A Study of Channel Estimation for OFDM Systems and System Capacity for MIMO-OFDM Systems

A Study of Channel Estimation for OFDM Systems and System Capacity for MIMO-OFDM Systems

Abstract of thesis entitled

“A Study of Channel Estimation for OFDM Systems and System Capacity for MIMO-OFDM Systems”

Submitted by Zhou Wen

For the degree of Doctor of Philosophy at the university of Hong Kong in July 2010

This thesis concerns about two issues for the next generation of wireless communications, namely, the channel estimation for orthogonal frequency-division multiplexing (OFDM) systems and the multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) system capacity

For channel estimation for OFDM systems over quasi-static fading channels having

resolvable mulitipath number L, a novel fast linear minimum mean square error (LMMSE)

channel estimation method is proposed and investigated The proposed algorithm deploys Fourier transform (FFT) and the computational complexity is therefore significantly

reduced to O(Nplog2(Np)), as compared to that of O(Np3) for the conventional LMMSE

method, where the notation O(·) is the Bachmann–Landau function and Np is the number of pilots for an OFDM symbol The normalized mean square errors (NMSE) are derived in closed-form expressions Numerical results show that the NMSE is marginally the same with that of the conventional LMMSE for signal to noise ratio (SNR) ranges from 0 dB to

25 dB For channel estimation for OFDM systems over fast fading and dispersive channels, a novel channel estimation and data detection method is proposed to reduce the inter-carrier interference (ICI) A new pilot pattern composed of the comb-type and the grouped pilot pattern is proposed.A closed-form expression for channel estimation mean square error (MSE) has been derived For SNR = 15 dB, normalized Doppler shift of 0.06,

and L = 6, both computer simulation and numerical results have consistently shown that

the ICI is reduced by 70.6% and 43.2%, respectively for channel estimation MSE and bit error rate (BER) The pilot number per OFDM symbol is also reduced significantly by 92.3%, as compared to the comb-type pilot pattern

A closed-form mathematic expression has been proposed for the capacity of the closed-loop MIMO-OFDM systems with imperfect feedback channel The lower threshold

of feedback SNR is derived For L = 6, numerical results show that the lower threshold of

feedback SNR is proportional to antenna numbers N′ and system SNR The increasing rate of the feedback SNR threshold increases from 0.82 to 1.01 when N′ increases from 2 to 16

The variance and mean of OFDM system capacity over Rayleigh channels and Ricean channels have been respectively investigated that the closed-form expression for the capacity variance has been proposed The resultant system capacity variances over the two channels are respectively evaluated by numerical method and also verified by computer simulation The joint probability density function (PDF) of two arbitrary correlated Ricean random variables has also been derived in an integral form Numerical results reveal that

the variance of OFDM system is proportional to SNR and inversely proportional to L for

the two channels respectively For the same two respective channels, the variance

marginally increases with a linear rate of 0.166 bit2/dB and 0.125 bit2/dB, when L = 2 and

SNR ranges from 0 dB to 15 dB The variance is reduced from 1.75 bit2 to 1.30 bit2 and from 1.48 bit2 to 1.26 bit2, when SNR = 10 dB and L ranges from 2 to 4

(Total words: 495)

A Study of Channel Estimation for OFDM Systems and System Capacity for MIMO-OFDM Systems

by

Zhou Wen

B Eng., M Eng., USTC, P R China

A thesis submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy at the university of Hong Kong in July 2010.

Declaration

I declare that this thesis represents my own work, except where due acknowledgement is made, and that it has not been previously included in a thesis, dissertation or report submitted t to this University or any other institution for a degree, diploma, or other qualifications

Signature: _ Zhou Wen

I also gratefully acknowledge Prof V.O.K Li, Prof G.L Li, Prof Y.C Wu, Prof S.C Chan, Prof T.S Ng and Prof Agnes S.L Lam for their interesting courses and helpful discussions I would like to thank the office staff and technical staff from the EEE department for their helpful administrative and facility supports Especially, Ms Julie Hung’s readiness to help students is very impressive I also appreciate the HKSAR government for the studentship support to the study in the University of Hong Kong

