Fundamental Performance Limits on Time of Arrival ...1182120/FULLTEXT01.pdf · In currently deployed Long Term Evolution (LTE) networks, Observed Time Di↵erence of Arrival (OTDoA)

IN DEGREE PROJECT ELECTRICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2017

Fundamental Performance Limits on Time of Arrival Estimation Accuracy with 5G Radio Access

DEHAN LUAN

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ELECTRICAL ENGINEERING

Abstract

5G radio access technology stimulates new use cases and emerging businessmodels, evolving the world to become a fully mobile and connected networksociety, expected to be operational by 2020. To enable ultra-high reliable andhighly precise features, there are more stringent positioning accuracy require-ments for location based services and E911 emergency calls, targeting at bothindoor and outdoor users which include humans, devices, vehicles and machines.

In currently deployed Long Term Evolution (LTE) networks, Observed TimeDi↵erence of Arrival (OTDoA) positioning is acting as one of the User Equip-ment (UE) localization techniques. The positioning accuracy in OTDoA methoddepends on various factors, e.g. network deployment, signal propagation con-dition and properties of Positioning Reference Signal (PRS). For a given de-ployment and propagation scenario, significant improvements of positioning ac-curacy is achievable by appropriately redesigning the PRS with 5G radio access.

In this thesis, fundamental performance limits (i.e. Cramer Rao Lower Bounds)on Time of Arrival (ToA) estimation are derived respectively considering fre-quency selective channel with additive white noise, carrier frequency o↵set andWiener phase noise. Particularly, the e↵ects of flexible bandwidth, subcarrierspacing and power allocation on ToA estimation accuracy have been investigatedin di↵erent settings based on corresponding performance bounds. Furthermore,the performance limits on ToA estimation can be translated into UE positioningaccuracy for a given deployment scenario. Overall, the thesis has built valuableinsights on PRS design and waveform optimization for 5G based positioning.

Sammanfattning

5G radioteknik stimulerar nya användaromr̊aden och nya a↵ärsmodeller samtutvecklar världen till att bli ett helt mobilt och anslutet nätverkssamhälle,vilket förväntas vara i drift till år 2020. För att möjliggöra ultrahög p̊alitlighetoch högprecisionsfunktionalitet krävs det strängare krav p̊a positionering förplatsbaserade tjänster och nödsamtal, som riktar in sig p̊a b̊ade inomhus- ochutomhusanvändare vilket inkluderar människor, enheter, fordon och maskiner.

I det nuvarande distribuerade LTE-nätverket fungerar Observed Time Di↵er-ence of Arrival-positionering (OTDoA)-positionering som en av lokalisering-steknikerna för användarutrustning (UE). Positioneringsnoggrannheten i OTDoA-metoden beror p̊a olika faktorer, t.ex. nätverksutbyggnad, signalutbrednings-förh̊allande och egenskaper för positions referenssignal (PRS). För ett visstinstallations- och utbredningsscenario kan betydande förbättringar av position-snoggrannheten uppn̊as genom att omforma PRS p̊a lämpligt sätt.

I detta examensarbetet undersöks de grundläggande prestandabegränsningarna(dvs Cramer Rao Lower Bounds) vid Ankomsttid (ToA) härlädda med avseendep̊a frekvensselektiva kanaler med additivt vitt brus, bärfrekvensförskjutning ochWienerfasbrus. I synnerhet, undersöks e↵ekten av flexibel bandbredd, subcar-rieravst̊and och e↵ektallokering p̊a ToA-estimeringsnoggrannhet undersökts iolika inställningar baserat p̊a prestandabegränsningarna. Vidare kan prestand-abegränsningarna för ToA-estimering översättas till UE-positionsnoggrannhetför ett givet implementationsscenario. Slutligen s̊a har detta examensarbetetgett viktiga insikter om positionering av referenssignaldesign och v̊agformsopti-mering för 5G-baserad positionering.

Acknowledgment

First of all, I’d like to show my gratitude to master thesis supervisor, Mr. AliZaidi, from Ericsson Research, who has provided me opportunity and guid-ance on this promising topic. This working thesis has helped me to sort outknowledge pieces and sew them together. From working in the industry, I alsobroadened my sights and gained experience.

Also, I’d like to extend my gratitude to my examiner, Mr. Tobias Oechtering,Associate Prof. from KTH University, who has given me helps and encourage-ments during these months.

Additionally, I would like to thank Navneet Agrawal, Maria Edvardsson, ZhaoWang, Vicent Moles Cases, Hieu Do, Wanlu Sun, Yiqing Wang from Ericssonwho havs ever inspired me and provided great helps.

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Formulation and Methodology . . . . . . . . . . . . . . 21.3 Societal and Ethical Aspects . . . . . . . . . . . . . . . . . . . . 31.4 Pevious Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Background 6

2.1 Positioning in Future 5G Networks . . . . . . . . . . . . . . . . . 62.1.1 OTDoA Positioning . . . . . . . . . . . . . . . . . . . . . 62.1.2 ToA Estimation . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 Positioning Reference Signal . . . . . . . . . . . . . . . . . 9

2.2 Fundamental Performance Limit . . . . . . . . . . . . . . . . . . 112.2.1 Minimum Variance Unbiased Estimation . . . . . . . . . . 112.2.2 Cramer Rao Lower Bound . . . . . . . . . . . . . . . . . . 11

3 Derivatons - Cramer Rao Lower Bounds on ToA Estimation 14

3.1 AWGN Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Frequency Selective Channel . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 CSIR - Channel State Information at Receiver . . . . . . 173.2.2 CSIT - Channel State Information at Transmitter . . . . 173.2.3 No CSI - No Channel State Information . . . . . . . . . . 17

3.3 Carrier Frequency O↵set/Doppler Shift . . . . . . . . . . . . . . 193.4 Wiener Phase Noise . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Simulations and Result Analysis - Flexible Parameter Exploit 22

4.1 Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Subcarrier Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Conclusions and Future Work 33

5.1 Main Findings on ToA Performance . . . . . . . . . . . . . . . . 335.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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List of Figures

1.1 ToA Estimation Uncertainty in OTDoA Positioning . . . . . . . 2

2.1 Dense Urban and Indoor Scenario . . . . . . . . . . . . . . . . . . 72.2 OTDoA Multilateral Positioning Method . . . . . . . . . . . . . 82.3 PRS in Physical Resource Block . . . . . . . . . . . . . . . . . . 92.4 Frequency Spectrum of OFDM Waveform . . . . . . . . . . . . . 102.5 PRS in Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 Cramer Rao Lower Bound on Variance of Unbiased Estimator . . 11

4.1 Bandwidth Exploit on ToA Estimation in AWGN Channel . . . 234.2 Specific Channel Frequency Response . . . . . . . . . . . . . . . 244.3 Bandwidth Exploit on ToA Estimation in Multipath Channel . . 254.4 Bandwidth Exploit on ToA Estimation with Wiener Phase Noise

in AWGN Channel . . . . . . . . . . . . . . . . . . . . . . . . . . 254.5 Subcarrier Spacing Exploit on ToA Estimation in AWGN Channel 264.6 Subcarrier Spacing Ratio in AWGN Channel . . . . . . . . . . . 274.7 Subcarrier Spacing Exploit on ToA Estimation in Multipath Chan-

nel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.8 Subcarrier Spacing Exploit on ToA Estimation withWiener Phase

Noise in AWGN Channel . . . . . . . . . . . . . . . . . . . . . . 284.9 Power Allocation Exploit on ToA Estimation in AWGN Channel 294.10 Monte Carlo Simulation for Power Optimization . . . . . . . . . 304.11 Power Allocation Exploit on ToA Estimation with No CSI in

Multipath Channel . . . . . . . . . . . . . . . . . . . . . . . . . . 304.12 Wiener Phase Noise Variance Exploit on ToA Estimation in AWGN

Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.13 Power Allocation Exploit on ToA Estimation with Wiener Phase

Noise in AWGN Channel . . . . . . . . . . . . . . . . . . . . . . 32

5.1 Main Findings on ToA Performance . . . . . . . . . . . . . . . . 33

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Abbreviations

3GPP 3rd Generation Partnership Project

5G 5th Generation Wireless System

A-GNSS Assisted-Global Navigation Satellite Systems

AWGN Additive White Gaussian Noise

CFO Carrier Frequency O↵set

CIR Channel Impulse Response

CP Cyclic Prefix

CRLB Cramer Rao Lower Bound

CRS Cell-specific Reference Signal

CSI Channel State Information

CSIR Channel State Information at Receiver

CSIT Channel State Information at Transmitter

ECID Enhanced Cell ID

FCC Federal Communications Commission

FIM Fisher Information Matrix

GPS Global Positioning System

i.i.d. independent and identically distributed

ICI Inter Carrier Interference

ICT Information and Communication Technologies

IFFT Inverse Fast Fourier Transform

IoT Internet of Things

ISI Inter Symbol Interference

LBS Location-Based Services

LO Local Oscillator

LTE Long Term Evolution

MAP Maximum A Posteriori

mIoT massive Internet of Things

ML Maximum Likelihood

mMTC massive Machine Type Communication

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MSE Mean Square Error

MU-MIMOMulti-User Massive Input Massive Output

MVU Minimum Variance Unbiased

NB IoT Narrow Band Internet of Things

NLoS Non Line of Sight

NR New Radio

OFDM Orthogonal frequency Division Multiplexing

OTDoA Observed Time Di↵erence of Arrival

PDF Probability Density Function

PHN Phase Noise

PSD Power Spectral Density

QPSK Quadrature Phase Shift Keying

RBLS Rao Blackwell Lechman Sche↵e

RSTD Reference Signal Time Di↵erence

SMLC Serving Mobile Location Center

ToA Time of Arrival

UE User Equipment

URLLC Ultra Reliability Low Latency Communication

ZP Zero Padding

ZZB Ziv Zakai Bound

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Chapter 1

Introduction

1.1 Motivation

Nowadays a telecom evolution is taking place in this Science and Technology era,transforming our world into a super-connected network society. 5G New Ra-dio (NR) access technology triggers this digital transformation and enables newuse cases [1], such as Low Latency Ultra Reliability Communication (URLLC),massive Internet of Things (mIoT) and massive Machine Type Communication(mMTC).

As mentioned by 3rd Generation Partnership Project (3GPP) Release 14 [2],5G mobile communication networks are expected to enable highly accurate ande�cient User Equipment (UE) positioning. Compared with existing positioningtechniques whose accuracy can typically achieve tens of meters, future 5G posi-tioning is expected to achieve less than one meter or even below for both indoorand outdoor users. Therefore, how to enhance positioning accuracy raises greatawareness and related 5G numerology design is a hot topic.

Considering the determinants such as ue complexity, standardization and net-work impacts, 3GPP Release 14 [2] puts the focus on enhancements of ObservedTime Di↵erence of Arrival (OTDoA) positioning. OTDoA is a ue assisted mul-tilateral positioning technique, firstly introduced in 3GPP Release 9 [3] sup-porting Long Term Evolution (LTE) networks. In OTDoA, ue measures Timeof Arrival (ToA) of specific positioning reference signals (PRS) received frommultiple base stations, calculates Time Di↵erence of Arrival (TDOA) valuesand reports to Serving Mobile Location Center (SMLC). During this process,the properties of PRS structure directly a↵ect the accuracy of ToA estimation,which could be translated into OTDoA positioning accuracy according to themultilateral algorithm. This thesis focuses on the study of fundamental per-formance limits on ToA estimation accuracy and flexible parameters such asbandwidth, subcarrier spacing and power allocation are investigated. Based onthat, some important insights regarding positioning reference signal design andwaveform optimization are provided for 5G NR based positioning.

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1.2 Problem Formulation and Methodology

According to OTDoA positioning algorithm, one time di↵erence measurementdetermines a certain hyperbola line and di↵erent hyperbolas would intersect atone point ideally in geometry, which is the desired UE’s position. However inpractice, due to various factors such as noise e↵ect in the environment, low-precision receiver, improper estimating algorithm and the existing interferencefrom neighbouring cells, there will be uncertainty in ToA measurements, whichforms an intersection area instead of a certain point indicating UE’s position.This causes a degradation on OTDoA positioning accuracy, as seen in Figure1.1

Figure 1.1: ToA Estimation Uncertainty in OTDoA Positioning

Here we can see ToA estimation accuracy could be influenced by a bunch ofdominants and the complexity will formulate problems as below:

• Under this complex environment, how to separate di↵erent concerningfactors for investigation?

• What is the relationship between properties of PRS and ToA estimationaccuracy, i.e. which parameters of PRS are associated with ToA estima-tion accuracy?

• Di↵erent estimation algorithm has its own characteristics and applicableconditions, so which estimator can we trust to indicate the best achievableaccuracy?

2

Corresponding to the problem formulation, some research methodologies areused:

• System modelling method is applied under di↵erent cases to clarify eachconcerning factors, which are frequency selective channel with additivewhite noise, carrier frequency o↵set and Wiener phase noise

• Necessary assumptions are made to simplify our problem, e.g. we targeton one path ToA and assume no interference from neighbouring cells,whichmeans the focus is flexible parameter investigation regarding PRS propertyinstead of PRS pattern, which is out of our scope. Underlying assumptionis fairly comparison when investigating the flexible parameters.

• In order to indicate the best achievable accuracy, Hypothetico-deductivemethodology [4] is applied. To be more specific,

(a) Firstly, a hypothesis is proposed as: a minimum variance unbiased(MVU) estimator is given.

(b) Then, following predictions are deduced based on the hypothesis,which is Cramer Rao Lower Bound (CRLB) derivation part in thisthesis and fundamental performance limits are obtained, i.e. achiev-able ToA estimation accuracy.

(c) Lastly, looking for evidence or observations that conflicts with the hy-pothesis, i.e. developing estimators that can achieve the fundamentalperformance lower bounds.

1.3 Societal and Ethical Aspects

Location-Based Services (LBS) and E911 emergency call positioning drives thedevelopment of localization techniques in LTE wireless networks, which are thebaseline of evolved 5G positioning.

From societal point of views, enhanced positioning technique will create hugeeconomical and commercial values in this ICT era. More accurate UE’s positioninformation brings better user experience to the society and great convenienceto people’s daily life. From ethical perspectives, the OTDoA positioning is in-troduced motivated by the requirements of emergency calling services (E911)by US Federal Communications Commission (FCC). In dense urban and indoorscenarios, location information becomes extremely important for life safety. Forthe upcoming 5G future networks, enabling technologies for highly e�cient 5Gpositioning is classified as one of the promising goals and raises great socialawareness.

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1.4 Pevious Work

Minimum variance unbiased estimation, Fisher Information Matrix (FIM) con-cept, Cramer Rao Lower Bound theorem were firstly introduced by Kay [5],in which the general CRLB expression for signals in Additive White GaussianNoise (AWGN) model was given as the basis for following studies. Then, therewere several studies [6, 7, 8] exploring CRLB for ranging estimation based oncontinuous time Fourier Transform, which referred to the mathematical expres-sion in Kay [5]. In [9], under the approximation that considering mean squarebandwidth of OFDM signal as a rectangular power spectral density (PSD),Del Peral-Rosado et al. gave a deformation expression of CRLB on time delayestimation in AWGN channel. In [10], W.Xu and M. Huang et al. derivedout a closed form CRLB expression based on discrete-time Fourier Transformin AWGN channel. In [11], from pilot optimization point of view, D. Larsenconsidered multipath channel and time delay jointly estimation, who firstlyproposed the relationship between frequency selective fading channels and ToAestimation.

