location information - aalborg...
TRANSCRIPT
WING Group DeliverableWING Group, AAU, 2005-00-00 V1.0
Location Information
WING Group, KOM Department, Aalborg University, Denmark
Prepared byJoao FigueirasCenter for TeleinfrastructuresAalborg University, Niels-Jernes-Vej 12, A6-2049220 Aalborg, [email protected]
Issued by Aalborg University operated for the WANDA project.
WANDA Project: This report was prepared as an account of work sponsored bythe WANDA project.The WANDA project is a joint research activity in a consortium consisting of TexasInstruments, Siemens, RF Micro devices, BLIP Systems, Technological Institute, andAalborg University. The overall research area is future wired and wireless networkswith emphasis on devices, protocols/architectures, and applications. Here it is expectedthat MIMO and OFDM will be key words for the RF-oriented research, whereas networkoptimization based on positioning information is the main focus of the network relatedresearch.
WING Group, AAU, 2005-00-00 V1.0
Location Information
Joao FigueirasCenter for TeleInFrastruktur
Aalborg University, Niels-Jernes-Vej 12, A6-2049220 Aalborg, Denmark
3
Abstract
This is the abstract
4
Acknowledgment
Thanks to....
5
Contents
Preface 12
Summary 13
Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
1 Introduction 16
1.1 A Taxonomy For Localization Purposes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Localization Requirements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
2 Bluetooth survey for location purposes 19
2.1 Bluetooth General Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
2.2 The Inquiry procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
2.2.1 Inquiry Substate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21
2.2.2 Inquiry Scan Substate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
2.2.3 Inquiry Response Substate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.4 Packets used in Inquiry Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 The Frequency Generator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24
3 Localization Methods, Solutions and Systems 26
3.1 Motivation for Location Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Location Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
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3.2.1 Triangulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
3.2.1.1 Lateration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
3.2.1.2 Angulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30
3.2.2 Proximity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32
3.2.2.1 Physical Contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.2.2 Cell Identification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.3 Environment Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32
3.2.3.1 Database Correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.3.2 Video Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Location Solutions and Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 A Survey on Networking Optimization based on location information 37
5 Timing Aspects for Location in Bluetooth Networks 38
5.1 Scenarios and General Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2 Simulation Results of Inquiry Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 Theoretical approach of the Inquiry Procedure in Bluetooth. . . . . . . . . . . . 44
5.3.1 Inquiry Time as a ball-urn problem. . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3.1.1 The rules of the game. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3.1.2 The relation between the game and the Inquiry Procedure46
5.3.2 Approach for the Inquiry Time distribution as a ball-urn problem. . 47
5.3.3 Inquiry time in a scenario with one Access Point. . . . . . . . . . . . . . . 47
5.3.4 Inquiry time in a scenario with two Access Points. . . . . . . . . . . . . . 48
5.4 Experimental validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49
6 Accuracy Aspects for Location in Bluetooth Networks 50
6.1 Static Devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50
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6.2 Moving Devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50
7 Access Point placement to enhance accuracy on device localization 51
8 Bayesian Filtering for Location Estimation 52
8.1 Bayes Filters Implementations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53
8.1.1 The Kalman Filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54
8.1.2 The Extended Kalman Filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8.2 Kalman Filters Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54
8.2.1 The static case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55
8.2.2 The dynamic case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57
9 Conclusion 61
9.1 sfgsdfg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61
References 63
Appendix
A sdfsdfs 64
A.1 XSV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64
B Some Other Appendix 65
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List of Figures
2.1 Bluetooth Scatternet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
2.2 Inquiry Procedure in a macro point of view. . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 The Inquiry Substate in detail. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
2.4 Frequency shifting that occurs in the master node. . . . . . . . . . . . . . . . . . . . . 25
3.1 Location techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
3.2 Lateration Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
3.3 Time Difference of Arrival Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Angulation Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31
3.5 Cell Identification Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33
3.6 Database Correlation Technique: a) calibration procedure; b) localizationprocedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33
3.7 Video Analysis Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
5.1 Scenarios - inquiry time for: a) 1 AP and 1 device; b) 2 APs and 1 device;c) n APs and 1 device.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39
5.2 Inquiry Time definition for scenarios with: a) 1 AP and 1 Device; b) n APsand 1 Device.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39
5.3 Inquiry TimePDF for aSingleAP scenariowith Bluetooth specificationsV1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40
5.4 Inquiry TimePDF for aSingleAP scenariowith Bluetooth specificationsV1.1. [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40
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5.5 Inquiry Time PDF to aMultiple Access Point Scenario(n = 2,4,7) . . . . . . . 41
5.6 Simulated and approximated Inquiry Time CDF to a2, 4, 7 AP scenario(obtained simulating 106 values of inquiry time). . . . . . . . . . . . . . . . . . . . . . 42
5.7 Simulated Inquiry time CDF toMultiple Access Point Scenario(n=1 up to20) and 1 MD (obtained simulating 106 values of inquiry time). . . . . . . . . . 43
5.8 Inquiry Time duration depending on the number of APs for different pen-centiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43
5.9 Ball-urn game. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45
5.10 Inquiry Distribution time for a scenario with one AP and one device. . . . . . 48
5.11 Inquiry Distribution time for a scenario with two APs and one device. . . . . 49
8.1 Bayesian filtering cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53
8.2 Properties of the most common implementations of Bayes filters for loca-tion estimation [?] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54
8.3 Kalman Filter Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55
8.4 Extended Kalman Filter Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
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List of Tables
2.1 Set of frequencies used in the Inquiry procedure: sorted in ascendant man-ner and in the way they appear in the sequence. . . . . . . . . . . . . . . . . . . . . . . 24
5.1 MeanInquiry Timefor aMultiple AP Scenariowith n Access Point (AP)s . 44
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Preface
This document reports the work related to localization issues inside the scope of the projectWANDA. The document presents some obtained results and also some on going tasks thatstill have no results. However the main goal is to obtain a document that relates what wasdone and identifies the possible tracks for future directions.
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Summary
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Acronyms
2D 2-Dimensional
3D 3-Dimensional
ACL Asynchronous Connection Oriented
AP Access Point
BS Base Station
CDF Cumulative Distribution Function
DIAC Dedicated Inquiry Access Code
E911 Enhanced 911
FCC Federal Communications Commission
FHS Frequency Hop Synchronization
FHSS Frequency Hopping Spread spectrum
GFSK Gaussian Frequency Shift Keying
GIAC General Inquiry Access Code
GPS Global Positioning System
GSM Global System for Mobile communications
i.i.d. Independent and Identically Distributed
IAC Inquiry Access Code
ID IDentity
ISM Industrial, Scientific and Medical
LAP Lower Part Address
MD Mobile Device
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MS Mobile Station
PDF Probability Density Function
PF Particles Filter
SCO Synchronous Connection Oriented
SIG Special Interest Group
ToA Time of Arrival
UMTS Universal Mobile Telecommunications System
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Chapter 1
Introduction
Las years, wireless short-range communications have become more and more available inindoor scenarios. Mainly used for communication, these networks tend to be used also forlocalization purposes. In fact, the topic of localization of mobile devices in wireless short-range communications has become a hot issue among the scientific community.
Location information is intended, inside this scope, as the entire information obtained aboutdevices in the network. This information includes space/time results of localization of thedevices, the position of the access points and the Bluetooth address of the intervenientnodes.
The motivation for obtaining location information is based in three main bullets:- Laws enforcement- In some countries, some laws encourage wireless providers to im-plement localization solutions on their networks. An good example of such laws is theEnhanced 911 (E911) Service defined by the Federal Communications Commission (FCC)governmental organization in United States of America. The purpose is to obtain locationinformation of the caller, when the emergency number 911 is dialled.- Navigation/Tracking Applications - Recent applications run based on location informa-tion of the user. For example navigation or tracking applications make use of the localiza-tion of the devices. A use-case for such applications may be the tracking of football playersin the football field. This information can be later used for game planning.- Context-Sensitive Networking- The new generation of networks, make extensively useof device discovery services for example for networking formation purposes. Such net-working optimization can pass also by considering the localization of some or every nodeexistent in the network.
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1.1 A Taxonomy For Localization Purposes
Throughout this document we make use of several words or expressions that are explainedin this section. The purpose is to identify clearly the concept in order to allow a completeunderstanding of this document.
Equipment definitions:
Node defines any equipment that belong to the network and is a terminal or a point ofconnectively among another equipments in the network.
Base Station (BS)is the radio equipment located at one fixed location which is able toserve and handle all incoming and outgoing traffic from several terminals.
Mobile Station (MS) is the terminal equipment which position may change in time.
