Performance and Complexity Comparison of Doppler Spread ...749319/FULLTEXT01.pdf · Performance and Complexity Comparison of Doppler Spread Estimation for WCDMA Systems ZIQI PENG

Performance and Complexity Comparisonof Doppler Spread Estimation for

WCDMA Systems

ZIQI PENG

Master’s Degree ProjectStockholm, Sweden 2014

XR-EE-KT 2014:009

Abstract

In cellular communication systems, the estimation of Doppler spread has a widerange of applications such as handoff, channel assignment scheme, adaptivetransmission, power control, etc. A great quantity of Doppler spread estimationalgorithms have been proposed in the literature. But there has been few investi-gations which gives a comprehensive comparison of these algorithms. Therefore,it is of great significance to compare and evaluate the performance of the existingalgorithms in the same simulation framework.

In this report, the uplink of WCDMA is considered. Four different types ofDoppler spread estimation algorithms are evaluated and compared in a link levelbaseband simulator. The performance and the ability to implement are consid-ered as the metrics for evaluation. Both Rayleigh and Rician fading channelmodel are applied, and the effect of speed, signal to noise ratio, Rician factorand the angle of arrived line of sight component are also tested. Moreover,the computational complexity is analysed to evaluate the practical value forimplementation.

Key words: Doppler spread estimation; WCDMA uplink systems; Rayleighfading model; Rician fading model; computational complexity.

Abstrakt

Estimatering av en mobils hastighet i form av Dopplerspridning har ett brettspektrum av tillmpningar i cellulra kommunikationssystem ssom frflyttningen avmobiler mellan celler, kanaltilldelningsschema, adaptiv sndning, effektstyrning,etc. En stor mngd algoritmer fr estimering av Dopplerspriding har frslagitsi litteraturen, men det r ovanligt med heltckande jmfrelser mellan med dessaalgoritmer. Drfr r det av stor betydelse att jmfra och utvrdera resultaten avbefintliga algoritmer inom ramen fr samma simuleringsvertyg.

I denna rapport anvnds upplnken fr WCDMA fr utvrdering av fyra olikatyper av algoritmer fr estimering av Dopplerspridning. Metriker fr utvrderin-gen r prestanda och implementeringsvnlighet. Bde Rayleigh och Rician fd-ningskanal modeller har utvrderas, samt effekten av mobilens hastighet, signaltill brus frhllande, Rician faktor och infallsvinkel i ppet flt scenario. Dessutomhar den berkningsmssiga komplexiteten analyseras fr att utvrdera den praktiskaanvndbarheten i riktiga system.

Acknowledgment

First, I would like to thank my supervisor Per Lofving in Ericsson for giving methis opportunity to work on this thesis and providing the insightful guidance formy work. And I would also like to express my appreciation to Henrik Sahlin,Magnus Nilsson and Lu Li for the continuous help, encouragement and feedbacksthroughout the entire thesis work.

Second, I would like to thank my thesis partner Hui Wen in KTH for thecooperation and inspiration.

And I would also like to thank my examiner Prof. Tobias Oechtering inKTH for his advice and time taken to supervise my thesis.

Finally I would like to thank my family for their constant support and loveduring my studies in Sweden.

Contents

List of Figures ii

List of Tables v

List of Symbols vi

List of Abbrevations viii

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background 42.1 Propagation Phenomena . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Path Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Shadow Fading . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Multipath . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Doppler Effect and Doppler Spread . . . . . . . . . . . . . . . . . 62.2.1 Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Doppler Spread . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Channel Response . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Autocorrelation Functions . . . . . . . . . . . . . . . . . . 82.3.3 Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Structure of DPCCH in WCDMA Uplink . . . . . . . . . . . . . 9

3 Estimation Algorithms 113.1 Level Crossing Rate Estimator . . . . . . . . . . . . . . . . . . . 113.2 Hybrid Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Curvature Estimator (small Doppler spread) . . . . . . . 133.2.2 First Zero Detection Estimator (large Doppler spread) . . 143.2.3 Hybrid Estimator . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Power Integration Estimator . . . . . . . . . . . . . . . . . . . . . 173.4 Frequency Domain Maximum Likelihood Estimator . . . . . . . . 19

i

4 Simulation Results 214.1 Simulation Environment . . . . . . . . . . . . . . . . . . . . . . . 214.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.1 Level Crossing Rate Estimator . . . . . . . . . . . . . . . 224.2.2 Hybrid Estimator . . . . . . . . . . . . . . . . . . . . . . . 244.2.3 Power Integration Estimation . . . . . . . . . . . . . . . . 264.2.4 Frequency Domain ML Estimator . . . . . . . . . . . . . 30

5 Performance Evaluation 345.1 Computational Complexity . . . . . . . . . . . . . . . . . . . . . 34

5.1.1 Level Crossing Rate Estimator . . . . . . . . . . . . . . . 345.1.2 Hybrid Estimator . . . . . . . . . . . . . . . . . . . . . . . 355.1.3 Power Integration Estimator . . . . . . . . . . . . . . . . 365.1.4 Frequency Domain ML Estimator . . . . . . . . . . . . . 365.1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . 385.2.1 Rayleigh Fading Channel . . . . . . . . . . . . . . . . . . 395.2.2 Rician Fading Channel . . . . . . . . . . . . . . . . . . . . 41

6 Conclusions 456.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2 Social and Ethical Aspects . . . . . . . . . . . . . . . . . . . . . . 456.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Appendices 47

A Whittle Approximation Log-likelihood Goal Function Deriva-tion 47

ii

List of Figures

2.1 The propagation environment. . . . . . . . . . . . . . . . . . . . 42.2 The pass loss model. . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Multipath fading. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 The effect of multipath propagation. . . . . . . . . . . . . . . . . 62.5 Illustration of Doppler effect. . . . . . . . . . . . . . . . . . . . . 72.6 The structure of the uplink Dedicated Physical Data CHannel

(DPDCH) and Dedicated Physical Control CHannel (DPCCH). . 9

3.1 Level crossing example. . . . . . . . . . . . . . . . . . . . . . . . 123.2 Illustration of linear interpolation. . . . . . . . . . . . . . . . . . 153.3 Illustration of power integration estimator. . . . . . . . . . . . . . 173.4 The Continuous and Discrete form of Theoretical Periodogram

PSD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1 The estimated maximum Doppler spread of Level Crossing Rate(LCR) method with different levels,versus the theoretical maxi-mum Doppler spread. . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 The NMSE of LCR method with different levels,versus the theo-retical maximum Doppler spread. . . . . . . . . . . . . . . . . . . 23

4.3 The NMSE of LCR method with different levels, versus SNR forfour speed values. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4 The comparison of two polynomial fitting algorithms. . . . . . . 254.5 The estimated maximum Doppler spread of Hybrid method,versus

the theoretical maximum Doppler spread. . . . . . . . . . . . . . 264.6 The NMSE of estimated maximum Doppler spread of Hybrid

method,versus the theoretical maximum Doppler spread. . . . . . 264.7 The NMSE of Hybrid method with four speed values, versus SNR. 274.8 The NMSE of Hybrid Estimator for different number of used

paths, versus theoretical maximum Doppler spread. . . . . . . . . 274.9 The NMSE of Power Integration Estimator with β = 1, 2, versus

threshold ψ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.10 The estimated maximum Doppler spread of Power Integration

method, versus the theoretical maximum Doppler spread. . . . . 284.11 The NMSE of estimated maximum Doppler spread of Power In-

tegration method,versus the theoretical maximum Doppler spread. 294.12 The comparison of the number of frames used in simulation for

Power Integration method, versus the theoretical maximum Dopplerspread. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

iii

4.13 The NMSE of Power Integration method with four speed values,versus SNR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.14 The normalized bias and normalized standard deviation of PowerIntegration method with four speed values, versus SNR. . . . . . 30

4.15 The estimated maximum Doppler spread of ML method,versusthe theoretical maximum Doppler spread. . . . . . . . . . . . . . 31

4.16 The comparison between estimated spectrum and theoretical spec-trum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.17 The NMSE of estimated maximum Doppler spread of ML method,versus the theoretical maximum Doppler spread. . . . . . . . . . 32

4.18 The NMSE of ML method with four speed values, versus SNR. . 33

5.1 The comparison of computational complexity. . . . . . . . . . . . 385.2 The comparison of mean value of estimated maximum Doppler

spread in Rayleigh fading model, versus the theoretical maximumDoppler spread. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3 The comparison of NMSE of estimated maximum Doppler spreadin Rayleigh fading model, versus the theoretical maximum Dopplerspread. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.4 The comparison of NMSE of estimated maximum Doppler spreadin Rayleigh fading model, versus SNR. . . . . . . . . . . . . . . . 41

5.5 The comparison of the mean value and NMSE of estimated max-imum Doppler spread, versus Rician factor. . . . . . . . . . . . . 42

5.6 The comparison of the mean value and NMSE of estimated max-imum Doppler spread, versus angle of arrival of LOS component. 43

iv

List of Tables

1.1 Tasks division for two persons . . . . . . . . . . . . . . . . . . . . 2

4.1 Pedestrian A Channel Model . . . . . . . . . . . . . . . . . . . . 21

5.1 Parameter setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Computational Complexity for Level Crossing Rate Estimator . . 355.3 Computational Complexity for Hybrid Estimator for low velocity 365.4 Computational Complexity for Hybrid Estimator for high velocity 365.5 Computational Complexity for Power Integration Estimator . . . 375.6 Computational Complexity for ML Estimator . . . . . . . . . . . 37

v

List of Symbols

a Polynomial coefficient vectoram Normalized amplitude of path mα Angle between the average scattering direction and the mo-

bile direction

β Positive exponent in Power Integration estimatorη0 First zero point of Bessel functionfd Normalized maximum Dopple spread

fd Estimated normalized maximum Dopple spreadfD Maximum Dopple spread

fD Estimated maximum Dopple spreadFs Sampling frequency of channel estimatefshift Doppler shiftγ SNRΓ Goal function in ML estimatorΓc Concentrated goal function in ML estimatorh Theoretical channel responsehm Theoretical multipath component channel responsehLOS Theoretical LOS component channel responseIn nth order modified Bessel function of the first kindJ0 Zero order Bessel function of the first kindK Rician factorkmin Smallest value make autocorrelation function is negative in

the Hybrid estimator

κ Beam widthL Maximum lag in autocorrelation functionsLc Level in the LCR estimatorLnc Normalized level in the LCR estimatorM Number of independent paths with same delayN Number of channel estimate samplesNfft Number of FFT pointNp Number of pilot symbolsNpc Number of positive crossing

Py Estimated PSDφ0 Phase of the LOS componentφm Phase of path mψ Threshold in Power Integration estimatorRh Autocorrelation functions of the theoretical channel

response

Ry Autocorrelation functions of the channel estimation samplesRy′′ Second derivative of autocorrelation functions

Ryq Autocorrelation functions for finger q

Sy Estimated power spectrumσ2

h Power of theoretical channel responseσ2

n Power of noiseσ2

y Power of channel estimate

T Threshold in the Hybrid estimatorTmeas Measurement time in the LCR algorithm

vi

Ts Sampling period of channel estimateτ0 First zero crossing point of the autocorrelation functionsθ0 Angle of the LOS componentθm Angle of path m

vii

List of Abbrevations

AFC Automatic Frequency Control

AGC Automatic Gain Control

AOA Angle Of Arrival

BCL Baseband Core Library

DFT Discrete Fourier Transform

DPCCH Dedicated Physical Control CHannel

DPDCH Dedicated Physical Data CHannel

DSP Digital Signal Processor

FBI FeedBack Information

FFT Fast Fourier Transform

LCR Level Crossing Rate

LOS Line Of Sight

ML Maximum Likelihood

NB Normalized Bias

NMSE Normalized Mean Square Error

NSD Normalized Standard Deviation

PDF Probability Distribution Function

PSD Power Spectral Density

SNR Signal to Noise Ratio

TFCI Transport Format Combination Indicator

TPC Transmit Power Control

UE User Equipment

WCDMA Wideband Code Division Multiple Access

ZCR Zero Crossing Rate

viii

Chapter 1

Introduction

In this chapter, a brief overview of this thesis report is given. The motivationincluding the general description of the research field and the previous workare presented. The main objective and methodology are discussed prior to thethesis outline.