Furthermore, Del Peral-Rosado et al. developed associated maximum likeli-hood estimation technique in [12] based on the CRLB in multipath channel asin [11]. In [13, 14], the cross e↵ects between Channel Impulse Response (CIR),Phase Noise (PHN) and Carrier Frequency O↵set (CFO) were investigated inOFDM systems and a hybrid CRLB was given based on Bayesian Theorem.Additionally, a general continuous-time localization waveform were proposed in[15] with shaping the power spectrum and related variance lower bounds suchas Cramer Rao Lower Bound (CRLB) and Ziv Zakai Bound (ZZB) were evalu-ated as indicators of positioning accuracy. In [16], Dardari D. and M.Z. Win.studied ZZB on ToA estimation with general continuous time waveform assum-ing statistical channel knowledge at the receiver side. In [17], Waterschoot etal. derived analytical expressions for Power Spectrum Density (PSD) of OFDMsignal employing a Cyclic Prefix (CP) or Zero Padding (ZP) guard time interval.

Overall, there are some existing studies which have given the CRLB expressionon ToA estimation in AWGN channel and multipath channel with discrete-timeOFDM waveform. But further flexible parameter exploit from PRS propertyperspective is not emphasized and further studied. Regarding carrier frequencyo↵set and phase noise, a hybrid CRLB was given but not from ToA estimationperspective for positioning purpose, and it is still unclear how the frequencyo↵set and phase noise will a↵ect ToA estimation. 5G NR is being standardizedand the numerology design is still an open question. The thesis outcomes wouldprovide valuable intuitions on how to redesign PRS and do waveform optimiza-tion to realize positioning enhancements with 5G Radio access, as required in3GPP Release 14 in[2].

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1.5 Goals

According to state-of-art work and the requirements of future 5G positioning,the goals of this thesis are set as below:

• Deriving fundamental bounds on achievable ToA estimation accuracy con-sidering respective possible cases:

(a) Frequency selective channel with complex white noise

(b) Carrier frequency o↵set (i.e. Doppler shift)

(c) Wiener phase noise

• Based on the fundamental bounds, exploit the available degrees of freedom(flexible bandwidth, subcarrier spacing and power allocation) to test novelPRS design

• Investigate how frequency selective channel, carrier frequency o↵set andphase noise give impacts on ToA estimation accuracy

This thesis work contains theoretical derivations in mathematics and also sim-ulation works in MATLAB, including sequence generation, OFDM modulation,PRS structure formulation, CRLB calculation and flexible parameter analysis.These results will provide important insights for positioning reference signaldesign and waveform optimization for 5G NR based positioning.

1.6 Thesis Outline

The master thesis is organized as follows: Chapter 2 introduces the theory andbackground of 5G positioning and fundamental performance limit, especiallythe ToA estimation in OTDoA positioning technique and Cramer Rao LowerBound, which are the basics for readers to understand. Chapter 3 points at theCramer Rao Lower Bound derivations regarding ToA estimation, which gives aclosed form bound expression based on respective cases considering frequencyselective channel, carrier frequency o↵set and Wiener phase noise. Within fre-quency selective channel cases, it is classified into three subsections based onthe Channel State Information (CSI) knowledge: Channel State Information atReceiver (CSIR), Channel State information at Transmitter (CSIT) and with-out Channel State Information (No CSI), corresponding to di↵erent bounds orsolutions on ToA estimation. Based on the derived performance bounds, Chap-ter 4 shows the simulations and result analysis which investigates the e↵ects offlexible bandwidth, subcarrier spacing and power allocation on ToA estimationaccuracy. Chapter 5 identifies the main findings and reflections on PRS design,gives conclusions of the thesis and future work.

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Chapter 2

Background

Positioning support in LTE networks was firstly introduced in 3GPP Release 9[3], enabling the operators to retrieve UE’s location information for Location-Based Services (LBS) and regulatory emergency calling services.

In this chapter, a brief discussion is provided on essential background which areused throughout the thesis. Section 2.1 introduces the algorithm for OTDoApositioning, the importance of ToA estimation and the dedicated PositioningReference Signal (PRS). Section 2.2 describes some estimation theory knowl-edge: Minimum Variance Unbiased (MVU) estimation and Cramer Rao LowerBound (CRLB) theorem which push forward the thesis flow.

2.1 Positioning in Future 5G Networks

2.1.1 OTDoA Positioning

As specified in [18], Assisted-Global Navigation Satellite Systems (A-GNSS)such as Global Positioning System (GPS) uses standalone satellite system withassistance of cellular networks to realize high accurate positioning technique foroutdoor scenarios. While for indoor and dense urban scenarios (see Figure 2.1)when Line of Sight (LoS) access is not good enough, fallback mobile radio cel-lular positioning techniques will be applied such as Observed Time Di↵erenceof Arrival (OTDoA) and Enhanced Cell ID (ECID).

Additionally, due to the limited bandwidth usage for satellite system, cellu-lar networked based positioning is subject to greater attention. Consideringthe features of 5G cellular networks, such as wide bandwidth, beamforming,Multi-User Massive Input Massive Output (MU-MIMO) and future dense mi-cro/pico sites deployment, OTDoA has many possibilities and 3GPP Release 14has pointed out the goal of enhancements on OTDoA positioning.

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Figure 2.1: Dense Urban and Indoor Scenario

From algorithm perspective, OTDoA is a downlink multilateral positioningmethod to retrieve UE’s location. As seen in Figure 2.2, one of cells is cho-sen as a reference cell, then the positioning process is divided into followingsteps:

(a) Multiple cells start to transmit downlink Positioning Reference Signals(PRS) to target User Equipment (UE)

(b) UE measures Time of Arrivals (ToA) based on received PRS from multiplecells, and subtracts the ToA from the reference cell (e.g. if C is chosenas the reference cell) with other measured ToA values (from cell A andB) respectively to get time di↵erence measurements, i.e. Reference SignalTime Di↵erence (RSTD). Geometrically, each RSTD will determine a hy-perbola (red and pink lines in Figure 2.2), the intersection point is targetUE’s position.

(c) RSTD measurements are reported to Serving Mobile Location Center(SMLC), then the center calculates UE’s position based on RSTD mea-surements and known base stations’ coordinates.

Note that at least three ToAs (i.e. three base stations) are required to achievehorizontal positioning accuracy of UE, the reason is that we need at least twoequations to solve two parameters equation set as below, i = a, b

RSTDi,ref. = 4ti,ref. =p

(xUE � xi)2 + (yUE � yi)2/c�q

(xUE � xref.)2 + (yUE � yref.)2/c+ (ni � nref )(2.1)

Three ToA measurements would get two RSTD measurements, in order to besu�cient to solve UE’s two dimensional coordinates (xUE , yUE) under the as-sumption that there is no timing o↵set from multiple cells. In theory, at leastfour ToA measurements (i.e. four base stations) are required to solve a three-dimension case. Since each ToA measurement has uncertainty, which refers tomeasurement noise as ni, nref.. Practically, more base stations would be usedto get more accurate positioning accuracy.

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Figure 2.2: OTDoA Multilateral Positioning Method

2.1.2 ToA Estimation

Time of Arrival (ToA) means the first arriving path at the receiver side, basi-cally refers to the LoS path. For a given deployment scenario, ToA estimationaccuracy can be translated to positioning accuracy considering OTDoA posi-tioning algorithm. Since each ToA measurement has a certain uncertainty, sothe hyperbolas as shown in Figure 1.1 has a scattering width, illustrating themeasurement uncertainty. As assumed in [9], RSTD measurement is consideredas a Gaussian distribution

c4t = 4t+ n, n ⇠ N(0,Cov) (2.2)

Noise vector is assumed to be Additive White Gaussian Noise (AWGN) withconstant covariance matrix as below, where �2A, �

2B , �

2C are ToA measurement

variances from cell A, B, C to target UE respectively

Cov =

�2c + �2a �

2c

�2c �2c + �

2b

�

(2.3)

OTDoA positioning accuracy is represented by the position error between trueUE’s position and estimated position

✏(xUE , yUE) =q

trace [(DTCov�1D)�1] (2.4)

where dUE,A, dUE,B , dUE,C are Euclidean distance from multiple cells to tar-get UE.