Access Point (AP) is the name given to the Base Station (BS) when the network is com-posed by short range communications technologies.
Mobile Device (MD) is the name given to the Mobile Station (MS) when the network iscomposed by short range communications technologies.
Location definitions:
Location means a precise place in space with well defined coordinates(x,y) for 2-Dimensional(2D) space or(x,y,z) for 3-Dimensional (3D) space.
Real Location means the true location where is possible to find theMS
Predicted Location is the location estimated by a Location Method
Localization is the abstract concept of predicting the Location of a device.
Location Method is related to all the procedures, algorithms, computations, processingand calculations used in order to get the predicted location.
Position Same definition as Location.
Placement Same definition as Location but usually related toBS.
Timing definitions:
Localization Time Delay is the interval of time between the time when location informa-tion is asked to the network and the time when that information is delivered to therequester.
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1.2 Localization Requirements
In order to obtain location information it is required the spacial localization, time delayuntil obtain location information, positioning of theAPs, and an identification of every in-tervenient node.In terms of performance and precision, there are no standard requirements at the moment,since location information in short-range communication is a recent issue of research.Thus, our goal is to obtain as precise values as possible.
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Chapter 2
Bluetooth survey for location purposes
In this section we present the Bluetooth features that are related to the procedures of gettinglocation information.
2.1 Bluetooth General Concepts
The Bluetooth technology stated being developed in 1994 by Ericsson Mobile Communi-cations, and later in 1998 expanded to a wider group, called Special Interest Group (SIG).The first specifications V1.0 were defined in 1999, and later reviewed for new publicationsof Bluetooth versions 1.1, 1.2 and 2.0.
Bluetooth operates in the Industrial, Scientific and Medical (ISM) frequency band, be-tween f = 2.4GHzand f = 2.4835GHz. In this band, Bluetooth uses a technique of spreadspectrum called Frequency Hopping Spread spectrum (FHSS), which makes use of a hopfrequency of 1600hops/samong 79 different frequencies with 1µs= 1MHzwide. Becausethe modulation used in Bluetooth is Gaussian Frequency Shift Keying (GFSK), the 1MHzfrequency corresponds to a capacity of 1Mb/s.
Besides the transmission power being dependent on the class of device, its value is con-siderable small if compared with other technologies operating in the same frequency band.For devices of class 1, the transmission power is defined as 100mW (20dBm), while fordevices of class 2 that value is about 2.5mW (4dBm). These transmissions power valueslimit the communication range to about 100m for class 1 and 10m for class 2 devices.
Bluetooth nodes transmit data among them in a serial fashion, using asynchronous links,Asynchronous Connection Oriented (ACL), or synchronous links, Synchronous Connec-tion Oriented (SCO).TheACL link can support symmetrical or asymmetrical. For symmetrical connections, the
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maximum bit rate is 433.9Kbpsin both directions, while for asymmetrical connections themaximum bit rate is 732.3Kbpsin one direction and 57.6Kbpsin the opposite direction.On the other hand, theSCOlink only supports symmetrical connections, using for that 3channels of 64Kbps each.
The Bluetooth Networks have a formation defined as scatternets, which structure is rep-
Figure 2.1. Bluetooth Scatternet
resented in Fig.2.1. As we can see, the network is formed by several cells, called piconets.In each piconet, Bluetooth is allowed to have up to 8 active nodes at the same time. Amongthese nodes, one is assigned to be the master during the inquiry procedure and is the onethat rules all the connection inside the piconet. The remaining nodes play the role of slaves,being allowed to communicate only when requested by the master. In order to establish theconnectivity among piconets there are some nodes called bridges that belong to more thanone piconet and have the function of transmitting the information from one piconet to theother.
2.2 The Inquiry procedure
TheInquiry Procedure(Fig. 2.2) is the mechanism existent in Bluetooth that allows nodesto establish a connection. This procedure is the first step, being then followed by thePageProcedurein order to finalize the connection establishment.
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In order to discover other devices in its range, a device, called master, shall enter periodi-
Master
Slave
f0 f1 f2 f3 f4 f5 fi-2 fi-1 fi fi+1 fi+2 fi+3
fi-1 fi
IDpacke
t
FH
Spack
et time
time
Figure 2.2. Inquiry Procedure in a macro point of view
cally in theInquiry Substate(Section2.2.1). In this substate, the device transmits inquirymessages in form of IDentity (ID) packets at different hop frequencies, as we can see inFig. 2.2. One of these packets shall be received by another device, called slave, whichone must be running theInquiry Scan Substate(Section2.2.2). In this substate, the slavenode must be scanning in the same frequency in order to receive theID packet sent by themaster node (frequencyfi in Fig. 2.2). When the packet is received, the slave may providean answer to the master, what is done entering in theInquiry Response Substate(Section2.2.3). However this substate may not be entered if the slave decides not to answer to themaster. As said in the Bluetooth specifications, the slave has the right to choose if answeror not.
2.2.1 Inquiry Substate
The Inquiry Substateis entered by the master when it wants to collect information aboutneighboring devices. This substate has a typical duration of 10.24s as advised by theBluetooth specifications. During this period, the master sends continuouslyID packets(Fig. 2.3) at different hop frequencies dictated by a pseudo-random generator (Section2.3). Shortly, frequencies are divided in 2 sets, called inquiry trains, each of which has 16frequencies. As we can see in Fig.2.3, every 1.28s there is a change of 1 frequency inthe set of used frequencies, and every 2.56s there is a interchange of frequency sets used.Section2.3addresses detailed this aspect.
The master may either search for any device, inquiring for the General Inquiry AccessCode (GIAC) or search for a certain type of devices, inquiring for the Dedicated InquiryAccess Code (DIAC). Despite the type of devices that the master is searching for, it al-ways follows the same hop frequency pattern, being the sequence determined based on theGIAC and the phase of the sequence determined by the Lower Part Address (LAP) of theBluetooth address.
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(from 0s to 1.28s)Ex: f = fk 0
time
(from 1.28s to 2.56s)Ex: f = fk 1
time
(from 2.56s to 3.84s)Ex: f = fk 2
fk-10fk-9
fk+8fk+9
fk+10fk+11
time
fk-14
fk-11
fk-15
fk-13
fk-16
fk-12
fk+12fk+13
fk+14fk+15
fk+15
fk-16
(from 3.84s to 5.12s)Ex: f = fk 3
time
train A used train B used
fk+5 fk+5
fk-2fk-1
fkfk+1
fk+2fk+3
fk-6
fk-3
fk-7
fk-5
fk-8
fk-4
fk+4fk+5
fk+6fk+7
fk+7
fk-8
fk+6
fk-2fk-1
fkfk+1
fk+2fk+3
fk-6
fk-3
fk-7
fk-5
fk-8
fk-4
fk+4fk+5
fk+6fk+7
fk+7
fk-8
fk+6 fk+14
fk-10fk-9
fk+8fk+9
fk+10fk+11
fk-14
fk-11
fk-15
fk-13
fk-16
fk-12
fk+12fk+13
fk+14fk+15
fk+15
fk-16
fk+14
fk+4 fk+4
Timeslot where2 ID packets are sent
in 2 different freq.
Timeslot wherenode listens 2 freqfor FHS packets arrivals
Figure 2.3. The Inquiry Substate in detail
During theInquiry Substate, the time is divided in between sending inquiry messages andlistening answer messages. In even timeslots the master sendsID packets while in oddtimeslots the master listens for Frequency Hop Synchronization (FHS) packets (Section2.2.4address characteristics of the packets). For each timeslot either 2ID packets are sent,one in each half of the timeslot, or 2 frequencies are listened forFHSpackages, one in eachhalf of the slot. In two consecutive slots the device listens to the same frequencies and withthe same order as used to previously send theID packets, as we can see in Fig.2.3.
2.2.2 Inquiry Scan Substate
Inquiry Scan Substate is entered by a device when it wants to check if there are any otherdevices in its range searching for it. This substate can be entered either from the standbyor connection state. Entering from the standby state means that there was no packet traffic
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in the device, while entering from the connection state implies that there was some packettraffic already existent. The existence or not of additional packet traffic in the device influ-ences some parameters in the Inquiry Scan Substate.
When no packet traffic exists in the network, as advised in the Bluetooth specifications,the Inquiry Scan Substate has a duration of 11.25s, called Scan Window, and an intervalbetween scans of 1.28s, called Scan Interval. The Scan Window equals the same dura-tion of 18 timeslots, what was done with the purpose of covering lacks of synchronizationbetween master and slave. During the Scan Window, the device searches forID packetscontinuously in the same frequency, changing the scan frequency only after one Scan Inter-val. The process of determining the scanning frequency is explained in Section2.3. Duringthe remaining period of the Inquiry Interval, the device returns to the standby or connectionstate, as it was before start scanning.
