Collaborative Power Control for Wireless Sensor Networks

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Collaborative Power Control for Wireless Sensor Networks

M A N U E L A C I P P I T E L L I

Master's Degree Project

Stockholm, Sweden 2005

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Abstract

Sensor nodes within networks are often grouped to participate to a com- mon processing task. Cooperative diversity is a technique that exploits groups of sensor node randomly placed to cooperatively relay a common re- ceived signal toward a destination with the goal to combat severe attenuation or disconnections of the signal strength.

In recent years, cooperative diversity has received attention for cellular radio systems and ad-hoc wireless networks. Such systems, however, are usually equipped with high processing capability. On the contrary, the nodes of a WSNs have limited memory and power capabilities and are usually deployed in unfriendly environment, where recharging and maintenance is not possible.

In this thesis, we investigate the problem of power control of nodes per-

forming cooperative diversity. Speciﬁcally, we study the problem of minimiz-

ing the power consumption of the sensor nodes transmitters while guaran-

teeing a minimum quality of the signal at the data collector. After studying

the most relevant algorithms existent in literature for had-hoc networks, we

propose an sub-optimal algorithm suitable for nodes equipped with low com-

putational capabilities. We implement a WSNs performing cooperative diver-

sity with Omnet++, where the network simulator includes the sub-optimal

solution. Numerical results show that for the set of parameters of practical

interest, our solution exhibits good performance for low correlated channel

links, while the increase of relaying nodes ensures a decreasing of total power

consumption.

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Acknowledgements

This research was carried out at the department of Signals, Sensors &

Systems (S3) at the Royal Institute of Technology (KTH), Stockholm, and was fully sponsored by the University degli studi di Siena. I would like to thank the S3-department for their support and for giving me the opportunity to perform this work at Stockholm.

There are a number of people I would like to thank for their assistance with the work on this thesis, either directly or indirectly. Please forgive me if I forget anyone.

I would like to express my sincerest gratitude to the following people:

Prof. Mikael Johansson at KTH for for giving me the opportunity to perform this work at KTH.

My supervisor Dr. Carlo Fischione at KTH for guidance and crucial feed- back on this work and for introducing me to the interesting topic of wireless sensor network. Thanks for always having time and being so enthusiastic about this work.

Prof. Andrea Abrardo in Siena, who is my examiner, for giving me the opportunity to perform this work at Stockholm.

Pablo Soldati, Youssef Chraibi, Colette, Michael Redmond and Andrea Fornara for all the great time we had together at KTH.

My friends Erika and Angelita for being in our long conversation on MSN.

My boyfriend David also deserves special thanks for his love and support during this work.

Last but not least, my dear parents and my little sister Claudia for always being there for me, supporting me and encouraging me whenever needed.

Manuela Cippitelli

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List of Figures

1.1 The design space of wireless sensor networks . . . . 8

1.2 Wireless sensor networks . . . . 12

2.1 Path Loss, Shadowing and Multipath versus Distance. . . 15

2.2 Illustration of multipath. . . 16

2.3 fading . . . 20

3.1 Energy pattern of beamformer using a Uniform Circular Array (UCA) with six transmitting antennas. . . . 29

3.2 Antenna array and Spatio-temporal ﬁltering . . . . 30

4.1 Wireless sensor network . . . . 48

4.2 Spatio-temporal ﬁltering . . . 49

4.3 Matched ﬁlter . . . . 51

4.4 distribution of g(t) . . . . 52

4.5 Complex weights . . . . 53

4.6 Outage probability . . . . 59

4.7 Sensors network with two sensor nodes. . . . 71

5.1 Wireless sensor network considered in the simulations. . . 76

5.2 Minimum channel attenuations values. . . 78

5.3 Retransmission power consumption considering a high value of ρ

_i,i

. . . 80

5.4 Retransmission power consumption considering a lower value of ρ

_i,i

. . . 82

5.5 Retransmission power consumption considering lower values of channel attenuations. . . . . 84

5.6 SINR at the increasing of the sensors number. . . . 85

5.7 Retransmission power at the increasing of the sensors number. 86

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5.8 SINR at the increasing of the sensors number and considering higher values of channel attenuations. . . . 89 5.9 Retransmission power at the increasing of the sensors number

and considering higher values of channel attenuations. . . . 90 5.10 OMNeT++ simulation environment. . . 91 5.11 Channel attenuations correlation in the time domain. . . 92 5.12 Sensors retransmission power variation channel attenuations

in the time domain. . . 93

B.1 Example of OMNeT++ simulation network. . . 101

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List of Tables

5.1 Values of the parameters used in the simulation in figure 5.3. . 79 5.2 Values of the parameters used in the simulation in figure 5.4. . 81 5.3 Values of the parameters used in the simulation in figure 5.5. . 83 5.4 Values of the parameters used in the simulation in figure 5.6

and 5.7. . . 87 5.5 Values of the parameters used in the simulation in ﬁgure 5.8

and 5.9. . . . 88

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Introduction

The reduced cost and dimensions of electronic devices, and the development of pervasive networking technology, makes possible to deploy Wireless Sensor Networks (WSNs)[2]. WSNs, are wireless communication networks composed by cheap nodes equipped with sensing and monitoring capabilities. WSNs are supposed to perform a multiplicity of tasks. They can be roughly diﬀer- entiated as in the following:

• monitoring space: ocean water, pollution,...

• monitoring things: robots, human body, tracking,..