D =

2

6

4

xUE�xCdUE,C

� xUE�xAdUE,AyUE�yCdUE,C

� yUE�yAdUE,A

xUE�xCdUE,C

� xUE�xBdUE,ByUE�yCdUE,C

� yUE�yBdUE,B

3

7

5

(2.5)

As we can see from the equations above, ToA measurement variance directlya↵ects the OTDoA positioning accuracy. So the focus of this thesis is put onachievable performance limit on ToA estimation accuracy enhancement.

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2.1.3 Positioning Reference Signal

Positioning Reference Signal (PRS), as the downlink transmitted signals in OT-DoA positioning process, is a pseudo-random Quadrature Phase Shift Keying(QPSK) sequence, being mapped into diagonal pattern within Physical ResourceBlock (PRB) with shifts in the frequency domain and time domain to realize lowinterference structure. Figure 2.3 shows the mapping of PRS within PRB in ex-isting LTE networks. Specifically, multiple cells use di↵erent frequency shift onPRS structure to realize low interference structure according to Orthogonal fre-quency Division Multiplexing (OFDM) waveform (OFDM frequency spectrumis shown in Figure 2.4) properties that di↵erent subcarriers are orthogonal toeach other. The properties of PRB are described by parameters, such as sub-carrier spacing, total number of subcarriers (i.e. bandwidth), OFDM symbolduration, pattern (i.e. power allocation).

Figure 2.3: PRS in Physical Resource Block

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Figure 2.4: Frequency Spectrum of OFDM Waveform

As standardized in [3] of LTE networks, one PRB consists of 7 OFDM symbolsand 12 subcarriers. PRS pattern is as shown in Figure 2.3 for normal CyclicPrefix (CP) example. With respect to 5G OFDM design numerology designconsiderations in [19], there will be many new freedoms and design in the physi-cal layer. For example, OFDM waveform is scalable in the sense that subcarrierspacing can be chosen as 15⇥ 2n kHz, where n is an integer and 15 kHz is thevalue used in LTE networks.

It is worth mentioning that the scope of this thesis focuses on enhancementsof ToA estimation restricted for one path between the cell and UE, so neigh-bouring cell interference is not considered, i.e. PRS pattern design is out of ourscope. In addition, there will be new updates on PRS patterns in upcoming3GPP Releases, and CRS (as seen in Figure 2.5) will be replaced. Therefore forsimulations in this thesis, PRS pattern is assumed as in Figure 2.4, aiming tooptimize bandwidth, subcarrier spacing and power allocation.

Figure 2.5: PRS in Simulations

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2.2 Fundamental Performance Limit

2.2.1 Minimum Variance Unbiased Estimation

As specified in [5], statistically, an estimator uses algorithms of estimating anunknown parameter based on observation data. the “quality” of an estimatoris quantified in several properties. Error calculation is a quite straight forwardway to describe the deviation between the true value and estimated error.

For a given sample x, the estimation error of estimator b✓ is defined as

✏(x) = b✓(x)� ✓(x) (2.6)

Another more common and useful way is to use Mean Square Error (MSE),defined as

MSE(✓) = Eh

(b✓(x)� ✓(x))2i

= Eh

(b✓(x)� E[b✓(x)])2i

+⇣

E[b✓(x)]� ✓(x)⌘2

= var(b✓) + bias2(b✓)

(2.7)

An estimator is unbiased, i.e. bias term in equation (2.7) is zero, when onaverage the estimator yields the true value of estimated unknown parameter, as

Eh

b✓(x)i

= ✓(x) (2.8)

Constrain an estimator to be unbiased, then the estimator which produces min-imum variance is termed as Minimum Variance Unbiased (MVU) estimator.However, it does not mean an MVU estimator always exists for a given set ofdata or given scenario. Sometimes it is possible that the estimator is not unbi-ased or even there is an unbiased estimator, the minimum variance is not easyto find.

2.2.2 Cramer Rao Lower Bound

One of the methods to find an MVU estimator is: to determine Cramer RaoLower Bound (CRLB) and try if there are some estimators that can achieve thebound. As shown in Figure 2.5, suppose that there are several unbiased estima-tors b✓1, b✓2, b✓3, CRLB allows us to determine a lower bound for the estimationerror variance.

Figure 2.6: Cramer Rao Lower Bound on Variance of Unbiased Estimator

11

If an estimator whose variance equals CRLB at each value of b✓, then an MVUestimator is found. While it may also happen that there is no estimator canachieve the bound, but a MVU estimator may still exist. An estimator is alsopossible to partially achieve the bound which is called asymptotically e�cientestimator. There are also other methods such as Rao Blackwell Lechman Sche↵e(RBLS) [5] which finds the MVU estimator from a statistical way.

While CRLB would provide a benchmark against the physical impossibilitythat no estimator can achieve variance less than this bound. Being able to finda CRLB on the variance of any unbiased estimator is extremely useful in signalprocessing feasibility studies and practical cases, such as DC level estimation,phase estimation of a sine wave and other applications. In this thesis, inves-tigating CRLB on ToA estimation will provide us important intuitions on thepossibility to enhance positioning accuracy. Some CRLB theorems which areapplied in this thesis are listed as below:

• Cramer Rao Lower Bound for Scalar Parameter

As introduced in [5], for a given sample x and b✓ is an unbiased scalar estima-tor. Assuming that the Probability Density Function (PDF) p(x; ✓) satisfies the“regularity” condition

E

⇢

@ ln [p (x; ✓)]

@ ✓

�

= 0, 8 ✓ (2.9)

where the expectation is taken with respect to p(x; ✓). After applying Cauchy-Schwartz inequality, Cramer Rao Lower Bound on the variance of any unbiasedestimator b✓ satisfies

var(b✓) � CRLB = 1�E

n

@2 ln[p(x;✓)]@ ✓2

o (2.10)

where the second derivative of log likelihood function is taken at the true valueof ✓ and the expectation is taken with respect to p(x; ✓). The denominator,termed as Fisher Information, is denoted by

I(✓) = �E⇢

@2 ln [p (x; ✓)]

@ ✓2

�

= �N�1X

n=0

E

⇢

@2 ln [p (x[n]; ✓)]

@ ✓2

�

(2.11)

which is non negative and the observations are independent and identicallydistributed (i.i.d.). CRLB is obtained through taking the inverse of FisherInformation.

