A Thesis Entitled Performance Evaluation of 2-D Pilot Aided OFDM

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A Thesis

entitled

Performance Evaluation of 2-D Pilot Aided OFDM System under Hyper-Rayleigh Fading Channel

Haobo Zhen

Submitted to the Graduate Faculty as partial fulfillment of the requirements

for the Master of Science Degree in Electrical Engineering

Dr. Junghwan Kim, Committee Chair

Dr. Ezzatollah Salari, Committee Member

Dr. Dong-Shik Kim, Committee Member

Dr. Patricia R. Komuniecki, Dean College of Graduate Studies

The University of Toledo August 2011

This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of Performance Evaluation of 2-D Pilot Aided OFDM System under Hyper-Rayleigh Fading Channel by

Haobo Zhen Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Science Degree in Electrical Engineering

The University of Toledo August 2011

Wireless radio propagation channel is widely known as the most hostile communication channel. Modeling and simulation of wireless fading channel has been an essential issue in the design of communication system. Rayleigh fading and Rician fading model are the most commonly used small scale models in wireless communication. However, recent research shows that some WSN applications where sensor nodes deployed within cavity environment suffer from more severe fading than

Rayleigh fading predicted, which is referred to as hyper-Rayleigh fading. Therefore, design of a more applicable model is necessary to describe hyper-Rayleigh fading.

Two-wave with diffuse power (TWDP) model is suggested to be the proper model to represent hyper-Rayleigh fading behavior. However, little effort has been made to evaluate the anti-interference capacity of OFDM system under hyper-Rayleigh fading channel. In the thesis, the characteristic of hyper-Rayleigh fading is explored and analyzed. Furthermore, performances of OFDM system with various pilot aided channel estimation techniques under hyper-Rayleigh fading are investigated.

Simulation results indicate hyper-Rayleigh fading exhibits worse fading phenomena

iii than Rayleigh fading when parameters ∆ and K exceed certain values. OFDM system with 2-D channel estimation demonstrates strong resistance to multipath fading, thus can be regarded as a promising candidate of WSN applications under hyper-Rayleigh fading.

Acknowledgements

First of all, I would like to thank my advisor Dr. Junghwan Kim for his guidance, his encouragement and support to me. The thesis would not have been possible without his help. I would also like to thank Dr. Ezzatollah Salari and Dr. Dong-Shik Kim for being my committee members.

I want to give my thanks to my friends and fellow students in University of Toledo, who make my life in America happy and memorable.

Finally, I want to express my deep appreciation and gratitude to my parents, whose love always accompany me through difficult times and days of joy.