• monitoring the interactions of things with each other and encompassing space

Nodes may be unreliable or deployed in an unfriendly environment. There- fore, the cooperation of nodes increase the performance of the application algorithms running over the WSNs. Moreover, in many situations, nodes of WSNs are grouped to perform a common task. This is the case of localiza- tion, distributed source coding and cooperative diversity.

Cooperative diversity, while minimizing the energy consumption of the sensor nodes, is a relevant problem for WSNs. In fact, disconnection events have strong inﬂuence on the performance of upper layers in the protocol stack, e.g.,[18, 26, 8]. An example is when a WSN is used to gather information for real-time control of a plant. The stability of the closed-loop control system may require rates of signal disconnections lower than a certain frequency related to the sampling frequency of the sensors and actuators as well as the dynamics of the plant.

Cooperative diversity uses STP algorithms to cooperatively increase the

performance of the system. In the last years, the investigation of STP for

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cellular radio system as well as ah-hoc wireless networks has been an area of intense research. The existing STP techniques, however, need further reﬁne- ments if implemented on WSNs, where the constraints on the energy con- sumption and reduced processing capabilities require the adoption of more suitable algorithms. Moreover, the (complex) design of WSN requires the adoption of a cross-layer approach, where the behavior of each communi- cation layer is properly abstracted in order to perform consisted design of control algorithms.

Problem deﬁnition

Because of the WSNs intrinsic nature, the existing spatio-temporal processing algorithms can not be easily applied in their entirety[3]. In this framework, an interesting activity of studying and researching is dedicated to the modelling of the properties of the physical layer of the communication stack, tacking into account all the relevant parameters that inﬂuence the quality of the transmission. We will consider all the attenuations in the link, such as path loss, shadowing, slow and fast fading. A second phase of studying is dedicated to the analysis of existing solutions of STP, with the aim to evidence the characteristics of suitable algorithms for WSN. The purpose is to deﬁne a STP algorithm with reduced processing requirements for WSNs performing cooperative diversity.

Research approach

The ﬁrst step of the research activity consist in the studying the contribu-

tions that can be found in literature about STP both for cellular system and

WSNs. An interesting investigation will be dedicated to the modeling of the

properties of the physical layer, tacking into account all the relevant para-

meters that inﬂuence the quality of the transmission. The research approach

will be focused on the deﬁnition of a quality of service metric that takes into

account the cross-layer requirements of WSNs. The study of optimization

algorithms will be also part of the activity.

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Expected results

The goal is the deﬁnition of a STP control technique suitable for WSNs. The expected solution of STP for WSNs shall be ﬁrstly centralized and then, if possible, distributed. It is supposed that the performance analysis of the algorithms that will be proposed should be tested with network simulation environments.

Outline

Introduction gives a brief introduction to the topics and lists the outline and contributions. A list of acronyms and notation used in this work is also given.

Chapter 1 Wireless sensor networks gives an overview of wireless sen- sor networks and a description of all the parameters of interest that characterize them.

Chapter 2 Overview on wireless channel models gives an overview on the most relevant wireless channel models.

Chapter 3 Spatio-temporal filtering gives a description of spatio-temporal filtering as an optimization technique, which consists on finding a weights vector such that a measure of the quality of the transmitted signal is maximized.

Chapter 4 Collaborative power control and spatio-temporal ﬁlter- ing In this chapter methods for collaborative power control and spatio- temporal ﬁltering for wireless sensor networks are investigated.

Chapter 5 Simulations Numerical results of the investigated algorithms are provided using Matlab and OMNeT++ simulation environments.

Chapter 6 Conclusion and future work The main results of this thesis project are summarized and future perspectives are envisaged.

Appendix A Dual problem gives a quick description of Lagrange dual method.

Appendix B OMNeT++ gives a description of OMNeT++ simulator

and of all the parameters of interest that characterize it.

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Notation

R The set of real scalars

X

^T

The transpose of the matrix X C The set of complex valued scalars ( ·)

^∗

The conjugate operation

( ·)

^T

The transpose operation ( ·)

^H

The Hermitian operation ( ·)

ij

The i, jth element of a matrix ( ·)

i

The ith element of a vector I

_n

An n × n identity matrix

· The l

₂

-norm operation

Γ Signal to Interference plus Noise Ratio (SINR) γ

₀

Quality of Service (QoS) requirement

β Scale factor in the optimal approach E[ ·] The expected value

N Total number of nodes

min The minimal value of an objective function O

_p

Outage probability

O

_t

Target outage probability L

_p

Path loss

T

_m

Delay time T

_s

Symbol time T

₀

Channel coherence f

₀

Coherence bandwidth σ

_τ

Delay spread

W Signal bandwidth f

_d

fading rate

a(θ, φ) Steering vector

f (θ, φ) Radiation diagram in direction (θ, φ) w Weights vector

R

_n

Covariance noise matrix

G

_ji

Link gain between transmitter j and receiver i N

_i

Noise power at the ith receiver

dBm dB mW

c(t) Spreading sequence

b

_i

(n) ith node information bit stream

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n

_i

(t) Thermal noise at the input of the ith sensor σ

_i

Variance of n

_i

(t)

r

_ij

(τ ) Channel correlation

w

_i

Beamforming weight at the ith sensor n(t) Thermal noise at the input of data collector σ

_N

Variance of n(t)

g(t) Pulse shaping ﬁlter impulse response ρ( ·) Spectral radius

σ( ·) Spectrum λ Eigenvalue T r[ ·] Trace

L( ·, ·) Lagrangian function

ξ Lagrange multiplier

g( ·, ·) Lagrange dual function

dom Domain of a function

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Acronyms

ADC Analogical to Digital Converter CIR Carrier-to-Interference Ratio CDMA Code Division Multiple Access DS/CDMA Direct Sequence CDMA