The device can be scanning for theGIAC and/or one or moreDIACs. Thus, when thedevice is triggered with anID packet, the device read the packet and decides, based on theInquiry Access Code (IAC), if it wants to go further with the connection or just dischargethe packet. In case theID packet includes theIAC searched, the device shall enter theInquiry Response Substate (Section2.2.3).
2.2.3 Inquiry Response Substate
This substate is entered by the slave as a way of providing an answer to the master, whenthe slave is triggered with anID packet.
After receiving theID packet, still in the Inquiry Scan Substate, the device right after625µs, and already in Inquiry Response Substate, shall send anFHS packet in the samefrequency which theID packet was received. When sent theFHSpacket, the device addsone unit in phase with step of 1.28s to a counter included in the frequency generator.When the answer is provided, it can happen that the device is synchronized with other de-vices in such a way that there is a time period where they search the same frequency. Inthis case, theFHSpackets would collide and the master wouldn’t receive any answer. Toavoid this problem, right after sending theFHS packet, the slave shall choose a randomnumber between 0 and 1023 timeslots (639.375ms) uniformly distributed, and leave theInquiry Procedure during that period. Because devices will wake up after different timeperiods, the synchronization, most probably, is lost. When the device wakes up, it returnsto the Inquiry Scan Substate.
2.2.4 Packets used in Inquiry Procedure
As seen already in previous sections, there are two different packets used in the InquiryProcedure:
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- ID packet This packet has a fixed length of 68bits. Since for Bluetooth specificationsV1.2, the data rate is 1Mb/s, the size of the packet represents, roughly, only 11%of the timeslot used to send theID packet. This packet includes, for the case of theInquiry Procedure, only theIAC.
- FHS packet This packet is sent in the slave-master direction and has synchronizationpurposes. It encapsulates the Bluetooth Address and the clock of the sender. Thispacket has a size of 366bits, meaning for the Bluetooth specifications a fraction ofaround 59% of the duration of one timeslot.
2.3 The Frequency Generator
The frequency generator existent in Bluetooth is relatively complex having several op-erations and several input parameters. In this section we explain conceptually how thisgenerator works and which outputs it does, disregarding detailed explanation on the imple-mentation.The frequency generator is a unique piece of hardware that is responsible to generate everyfrequency hopping for every possible case in Bluetooth protocols. Inquiry Procedure is oneout of all possible cases that use the frequency generator. In the inquiry Procedure, twoinput parameters are needed. The first is theGIAC, which defines the sequence used andthe native Bluetooth clock, that determines the phase of hopping. However, depending onthe substate of the Inquiry Procedure, the frequency hopping will behave slightly different.In general, for Inquiry Procedure, only 32 out of the 79 frequencies are used. From those32 frequencies, one different frequency,fk, is selected each 1.28s following the patternin Table2.1. For example if at timet0 is selected frequencyfk = 57, at timet0 + 1.28 isselected frequencyfk73. For Inquiry Scan Substate, the selected frequency represents the
Table 2.1. Set of frequencies used in the Inquiry procedure:sorted in ascendant manner and in the way they appear in the se-quence
Pattern55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 047 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
Sorted0 2 4 6 8 10 27 29 31 33 35 37 39 41 43 4547 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77
frequency that devices listen to. For the Inquiry Substate, the frequency used to sendIDpackets is more complex.In the Inquiry Substate, the selected frequency, is not the only used frequency. Besides hap-pens a frequency selection every 1.28s, the substate makes use of a set of 16 frequencies,called inquiry train. There are 2 inquiry trains: the inquiry trainA that is defined as the setof frequencies fromfk−8 to fk+7, i.e., includesfk and inquiry trainB that is the comple-mentary of inquiry trainA, i.e., includes frequencies fromfk−16 to fk−9 and from fk+8 to
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fk+15. Looking at Fig.2.4we can see the trains and that the frequency selection change ateach 1.28shas a consequence of interchanging one frequency between both trains. Besides
55 71 39 10 57 73 41 75
43
59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
55 71 39 10 57 73 41 75 43 59 27 77 45 61 29 0 47 63 31 2 49 65 33 4 51 67 35 6 53 69 37 8
0s
1.28s
2.56s
3.84s
5.12s
6.40s
7.68s
8.96s
10.24s
20.48s
30.72s
40.96s
XXM1XXX
XX 2XXXM
XX 3XXXM
XX 4XXXM
XX 5XXXM
XX 6XXXM
XX 7XXXM
XX 8XXXM
XX 9XXXM
XX AXXXM
XX BXXXM
XX CXXXM
XX DXXXM
XX EXXXM
XX FXXXM
XXN0XXX
XX 1XXXN
XX 2XXXN
XX 3XXXN
XX 4XXXN
XX 5XXXN
XX 6XXXN
XX 7XXXN
XX 8XXXN
XX 9XXXN
XX AXXXN
XX BXXXN
XX CXXXN
XX DXXXN
XX EXXXN
XX FXXXN
XXM0XXX
XXM1XXX
XXM2XXX
XXM0XXX
M=even; N=odd; X=any number
Bluetooth internal clock (hex)Time of frequency change occurrence (seconds)
55
71
43
8
Freq. 55 determined by the frequency generator and in use
Freq. 8 determined by the frequency generator and not in use
Freq. 43 not in use
Freq. 71 in use
Figure 2.4. Frequency shifting that occurs in the master node
that, Bluetooth specifications say that every 2.56s a change of used train must occur. Thisfact is observable in Fig.2.4at time instants multiples of 2.56s.As we can see from Fig.2.4 (in the right) the hopping phase is determined by the na-tive clock, being the clock bits from 12−16 responsible to determine the actual selectedfrequencyfk.
25
Chapter 3
Localization Methods, Solutions andSystems
This chapter presents the state-of-the-art in localization issues for wireless networks. Thelocalization problem is looked into detail, explaining some of the common location tech-niques and existing location systems.Location techniques are identified and grouped according to the main concept used for lo-calization. For each location technique more than one approach may be possible. Thesetechniques and some common approaches are further explained in more detail.The existing location systems are identified and linked to the location techniques that ituses.
3.1 Motivation for Location Solutions
Several are the bullets that motivate localization systems, and several are already the appli-cations running making use of location information.
Nowadays, government laws persuade companies that provide wireless solutions to imple-ment on their systems a method able to determine the absolute localization of their wirelessdevices. An example of this is theE911service [1] specified by theFCC, which is an inde-pendent government agency of the United States of America. This wirelessE911programspecifies that every call to the emergency number 911 must be processed by systems ableto determine the location of the caller.
Another important issue that motivates location systems are the existent and future ap-plications that make use of location information of the devices in the network. For examplein a outdoor scenario, the Global Positioning System (GPS) system is a good example of asystem that is used for tracking cars in the streets.
26
Research work on localization applications has been held in order to develop several appli-cation such as: tracking people in museums, football fields or amusement parks; trackinganimals in stables; tracking robots in arenas.
Context-sensitive networks can and will be sensitive also to location information. Researchhas been held in order to apply location information on network planning. Use-cases aretopology formation in Bluetooth network and routing protocols.
3.2 Location Techniques
Localization is a very wide and hot field of research. Several are the techniques usedin wireless communications and several are the systems already working based on thesetechniques. Research have been done addressing the general problem of localization [2, 5,6] resulting in different ways of grouping location techniques.As we can see in Fig.3.1, this document groups the location techniques based on the main
LocationTechniques
�������
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HHHHHHH
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Triangulation Proximity EnvironmentAnalysis
��
��
���
��
��
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��
��
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AAAAAAU
AAAAAAU
AAAAAAU
Lateration Angulation PhysicalContact
CellIdentification
DatabaseCorrelation
VideoAnalysis
Figure 3.1. Location techniques
localization concept that is used for localize devices:
27
Triangulation In this group are included techniques that use the trigonometry theory tocompute location information of theMSs based on the obtained measurements.
Proximity In this techniques, localization is not performed in terms of(x,y) position. In-stead, localization means, in these group of techniques, identifying theMSs as beingor not in the coverage range of theBSs.
Environment Analysis This group of techniques are those that analyzes the environmentwhere theMSs are included. The localization is the result of the correlation of ob-tained measurements with a-piori knowledge of the environment.