12

• Cramer Rao Lower Bound for Vector Parameter

When estimating a vector parameter, i.e. ✓ = [✓1 ✓2 ✓3 · ··]T . Assume the“regularity” condition

E

⇢

@ ln [p (x; ✓)]

@ ✓

�

= 0, 8✓ (2.12)

Then, the covariance matrix of any unbiased estimator b✓ satisfies

Cov b✓ � I�1(✓) � 0 (2.13)

Then the Fisher Information Matrix (FIM) I(✓) is given as

[I(✓)]ij = �E⇢

@2 ln [p(x;✓)]

@✓i@✓j

�

(2.14)

And the CRLB for i-th parameter is found as the [i, i] element of the inverse ofFIM, i.e. the diagonal terms of the matrix

var(b✓i) � CRLB =⇥

I�1(✓)⇤

ii(2.15)

• Bayesian Cramer Rao Lower Bound

When a prior distribution is considered on the parameter space, the BayesianCRLB theorem would be applied to find the bound. Suppose b✓ is an estimatorof ✓, where the PDF of x is p(x|✓). When a prior distribution is placed onthe parameter space and the density of prior is denoted as �. Let E✓ denotesthe expectation with respect to p(x|✓) and E denotes the expectation over thedensity of parameter �. Then according to Bayesian Theory in [20],

p(x, ✓) = p(x|✓) · �(✓) (2.16)

and

E(b✓) = EE✓h

b✓ (X)i

=

Z Z

b✓ (x) · p(x|✓) · �(✓)dxd✓ (2.17)

var(b✓) = E

⇢

E✓

⇣

b✓ (X)� ✓⌘2

��

(2.18)

The Bayesian CRLB as expressed in [21] is consisted of conditional FIM addingwith prior FIM

var(b✓) � CRLB = {E [I(✓) + I(�)]}�1 (2.19)

where

E [I(✓)] = E

(

Z

@ln [p(x|✓)]@✓

�2

p(x|✓)dx)

(2.20)

I(�) =

Z

@ln [�(✓)]

@✓

�2

�(✓)d✓ (2.21)

13

Chapter 3

Derivatons - Cramer RaoLower Bounds on ToAEstimation

In this chapter, the CRLBs on ToA estimation accuracy are derived respec-tively considering frequency selective channel with additive white noise, carrierfrequency o↵set and Wiener phase noise.

We will start from the AWGN channel as part of the results in [10], makingit easier for readers to understand from the straightforward case. Consideringa frequency selective channel for a point to point link, Channel State Informa-tion (CSI) knowledge is an essential proof to divide estimation scenario intothree cases: with channel knowledge at the receiver side (CSIR), with channelknowledge at the transmitter side (CSIR), without channel knowledge at eitherreceiver side or transmitter side (No CSI). General for an estimation process,it is mostly divided into CSIR and No CSIR two cases as in [22]. While poweroptimal allocation is possible to be investigated at the transmitter side (CSIT),the thesis has considered three cases as below. Additionally, system models andnecessary intermediate steps are specified in the following sections.

3.1 AWGN Channel

Let’s start the derivation from a simple AWGN channel case which means onlyLine of Sight (LoS) component is considered. The Channel Impulse Response(CIR) is

h(t) = h0�(t� ⌧d) (3.1)where � is the impulse function and ⌧d is the ToA that we are going to estimate.The received signal r(t) is obtained by taking the convolution of transmittedsignal x(t) and CIR h(t) with additive complex white Gaussian noise w(t)

r(t) = x(t) ⇤ h(t) + w(t) = h0x(t� ⌧d) + w(t) w(t) ⇠ CN(0,�2) (3.2)

After sampling with sampling rate fs, the system model is

r(nTs) = h0x(nTs � ⌧d) + w(nTs) (3.3)

14

Considering the samples are independent and identically distributed, the PDFcould be expressed as

p(r; ⌧d) =N�1Y

n=0

1

⇡�2exp

⇢

� 1�2

[r(nTs)� h0x(nTs � ⌧d)]2�

=N�1Y

n=0

1

⇡�2exp

⇢

� 1�2

[w(nTs)]2�

(3.4)

In the time domain, the nth sample for one OFDM symbol is obtained by takingInverse Fast Fourier Transform (IFFT) of QPSK sequence a(k)

xn = x(nTs) =1pN

X

k

a(k)ej2⇡kn/N n✏[0, N � 1] (3.5)

where N is the length of one OFDM symbol which equals to the number ofsubcarriers Nc The accumulated energy of OFDM signal is calculated as

E =N�1X

n=0

|xn|2 =N�1X

n=0

1

N

2

4

N2 �1X

k=�N2

a(k)ej2⇡kn/N ·N2 �1X

m=�N2

a⇤(m)e�j2⇡km/N

3

5 (3.6)

due to the orthogonality property of OFDM waveform, i.e.

N�1X

n=0

ej2⇡(k�m)n/N = 0, k 6= m (3.7)

Then the accumulated energy of OFDM signal equals to the total power ofQPSK sequences

E =N�1X

n=0

1

N

2

4

N2 �1X

k=�N2

a(k)a⇤(m)ej2⇡(k�m)n/N

3

5 =

N2 �1X

k=�N2

|a(k)|2 (3.8)

Substitute equation (3.4) into CRLB theorem for scalar parameter (2.10)

var(b⌧d) �1

�En

@2 ln[p(x;⌧d)]@ ⌧2d

o =1

I(⌧d)=

�2

2|h0|2PN�1

n=0 |@

@⌧dx(nTs � ⌧d)|2

(3.9)

Then, the derived CRLB on ToA estimation in AWGN channel is

var(b⌧d) � CRLB⌧d AWGN =�2

8⇡2|h0|24f2P

N2 �1k=�N2

k2|a(k)|2(3.10)

From equation (3.10), we can see some related parameters associated with thebound: bandwidth (N : total number of subcarriers), subcarrier spacing (4f),power allocation and SNR (�2, k2|a(k)|2)

15

3.2 Frequency Selective Channel

In multipath channel, frequency selective fading channels (L taps) are consideredwith LoS path and NLoS paths

h(t) =L�1X

i=0

hi�(t� ⌧d � iTs) = h0�(t� ⌧d) +M�1X

i=1

hi�(t� ⌧d � iTs) (3.11)

The received signal r(t) is obtained by taking the convolution of transmittedsignal x(t) and CIR h(t) with additive complex white Gaussian noise w(t)

r(t) = x(t)⇤h(t)+w(t) = h0x(t�⌧d)+L�1X

i=1

hix(t�⌧d�iTs) w(t) ⇠ CN(0,�2)

(3.12)Our case is restricted with zero Inter Carrier Interference (ICI) and Inter SymbolInterference (ISI), assuming that the CP is added transmitted side and removedperfectly at the receiver side with CP length larger than the delay spread ofmultipath channel. After sampling with sampling rate fs, the discrete-timereceived signal is

r(nTs) = x(nTs) ⇤ h(nTs) +w(nTs) = h0x(nTs � ⌧d) +M�1X

i=1

hix(nTs � ⌧d � iTs)

(3.13)For the simplicity of deriving CRLB in multipath channel, the system model isexpressed into matrix form

r = FH ·A · � · FL · h+w (3.14)where

r =

r(�N2) · · · r(N

2� 1)

�T

w =

w(�N2) · · · w(N

2� 1)

�T

h = [h0 · · · hL�1]T

A = diag

a(�N2) · · · a(N

2� 1)

�T

� = diagh

e�j2⇡4f⌧d(�N2 ) · · · e�j2⇡4f⌧d(N2 �1)

iT

(3.15)

FL is composed of the first columns of zero frequency centered DFT matrix, FH

means taking IFFT to convert frequency domain signals into the time domainsignals and w = e�j

2⇡N

FL =

2

6

6

6

6

6

6

6

6

6

4

1 w�N2 · · · w(�N2 )(L�1)

......

. . ....

1 1 1 1

1 w... wL�1

......

. . ....

1 wN2 �1 · · · w(N2 �1)(L�1)

3

7

7

7

7

7

7

7

7

7

5

(3.16)

16

3.2.1 CSIR - Channel State Information at Receiver

When there is channel knowledge at the receiver side, the estimated parameter is⌧d and CRLB of scalar parameter theorem is applied as equation (2.10). Takingthe second derivative of log likelihood function on ⌧d. The FIM for one OFDMsymbol is

I(⌧d) = hH · FHL ·AH ·D2 ·A · FL · h (3.17)

Taking the inverse of FIM and the CRLB on ToA estimation with CSIR inmultipath channel is expressed as

var(b⌧d) � CRLB⌧d CSIR =�2

2·⇥

h

H · FHL ·AH ·D2 ·A · FL · h⇤�1

(3.18)

where E[w ·wH ] = �2 ·I and D = 2⇡4f ·diag⇥

�N2 · · ·N2 � 1

⇤THere we can also

see the related parameter is bandwidth, subcarrier spacing and power allocation.