1.1 Motivation

In mobile cellular systems, the movement of the User Equipment (UE) or thesurrounding objects leads to time variations of the channel. Having an accu-rate estimation of the maximum Doppler spread, which indicates the rate ofchannel variation, is of great importance in many applications such as adaptivetransmission, channel assignment scheme, handover, power control, etc [1] [2].

Therefore the Doppler spread estimation has draw much attention in theresearch communities. Up until now, many different Doppler spread estimationalgorithms have been proposed in the literature. Based on their principle, theycan be classified into four types: crossing rate based estimators [1] [3] [4], cor-relation based methods [5] [6] [7], power spectrum based approaches [8] [9], andmaximum likelihood techniques [10] [11] [12].

Most of the references have been focusing on designing new algorithms andgive the simulation results with an ideal channel model. Whereas in [13], severalalgorithms are compared, but only for two types of estimators. However, it isvery interesting and practical to give a comparison study to evaluate four classesof algorithms with the same simulation environment. Moreover, the possibilityof implementation is another important aspect that should be considered.

1.2 Objective

This thesis aims at evaluating and comparing several algorithms for estimatingthe Doppler spread in Wideband Code Division Multiple Access (WCDMA)uplink systems. The investigation shall take both the performance and theability of implementation into consideration. The thesis is done by two personsduring 2014 at Ericsson Lindholmen, Gothenburg.

From the investigation, the following three research questions can be an-swered:

1

1. Which algorithm is the most accurate one in the same simulationframework?

2. If the simulation results are inaccurate, what is the reason causing theinaccuracy for each investigated algorithms?

3. Considering the computational complexity of each algorithm, whichone has the largest practical value for implementation?

1.3 Methodology

In order to achieve the objective of the thesis, the whole work is divided into fivesteps: literature investigation, algorithm selection, implementation of selectedalgorithms, performance comparison, and the analysis of computational com-plexity. Since the thesis is for two persons, the detailed work division is shownin Table 1.1. Noting that the implementation of algorithms are done parallelfor two persons, and for the complexity analysis, each person focuses on one’sown investigated algorithms.

Ziqi Peng(This Report)

Hui Wen(See [14])

Literature Investigation X XAlgorithm Selection X XLCR Estimator XHybrid Estimator XPower Integration Estimator XFrequency Domain ML Estimator XZCR Estimator XMoser’s Estimator XPSD Slope Estimator XTime Domain ML Estimator XComplexity Analysis X XPerformance Comparison X X

Table 1.1: Tasks division for two persons

After the literature investigation, we found that the existing Doppler spreadestimation algorithms can be classified into four categories as described in Sec-tion 1.1. In order to do the research in a wide range of algorithm types, wedecided to choose two algorithms from each type of Doppler spread estimator,and four algorithms are allocated to each person. The benefits of this selectionscheme is that the algorithms can be compared through not only different typesbut also different approaches in the same type. Thus four types of estimatorscan be compared with each other, and the two algorithms of same type can becompared as well. Moreover, the metric of algorithm selection is based on thepossibility of implementation in Baseband Core Library (BCL) and WCDMA,and the reliability of the algorithms, such as how many times the article hasbeen citied or referenced by other articles, how is the comment on the methodfrom other articles. The selected algorithms are listed in the Table 1.1. Inthis report, the LCR estimator, the hybrid estimator, the power integrationestimator and the frequency domain Maximum Likelihood (ML) estimator are

2

presented.Moreover, this thesis research focuses on the evaluation and comparison from

two perspectives: the performance and the computational complexity. In orderto answer the first two research questions, the algorithms should be implementedin the same simulation framework, with same parameter setting, such as Signalto Noise Ratio (SNR), how many simulated frames are tested, etc. For theevaluation part, the Rayleigh fading channel model is applied, and only affectedby two main factors, i.e. the velocity and the SNR. Here the velocity is consid-ered from walk speed to high vehicle speed, i.e. 2.7-162 km/h, correspondingto theoretical maximum Doppler spread of 5-300 Hz. The effect of noise is alsotaken into account with SNR setting of 0-15 dB for various level of speeds.

For the comparison part, another four estimation algorithms implementedby my partner [14] are also introduced. All the algorithms are implemented inBCL by C++. In this part, both Rayleigh fading and Rician fading channelare applied. The reason is that some previous research only consider the idealRayleigh fading channel, thus it is interesting to evaluate these estimators inRician fading model as well. The Rician fading model is closer to the real radiochannel and it is helpful to test the effect of more factors. In the Rician fadingcase, we concentrate on assessing the effect of Rician factor K with K from 0to 8 and the Angle Of Arrival (AOA) of Line Of Sight (LOS) component withθ0 = 0◦, 20◦, 40◦, · · · , 180◦.

Finally, the computational complexity is analysed as the criteria to evaluatethe ability of implementation. The number of multiplication operations requiredfor each algorithm is considered as main metric to evaluate the algorithmiccomplexity.

1.4 Outline

The rest of report is organized as follows: Chapter 2 gives a background in-cluding some concepts related to the propagation and Doppler spread. Channelmodels for Rayleigh and Rician fading are given as well. In Chapter 3, severalDoppler spread estimators are described. The simulation environment and theestimation results of each algorithm are presented individually in Chapter 4.In Chapter 5, we give the analysis of the computational complexity of all al-gorithms, along with the performance comparison based on the four describedestimators together with another four estimators implemented by my thesispartner [14]. Finally, we summarize some conclusions and discuss the social andethical aspect and the future work directions in Chapter 6.

3

Chapter 2

Background

In this chapter, the background for the thesis research is presented. First,the propagation in a radio channel is introduced. Then the concepts of theDoppler effect and Doppler spread are described. Next, the channel modelsof Rayleigh fading and Rician fading as well as some channel properties arepresented. Finally, we provide the structure of the uplink DPCCH in WCDMA.

2.1 Propagation Phenomena

In general, the transmitted signal is affected by the propagation phenomenaduring its travel from the transmitter antenna to the receiver antenna. Theseeffects depend on the distance between the transmitter and receiver and theenvironment around the path, such as buildings and trees, see Figure 2.1. Thethree major propagation phenomena are path loss, shadow fading, and multi-path as introduced below.

Figure 2.1: The propagation environment.

4

2.1.1 Path Loss

Since the transmitted signal is spread spherically, the energy attenuates withrespect to the distance. This loss of energy is called path loss. In the free spaceLOS channel, where no obstacles lie between the transmitter and the receiveror around the path between them, the power falls off proportional to square ofthe range. But due to the reflection from the ground, there is not only the LOSpath for long term propagation. In this case, the reflected waves may reducethe received power and result that the power falls off in proportional to fourthpower of the distance [15]. These two cases are shown in Figure 2.2.

Free Space Loss Plane Earth Loss

Figure 2.2: The pass loss model.

2.1.2 Shadow Fading

Shadow fading is caused by the obstacles between or around the transmissionpaths. These obstacles change the direction of the transmitted signal or evenlose the signal through absorption, reflection, scattering, and diffraction.

2.1.3 Multipath

Direct Signal

Reflected Signal

Combined Signal

Figure 2.3: Multipath fading.

5

Multipath phenomenon is caused by the reflection or refraction from theterrestrial objects as mentioned above. As a result, the receiver receives signalsfrom different path, with different delay and angles of arrival. In addition,the combination of the direct wave and the out-of-phase reflected wave yieldsattenuated signals as shown in Figure 2.3, this may result in the well knownRayleigh fading or Rician fading.

See Figure 2.4 for a presentation of the effect of multipath. Assuming aDirac pulse is transmitted, the received signal is a sum of pulses with differentamplitudes at different times. The channel response can be expressed as

h(t) =

N∑n=1

ρnejφnδ(t− τn), (2.1)

note that ρnejφn represents the signals received from different paths have dif-

ferent amplitude and phase.

�

ℎ(�)

�

ℎ(�)

�2

�1

��

�1 �2 ��

transmitted signal received signal

Figure 2.4: The effect of multipath propagation.

2.2 Doppler Effect and Doppler Spread

2.2.1 Doppler Effect

The Doppler effect is the change in frequency of the signal due to the relativemotion between the transmitter and the receiver. It means when a UE (orreflectors in its environment) is moving, a frequency shift occurs at the receivedsignal. This shift is known as the Doppler shift. The relationship between theDoppler shift and the velocity v of transmitter or receiver is given by

fshift =v

cfc cos θ, (2.2)

in Hz, where c is the speed of light, fc is the carrier frequency, and θ is theangle difference between direction of the transmitted signal and the speed, asillustrated in Figure 2.5. When the movement has the same direction as thedirection towards the receiving antenna, then Equation (2.2) reduced to

fshift =v

cfc. (2.3)

6

vθ

Figure 2.5: Illustration of Doppler effect.

2.2.2 Doppler Spread

As described in Section 2.1.3, signals from different transmission paths are re-ceived at the receiver due to the multipath phenomenon. The signals propagatethrough different paths with different angles which give rise to different Dopplershifts. This difference in Doppler shifts contributes to the spread in the spec-trum. The spectrum is spread to the range of [fc − fD, fc + fD], where fDis known as the maximum Doppler spread, which is also called the maximumDoppler frequency in some papers, corresponding the maximum speed v withθ = 0◦:

fD =v

cfc. (2.4)

2.3 Channel Model

2.3.1 Channel Response

The complex base band channel impulse response h(t) can be modelled as thecombination of a diffuse component and a LOS component [16] [17]

h(t) =

√σ2

h

K + 1hm(t) +

√Kσ2

h

K + 1hLOS(t), (2.5)

where σ2h is the average power of received signal, K is Rician factor describing

the ratio of the LOS power to the scattering power.Here, the term finger is used to denote a sum of the path responses with

the same delay. In a finger, the channel response for the scattered and the LOScomponent are given by [18] [19]

hm(t) = limM→∞

1√M

M∑m=1

amej(2πfDt cos θm+φm) (2.6)

andhLOS (t) = ej(2πfDt cos θ0+φ0), (2.7)

where M is the number of independent paths, fD is the maximum Dopplerspread, {am}Mm=1 are the normalized amplitude of path that satisfy limM→∞

1√M

∑Mm=1|am|2 = 1, {θm}Mm=1 are identically distributed arrival angles of the

paths, {φm}Mm=1 are the uniformly distributed path phase, and θ0 and φ0 aredeterministic constants representing the angle and phase of the LOS component,respectively.

7

Note that when M is large, hm(t) is a zero-mean unit-power complex Gaus-sian process by the central limit theorem. The envelop |h(t)| has a RicianProbability Distribution Function (PDF), which reduces to the Rayleigh distri-bution when there is no LOS component (K = 0).

In wireless transmission systems, a frequency offset f0 between the terminaloscillator and the base station oscillator affects the channel response. It canhave a great impact to the Doppler spread estimation. The channel responseincluding the frequency offset is given by

h(t) =

√ σ2h

K + 1hm(t) +

√Kσ2

h

K + 1hLOS(t)

ej2πf0t. (2.8)

To mitigate the effect of the frequency offset, the Automatic Frequency Control(AFC) and the frequency offset compensation are used. The AFC responsiblefor keeping the frequency offset between ±50 Hz, and the remaining frequencyoffset will be reduced by frequency offset compensation.