EGC Equal Gain Combining

FDMA Frequency Division Multiple Access FH/SS

FSK Frequency-Shift Keying GNED Graphical Network Editor ISI Inter Symbol Interference LAN

LCMV Linear Constraint Minimal Variance

LS Least Squares

MAI Medium Access Interference

MLSE Maximum Likelihood Sequence Estimation MMSE Minimum Mean Square Error

MRC Maximal Ratio Combining NED NEtwork Description

OFDM Orthogonal Frequency Division Multiplexing OMNeT Objective Modular Network Testbed

PLL Phase-Locked-Loop

QoS Quality of Service

RF Radio Frequency

SC Selective Combining

SNR Signal to Noise Ratio STF Spatio-Temporal Filtering STP Spatio-Temporal Processing TDMA Time Division Multiple Access UCA Uniform Circular Array

WSN Wireless Sensor Network

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Chapter 1 Wireless sensor networks

In the past, a number of early, mostly U.S.-based research projects estab- lished a de facto deﬁnition of a wireless sensor network as a ”large-scale, ad hoc, multihop, unpartitioned network of largely homogeneous, tiny, resource- constrained, mostly immobile sensor nodes that would be randomly deployed in the area of interest ” [12].

A giving computing capacity becomes exponentially smaller and cheaper with each passing year [6]. This trend has advanced with the prolonged ex- ponential growth in the underlying semiconductors technology. Therefore, today we have the resources to develop the nodes of a WSN. However, physi- cal limitations (reduced dimension, cost, etc.) make the individual devices or nodes in a wireless sensor network (WSN) inherently resource constrained:

they have limited processing speed, storage capacity and communication bandwidth. In most settings, the network must operate for long periods of time and the nodes communicate wireless, so the available energy resources (wether batteries, energy harvesting or both) limit their overall operation.

To minimize energy consumption, most of device’s components, including the radio, will likely be turned oﬀ most of the time. Because they are so closely coupled to a changing physical world, the nodes forming the network will experience wide variations in connectivity and will be subject to potentially harsh environmental conditions. Their devices deployment generally means that there will be a high degree of interaction between nodes, both positive and negative. Each of this factors should be carefully taken into account for the design of networking protocols.

The nodes must organize themselves and provide a mean of program-

ming and managing the network as an ensemble, rather then administrating

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individual devices.

wireless sensor network

lifetime

network size connectivity

coverage topology

infrastructure

communication modality

heterogeneity

cost, size, resources and energy

deployment

QoS

requirements

mobility

Figure 1.1: The design space of wireless sensor networks

Figure 1.1 shows the fundamental aspects of a wireless sensor network.

1.1 Design space for WSNs

Initial research into wireless sensor networks was mainly motivated by mili-

tary applications. More recently, other civilian application domains of wire-

less sensor networks have been considered, such as environmental and species

monitoring, agriculture, production and delivery and healthcare. Although

computer-based instrumentation has existed for a long time, the density of

instrumentation made possible by a shift to mass-produced intelligent sen-

sors and the use of pervasive networking technology gives WSNs a new kind

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of scope that can be applied to a wide range of uses. They can be roughly diﬀerentiated into:

• monitoring space

• monitoring things

• monitoring the interactions of things with each other and encompassing space.

Many initial WSNs have been deployed for environmental monitoring, which involve collecting readings overtime across a volume of space large enough to exhibit signiﬁcant internal variation. Researches are using WSNs to monitor nesting seabird habitats and microclimate chaparral transects and to conduct analogous studies of contaminant propagation, building comfort and intrusion detection.

1.2 WSNs characteristics

In this section, the most relevant aspects to consider for the design of algo- rithm and protocol for WSNs are evident as they where discussed in [12].

1.2.1 Deployment

Nodes may be deployed at random (e.g.,in monitoring space applications by dropping them from an aircraft) or installed at deliberately chosen spots.

However, deployment may also be a continuous process, with more nodes being deployed at any time during the use of the network, for example, to replace interesting locations.

1.2.2 Mobility

Sensor nodes may change their location after initial deployment. Mobility

can result from environmental inﬂuences such as wind or water, sensors nodes

may be attached to or carried by mobile entities, and sensors nodes may

possess automotive capabilities. So mobility may be an incidental side eﬀect,

a desired property of the system, active or passive. The degree of mobility

may also vary from occasional movement with long periods of immobility

in between to constant travel. Mobility inﬂuences the design of networking

protocols and distributed algorithms.

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1.2.3 Cost, Size, Resources and Energy

Depending on the actual needs of the application, the form factor of a sin- gle sensor node may vary from the size of a shoebox to a microscopically small particle. Similarly, the cost of a single device may vary from hundreds of to a few cents of euros. Since sensor nodes are untethered autonomous devices, their energy and other resources are limited by size and cost con- straints. Varying size and cost constraints directly result in corresponding varying limits on the energy available. These resource constraints limit the complexity of the software executed on sensor nodes.

A sensor network node’s hardware consists of a microprocessor, data stor- age, sensors, analog-to-digital converters (ADCs), a data transceiver, con- trollers that tie the pieces together and an energy source. As semiconductor circuits become smaller, they consume less power for a given clock frequency and ﬁt in a smaller area. Most of the circuits can be powered oﬀ, so the standby power can be about one microwatt. Limited amount of memory consumes most of the chip area and much of the power budget.