Table of Contents

Abstract iii

Acknowledgements v

Table of Contents vi

List of Tables viii

List of Figures ix

1 Introduction………………………………………………………………………… 1

1.1 Problem Statement…………………………………………………………… 1

1.2 Thesis Contribution………………………………………………………… 3

1.3 Thesis Outline…………………………………………………………….. 4

2 Principle and Model of OFDM System………………………………………...… 6

2.1 Principle of OFDM system…………………………………………….…… 6

2.2 Model of OFDM system……………………………………………………… 9

2.2.1 Convolutional Coding………………………………………………….. 10

2.2.2 Modulation Schemes………………………………………………….. 14

2.2.3 IFFT and FFT………………………………………………………….. 23

2.2.4 Cyclic Prefix………………………………………………………….. 23

3 Characteristic and Modeling of Hyper-Rayleigh Fading Channel……………… 26

3.1 Small Scale Fading…………………………………………………………... 26

3.1.1 Multipath Fading……………………………………………………….. 28

3.1.2 Doppler Effect………………………………………………………….. 29

3.1.3 Expression of Received Signal……………………………………….. 30

3.2 Typical Small Scale Fading Models…………………………………………32

3.2.1 Rayleigh Fading Model……………………………………………….. 32

3.2.2 Rician Fading Model………………………………………………….. 36

3.3 Hyper-Rayleigh Fading Model………………………………………………40

4 2-D Pilot Aided Channel Estimation in OFDM System………………………… 50

4.1 Procedure of Channel Estimation………………………………………..… 50

4.2 2-D Pilot Arrangement…………………………………………………….…. 52

4.2.1 1-D Pilot Pattern……………………………………………………….. 52

4.2.2 2-D Pilot Pattern……………………………………………………….. 56

4.3 Channel Estimation Algorithms…………………………………………….. 59

5 Simulation Results and Performance Analysis………………………………….. 63

5.1 Simulation Results with Different Modulation Schemes…………………………...63

5.2 Simulation Results under Rayleigh Fading Channels……………………… 65

5.3 Simulation Results under Rician Fading Channels………………………… 67

5.3 Simulation Results under Rician Fading Channels………………………… 70

6 Conclusion and Future Work…………………………………………………..… 76

References…………………………………………………………………………... 78

vii

List of Tables

2-1 Simulation specification of OFDM system……………………………………..13

3-1 Fading scenario characterized by specular components……………………….32

3-2 Comparison between four fading models……………………………………….43

5-1 Specification of OFDM system simulation under Rayleigh fading…………….65

viii

List of Figures

1-1 Wireless communication environment...…………………………………………. 2

2-1 Frequency spectrum of OFDM system sub-carriers…………………………….. 7

2-2 Model of OFDM system………………………………………………………… 9

133 145 175 2-3 Rate convolutional encoder……………………………… 11

133 171 2-4 Rate convolutional encoder ………………………..………… 11

2-5 BER performance curves of coded and uncoded OFDM systems……………. 13

2-6 BPSK constellation…………………………………………………………..… 16

2-7 8PSK constellation with Gray mapping………………………………………… 17

2-8 16PSK constellation with Gray mapping………………………………………. 18

2-9 The BER performances of OFDM system with different MPSKs……………. 19

2-10 16QAM constellation with Gray mapping……………………………………. 21

2-11 BER curves of OFDM with M-ary QAMs under AWGN……………………. 22

2-12 Cyclic prefix adding………………………………………………………….. 24

3-1 Large scale fading and small scale fading…………………………………….. 27

3-2 Illustration of Doppler shift in time varying channel…………………………… 30

3-3 Illustration of specular components and diffuse components…………………. 31

3-4 Illustration of Rayleigh fading scenario………………………………………… 33

3-5 PDFs of Rayleigh fading with different variances……………………………… 34

3-6 CDFs of Rayleigh fading with various variances………………………………. 35

3-7 Illustration of Rician fading scenario……………………………………………37

3-8 PDFs of Rician fading with various V…………………………………………. 38

3-9 CDFs of Rician with various V…………………………………………………39

3-10 Illustration of hyper-Rayleigh fading scenario……………………………… 41

3-11 PDFs of TWDP fading with K=3……………………………………………… 44

3-12 PDFs of TWDP with ∆ 1…………………………………………………… 45

3-13 Comparison of PDFs among Rayleigh, Rician and TWDP…………………… 45

3-14 Illustration of range of TWDP CDFs………………………………………… 47

3-15 CDFs of the three traces……………………………………………………… 48

4-1 Block type pilot pattern……………………………………………………… 53

4-2 Illustration of block type pilot channel estimation………………………….. 54

4-3 Comb type pilot pattern……………………………………………………….. 55

4-4 Illustration of comb type pilot channel estimation……………………………… 56

4-5 Rectangular type pilot pattern…………………………………………………. 57

4-6 Procedure of rectangular type pilot channel estimation………………………… 58

5-1 Comparison of different modulation schemes under AWGN………………… 64

5-2 Comparison of different modulation schemes with coding…………………… 64

5-3 Block type BER performances with LS and LMMSE under Rayleigh……….. 65

5-4 Comb type BER performances with LS and LMMSE under Rayleigh……….. 66

5-5 Rectangular type performances with LS and LMMSE under Rayleigh……….. 67

5-6 BER performances under Rician fading with various K-factors……………….. 68

5-7 Block type BER performances with LS and LMMSE under Rician………….. 69

5-8 Comb type BER performances with LS and LMMSE under Rician………….. 69

5-9 Rectangular type BER performances with LS and LMMSE under Rician…….. 70

5-10 BER performances under TWDP model with various K…………………….. 71

5-11 Block type BER performances under TWDP model with ∆ 1 , K 6..…… 71

5-12 Comb type BER performances under TWDP model with ∆ 1 , K 6……. 72

5-13 Rectangular type BER performances under TWDP model with ∆ 1 , K 6 .72

5-14 Comparison of BER performances under TWDP with K=3………………….. 73

5-15 Comparison of BER performances under TWDP with K=6………………….. 74

5-16 Comparison of BER performances under various fading channels………….. 74

Chapter 1

Introduction

1.1 Problem Statement

Fading and interference are the major performance degrading factors in wireless/mobile communications. In order to improve and testify the system’s effectiveness to resist fading, modeling and simulation of communication system under fading channel is of great significance in the design of communication system. For different propagation environment, the characteristic of fading channel is diverse and complex. Therefore, design of proper fading model in particular communication circumstance is essential in this regard.

Rayleigh fading and Rician fading model are the most commonly used small scale models in wireless communication [1]. However, as wireless sensor networks (WSN) migrate into vastly different applications, conventional Rayleigh and Rician channel model don’t fit in every WSN environment. Recent research [2] [3] shows that some

WSN applications where sensor nodes deployed within cavity environment suffer from more severe fading than Rayleigh fading predicted, which is referred to as hyper-Rayleigh fading. Herein, development of a more applicable fading model which 1 fits in some particular WSN circumstance has become an important issue.

Figure 1-1 Wireless communication environment.

Another problem lies in the effectiveness of anti-interference technologies. For wireless communication, OFDM is good multi-carrier scheme due to its nature of strong resistance to interference and high spectra efficiency, high data rate transmission

[4]. Channel estimation technologies are implemented in order to estimate the effect of propagation delay and channel synchronization. Channel estimation methods can be classified into two categories: blind channel estimation and pilot-aided channel estimation. The channel estimation techniques studied in the thesis are all pilot-aided, for pilot-aided channel estimation are more applicable in fast-fading frequency selective radio propagation channel. Different pilot insertion patterns results in diverse

BER performances. 2-D pilot channel estimation is proven to have better performance comparing to 1-D pilot channel estimation

Bit-error-rate (BER) is a key factor to measure the capacity and performance of communication system. Much effort has been made to explore the characteristic and

BER performance of hyper-Rayleigh fading [5] [6]. However, there is little work on evaluating BER performance of OFDM system under such radio propagation environment. Herein, the thesis is focused on the investigation of OFDM system performance under various fading environment, especially hyper-Rayleigh fading. 2-D pilot-aided channel estimation, convolutional coding and cyclic prefix are also implemented in OFDM system. The performance of OFDM system can be determined by evaluating system’s BER.

1.2 Thesis Contribution

The thesis aims to explore the BER performance of OFDM system under hyper-Rayleigh fading environment and investigate anti-fading capabilities of 2-D pilot channel estimation methods. The contributions of the thesis are as follows:

Explain the basic principle of OFDM system and capacity to overcome ISI

and ICI caused by multipath. Coding and modulation schemes are also

introduced and analyzed to further enhance the performance of OFDM

system;

Develop fading models to represent the actual small scale fading situation

exist in wireless communication. Apart from conventional small-scale fading

models, a recently proposed fading model, known as hyper-Rayleigh model,

is developed and investigated;

Present pilot-aided channel estimation techniques to address small scale

fading problem in wireless communications. Two 1-D pilot pattern, block

type and comb type, are investigated. Rectangular 2-D pilot pattern, which is

more applicable in frequency selective and time-variant fading channel, is

presented and analyzed. LS and LMMSE channel estimation is also

introduced;

Investigate BER performance of OFDM system under various small-scale

fading environments and validate effectiveness of applying 2-D pilot channel

estimation to OFDM system under hyper-Rayleigh fading using MATLAB

simulation.

1.3 Thesis Outline

The thesis presents modeling of various fading environments and techniques to improve BER performance of OFDM system. Chapter 1 gives a brief introduction to the signal fading problems in wireless communication and the motivation of the research is highlighted. In Chapter 2, principle of OFDM system is presented. Coding, modulation schemes and other techniques are introduced in order to improve OFDM system performance. In Chapter 3, characteristic of various fading environments is

4 investigated. Fading models, Rayleigh, Rician and hyper-Rayleigh, are proposed and simulated to represent propagation phenomenon during transmission. In Chapter 4, 2-D and 1-D pilot patterns are presented and analyzed. LS and LMMSE estimation methods are also discussed in this chapter. In Chapter 5, BER simulation results of OFDM system with different technologies under various fading environments are given. BER is utilized as the key factor to evaluate the performance of OFDM system. Chapter 6 draws conclusion and shows the course of future work.

Chapter 2

Principle and Model of OFDM System

Orthogonal Frequency Division Multiplexing (OFDM), which is also referred to as Discrete Multi-tone Modulation (DMT), is a multi-carrier transmission technique, is widely applied to wireless communications, such as digital audio broadcasting, digital video broadcasting and wireless local area network (WLAN). OFDM is also regarded as one of the most promising technologies for the fourth generation (4G) mobile communication system. OFDM technology has distinctive advantages on high data transmission, anti-interference and low equipment complexity [4]. Chapter 2 gives a detailed explanation of principle and model of OFDM system. The discussion and analysis of coding and modulation schemes involved in OFDM signal generation are also included in this chapter.

2.1 Principle of OFDM system

The idea of OFDM is to divide the original data steam into several parallel narrowband low-rate streams modulated on corresponding orthogonal sub-carriers [7].

To be specific, each sub-carrier has integer periods in OFDM symbol duration. 6

Neighboring sub-carriers have one period difference to maintain orthogonality.

Suppose T is denoted as OFDM symbol width, and are the frequencies of two sub-carries, we have:

(2.1) 1 exp ∙ exp 0

This orthogonality characteristic of OFDM system can also be understood in the view of frequency domain. As shown in Figure 2-1, all sub-carriers are controlled to maintain orthogonality by making the peak of each sub-carrier signal coincide with the nulls of other signals.

Figure 2-1 Frequency spectrum of OFDM system sub-carriers.

The orthogonal characteristics of sub-carriers enable OFDM system to have higher spectral efficiency than conventional multi-carrier technique. For conventional multi-carrier techniques, guard intervals are inserted between sub-carriers so that

7 sub-carrier signal can be separated from other signal by corresponding filter at the receiver. In the case of OFDM system, however, sub-carriers overlap each other and can be demodulated without guard interval.