Moreover, in order to capture the effects of directional scattering, a para-metric model for the non uniform AOA distribution can be applied. It is calledvon Mises distribution:

p(θ) =1

2πI0(κ)eκ cos(θ−α), (2.9)

where In(κ) is the nth order modified Bessel function of the first kind, κ isthe beam width and α denotes the angle between the average scattering direc-tion and the mobile direction. More detailed explanation can be found in [16][20] [21].

2.3.2 Autocorrelation Functions

Without loss of generality, the following derivation will be based on the Ricianfading channel model with frequency offset and non isotropic distribution ofAOA. The autocorrelation functions of the channel respond is given by [20]:

Rh(τ) = E[h(t)h∗(t− τ)]

=

(σh

2

K + 1Rhm

(τ) +Kσh

2

K + 1RhLOS

(τ)

)ej2πf0τ ,

(2.10)

with

Rhm(τ) =J0

(√−κ2 + (2πfDτ)2 − 4jκ cos(α)πfDτ

)I0(κ)

(2.11)

andRhLOS

(τ) = exp(j2πfDτ cos(θ0)), (2.12)

where J0(·) is the zeroth order Bessel function of the first kind. Note thatEquation (2.10) is complex valued in general, but it is real whenK = κ = f0 = 0,in which case the AOA of scatted component is omnidirectional and uniform,no frequency offset, and results in the Rayleigh model with [16] [8]

Rh(τ) = σh2J0(2πfDτ). (2.13)

8

Moreover, the zeroth order Bessel function of the first kind J0(z) is given by

J0(x) =1

2π

∫ 2π

0

eix cos τdτ =

∞∑k=0

(−1)k

(k!)2

(x2

)2k

. (2.14)

2.3.3 Power Spectrum

The Power Spectral Density (PSD) of channel response can be calculated by theFourier transformation of the correlation functions, which is also called Dopplerspectrum, which is given in [17] and [20]:

Ph(f) =σh

2

K + 1Phm

(f − f0) +Kσh

2

K + 1PhLOS

(f − f0) (2.15)

with

Phm(f) =

exp(κf cos(α)fD

) cosh(κ sin(α)√

1− ( ffD

)2)

πI0(k)√

1− ( ffD

)2(2.16)

andPhLOS

(f) = δ(f − fD cos(θ0)). (2.17)

When K = κ = f0 = 0, the spectrum has a high density at plus and minusthe maximum Doppler spread, which can be written as

Ph(f) = σh2 1

π√

1− ( ffD

)2. (2.18)

It is called the Klarke’s or Jakes’ spectrum.

2.4 Structure of DPCCH in WCDMA Uplink

Since the investigation in this report focuses on the WCDMA uplink, the back-ground about the uplink dedicated channel is presented in this section.

Figure 2.6: The structure of the uplink DPDCH and DPCCH.

The structure of the uplink dedicated channels are shown in Figure 2.6. Itconsists of the DPDCH and the DPCCH, which are in-phase (I)/ quadrature

9

(Q) code multiplexed. The DPDCH carries higher layer information includ-ing the user data, whereas the DPCCH is used to carry physical layer controlinformation.

Each frame of length 10 ms is divided into 15 slots. There are 10 bitsper DPCCH slot and 10 to 640 bits per DPDCH slot. Each DPCCH slot hasfour fields to be used for pilots bits, the Transport Format Combination Indi-cator (TFCI), the FeedBack Information (FBI) and the Transmit Power Con-trol (TPC). The pilot bits are used for the channel estimation in the receiver andthe TFCI indicates the transport format combination of the transport channelsmapped to DPDCH. The FBI bits are used for transmit diversity and the TPCbits carry the power control commands. The exact number of bits of differentfields depends on the slot format, which can be found in [22].

In this thesis, the pilot bits are used to estimate the channel response asthe input of all the selected algorithms, the detailed estimation is given in nextchapter.

10

Chapter 3

Estimation Algorithms

In this chapter, four Doppler spread estimation algorithms will be described.All of these methods require the estimation of the channel response as input,thus the following assumption is applied.

As described in Chapter 2, the pilot symbols in uplink DPCCH are usedfor the channel estimation in the receiver. However, it is difficult to use thecontinuous pilot symbols with more than one slot length in the Doppler spreadestimation since the pilot bits are not continuously, and only 3 to 8 pilot bitsexist in each slot. Therefore the average received pilot symbols over one slot isused as one sample of channel response estimation, denoted by

y[n] = h[n] =1

Np

Np∑i=1

yp[i], (3.1)

where yp are the pilot bits in each slot and Np is the number of pilot symbols inone slot. Then the sampling period Ts for the channel impulse response estima-tion is equal to the slot length, Ts = 2

3 ms≈ 6.7×10−4 s. Moreover, assuming theobtained channel estimation is a block of N samples, i.e. y[0], y[1], · · · , y[N−1].And the value of maximum Doppler spread is assumed to be stable for these Nsamples. Noting that [·] is used to represent time discrete symbols.

One thing needs to mention here is that all the estimators are designed basedon Rayleigh fading channel.

3.1 Level Crossing Rate Estimator

The level crossing rate estimator calculates the Doppler spread by using theLCR of the channel response. The LCR is defined as the number of timesthat the envelope of channel response crosses a certain level Lc with positive ornegative slope during a time interval Tmeas . It is well known [1] [3] that LCRis proportional to the maximum Doppler spread. In case of a Rayleigh fadingchannel, the relation between LCR and the maximum Doppler spread is givenby [16]

LCR =Npc

Tmeas= fD

√2πLnce

−L2nc , (3.2)

11

where Npc is the number of positive crossing of the channel response envelope,Lnc ∈ [0, 1] is the normalized level. The above equation is a function of normal-ized level Lnc , which is normalized to the root of the average power, i.e.

Lnc =Lc√∑N−1

n=0 |y[n]|2N

, (3.3)

where y[n] is the channel response estimation and N is the number of channelestimation samples.

xxxxxx xx x

Figure 3.1: Level crossing example.

It can be calculated from Equation (3.2), the maximum level crossing rateis obtained for Lnc = 1√

2. And the author in [3] proposed that the normalized

level equals to one is roughly independent of Rician factor. So both of these twolevels tested for comparison in this report, i.e. one level is to use the square rootof the half average power as level, the other one uses the root mean square valueof the received signal as level. Figure 3.1 shows the envelop of channel responsewith two different levels. If take the red line as an example, the green circlesmarked as the crossings with positive slope, while the orange crossed representthe crossings with negative slope.

Then the maximum Doppler spread can be estimated by

fD =

{Npc

Tmeas

√eπ Lnc = 1√

2Npc

Tmeas

e√2π

Lnc = 1, (3.4)

the corresponding level Lc is given by

Lc =

√∑N−1

n=0 |y[n]|22N Lnc = 1√

2√∑N−1n=0 |y[n]|2

N Lnc = 1. (3.5)

12

Here counting the number of crossing is achieved by compare the absolute valueof the channel response samples with the level. If the value of sample smallerthan the level, and the value of next sample is larger than the level, it can beseen as one time crossing with positive slope. On the contrary, if the value ofsample larger than the level, and the value of next sample is smaller than thelevel, then it is considered as a crossing with negative slope.

3.2 Hybrid Estimator

A hybrid Doppler spread estimation algorithm is proposed in [23]. It is basedon autocorrelation functions of the channel estimate, and consists of two steps.The first step is to determine whether the Doppler spread is “small” or “large”.The second step is to estimate the exact value of the maximum Doppler spreadusing either a curvature method (for small Doppler spread) as proposed in [20]or a first zero detection method (for large Doppler spread) as proposed in [24].The reason behind using a two step approach depends on the fact that thecurvature estimator is accurate only at low speeds but cannot achieve enoughaccuracy at high speeds, and the first zero detection estimator is more suitablefor high speeds than low speeds.

The following sections will begin with the description of the two estimationmethods in Section 3.2.1 and 3.2.2, then the hybrid estimation method will bepresented in Section 3.2.3.

3.2.1 Curvature Estimator (small Doppler spread)

In [20], the author proposes that the Doppler spread can be calculated as:

fD =1

2π

√−2Ry ′′[0]

Ry[0]. (3.6)

It requires the knowledge of autocorrelation functions Ry and the secondderivative of autocorrelation functions Ry

′′ at lag zero, i.e. Ry[0] and Ry′′[0].

Here Ry[0] can be estimated directly from the channel estimate, and Ry′′[0] can

be obtained by the polynomial fitting of Ry. The polynomial fitting algorithmis described as below.

Polynomial Fitting Algorithm

Polynomial fitting algorithm models variables as an nth order polynomial. Thepolynomial coefficients can be solved by using the least-squares method.

In the curvature estimator [23], both quadratic and fourth order fitting areconsidered. The goal is to minimize

arg minai

L∑k=1

|Ry[k]−∑i

aiki|2, (3.7)

with respect to ai, where i = 0, 1, 2 for a second order polynomial and i = 0, 2, 4for a fourth order polynomial, and L is the maximum lag of autocorrelationfunctions. More detailed explanation can be found in [20].

13

The polynomial fitting algorithm can also be described by a matrix. As-suming the autocorrelation vector r = [r1, r2, · · · , rL]T, where rk = Ry[k], (·)T

denotes matrix transpose, a is coefficient vector with elements ai and H is amatrix such that Ha =

∑i aik

i for all k from 1 to L. Then Equation (3.7) canbe written as ‖r−Ha‖2.

When the second degree polynomial is applied, define the vector a = [a0, a1, a2]T,H is given by

H =

1 1 11 2 22

......

...1 L L2

.

When the fourth degree polynomial is applied, the vector a becomes [a0, a2, a4]T,and H should have the following form instead [23]:

H =

1 1 11 22 24

......

...1 L2 L4

.

For these two approximations, the polynomial coefficients vector a is calcu-lated by [25]

a = (HTH)−1HTr. (3.8)

Since the nth order derivative of autocorrelation functions can be obtainedby [20]

R(n)y [0] = n!an/T

ns , n = 0, 2, (3.9)

only a0 and a2 are needed to estimate the maximum Doppler spread, such that(3.6) can be simplified to

fD =1

Tsπ

√−a2

a0. (3.10)

3.2.2 First Zero Detection Estimator (large Doppler spread)

In [24], first zero detection estimator detects the first zero crossing point ofthe autocorrelation functions, and relates this point with the Bessel function toestimate the maximum Doppler spread.

As described in Section 2.3.2, in Rayleigh fading model, the autocorrelationfunctions can be expressed as

Ry [k] = σ2yJ0 [2πfDk], (3.11)

where σ2y is the power of channel estimate, J0 is the zeroth order Bessel function

of the first kind. If the Equation (3.2.2) is setted equal to zero, i.e.

Ry[τ0 ] = J0[2πfdτ0 ] = J0 [η0 ] = 0, (3.12)

14

0 10 20 30 40 50 60 70

−400

−200

0

200

400

k

Ryy[k

]

8.5 9 9.5 10 10.5 11 11.5 12 12.5−150

−100

−50

0

50

100

150

( k1, Ryy[k1] )

( k2, Ryy[k2] )

τ0/Ts

k

Ryy[k

]

Figure 3.2: Illustration of linear interpolation.

where τ0 is the first zero crossing of autocorrelation functions and η0 is definedas the first zero point of Bessel function, therefore the relation between themaximum Doppler spread fD and τ0 can be represented by

2πfDτ0 = η0 , (3.13)

fD =η0

2πτ0.