1.2.4 Heterogeneity

In many prototypical systems available today, sensor networks consist of a variety of different devices. Nodes may differ in the type and number of attached sensors, they may collect, process, and route sensory data from many more limited sensing nodes. The degree of heterogeneity in a sensor network is an important factor since it affects the complexity of the software executed on the sensor nodes and in the management of the whole system.

1.2.5 Communication modality

The most common communication modality is radio waves which do not re-

quire a free line of sight and communication over medium ranges can be im-

plemented with relatively low power consumption and relatively small anten-

nas. Using light beams for communication requires a free line of sigh and may

interfere with ambient light and daylight, but allows for much smaller and

more energy-eﬃcient transceivers than does radio communication. Sound

or ultrasound is typically used for communication under water or to mea-

sure distances based on time-of-ﬂight measurements. Sometimes, multiple

modalities are used by a single sensor network system. The communication

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modality used obviously inﬂuences the design of medium access and commu- nication protocols, but also aﬀects other properties that are relevant to the application.

1.2.6 Infrastructure

The various communication modalities can be used in diﬀerent ways to con- struct an actual communication network. Two common forms are so-called infrastructure-based networks on one hand and ad hoc networks on the other hand. Communication between sensor nodes is relayed via the base station.

In ad hoc networks, nodes can directly communicate with each other without an infrastructure and so they are preferred for many applications.

1.2.7 Network topology

One important property of a sensor networks is its diameter, that is, the maximum number of hops between any two nodes in the network. In its simplest form, a sensor network forms a single-hop network, with every sensor node able to directly communicate with every other node. An infrastructure- based network with a single base station forms a star network with diameter of two. A multihop network may form an arbitrary graph, but often an overlay network with a simple structure is constructed such as a tree or a set of connected stars. The topology aﬀects many network characteristics such as latency, robustness, and capacity. The complexity of data routing and processing also depends on the topology.

1.2.8 Coverage

The eﬀective range of the sensors attached to a sensor node deﬁnes the cover-

age area of a sensor node. Network coverage measures the degree of coverage

of the area of interest by sensor nodes. The degree of coverage also inﬂuences

information processing algorithms. High coverage is key to robust systems

and may be exploited to extend the network lifetime by switching redundant

nodes to power-saving sleep modes.

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Figure 1.2: Wireless sensor networks

1.2.9 Connectivity

A network consists of many nodes, each with multiple links connecting to other nodes. Information moves hop to hop along a route from the point of production to the point of use. In WSNs, each node has a radio that pro- vides a set of communication links to nearly nodes. By exchanging informa- tion, nodes can discover their neighbors and perform a distributed algorithm to determine how the route data according to the application’s needs. Al- though physical placement primarily determines connectivity, variables such as obstructions, interference, environmental factors, antenna orientation and mobility make determining concertize a priori diﬃcult.

The communication ranges and physical location of individual sensor nodes deﬁne the connectivity of a network. If there is always a network connection between any two nodes, the network is said to be connected.

Connectivity is intermittent if the network may be occasionally partitioned.

If nodes are isolated most of the time and enter the communication range

of other nodes only occasionally, communication is sporadic. Connectivity

mainly inﬂuences the design of communication protocols and methods of data

gathering.

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1.2.10 Network size

The number of nodes is mainly determined by requirements relating to net- work connectivity, coverage and by the size of the area of interest. The net- work size determines the scalability requirements with regard to protocols and algorithms.

1.2.11 Lifetime

Depending on the application, the required lifetime of a sensor network may range from some hours to several years. The necessary lifetime has a high impact on the required degree of energy eﬃciency and robustness of the nodes.

1.2.12 Other QoS requirements

A sensor network must support certain quality-of-service (QoS) aspects such as real-time constraints, robustness, eavesdropping resistance and unobtru- siveness or stealth. These requirements may impact other dimensions of the design space such as a coverage and resources.

However, research into software support for WSNs is still at an early stage,

and signiﬁcant advances will be required to approach the goal of easy and

consistent programmability, testing, and deployment of applications across

the design space.

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Chapter 2 Overview on wireless channel models

In this chapter we report on channel models for wireless transmission. The physical link between the transmitting antenna and the receiving antenna is referred to as the communication channel, or just the channel. When the signal s(t) is transmitted over the channel, it is corrupted and the received signal r(t) will diﬀer from the transmitted signal. The channel attenuations are dependent on time and depending on the surroundings, may change fast or slow. We considers in this chapter the most relevant models for description of wireless channels.

2.1 Characterization

A wireless channel can be characterize substantially by three parameters:

1. Path loss: describes the attenuation of the signal radiated by the trans- mitter in a free space propagation situation, due to isotropic power spreading.

2. Large scale fading: represents the average signal power attenuation due to motion over large areas. This phenomenon is aﬀected by terrain contours such as hills, forests, clumps of buildings, etc. between the transmitter and the receiver.

3. Small scale fading : Small scale fading refers to the dramatic changes

in signal amplitude and phase that can be as result of small changes

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as small as half wavelength in the spatial separation between a receiver and a transmitter.

Figure 2.1: Path Loss, Shadowing and Multipath versus Distance.

2.1.1 Path loss

The model of free space treats the region between the transmit and receive antennas as being free of all objects that might absorb or reﬂect radio fre- quency (RF) energy [24]. In this idealized free-space model, the attenuation of RF energy between the transmitter and the receiver behaves according to an inverse-square law. The received power expressed in term of transmitted power is attenuated by a factor, L

_s

(d), where this factor is called pathloss or f reespaceloss. When the receiving antenna is isotropic, this factor is expressed as:

L

_s

(d) =

4πd λ

₂

where d is the distance between the transmitter and the receiver, and λ is

the wavelength of the propagation signal.