3.2.1 Triangulation
Triangulation is the general name that is given to all the techniques that make use oftrigonometry theory, specially the properties of triangles, in order to localizeMSs. Anyof the techniques in this group allows to perform localization in terms of single predictedposition. This predicted position may be, in fact, not precise, depending on several factors,being the most important the propagation conditions of the transmission signal.This group of techniques can be subdivided in two different groups depending on howtrigonometry is used: Lateration is the name given to the techniques that make use of dis-tances amongBSs andMSs for localization purposes. Angulation techniques are those thatallow localization ofMS based on measured angles amongBSs andMS.
3.2.1.1 Lateration
Lateration is a subgroup of Triangulation techniques that includes those techniques whichperform localization making use of distances amongBSs andMSs. The name is due to therelation that exists between the distances and the sides of a triangle. The calculation of thepredicted location can be in fact considered as the problem of determining the vertex of atriangle knowing the sizes of the three sides an the location of the other two vertices. Thesetwo vertices can be the position of theBSs and the third the position of theMS. However,with only two BSs, the predicted location may result in two points, for a2D space, or acircumference for the3D space, due to the symmetry around the line that connects bothBSs. For this reason, the common minimal number ofBSs necessary are 3, for a2D space,and 4, for a3D space.Lateration techniques have several approaches. These approaches depend mainly on howdistances are measured and some minimal changes in the localization procedure as the re-sult of the measurement procedure.
Direct LaterationThis approach uses physical action or movement to predict location. For example a robotthat can measure distances during the movement of its feet or wheels can extrapolate itslocation, or, when it touches something, its localization may be corrected if there is a-priori
28
knowledge about the scenario.
AttenuationFor this approach of the Lateration Technique, the distance betweenBSs andMS is cal-culated making use of a relation between distance and attenuation. This relation may bethe result of free space pass loss, multipath [8] or even a result of a more complex channelcharacterization [4]. The received power is measured by one of the devices, and knowingthe transmission power it is possible with a simple subtraction (indBdomain) to obtain theattenuation. Then, using the relation attenuation/distance is possible to predict the distancebetween theMS and theBS.After calculating the distance, the approach is done is the following way: for eachBS,knowing the distance, is possible to describe a circumference around it where theMS islocalized. As we can see in Fig.3.2, for the2D space, intercepting the circumferences of
Base Station
Mobile Station
Figure 3.2. Lateration Technique
the 3BSs is possible to localize theMS.This approach is very sensible to the propagation conditions, which has a considerable re-flection in the accuracy of the localization.
Time of Arrival (ToA)
This approach follows the same procedures as attenuation approach in Fig.3.2. Theonly difference is in the way distance is calculated. For this case, the distance betweentheBS and theMS is direct related to the time used by a signal to travel from the sourceuntil the destination and the speed of that signal. Usually is possible to calculate Time of
29
Arrival (ToA) with ultrasound pulses, light or radio pulses. Moreover this pulses may bepart of the communication or included in a different channel as a control channel. ThisToAapproach has an important drawback related to the clocks of the intervenient devices. Theclocks must have a very high resolution because theToA values are typically of the order ofnanoseconds or microseconds. Besides that, ifToA is measured in a device different fromthe one that produced the signal, both clocks must be synchronized with a higher precisionthan the typical value ofToA.Another common problem inToA approach concerns the multipath effect. This effect bydefinition is the effect of reflections in environment objects, resulting in different paths thatthe signal take. These different paths, because of having different lengths, provoke severalpulses arriving at the receiver, what makes the problem of calculating theToA even morecomplicated. There are filtering techniques such Bats, that try to solve this problem.
Time Difference of Arrival (TDOA)This approach brings a solution to one of the drawbacks present in theToA approach. Theproblem is due to the necessary synchronization between transmitters and receivers. Thissynchronization, besides being sometimes impossible to establish, requires very preciseclocks in theBS and in theMS, what may be expensive to the common user (MS) anddifficult to implement.In this approach, theMSs and theBSs do not need to be synchronized. Instead, only theBSs are synchronized with each others. The consequence of this is that the parameter timethat represents the error between the receiver clock and the synchronized transmitter clocksmust be calculated. According to [3, 7], for a2D space, it is possible to obtain the variables(x,y, t) from a system of equations with 4 transmitted signals. In fact, generally for a systemof 3 unknowns, 3 equations are enough. However, for the same reason that we need thethird BS to have a unique solution in the previous approaches (with 2 unknowns(x,y)), tothis approach we need the forthBS.This approach of localization can be considered as divided in two different stages [3]: thefirst stage determines, for each pair ofBSs, the differences of the signal time of arrival.Then, in the second stage, it is necessary to transform this time differences in distancedifferences, which describes a hyperbola as possible location for theMS (Fig. 3.3). In fact,by definition, for a2D space, the locus of all the points which have a constant difference ofdistances to two fixed points (BS) represent the conic section called hyperbola. Then, witha smart algorithms to make the interception of three hyperbolas, the location of theMS ispossible to predict.
Upload Time Difference of Arrival (U-TDOA)WRITE HERE
3.2.1.2 Angulation
This techniques as a Triangulation technique, makes use of the properties of triangles, spe-cially angles, in order to localizeMSs. For a2D space, twoBSs at a known position areenough to predict the location of theMS (Fig 3.4). For a3D space it is possible to use this
30
Base Station
Mobile Station
Figure 3.3. Time Difference of Arrival Technique
Base Station
Mobile Station
α
β
00
Figure 3.4. Angulation Technique
technique if another measurement of azimuth is obtained.In this technique is common to pre-define one of the directions of the vector that connectstheBSs as the magnetic north. This vector defines the reference at 0o and, consequently,the called magnetic south at 180o.In this techniques, the intervenientBSs must be able to identify the angle from which theemission was originated. Several methods can be used to measure this angle such as direc-tive antennas or methods making use of the signal phase properties. After measuring the
31
angles and knowing the location of theBSs, it is possible to predict the location of theMS.This techniques is very dependent on the propagation aspects such as multipath and shad-owing. Multipath for example may originate several reflections of the signal with differentangles, forcing the use of filtering techniques to identify the direct ray which has the correctangle. Shadowing may result in totally wrong measurements of angles, since the direct rayis not received. however, having some knowledge about the scenario is possible to includesome corrections in the measurements.
3.2.2 Proximity
In proximity techniques, the idea is identify the Mobile Device (MD) when it is ”near” theBS. TheBS with a physical phenomenon such as light or radio waves, is able to cover acertain area, identifying theMD when it is inside that area. For this group of localizationtechniques the localization can not be predicted in terms of exact location(x,y) but in termsof an area where theMD has an uniform probability to be found in.
3.2.2.1 Physical Contact
This is the most basic technique, which one identifies theMD when it establish a physicalcontact with theBS. Here the definitionMD and BS may be a bit abusive, since thistechniques relies basically in sensors of pressure or touch.
3.2.2.2 Cell Identification
This location technique assumes that the location of theBSs and the size of the cell arewell known. This technique consists only in determining to whichBStheMS is connectedto. The predicted location equals the area defined by the position of theBS and the size ofthe cell (Fig.3.5).In this Cell Identification technique, the location can be predicted by theBS, but also bythe MS if the system delivers, through a control channel, information about the locationand the range of the cells.
3.2.3 Environment Analysis
This group of techniques perform localization ofMDs, based on information obtained fromthe environment where theMDs are included. For these techniques, it is important to havea good knowledge about the environment in order to correlate the information obtainedduring the location method. This requires a big effort in the calibration part of the systems,what may not be desirable or not implementable for several applications.
32
(x ,y )0 0
Cell SizeBase Station
Mobile Station
Figure 3.5. Cell Identification Technique
3.2.3.1 Database Correlation
On this technique the most important issue to have an accurate location prediction is thedatabase. The database shall be as much complete as possible with as much variety ofmeasurement as possible, such as received power, time-of-flight, angle of arrival. As we
Base Station
Mobile Station
UMTSGSM
X
Y
Measurements& location
UMTSGSM
Location
Measurements
DataBaseServer
a) b)
WLAN WLAN
Figure 3.6. Database Correlation Technique: a) calibration pro-cedure; b) localization procedure
can see in Fig.3.6, it is also possible to consider measurements from several systems such
33
asGPS, Global System for Mobile communications (GSM) or Universal Mobile Telecom-munications System (UMTS).As bigger is the amount of information in the database, more complex are the algorithmsfor localizeMSs and also bigger is the problem to manage the database. The reason of thisis because this techniques is very dependent on the scenario. If something changes on thescenario, as for example the disposition of one room that was changed, the database shallbe calibrated all over again.As we can see in Fig.3.6 during the calibration method, the measurements shall be per-formed with a certain step all over the possible area. The location and those measurementsare then passed into the database for further processing. During the localization, theMSshall be able to measure some of the possible signals and relate them to a central server.This server then after correlating this measurements with the information included in thedatabase shall determine the predicted location.