Substituting η0 = 2.40 and π = 3.14 into Equation 3.2.2 yields

fD ≈0.38

τ0. (3.14)

Since the autocorrelation function estimated from simulation is discrete intime, τ0 can be obtained by the linear interpolation as described in the followingsection.

15

Linear Interpolation Method

Linear interpolation calculates the zero-crossing point of autocorrelation func-tions using first two adjacent samples of the Ry that have positive and negativesign, respectively. A realization of the autocorrelation functions is shown in theupper figure of Figure 3.2 and the lower one is the enlarged samples aroundfirst zero crossing point (in the red rectangular), where the two blue pointsare samples of the autocorrelation function with coordinates (k1, Ry[k1]) and(k2, Ry[k2]). The straight line represents the linear interpolation between thesetwo points, and the red dot in the middle corresponds to the first zero crossingwith coordinates (τ0 , Ry[τ0/Ts ]). With a straight line approximation, the valueof τ0 can be calculated from

Ry[k1]−Ry[k2])

k2 − k1=Ry[τ0/Ts ]−Ry[k2]

k2 − τ0/Ts. (3.15)

Substituting Ry[τ0/Ts ] = 0 into (3.15) and solve the equation, gives

τ0 = k2Ts +(k2 − k1)Ry[k2]Ts

Ry[k1]−Ry[k2]. (3.16)

It shows that τ0 can be more accurate if the resolution of autocorrelationsamples around zero crossing points is increased.

3.2.3 Hybrid Estimator

The hybrid estimation algorithm is based on estimated autocorrelation func-tions to decide which method will be used, i.e. curvature estimator or first zerodetection estimator. So the first step of this method is to estimate the autocor-relation functions. The real part of the autocorrelation functions is considered,since the correlation function in Equation (3.2.2) is real [23]. In order to im-prove the accuracy, the author in [23] suggests to estimate the autocorrelationfunctions using all detected fingers instead of only one finger:

Ryq [k] =1

N − k

N−k∑n=1

R{y∗q [n]yq[n+ k]}, (3.17)

where Ryq denotes the autocorrelation functions for path q, k = 0, 1, · · · , L, andR{·} denotes the operation for taking the real part.

Then sum Ryq [k] for all fingers to form a composite autocorrelation func-tions:

Ry[k] =∑q

Ryq [k]. (3.18)

In order to decide which estimator should be used, the smallest kmin isneeded to obtain such that Ry[kmin] is negative, and compare it with a prede-fined threshold T , where T ≤ L.

If kmin > T or if Ry[k] is positive for all k, then choose curvature estimatoras described in Section 3.2.1.

If kmin < T , then choose first zero detection estimator as described inSection 3.2.2.

16

3.3 Power Integration Estimator

The method proposed in [8], referred to as power integration estimator in therest of the paper, is based on a nonparametric estimation of the power spectrumdensity. The Doppler spectrum is bandlimited in principle, since the maximumDoppler frequency is proportional to the finite mobile velocity. However, thetotal spectrum also contains noise spectrum in the simulator. Therefore, ifthe PSD of the received signal is obtained, it can be expected that a highpercentage of the total power would lie in the frequency range [−fD, fD]. Itmeans the integral of the estimated PSD within the frequency range [−fD, fD]should occupy a certain percentage of the total power, as shown in Figure 3.3.In [8], fD is calculated as the minimum frequency fD that satisfies∫ fD

−fDPyy(f)βdf∫ 1/2Ts

−1/2TsPyy(f)βdf

> ψ, (3.19)

where Pyy(f) is the estimated PSD of received signal, β is a positive exponent,ψ is a threshold that satisfies 0 < ψ < 1. The practical implementation ofthe algorithm is described in the following. A periodogram-based method is

-1/2Ts 1/2Ts-fD fD

Figure 3.3: Illustration of power integration estimator.

proposed in [8] to estimate PSD based on a block of observed channel samples.For a length-N block, this estimator is given by

Py(f) =TsN

∣∣∣∣∣N−1∑n=0

y[n]exp(−j2πfnTs)

∣∣∣∣∣2

. (3.20)

In practice, Pyy(f) can be evaluated by using a finite number of frequency

points, which can be seen as a sampling of Pyy(f). The periodogram estimator

17

0 200 400 600 800 1,000 1,200 1,4000

50

100

150

200

250

300

350

Frequency

Continuous PSD

Discrete PSD

Figure 3.4: The Continuous and Discrete form of Theoretical Periodogram PSD.

considered in the report is Nfft -point PSD estimate

Pyy[fk] =Ts

N

∣∣∣∣∣∣Nfft−1∑n=0

y[n]exp(−j2πfknTs)

∣∣∣∣∣∣2

=Ts

N

∣∣∣∣∣∣Nfft−1∑n=0

y[n]exp(−j2πnk/M)

∣∣∣∣∣∣2

, (3.21)

where fk = k/(NfftTs), k = 0, 1, · · · , Nfft − 1. The continuous and discrete formof periodogram estimated PSD is shown in Figure 3.4.

The Discrete Fourier Transform (DFT) of N -point y[n] can be given by

Y [k] = DFT{y[n]} =

N−1∑n=0

y[n]exp(−j2πnk/N), (3.22)

which can be computed efficiently using an Nfft -point Fast Fourier Transform(FFT), and where Nfft is a power of two and larger than the block size N .In this case, y[n] must be zero-padded with Nfft − N zeros prior to the FFTprocessing. Equation (3.21) can be seen as the squared modulus of a Nfft -pointFFT

Pyy[fk] =TsN|FFT{y[n], Nfft}|2 . (3.23)

Considering the periodicity property of the periodogram estimator and usingthe discrete samples of PSD, Equation (3.19) can be transformed to the followingform. The goal is to find the minimum value of the index p that satisfies

Pyy[f0] +∑pk=1

[Pyy[fk] + Pyy[fNfft−k]

]∑Nfft−1k=0 Pyy[fk]

> ψ, (3.24)

18

and the estimation of Doppler spread fD can be obtained by

fD =p

NfftTs. (3.25)

3.4 Frequency Domain Maximum Likelihood Es-timator

An ML Doppler spread estimation algorithm in frequency domain is describedin [10]. The basic idea is to use an estimated spectrum to approximate theshape of theoretical Jakes’ spectrum. Furthermore, the Whittle approximationis applied in the likelihood function to estimate the maximum Doppler spread.

For better explanation, the estimated power spectrum from the fading chan-nel and ideal Jakes’ spectrum model are given as follow.

Based on the channel estimate, power spectrum Sy[fn] can be estimated byapplying periodogram approach as described in Section 3.3:

Sy [fn] = Pyy[fn]× Fs =Pyy[fn]

Ts

=1

N|FFT{y[n], Nfft}|2 , (3.26)

where fn = fkFs

= nNfft

, n = 0, 1, · · · , Nfft − 1 is the normalized frequency with

sampling frequency Fs for an Nfft -point FFT, Ts is the sampling period.The ideal Jakes’ spectrum is given by [10]

c(f ; fD) =

1√

f2D−f2

0 ≤ f < fD

1√f2D−(1−f)2

1− fD < f ≤ 1. (3.27)

Taking the power of channel estimate and noise into consideration, the powerspectrum can be calculated from

SJ [fn; fd] = σ2yc[fn; fd ] + σ2

n

= σ2n(γc[fn; fd] + 1), (3.28)

where fd = fDFs

is the normalized maximum Doppler spread, σ2y and σ2

n denotethe power of received signal and noise respectively, and the SNR is given by

γ =σ2

y

σ2n.

According to [10], Whittle approximation can be applied to the log-likelihoodwhen Nfft is large. The ML function proposed in [10] is given by:

Γ ≈ −Nfft lnπ −Nfft−1∑n=0

[ln(SJ [fn; fd]) +

Sy [fn]

SJ [fn; fd]

]. (3.29)

One can see that except for the estimated power spectrum of channel esti-mates, the knowledge of received SNR γ and noise variance σ2

n are also needed

19

in Equation (3.29). From (3.28), the noise variance σ2n can be written as

σ2n =

SJ [fn; fd]

γc[fn; fd] + 1

≈ 1

Nfft

Nfft−1∑n=0

(Sy [fn]

γc[fn; fd] + 1

).

(3.30)

Substituting the above equation into the goal function (3.29) and neglectingconstant terms yields the concentrated likelihood function (the detailed deriva-tion is shown in Appendix A)

Γc = −Nfft ln

Nfft−1∑n=0

Sy [fn]

γc[fn; fd] + 1

− Nfft−1∑n=0

ln [γc[fn; fd] + 1] (3.31)

to be maximized with respect to fd , where fd ∈ (0, 1/2] is the normalizedfrequency.

Finally, the estimation of maximum Doppler spread is obtained by

fD = fdFs . (3.32)

The selection of the suitable choice for the parameters in all the algorithmsare discussed in next chapter, and the analysis of their effects are presented aswell.

20

Chapter 4

Simulation Results

In this chapter, the simulation results for four Doppler spread estimation algo-rithms will be presented. Section 4.1 gives a general description of the simulationenvironment. In Section 4.2, the simulation results for each algorithm will beprovided and analysed individually.

4.1 Simulation Environment

The simulation framework is implemented by Ericsson. It consists of user equip-ment, radio channel and antennas, demodulator and decoder. Since the inves-tigation in this report focuses on the uplink in WCDMA systems, the proposedalgorithms are implemented in the BCL which is a reference model of WCDMAuplink baseband processing. The simulation is executed by Linux servers.

Both Rayleigh fading and Rician fading channel model are considered inthis report. The common configuration for these two scenarios is to turn off theTPC and the Automatic Gain Control (AGC). The number of channel samplesN = 150 is used in each algorithm, and 8000 frames are simulated, the length ofFFT is set to 512 points. As described in the previous chapter, the maximumvalue of Doppler spread is assumed to be stable for each 150 slots, which impliesthe channel is also stable for each 150 slots. The empirical channel model usedto model the Rayleigh fading channel in the simulation is Pedestrian A” model,which is specified in ITU-R recommendation M.1225 [26]. The specific delayand power for each multipath component are listed in Table 4.1. Most of thesimulations in this report only depend on the first tap.

TapRelative delay

(ns)Average power

(dB)1 0 02 110 -9.73 190 -19.24 410 -22.8

Table 4.1: Pedestrian A Channel Model

Noting that in this chapter, only the simulation results based on Rayleighfading channel are presented. The results under Rician fading scenario will be

21

discussed in the next chapter for comparison.

4.2 Simulation Results

The four estimation algorithms described in this report are evaluated in thissection. The evaluation concentrates on the performance of every estimator andthe selection of optimum parameters used in further comparisons. To investigatethe accuracy of the estimation results over the velocity range of 2.7-162 km/h(corresponding the theoretical Doppler spread is 5-300 Hz), the mean value andthe Normalized Mean Square Error (NMSE) of estimated maximum Dopplerspread are shown as a function of theoretical Doppler spread with SNR = 15dB. The reason of choosing mean value and NMSE as the metrics is that themean value shows an intuitive estimation result compared to the true value ofmaximum Doppler spread, while the NMSE not only indicates the bias andvariation of the estimation results, but also shows the relative error comparedto the theoretical value.

NMSE{fD} = E{( fD − fDfD

)2} (4.1)

In addition, the effect of the noise is also evaluated for several level of veloc-ities. Here the value of SNR is from 0 to 15 dB, and fD = 10, 80, 160, 240 Hz isused in the simulation. The quality of fD is mainly measured by NMSE.