3.2.2 CSIT - Channel State Information at Transmitter

Channel State Information at Transmitter (CSIT) is also classified as one casemainly considering power optimal allocation on the transmitter side, which couldbe taken into account in this case. According to OTDoA which is a downlinkpositioning method, the CRLB with CSIT is the same as CRLB with CSIR.

var(b⌧d) � CRLB⌧d CSIT = CRLB⌧d CSIR (3.19)

Simulations about power allocation exploit are specified in Chapter 4.

3.2.3 No CSI - No Channel State Information

Where there is no Channel State Information at either transmitter or receiverside, jointly estimation of ToA and CIR should be considered and the e↵ect offrequency selective channel on ToA estimation is investigated. The estimatedparameter extends to a vector and CRLB of vector parameter theorem is appliedas equation (2.15)

✓ =⇥

⌧d Re�

h

T

Im�

h

T ⇤T

(3.20)

where CIR is divided into real part and imaginary part respectively. The FIMis calculated as

17

I(✓)

=

2

6

6

6

6

6

6

6

4

�En

@2 ln[p(r;⌧d)]@ ⌧2d

o

�En

@2 ln[p(r;⌧d)]@ ⌧d@Re{hT }

o

�En

@2 ln[p(r;⌧d)]@ ⌧d@Im{hT }

o

�En

@2 ln[p(r;⌧d)]@Re{hT }@ ⌧d

o

�En

@2 ln[p(r;⌧d)]@ Re{hT }2

o

�En

@2 ln[p(r;⌧d)]@ Re{hT }@Im{hT }

o

�En

@2 ln[p(r;⌧d)]@Im{hT }@ ⌧d

o

�En

@2 ln[p(r;⌧d)]@Im{hT }@ Re{hT }

o

�En

@2 ln[p(r;⌧d)]@ Im{hT }2

o

3

7

7

7

7

7

7

7

5

=2

�2Re

2

6

6

6

6

6

4

@uH

@⌧d· @u@⌧d

@uH

@⌧d· @u@Re{hT }

@uH

@⌧d· @u@Im{hT }

@uH

@Re{hT } ·@u@⌧d

@uH

@Re{hT } ·@u

@Re{hT }@uH

@Re{hT } ·@u

@Im{hT }

@uH

@Im{hT } ·@u@⌧d

@uH

@Im{hT } ·@u

@Re{hT }@uH

@Im{hT } ·@u

@Im{hT }

3

7

7

7

7

7

5

(3.21)

Taking the inverse of FIM and the CRLB on ToA estimation with No CSI inmultipath channel is expressed as

var(b⌧d) � CRLB⌧d NoCSI = [I(✓)]�1(1,1) (3.22)

With further simplification

CRLB⌧d NoCSI =�2

2

"

h

HF

HL A

HD

?Y

BFL

DAFLh

#�1

?Y

BFL

= I �AFL(FHL AHAFL)�1FLA

(3.23)

Comparing with the equation (3.18) and (3.23), �AFL(FHL AHAFL)�1FLAoccurs as a penalty term resulting from the fact that channel coe�cients are un-known. So, the channel knowledge will determine di↵erent performance boundson ToA estimation.

18

3.3 Carrier Frequency O↵set/Doppler Shift

When the receiver has relative motion relative to the transmitter, the DopplerShift will occur. In this thesis, Doppler Shift, also termed as Carrier frequencyO↵set is considered, the estimated parameter extends to

✓ = [⌧d WCFO]T (3.24)

where WCFO = 2⇡4fCFO and 4fCFO is the Carrier frequency O↵set.

The discrete-time system model in AWGN channel is expressed as

r(nTs) = h0 · x(nTs � ⌧) · ✏jWCFO·nTs +w(nTs), w(nTs) ⇠ CN(0,�2) (3.25)

Follow the CRLB theorem for vector parameter as in equation (2.15). The FIMis calculated as

I(✓)

=

2

6

6

4

�En

@2 ln[p(r;✓)]@ ⌧2d

o

�En

@2 ln[p(r;✓)]@⌧d@WDS

o

�En

@2 ln[p(r;✓)]@WDS@⌧d

o

�En

@2 ln[p(r;✓)]@ W 2DS

o

3

7

7

5

=2

�2|h0|2Re

2

6

4

�

�

�

@x(nTs�⌧d)@⌧d

�

�

�

2jnTs

@x⇤(nTs�⌧d)cd x(nTs � ⌧d)

�jnTs @x(nTs�⌧d)cd x⇤(nTs � ⌧d) |nTsx(nTs � ⌧d)|2

3

7

5

(3.26)

Within the derivation, Carrier Frequency O↵set factor is all cancelled by itsconjugates, and the bound will have the same result as in AWGN channel asequation (3.10)

var(b⌧d) � CRLB⌧d CFO = CRLB⌧d AWGN

=�2

8⇡2|h0|24f2P

N2 �1k=�N2

k2|a(k)|2(3.27)

Therefore, Carrier Frequency O↵set (i.e. Doppler Shift) has no e↵ect on ToAestimation from CRLB point of view.

19

3.4 Wiener Phase Noise

In signal processing [23], there are rapid, shot-term and random fluctuations inthe phase of a waveform caused by the instability of an oscillator, is called phasenoise. Generally, phase noise increases with frequency of the Local Oscillator(LO).

As mentioned in [19], from 5G OFDM waveform numerology considerations,Inter Carrier Interference (ICI) decreases as a function when increasing subcar-rier spacing. It means choosing a larger subcarrier spacing is robust againstICI in the OFDM communication system. In this section, the phase noise isconsidered from a ToA estimation point of view to investigate how phase noisewill a↵ect the positioning accuracy and which option of parameter settings arerobust against phase noise in order to enhance positioning accuracy.

First, the estimator vector consists of ToA as usual and also phase noise factors

✓ = [⌧d ']T = [⌧d '1 · · ·'N�1]T (3.28)

' is the phase noise samples. Define � = 'n �'0, n = 0 · · ·N � 1, which helpsto distinguish the phase noise and the channel phase for the first sample. Andthe phase noise is modeled as a Wiener process due to industry consideration

'n = 'n�1 + �n, �n ⇠ N(0,�2� ) (3.29)

The discrete-time system model with considering Wiener phase noise is

r(nTs) =�

x(nTs) ⇤ h(nTs)

·ej'(nTs)+w(nTs), w(nTs) ⇠ CN(0,�2) (3.30)

In matrix form

r = P · FH ·A · � · FL · h+w = u+w (3.31)

where

r =

r(�N2) · · · r(N

2� 1)

�T

P = diag[ej'0 · · · ej'N�1 ]T

w =

w(�N2) · · · w(N

2� 1)

�T

h = [h0 · · · hL�1]T

A = diag

a(�N2) · · · a(N

2� 1)

�T

� = diagh

e�j2⇡4f⌧d(�N2 ) · · · e�j2⇡4f⌧d(N2 �1)

iT

(3.32)

where P is the NxN phase noise matrix and FL is composed of the first columnsof zero frequency centered DFT matrix, same as equation (3.16). FH meanstaking IFFT to convert frequency domain signals into the time domain signals.