3.2.3.2 Video Analysis
This technique makes use of video in order to locate devices. The video is analyzed andbased on a-priori knowledge of the environment is possible to infer the position of theMS.
Figure 3.7. Video Analysis Technique
As we can see in Fig.3.7, the extrapolation of the shape of the horizon silhouette is givingsome information about the location of theMS.This technique besides being conceptually easy to understand and the closest one to thesense of orientation of the Human Being, it may be extremely complex to implement dueto the huge computation and the complex algorithms that are involved in the procedure.However, this technique has the strong advantage of being still operative even if theMSsare in idle mode, because no emission signals are required.This technique considers two approaches: The static scene, where only one image is ana-lyzed at a time, and the differential scene, where the analysis is done over the differences
34
observed in sequential images. For the static scene, it is necessary to have a predefineddatabase that maps the observed features in the image to the position of theMSs. On theother hand, for the differential scene case, the differences in the scene are related to themovement of theMS, what allows theMS to compute its location if the position of thefeatures in the image are known.
3.3 Location Solutions and Systems
The systems and solutions
Global Positioning System (GPS)WRITE HERE
Active BadgesWRITE HERE
Active BatsWRITE HERE
MotionStarWRITE HERE
VHF Omni Ranging (VOR)WRITE HERE
CricketWRITE HERE
MSR RADARWRITE HERE
PinPoint 3D-iDWRITE HERE
Avalanche TransceiversWRITE HERE
Easy LivingWRITE HERE
Smart FloorWRITE HERE
Automatic ID SystemWRITE HERE
35
Wireless AndrewWRITE HERE
E911WRITE HERE
SpotONWRITE HERE
True PositionWRITE HERE
Bruel & KjærWRITE HERE
BLIP ZoneWRITE HERE
EkahauWRITE HERE
36
Chapter 4
A Survey on Networking Optimizationbased on location information
37
Chapter 5
Timing Aspects for Location inBluetooth Networks
This Chapter presents some timing aspects for localization purposes for the Bluetooth tech-nology V1.2 [11].As already stated in Section2.2, Bluetooth technology has a mechanism of measuring re-ceived power level, which can be used to get location information. This mechanism isincluded in the inquiry procedure which is evaluated above.In Section5.1 some definitions used all over the Chapter are presented. This section alsopresents the scenarios that are later used in following Sections. In Section5.3 a theoreti-cal approach of the scenarios is done in an engineering point of view, suppressing severalaspect that can be simplified. Later in Section
5.1 Scenarios and General Definitions
In this Chapter the following definitions are used:
• Inquiry Time - It is described as the time spent since the inquiry procedure is starteduntil all the APs have discovered the existent Devices in the neighborhood.
In this Chapter the considered scenarios are shown in Figure5.1. The scenario in Figure5.1.a considers 1 AP and 1 device, scenario of Figure5.1.b considers 2 APs and 1 Deviceand Figure5.1.c considers n APs and 1 Device. In all the scenarios the inquiry time shall beexplained in more detail. The inquiry time is defined in Figure5.2.a for 1 AP and 1 Deviceand in Figure5.2.b for n APs and 1 Device. As we can see on Figure5.2the inquire time isdefined as the time since the first AP starts inquiring until all the APs have acknowledgmentof the existence of the Device.
38
M M M M MM M
a) b) c)
Figure 5.1. Scenarios - inquiry time for: a) 1 AP and 1 device;b) 2 APs and 1 device; c) n APs and 1 device.
Slave
Master 1
Master 2Master 3
Master n
Slave
Master
a) b)
Figure 5.2. Inquiry Time definition for scenarios with: a) 1 APand 1 Device; b) n APs and 1 Device.
5.2 Simulation Results of Inquiry Time
In order to evaluate theInquiry Timea simulator of theInquiry Procedurewas built inC/C++. The following assumptions were considered:
• There is perfect time slot synchronization between APs and the MD.
• There is no additional data packet traffic in the network; only the inquiry ID and FHSpackets [?].
• There is no interference with other technologies in the same frequency band (asWLAN or microwaves).
• Collisions among ID packets are treated as lost packets. Collisions between ID andFHS packets are not considered (FHS packet is never destroyed).
• Whenever an AP enters theInquiry Substate, it is delayed for a period between 0 upto 31 equally probable timeslots, in order to avoid AP synchronization.
• The MD starts scanning in a period uniformly distributed between 0s and 1.28s.
• Each time an FHS packet is received, the corresponding AP stops sending ID packets.
39
• The Inquiry Time is defined as the time since the first AP starts inquiring until thelast AP discovers the MD.
The evaluation of theInquiry Timeconsidering first the simplest case, theSingle AP sce-nario. For this scenario, using Bluetooth specifications V1.2 an the above assumptions, wecomputer the Probability Density Function (PDF) of the Inquiry Timeand the result wasplotted in Fig.5.3. To support this simulation results, we included the Fig.5.4, which canbe found in the literature [9]. Figure5.4 besides representing the samePDF for a SingleAP scenario, it was obtained for Bluetooth specifications V1.1.
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
Inquiry Time (s)
PD
F
Figure 5.3. Inquiry TimePDF for aSingleAP scenariowith Bluetooth specificationsV1.2
Figure 5.4. Inquiry TimePDF for aSingleAP scenariowith Bluetooth specificationsV1.1. [9]
Looking only to Fig.5.3we can see that there basically two peaks: one starts at timet = 0sand the other starts at timet = 2.56s. If we look into the Bluetooth specifications we cansee that in fact, the timet = 2.56scorrespond to the exchange of inquiry trains from A to Bthat happens in the master. This means that, the set of 16 frequencies that have been usedsincet = 0s to sendID packets, is exchanged att = 2.56swith the set of 16 frequencies thathad not been used until the moment. On the other hand, since the slave can only scan onefrequency each 1,28s, the probability of this frequency being either in train A or B is 50%,what provokes the 2 peaks with the same height. Besides that we know that the scan inter-val for the slave is 1.28s with a scanning window of only 11.25ms. Thus, since master andslave clocks are totally independent, when the master starts inquiring, the time at which theslave starts scanning is uniformly distributed between 0sand 1.28s−11.25ms= 1.26875s.This is the reason of the width of each peak being approximately 1.28s. The fluctuations inthe peaks are due to probabilistic errors, which are consequence of the non-infinite numberof samples.Comparing now Fig.5.3 with Fig. 5.4 we can see that besides the curves look similar,several differences are observed. In Fig.5.4the slope of the peaks is tilted and some smalladditional peaks appear for values of time higher thant = 6s. Both effects are due to thespecification of Bluetooth that specifies that after the slave receives the firstID packet:
• in V1.2, the device first answers to theID packet and then enters in back-off• in V1.1, the device enters in back-off and then answers to the nextID packet received
40
As stated in Section2.2 the back-off time is uniformly distributed between 0s and approx-imately 639ms. The fact that the back-off is before the answer, as in V1.1, influences theInquiry Timein the way that this back-off time has also to be included. The sum of anotherrandom variable (back-off time) represents in the distribution a convolution, resulting inthe slopes observed in Fig.5.4. On the other hand, the additional peaks are due to possiblecase of the slave wake up from back-off and the scanning frequency is not in the currenttrain anymore.
For further understanding theInquiry Timebehavior in a common network, we extendedour scenario to theMultiple AP scenario. For this scenario we definen as the number ofAPs. The evaluation consisted in determining thePDF of the Inquiry Timeas defined inSection5.1, for n = 1,2,3,4 and considering the assumptions stated in the beginning ofthis section.
0 2 4 6 8 10 12 14 160
0.1
0.2
0.3
0.4
0.5
0.6
Inquiry Time (s)
PD
F
n = 1 APn = 2 APn = 3 APn = 4 APn = 7 AP
Figure 5.5. Inquiry Time PDF to aMultiple Access Point Sce-nario (n = 2,4,7)
As we can see in Fig.5.5, the Inquiry Timedepends strongly on the number ofAPs: asbigger the number ofAPs, larger are the times ofInquiry Time. This provokes that thePDFtends to be wider and wider. Furthermore, two other effects are observed in Fig.5.5: thefirst is that the synchronization existent for theSimple AP ScenariobetweenInquiry Timepeaks and inquiry trains fades away when the number ofAPs grows. The second effect isthe additional peaks that start to appear in thePDFfor larger values ofInquiry Time. Thisenlargement of theInquiry Timelet us conclude thatAPs influence each other.In order to understand the influence that oneAP has in the remainingAPs, we compared
41
thePDFof Fig. 5.5with the distribution of the random variable of Eq. (5.1).