4.2.1 Level Crossing Rate Estimator

The simulation result of LCR estimator is presented in this section. In thesimulation, two levels are tested, i.e. Lnc = 1√

2and Lnc = 1. Since the

estimation result of the Doppler spread is not only related to the number ofcrossing, it is also related to the measurement time. It means if the measurementtime is very short, even though the different between the number of cross withpositive slope and negative slope is only one or zero, one different crossing willresults in a large difference in the estimation result. So for each level, notonly the crossing with positive slope, but also the crossing with negative slope,and the average number of crossings of the two previous cases are considered.Therefore, there are totally six estimators to estimate the maximum Dopplerspread.

Effect of Speed

Figure 4.1 demonstrates the mean value of the estimated maximum Dopplerspread. There should be six lines in the figure, but they overlap with each otherfor the same level, so only two curves can be observed. It implies there is nobig difference between different crossing count methods when the large numberof simulation applies, which shows an agreement of the theory. Moreover, thecurve with normalized level Lnc = 1√

2is more accurate than the other one,

especially for the low speed values as shown in Figure 4.1 (b).However, both two estimators have larger bias at low velocities. In this

report, each crossing increased the estimate by 9.3 Hz for Lnc = 1√2

and 10.8

Hz for Lnc = 1. Comparing to high speed, one more miscount will cause larger

22

percentage of error for low speed. This is one of the reason that why thismethod is biased at low velocities. Another reason is that the channel responsefor the low velocity will change slowly, it is easier influenced by noise and thenintroduces additional crossing.

0 50 100 150 200 250 30050

100

150

200

250

300

True maximum Doppler spread [Hz]

Mea

nes

tim

ate

maxim

um

dopple

rsp

read

[Hz]

Lnc=1/√

2 posi

Lnc=1/√

2 nega

Lnc=1/√

2 mean

Lnc=1 posi

Lnc=1 nega

Lnc=1 mean

(a) Regular speed

6 8 10 12 14 16 18 20 22 24

60

70

80

90

100


Mea

nes

tim

ate

maxim

um

dopple

rsp

read

[Hz]

Lnc=1/√

2 posi

Lnc=1/√

2 nega

Lnc=1/√

2 mean

Lnc=1 posi

Lnc=1 nega

Lnc=1 mean

(b) Low speed

Figure 4.1: The estimated maximum Doppler spread of LCR method with dif-ferent levels,versus the theoretical maximum Doppler spread.

Figure 4.2 illustrates the NMSE of fD over the range of 5-300 Hz. It can beseen that estimator with Lnc = 1√

2has lower NMSE, providing more reliable

Doppler spread estimate than the estimator with Lnc = 1. Consequently, onlythe mean crossing count method with Lnc = 1√

2is chosen for comparison.

0 50 100 150 200 250 30010−2

10−1

100

101

102

103


Norm

alize

dM

SE

Lnc=1/√

2 posi

Lnc=1/√

2 nega

Lnc=1/√

2 mean

Lnc=1 posi

Lnc=1 nega

Lnc=1 mean

(a) Regular speed

6 8 10 12 14 16 18 20 22 24100

101

102

103


Norm

alize

dM

SE

Lnc=1/√

2 posi

Lnc=1/√

2 nega

Lnc=1/√

2 mean

Lnc=1 posi

Lnc=1 nega

Lnc=1 mean

(b) Low speed

Figure 4.2: The NMSE of LCR method with different levels,versus the theoret-ical maximum Doppler spread.

Effect of Noise

Figure 4.3 shows the NMSE of estimated maximum Doppler spread with twodifferent levels of various Rayleigh fading channel. As expected, it is found thatfor all scenarios the NMSE becomes lower as the SNR increases, which meansLCR method is sensitive to the noise. In addition, one can draw the sameconclusion with the previous part that LCR method is biased at low velocities.Moreover, the results with Lnc = 1√

2always provides better performance than

23

the other one with Lnc = 1 when the SNR is above 11 dB. It can explain whynormalized level to 1√

2is chosen in the following comparison.

0 2 4 6 8 10 12 1410−2

10−1

100

101

102

103

SNR [dB]

Norm

alize

dM

SE

Lnc=1/√

2 fD=10 Hz

Lnc=1/√

2 fD=80 Hz

Lnc=1/√

2 fD=160 Hz

Lnc=1/√

2 fD=240 Hz

Lnc=1 fD=10 Hz

Lnc=1 fD=80 Hz

Lnc=1 fD=160 Hz

Lnc=1 fD=240 Hz

Figure 4.3: The NMSE of LCR method with different levels, versus SNR forfour speed values.


The hybrid estimator uses either curvature method or first zero detection methodfor estimation. In the implementation process, some modification for these twomethods are made.

For curvature estimator, when the value under the square root is negativein Equation (3.10), it’s invalid to calculate the estimated Doppler spread. So inthis case, the result is setted to equal to the previous calculated value. If thevalue under the square root is negative at the first time calculation, the initialestimated maximum Doppler spread value is set to zero. This method is appliedto avoid the potential error.

For the first zero detection estimator, the first zero crossing is required, i.e.the first negative value point of autocorrelation functions. However, the firstzero crossing point will be large when the speed becomes low, which impliesthat if the maximum lag L is too small, the crossing point can not be found. Inorder to evaluate the hybrid estimator, the performance of curvature methodand first zero detection method are also presented in the report, over the entirevelocity range. Theoretically when the speed under 30 km/h, correspondingmaximum Doppler spread is 56 Hz, the first negative point of autocorrelationfunctions is larger than 10, which is the maximum lag used in the simulation.In this case, kmin can not be found, and it is impossible to calculate τ0 , thenthe estimated maximum Doppler spread by first zero detection is set to be equalto the previous estimated value. Moreover, if kmin can not be found at the firsttime, then set fD = 0.

In this report, we fix the parameters to L = 10, T = 7. Applying thetheoretical derivation, if τ0 = 7Ts , the maximum Doppler spread fD ≈ 81 Hz,

24

which implies the switching of algorithms at 81 Hz. In order to compare thismethod with the others, we consider only one finger in this algorithm, which isdifferent compared to [23].

As described in Section 3.2.1, there are two kinds of polynomial fitting.Therefore, these two approximation approach are discussed as shown in Fig-ure 4.4. Figure 4.4 (a) shows that the fourth degree approach has a largerrange than the second degree approach. When the maximum Doppler spreadis smaller than 100 Hz, fourth degree gives a good estimation, while both ofthese two lines are accurate only when fD < 40 Hz. However, if we focus onFigure 4.4 (b), where the results for low speed are presented. We find that thesecond order approximation is more accurate than the fourth order one at lowvelocities. Since the threshold is set to around 81 Hz, fourth degree polynomialfitting is chosen to use as curvature estimator.

0 50 100 150 200 250 3000

50

100

150

200

250

300


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imate

maxim

um

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rsp

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[Hz]

degree2

degree4

(a) Regular speed

6 8 10 12 14 16 18 20 22 24

5

10

15

20


Est

imate

maxim

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Dopple

rsp

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[Hz]

degree2

degree4

(b) Low speed

Figure 4.4: The comparison of two polynomial fitting algorithms.

Effect of Speed

Figure 4.5 illustrates the mean value of estimated maximum Doppler spreadwith different speeds for previous described three algorithms. It can be observedthat the hybrid estimator gives an unbiased estimation result. We can see fromFigure 4.5, it chooses curvature estimator when fD ≤ 60 Hz, otherwise it choosesfirst zero detection estimator. It confirms switching algorithms in a proper way.It is worth mentioning that the first zero detection method is not able to givereasonable result at low velocities, since it results in the initial value as describedearlier.

Again, as shown in Figure 4.6, where the NMSE of estimated maximumDoppler spread is presented, the Hybrid estimator is a good combination ofthese two methods since it always results in lower NMSE.

Effect of Noise

The estimated results for different SNR are shown in Figure 4.7. It shows therobustness against noise especially for the higher velocity values. As the speedincreases, the performance goes better. However, there is no big differencebetween the NMSEs of the medium and high speeds.

25

0 50 100 150 200 250 3000

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(a) Regular speed

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[Hz]

degree4

first zero

hybrid

(b) Low speed

Figure 4.5: The estimated maximum Doppler spread of Hybrid method,versusthe theoretical maximum Doppler spread.

0 50 100 150 200 250 30010−3

10−2

10−1

100

101


Norm

alize

dM

SE

degree4

first zero

hybrid

(a) Regular speed

6 8 10 12 14 16 18 20 22 2410−2

10−1

100

101

102


Norm

alize

dM

SE

degree4

first zero

hybrid

(b) Low speed

Figure 4.6: The NMSE of estimated maximum Doppler spread of Hybridmethod,versus the theoretical maximum Doppler spread.

Figure 4.8 shows the comparison between different numbers of received pathsused in the calculation of the autocorrelation functions. It can be observed thatthe usage of two paths gives a better performance. It is more obvious for thehigh speed values. However, in order to have the same parameter setting withother algorithms, the single path case is considered in the following report.

4.2.3 Power Integration Estimation

In order to demonstrate the effect of the value of exponent β and threshold ψ,Figure 4.9 is presented. It shows the NMSE of four speed values with β = 1, 2.On the one hand, β = 2 provides better estimation performance. The differencebetween β = 1 and β = 2 for the same speed becomes smaller as speed increases.On the other hand, the optimum choice of threshold varies with the actualDoppler spread. For instance, ψ = 0.75 is the best when fD = 10 Hz, ψ = 0.9is the best when fD = 240 Hz. For the rest of the simulations, the parametersare fixed to β = 2, ψ = 0.9 as suggested in [8].

26

0 2 4 6 8 10 12 1410−3

10−2

10−1

100

SNR [dB]

Norm

alize

dM

SE

fD = 10 Hz

fD = 80 Hz

fD = 160 Hz

fD = 240 Hz

Figure 4.7: The NMSE of Hybrid method with four speed values, versus SNR.

0 50 100 150 200 250 30010−3

10−2

10−1

100

101


Norm

alize

dM

SE

one path

two paths

Figure 4.8: The NMSE of Hybrid Estimator for different number of used paths,versus theoretical maximum Doppler spread.

Effect of Speed

Figure 4.10 illustrates the mean value of fD under various Doppler spreads. Itprovides a reliable and unbiased estimation result. In theory, the high speedpart should have the better performance than low speeds since the chosen valueof parameter is more suitable to high speeds, Figure 4.10 shows good agreementwith it. The enlarged figure for low velocities has a correct tendency but withvariance. It is presented more clearly in Figure 4.11, considering the fact ofaccuracy mean value, the reason for the fluctuation of the curves in two figuresis their high variance.

In order to find out the reason of this high variance, the estimation results

27

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.9510−3

10−2

10−1

100

101

102

Threshold

Norm

alize

dM

SE

fD = 10Hz, β =1

fD = 10Hz, β =2

fD = 80Hz, β =1

fD = 80Hz, β =2

fD = 160Hz, β =1

fD = 160Hz, β =2

fD = 240Hz, β =1

fD = 240Hz, β =2

Figure 4.9: The NMSE of Power Integration Estimator with β = 1, 2, versusthreshold ψ.

0 50 100 150 200 250 3000

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[Hz]

(b) Low speed

Figure 4.10: The estimated maximum Doppler spread of Power Integrationmethod, versus the theoretical maximum Doppler spread.

with different number of frames are simulated, as shown in Figure 4.12. Asexpected, the usage of longer simulation provides better performance in termsof mean value. But the NMSE does not give a reasonable result. That is becauseof the outlier during the simulation. However, we haven’t been able to give areasonable explanation from analysis due to the time constraint. It will be putinto the future work of the investigation.

