Statistical Analysis and Reduction of Multiple Access Interference in MC-CDMA Systems

Transcription

1 Statistical Analysis and Reduction of Multiple Access Interference in MC-CDMA Systems Xuan Li Faculty of Built Environment and Engineering Queensland University of Technology A thesis submitted for the degree of Doctor of Philosophy 2008

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3 Abstract Multicarrier code division multiple access (MC-CDMA) is a very promising candidate for the multiple access scheme in fourth generation wireless communication systems. During asynchronous transmission, multiple access interference (MAI) is a major challenge for MC-CDMA systems and significantly affects their performance. The main objectives of this thesis are to analyze the MAI in asynchronous MC-CDMA, and to develop robust techniques to reduce the MAI effect. Focus is first on the statistical analysis of MAI in asynchronous MC-CDMA. A new statistical model of MAI is developed. In the new model, the derivation of MAI can be applied to different distributions of timing offset, and the MAI power is modelled as a Gamma distributed random variable. By applying the new statistical model of MAI, a new computer simulation model is proposed. This model is based on the modelling of a multiuser system as a single user system followed by an additive noise component representing the MAI, which enables the new simulation model to significantly reduce the computation load during computer simulations. MAI reduction using slow frequency hopping (SFH) technique is the topic of the second part of the thesis. Two subsystems are considered. The first subsystem involves subcarrier frequency hopping as a group, which is referred to as GSFH/MC-CDMA. In the second subsystem, the condition of group hopping is dropped, resulting in a more general system, namely individual subcarrier fre-

4 quency hopping MC-CDMA (ISFH/MC-CDMA). This research found that with the introduction of SFH, both of GSFH/MC-CDMA and ISFH/MC-CDMA systems generate less MAI power than the basic MC-CDMA system during asynchronous transmission. Because of this, both SFH systems are shown to outperform MC-CDMA in terms of BER. This improvement, however, is at the expense of spectral widening. In the third part of this thesis, base station polarization diversity, as another MAI reduction technique, is introduced to asynchronous MC-CDMA. The combined system is referred to as Pol/MC-CDMA. In this part a new optimum combining technique namely maximal signal-to-mai ratio combining (MSMAIRC) is proposed to combine the signals in two base station antennas. With the application of MSMAIRC and in the absents of additive white Gaussian noise (AWGN), the resulting signal-to-mai ratio (SMAIR) is not only maximized but also independent of cross polarization discrimination (XPD) and antenna angle. In the case when AWGN is present, the performance of MSMAIRC is still affected by the XPD and antenna angle, but to a much lesser degree than the traditional maximal ratio combining (MRC). Furthermore, this research found that the BER performance for Pol/MC-CDMA can be further improved by changing the angle between the two receiving antennas. Hence the optimum antenna angles for both MSMAIRC and MRC are derived and their effects on the BER performance are compared. With the derived optimum antenna angle, the Pol/MC-CDMA system is able to obtain the lowest BER for a given XPD.

5 Contents 1 Introduction Need for multicarrier code division multiple access Frequency division multiple access Time division multiple access Code division multiple access Orthogonal frequency division multiplexing Multicarrier code division multiple access Multiple access interference problem in MC-CDMA Thesis objectives and contributions Organization of this thesis Statistical analysis of MAI in asynchronous MC-CDMA systems Problems with the existing statistical analysis of MAI in asynchronous MC-CDMA systems Asynchronous MC-CDMA model Channel Receiver Derivation of test statistic iii

6 CONTENTS 2.4 Statistical modelling of MAI and its power Mean and variance of MAI power with different distributions of timing offset Applications of the new developed MAI model: A tool for analyzing the effect on BER Theoretical derivation of BER Simulation results of the effects on BER Application of the new developed MAI model: computer simulations 43 3 Multiple access interference reduction techniques Spreading sequence Multiuser detection Minimum mean-square error detector Subtractive interference cancellation Slow frequency hopping Group subcarrier frequency hopping MC-CDMA Asynchronous GSFH/MC-CDMA model MAI analysis in different detection scenarios Scenario A Scenario B Scenario C Scenario D Average overall MAI Power Simulation Results Discussions iv