Another significant feature of OFDM system is that modulation and demodulation procedure is implemented by IFFT and FFT respectively, which will greatly reduce the complexity of equipment and structure. Suppose is sub-carrier 1,2,…, frequency, the modulated signal at chip duration is denoted as: th (2.2) ∑ exp 2

Where carries the data information at chip duration and determines the amplitude and phase of signal . At the receiver, the spectrum component of signal is calculated by th -point DFT (Discrete Fourier Transform). Assume the sampling frequency is , and frequency interval of adjacent sub-carriers is . Therefore, is ∆ / ∆ written as:

(2.3) ∆ ∑ / exp 2/

Since , equation (2.2) is taken in equation (2.3): / 2 2 ∆ exp exp (2.4) ∑ Where,

0 , 1 From the equation (2.4), it is known that if is integer times of , the ∆ th spectrum component of signal is obtained at the receiver.

2.2 Model of OFDM system

In this section, model of OFDM system is presented, where some significant

functions are analyzed. Coding and modulation schemes are essential in developing a

feasible OFDM communication system. Moreover, cyclic prefix is considered as an

indispensible part of OFDM system to combat inter-carrier interference (ICI), since

OFDM system is particularly vulnerable to ICI. Radio propagation channel and pilot

aided channel estimation are the major research objects of the thesis, which will be

discussed in detail in chapter 3 and chapter 4 respectively. Figure 2-2 shows a typical

model of OFDM system. input

Pilot Coding Modulation S/P IFFT P/S Cyclic Prefix Insertion

Physical Channel output

Channel Remove Decoding Demodulation P/S FFT S/P Estimation Cyclic Prefix

Figure 2-2 Model of OFDM system.

2.2.1 Convolutional Coding

Despite the fact that OFDM system has inherent resistance to fading, in multipath fading channel, however, some sub-carriers may suffer the deep fades, which will results in degradation of BER. Herein, forward-error correction coding is essential.

Convolutional codes have strong error correcting capability and are widely applied to communication practices. Therefore, Convolutional code is selected as FEC channel coding scheme in the thesis.

Block codes and convolutional codes are the most commonly used coding scheme.

Unlike block codes, the encoder of convolutional codes contains memory and the encoder outputs at any given time unit depend not only on the inputs at that time unit but also on m previous input blocks [8]. A convolutional encoder can be , , implemented with a k-input, n-output linear sequential circuit with input memory m, which means encoded bits are generated for each information bits. Therefore, the code rate is defined as . / A convolutional encoder consists of a number of shift registers and modulo-2 adders. Suppose the constraint length K is set to be 7, therefore 6-stage shift register is required in a convolutional encoder since is decided by the equation . In 1 the thesis, the generator polynomial is set to be for convolutional 133 145 175 code with coding rate. For the convolutional code with coding rate, the 1/3 1/2 generator polynomial is chosen. Figures 2-3 and 2-4 show the 133 171 encoder and encoder respectively: 133 145 175 133 171 10

Figure 2-3 rate convolutional encoder. 133 145 175

Figure 2-4 rate convolutional encoder. 133 171

A convolutional encoder generates encoded bits for each information bits, and is called the code rate. / A convolutional encoder works by performing convolutions on the incoming input data. Let denote the input sequence. For each path of the encoder, , , , ⋯ the output sequence can be written as,

(2.5) ∑ , 0,1,2,⋯

Where denotes a certain path of the encoder, and =0 when . After convolution, the encoder generates the coded sequence by combining all the output sequences of each path.

The signal received at the decoder is distorted by noise and interference. Maximum-likelihood decoder is applied to error correcting. In the specific case of binary symmetric channel (BSC), the log-likelihood function can be written as,

(2.6) ln ,ln ln1

where is the transition probability, is the Hamming distance between , .Note that is a constant, thus maximum-likelihood is equivalent and 1 with minimum distance. That is, the maximum-likelihood decoder reduces to a minimum distance decoder, by which the decoding process is to choose a path in the trellis whose coded sequence most resembles the received sequence. By computing the metric for each path, the survivor thus can be found and retained by the algorithm. The metric for a certain path is defined as the Hamming distance between the coded

12 sequence and the received sequence.

It can be expected that bit-error-rate (BER) of system with 1/3 coding rate is lower than that with 1/2 coding rate. Figure 2-5 shows BER performance curves of coded and uncoded OFDM systems with BPSK under AWGN:

Table 2-1 Simulation specification of OFDM system

Parameter Specification

FFT Size 512

Cp Length 128

Channel AWGN

Modulation BPSK

Figure 2-5 BER performance curves of coded and uncoded OFDM systems.

It can be seen from Figure 2-5 that signal-noise-ratio (SNR), by convolutional coding, obtained 5 dB of coding gain, thus BER performance is enhanced. It is also known from Figure 2-5 that better BER performance can be acquired by having lower coding rate. Of course, coding gain is obtained with the sacrifice of reducing transmission efficiency. For example, in the case of rate 1/3 133 145 175 convolutional, every 3 coded bits only carries 1 data bit.

2.2.2 Modulation Schemes

In practical wireless communications, baseband signal cannot be transmitted without modulation. Information of baseband signal is transmitted in the way that parameter of high frequency carrier wave, such as amplitude or phase, is modulated by baseband signal, hence conveys the information that can be restored to original signal at the receiver. Selection of proper modulation scheme is essential to communication system design. This section presents coherent M-PSK and M-QAM schemes and compares their performances.

M-PSK

The idea of Phase-shift keying (PSK) modulation scheme is that information of baseband signal is conveyed by changing of carrier wave’s phase. Family of coherent

M-PSK includes BPSK, QPSK, 8PSK and 16PSK, where BPSK, 8PSK and 16PSK are in discuss in this section.

BPSK is a binary digital modulation scheme, which is also the simplest form of

M-PSK. Binary data (“0” and “1”) are represented by two carrier waves with phases of

and respectively, which has the following form: 0

(2.7) 2, 0

(2.8) 2 2 , 0, where is a constant amplitude, is carrier frequency and is the bit duration. Suppose bit energy is denoted as , equation (2.7) and (2.8) can be expressed as:

(2.9) 2, 0

(2.10) 2, 0

The constellation of BPSK is shown in following Figure:

Figure 2-6 BPSK constellation.

The bit error probability of BPSK in AWGN is given as:

2 1 (2.11) 0 2 0

Compared to binary modulation, multi-level modulation has higher spectra utilization, thus increase transmission speed. In 8PSK, for instance, a symbol can represent 3 bits.

That is to say, bandwidth efficiency increase to 3 times compared to BPSK. Signal modulated by 8PSK has the following form:

(2.12) cos2 , 0 , 0 7

where is the symbol energy and is the symbol duration. is defined as:

(2.13) , , 0 7

The constellation of 8PSK is shown as:

Figure 2-7 8PSK constellation with Gray mapping.

The bit error probability of 8PSK under AWGN is given as:

(2.14) sin

Another case of M-PSK is 16PSK, whose modulated signal is similar to that of

8PSK:

(2.15) cos2 , 0 , 0 15 where in 16PSK is defined as:

(2.16) , , 0 15

The constellation of 16PSK is shown as:

Figure 2-8 16PSK constellation with Gray mapping.

The bit error probability of 16PSK under AWGN is given as:

(2.17) sin

BPSK has the lowest BER by comparison with their bit error probability equations.

It can be expected that BER performance becomes worse when gets higher. The BER performance curve of OFDM system with different MPSK schemes is shown below:

Figure 2-9 The BER performances of OFDM system with different MPSKs.