20

As can be seen from the system model, there are two distributions and BayesianCRLB theorem will be applied as equation (2.19). The FIM consists of twoparts: conditional FIM and prior FIM

I(✓) = tc +tp (3.33)

• The NxN conditional FIM (assume the prior distribution is known) istc = E Er|� {�4ln [p(r | �, Re {h}), Im {h} , ⌧d)]}

= E

2

�2Re

⇢

@uH

@�m· @u@�n

��

=2

�2Re

⇢

@uH

@�m· @u@�n

�

m,n = 1, 2 · · ·N

=2

�2Re

2

6

4

@uH

@⌧d· @u@⌧d

@uH

@⌧d· @u@ i

@uH

@ i· @u@⌧d

@uH

@ i· @u@ j

3

7

5

i, j = 1, 2 · · ·N � 1

=2

�2Re

2

4

q

H · q �qH ·Q · bi

�bHi ·QH · q QH ·Q(2:N, 2:N)

3

5

(3.34)

where

q = FHDAlFLh

Q = diag⇥

F

HAlFLh

⇤

bi = [0, 01⇥(i�1), 1, 01⇥(N�i�1)]T , i = 1, 2...N � 1

(3.35)

• The NxN prior FIM (considering the Wiener phase noise distribution) is

tp = E �

�4✓✓ ln [p( | Re {h} , Im {h} , ⌧d)]

=

2

4

E �

�4⌧d⌧d ln [p( | Re {h} , Im {h} , ⌧d)]

E n

�4�i⌧d ln [p( | Re {h} , Im {h} , ⌧d)]o

E n

�4⌧d�i ln [p( | Re {h} , Im {h} , ⌧d)]o

E n

�4�i⌧d ln [p( | Re {h} , Im {h} , ⌧d)]o

3

5

= � 1�2�

2

6

6

6

6

6

6

6

6

6

4

0 0 0 0 · · · 0 0 00 �1 1 0 · · · 0 0 00 1 �2 1 · · · 0 0 0...

......

.... . .

......

...0 0 0 0 · · · �2 1 00 0 0 0 · · · 1 �2 10 0 0 0 · · · 0 1 �1

3

7

7

7

7

7

7

7

7

7

5

(3.36)

Then, the CRLB on ToA estimation considering Wiener phase noise is expressedas

var(b⌧d) � CRLB⌧d PHN = [I(✓)]�1(1,1) = [tc +tp]

�1(1,1) (3.37)

21

Chapter 4

Simulations and ResultAnalysis - FlexibleParameter Exploit

From derived CRLBs in di↵erence cases above, we can see that some parametersare quite relevant to the bound on ToA estimation accuracy. In this chapter,the e↵ects of flexible bandwidth, subcarrier spacing and power allocation onToA estimation accuracy are investigated in di↵erent settings based on the per-formance bounds.In the simulation, SNR is defined as

SNR = 10log10

Es�2

�

= 10log10

"

E�

|a(k)|2

�2

#

(4.1)

Note that, he standard variance of CRLB is used in all plots,converting thebound into meters to evaluate positioning accuracy is the standard required toevaluate positioning accuracy (C is speed of the light)

�CRLB = C ·pCRLB (4.2)

The underlying principle in simulations: when exploiting di↵erent freedoms ofPRS signals, (e.g. bandwidth, subcarrier spacing and power allocation), thetotal symbol length, accumulated energy and average transmitted power are allguaranteed as the same to realize fairly comparison. For example, when sub-carrier spacing is evaluated, more OFDM symbols are transmitted for largersubcarrier spacing. When bandwidth is evaluated, more OFDM symbols areconsidered for smaller bandwidth case. When power allocation is evaluated,the amplitude of windowing on spectrum will be changed according to di↵erentwindow types.

Regarding the Carrier Frequency O↵set (i.e. Doppler Shift) case, as shownin equation (3.27). the results are the same as in AWGN channel. So the sim-ulation part for AWGN channels can also represent the performance of CRLBwith CFO case.

22

4.1 Bandwidth

• AWGN Channel

10MHz, 20MHz, 40MHz, 80MHz bandwidth are evaluated based on the de-rived performance bound as in equation (2.10)

As shown in Figure 4.1, when increasing bandwidth, the bound gets lower whichmeans the positioning accuracy is improved. Also, the improvement performsa root of n times relationship, e.g. the bound for 40MHz is

p2 lower than

20MHz.

Figure 4.1: Bandwidth Exploit on ToA Estimation in AWGN Channel

Mathematically, we try to prove this multiple times relationship and every PRSsignal is assumed to have the same power. The original CRLB is

CRLB⌧d =�2

8⇡2 | h0 |2 4f2Nsymbk2a21

=3�2

4⇡2 | h0 |2 4f2a21NsymbN·

1N2

2 + 1

! (4.3)

When the bandwidth is increased by n times

CRLB⌧d n timesBW =n�2

8⇡2 | h0 |2 4f2Nsymbn k2 · a21

=3�2

4⇡2 | h0 |2 4f2a21NsymbN·

1N2

2 · n+1n

! (4.4)

23

After calculating the ratio

�CRLB⌧d n timesBW�CRLB⌧d

=

r

CRLB⌧d n timesBWCRLB⌧d

=

s

N2

2 + 1N2

2 · n+1n

⇡ 1pn

when N is large

(4.5)

So equation (4.5) has proved that: when bandwidth (total number of subcarri-ers) is large, increasing bandwidth by n times, CRLB on ToA estimation accu-racy improves

pn times.

• Frequency Selective Channel

For multipath channel, it is divided into two cases: CSIR and No CSI. A specificfrequency selective channel is referred as in [11], with 4 taps channel impulseresponse

h = [0.38 + 0.23j, 1.3� 0.92j, �1.6� 0.31j, 0.61 + 0.24j]T (4.6)

The frequency response is shown in Figure 4.2

Figure 4.2: Specific Channel Frequency Response

When there is channel knowledge at the receiver side (CSIR), equation (3.18)is applied. When there is no channel knowledge (No CSI), equation (3.23) isapplied. As shown in Figure 4.3, there are some observations:

24

(a) CSIR (b) No CSI

Figure 4.3: Bandwidth Exploit on ToA Estimation in Multipath Channel

(a) In a frequency selective channel, the multiple times improvement still re-mains: increasing bandwidth by n times, CRLB on ToA estimation accu-racy improves

pn times.

(b) Unknown channel knowledge (No CSI) introduces more uncertainty tothe ToA estimation accuracy comparing with the case with CSIR, whichis also reflected in the penalty term as equation (3.23)

• Wiener Phase Noise

For phase noise case, Bayesian CRLB theorem is applied as equation (2.19).The simulation is done in AWGN channel and bandwidth exploit is plotted asFigure 4.4 in linear scale

Figure 4.4: Bandwidth Exploit on ToA Estimation with Wiener Phase Noisein AWGN Channel

25

The performance is quite di↵erent comparing with the cases without phase noise.Observations are

(a) With the existence of Wiener Phase Noise, increasing bandwidth will onlyimprove the bound for low SNR region.

(b) For high SNR, the phase noise is dominating and bandwidth does nota↵ect too much, which means phase noise will give a degradation on ToAestimation.

4.2 Subcarrier Spacing

With respect to subcarrier spacing, from 5G OFDM design numerology designconsiderations in [19], OFDM waveform is scalable in the sense that subcarrierspacing can be chosen as 15⇥ 2n kHz, where n is an integer and 15 kHz is thevalue used in LTE networks. The simulations have investigated the flexibilityof subcarrier spacing from positioning point of view.

• AWGN Channel

15 kHz, 30 kHz, 60 kHz, 120 kHz subcarrier spacing are evaluated based on thebound as in equation (2.10). As shown in Figure 4.5, there is no much di↵erencebetween the bounds with di↵erent subcarrier spacing.