Y =n
maxi=0
Xi (5.1)
In Eq. (5.1), the random variableY represents the maximum ofn independent randomvariablesX, which one representing theInquiry Timefor theSingle AP Scenario. Assum-ing that then random variablesX are Independent and Identically Distributed (i.i.d.), therandom variableY represents theInquiry Timein aMultiple AP ScenariowhenAPs do notinfluence each others. The distribution of Eq. (5.1) was compared with the distribution inFig. 5.5and the results were plotted in Fig.5.6.
0 2 4 6 8 100
0.5
1
Inquiry Time(s)
CD
F
CDF for 2 APs and 1 MD
0 2 4 6 8 100
0.5
1
Inquiry Time(s)
CD
F
CDF for 4 APs and 1 MD
0 2 4 6 8 100
0.5
1
Inquiry Time(s)
CD
F
CDF for 7 APs and 1 MD
simulated CDFapproximated CDF
simulated CDFapproximated CDF
simulated CDFapproximated CDF
Figure 5.6. Simulated and approximated Inquiry Time CDF toa 2, 4, 7 AP scenario(obtained simulating 106 values of inquirytime)
As we can see in Fig.5.6the approximation of Eq. (5.1) follows close the simulated distri-bution. However, with the growth of the number osAPs, both distribution start to divergemore and more. This fact let us conclude that theAPs influence each others, being thisinfluence more and more considerable when the number osAPs grow. There are two mainreasons for this fact: the first because packets sent by theAPs may collide with each oth-ers or even withFHSsent by the slave; the second reason is the back-off time. When thedevice receives anID, it answers with anFHSpackets and than enters in a back-off time.This period makes the device totally unavailable to be discovered by anotherAPs and is thebasis of influence amongAPs.
Figure5.7shows the Cumulative Distribution Function (CDF) for aMultiple AP Scenario,where the number ofAPs varies betweenn = 1 andn = 20. We can see that theInquiryTimedepends strongly on the number ofAPs. For scenarios withn = 1,2,3,4 APs, it ispossible to obtain a considerable percentage ofInquiry Timevalues in the first inquiry train
42
0 5 10 15 200
0.2
0.4
0.6
0.8
1
inquiry time (s)
CD
F
Number of APsincreases
Figure 5.7.Simulated Inquiry time CDF toMultiple Access PointScenario(n=1 up to 20) and 1 MD (obtained simulating 106 valuesof inquiry time)
(t < 2.56s). For scenarios with moreAPs, theInquiry Timevalues are ignorable those ob-tained in the first inquiry, but not in the second inquiry train (2.56s < t < 5.12s), as forexample those scenarios withn = 5,6,7,8,9 APs, where theInquiry Timeis mainly ob-tained in the second inquiry train. For scenarios withn larger than 10APs, more than 50%of the total number ofInquiry Timevalues are longer than the end of the second inquirytrain, i.e., longer than 5.12s.
A different representation of the time behavior for localization procedures is presentedin Fig. 5.8 , which shows theInquiry Timepercentiles of 50%, 75%, 90% and 95% for a
0 2 4 6 8 10 12 14 16 18 200
5
10
15
20
25
30
Number of AP
Inqu
iry T
ime
(s)
Percentile of 50%Percentile of 75%Percentile of 90%Percentile of 95%
Figure 5.8. Inquiry Time duration depending on the number ofAPs for different pencentiles
Multiple AP ScenarioAs we can see from Fig.5.8, for n≤ 3 APs, theInquiry Timeis lowerthan 5s even considering a high percentile such as 95%, i.e. the inquiry time is obtainedbefore the end of the second train (5.12s). For n = 4 the percentile of 95% is higher thanthe duration of the first 2 trains. As an example, if it is a requirement that the localization
43
procedure must be faster than 5s for 90% of the cases, it is advisable to not use more than4 APs for localization purposes.Table I summarizes the mean inquiry time to the Multiple Access Point Scenario with nbetween 1 and 5, 10, 15, 20.
Table 5.1.MeanInquiry Timefor a Multiple AP Scenariowith nAPs
n 1 2 3 4 5 10 15 20mean time (s) 1.9 2.8 3.5 4.0 4.5 7.8 11.4 14.5
5.3 Theoretical approach of the Inquiry Procedure in Blue-tooth
This section presents an engineering approach for the distribution of the inquiry time forBluetooth specifications V1.2 [11].
5.3.1 Inquiry Time as a ball-urn problem
As we have seen until the moment, the Inquiry procedure is a very complex process, whichdoes not have a trivial theoretical approach. Thus, in order to determine the theoreticalPDFof the Inquiry Time, we shall proceed with some simplifications to the problem and,if possible, give an alternative problem definition.
The Inquiry Timecan be approached by a ball-urn problem. In order to understand howcan the problem be connected with the game, we decided to follow a process of reverse-engineering and present first the rules of the game and then make the connections with theBluetooth Inquiry Procedure.
5.3.1.1 The rules of the game
For this game we needn balls, 2 urns, and a counter with a cycle time ofa seconds. Thegame is then divided in 2 main parts.
In the first part,nA balls out ofn will be picked up and inserted in the urn A, while theremaining balls (nB = n−nA) are inserted in urn B. The variablenA shall be randomly de-termined by a binomial distribution with bernoulli probabilityp = 1/2.
44
The second part of the game starts by setting the counter and the clock cycle to 0. Thisclock cycle and subsequently the counter are not touched anymore, meaning that they willrun freely all over the game. Then, we shall wait for a period of timeti randomly determineby a uniform distribution between 0 andTi . After waiting that period of time we look atthe counter and if the counter is even we pick up a ball from urn A, if is odd we pick upa ball from urn B as we can see in Fig.5.9. Right after this waiting period, considering
0s
0 21 3 4 5 6 87 9 10 11 12 13Counter =
11
9
Urn A =
Urn B =
t = as 2as 3as 4as 5as 6as 7as 8as 9as 10as 11as 12as 13as
10
9
10
8
10
7
9
7
8
7
8
6
7
6
6
6
6
5
5
5
5
4
4
4
3
4
3
3
3
2
2
2
1
2
1
1
0
1
0
0
ti tb tb tb tb tb tb tb tb tb tb tb tb tb tb tb tb tb tb tbWaiting period =
6
9
Urn A =
Urn B =
5
9
5
8
5
7
4
7
3
7
3
6
2
6
1
6
1
5
0
5
0
4
0
4
0
4
0
3
0
2
0
2
0
2
0
1
0
1
0
0
ti tb tb tb tb tb tb tb tb tb tb tb Tf tb tb tbWaiting period = Tf Tf Tf Tf
CASE
1
CASE
2
0st = as 2as 3as 4as 5as 6as 7as 8as 9as 10as 11as 12as 13as
Figure 5.9. Ball-urn game
that it takes 0s to pick up a ball, we shall wait a period of timetb randomly determine by auniform distribution between 0 andTb. Then, we look again to the counter and, again, if itis even we pick up another ball from urn A and if it is odd we pick up from urn B. Rightafter and so forth we continue waiting for a new period of timetb and picking up balls inthe same manner until one of the urns gets empty. Then, as in case 2 of Fig.5.9, whileone urn is empty and the other has still balls, and if the counter says that is time to pick upa ball from the empty urn, the game follows without picking up a ball and waiting a fixedperiod of time ofTf . On the other hand if the counter says that is time to pick up a ballfrom the urn that still has balls, the procedure follows as primarily described, waiting therandom period of timetb. The game ends when both urns are empty.As we can see comparing both cases in Fig.5.9it is also possible that none fixed periodTf
is used (case 1). To let it happen we just need to have a game where balls finish in a certaincycle for a certain urn and in the consecutive cycle, the other urn gets empty.
After finishing the game the distribution of the time used until pick up all then balls ap-proaches the Inquiry Time distribution.