M-QAM

Quadrature amplitude modulation (QAM) is the most commonly used type of modulation technique in OFDM [7]. As a multi-level modulation scheme, QAM modulation scheme acquires higher data rate, that is, higher bandwidth efficiency, by sacrificing power utilization. It implies that higher SNR is required for QAM if we intend to maintain low bit error rate.

QAM can be regarded as the combination of two modulations on in-phase (real) and quadrature (imaginary) branches since the carrier wave experiences amplitude as well as phase modulation [10]. In the case of 16QAM, the coordinates of the th message point is , where of QAM is an element of the , , following matrix, 4 4  −− )3,3()3,1()3,1()3,3(    −− )1,3()1,1()1,1()1,3( ba },{ =   ii  −−−−−− )1,3()1,1()1,1()1,3(    −−−−−−  )3,3()3,1()3,1()3,3( 

The constellation of 16QAM is shown in Figure 2-10,

Figure 2-10 16QAM constellation with Gray mapping.

In QAM modulation scheme, the in-phase and quadrature components are independent. Herein, the probability of correct detection can be written as: (2.18) 1

where is the probability of symbol error for either in-phase component or quadrature component. Thus, the probability of symbol error for 16 QAM is given by:

(2.19) 1

The equation of is written as: (2.20) 1 √ 2 1 √ , 16

The probability of symbol error is therefore written as:

(2.21) √ 1 1 √ , 16

There are four bits per symbol for 16 QAM modulation scheme. The bit error probability for 16 QAM is denoted as:

(2.22)

System modulated by M-ary QAM scheme acquires higher data transmission rate by suffering BER degradation. Figure 2-11 shows BER curves of OFDM systems with

M-ary QAM under AWGN:

Figure 2-11 BER curves of OFDM with M-ary QAMs under AWGN.

2.2.3 IFFT and FFT

As stated in Chapter 1, the modulation and demodulation of OFDM baseband signal can be implemented by inverse discrete Fourier transform (IDFT) and discrete

Fourier transform (DFT). Let be the OFDM signal. By sampling signal with the rate , is written as, / , 0,1,2,⋯,1 (2.23) ∑ exp 01

where is the original data information. In the same manner, is restored at the receiver by performing reverse calculation, i.e. DFT,

(2.24) ∑ exp 0 1

In practical OFDM applications, fast Fourier transforms (FFT/IFFT) are implemented to reduce computing complexity.

2.2.4 Cyclic Prefix

Inter symbol interference (ISI) is one of the most important issue in mobile communication. Due to the effect of multipath channel, transmitted wireless signals are propagated through various paths in the environment, which arrive at the receiver with different phase, resulting in time dispersion. If the symbol width is smaller than maximum spread delay, the performance is degraded due to ISI thus restrict high rate

23 transmitting.

One of the most important features of OFDM is inherent resistance to ISI caused by time dispersion. By splitting the input data stream into N sub-streams, period of data symbol in each sub-channel is expanded by N times compared to original data. Herein, delay spread is less likely to cause ISI.

To further eliminate ISI, guard interval is applied to OFDM system. In conventional communication system, null sequences are inserted among symbols.

Length of guard interval is set to be larger than maximum delay spread in wireless communication. However, such guard interval will damage the orthogonality of OFDM system and introduce inter-carrier interference (ICI), since sub-carriers cannot maintain integral periodic inequality due to multipath.

The cyclic prefix (CP) is introduced to combat ISI and ICI, which is a copy of the last part of the OFDM symbol adding to the start of the OFDM signal.

Figure 2-12 Cyclic prefix adding

The adding of cyclic prefix can effectively eliminate the affect of ISI and ICI. The length of cyclic preifx is chosen larger than maximum delay spread. Each OFDM symbol is preceded a copy of the last part of the OFDM symbol. Theoretically, cyclic prefix can completely keep the signal free from ISI and ICI, as long as the maximum delay is smaller than the length of cyclic prefix.

Chapter 3

Characteristic and Modeling of Hyper-Rayleigh Fading Channel

Analysis and modeling of radio propagation is a most important issue in wireless communication. Only by properly measuring the characteristic of fading channel, communication system can be correctly developed. Recent studies show propagation channel in some WSN applications where sensor nodes deployed within cavity environment exhibits worse behavior than Rayleigh fading channel, which is referred to as hyper-Rayleigh fading channel [12]. Therefore, modeling of hyper-Rayleigh fading channel is the central issue. Chapter 3 describes the cause and effect of small scale fading. Characteristic and modeling of hyper-Rayleigh fading channel is presented and comparison is made between conventional small scale fading model and hyper-Rayleigh fading model.

3.1 Small Scale Fading

There are generally two types of fading in wireless communication: large scale and small scale fading. Large scale fading, which is the major concern in microwave 26 communications, is mainly caused by long propagation distance and large obstacles like mountains and buildings, where signal power attenuates with the increase of distance. Other situations, such as the change of climate, also cause large scale fading.

The affect of large scale fading is not taken into consideration when the area of communication is relatively small. Therefore, we focus our study on modeling of small scale fading propagation. Figure 3-1shows illustration of large scale fading and small scale fading:

Figure 3-1 large scale fading and small scale fading.

Wireless communication channel, especially mobile communication channel, is the most complex and most hostile type of channel. Apart from additive white Gaussian noise (AWGN), wireless signal is attenuated and distorted by many other kinds of fading and interference. Wireless signal is propagated through various paths in the unpredictable environment. At the receiver, line of sight (LOS) waves, reflected waves

27 and scattering waves cause severe distortion of the original signal. The received waves can be categorized into two classes: specular waves and diffuse waves.

3.1.1 Multipath Fading

The main characteristic of wireless fading channel is multipath. In the wireless channel environment, transmitting signals are propagated through various paths, which arrive at the receiver with different phases and amplitudes, resulting in fading signal.

We describe this kind of situation as multipath fading, which will cause amplitude and phase fluctuations and time dispersion in the received signals.

During signal propagation, some received signal waves often spread to other signals because of delay spread, which causes inter-symbol interference (ISI).

Maximum delay spread is used to measure multipath fading in specific propagation environment.

In the view of frequency domain, delay spread could result in frequency selective fading [11]. For different frequency component of the signal, wireless channel exhibits diverse random response. Signal wave will suffer distortion after fading. Coherence bandwidth is introduced to measure frequency selective fading. In practice, coherent bandwidth is defined as:

(3.1)

28 where is maximum delay spread in fading circumstance. If the signal transmission rate is so high that signal bandwidth exceeds coherence bandwidth of wireless channel, frequency selective fading is occurred. Otherwise, when signal bandwidth is smaller than , signal is consider to experience flat fading.

3.1.2 Doppler Effect

Doppler Effect is occurred when mobile station is receiving signal in a move. The frequency of signal is changing depends on the speed and direction of mobile station.

Such characteristic of wireless channel is referred to as time-variance.

In a time-varying channel, the transfer function is varying with time, i.e. diverse signals are received when the same signal is transmitting at different time. Doppler shift is the reflection of time-variance in mobile communication system. At the receiver, transmitted single frequency signal becomes signal with bandwidth and envelop after time-varying channel. This phenomenon is also known as frequency dispersion, as shown in Figure 3-2:

Figure 3-2 Illustration of Doppler shift in time varying channel.