I am extremely grateful to all my friends and classmates who have kindly provided me assistance and companionship in the process of preparing this thesis: Dr Zhi Zhang, Dr Zhiqiang Chen, Dr Mingxiang Xiao, Dr Fei Mai, Mr Xueyong Liu, Mr Xiaoguang Dai, Ms Ziyun Shao, Mr Ka-Chung Leung, Mr Peng Zhang, Dr Yanhui Geng, Ms Qiong Sun, Mr Haoling Xiahou, Mr Zhibo Ni, Mr Jun Zhang, Mr Xiaolei Sun, Mr Chengwen Xing They have made the life during the past four years an enjoyable and memorable experience

Finally, I wish to express my hearty gratitude to my parents, for their encouragements and

love in all my endeavors

1.2 Organization and contributions of the thesis 5

Chapter 2: OFDM systems and MIMO systems 9

2.1 Wireless Channel 10

2.1.1 Large scale propagation 11

2.1.2 Small scale propagation 13

2.1.3 Typical wireless channel models 17

2.2 OFDM systems 20

2.2.1 Basic principles and characteristics for OFDM systems 21

2.2.2 Peak-to-Average (PAR) of OFDM systems 30

2.2.3 Channel estimation for OFDM systems 33

2.2.4 Synchronization of OFDM systems 38

2.2.5 Advantages and disadvantages of OFDM systems 39

2.3 MIMO systems 40

2.3.1 Basic MIMO system model 40

2.3.2 Functions of MIMO systems 42

2.3.3 Overview of Space Time codes 45

2.3.4 Capacity of MIMO systems 52

3.3 The Proposed Fast LMMSE Algorithm 63

3.3.1 Properties of the channel correlation matrix in frequency domain 63

3.3.2 The proposed fast LMMSE channel estimation algorithm 65

3.3.3 Computational complexity comparison between the proposed method and the conventional LMMSE method 69

3.4 Analysis of the Mean Square Error (MSE) of the Proposed Fast LMMSE Algorithm 70

3.4.1 MSE analysis of the conventional LMMSE algorithm 71

3.4.2 MSE analysis for the proposed fast LMMSE algorithm 72

3.5 Numerical and Simulation Results 75

4.3 The Proposed Channel Estimation and Data Detection 92

4.3.1 The proposed pilot pattern 92

4.3.2 Channel Estimation and data detection for the first M1 OFDM symbols of

each block 94

4.3.3 Channel estimation and data detection for the last M2 OFDM symbols of each block 95

4.3.4 Summary of the proposed channel estimation and data detection 98

4.4 Analysis of MSE of the proposed channel estimation method 99

4.4.1 MSE analysis of channel estimation for the first M1 OFDM symbols 100

4.4.2 MSE analysis of channel estimation for the last M2 OFDM symbols 103

4.4.3 MSE analysis of channel estimation for one OFDM block 105

4.5 Numerical and Simulation Results 106

4.6 Conclusion 112

Chapter 5: MIMO-OFDM system capacity with imperfect feedback channel 118

5.1 The open-loop and closed-loop capacity for MIMO Systems 119

5.1.1 MIMO system model 119

5.1.2 MIMO system capacity 120

5.1.3 Numerical Results and discussion 124

5.2 The closed-loop capacity with imperfect feedback channel for MIMO-OFDM systems 127

5.2.1 System Model 128

5.2.2 Closed-Loop Capacity and Feedback SNR for MIMO-OFDM Systems 1305.2.3 Numerical Results 136

5.3 Summary 142

channels 144

6.1 Introduction 145

6.2 OFDM System Model 147

6.3 OFDM System Capacity 148

6.3.1 OFDM system capacity over Rayleigh fading channels 148

6.3.2 OFDM system capacity over Ricean fading channels 153

6.4 Numerical and Simulation Results 157

APPENDIX B: The derivation of equation (3-20) in Chapter 3 171

APPENDIX C: The derivation of the joint PDF of two arbitrary correlated Ricean random variables 173

Appendix D: List of Abbreviations 176

REFERENCES 179

Publications 191

List of Figures

Fig 1.1: Organization of the thesis 6

Fig 2.1: Path Loss, shadowing and multipath versus distance 11

Fig 2.2: The Doppler power spectrum function expressed by (2-4) 14

Fig 2.3: The multi-path effect between the transmitter and the receiver in wireless communication 14

Fig 2.4: Time varying impulse response of a wireless channel, for the path number N = 3, 4, and 5 15

Fig 2.5: Four kinds of small scale propagations 16

Fig 2.6: PDFs for Rayleigh fading with the variance σ2 = 0.5, 2, and 5, respectively 17