Effect of Noise

Based on Figure 4.13, one can see that the curves are unstable for four speedvalues, but it becomes better as the velocity increases.

If we investigate the Normalized Bias (NB) and the Normalized Standard

28

0 50 100 150 200 250 30010−3

10−2

10−1

100


Norm

alize

dM

SE

(a) Regular speed

6 8 10 12 14 16 18 20 22 2410−2

10−1

100

101


Norm

alize

dM

SE

(b) Low speed

Figure 4.11: The NMSE of estimated maximum Doppler spread of Power Inte-gration method,versus the theoretical maximum Doppler spread.

0 50 100 150 200 250 3000

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Mea

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dopple

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[Hz]

8000 frames

1000 frames

(a) Regular speed

0 50 100 150 200 250 30010−3

10−2

10−1

100

101


Norm

alize

dM

SE

8000 frames

1000 frames

(b) Low speed

Figure 4.12: The comparison of the number of frames used in simulation forPower Integration method, versus the theoretical maximum Doppler spread.

Deviation (NSD), which are given by

NB = E[fD/fD − 1] (4.2)

NSD = E[(fD/E[fD ]− 1)2]12

, (4.3)

as shown in Figure 4.14. NB reflects the error between the estimated result andthe theoretical value, and NSD indicates the variance or dispersion from theaverage value of the estimation. It can be seen that the NB for the low Dopplerspread is degraded when the SNR is below 3 dB, whereas it is approximatelystable for SNR > 3 dB. While the NBs for the medium and high Dopplerspread are close to zero, which implies the errors for this area are at a low level.Moreover, it is not hard to observe that the shape of curves in NSD is verysimilar to that in Figure 4.13. Thus, the dominant factor for the unacceptableresults of Figure 4.13 should be the high variance of the estimation. Comparingto this high variance, noise does not give a significant effect to the results.

29

0 2 4 6 8 10 12 1410−3

10−2

10−1

100

101

102

SNR [dB]

Norm

alize

dM

SE

fD = 10 Hz

fD = 80 Hz

fD = 160 Hz

fD = 240 Hz

Figure 4.13: The NMSE of Power Integration method with four speed values,versus SNR.

0 2 4 6 8 10 12 14

0

0.1

0.2

0.3

0.4

SNR [dB]

Nor

mal

ized

Bia

s

fD = 10 HzfD = 80 HzfD = 160 HzfD = 240 Hz

(a) Normalized Bias

0 2 4 6 8 10 12 1410−2

10−1

100

101

SNR [dB]

Nor

mal

ized

Sta

ndar

ddev

iati

oner

ror

fD = 10 HzfD = 80 HzfD = 160 HzfD = 240 Hz

(b) Normalized Standard Deviation

Figure 4.14: The normalized bias and normalized standard deviation of PowerIntegration method with four speed values, versus SNR.

4.2.4 Frequency Domain ML Estimator

Among all the factors, the accuracy of estimation results of this algorithmmainly depends on two factors. The first one is the shape of the estimatedDoppler spectrum. The other factor is the resolution of frequency estimation,i.e. the interval between two hypothesis values of Doppler spread fd. Theo-retically more accurate result can be obtained by increasing the resolution, butthe computational complexity will be increased as well. There is a trade-offbetween accuracy and computational cost. For instance, if assuming the sam-pling frequency is set to be 1000 Hz, and 5 Hz is selected as the interval length,the algorithm needs to calculate the concentrated likelihood function for 200times. Whereas if 20 Hz is selected, only 50 times calculation are required.Thus, several resolution values are tested in the simulation in order to get rel-atively reliable estimation result without increasing too much computational

30

complexity.

Effect of Speed

As mentioned in Section 4.2.1, the estimation resolution of the LCR method isaround 10 Hz for both of two different levels. And the resolution of the powerintegration estimator can be calculated from Equation 3.25. The minimumpossible interval can be distinguished by the method is 2.93 Hz. Moreover,in order to consider a wider range, four values between 2 Hz and 20 Hz arechosen for evaluation, which are 2 Hz, 5 Hz, 10 Hz and 20 Hz. Figure 4.15demonstrates the estimation results with four resolutions. It is shown that thesmaller interval does not result in a significant improvement of performance.One can also observe that the simulation results have a bias around 40 Hz.Since the error is even larger than the maximum tested resolution value, i.e. 20Hz, then the conclusion can be drawn that the overestimate is not caused bythe inappropriate selection of resolution.

0 50 100 150 200 250 30050

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resolution=20Hzresolution=10Hzresolution=5Hzresolution=2Hz

(a) Regular speed

6 8 10 12 14 16 18 20 22 2456

58

60

62

64


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imum

Dop

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[Hz]


(b) Low speed

Figure 4.15: The estimated maximum Doppler spread of ML method,versus thetheoretical maximum Doppler spread.

If we turn to consider another factor, see Figure 4.16, where the left plotshows the estimated spectrum based on the channel response and the right plotrepresents the ideal Jakes’ spectrum when fD = 210 Hz. It is clear to seethat the estimated spectrum does not have the ideal shape, which is affectedby noise seriously. Therefore, there are two possible reasons for the bias ofML estimation result. The first one is the inaccuracy of the estimation forthe Doppler spectrum. This can be improved by increasing the number ofobtained channel estimate or increasing the FFT length. The second reasonis that BCL models the practical environment, and it is not suitable to usepractical spectrum approximate the theoretical one. If the ideal channel modelis used in the simulation, it will result in a more accurate performance.

Figure 4.17 shows the NMSE of the ML estimation with four resolution val-ues. It provides almost the same results for different resolutions, which impliesthat not only the mean value but also the variance of estimation results areclosed to each other. It also confirms the conclusion that the inaccuracy of es-timation is independent of current selection of resolution. Consequently, 20 Hzis used in the following simulations.

31

0 200 400 600 800 1,000 1,200 1,4000

0.2

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0.6

0.8

1

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·108

Frequency [Hz]

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iodogra

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(a) Periodogram estimated spectrum

0 200 400 600 800 1,000 1,200 1,4000

1,000

2,000

3,000

4,000

5,000

Frequency [Hz]

Model

Jake

ssp

ectr

um

(b) Jakes’ spectrum

Figure 4.16: The comparison between estimated spectrum and theoretical spec-trum

0 50 100 150 200 250 30010−2

10−1

100

101

102

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Nor

mal

ized

MSE


(a) Regular speed

6 8 10 12 14 16 18 20 22 24100

101

102

103


Nor

mal

ized

MSE


(b) Low speed

Figure 4.17: The NMSE of estimated maximum Doppler spread of ML method,versus the theoretical maximum Doppler spread.

Effect of Noise

In Figure 4.18, we consider the effect of noise and show the NMSEs of the MLestimator as functions of SNR. The NMSEs slowly decrease with the increasingof the SNR except the result at low speed. Specifically, the value of NMSEs tendsto be stable when the SNR is above 8 dB. Thus, this method does not requirevery high SNR to guarantee the performance for medium and high velocities.On the other hand, the NMSE is fluctuate within 20 to 30 when the velocityis low, the corresponding error is 45 Hz to 55 Hz. It means the estimator isseriously biased, and the noise is not the dominant reason for the bias.

32

0 2 4 6 8 10 12 1410−2

10−1

100

101

102

SNR [dB]

Norm

alize

dM

SE

fD = 10 Hz

fD = 80 Hz

fD = 160 Hz

fD = 240 Hz

Figure 4.18: The NMSE of ML method with four speed values, versus SNR.

33

Chapter 5

Performance Evaluation

In this chapter, the evaluation and comparison of proposed algorithms are car-ried out . In order to implement the estimator in practical applications, it isof great importance to consider the computational complexity for each method.In Digital Signal Processor (DSP), if a certain number of multiplications arecalculated, the same number of accumulations can be computed synchronouslyin each clock cycle. In this case, only the number of multiplications or ac-cumulations is needed to represent the computational complexity. Since themultiplication is the limitation factor, the number of multiplication operationsis considered as the main metric to access the algorithmic complexity. More-over, performance comparison under both Rayleigh and Rician fading channelmodel are given.

5.1 Computational Complexity

In this section, the computational complexity of four types of Doppler spreadestimator algorithms will be discussed. After a detailed analysis, a summarytable for each method will be presented. The table includes the theoretical as-sessment and an example case with parameter setting as shown in Table 5.1(thesame as Chapter 4). In order to analyse without loss of generality, the worstcase scenario is considered in this section, i.e. the maximum complexity costrequired for each algorithm.

Symbol Description ValueN Number of channel estimate samples 150L Maximum lag of Ry , i.e. the length of Ry is L+ 1 10Nfft Number of FFT point 512

Table 5.1: Parameter setting

5.1.1 Level Crossing Rate Estimator

The LCR estimator is directly based on the channel estimate, as it counts thenumber of times that the channel estimate crosses a given level. The value ofthe level is calculated by Equation (3.5). One thing worth mentioning is that

34

one multiplication operation between two complex valued (i.e. consists of realpart and image part) equals to the complexity of four real valued multiplica-tion. However, one multiplication operation between the complex value and itsconjugate value costs the complexity of two real valued multiplication. There-fore in order to calculate the level, 2N + 1 multiplications and one square rootare required. The next step is to compare the channel response to the levelto detect the crossing. Note that in this report, one comparison is consideredas one multiplication. After the number of crossing is obtained, the maximumDoppler spread can be estimated according to Equation (3.4). Assuming theconstant term is already pre-calculated, the cost for the final estimation is onlyone multiplication.

Consequently, the computational complexity for the LCR algorithm consistsof the calculation of level, comparison between the channel response and the leveland the final Doppler spread estimation. The summary is shown in Table 5.2.

Description Multiplication Example Others

Level Calculation 2N+1 301 Square root: 1Comparison N 150

fD 1 1Sum 3N+2 452 Square root: 1

Table 5.2: Computational Complexity for Level Crossing Rate Estimator


As described in Section 3.2.3, the hybrid estimator uses either the curvaturemethod (for “low” Doppler spread) or the first zero detection method (for “high”Doppler spread). Thus, in this section the computational complexity for twocases will be analysed: low velocity and high velocity.

However, there are some common cost for these two cases since there is ajudgement to decide which method will be used. The judgement is based onthe estimated autocorrelation functions as shown in Equation (3.17). For lagk, it requires [4 ∗ (N − k) + 1] times of multiplication operations. If consider

k from 0 to L, it costs∑Lk=0[4 ∗ (N − k) + 1] = (L + 1)(4N − 2L + 1) times

of multiplications for one finger. In [23] the author suggests to combine alldetected fingers (for instance, totally q fingers are detected) to calculate theautocorrelation functions to increase the accuracy. It will results in q timesas much as the cost for only one finger. But note that only one finger case isconsidered in this part in order to compare the complexity with other algorithmswith the same parameter setting. To make the judgement, the smallest kmin

needs to be detected such that Ry [kmin ] is negative, and then compare it withthe predefined threshold T . The worst case is that Ry [k] is positive for all k, itcosts L + 1 comparisons between Ry [k] and 0, and one additional comparisonbetween kmin and T is required.

For the curvature estimator, Equations (3.8) and (3.10) are applied to esti-mate the maximum Doppler spread. Assuming (HTH)−1HT is pre-calculatedin Equation (3.8), then this equation becomes matrix multiplication with size(3× 1) = (3×L) · (L× 1), which costs 3L multiplications to solve this equation.

35

Note the elements in matrix are real valued. After obtaining the vector a, twomultiplications and one square root operation are needed in (3.10).