3.1.3 Expression of Received Signal

In wireless communication, transmitting signal is propagated through various paths. Herein, received signal is combination of multipath waves that arrive at the receiver with different phases and amplitudes. The complex baseband voltage of the received signal, , is expressed as [5]: (3.2) ∑ exp

Where is denoted as the number of multipath waves, while and are the corresponding amplitudes and phases respectively. The propagation waves can be classified as specular waves and diffuse waves. Specular waves are characterized as strong waves, such as LOS wave and reflected waves. The diffuse waves are made up of many faint waves with random magnitudes and phases. Therefore, (3.2) has the

30 following form:

(3.3) ∑ exp ∑ exp

where is the number of specular waves and is the number of diffuse components.

Figure 3-3 Illustration of specular components and diffuse components

The in-phase and quadrature parts of diffuse voltage are proved to be independent, zero-mean Gaussian random variables, with identical variance . The equation (3.3) can be written as:

(3.4) ∑ exp

Equation (3.4) is a general expression of received signal, from which specific expressions under various fading environments can be derived. Three small scale fading environments, Rayleigh fading, Rician fading and hyper-Rayleigh are discussed in the thesis. The major distinction between these fading scenarios is the existence of specular components, as shown in Table 3-1:

Table 3-1 Fading scenario characterized by specular components

Fading scenario Specular components

Rayleigh fading Not exist

Rician fading 1 specular wave

Hyper-Rayleigh fading 2 specular waves

The characteristic and behavior of three fading channels will be discussed in detail in the next section.

3.2 Typical Small Scale Fading Models

In this section, two typical small-scale fading channel models, Rayleigh and

Ricean, are presented and investigated. Modeling of radio propagation is essential to wireless communication systems since it enables one to adopt appropriate means to reduce signal attenuation and distortion. Rician and Rayleigh models are most commonly used to describe wireless propagation fading channel, especially mobile communication channel.

3.2.1 Rayleigh Fading Model

Rayleigh fading environment is characterized by many multipath components, each with relatively similar signal magnitude, and uniformly distributed phase, which means there is no line of sight (LOS) path between transmitter and receiver. Therefore,

Rayleigh fading is often considered as a worst-case scenario for mobile communications within urban environments. Figure 3-4 shows the Rayleigh fading scenario:

Figure 3-4 Illustration of Rayleigh fading scenario.

For Rayleigh fading channel, there exist a large number of multipath components, each with random amplitude and phase. According to central limit theorem, the real and imaginary components of the complex envelope comply with Gaussian distribution.

Suppose and denoted real and imaginary components respectively. The y possibility density function (PDF) of the two is respectively expressed as:

(3.5) √ (3.6) √

where σ represents the standard deviation of the envelope amplitude (also known as the rms value of the envelope). Since both random variables are independent and identically distributed, the joint distribution can be written as

(3.7) , ∙

Therefore, the PDF for the received signal magnitude under Rayleigh fading is:

(3.8) exp

The PDFs of Rayleigh fading with different variances is shown in Figure 3-5:

Figure 3-5 PDFs of Rayleigh fading with different variances

The CDF is introduced to describe the statistical characterization of fading channel. CDF plays an essential role in characterizing fading scenarios. By integrating the PDF shown above, the CDF of Rayleigh fading can be written as:

(3.9) 1

The CDFs of Rayleigh fading with various variances is shown in Figure 3-6:

Figure 3-6 CDFs of Rayleigh fading with various variances

Jakes’ model [17] is extensively applied to the modeling of Rayleigh fading channel. By summing a finite number of sinusoids, the Rayleigh fading function of ith path is generated by Jakes’ model. The envelope fluctuation follows a Rayleigh

35 distribution, and the phase fluctuation follows a uniform distribution on the fading in the propagation path. Therefore, the simulation equation of Rayleigh channel can be written as follows [18]:

2 2 2 2 2 1 1 (3.10) ∑ 2

where is the maximum Doppler frequency and is the number of waves applied to generate Rayleigh fading signal.

3.2.2 Rician Fading Model

Rician fading model is another most commonly employed model, which is adopted when there is a dominant LOS path and a number of weak multipath components in propagation environment. In mobile communication, Rician fading model can be used when mobile station is moving across the open ground, such as suburb and rural area, where LOS signal can be received. Figure 3-7 shows Rician fading scenario:

Figure 3-7 Illustration of Rician fading scenario.

The fading amplitude at the time instant can be represented as r ith (3.11)

Where is the amplitude of the specular component, and are samples of x y zero-mean stationary Guassian random processes each with variance . The ratio of σ specular to diffuse energy is known as Rician K-factor, which is given by

(3.12) /2

K-factor reflects extent of LOS signal in Rician fading. If , the LOS signal ∞ is so strong that diffuse waves can be regarded as white Gaussian noise. The fading

37 behavior will follow Gaussian distribution. If , on the other hand, there is no LOS 0 path in propagation channel, and the fading behavior will show the characteristic of

Rayleigh fading. The Rician PDF is shown as below,

(3.13)

where is 0th order modified Bessel function of the first kind. Suppose ∙ variance , the PDFs of Rician fading with different value is shown in Figure 1 3-8:

Figure 3-8 PDFs of Rician fading with various .

The Rician CDF has the following form:

(3.14) 1 , 38 where is the Marcum-Q-function. Figure 3-8 shows the CDFs of Rician fading with various :

Figure 3-9 CDFs of Rician fading with various .

There is no closed-form expression of mean value of Rician distribution; however, mean-squared value can be derived as

(3.15) 2

In system simulation, it is often required a Rician distribution with unit mean-squared value, i.e., , so that the signal power and the signal-to-noise ratio (SNR) Er 1 coincide. Therefore, we can get the following expressions

(3.16) 2 39

(3.17)

Therefore, the fading amplitude at the h time instant can be written in the form rt i

√ (3.18)

3.3 Hyper-Rayleigh Fading Model

Wireless sensor networks (WSN) have gained much interest lately as an effective means to monitor industrial, military and natural environments. As applications of

WSN being widely spread, selection of proper propagation model for particular WSN application has become a vital problem. Argument of anti-interference technologies’ feasibility is based on correct modeling of the fading circumstance. Large scale model, such as two-wave and log-shadowing models, are employed to represent radio propagation phenomena. Although sensor nodes are statically equipped in most cases, it is often found necessary to consider small scale fading in some WSN applications.

Rayleigh fading model was found a proper model to represent small scale fading behavior occurred in WSN environments. However, for some particular WSN applications where sensor nodes are deployed within cavity environment, such as airframe and shipping containers, transmitting signal experiences severe multipath fading which is worse than fading behavior predicted by Rayleigh fading model [2] [12]

[13]. Therefore, a more applicable fading model is required to employ to represent such

40 small scale fading circumstance, which is referred to as hyper-Rayleigh fading channel.

Recent researches suggest that two-wave with diffuse power (TWDP) model is a good candidate of hyper-Rayleigh fading model [2] [13] [14].

Two-wave with diffuse power model is considered as the most promising candidate model for hyper-Rayleigh fading scenario [2] [3]. Based on in-vehicle data collection in airframe and bus, research shows that propagation behavior is very similar to the characteristic of TWDP model. Take airframe environment for example, specular waves are expected to receive for each sensor node. Therefore, TWDP model instead of

Rayleigh model is an applicable candidate in this occasion. A typical hyper-Rayleigh fading scenario is shown in Figure 3-10:

Figure 3-10 Illustration of hyper-Rayleigh fading scenario.

The TWDP model is characterized by two strong specular components with

41 constant amplitude and random phases, along with a number of scattering waves.