Fig 2.7: PDFs for Ricean fading with Ricean factor Kr = 0 dB, 10 dB, and 20 dB, respectively 18

Fig 2.8: PDFs for Nagakami-m fading with m = 0.5, 1, and 10 19

Fig 2.9: The continuous OFDM system model 22

Fig 2.10: The waveform ofG wk( ) 24

Fig 2.11: Equivalent transmitter for OFDM systems 25

Fig 2.12: Equivalent receiver for OFDM systems 25

Fig 2.13: CP for an OFDM symbol 26

Fig 2.14: SNRloss versus CP length 27

Fig 2.15: The inter-symbol interference of OFDM systems without CP 28

Fig 2.16: Extraction of the data in frequency domain 28

Fig 2.17: The discrete baseband OFDM system model 29

Fig 2.18: The output power versus the input power for a power amplifier 31

Fig 2.19: The power spectrum comparison between the input signal and the output signal passing through an amplifier 32

Fig 2.20: Two kinds of pilot patterns (black dot: pilot, white dot: user data) 34

Fig 2.21: Pilot-aided channel estimation for OFDM systems 35

Fig 2.22: The basic MIMO system model 40

Fig 2.23: Received signal after diversity operation 43

Fig 2.24: Diversity-multiplexing tradeoff, d*(r) versus r 44

Fig 2.25: The Alamouti STBC diagram for 2×2 MIMO systems 46

Fig 2.26: V-BLAST system diagram 49

Fig 2.27: The baseband MIMO-OFDM system model 55

Fig 3.1: Baseband OFDM system 61

Fig 3.2: Channel estimation based on comb-type pilots 62

Fig 3.3: The first row of the channel autocorrelation matrixppH HR ,A 82

Fig 3.4: The first row of the LMMSE matrix1SNRβ −⎛ + ⎞⎜ ⎟⎝ ⎠ppppH HH HRRI with different SNRs 83

Fig 3.5: Normalized Mean square error (NMSE) of channel estimation of LMMSE algorithm versus that of the proposed fast LMMSE algorithm by computer simulation and numerical method 83

Fig 3.6: NMSE of LMMSE algorithm with matched SNR and mismatched SNRs versus SNR, by simulation and numerical method, respectively 84

Fig 3.7: NMSE of the proposed fast LMMSE algorithm with matched SNR and mismatched SNRs versus SNR, by simulation and numerical method, respectively 84

Fig 3.8: Bit error rate (BER) of the LS, LMMSE, the proposed fast LMMSE and perfect channel estimation versus SNR 85

Fig 3.9: BER comparison between LMMSE channel estimation with matched SNR and LMMSE channel estimation with designed SNRs 85

Fig 3.10: BER comparison between the proposed fast LMMSE channel estimation with estimated SNR and the proposed fast LMMSE channel estimation with designed SNRs 86

Fig 4.1: Pilot pattern (gray circle: user data, black circle: pilot) 114

Fig 4.2: The normalized mean square error (NMSE) of channel estimation for the first M1OFDM symbols, forf Td = 0.01, 0.06 and 0.1, respectively 114Fig 4.3: The NMSE of channel estimation based on equi-spaced and grouped pilot pattern,

for the polynomial orderQ=1, 2, 3 and the normalized Doppler shift f Td = 0.01 and 0.1, respectively 115Fig 4.4: The NMSE of channel estimation based on grouped pilot pattern, forc=1, 2, 3 and

the normalized Doppler shift f Td = 0.01 and 0.1, respectively 115Fig 4.5: The NMSE of channel estimation based on grouped pilot pattern, for the number

of pilot groupsNgroup=18, 36, 72 and the normalized Doppler shiftf Td = 0.01 and 0.1, respectively 116Fig 4.6: The NMSE of channel estimation for the proposed algorithm and LS algorithm by

numerical method and simulation atf Td = 0.01 and 0.06, respectively 116Fig 4.7: Bit error ratio (BER) of LS, the proposed algorithm and the algorithm in [29], for

normalized Doppler shift f Td = 0.01 and 0.06, respectively 117Fig 5.1: The eigenmode transmission of MIMO systems 123Fig 5.2: The MIMO system open-loop capacity versus the number of transmitter

antennasNT, for the number of receiver antennasNR=1 124Fig 5.3: The MIMO system open-loop capacity versus the number of receiver antennasNR,

for the number of transmitter antennasNT =1 124

Fig 5.4: The capacities of the N by 1 MISO system, the 1 by N SIMO system, and the N by N MIMO system as a function of N, for SNR = 5 dB 125