45
5.3.1.2 The relation between the game and the Inquiry Procedure
In this section we explain how the Bluetooth Inquiry Procedure introduced in Section(reffff) can be approcimated by the ball-urn game presented in Section5.3.1.1.As previously introduced in Section5.1theInquiry Timeis defined, for aMultiple AP Sce-nario, as the time since the firstAP starts inquiring until the lastAP notices that there is aMD around. Additionally and as it was assumed, theAPs are forced to start inquiring atthe same time, which is defined as the starting time of the process (timet = 0 in Fig. 5.9).Considering that there is a change of frequency each 1.28safter starting inquiring, theAPswill suffer, all at the same time, a change in its trains every 1.28s. Because the sequenceof frequencies is pre-determined and unique in the Inquiry procedure, all theAPs will fol-low the same pattern, meaning that theAPs will have a synchronization in terms of thechanging frequency. On the other hand, in the device point of view, the frequency scanned1.28s after a certain scan also follows the same pattern, meaning that all the system canbe considered as if there was synchronization on the changed frequency. Because of thissynchronization we can build 2 groups ofAPs: the group A that includes theAPs that havein its train A the frequency scanned by theMD; and the group B that includes those thathave the scanning frequency in their tain B. This corresponds in our game to the existenceof 2 urns: A and B.As all the APs and theMD have a free running clock, the probability of the scanningfrequency be in train A is 1/2, what is known as a Bernoulli distribution with parameterp = 1/2. For a number of casesn, the probability of havingnA cases succeeding in trainA follows, by definition, a binomial distribution, which corresponds to the first part of ourgame.As theAPs interchanges the inquiry train every 2.56s, we can say that if the scanning fre-quency was belonging previously to the group A, after the interchange it starts to belong tothe group B, corresponding this, in our game, that instead of picking up a ball from the urnA, we start picking up from urn B, being the cyclea equal to 2.56s.When the Inquiry Procedure starts, theMD may or not be in inquiry scan substate. If it isnot already in this substate, theMD will start scanning in a period of time uniformly dis-tributed between 0s and 1.28s. Correlating with the game, the period of time correspondsto the random variableti and the time limit of 1.28s corresponds toTi .After the MD starts scanning, theMD searches for a determined frequency for 11.25ms.During this period of time, if theMD is triggered with anID packet, theMD enters in aback-off time uniformly distributed between 0 and 1023 time slots, what means roughlybetween 0s and 0.639s. In fact, correlating with our game, the period of time of 11.25msthat theMD is scanning corresponds to the time spend picking up a ball, that for simpli-fication we ignore because it is much lower that the remaining time values existent in theprocedure. The back-off time correspond to the variabletb and the maximum back-off timeof 1023 times slots corresponds toTb.If all the APs in group A already discovered the device and if the device scans for a fre-quency that does not belong to used train of any of the remainingAPs, we have the case 2in Fig. 5.9, whereTf is the fixed time of the scan window of 1.28s.In the Inquiry Procedure, when all theAPs discovered the device we have the same solution
46
as when we picked up all the balls in our game.
5.3.2 Approach for the Inquiry Time distribution as a ball-urn prob-lem
5.3.3 Inquiry time in a scenario with one Access Point
This section regards the simplest scenario which is shown in Figure5.1.a. Only one APstarts inquiring the only existent Device. When the AP starts inquiring, the Device may beor not in the scan window, meaning that the AP may has to wait a period of timeTs thatis uniformly distributed between 0s and 1.26875s if is given that the scanning frequencyis the train A. This time of 1.26875s correspond to the time difference between the scaninterval (1.28s) and the scan window (11.25ms).
fTs = Uni f (0,1.28)[s] (5.2)
Once the scan window starts or if the inquired started during the scan window, an additionaltime must be considered. This time, defined asTsw is the time spent since the scan windowstarts until the frequency scanned by the slave is used by the master. As presented inprevious work [10] this distribution is not uniform. However according to literature [reffff]it can be considered as a uniform time between 0s and 11.25ms.
fTsw = Uni f (0,11.25)[ms] (5.3)
The previous description was made given that the scanning frequency belong to the trainA. However, easily can be extended to the general case because the probability of thefrequency being either in train A or B is the same and equals 1/2. Thus, this distribution isa discrete function as described in eq.(5.4).
fTtr =12
δ(t)+12
δ(t−2.56)[s] (5.4)
The variableTtr can have one of two values: either 0s or 2.56s.With the three variablesTs, Tsw andTtr and the respective distribution in eq.(5.2), eq.(5.3)and eq.(5.4) we can define the inquiry timeTt as the sum of the three variables (eq.(5.5))
Tt = Ts+Tsw+Ttr (5.5)
The distribution of a variable likeTt in eq.(5.5) that is the sum of random variables equalsthe convolution of the distributions of all the variables.
fTt = fTs ∗ fTsw∗ fTtr (5.6)
47
fTt =12
[Uni f (0,1.28)+Uni f (2.56,3.84)
]∗Uni f (0,0.01125) (5.7)
The plot of eq.(5.7) is show in Figure5.10
0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4Inquiry time PDF
Time [s]0 1 2 3 4 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Inquiry time CDF
Time [s]
Figure 5.10. Inquiry Distribution time for a scenario with oneAP and one device
5.3.4 Inquiry time in a scenario with two Access Points
In this scenario the problem gets more complicated. The distribution of the inquiry timeequals the distribution of the time since the first AP starts inquiring until the second APdiscovers the device.To simplify the problem we can subdivide it in 4 different cases:
case 1Scan frequency belongs to train A in AP 1 and in AP 2
case 2Scan frequency belongs to train A in AP 1 and train B in AP 2
case3Scan frequency belongs to train B in AP 1 and train A in AP 2
case 4Scan frequency belongs to train B in AP 1 and in AP 2
Either in case 2 or 3, the distribution is easy to obtain because the inquiry time of thesecond AP is totally independent on the inquiry time of the first and is always longer in anyinstance. The problem is much harder to sold in cases 1 and 4. In these cases both APs canstart
48
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7Inquiry time PDF
Time [s]0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4Inquiry time CDF
Time [s]
Figure 5.11. Inquiry Distribution time for a scenario with twoAPs and one device
5.4 Experimental validation
49
Chapter 6
Accuracy Aspects for Location inBluetooth Networks
6.1 Static Devices
6.2 Moving Devices
50
Chapter 7
Access Point placement to enhanceaccuracy on device localization
khhb
51
Chapter 8
Bayesian Filtering for LocationEstimation
Bayesian filtering techniques are techniques that estimate the state of a dynamic systembased on noisy sensors measurements. These techniques are used widely within severalcontexts, such as Signals & Systems, Image Processing or Mobile Tracking. To the speficiccase of location information estimation in scenarios with static or moving devices, theBayesian filtering techniques are being used in order to optimize accuracy [?].The Bayes filter is an abstract concept that describes a probabilistic framework for recursivestate estimation. Its main idea consists in estimate the distribution of the uncertainty, calledbelief, over the possible state space given the sensor measurements obtained until the timeof the estimation. Defining the variablei as thei− th estimation,ti as the discretized timewhen estimation was performed,xti as ther.v.! (r.v.! ) that represents the state space at timeti andz[t0,ti ] as the set of all the sensor measurements until timeti , thebelief is defined as:
Bel(xti) = p(xti |z[t0,ti ]) (8.1)
This Eq. (8.1) has a drawback in terms of computing time, which grows considerably de-pending on the number of sensor measurements. For this reason bayes filters are consideredMarkov chains which are memoryless, i.e. all the necessary information is included in thecurrent state.The biggest picture of a bayes filter is described in Fig.8.1. As we can see the Baesyianfiltering consists in two main steps:
• Prediction: At each time, the current statext is updated based on previous statesxti−1:
Bel−(xti) =∫
p(xti |xti−1)Bel(xti−1)dxti−1 (8.2)
52
Time Update(”Prediction”)
Measurement Update(”Correct”)
Figure 8.1. Bayesian filtering cycle
In Eq.(8.2) the termp(xti |xti−1) describes thesystem dynamics, i.e. how the systemschanges from the statexti−1 to the current statexti . The belief for state space at timet0is usually defined as an uniform distribution in case no a-priori knowledgeless exists.
• Correction: The predicted belief is corrected everytime a new sensor measurementis obtained:
Bel(xti) = αti p(zti |xti)Bel−(xti) (8.3)
In Eq.(8.3) the termαti represents a normalization constant to ensure that∫
Bel(xti)dxti =1, while p(zti |xti) represents theperceptual model. This term described the probabil-ity of having a sensor measurementzti given that the system is in statexti .
For localization purposes, the termp(xti |xti−1) in Eq.(8.2) is determined by the motionmodel, while the termp(zti |xti) in Eq.(8.3) is dependent on the technology of the sensorand the location method.