Specifically, the received voltage of complex envelop is dominated by two V specular waves with constant amplitude ( and ) and random phase (φ and φ ), V V while the remaining components are categorized as non-specular or diffuse components.

The diffuse components are made up of many waves of random magnitude and phase, the latter being uniformly distributed over [0, 2π). The diffuse components are the same with the Rayleigh envelope component.

If no specular components existed in propagation environment, the fading channel is characterized as Rayleigh fading. By adding a LOS signal to the channel, Rician fading model becomes a proper candidate to represent fading behavior. TWDP model is similar to the Rician fading model but with two specular components [15],

(3.19) ∑

where N is the number of waves, represent two specular and components, is denoted as diffuse components. ∑ In Rician fading channel, ratio of specular to diffuse energy is introduced to demonstrate reflects extent of LOS signal in Rician fading. Similarly, ratio of average specular power to diffuse energy is utilized in TWDP model to indicate relative weights between specular components and diffuse components. As there are two specular components in TWDP model, peak to average specular power ratio is also an important parameter. Therefore, two parameters ( ) are introduced to ∆ and characterize TWDP model. are given as: ∆ and 42

(3.20) ∆ 1 σ (3.21) K

The relation between TWDP model and other fading models can be defined by . For Rayleigh fading, there is no specular wave, thus is not applicable ∆ and ∆ and . In the case of Rician fading, peak specular power and average specular 0 power are identical, herein is equal to 0. The diffuse components are trivial compared ∆ with two strong specular waves in Two-ray model, which make approach infinity. TWDP model describes the fading situation between Rician fading and two-ray fading.

To sum up, the comparison of four fading models, Rayleigh, Rician, two-ray and hyper-Rayleigh fading, can be made by the parameters , as Table 3-2 shows: ∆ and Table 3-2 Comparison between four fading models

Fading scenario ∆ Rayleigh fading Not applicable 0 Rician fading ∆ 0 0 Two-ray fading ∆ 0 ≫ 0 Hyper-Rayleigh fading ∆ 0 0

There is no closed-form expression of TWDP PDF. [5] presents approximate of

TWDP PDF:

(3.22) exp ∑ , , Δ

43 where is the order of approximate, is the corresponding value, is the function defined as:

, , exp 2 1 exp 2 1 (3.23) where is 0th order modified Bessel function of the first kind. ∙ With the increase of , the approximate PDF becomes more accurate. In the thesis, the order of is employed and the values of three coefficients are 3

[5]. Figures 3-11, 3-12 and 3-13 show the approximate , , PDFs of TWDP with various : ∆ and

Figure 3-11 PDFs of TWDP fading with . 3

Figure 3-12 PDFs of TWDP fading with . ∆ 1

Figure 3-13 Comparison of PDFs among Rayleigh, Rician and TWDP.

The above figures indicate that PDFs of TWDP fading is similar to Rician PDFs when are small. When the value of exceeds 3, TWDP PDFs exhibit ∆ and poorer performance than Rician and Rayleigh PDFs.

There is no closed-form equation for TWDP CDF. However, the range of TWDP

CDF can be determined by two-ray fading and Rayleigh fading, which are served as the upper bound and lower bound respectively.

Two-ray fading can be viewed as the special case of TWDP fading, where diffuse powers are negligible comparing to the two strong specular components with the similar magnitude respectively. Therefore, the PDF can be written as [5]: and

(3.24) , |V V| r V V

It has been proven that two-ray fading has the worst situation when the two components have the same magnitude . For example, two waves arrive at the receiver with opposite phases , thus the waves will cancel out each other θ and θ and result in severe performance degradation. Suppose , the PDF of the 1 received envelope with ∞ is ∆ 1 and K

(3.25) √ , 0 2

The two-ray CDF is shown as follows,

(3.26) 1 , 0 2

Therefore, the range of TWDP CDFs can be determined:

Figure 3-14 Illustration of range of TWDP CDFs

Rayleigh CDF can be regarded as the special case of TWDP when . TWDP 0 CDF reaches the upper limit when approaches to infinity, where TWDP fading exhibits two-ray propagation behavior.

A test was carried out in MD-90 aircraft, where a portable signal generator (PSG) was placed at the aisle floor (denoted as A), on seat backs (B) and inside open stowage bins [2]. By plotting the CDF for each of the three records, Figure 3-13 shows that

Trace A exhibits Rician fading behavior, Trace B Rayleigh and Trace C hyper-Rayleigh:

Figure 3-15 CDFs of the three traces [2].

In practice, the amplitude of the two specular components cannot be strictly identical in hyper-Rayleigh fading. The diffuse power is various depending on the specific sensors deployments. However, the value of is likely to go below 2.Therefore, hyper-Rayleigh fading channel can be represented by TWDP model with parameters . ∆ 1 and K 0 Similar to Rician fading, is variance of diffuse components, which comply σ with zero-mean stationary Guassian random processes. Mean-squared value of TWDP distribution can be derived as

(3.27) 2

By substituting with , we have V V ∆ and K (3.28) ∆

The resultant of the two specular paths can be written as

(3.29) V V Vexpjθ θ

where is a random variable uniformly distributed over the interval . θ θ θ 0,2π

Chapter 4

2-D Pilot Aided Channel Estimation in OFDM System

In mobile communication, radio signal transmitted through the wireless channel is suffered from time dispersion and frequency dispersion caused by multipath propagation and Doppler shift, which results in severe performance degradation of the communication system. Though OFDM system can significantly reduce the effect of multipath fading, OFDM signal is very sensitive to Doppler shift, since Doppler shift may bring about frequency offset and impair the orthogonality of OFDM sub-carriers

[15]. Channel estimation is designed to overcome fading and interference in OFDM system by finding out the frequency response of the fading channel. In Chapter 4, three pilot-aided channel estimation techniques are presented and analyzed.

4.1 Procedure of Channel Estimation in OFDM System

Pilot-based channel estimation has been proven to be a feasible and effective method for OFDM systems. The idea of pilot-aided channel estimation is to insert known pilot sequence into data symbol complying with specific pilot pattern, so that the channel information can be obtained at the receiver by estimation techniques and 50 interpolation methods.

Anti-interference capacity and transmission efficiency may vary with various pilot patterns, which will be further discussed in the next section. In this section, the procedure of channel estimation in OFDM system is presented. After pilot insertion and

IFFT, the modulated signal will be transmitted through wireless communication channel, which is set to be a frequency selective time varying fading channel with AWGN. Hence, the received signal is given as follows: (4.1)

where is additive with Gaussian noise, and is the channel impulse response due to multi-path delay, which is expressed as [16]:

(4.2) ∑

where is the total number of propagation paths, h is the complex impulse response γ of the th path, is the th path Doppler frequency shift which causes ICI of the i f i received signals. is delay spread index, is the sample period and is the th path λ T τ i delay normalized by the sampling time.

At the receiver, signal is sent to FFT. After FFT processing, the signal is given as:

(4.3) , 0,1,⋯,1

where is ICI due to Doppler frequency. Suppose there is ICI occurred because of cyclic prefix. Equation (4.3) can be written as:

(4.4) , 0,1,⋯,1

The received pilot signals are extracted to obtain the channel impulse response at the pilots, which is given as:

(4.5)

The channel transfer function can be estimated based on . With the estimated channel transfer function , the data signal is recovered:

(4.6)

4.2 2-D Pilot arrangement

Pilot arrangement is an essential issue in pilot-based channel estimation. Under various fading environments, it is important to utilize proper pilot pattern to achieve optimum BER performance. Pilot patterns can be grouped into two categories: one-dimensional (1-D) pilot patterns and two-dimensional (2-D) pilot patterns [20].