7 CONTENTS 5 Individual subcarrier frequency hopping MC-CDMA Asynchronous ISFH/MC-CDMA model MAI analysis for ISFH/MC-CDMA Special case: GSFH/MC-CDMA MAI power comparison between asynchronous MC-CDMA and asynchronous ISFH/MC-CDMA Bit error rate analysis Simulation results MAI power ratio Bit error rate Discussion Asynchronous MC-CDMA with base station polarization diversity Polarization diversity versus other diversity techniques The mechanism of polarization diversity Two factors affecting polarization diversity Correlation coefficient Cross polarization discrimination (XPD) Diversity combining technique Selection combining Maximal ratio combining Equal gain combining Base station polarization diversity reception model and its applications v

8 CONTENTS 6.6 System model for Pol/MC-CDMA system Derivation of the test statistic for Pol/MC-CDMA Maximal signal-to-mai ratio combining Performance analysis of Pol/MC-CDMA in the presence of both AWGN and MAI Results and Discussions for MSMAIRC Optimum antenna angle Optimum antenna angle for MRC Optimum antenna angle for MSIRC Special Case Special Case Results and discussion for the antenna angle effect Conclusions and future works Conclusions Future works Appendices 136 A Statistical modeling of sum of correlated Rayleigh random variables 139 B Proof of (5.15) 145 C Proof of (5.20) 147 D Proof of (6.27) and (6.28) 151 vi

9 List of Figures 1.1 Frequency division multiple access (FDMA) [1] Time division multiple access (TDMA) [1] Code division multiple access (CDMA) [1] MAI generation in asynchronous MC-CDMA systems Effect of MAI on the performance of asynchronous MC-CDMA system Asynchronous MC-CDMA transmitter and receiver model Effective timing offset scenarios PDF of MAI in asynchronous MC-CDMA (with N=K=16, uniformly distributed τ k ) PDF of MAI in asynchronous MC-CDMA (with N=K=16, exponentially distributed τ k ) PDF of MAI in asynchronous MC-CDMA (with N=K=16, Gaussian distributed τ k ) PDF of MAI power for asynchronous MC-CDMA (N=K=8, uniformly distributed τ e ) vii

10 LIST OF FIGURES 2.7 PDF of MAI Power for asynchronous MC-CDMA (N=32 K=8, Gaussian distributed τ e ) BER comparison between uniformly distributed τ k and exponentially distributed τ k with SNR = 10dB BER comparison between uniformly distributed τ k and Gaussian distributed τ k with SNR = 10dB Proposed asynchronous MC-CDMA simulation model MAI noise generator for asynchronous MC-CDMA Asynchronous GSFH/MC-CDMA transmitter and receiver model Frequency spectrum of asynchronous GSFH/MC-CDMA signals Asynchronous GSFH/MC-CDMA detection interval with length of 2N h symbols Asynchronous GSFH/MC-CDMA detection scenario A Asynchronous GSFH/MC-CDMA dectection scenario B Asynchronous GSFH/MC-CDMA detection scenario C Asynchronous GSFH/MC-CDMA detection scenario D GSFH/MC-CDMA MAI power ratio for different number of subcarrier frequency groups (N=K=16) GSFH/MC-CDMA MAI power ratio for different spreading factors Asynchronous ISFH/MC-CDMA transmitter and receiver Model MAI power ratios (as a percentage) for asynchronous ISFH/MC- CDMA systems with different values of Q and N MAI power ratio for asynchronous GSFH/MC-CDMA and ISFH/MC- CDMA systems for N=K= viii