Fig 5.5: The open-loop and closed-loop capacity for MIMO systems, versus SNR 126Fig 5.6: The closed-loop MIMO-OFDM system model 128Fig 5.7: The open-loop and closed-loop system capacity for MIMO-OFDM systems

having different transmitter antenna and receiver antenna numbers 138Fig 5.8: The capacity gain of the closed-loop capacity with imperfect feedback over that of

the open-loop capacity versus feedback channel SNR, forNT =NR =4 140Fig 5.9: The capacity gain of closed-loop capacity with imperfect feedback over that of the

open-loop capacity versus feedback channel SNR, for system SNR = 10 dB 140

for different antenna pairs 142Fig 6.1: The PDF of the capacity at a certain subcarrier,

k andk2 163

Fig 6.4: The variance of OFDM system capacity for the number of channel paths L = 2, 4,

and 8, over the Rayleigh fading channel 163Fig 6.5 The variance of OFDM system capacity versus the CP of an OFDM symbol in unit

of sample point, over the Rayleigh fading channel 164Fig 6.6: The variance of OFDM system capacity versus the number of subcarriers of one

OFDM symbol, for Rayleigh fading channels 164

Fig 6.7: The variance of OFDM system capacity over Ricean fading channels for L = 2, 4,

8, respectively 165Fig 6.8: The mean value of OFDM system capacity for Rayleigh fading channel and

Ricean fading channel, by numerical method 165Fig 6.9: The variance of OFDM system capacity for Rayleigh fading channel and Ricean

fading channel, by computer simulation and numerical method 166

Chapter 1: Introduction

The research on wireless communication systems with high data rate, high spectrum efficiency and reliable performance is a hot spot There are several advanced communication technologies or protocols proposed recently, including Orthogonal frequency division multiplexing (OFDM) [1], multiple input multiple output (MIMO) [2], Ultra-Wideband (UWB) technology [3], cognitive radio [4], World Interoperability for Microwave Access (WiMAX) [92], and 3GPP Long Term Evolution (LTE) [92], [93]

OFDM is an efficient high data rate transmission technique for wireless communication OFDM presents advantages of high spectrum efficiency, simple and efficient implementation by using the fast Fourier Transform (FFT) and the inverse Fast Fourier Transform (IFFT), mitigration of inter-symbol interference (ISI) by inserting cyclic prefix (CP) and robustness to frequency selective fading channel MIMO is the use of multiple antennas at both the transmitter and receiver to improve communication performance It is one of several forms of smart antenna technology MIMO technology has attracted attention in wireless communications, because it increases in data throughput without additional bandwidth or transmit power It achieves this by higher spectral efficiency and link reliability or diversity The combination of MIMO with OFDM technique is a promising technique for the next generation wireless communication A new protocol draft employing the MIMO-OFDM as the physical layer technology, IEEE 802.11n, as an amendment to IEEE 802.11 standards has been proposed [53] Wireless LAN technology

has seen rapid advancements and MIMO-OFDM has gradually been adopted in its standards The following table shows the existing IEEE 802.11 WLAN protocols

Table 1.1 Existing 802.11 WLAN Standards

2003

Released in 2009

MHz Frequency Band of

DSSS, CCK, OFDM,

MIMO

UWB is a technology for transmitting data spread over a large bandwidth (usually larger than 500 MHz) that shares among users UWB was traditionally applied in non-cooperative radar imaging Most recent applications include sensor data collection, precision locating, and tracking applications The concept of cognitive radio was first proposed by Dr J Mitola and Prof G Q Maguire [4] in 1999 and was an extension to the concept of software radio Cognitive radio is an intelligent communication system that could detect and track the communication environments It would adjust the transmitter and the receiver’s parameters adaptively according to the changes of environment parameters such as the mobile velocity of the user, so that the system stability could be