Pilots are inserted in either time domain or frequency domain, while 2-D pilots are inserted in both time domain and frequency domain. This section gives an extensive analysis and comparison between 1-D pilot patterns and 2-D pilot patterns.

4.2.1 1-D Pilot Pattern

There are two typical types of 1-D pilot pattern: block type and comb type [21]. In

block type pilot pattern, pilot symbols are inserted periodically in time domain. In

comb type pilot pattern, pilot symbols are inserted in frequency domain.

In block type pilot pattern, OFDM channel estimation symbols are inserted

periodically in time domain, where all sub-carriers are used as pilots. The block type

pilot pattern is illustrated in Figure 4-1:

Figure 4-1 Block type pilot pattern.

Given that is pilot insertion interval and is the number of symbols transmitted, the signal sequence can be divided into blocks. Each block has one / pilot symbol, where impulse response at the pilot symbol is 1, 2, ⋯ , served as impulse response of the th block:

Figure 4-2 Illustration of block type pilot channel estimation.

Since there are pilot signals in every sub-carrier, block type pilot pattern is an appropriate selection under frequency selective slow fading channel. If the channel impulse is constant during each block, there will be no channel estimation error since every OFDM symbol in the th block has the same channel transfer function with . However, block type pilot channel estimation is vulnerable to fast fading channel, since channel transfer function may vary rapidly even in one block.

It has been proved that comb type pilot pattern has better performance in fast fading channel, since part of the sub-carriers are always reserved as pilot for each symbol so that pilots are inserting through time domain to track fast fading [22].

Suppose is number of carriers in OFDM system and pilot signals are 54

uniformly inserted into according to the following equation: (4.7)

where . If , , where is the th pilot / 0 carrier value. The comb type pilot arrangement is shown in Figure 4-3:

Figure 4-3 Comb type pilot pattern

Unlike block type pilot pattern, each OFDM symbol is inserted with pilots, where channel impulse is only known at pilot tones. Therefore, channel transfer function of each OFDM symbol is obtained by interpolation methods. In the thesis, linear interpolation is applied. Figure 4-4 illustrates the processes of calculating channel transfer function for each OFDM symbol:

Figure 4-4 Illustration of comb type pilot channel estimation.

Comb type pilot pattern is widely recognized as a proper pilot arrangement for

OFDM system, due to its good performance against fast fading channels. The main disadvantage of comb type pilot channel estimation is its sensitivity to frequency selective fading, though cyclic prefix is added to prevent ISI.

4.2.2 2-D Pilot Pattern

Though block type pilot pattern and comb type pilot pattern have their advantage under certain fading scenarios, neither block type pilot pattern nor comb type pilot pattern is able to adapt to changing wireless radio propagation environment, which is always the case in mobile communications, where wireless channel exhibits frequency

56 selective property and time varying behavior. Moreover, there is another drawback of the 1-D pilot pattern, which is transmission inefficiency. For example, in an OFDM system with pilot interval , there are 12.5 percent of total transmitting signals are 8 occupied by pilot tones.

2-D pilot patterns show strong adaptability to changing communication environment, which is often the case in mobile communication. In 2-D pilot patterns, pilot tones are inserted in both frequency domain and time domain, which enable the system to combat fast fading frequency selective channel. A number of 2-D pilot patterns are proposed [23] [24], and rectangular pilot pattern is applied to OFDM system in the thesis for its easy implementation. Rectangular pilot arrangement is shown in Figure 4-5:

Figure 4-5 Rectangular type pilot pattern. 57

Rectangular pilot pattern is the typical type of 2-D channel estimation, which is extensively applied due to its easy implementation and strong resistance to frequency selective time varying channel. Pilot tones are periodically inserted into both frequency domain and time domain. The procedure of rectangular type pilot channel estimation is to implement channel estimation method to the OFDM sub-carriers in which pilot tones are inserted, in order to calculate channel impulse response for every OFDM sub-carrier.

Then channel transfer function of each OFDM symbol can be obtained by implementing channel estimation method again in time domain. Figure 4-6 shows the procedure of rectangular type pilot channel estimation:

Figure 4-6 Procedure of rectangular type pilot channel estimation.

Rectangular type pilot pattern can better track frequency selective time varying fading channel, since there are pilot tones in both frequency domain and time domain.

The performance comparison between rectangular type pilot pattern and 1-D pilot pattern is investigated in chapter 5.

Another advantage of 2-D pilot pattern is 2-D pilot pattern can significantly enhance transmission efficiency compare to 1-D pilot patterns. For 1-D pilot patterns, pilot symbols will take up 12.5% of transmitting symbols if pilot interval is set to be 8.

However, in the case of rectangular pilot pattern, the pilot ratio is only 6.25% when pilot interval is 4, which means for 512 16 input matrix, only 512 symbols are reserved for pilot insertion.

The shortcoming of 2-D pilot patterns is that channel estimation method and interpolation needs to be performed twice, both in frequency domain and time domain, which will increase computational complexity. That also means performances of 2-D pilot patterns are more rely on the accuracy and effectiveness of channel estimation methods and interpolation methods.

4.3 Channel Estimation Algorithms

In this section, two channel estimation methods, LS algorithm and LMMSE algorithm are introduced.

Least Square (LS) Algorithm

LS algorithm is simple but effective channel estimation method, which does not require the information of fading channels [25]. The estimate of the channel transfer function is defined as: (4.8)

where is the transmitted data matrix, and is the received information sequence. For comb type pilot channel estimation in OFDM system, the channel transfer functions at pilot symbols is obtained by LS algorithm. Thus, linear interpolation method is applied to determine the channel impulse response of the OFDM symbol:

(4.9) 1

The LS estimate of is susceptible to Gaussian noise and inter-carrier interference (ICI). For applications that require higher accuracy, LMMSE algorithm described below is applied to the system instead.

LMMSE (Linear Minimum Mean Squared Error) Algorithm

MMSE algorithm is proved to provide more dB gain in SNR over LS estimation by increasing computational complexity. The computational complexity of the MMSE estimator can be reduced by using a simplified linear minimum mean-squared error

(LMMSE) estimator.

LMMSE algorithm requires the knowledge of auto-correlation matrix of the channel frequency response , which is given as: (4.10) 60

where is channel frequency response at the pilot locations, the superscript ∙ denotes the Hermitian transpose.

For an exponentially decaying multipath power-delay profile , exp the correlation between th and th sub-carriers is given as:

(4.11) ,

where is RMS delay spread factor of the channel, is the length of cyclic prefix. τ Therefore, the channel correlation matrix has the following form: 0,0 0,1 ⋯ 0,1 1,0⋮ 1,1 ⋮ ⋯ ⋱ 1,1 ⋮ (4.12) ⋮ ⋮ ⋮ 1,0 1,1 ⋯ 1,1

LMMSE estimate can be viewed as a weighted combination of LS estimate , which is given: (4.13)

The weighting coefficient is expressed as: (4.14)

where is a constant depending on the signal constellation. In the case of 16QAM scheme, . 61

BER Performance Comparison

LS algorithm is the simplest channel estimation and the basis of other channel estimation technologies. The main drawback of LS algorithm is vulnerability to

Gaussian noise and ICI. The BER performance can be elevated by LMMSE algorithm, where weighting matrix is added to . However, LMMSE algorithm increases computational complexity of the system. Moreover, knowledge of channel information is required in LMMSE algorithm, which gravely restricts the implementation of

LMMSE algorithm, because of the changing propagation environment in mobile communications.