11 LIST OF FIGURES 5.4 Bit error rate performance for asynchronous ISFH/MC-CDMA with (N = K = 16 and the correlation coefficient for subcarrier fading is 0.7) Two-branch receiver model for base station MC-CDMA receiver with polarization diversity BER comparison between Pol/MC-CDMA with MRC and MC- CDMA (with α = π/4; δ r = 0 and N = K = 16) BER comparison between Pol/MC-CDMA with MSMAIRC and MC-CDMA (with α = π/4; δ r = 0 and N = K = 16) BER comparison between Pol/MC-CDMA with MSMAIRC and MRC (with α = π/4; δ r = 0 and N = K = 16) Threshold XPD for Pol/MC-CDMA with MSMAIRC and MRC (with α = π/4; δ r = 0 and N = K = 16) Threshold XPD for Pol/MC-CDMA with MSMAIRC and MRC given different number of users (α = π/4; δ r = 0 and N = 16) Threshold XPD for Pol/MC-CDMA with MSMAIRC and MRC for 2 users (α = π/4; δ r = 0 and N = 16) Threshold XPD for Pol/MC-CDMA with MSMAIRC and MRC for 4 users (α = π/4; δ r = 0 and N = 16) Threshold XPD for Pol/MC-CDMA with MSMAIRC and MRC for 8 users (α = π/4; δ r = 0 and N = 16) Effect of antenna angle when E b /N 0 = 10dB Effect of antenna angle when E b /N 0 = 30dB ix

12 LIST OF FIGURES A.1 Approximation of a sum of correlated Rayleigh random variables (N=16 and correlation coefficient = 0.7) A.2 Approximation of a sum of Rayleigh random variables (8 correlated Rayleigh random variables + 8 independent Rayleigh random variables and correlation coefficient = 0.7) x

13 List of Tables 2.1 Distribution fitting for conditional MAI power in asynchronous MC-CDMA: Kullback-Leibier divergence values Pair T-test results for BER samples obtained in the mathematical model and the proposed model (System 1) Pair T-test results for BER samples obtained in the mathematical model and the proposed model (System 2) A.1 Table of Fitting Distributions A.2 Fitting Result of Correlated B (Kullback-Leibier divergence values) 141 xi

14 Acronyms 4G The fourth generation of mobile communication systems AWGN Additive white Gaussian noise BER bps BPSK BS Bit error rate Bits per second Binary phase shift keying Base station CDMA Code division multiple access DS-CDMA Direct-sequence CDMA EGC Equal gain combining

15 FDMA FFT Frequency division multiple access Fast Fourier transform GSFH/MC-CDMA Group subcarrier frequency hopping MC-CDMA Hpol Horizontal polarization ICI i.i.d. Inter-channel interference Independent and identically distributed ISFH/MC-CDMA Individual subcarrier frequency hopping MC-CDMA ISI Inter-symbol interference LOS Line of sight MAI MC-CDMA MC-DS-CDMA MMSE Multiple access interference Multicarrier CDMA Multicarrier DS-CDMA Minimum mean-square error

16 MRC MS MSMAIRC MUD Maximal ratio combining Mobile subscriber Maximal signal-to-mai ratio combining Multiuser detection NLOS No line-of-sight OFDM Orthogonal frequency division multiplexing PDF Probability density function SFH SMAIR SMAINR SNR Slow frequency hopping Signal-to-MAI ratio Signal-to-MAI-plus-noise ratio Signal-to-noise ratio TDMA Time division multiple access Vpol Vertical polarization

17 Authorship The work contained in this thesis has not been previously submitted to meet requirements for an award at this or any other higher education institution. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made. Signed: Date:

18 List of Publications Conference papers X. Li and B. Senadji, Statistical Analysis of Interference In Asynchronous MC-CDMA Systems, The 1st Australian Conference on Wireless Broadband and Ultra Wideband Communications, Sydney, X. Li, D. Carey and B. Senadji, Interference Reduction and Analysis for Asynchronous MCCDMA Using a Dual Frequency Switching Technique, The 5th Workshop on the Internet, Telecommunications and Signal Processing, Hobart, X. Li and B. Senadji, Multiple access interference analysis in asynchronous GSFH/MC-CDMA systems, Wireless Communication and Networking Conference, IEEE, pp , March X. Li, Y.C. Huang and B. Senadji, MAI Analysis of an Asynchronous MC-CDMA System With Polarization Diversity, The 1st International Conference on Signal Processing and Communication Systems (ICSPCS 2007), Gold Coast, Papers submitted to journals X. Li, D. Carey and B. Senadji, Statistical Analysis of Multiple Access Interference In Asynchronous MC-CDMA Systems, IEEE Transactions of Vehicular Technology, [Submitted].