ensured, the system performance could remain a good condition, and the spectrum efficiency could be improved WiMAX is a telecommunications protocol that provides fully mobile Internet access The name "WiMAX" was created by the WiMAX Forum, which was founded in 2001 The forum refers to WiMAX as a standards-based technology enabling the delivery of last mile wireless broadband access as an alternative to cable and digital subscriber line (DSL) The basis of WiMAX is IEEE 802.16 standard which is sometimes referred to as “WiMAX” equivalently The current WiMAX revision is based on IEEE 802.16e, which was approved in December 2005 The physical layer of WiMAX adopts a lot of advanced technologies such as scalable orthogonal frequency division multiplexing access (OFDMA), MIMO, adaptive antenna array and so on Current WiMAX that is based on the IEEE 802.16e protocol belongs to 3G family Future WiMAX is based on IEEE 802.16m, which has been submitted to the International Telecommunication Union (ITU) for International Mobile Telecommunication Advanced (IMT-Advanced) standardization Future WiMAX, or the proposed WiMAX release 2, is considered as a candidate of 4G family LTE is the latest standard in the mobile communication systems The current generation of mobile communication system is collectively known as 3G Although LTE is often referred to as 4G, the first released LTE is actually a 3.9G technology as it does not completely meet the 4G requirements The main advantages of LTE include high throughput, low latency, plug and play, a simple architecture resulting in low power consumption, supporting seamless passing by base stations with former wireless networks such as Global System for Mobile Communications

(GSM), Universal Mobile Telecommunications System (UMTS), and CDMA2000 LTE also adopts OFDM and MIMO technologies in the physical layer It uses a 2 by 2 MIMO system as the basic configuration, that is, both the base station and the mobile end equip 2 antennas The next step for LTE evolution is LTE Advanced and is currently being standardized by 3rd Generation Partnership Project (3GPP) organization

The thesis studies two issues: channel estimation for OFDM systems and MIMO-OFDM system capacity The chapter is organized as follows Section 1.1 describes the research motivation Section 1.2 provides the thesis contributions and the overall organization of the thesis

1.1 Research motivation

Earlier OFDM systems such as the digital audio broadcasting (DAB) system in Europe does not require channel estimation module It only uses DPSK demodulation for the sake of reducing the complexity of the receiver However, with increasing demands of high data transmission rate and reliable communication quality, channel estimation has become a necessary part in the OFDM system For example, the digital video broadcasting (DVB) system adopts the channel estimation module In a broadband wireless environment, the channel is often time varying and frequency selective, which distorts the transmitted signal significantly, so that accurate and real time channel estimation is the challenging topic in the OFDM system Channel state information can be used for the detection of the received signal, improving the capacity of the system throughput by adjusting the modulation at the

transmitter through the feedback Therefore, one issue of the thesis studies channel estimation for OFDM systems over time varying and frequency dispersive fading channels

With the increasing number of mobile phone users and higher demands for wireless services, future communication systems should have higher system capacity MIMO technique is a breakthrough of improving system capacity Telatar [46] and Foschini [47] have firstly formulated the system capacity of the MIMO systems assuming independent and identically distributed fading at different antennas They have proved that the MIMO

system capacity for n transmitter antennas and n receiver antennas increases linearly with n at a fixed transmitter power That is, MIMO systems can improve the system capacity

significantly without increasing the system bandwidth A number of MIMO techniques known as layered space time architectures or Bell Laboratories layered space time (BLAST) architectures [5]–[8] have been proposed Many studies on MIMO system capacity have been conducted Since the combination of MIMO with OFDM is a trend, a lot of research work has been done on the MIMO-OFDM system capacity However, the research on MIMO-OFDM system capacity with imperfect feedback channel is not mature and corresponding work is not much Therefore, the second issue of the thesis is to study the MIMO-OFDM system capacity with imperfect feedback channels

1.2 Organization and contributions of the thesis

Firstly, we briefly describe the organization of the thesis Chapter 1 gives a general

introduction, research motivation, organization, and contributions of the thesis Chapter 2 describes the basic OFDM system model and MIMO system model The wireless channel model, the principles of OFDM and MIMO systems, the combination of MIMO and OFDM, that is, MIMO-OFDM system is also introduced in Chapter 2 Next, as depicted in Fig 1.1, the thesis begins with the first issue, that is, channel estimation The proposed channel estimation method in Chapter 3 is based on quasi-static fading channels and that in Chapter 4 is based on fast fading channels Then, the thesis switches to next issue, that is system capacity In Chapter 5, the MIMO-OFDM system capacity with imperfect feedback channel is investigated In Chapter 6, the capacity variances for OFDM systems over Rayleigh and Ricean fading channels are derived, respectively Finally, Chapter 7 concludes the thesis and discusses future research works