Figure 4.5: Subcarrier Spacing Exploit on ToA Estimation in AWGN Channel

Mathematically, it is able to explain this observation. The original CRLB is

CRLB⌧d =�2

8⇡2 | h0 |2 4f2Nsymb k2 · a21

=3�2

4⇡2 | h0 |2 4f2a21NsymbN·

1N2

2 + 1

! (4.7)

26

When the subcarrier spacing is increased by n times

CRLB⌧d n times SCS =�2

8⇡2 | h0 |2 (n4f)2 · (nNsymb) · k2 · a21

=3�2

4⇡2 | h0 |2 4f2 · a21NsymbN·

1N2

2 + n2

! (4.8)

After calculating the ratio and plotting it in Figure 4.6

�CRLB⌧d n times SCS�CRLB⌧d

=

r

CRLB⌧d n times SCSCRLB⌧d

=

s

N2

2 + 1N2

2 + n2⇡ 1 when N is large

(4.9)

Figure 4.6: Subcarrier Spacing Ratio in AWGN Channel

As shown from Figure 4.6, for small bandwidth (total number of subcarriers),large subcarrier spacing will give a lower bound referring to the ratio smallerthan one. Related with the real application, Narrow Band Internet of Things(NB IoT) would be the case. However, for large bandwidth, changing subcarrierspacing will not give much influence on the CRLB.

27


For the multipath channel case, the specific channel impulse response is applied,as in equation (4.6). Figure 4.7 also shows that subcarrier spacing has slightinfluence on CRLB for either CSIR case or No CSI case.

(a) CSIR (b) No CSI

Figure 4.7: Subcarrier Spacing Exploit on ToA Estimation in Multipath Chan-nel


As shown in Figure 4.8, larger subcarrier spacing improves CRLB on ToA esti-mation significantly with the existence of phase noise. The simulation result isvaluable showing that larger subcarrier spacing are more robust against phasenoise and beneficial to enhance positioning accuracy.

Figure 4.8: Subcarrier Spacing Exploit on ToA Estimation with Wiener PhaseNoise in AWGN Channel

28

4.3 Power Allocation

For current LTE networks, PRS signals are uniformly power allocated accord-ing to the existing structure. In this chapter, non-uniform power allocation isinvestigated to enhance the ToA estimation accuracy.

• AWGN Channel

Recall the equation (3.10), the optimal power allocation solution is to put allpower to two edges of the subcarriers as shown in Figure 4.9

(a) Optimal Power Allocation Scheme(b) CRLB Comparison for Power

Allocation in AWGN

Figure 4.9: Power Allocation Exploit on ToA Estimation in AWGN Channel


The optimal power allocation in AWGN channel, which is putting all power tothe two edges on band is not feasible for frequency selective channels. For CSITcase, when the transmitter has channel knowledge and power optimization ispossible to be implemented.

As studied in [11], a Monte Carlo simulation method is done in order to findoptimal pilots allocation (i.e. power optimization) for a referred specific chan-nel, in which 32 subcarriers and 5 pilots (minimum required number of pilots isas per convex solution) are used.

29

Figure 4.10: Monte Carlo Simulation for Power Optimization

Figure 4.10 shows a non-uniform power allocation solution that pushing powertowards the edges on band. Motivated by this result, a windowing methodapplied on power spectrum of the transmitted signal is used to investigate non-uniform power optimization solution for the case with No CSI knowledge.

Based on the specific frequency selective channel, rectangular windowing, in-verse triangular windowing, inverse Gaussian windowing and inverse Cheby-shev windowing are applied on the power spectrum of transmitted signal andcorresponding CRLBs as equation (3.23) are used. Note that under di↵erentwindowing types, the average transmitted power is guaranteed as the same torealize fairly comparison.

(a) (b)

Figure 4.11: Power Allocation Exploit on ToA Estimation with No CSI inMultipath Channel

Observations are

(a) Based on the performance order, inverse Chebyshev windowing gives thebest improvement than inverse Gaussian windowing, than inverse trianglewindowing and rectangular windowing (i.e. uniform power allocation).It conveys an intuition that non-uniform power allocation is better thanuniform power allocation.

30

(b) Considering the inverse Chebyshev windowing spans more power towardsthe edges, we can deduce that pushing power towards the edges on bandgives more improvement on the bound, which is in accordance with theconclusion in [11].

(c) The AWGN optimal power allocation method is plotted as green line inFigure 4.11 (a), which shows the worst performance, which means allocat-ing all power only at two edges is not feasible.


Firstly, the variance of Wiener phase noise is investigated in Figure 4.12, referto same variance values as in [14]. As illustrated, there is a degradation onthe bound with existing Wiener phase noise and the bounds become closerwith increasing SNR. While there is still a big degradation gap at high SNR,comparing with the case without any phase noise.

Figure 4.12: Wiener Phase Noise Variance Exploit on ToA Estimation inAWGN Channel

A windowing method is also applied on the case when considering Wiener phasenoise. The Bayesian CRLB theorem as equation (2.19) is applied and CRLBsare plotted as in Figure 4.12

31

Figure 4.13: Power Allocation Exploit on ToA Estimation with Wiener PhaseNoise in AWGN Channel

When Wiener phase noise exists, non-uniform power allocation still gives abetter performance. Inverse Chebyshev windowing shows the best performancewhich means pushing more power towards the edges on band is a better poweroptimization solution.

32

Chapter 5

Conclusions and FutureWork

5.1 Main Findings on ToA Performance

Based on the simulation results, there are some main findings, as listed in Figure5.1

Figure 5.1: Main Findings on ToA Performance

From this table, there are new observations comparing with state-of-art works.From the CRLB on ToA estimation point of view, the main findings could besummarized as:

(a) Increasing bandwidth will often improve the ToA estimation accuracy withsquare root of N times relation. However, when there is phase noise, theimprovement degrades, depending on the phase noise variance and alsoSNR region.

(b) Changing subcarrier spacing has quite slight e↵ect on the bound withoutphase noise. When there occurs Wiener phase noise, larger subcarrier

33

spacings are more robust and gives significant improvement on ToA esti-mation accuracy.

(c) Non-uniform power allocation always performs a better performance inall cases and pushing more power towards the edges on band is a poweroptimization solution.

5.2 Conclusions

In this thesis, fundamental performance bounds (i.e. Cramer Rao Lower Bounds)on Time of Arrival (ToA) estimation are derived respectively considering fre-quency selective channels with additive white noise, carrier frequency o↵set andWiener phase noise. In particular, the e↵ects of flexible bandwidth, subcarrierspacing and power allocation on ToA estimation accuracy have been investi-gated in di↵erent settings based on the performance bounds. Based on theresults, some important intuitions are provided such as the existence of phasenoise will dominate and give significant degradation on ToA performance, largersubcarrier spacing is more robust and beneficial from practical consideration,non-uniform power optimization and so on. Constructive conclusions are builton how to redesign positioning reference signal design and waveform optimiza-tion for 5G based positioning.

5.3 Future Work

The future work could be classified into several parts:

First, as discussed in the background part of this thesis. Determining a variancebound is the first step of finding a MVU estimator. Trying estimators such asMaximum Likelihood (ML) estimator and Maximum A PosteriorI (MAP) esti-mator, finding a way to achieve the derived bound will be a next step. Apartfrom Cramer Rao Lower Bound, the author has also studied and tried to deriveZiv Zakai Bound (ZZB) on ToA estimation. However, CRLB is much easier toobtain a closed form and do simulations. Every bound has its own advantagesand limitations, e.g. CRLB is accurate for high SNR region but not tightenenough for low SNR case. Therefore, the trade-o↵ is quite worthy to explorefurther.

Secondly, the simulations are based on a specific channel impulse response.5G channel models such as [24] could be applied into the simulations. Takingbeamforming into account and do link level simulations are necessary.

Last but not the least, with new updates from 3GPP releases, di↵erent types ofPRS patterns could be taken into considerations and do evaluations about thee↵ect of neighbouring cell interference on ToA estimation accuracy.

34

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Fundamental Performance Limits on Time of Arrival ...1182120/FULLTEXT01.pdf · In currently deployed Long Term Evolution (LTE) networks, Observed Time Di↵erence of Arrival (OTDoA)