8.1 Bayes Filters Implementations
In order to implement Bayes filters it is useful to specify the modelsp(xti |xti−1), p(zti |xti)and a representation of the beliefBel(xti). This representation of the belief is where maindifferences exist in different representation of Bayes filters8.2. The most famous andwidely used implementations are the Kalman filter and the Particles Filter (PF).The Kalman filter assumes the belief distribution as a unimodal gaussian distribution wherethe mean represents the expected location and the variance represents the uncertainty. Thisfilter has robust performances in case localization uncertainty is not too high. The mainadvantage of this filter is the computational efficiency.ThePF in opposite to the kalman filter is able to represent any kind of distribution of thebelief, since it represents the belief as a set of weighter samples according to the beliefof the device being in that sample. The focus of the filter falls into the areas where the
53
Figure 8.2. Properties of the most common implementations ofBayes filters for location estimation [?]
probability is high. A new approach on filtering considers a combination ofPFand Kalmanfilters, called Rao-Blackwellised particle filters, which has been used successfully.MHT! (MHT! ) is a combination of Kalman filters since it considers multi-modal beliefs.Each hypothesis is treated as a Kalman filter, which are weighted depending on how wellthey predict the sensor measurements.Grid-based approaches are similar toPFis the aspect they consider belief. This distributionof the belief is considered as a whole, instead of considering just the most probable areas.This may get difficult or impracticable to implement, since the computer complexity riseswith the increase of the size of the area and the decrease of the size of the samples.The topological approach makes use of graphs to represent the motion of the devices. In anindoor scenario with imprecise sensors may be a typical scenario for this kind of approach,since possible path are modelled by the graph.
8.1.1 The Kalman Filter
8.1.2 The Extended Kalman Filter
8.2 Kalman Filters Application
In this section we develop a filter for tracking of devices in the Bluetooth network. In orderto make easier the modelling of the filter we start by considering a static model where theMDs do not move. Then we go further in considering movingMDs.
54
Inicial estimates for ˆxk−1 andPk−1
?
-
�
Time Update (Prediction) Measurements Update (Correction)
a) Project the state ahead:x−k = Axk−1 +Buk−1
b) Project the error covariance aheadP−k = APk−1aT +Q
c) Compute the Kalman gainKk = P−k HT(HP−k HT +R)−1
d) Update estimate with measurementzk
xk = x−k +Kk(zk−Hx−k )e) Update the error covariance
Pk = (I −KkH)P−k
Figure 8.3. Kalman Filter Equations
Inicial estimates for ˆxk−1 andPk−1
?
-
�
Time Update (Prediction) Measurements Update (Correction)
a) Project the state ahead:x−k = f (xk−1,uk−1,0)
b) Project the error covariance aheadP−k = APk−1aT +WkQWT
k
c) Compute the Kalman gainKk = P−k HT(HP−k HT +VkRVT
k )−1
d) Update estimate with measurementzk
xk = x−k +Kk(zk−h(x−k ,0))e) Update the error covariance
Pk = (I −KkH)P−k
Figure 8.4. Extended Kalman Filter Equations
8.2.1 The static case
As we could see in the previous section, the kalman filter can have non-linear equations forthe prediction, correction or even both. For this simpler case where theMDs are assumedto be stopped, the only non-linear equation in the model is in the correction part. Thisnon-linearity is due to the propagation equation that relates the measured received powerwith the distance, and subsequently with the space coordinates.
55
As we assume that theMDs are static, the state vectorxk includes only the space coordinatesx andy:
xk =[
xy
](8.4)
and of course, because the previous state is always the same as the same as the previousone, the matrixA is the identity matrix:
A = I =[
1 00 1
](8.5)
For the process as we do not have any external input we consider the matrixB as beingthe null matrix. The matrixQ could be considered 0 because is assumed that there is noprocess noise. However, according with the literature [reffff] it is advisable to considerQnon-zero but with a low value of order of 1E−5.
B =[
0 00 0
]Q = 1E−5×
[1 11 1
](8.6)
Defined the prediction part of the filter, it is necessary to define the non-linear correctionpart. The first step start defining the functionh from Eq.(refffff). This relation between themeasured power and the predicted location is determined by the propagation equation.According to the literature [reffff], the distribution of the differences between the receivedpower and the mean received power can be approximated by a normal distribution withmeanµ= 0.55dBand standard deviationσ = 2.42dB.
1
σ√
2πexp
{− (G(d)−µ)2
2σ2
}(8.7)
G(d) = P(D)−Pm (8.8)
P(d) = PTX +GTX +GRX+40−20log(d) (8.9)
d =√
(x−xm)2 +(y−ym)2 (8.10)
Assuming that the predicted location is dictated by the maximum of Eq.(8.7), we can easilyderive and equalize to zero in order to get the relation between the measured power and theexpected location.
Pm = PTX +GTX +GRX−40−10log((x−xm)2 +(y−ym)2
)−µ (8.11)
56
To build the matrixH is useful to determine the derivatives ofPm in Eq.(8.11) in functionof the space coordinatesx andy:
∂Pm
∂x=−20
x−xm
d2
∂Pm
∂y=−20
y−ym
d2 (8.12)
H =−20d2 ×
x−x1 y−y1
x−x2 y−y2...
...x−xm y−ym
(8.13)
8.2.2 The dynamic case
In this section we determine the parameters to implement a Kalman Filter in an example oftracking mobiles.Consider the simple dynamic movement model obeying the constant-coefficient first-orderlinear differential equation.
S(t) = FS(t)+G(t) (8.14)
where
S(t) =
x(t)y(t)x(t)y(t)x(t)y(t)
F =
0 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0
G(t) =
0000
gx(t)gy(t)
(8.15)
The general solution:
S(t) = Φ(t, t−T)S(t−T)+W(t,T) (8.16)
where
W(t,T) =∫ t
t−TΦ(t,λ)G(λ)dλ (8.17)
Conditions:
Φ(t, t) = I (8.18)
57
Φ(t,λ) = Φ(t,µ)Φ(µ,λ) (8.19)
∂Φ(t,λ)∂t
= FΦ(t,λ) (8.20)
Equation (8.20) has as solution:
Φ(t,λ) = exp[(t−λ)F ] (8.21)
which can be expanded in:
Φ(t,λ) = I +(t−λ)F +(t−λ)2F2
2!+
(t−λ)3F3
3!+
(t−λ)4F4
4!(8.22)
F =
0 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0
F2 =
0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
Fn =
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
∀n≥ 3
(8.23)
Φ(t,λ) =
1 0 t−λ 0 1
2(t−λ)2 00 1 0 t−λ 0 1
2(t−λ)2
0 0 1 0 t−λ 00 0 0 1 0 t−λ0 0 0 0 1 00 0 0 0 0 1
(8.24)
Φ(t,λ) =
1 0 T 0 1
2T2 00 1 0 T 0 1
2T2
0 0 1 0 T 00 0 0 1 0 T0 0 0 0 1 00 0 0 0 0 1
(8.25)
58
If W in Eq.() follows the same carcateristics as in Eq.() of the Kalman filter equations,we can consider W the process noise, since we do not have (see reference)
The covariance ofg(t), with varianceσ, is:
cov(g(λ),g(µ)) = E[g(λ)g(µ)] = σ2δ(λ−µ) (8.26)
The covariance ofQ is:
Q= E[W(t,T)W(t,T)T ] = E[∫ t
λ=t−T
∫ t
µ=t−TΦ(t−λ)G(λ)G(µ)TΦ(t−µ)Tdλdµ](8.27)
Since the expectation is a operator linear, we can write:
Q =∫ t
λ=t−T
∫ t
µ=t−TΦ(t−λ)E[G(λ)G(µ)T ]Φ(t−µ)Tdλdµ (8.28)
Q = σ2∫ t
λ=t−TΦ(t−λ)
001
(0 0 1
)Φ(t−λ)Tdλ (8.29)
Settingt−λ = γ and changing the integration variable toγ:
Q = σ2∫ T
γ=0Φ(γ)
001
(0 0 1
)Φ(γ)Tdγ (8.30)
Q = σ2∫ T
γ=0
12γ2
γ1
(12γ2 γ 1
)dγ (8.31)
Expanding we can get:
Q = σ2T
120T4 1
8T3 16T2
18T3 1
3T2 12T
16T2 1
2T 1
(8.32)
This kind of noise is commonly used in tracking problems when the acceleration exci-tation is unknown.The Measurement equation is:
y(t) = HS(t)+v(t) (8.33)
59
(y1(t)y2(t)
)=
(1 0 0 0 0 00 1 0 0 0 0
)
x(t)y(t)x(t)y(t)x(t)y(t)
+(
v1(t)v2(t)
)(8.34)
60
Chapter 9
Conclusion
conclusion sdfgas gsafdgvasdfvs dfvsdfvsdfvsdf
9.1 sfgsdfg
sdfvsdfg
61
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Appendix A
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A.1 XSV
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Appendix B
Some Other Appendix
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65
Glossary
Base Station definition, 16
Location Information definition, 15Location Information motivation, 15
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