Fig 1.1: Organization of the thesis

Secondly, the major contributions of this thesis are summarized as follows

9 A fast linear minimum mean square error (LMMSE) channel estimation method for OFDM systems over slow fading channels has been proposed Unlike the conventional method, the channel state information is not needed in advance Almost the same performance with the conventional LMMSE channel estimation in terms of the normalized mean square error (NMSE) of channel estimation and bit error rate (BER) could be achieved for the proposed method The computational complexity can be reduced significantly since the proposed method replaces the inverse operation with FFT operation (Chapter 3)

9 A new pilot pattern and corresponding channel estimation method and data detection for OFDM systems over fast fading channels have been proposed The proposed channel estimation and data detection based on the proposed pilot pattern can eliminate inter-carrier interference (ICI) effect effectively And the number of required pilots is also reduced significantly, compared with the conventional least square (LS) method MSE analysis for the channel estimation based on the grouped pilot pattern is provided, too (Chapter 4)

9 The closed-loop MIMO-OFDM system capacity with imperfect feedback channel has been formulated We use the feedback SNR to measure the closed-loop capacity Since low feedback SNR may not yield positive gain of closed-loop capacity over

open-loop capacity, there exists a lower threshold of feedback SNR The lower thresholds for different antenna pairs are further investigated by numerical method (Chapter 5)

9 The variances of OFDM system capacity over Rayleigh fading channels and Ricean fading channels have been derived The effects of SNR, the number of channel paths, power profile of fading channel, the delay of the channel on the variance have been thoroughly investigated, for both multipath Rayleigh channels and multipath Ricean channels The joint PDF of two arbitrary correlated Ricean random variables has been provided in an integral form (Chapter 6)

Chapter 2: OFDM systems and MIMO systems

Since the thesis studies the channel estimation for OFDM systems and system capacity for MIMO-OFDM systems, this chapter briefly introduces the background of OFDM systems and MIMO systems

OFDM is an effective technology which provides high spectrum efficiency, high data transmission rate and is robust to multi-path fading [1] OFDM has already been widely put into practice in DAB system, DVB system and WLAN MIMO systems which employ multi-element antenna arrays at the transmitter and receiver ends are capable of high data rate transmission A number of MIMO techniques known as layered space time architectures or BLAST architectures have been proposed [5]-[7], [9]

As OFDM technique can mitigate the ISI and transform the frequency selective fading channel into a set of flat fading channels, the combination of MMO with OFDM technique is a trend for future wireless communication A new protocol draft, IEEE 802.11n, as an amendment to IEEE 802.11 standards has been proposed and investigated [53] The draft proposes MIMO-OFDM as the physical layer technique and is to be approved by IEEE The chapter is organized as follows Section 2.1 provides an introduction to wireless channel in communication systems Section 2.2 describes the basic OFDM system model, principles, and related key technologies in OFDM systems Section 2.3 introduces the basic MIMO system model and briefly overviews the existing MIMO systems Section 2.4 presents the MIMO-OFDM system model The last section 2.5 summarizes the chapter

2.1 Wireless Channel

Since there exist reflections, scattering, and diffraction in the transmission of electromagnetic wave, the spatial environments such as the landscape of a city, obstructions and so on will make complicated impacts on the transmission of electromagnetic wave There are two kinds of propagation, including the large-scale propagation and the small scale propagation The signal variations due to path loss or shadowing occur over relatively large distances, this variation is referred to as the large scale propagation effects Path loss is a major component in the analysis and design of the link budget of a telecommunication system The small scale propagation refers to the phenomena that the amplitude of the received wireless signal varies very fast in a short time period or a short distance The sources of small scale propagation include the Doppler shift effect and multi-path effect We begin with the introduction of the large scale propagation

2.1.1 Large scale propagation

Fig 2.1: Path Loss, shadowing and multipath versus distance

Fig 2.1 plots the ratio of the received-to-transmit power in dB versus the distance for the combined effects of path loss, shadowing, and multipath Observe that the free space path loss is linearly proportional to the log-distance between the transmitter and receiver The shadowing loss has slower variations compared to that of the multipath effect The large scale path loss model has many successful types including the free space path loss model, the Hata model, and the Okumura model and so on Most of them are obtained by a combination of the analytical and empirical methods We next briefly introduce the free space path loss model as an example of path loss effect

9 Free space path loss model: An example of path loss effect

Consider a signal transmitted through free space to a receiver located at distance d from

the transmitter It is assumed that there does not exist any obstructions between the

transmitter and receiver The channel model with this kind of transmission method is called a LOS channel, and the corresponding received signal is named the LOS signal