For the first zero detection estimator, the value of τ0 can be calculatedby using linear interpolation method according to Equation (3.16).One scalingand one division are required to get τ0 and one multiplication to estimate themaximum Doppler spread in Equation (3.14). Here one division is consideredas having the same cost as one multiplication. The results are presented inTable 5.3 and Table 5.4.

Description Multiplication Example OthersAucocorrelation Function (L+1)(4N-2L+1) 6391Comparison L+2 12Matrix Multiplication 3L 30

fD 2 2 Square root: 1Sum (L+1)(4N-2L+5) 6435 Square root: 1

Table 5.3: Computational Complexity for Hybrid Estimator for low velocity

Description Multiplication Example OthersAucocorrelation Function (L+1)(4N-2L+1) 6391Comparison L+2 12Linear Interpolation 2 2

fD 1 1Sum (L+1)(4N-2L+2)+4 6406

Table 5.4: Computational Complexity for Hybrid Estimator for high velocity

5.1.3 Power Integration Estimator

The power integration estimator based on PSD is described in Section 3.3. Ac-cording to Equation (3.23), PSD is estimated by periodogram method and FFToperation is applied in this approach. Since FFT can be efficiently calculated bysome DSP chips, the number of FFT operation is listed separately. In order tocalculate one point of Pyy[fk], only two multiplications are required for squareoperation, since the scaling is cancelled out in Equation (3.24). Thus for Nfft

point PSD, it results in 2Nfft multiplication operations in total. Furthermore, tofind the index p, one division and one comparison with threshold are needed forone value of p. Considering the worst case, the selection of p should begin from1 to bNfft−1

2 c, where b·c denotes to find the nearest smaller integer. After theindex p is obtained, only one operation is needed in Equation (3.25) to estimatethe maximum Doppler spread. See Table 5.5 for a summary of the result.

5.1.4 Frequency Domain ML Estimator

The goal of the ML estimator is to maximized Equation (3.31) with respect to

fD . Assuming the ideal Jakes’ spectrum is pre-calculated, periodogram esti-mated Power Spectrum Syy[fn] has the same cost with estimated PSD as pre-sented in the previous section. For one loop, the calculation of the concentrated

36

Description Multiplication Example OthersPeriodogram Estimation 2Nfft 1024 FFT: Nfft

Division bNfft−12 c 255

Comparison(with φ) bNfft−12 c 255

fD 1 1

Sum 2 +Nfft + 2bNfft−12 c+ 1 1535 FFT: Nfft

Table 5.5: Computational Complexity for Power Integration Estimator

likelihood function in Equation (3.31) requires (Nfft +1) multiplications and onelogarithm in the first term. Note that the second term can be pre-calculated.

Moreover, the whole computational complexity depends on the resolutionof selected fd, i.e. the interval between the two adjacent frequency. If thenormalized frequency resolution is defined as ∆f , since the range of normalizedDoppler spread is fd ∈ (0, 1/2], the tested hypothesis of normalized Doppler

spread is fd = ∆f, 2∆f, · · · , b 12∆f c∆f , the number of loops is b 1

2∆f c. For

instance, if we set ∆f = 20Fs

, totally b 12∆f c = b Fs

2×20c = 37 times of calculationare needed. Furthermore, the complexity also depends on another two factors:the sampling period of the channel estimation Ts and the number of FFT Nfft .First, the sampling period of the channel estimation directly decides the periodof the spectrum, larger period means larger frequency range in spectrum, whichresults in more points need to be calculated under the same resolution. Second,the number of FFT points can be seen as how many points are used to samplethe PSD from the continuous PSD. For instance, if Nfft = 128, on the one hand,only 128 points need to be calculated in Equation (3.31) for one loop. On theother hand, a smaller Nfft may reduce the number of loops. Thus, if we onlyconsider the perspective of the computational cost, increasing of the samplingperiod, decreasing the number of FFT points or increasing the interval betweentwo adjacent hypothesis frequency can lower the complexity of the frequenciesdomain ML estimator.

Description Multiplication Example OthersPeriodogram 2Nfft 1024 FFT: Nfft

Goal Function b 12∆f c (Nfft +1) 18981 Logarithm: b 1

2∆f cComparison b 1

2∆f c − 1 36

fD 1 1Sum b 1

2∆f + 2c(Nfft + 1)− 1 20042 Logarithm: b 12∆f c

FFT: Nfft

Table 5.6: Computational Complexity for ML Estimator

5.1.5 Summary

In order to give an intuitive comparison, the number of multiplication is used torepresent the total cost of each algorithm. Here (4× N

2 logN) times multiplica-tion are used to approximate an N points complex valued FFT operation. Onetime comparison, square root, or logarithm is considered to have the same cost

37

as one multiplication. Based on these assumptions, the comparison are shownin Figure 5.1.

Figure 5.1: The comparison of computational complexity.

As one can see, the LCR estimator has the lowest computational cost amongall algorithms. In contrast, the frequency domain ML estimator has the high-est complexity of computation due to the large number of loops required totest the hypothesis values in ML algorithm. Moreover, for the hybrid and thepower integration estimators, the computational cost are relatively low com-pared to others, with the hybrid estimator showing lower complexity. However,if the FFT can be calculated efficiently, then the cost for the power integrationestimator can be further reduced.

5.2 Performance Comparison

In this section, the four Doppler spread estimators will be compared togetherwith another four algorithms, which are implemented in the same simulatorby my partner [14]. The first one is the Zero Crossing Rate (ZCR) estimator.It is similar to the LCR method since both of them are based on the channelresponse directly. The second one is the Moser’s estimator, it depends on theautocorrelation functions of channel estimate. The third one is the PSD slopeestimator, it detects the slope of PSD which is estimated by Periodogram es-timator. The last one is the time domain ML estimator, with the objective tomaximize the likelihood function based on the approximation of the correlationfunctions. More detailed description can be found in Hui Wen’s report or see[1] [27] [9] [12], respectively.

For a more comprehensive performance evaluation, both Rayleigh and Ricianfading channel model are considered. Section 5.2.1 discusses the assessment forthe effect of speed and noise. Section 5.2.2 focuses on the effect of the Ricianfactor K and the AOA of the LOS component.

Note that in the following figures, four different colours of plots refer to fourtypes of estimator introduced in this report, and the curves with triangle denotethe estimators implemented by Hui Wen [14].

38

5.2.1 Rayleigh Fading Channel

Effect of Speed

Figure 5.2 illustrates the mean value of fD of eight algorithms over the speedrange of 2.7-162 km/h (corresponding theoretical Doppler spread is 5-300 Hz).Generally, most of the estimators provide reliable estimate for the whole rangeof the Doppler spread, except for the frequency domain ML and crossing-basedmethods.

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ZCR

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ML in Time Domain

(a) Regular speed

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Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(b) Low speed

Figure 5.2: The comparison of mean value of estimated maximum Dopplerspread in Rayleigh fading model, versus the theoretical maximum Dopplerspread.

It is clearly seen that both LCR and ZCR algorithms have a high bias atlow velocities, as shown in the lower plot in Figure 5.2. The LCR estimatorgives even worse results than the ZCR estimator in terms of error. However,Figure 5.3 demonstrates that LCR has a lower NMSE at both low and highspeeds, and the curve of LCR estimator is more smooth, which means LCR ismore stable. The reason is that the results of LCR are based on the number ofcrossing count directly, and the count depends on the value of level which is afunction of the received signal’s power. Thus, the level can adapt to the changeof the channel such that provide a stable result. Moreover, it is believed that amore suitable level can be found to reduce the error after statistical analysis.

Now the autocorrelation functions based algorithms are compared, i.e. theHybrid and the Moser’s estimator. It is hard to distinguish the two curves ofthe mean estimation values over the entire speeds, since the performance is veryclose to each other. So the analysis concentrates on the comparison of NMSEs.As one can clearly see that the hybrid estimator performs better at low velocities,it confirms that the hybrid estimator has a wide detection range. And both ofthese two algorithms performs good when the speed is high. Furthermore, theautocorrelation based algorithms provide the best estimation results among allmethods in the current scenario.

Next, the performance of the PSD based algorithms is taken into considera-tion. The difference between the power integration estimator and the PSD slopeestimator is that the former one integrates over entire PSD and the latter onefocus on the peak slope of PSD. Even though the curve of NMSE for the powerintegration method at low speed values is very unstable, it is still lower than

39

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10−1

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LCR

Hybrid

Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(a) Regular speed

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10−1

100

101

102

103


Norm

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dM

SE

LCR

Hybrid

Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(b) Low speed

Figure 5.3: The comparison of NMSE of estimated maximum Doppler spreadin Rayleigh fading model, versus the theoretical maximum Doppler spread.

the NMSE of the PSD slope method. For the whole velocity range, the PSDslope provides a smooth and relatively stable result comparing to the powerintegration estimator.

Finally, two ML based estimators are compared. It is not difficult to seethat the time domain ML method gives much more accurate results than thefrequency domain ML method. As mentioned earlier, the possible reason arethe mismatch between the received spectrum and the Jakes’ spectrum or theinstability of received spectrum. Whereas, the time domain ML estimator isbased on the approximation of the autocorrelation functions. Hence, it is canbe seen that the autocorrelation is more reliable for the estimation in practice.

Effect of SNR

Figure 5.4 demonstrates the NMSE of all the algorithms with different receivedSNRs for four Doppler spread values. In general, for the low and mediumDoppler spread, autocorrelation functions based methods are robust to noise,whereas the crossing rate based methods are relatively more sensitive to noise.The power integration estimator flucturates with the increase of SNR due tothe outlier of simulation. And the PSD slope is relatively independent to noise.In addition, frequency domain ML is robust to the effect of noise even thoughit suffers high bias. On the contrary, the time domain ML shows its sensitivityto the noise.

When the Doppler spread is high, see (c) and (d) in Figure 5.4, crossingbased estimators are still sensitive to noise, but it can be improved with theincrease of speed. For the autocorrelation based estimator, PSD slope estimatorand frequency domain ML estimator, the performance for low velocities andhigh velocities are similar. Whereas, for the power integration method, thevariation of the NMSE decreases when the velocity increases, specifically whenfD = 240 Hz, the NMSE is even lower than that of the hybrid method. It alsodemonstrates the robustness against noise. And the NMSEs of the time domainML estimator are relatively stable over all values of SNR.

Therefore, from the perspective of robustness to noise, the Hybrid estimatorshows the best performance among all the algorithms.

40

0 2 4 6 8 10 12 1410−2

10−1

100

101

102

103

SNR [dB]

Norm

alize

dM

SE

LCR

Hybrid

Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(a) fD = 10 Hz

0 2 4 6 8 10 12 1410−2

10−1

100

101

SNR [dB]

Norm

alize

dM

SE

LCR

Hybrid

Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(b) fD = 80 Hz

0 2 4 6 8 10 12 1410−3

10−2

10−1

100

SNR [dB]

Norm

alize

dM

SE

LCR

Hybrid

Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(c) fD = 160 Hz

0 2 4 6 8 10 12 1410−3

10−2

10−1

100

SNR [dB]

Norm

alize

dM

SE

LCR

Hybrid

Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(d) fD = 240 Hz

Figure 5.4: The comparison of NMSE of estimated maximum Doppler spreadin Rayleigh fading model, versus SNR.

5.2.2 Rician Fading Channel

In this section, the simulation results in Rician fading channel are presented.Here two theoretical Doppler spread values are used in the simulation, i.e. fD =20 Hz and fD = 120 Hz to represent the low speed scenario and high speedscenario. The criterion for performance evaluation is the same as above, i.e.themean estimation value and the NMSE.

Effect of Rician Factor

Figure 5.5 demonstrates the impact of Rician factor K. K represents the ratioof the LOS power to the diffuse power. The upper plots are the mean value andlower plots are the NMSE for 20 Hz (left) and 120 Hz (right), and the blackdashed line represents the theoretical value of Doppler spread.