19 X. Li and B. Senadji, Performance Analysis of Asynchronous MC-CDMA with Subcarrier Frequency, IEEE Transactions of Vehicular Technology, [Submitted]. X. Li, Y.C. Huang and B. Senadji, Maximal Signal-to-MAI Ratio Combining for MC-CDMA with Base Station Polarization Diversity, IEEE Transactions of Vehicular Technology, [Submitted].

20 Acknowledgements I would like to thank my principal supervisor, Dr. Bouchra Senadji, for her continuous support in my PhD program. Bouchra has guided me throughout the entire PhD program. She is always there, giving advice to help me solve difficult problems. Especially I would like to thank Bouchra for her patience in helping me to improve my fluency in the English needed for this subject area. I would like to thank my associate supervisor, Prof. Sridha Sridharan, for giving me opportunities to work for him during the past four years. It is my great honor to work for Prof. Sridharan, who has given me superb opportunities to broaden my knowledge in the area of telecommunication. I would also like to thank him for his support in my PhD program. I would like to thank my fellow PhD student, Brian Huang, for his contribution in the later part of my PhD candidature. Brian is smart and hardworking. It was a great pleasure for me to work with him. Finally, I would like to thank Patrick Lau, one of my best friends over the last seven years, for his advice on my thesis writing. I am so lucky to meet these wonderful people. Without them I could not have come this far. Thank you all.

21 Chapter 1 Introduction In telecommunication systems, communication resources refer to the time and frequency bandwidth that is available in a given system. With multiple users on a system, resources (time/bandwidth) need to be shared among the users in order to establish communication links between the mobile subscriber (MS) and the base station (BS). However, the available resources are often limited for any given user, as the total bandwidth on a system is restricted. In order to improve the efficiency of resource allocations, multiple access techniques have been developed, with an ideal system illustrating the following qualities [2]: 1) available resources are fully utilized; 2) all resources are shared equally among users; 3) interference is not introduced between users i.e. no multiple access interference (MAI), and 4) the capacity of the system is maximized. 1

22 1. INTRODUCTION 1.1 Need for multicarrier code division multiple access Three major multiple access schemes exist: frequency division multiple access (FDMA), time division multiple access (TDMA) and code division multiple access (CDMA) [1]. In the following sections, the advantages and disadvantages of these techniques will be reviewed, and the need for multicarrier code division access (MC-CDMA) will be discussed Frequency division multiple access Figure 1.1: Frequency division multiple access (FDMA) [1] Frequency division multiple access (FDMA) was the first multiple access technique, developed in the early 1900s [2]. With FDMA, the total frequency bandwidth is divided into frequency channels that are assigned to each user perma- 2

23 1.1 Need for multicarrier code division multiple access nently, resulting in multiple user signals that are both spectrally separated and simultaneously transmitted and received. This has been graphically represented in Fig The FDMA systems requires a relatively simple algorithm and implementation compared to TDMA and CDMA [1], but there are several drawbacks. Firstly, due to the permanent assignment of FDMA channels, unused channels cannot be utilized by other users, resulting in wasted communication resources. Secondly, nonlinearities in the power amplifier can cause signal spreading in the frequency domain, causing inter-channel interference (ICI) in other FDMA channels. Finally, the capacity of an FDMA system is limited by the number of channels available Time division multiple access Figure 1.2: Time division multiple access (TDMA) [1] 3

24 1. INTRODUCTION Time division multiple access (TDMA) has been developed with a similar concept to FDMA, but with TDMA, multiple user signals are separated in the time domain rather than in the frequency domain. Fig. 1.2 shows a TDMA system with the transmission time divided into a number of cyclically repeating time slots that can be assigned to individual users, allowing all users access to all of the available bandwidth. Compared to FDMA systems, TDMA systems offer more flexibility in the assignment of time slots whereby different numbers of time slots can be allocated to different users depending on the service demanded. Furthermore, because TDMA users can transmit signals only in their own time slots, the transmission of TDMA signal is noncontinuous and occurs in bursts, resulting in less battery power consumption. However, the TDMA signal requires a large synchronization overhead due to its non-continuous transmission. Inter-symbol interference (ISI), caused by multipath propagation, is also a major problem for TDMA, especially during high data rate transmissions Code division multiple access Over the last decade, code division multiple access (CDMA) has been developed to overcome the disadvantages of other multiple access techniques such as TDMA and FDMA [3]. Fig. 1.3 demonstrates multiple CDMA users signals that are separated by spreading sequences. In particular, each user signal is spread using a pseudorandom sequence which is orthogonal to the sequence of other users. As a result, only the intended user-receiver can despread and receive the information cor- 4