The average received power PR expressed in dB form is given by

PR = PT + 10 log10 (Gl ) + 20 log10 (λ) − 20 log10 (4π) − 20 log10 (d) dB (2-1) where Gl is the product of the transmit and receive antenna gains in the LOS direction, PT

is the transmitter signal power, d is the distance between the transmitter and the receiver, λ

is the wave length in unit of meter

9 Shadow fading

The signal fading due to shadowing from obstacles affecting the wave propagation is referred to as the shadow fading A signal will typically experience random variation due to blockage from objects in the transmission path Reflecting surfaces and scattering objects may produce random variation of the received signal, too Thus, a model for the random attenuation due to these effects is also needed Statistic models have been used to characterize this attenuation since the parameters such as the locations, sizes of the blocking objects or scattering objects are generally unknown The most common model for this additional attenuation is the log-normal shadowing fading This model has been demonstrated to accurately model the shadow fading in both outdoor and indoor radio propagation environments The path loss is a random

variable X with a log-normal distribution expressed by

xx

where β = 10/ ln10, µx is the mean value of xdB, xdB = 10 log10 x in unit of dB, and

σxis the standard deviation of xdB , also in unit of dB

2.1.2 Small scale propagation

The small scale fading of a signal is a more rapid fluctuation, which is usually caused by constructive and destructive interference between two or more versions of the same signal (multi-path effect) or Doppler effect due to moving terminals or surroundings objects We briefly introduce the multi-path effect and the Doppler shift effect

2.1.2.1 Doppler shift effect

The Doppler shift effect, named after Austrian physicist Christian Doppler who proposed it in 1842, is the change in frequency of a wave for an observer moving relative to the source of the wave When the mobile handset moves away from the base station, the carrier frequency is decreased and rather than vice versa The Doppler shift between the base

station and the mobile handset, fd, is expressed as cos( )

Where v is the velocity of the mobile handset, λ is the carrier wave length, θ is the angle

between the velocity of the mobile handset in the radial direction of the mobile handset and the base station Assuming that the arrival angles of the signal is uniformly distributed

in the range (-π, π), the power spectrum density function (PSD) is given by [36]

where Pav is the average power of the received signal and fc is the carrier frequency

f − ffcfc+ fd

( )

S f

Fig 2.2: The Doppler power spectrum function expressed by (2-4)

Fig 2.2 shows the classic U-shape power spectrum Note that no components fall outside

the interval [fc - fd, fc + fd] and the power of the transmitted signal spreads between (fc - fd)

reaching the receiving antenna by two or more paths Causes of multi-path include atmospheric ducting, ionospheric reflection and refraction, and reflection Fig 2.3 shows the multi-path effect in wireless communication

where ai is the magnitude of the i-th path and τi is the excess delay of the i-th path, N is the

number of resolvable paths As in Fig 2.4, the received signal is a series of pulses and the

number of pulses is the number of channel paths Note that usually the amplitude ai is time

varying and it is a function of time t

Fig 2.5: Four kinds of small scale propagations

Thus, based on Doppler shift effect and multi-path effect, T S Rappaport divided wireless fading channels into four categories, as shown in Fig 2.5 The first one is the fast fading channel, referred to the kind of channel when the coherence time of the channel, which is a function of Doppler shift, is small relative to the symbol duration of the transmitted signal The second is the slow fading channel, referred to the kind of channel when the coherence time of the channel is large relative to the symbol duration of the transmitted signal The third one is the flat fading channel, referred to the kind of channel when the coherence bandwidth of the channel, which is a function of the maximum delay of the channel, is larger than the bandwidth of the signal The fourth one is the frequency selective fading channel, referred to the channel when the coherence bandwidth of the

channel is less than the bandwidth of the signal

2.1.3 Typical wireless channel models

There are mainly three kinds of wireless channel models, as described as follows

(1) Rayleigh fading channel

σ2 = 0.5

σ2 = 2

σ2 = 5

Fig 2.6: PDFs for Rayleigh fading with the variance σ2 = 0.5, 2, and 5, respectively

Rayleigh fading is a reasonable model when there are many objects in the environment that scatter the radio signal before it arrives at the receiver The envelope of the channel impulse response is Rayleigh distributed, as expressed by

where r is the amplitude of the channel impulse response Fig 2.6 depicts several PDFs for