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For most practical channels, where signal propagation takes place in the atmosphere and near the ground, the free space propagation model is inad- equate to describe the channel and predict system performance. The eﬀect can cause ﬂuctuations in the received signal’s amplitude, phase and angle of arrival, giving rise to the terminology multipathf ading.

SENSOR

DATA

COLLECTOR

Figure 2.2: Illustration of multipath.

2.1.2 Shadow Fading and Rapid Fading

There are two types of fading eﬀects that characterize mobile communica-

tions: shadow and rapid fading. Shadow fading phenomenon is caused by

obstacles (hills, forests, buildings, walls...) between the transmitter and re-

ceiver. The receiver is often represented as being ”shadowed” by such ob-

stacles. Rapid fading refers to the dramatic changes in signal amplitude and

phase that can be experienced as a result of small changes in the spatial

separation between a receiver and transmitter. Rapid fading is also called

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Rayleighf ading since the multiple reﬂective paths are large in number and there is no line-of-sight signal component, the envelope of the received signal is statistically described by a Rayleigh pdf. There are three basic physi- cal mechanisms that impact signal propagation in a mobile communication system:

Reﬂection : when a propagating electromagnetic wave impinges on a smooth surface with very large dimensions compared to the RF signal wave- length (λ).

Diﬀraction : when the radio path between the transmitter and the receiver is obstructed by a dense body with large dimensions compared with λ, causing secondary waves to be formed behind the obstructing body.

Scattering : when a radio wave impinges on either a large rough surface or any surface whose dimensions are on the order of λ or less, causing the reﬂected energy to spread out (scatter) in all directions.

2.2 Channel Model

A received signal r(t) is generically described in terms of transmitted signal s(t) convolved with the impulse response of the channel h

_c

(t). Neglecting the degradation due to additive noise,

r(t) = s(t) ⊗ h

c

(t)

In the case of wireless channels, h(t) can be partitioned as:

h(t) = g(t)

G(t)L(t)

where g(t) is called the small scale fading component, G(t) is the path-loss, and L(t) is the shadow fading component.

2.2.1 Path Loss

The path loss G(t) is a function of distance d(t) between the transmitter and the receiver, and is proportional to:

G(t) = d

ⁿ

(t)

d

₀

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or in dB:

G(t)(dB) = L

_s

(d

₀

)(dB) + 10n log( d(t) d

₀

)

The reference distance d

₀

corresponds to a point located in the far ﬁeld of the antenna.

The value of the exponent n depends on the frequency, antenna heights and propagation environment. In free space we usually consider n=2, where obstructions are present n is larger.

2.2.2 Shadow Fading

The shadowing component L

_i

(t) is characterized with a log-normal distribu- tion

L(t) = e

^B^(t)

where B(t) is a Gaussian random process having average m

_B

and variance σ

_B²

. The shadowing component exhibits a spatial correlation structure that is dependent on the physical properties of the propagation scenario.

2.2.3 Rapid Fading

Rapid fading occurs due to the non-coherent superposition of a great number of multipath components, each having a diﬀerent phase variation over time or frequency that result in a Gaussian process g(t). The basic model for the envelope of the rapid fading is the Rayleigh distribution, that describes the time-variant ﬂuctuations of the amplitude and phase values by statistical distribution functions. The rapid fading component is hence expressed as follows:

g(t) = |g(t)|e

^jφ^(t)

where φ(t) is the phase of the fast fading. The envelope amplitude has a Rayleigh pdf:

p(r) =

⎧ ⎨

⎩

r

σ²

exp[ −

_2σ^r²2

] for r ≥ 0;

0 otherwise

where r = |g(t)|, and 2σ

²

is the pre-detection mean power of the multipath signal. The Rayleigh pdf results from having no specular

¹

component of the signal.

1The nonfading component is called the specular component.

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Rapid fading manifests itself in two mechanisms:

1. Time-spreading of the underlying digital pulses within the signal. It can be characterized in the time-delay domain as a multi- path delay spread, and in the frequency domain as a channel coherence bandwidth. Bello in 1963 was able to deﬁne functions that apply for all time and all frequencies: a multipath-intensity proﬁle, S(τ ), describes how the average received power vary as a function of time delay, τ

²

. It represents the signals propagation delay that exceeds the delay of the ﬁrst signal arrival at the receiver.

2. A time-variant behavior of the channel due to motion. It can be characterized in the time domain as a channel coherence time, and in the Doppler-shift (frequency) domain as a channel fading rate or Doppler spread. For wireless applications, the channel is time-variant because motion between the transmitter and receiver results in propa- gation path changes. The radio receiver sees variations in the signal’s amplitude and phase. Since the channel characteristics are dependent on the positions of the transmitter and receiver , time variance in this case is equivalent to spatial variance.

2.3 Degradation Categories

Two degradation categories are deﬁned for fading rapidity, fast-fading and slow fading [24].

