposed the Type-Based Random Access (TBRA) to improve the performance of TBMA over the noncoherent channel by control-ling the rate of random access [7]. These works mostly assumed conditionally-independent sensor observations, and not much has been explored along the line for correlated observa-tions, which would arise when detecting a random spatially-correlated signal in noise, or a deterministic signal in noise where the noise samples are correlated due to the proximity of sensors.

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net-H

₀

/ H

₁

U(.)

FC

r

r

n

PAC

u

₀

/ u

₁

(a)

(b)

Figure 3.1: Distributed detection fusion schemes: (a) PAC fusion; (b) MAC fusion

Distributed and Collaborative Processing in Wireless Sensor Networks

Contents

List of Figures

List of Tables

List of Abbreviations and Acronyms

Chapter 1

Introduction

1.1

Wireless Sensor Networks

1.2

Optimal Throughput and Energy Efficiency in WSN

1.3

Distributed Detection in WSN

1.4

Distributed Consensus in WSN

1.5

Dissertation Organization

Chapter 2

Optimal Throughput and Energy

Efficiency of WSN

2.1

Introduction

2.2

System Description

2.3

Linear Multiuser Detectors in Rayleigh Fading Channels

2.4

Throughput and Energy Optimizations

2.5

Numerical Results and Discussions

2.6

Multiuser Scheduling Under Shadow Fading

2.7

Summary

Chapter 3

Distributed Detection in WSN

over A Multi-Access Channel

3.1

Introduction

net-H

/ H

U(.)

U(.)

U(.)

FC

x

x

x

y

y

y

r

MAC

u

/ u

H

/ H

U(.)

U(.)

U(.)

FC

x

x

x

y

y

y

r

r

r

PAC

u

/ u

(a)

(b)

H

/ H

U(.)

U(.)

U(.)

_y

_y

_y