Chapter 5

Simulation Results and Performance Analysis

BER performance is the key factor in development of communication system, which represents systems’ robustness against fading and error-correcting capability. In this chapter, OFDM systems under various small scale fading channels are implemented and BER performances are simulated in MATLAB 2008. All simulation results are acquired under 512-point OFDM with CP length of 128.

5.1 Simulation Results with Different modulation Schemes

The BER performances of OFDM system with BPSK, 8PSK, 16PSK and 16QAM are investigated. It is expected that BPSK has the best BER performance, while 16PSK has the worst. It is important to note that 8PSK and 16QAM have the similar BER performance, but 16QAM has higher transmission efficiency. Therefore, 16QAM is regarded to have better performance than 8PSK.

Figure 5-1 shows BER performances of OFDM system with different modulation schemes under AWGN, and Figure 5-2 shows BER performances of different modulation schemes with coding rate 1/2 convolutional code: 133 171 63

Figure 5-1 Comparison of different modulation schemes under AWGN.

Figure 5-2 Comparison of different modulation schemes with coding.

5.2 Simulation Results under Rayleigh Fading Channel

BER performance of OFDM system under Rayleigh fading is investigated.

Table 5-1 Specification of OFDM system simulation under Rayleigh fading

Parameter Specification

Pilot Pattern Block type, Comb type, Rectangular type

Coding Scheme K=7, Convolutional Code

Maximum Doppler Frequency 200Hz

Channel Model Rayleigh Fading

Estimation Algorithm LS, LMMSE

Figure 5-3 Block type BER performances with LS and LMMSE under Rayleigh

Figure 5-4 Comb type BER performance with LS and LMMSE under Rayleigh.

It is well known that OFDM system is resistant to frequency selective fading yet sensitive to fast fading. Comb type pilot pattern caters for the demand of tracking fast fading, thus serves as a more appropriate pilot pattern for OFDM system than block type, which is evidently shown in Figure 5-3 and Figure 5-4. Note that LMMSE algorithm has better performance than LS algorithm. However, LMMSE algorithm will result in the increase of computational complexity. Moreover, LMMSE require the knowledge of fading channel, which is sometimes not available.

Theoretically, channel estimation with 2-D pilot insertion can achieve better BER performance compared to that with 1-D pilot pattern. In the thesis, a typical 2-D pilot pattern, rectangular type, is implemented into OFDM system. Figure 5-5 shows BER performance of rectangular type OFDM system with LS estimation and LMMSE 66 estimation:

Figure 5-5 Rectangular type performances with LS and LMMSE under Rayleigh.

Figure 5-5 demonstrates system’s strong resistance to frequency selective fast fading channel, when implementing rectangular type pilot channel estimation.

However, the hinder factor of 2-D pilot pattern is the increase of equipment complexity.

5.3 Simulation Results under Rician Fading Channel

Rician fading model is applied when there is LOS signal detected in communication environment. The propagation attenuation of Rician fading largely depends on the K-factor, which ranges from 0 to ∞. In worst case, when K goes to 0, the channel will show the characteristic of Rayleigh fading. When K approaches ∞, 67 the channel can be viewed as Gaussian channel. Figure 5-6 shows BER performances under Rician fading with various K-factors:

Figure 5-6 BER performances under Rician fading with various K-factors.

It can be seen from Figure 5-6 that BER performance deteriorates as K-factor goes high. Rayleigh fading curve and AWGN curve are served as the upper bound and lower bound of Rician fading, respectively.

The following figures show the BER performance of OFDM system with different pilot arrangements and channel estimation methods under Rician fading with K-factor equals to 6. Other parameters are the same with Table 5-1.

Figure 5-7 Block type BER performances with LS and LMMSE under Rician.

Figure 5-8 Comb type BER performances with LS and LMMSE under Rician.

Figure 5-9 Rectangular type BER performances with LS and LMMSE under Rician.

To sum up, BER performance is better under Rician fading than that under

Rayleigh fading, since there is dominant LOS signal at the receiver in Rician fading channel.

5.4 Simulation Results under Hyper-Rayleigh Fading Channel

TWDP model is employed to describe hyper-Rayleigh fading behavior. The characteristic of TWPD model is determined by two parameters, . With ∆ and K become higher, BER performance gets worse. Note that hyper-Rayleigh ∆ and K fading is a slow fading channel, since sensor nodes are placed statically.

Figure 5-10 BER performances under TWDP model with various K

Figure 5-11 Block type BER performances under TWDP with . ∆ 1, K 6 71

Figure 5-12 Comb type BER performances under TWDP with . ∆ 1 , K 6

Figure 5-13 Rectangular type BER performances under TWDP with . ∆ 1 and K 6 72

Unlike simulation results under Rayleigh fading, comb type pilot channel estimation and block type pilot channel estimation have similar performance under

TWDP fading. Comb type pilot pattern is designed to overcome Doppler Effect, while hyper-Rayleigh fading scenario is a slow fading channel. Hence, comb type pilot pattern doesn’t fit in TWDP fading. Rectangular type pilot pattern has similar BER performance with block type, but with higher transmission efficiency. Therefore, rectangular type pilot pattern is still the first priority of pilot patterns.

TWDP fading channel exhibits diverse propagation behavior with different values of The following figures show BER performance under TWDP fading with ∆ and K various , utilizing LS algorithm. ∆ and K

Figure 5-14 Comparison of BER performances under TWDP with . K 3 73

Figure 5-15 Comparison of BER performances under TWDP with . K 6

Figure 5-16 Comparison of BER performances under various fading channels.

From the above figures, it is important to note that TWDP fading model with

results in more severe degeneration of OFDM system performance ∆ 1 and K 6 than Rayleigh fading channel. BER Performances under TWDP fading become deteriorate as the value of are high. The product of is utilized to ∆ and K ∆ and K determine the propagation situation of TWDP fading channel.

Chapter 6

Conclusion and future work

The object of thesis is to explore the performances of OFDM system under recently proposed hyper-Rayleigh fading with 2-D pilot channel estimation. Three small scale fading channel channels, Rayleigh, Rician and hyper-Rayleigh, are analyzed and simulated. Among the three fading channels, hyper-Rayleigh fading behavior exists in WSN applications and its propagation characteristic can be represented by TWDP model. Channel estimation techniques to overcome frequency selective fading and fast fading channel are also investigated. The BER performance results indicate that 2-D pilot patterns that insert pilots in both frequency domain and time domain is adaptable to fast changing wireless communication channels.

Due to the limited time, issue of Synchronization is not included in the thesis, which is, however, an essential issue in developing OFDM system. Accurate synchronization is necessary for OFDM system, since sub-carriers need to be kept strictly orthogonal. The thesis also fails to investigate other 2-D pilot patterns, such as diamond type 2-D pilot patterns and tile type 2-D pilot patterns. In addition to two-wave with diffuse power model, the characteristic and applicability of three-wave

76 with diffuse power model has gained more and more attention, which may probably better represent the propagation situation of hyper-Rayleigh fading. Future work will focus on these aspects.

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