Rayleigh fading channel with the variance σ2 = 0.5, 2, and 5 The PDF curve with larger

variance has a broader spreading range Usually, the normalized autocorrelation function of a Rayleigh fading channel is a 0-th order Bessel function of the first kind, as expressed by

(2) Ricean fading channel

Ricean factor Kr = 0 dBRicean factor Kr = 10 dBRicean factor Kr = 20 dB

respectively

Ricean fading occurs when one of the channel paths, typically a line of sight signal (LOS), is much stronger than the others In a Ricean fading channel, the amplitude gain is

characterized by a Ricean distribution, which is formulated by

m = 1m = 0.5m = 10

Fig 2.8: PDFs for Nagakami-m fading with m = 0.5, 1, and 10

Both Rayleigh and Ricean distributions can be obtained by using mathematics to

describe the physical properties of the channel models However, a few channels can not be characterized well by the two previous models So a more general fading distribution was created whose parameters could be adjusted to characterize more channel models The

distribution is the Nakagami fading distribution, or Nakagami-m fading distribution, first

proposed by M Nakagami, in 1960 [10] The Nakagami fading distribution is given by

Where r is the amplitude of the channel impulse response, P is the average power of the

received signal, ( )Γ ⋅ is the Gamma function, defined as 10

about how to modulate/demodulate band signal by Discrete Fourier Transformation (DCT) [1] To suppress ISI, they proposed the empty guard interval between two adjacent symbols, but the orthogonality between two subcarriers over a frequency selective channel can not be ensured Another major contribution was made by A Peled and A Ruiz [11], who introduced the concept of CP in 1980, which ensured the orthogonality among subcarriers of an OFDM symbol The CP is copied from the end of the OFDM symbol and it is transmitted followed by each OFDM symbol When the length of CP is larger than that of the impulse response of the fading channel, ICI could be avoided

2.2.1 Basic principles and characteristics for OFDM systems

(a) The transmitter for continuous OFDM systems

(b) The receiver for continuous OFDM systems

(c) The frequency division for OFDM systems Fig 2.9: The continuous OFDM system model

As shown in Fig 2.9(a), the transmitted signal xi(t) for OFDM systems is given by

1 ,0

frequency fo = 0

The orthogonality between two subcarriers of an OFDM symbol

The orthogonality between two arbitrary subcarriers is described in the following

9 Two arbitrary subcarriers are orthogonal in the time domain since the following equation holds

∫g t g t dt T k lk( ) ( )l* = δ( − ) (2-15) 9 Two arbitrary subcarriers are also orthogonal in the frequency domain

The Fourier transform of gk(t) is expressed by

⎣ ⎦ ⊗δ(w−2πfk) (2-16)

where⊗denotes convolution Fig 2.10 shows the waveform of Gk(w)

( )

G w

Fig 2.10: The waveform ofG w k( )

The transmitted signals for the k1-th subcarrier and k2-th subcarrier in frequency domain

Thus, two arbitrary subcarriers are also orthogonal in the frequency domain

Using FFT/IFFT to implement the transmitter modulation and receiver demodulation

transmitted signal can be expressed as

= +=

Equation (2-18) can be further derived as

Fig 2.11: Equivalent transmitter for OFDM systems

And the receiver shown in Fig 2.9(b) is also equivalent to the following diagram

Fig 2.12: Equivalent receiver for OFDM systems

Since FFT and IFFT are easy to implement, the computational complexity can be reduced significantly compared to other systems

Cyclic Prefix

A Peled and A Ruiz [11] firstly introduced the concept of cyclic prefix in 1980, which not only eliminates ISI, but also ensures the orthogonality between two subcarriers of an OFDM symbol Cyclic prefix is a duplicate of last part of an OFDM symbol, shown as

following

C y c lic p re fix (C P )

tim eA n O F D M sy m b o l

Fig 2.13: CP for an OFDM symbol

The insertion of CP has two effects On one hand it eliminates the interference between OFDM symbols, however, on the other hand it also leads to the loss of SNR since it is necessary to provide the transmitter more power

(1) Loss of SNR:

At the receiver the SNR loss is given by

SNRloss =−10log10(1−γ) (2-20) whereγ =TCP/T, TCP is the CP length and T is the OFDM symbol length The longer the

CP is, the greaterSNR is, as shown in Fig 2.14 Observe that the SNRlossloss increases with the increase of CP length and whenγ <0.2,SNRloss <1 dB