2.3.1 Degradation Categories viewed in the time-delay domain

• A channel is said to exhibit frequency-selective fading if T

_m

> T

_s

That is, the delay time is greater than the symbol time. This condition occurs whenever the received multipath components of a symbol extent

2The term time delay is used to refer to the excess delay, i.e. the delay a signal experiences from the transmitter to the receiver

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Frequency-selective fading

(ISI distortion, pulse mutilation, irreducible BER)

Multipath Delay spread >

Symbol time

Fast fading

(high Doppler, PLL failure, irreducible BER)

Channel fading rate <

Symbol rate Time-spreading

mechanism due to multipath

Time-variant mechanism due to motion

D u a l m e c h a n i s m s

Flat fading

(loss in SNR)

Multipath delay spread

<

Symbol time

Slow fading

(low Doppler, loss in SNR)

Channel fading rate <

Symbol rate Time-delay

domain

Doppler shift domain

Frequency-selective fading

(ISI distortion, pulse mutilation, irreducible BER)

Channel coherence BW

<

Symbol rate

Fast fading

(high Doppler, PLL failure, irreducible BER)

Channel coherence time

<

Symbol time

D u a l m e c h a n i s m s

Flat fading

(loss in SNR)

Channel coherence BW

>

Symbol rate

Slow fading

(low Doppler, loss in SNR)

Channel coherence time >

Symbol time Frequency

domain

Time domain

Figure 2.3: fading

beyond the symbol’s time duration, thus causing channel-induced in- tersymbol interference (ISI). For a single transmitted impulse, the time T

_m

between the ﬁrst and the last received components represents the maximum excess delay during which the multipath signal power falls to some threshold level below that of the strongest component.

• A channel is said to exhibit frequency-nonselective or ﬂat fading if T

_m

< T

_s

In this case, all of the received multipath components of a symbol arrive

within the symbol time duration, hence the components are not resolv-

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able. In this case there is no channel-introduced ISI distortion, since the signal time spreading does not result in signiﬁcant overlap among neighboring received symbols. There is still performance degradation since the unresolvable phasor components can add up destructively to yield a substantial reduction in signal-to-noise ratio (SNR).

• A channel is referred to a fast fading whenever T

₀

< T

_s

where T

₀

is the channel coherence time and T

_s

is the symbol time. Fast fading describes a condition where the time duration for which the channel behaves in a correlated manner is short compared to the time duration of a symbol. Therefore it can be expected that the fading character of the channel will change several times during the time a symbol is propagating. This leads to distortion of the baseband pulse shape, because the received signal’s components are not all highly cor- related throughout time. Hence, fast fading can cause the baseband pulse to be distorted, resulting in a loss of SNR that often yields an irreducible error rate. Such distorted pulses typically cause synchro- nization problems, such as failure of phase-locked-loop (PLL) receivers.

The Doppler spread and the coherence time are related:

T

₀

≈ λ/2 V ≈ 0.5

f

_d

That is the time required to traverse a distance λ/2 at a constant velocity V.

• A channel is generically referred to as introducing slow fading if T

₀

> T

_s

In this case the time duration for which the channel behaves in a corre-

lated manner is long compared to the symbol time. Thus, the channel

state remains virtually unchanged during the time a symbol is trans-

mitted.

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2.3.2 Degradation Categories viewed in frequency do- main

• A channel is referred to as frequency-selective if f

₀

< 1

T

_s

W where the symbol rate,

_T¹

s

is nominally taken to be equal to the signal bandwidth W . The coherence bandwidth, f

₀

, is a statistical measure of the range of frequencies over which the channel passes all spectral components with approximately equal gain and linear phase. A mea- surement of delay spread is often characterized in terms of the root mean squared (rms) delay spread σ

_τ

:

σ

_τ

=

τ

²

− (τ)

²

The coherence bandwidth is approximately

f

₀

≈ 1 50σ

_τ

• A channel is referred to as Frequency-nonselective or ﬂat fading when- ever

f

₀

> W

In this case all of the signal’s spectral components will be aﬀected by the channel in similar manner. In order to avoid ISI distortion caused by frequency-selective fading, the channel must be made to exhibit ﬂat fading by ensuring that the coherence bandwidth exceeds the signaling rate.

• A channel is referred to as fast fading if the symbol rate

_T¹_s

, or the signal bandwidth W , is less than the fading rate

_T¹

0

or f

_d

: W < f

_d

• A channel is referred to as slow fading if the signaling rate, or band- width W , is greater than the fading rate

_T¹

0

: W > f

_d

In order to avoid signal distortion caused by fast fading, the channel

must be made to exhibit slow fading by ensuring that the signaling rate

exceeds the channel fading rate.

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2.3.3 Key parameters

Combining equations in (2.3.2) and (2.3.2), the conditions that must be met so that the channel does not introduce frequency-selective and fast-fading distortion are:

f

₀

> W > f

_d

or

T

_m

< T

_s

< T

₀

We want the channel coherence bandwidth to exceed our signaling rate, which in turn should exceed the fading rate of the channel.

2.4 Radio resource management

If the channel introduces signal distortion as a result of fading, the system performance exhibit an high error rate. In such cases, the general approach for improving performance is to use some form of resource management to remove or reduce the distortion. The resource management method depends on whether the distortion is caused by frequency-selective or fast fading.

1. Resource management to combat frequency-selective distor- tion

• Equalization can compensate for the channel-introduced ISI that is seen in frequency-selective fading. The process of equalizing the IS1 involves some method of gathering the dispersed sym- bol energy back together into its original time interval. In effect, equalization involves insertion of a filter to make the combination of channel and filter yield a flat response with linear phase. The phase linearity is achieved by making the equalizer filter the com- plex conjugate of the time reverse of the dispersed pulse.Because in a mobile system the channel response varies with time, the equal- izer filter must also change or adapt to the time-varying channel.

Such equalizer ﬁlters are therefore called adaptive equalizers.

• The maximum likelihood sequence estimation (MLSE) equalizer

tests all possible data sequences (rather than decoding each re-

ceived symbol by itself) and chooses the data sequence that is the

most probable of the candidates. The MLSE is optimal in the

sense that it minimizes the probability of a sequence error.