We will start with focusing on the analysis for the low speed. As one cansee that the frequency domain ML estimator and the LCR estimator are fairlyinsensitive to the Rician factor K, with the frequency domain ML estimatorshowing much smaller errors within 5 Hz. However, both of these two methodshave very high NMSEs compare to other algorithms due to their large bias to theactual value. Moreover, the PSD slope estimator also results in a convex curve,which reflects that the estimation results changes slowly when K is samll, i.e. 0 <K ≤ 4 and changes rapidly when K is large, i.e. 4 < K ≤ 8. On the contrary,

41

0 1 2 3 4 5 6 7 80

20

40

60

80

Rician factor

Mea

nes

tim

ate

maxim

um

Dopple

rsp

read

[Hz]

LCR

Hybrid

Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(a) fD = 20 Hz, Mean value

0 1 2 3 4 5 6 7 80

50

100

150

200

250

Rician factor

Mea

nes

tim

ate

maxim

um

Dopple

rsp

read

[Hz]

LCR

Hybrid

Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(b) fD = 120 Hz, Mean value

0 1 2 3 4 5 6 7 810−2

10−1

100

101

Rician factor

Norm

ailze

dM

SE

LCR

Hybrid

Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(c) fD = 20 Hz, NMSE

0 1 2 3 4 5 6 7 810−3

10−2

10−1

100

Rician factor

Norm

ailze

dM

SE

LCR

Hybrid

Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(d) fD = 120 Hz, NMSE

Figure 5.5: The comparison of the mean value and NMSE of estimated maxi-mum Doppler spread, versus Rician factor.

the rest five algorithms show concave behaviour as Rician factor K increases,and the estimation results of them are all have a tendency of convergence. Thebias for the autocorrelation based algorithms is relatively most less. Whereas,the ZCR estimation error change significantly. One thing worth mentioning isthat considering the high bias for the ZCR estimator in Rayleigh fading channel(K = 0), the ZCR estimation result shows a trend to the true value as theincreasing of K mainly because of the attenuation in Rician fading channel.Overall, Hybrid estimator has the best performance for the low speed.

On the other hand, see Figure (b) and (d) for the simulation result at highspeed. Frequency domain ML estimator is overestimated since the estimatedpower spectrum consists LOS component, which is different from the ideal Jakes’spectrum. Whereas, all the rest algorithms are underestimated. It also can beobserved that the LOS component affects the simulation results significantly.Among all the algorithms, the Moser’s estimator performs relatively best dueto its lowest NMSE.

Effect of Angle of Arrival

Figure 5.6 depicts the mean value and the NMSE of eight algorithms over theAOA range of 0◦ − 180◦, with Rician factor K = 2, SNR = 15 dB.

As one can observe that almost all the algorithms present the similar per-formance, which shows that they are affected by the AOA but with different

42

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

Angle of arrival (degree)

Mea

nes

tim

ate

maxim

um

Dopple

rsp

read

[Hz]

LCR

Hybrid

Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(a) fD = 20 Hz, Mean value

0 20 40 60 80 100 120 140 160 1800

50

100

150

200

250


Mea

nes

tim

ate

maxim

um

Dopple

rsp

read

[Hz]

LCR

Hybrid

Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(b) fD = 120 Hz, Mean value

0 20 40 60 80 100 120 140 160 180

10−0.5

100

100.5


Norm

ailze

dM

SE

LCR

Hybrid

Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(c) fD = 20 Hz, NMSE

0 20 40 60 80 100 120 140 160 18010−2

10−1

100


Norm

ailze

dM

SE

LCR

Hybrid

Power Integration


ZCR

Moser

PSD Slope

ML in Time Domain

(d) fD = 120 Hz, NMSE

Figure 5.6: The comparison of the mean value and NMSE of estimated maxi-mum Doppler spread, versus angle of arrival of LOS component.

levels of effect. The reason is that most of the algorithms are proposed based onthe Rayleigh fading channel model, and are not suitable for the Rician fadingchannel. Moreover, it is difficult to design a method that can always performaccurate in Rician channel, since there are different kinds of variation in Ricianchannel.

It can also be seen that the results are symmetrical about 90◦, where theworst performance is given by most of estimators. Noting that the NMSEof frequency domain ML at high speed shows a opposite trend to the otheralgorithms, since it has a high bias at 0◦ and the mean value is reduced as theangle increases to 90◦, which results in less bias to the actual value, and decreaseof NMSE. The symmetry can be explained by Figure 2.5 and Equation (2.2).For example, if the car in the figure drive to the opposite direction, the anglebetween the drive direction and the signal transmission is larger than 90◦, whichresults in negative frequency shift. However, the absolute values of the Dopplershift with angle θ and (180◦− θ) are the same, which means the frequency shiftin the spectrum are with the same value but with opposite direction.

Moreover, according to Equation (2.7), the AOA of LOS component givesrise to the frequency offset to the channel response except for the case withθ0 = 90◦. When θ0 = 90◦, Equation (2.7) is a real value, it only results ina Dirac pulse in the spectrum without frequency offset. Theoretically, whenθ0 = 90◦, more accurate results can be given compared to other values of angle.

43

However, in this thesis, the frequency offset compensation is turned on, whichcan reduce the effect of the frequency offset. It might be one of the reasons thatcan explain why the estimators have better performance when θ is low.

Furthermore, it can be seen that the hybrid estimator and the LCR estimatorare least sensitive to the effect of AOA of LOS component in term of NMSE atlow speed and high speed, respectively.

44

Chapter 6

Conclusions

6.1 Summary

This report investigates four Doppler spread estimation algorithms. The evalu-ation and conclusion is presented from the perspective of the performance andthe computational complexity.

The simulation shows that the hybrid estimator demonstrates the best per-formance among all the estimators under both Rayleigh and Rician models.The inaccuracy of LCR estimator is caused by two reasons: First, the rapidvariation of the channel leads to the variance of the energy, which will influencethe value of threshold level. As a result, the current level used may not beoptimal. Second, due to the property of the level crossing rate, the number ofcrossing is easily affect by noise. For the power integration estimator, the biasof estimation results in that it is difficult to find the optimal parameters for theentire range of velocity. Furthermore, the simulation results of the frequencydomain ML estimator show a large bias compared to the theoretical value. Thisis because of the mismatch between the estimated power spectrum and the idealJakes’ spectrum.

On the other hand, the analysis and discussion of computational complexityshow that the LCR estimator has the lowest cost. Whereas, due to the largenumber of loops required, the computational cost is heavy for the frequencydomain ML. In addition, the hybrid and the power integration have the relativelylower computational complexity compared to the other estimators.

In summary, with relatively less complexity and high accuracy, the hybridestimation algorithm has the largest practical value for implementation in realapplications.

6.2 Social and Ethical Aspects

It is important to consider the possible influence of a research from the socialand ethical aspects. In this thesis, the objective is to compare several Dopplerspread estimation algorithms to find the best algorithm for practical implemen-tation. The increasing of accuracy can bring benefits to many applications. Forinstance, it can be applied to the optimal tuning of system parameters withrespect to changing channel conditions in adaptive transmission systems. It can

45

potentially increase the energy efficiency of data transmission. Economically,less energy means less cost for operators. On the other hand, consider from theethical point of view, the Doppler spread estimation will not involve any ethicalissues if used to improve the estimate of WCDMA baseband algorithms. How-ever, if the focus is on the estimation value of the Doppler spread, which directlyindicates the speed of the user, then it can be used to collect information aboutthe behaviour of each individual.

6.3 Future Work

The results of this thesis serves as motivation to continue the research, andthe topic can be further investigated in several directions. First, the simulationenvironment can be further extended in order to test more influencing factors.For instance, applied the von Mises’s distribution to investigate the effect ofnonisotropic scattering, and test different length of channel estimation samples.Second, one can attempt to increase the accuracy of the proposed algorithms,such as finding optimum level for the LCR estimator, fixing the problem withthe simulation length of the power integration estimator, or reducing the biasin the frequency domain ML estimator by modifying the shape of theoreticalDoppler spectrum. Moreover, since the aim of this thesis is to compare andevaluate several algorithms, the selection of parameter setting has to be rela-tively suitable to all algorithms. However, the optimum setting can be selectedfor each algorithm, such as to turn on the function of TPC, AGC, to adjustthe length of the channel estimation, to combine all the detected fingers, etc.All this will help to further improve the accuracy of the estimation result andreduce the computational complexity.

46

Appendix A

Whittle ApproximationLog-likelihood GoalFunction Derivation

In this section, the derivation from Equation (3.29) to Equation (3.31) of theML estimator is presented.

To clearly present the derivation, here Equations (3.29) and (3.30) are givenagain, respectively:


[ln(SJ [fn; fd]) +

Sy [fn]

SJ [fn; fd]

], (3.29)

and

σ2n =

SJ [fn; fd]

γc[fn; fd] + 1

≈ 1

Nfft

Nfft−1∑n=0

(Sy [fn]

γc[fn; fd] + 1

). (3.30)

Expanding Equation (3.29) with the first line of Equation (3.30) to replaceSJ [fn; fd] by σ2

n(γc[fn; fd] + 1) results in


{ln[σ2

n(γc[fn; fd] + 1)] +Sy [fn]

σ2n(γc[fn; fd] + 1)

}, (A.1)

and then replacing σ2n in the above equation by the relation in second line of

47

Equation (3.30), gives

Γ ≈−Nfft lnπ

−Nfft−1∑n=0

ln

1

Nfft

Nfft−1∑m=0

(Sy [fm]

γc[fm; fd] + 1

)(γc[fn; fd] + 1)

−Nfft−1∑n=0

Sy [fn]

1Nfft

∑Nfft−1m=0

(Sy [fm]

γc[fm;fd]+1

)(γc[fn; fd] + 1)

.

(A.2)

Use Γ1,Γ2,Γ3 to present the three terms in the above equation for simplicity,

Γ1 = −Nfft lnπ

Γ2 = −Nfft−1∑n=0

ln

1

Nfft

Nfft−1∑m=0

(Sy [fm]

γc[fm; fd] + 1

)(γc[fn; fd] + 1)

= −

Nfft−1∑n=0

[ln

(1

Nfft

)]−Nfft−1∑n=0

ln

Nfft−1∑m=0

(Sy [fm]

γc[fm; fd] + 1

)−Nfft−1∑n=0

[ln(γc[fn; fd] + 1)]

= −Nfft

[ln

(1

Nfft

)]−Nfft

ln

Nfft−1∑m=0

(Sy [fm]

γc[fm; fd] + 1

)−Nfft−1∑n=0

[ln(γc[fn; fd] + 1)]

Γ3 = −Nfft−1∑n=0

Sy [fn]

1Nfft

∑Nfft−1m=0

(Sy [fm]

γc[fm;fd]+1

)(γc[fn; fd] + 1)

= −

Nfft−1∑n=0

[Sy [fn]

(γc[fn; fd] + 1)

]/

1

Nfft

Nfft−1∑m=0

(Sy [fm]

γc[fm; fd] + 1

)= −Nfft

Nfft−1∑n=0

(Sy [fn]

γc[fn; fd] + 1

) /Nfft−1∑m=0

(Sy [fm]

γc[fm; fd] + 1

)= −Nfft .

(A.3)

Neglect the constant terms, (A.2) results the concentrated likelihood func-tion:

Γc = −Nfft ln

Nfft−1∑n=0

Sy [fn]

γc[fn; fd] + 1

− Nfft−1∑n=0

ln [γc[fn; fd] + 1]

as shown in Equation (3.31).

48

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