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• Spread-spectrum techniques can be used to mitigate frequency- se- lective IS1 distortion because the hallmark of any spread-spectrum system is its capability to reject interference, and IS1 is a type of interference.

• Pilot signal is the name given to a signal intended to facilitate the coherent detection of waveforms. Pilot signals can be implemented the frequency domain as an in-band tone or in the time domain as a pilot sequence which can also provide information about the channel state and thus improve performance in fading.

2. Resource management to combat fast-fading distortion

• Error-correction coding and interleaving can provide resource man- agement, because instead of providing more signal energy, a code reduces the required

_N^E^b

0

. For a given

_N^E^b

0

, with coding present, the error ﬂoor will be lowered compared to the uncoded case.

• For fast fading distortion, robust modulation can be used, that does not require phase tracking and reduce the detector integra- tion time.

• If possible, increase the symbol rate W

_T¹_s

, to be greater than the fading rate f

_d

_T¹₀

, by adding signal redundancy.

• An interesting ﬁltering technique can provide resource manage- ment in the event of fast-fading distortion and frequency-selective distortion occurring simultaneously. The frequency-selective dis- tortion can be mitigated by the use of an OFDM

³

signal set. Fast fading, however, will typically degrade conventional OFDM be- cause the Doppler spreading corrupts the orthogonality of the OFDM subcarriers.

3. Resource management to combat loss in SNR Performance im- provement of communications over wireless channels can be obtained through Diversity techniques. The term diversity is used to denote the various methods available for providing the receiver with possible uncorrelated replicas of the same signal. Diversity can be provided in various domains of the communications and, hence, various schemes of diversity are possible:

3Orthogonal frequency-division multiplexing (OFDM) is a method of digital modulation in which a signal is split into several narrowband channels at diﬀerent frequencies.

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• Time diversity:transmit the signal on L diﬀerent time slots with the separation of at least T

₀

. Interleaving is a form of time diver- sity.

• Frequency diversity: transmit the signal on L diﬀerent carriers with frequency separation of at least f

₀

. Bandwidth expansion is a form of frequency diversity.

• Spread spectrum: spread spectrum signals experience multipath propagation. The Rake receiver combines the energy from each of the multipath components arriving along diﬀerent paths, thus it implement a multipath diversity.

• Spatial diversity: use of multiple receive antennas separated by a distance of at least 10 wavelengths from the transmitter.

• Error correction coding: represents a unique mitigation technique, because instead of providing more signal energy it reduces the required

_N^E^b

0

in order to accomplish the desired error performance.

2.5 Spatial diversity

One of the most powerful techniques to mitigate the eﬀects of fading is to use diversity-combining of independently fading signal paths, where multi- ple copies of the same information/signal are transmitted and/or received.

Diversity-combining uses the fact that independent signal paths have a low probability of experiencing deep fades simultaneously. Hopefully the proba- bility of all copies simultaneously getting into deep fades will become smaller as more copies are received. Thus, the idea behind diversity is to send the same data over independent fading paths. These independent paths are com- bined in some way such that the fading of the resultant signal is reduced.

There are many ways of achieving independent fading paths in a wire- less system. The most typical diversity technique is spatial diversity : this method consists on using multiple transmit or receive antennas, also called an antenna array, where the elements of the array are separated in space.

In receive diversity multiple receive antennas are used to generate multiple

copies of the same information signal. If the antennas are spaced suﬃciently

far apart, it is unlikely that they both experience deep fades at the same

time.

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2.5.1 Recombination techniques

The signals received by the diversity techniques have to be properly combined in order to extract the useful signal.

A diversity combiner is a circuit for combining two or more signals car- rying the same information received via separate paths or channels. The objective is to provide a single resultant signal that is superior in quality to any of the contributing signals. The combining can be done in several ways which vary in complexity and overall performance. Most combining tech- niques are linear: the output of the combiner is just a weighted sum of the different fading paths. Specifically, when all but one of the complex weights are zero, only one signal is passed to the combiner output. When more than one of the weights is nonzero, the combiner adds together multiple versions of the same signal, where each version may be weighted by different value.

The recombination techniques influences the quality of the signal accord- ing to the criteria design, and different criteria lead to different recombination algorithms. The implementation may requires some information about chan- nel conditions. Therefore, the accuracy and availability of such information influences the choice of recombination techniques. The main options used in combining the different diversity branches are briefly described below [10]:

• Maximal Ratio Combining (MRC): the signals from N branches are ﬁrst co-phased to bring mutual coherence and then summed after weighting.

The weights are chosen to be proportional to the respective signals level to maximize the combined signal-to-noise ratio Γ

_s

= S/N . The gain from MRC is directly proportional to the number of branches, that is the signal-to-noise ratio at the combining node Γ is:

Γ = N Γ

_s

Maximal Ratio Combining has good performance, but requires a perfect knowledge of the channel.

• Equal Gain Combining (EGC): although optimal, MRC is expensive to implement. A simpler alternative is given by equal gain combining which consists in summing the co-phased signals using weights that have all same unitary modulus and perform only a phase correction.

With respect to MRC, this scheme is simpler and has little performance

worsening.

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• Selective Combining (SC): After the demodulation, the signal with maximum SINR is selected.

The ﬁltering techniques we have summarized in the previous section can be applied in cooperative diversity. However, some advantages and disadvan- tages are present. For example, traditional combining techniques (MRC, EGC) require large amount of information to be sent to the combining node (data collector).

In the next chapter, we will discuss the most relevant aspects of spatio- temporal ﬁltering.

2.6 Cooperative diversity

Recently a new form of realizing spatial diversity called cooperative diversity

or node cooperation diversity has been introduced [13]: the main idea is to use

multiple nodes as a virtual macro antenna array, realizing spatial diversity in

a distributed fashion. In fact, an antenna array is inherently present in any

wireless network: diﬀerent nodes in the network can act like elements of an

antenna array. In such a network several nodes serve typically as relays for an

active source/destination pair, however, they are not physically connected.

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Chapter 3 Spatio-temporal ﬁltering

Spatio-temporal processing combines spatial diversity with time diversity as a powerful technique for the reduction of the interference in MAI limited sys- tems [4]. Spatio-temporal processing is a recombination technique. Systems that implement spatio-temporal processing are usually referred to a Smart Antennas in third generation wireless system. A smart antenna is an array of antenna-element where the output of each element is weighted and com- bined to shape a desired equivalent radiation diagram. When a transmitter is equipped with multiple antennas it is possible to control the energy pattern that is emitted from the transmitter. The signals sent from each antenna ele- ment can create beams in a direction of interest towards the receiver, leading to a factor of n gain in signal-to-noise ratio, if n is the number of the antenna elements[1]. By transmitting the signal with diﬀerent phase and amplitude on the diﬀerent antennas, the waves are made to add constructively in some directions and destructively in other directions [23]. Hence, the weights {w

i

} can be designed such that signals coming from desired directions are retained, while those coming from undesired direction are strongly attenuated.

Often, such transmit spatio-temporal ﬁltering from the transmitter to

the receiver could be preceded by training a receive spatio-temporal ﬁlter at

the transmitter using a transmission from the receiver to the transmitter. If

the same frequency band is used in both directions, then reciprocity can be

used to infer the transmit spatio-temporal ﬁltering weights from the receive

spatio-temporal ﬁltering weights.

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Figure 3.1: Energy pattern of beamformer using a Uniform Circular Array (UCA) with six transmitting antennas.

3.1 Idea of Spatio-temporal ﬁltering

Spatio-temporal ﬁltering is an optimization technique, which consists on ﬁnd- ing a weights vector such that a measure of the quality of the transmitted signal is maximized.

To understand the issues involved, consider first the operation of cen- tralized receive spatio-temporal filtering. Assuming that the channel from the transmitter to the receiver is frequency nonselective, first, we have to estimate the spatial channel {w

i

} and then a receive spatio-temporal ﬁlter corresponds to multiplying the complex baseband signal for antenna i by the complex conjugate weight w

^∗_i

. This corresponds to a spatial matched ﬁlter.

Once the receive spatio-temporal ﬁltering weights {w

i

} have been esti-

mated, if reciprocity applies, they can be used for transmit spatio-temporal

ﬁltering.

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3.1.1 System description

An adaptive antenna array consists of a set of antennas, designed to receive signals radiating from some speciﬁc directions and attenuate signals radiat- ing from other directions of no interest. The outputs of array elements are weighted and added by a spatio-temporal ﬁlter to produce a directed main beam and adjustable nulls. Now, consider a cochannel set consisting of M transmitter and receiver pairs, and assume antenna arrays with K elements used at the receivers. Denote the array response to the direction of arrival θ by:

v(θ) = [v

¹

(θ), v

²

(θ), ..., v

^K

(θ)]

where v

^k

(θ) is the response of the kth antenna element at the direction θ.

Σ

w¹* w0*

wk* Beamformer Weights

Adaptive Beamformer

. . .

Matched Filter

t = nT

. . .

Array elements

Figure 3.2: Antenna array and Spatio-temporal ﬁltering We make some assumptions:

1. The propagation delay in diﬀerent paths is much smaller than a fraction

of a symbol.

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2. We consider slow fading channels in which the channel response can be assumed constant over several symbol intervals.

So, the received vector at the ith array can be written as:

x

_i

(t) =

M j=1

P

_j

G

_ji

a

_ji

s

_j

(t − τ

j

) + n

_i

(t) (3.1)

where s

_j

(t) is the message signal transmitted from the jth node, τ

_j

is the corresponding time delay, n

_i

(t) is the thermal noise vector at the input of antenna array at the ith receiver, P

_j

is the power of the jth transmitter and a

_ji

is a vector K × 1 that represents the attenuation due to shadowing, it is called the array response of the ith antenna array to the jth source:

a

_ji

= α

_ji

v

_j

(θ)

The expression of the received vector at the ith array in (3.1) is a general expression, in fact we should deﬁne two diﬀerent message signal transmitted s

_i

(t) in case of nonspectrum system or spread spectrum system:

• In nonspectrum systems, the transmitted signal is given by:

s

_i

(t) =

n

b

_i

(n)g(t − nT )

where b

_i

(n) is the ith node information bit stream and g(t) is the pulse- shaping ﬁlter impulse response. The matched ﬁlter is given by g

^∗

( −t)

¹

. The output of the matched ﬁlter is sampled at t = nT

x

_i

(n) = x

_i

(t) ⊗ g

^∗

( −t)|

t=nT2

Hence, the received signal at the output of the matched ﬁlter is given by

x

_i

(n) =

M j=1

P

_j

G

_ji

a

_ji

+ b

_j

(n)n

_i

(n) (3.2)

where

n

_i

(n) = n

_i

(t) ⊗ g

^∗

( −t)|

t=nT 1where∗ indicates the complex conjugate.

2where⊗ indicates the convolution.

Collaborative Power Control for Wireless Sensor Networks