25 1.1 Need for multicarrier code division multiple access Figure 1.3: Code division multiple access (CDMA) [1] rectly; other users on the system perceive the signal as noise, resulting in multiple user signals that can be transmitted within the same bandwidth simultaneously. The main advantage with CDMA is that the system capacity is limited only by the amount of interference; with a lower level of interference the system can support a higher number of users [1]. CDMA systems are also robust to narrow band jamming as the receiver signal can spread the jamming signals energy over the entire bandwidth making it insignificant in comparison to the signal itself [2]. If the spreading sequence is perfectly orthogonal, it is possible to transmit multiple CDMA signals without introducing multiple access interference (MAI) during synchronous transmission [3] Various types of CDMA such as direct-sequence CDMA (DS-CDMA) and wideband CDMA (W-CDMA), have been developed and utilized in both 2G and 3G systems similar to cdmaone (IS-95), UMTS and CDMA2000 [4]. These 5

26 1. INTRODUCTION techniques are considered to be single-carrier CDMA systems. Unfortunately when moving into the fourth generation of wireless communication systems (4G), in which data is transmitted at a rate as high as 1 Giga bits-per-second (bps) [5], single-carrier CDMA systems are not suitable. This is because 1. With high data rates the symbol duration will become shortened, resulting in the channel delay spread exceeding the symbol duration causing ISI [6]. 2. When data rate goes beyond a hundred Mega bps, it becomes difficult to synchronize, as the data is sequenced at high speeds [7]. 3. Due to multipath propagation, signal energy is scattered in the time domain: in single-carrier CDMA systems such as DS-CDMA, RAKE receivers are often used to combine the multipath signals. However, not all paths of signals can be successfully received. If the number of fingers in the RAKE receiver is less than the number of resolvable paths, some of the received signal energy can not be combined, thus a portion of the signal energy is lost [8]. But if the number of fingers in the RAKE receiver is more than the number of resolvable paths, noise will be enhanced. Therefore a conventional single-carrier CDMA such as DS-CDMA is not practical for 4G systems where a high data rate is required Orthogonal frequency division multiplexing Orthogonal frequency division multiplexing (OFDM) proposed in [9] has the ability to support higher data rate transmission. When using OFDM, the channel bandwidth is divided into a number of equal bandwidth subchannels, with each 6

27 1.1 Need for multicarrier code division multiple access subchannel utilizing a subcarrier to transmit a data symbol. The frequency separation of adjacent subcarriers is chosen to equal the inverse of the symbol duration, resulting in all the subcarriers being orthogonal to one another over one symbol interval. Hence, OFDM technique can transmit a large number of different data symbols over multiple subcarriers simultaneously, enabling this technique to support a higher data rate transmission. In addition the bandwidth of each subchannel is designed to be so narrow that the frequency characteristics of each subchannel are constant, making OFDM signals robust to frequency selective fading [10]. The other advantage of OFDM is that the signal can be easily and efficiently modulated and demodulated using fast Fourier transform (FFT) devices [11]. As FFT can be easily implemented, the receiver complexity does not increase substantially while transmission rate can be largely increased. Despite all these advantages, OFDM still have some drawbacks due to its implementation of multicarrier modulation. OFDM suffers a high peak-to-average power ratio that occurs when all the signals in the subcarriers are added constructively [12, 13]. This results in the saturation of the power amplification at the transmitter, causing inter-modulation distortion. OFDM is very sensitive to frequency offset, as the spectrums of the subcarriers are overlapping [14, 15]. Any frequency offset can lead to ICI, which suggests that OFDM requires a high degree of synchronization of subcarriers. Besides, the conventional OFDM systems can support only a single user, raising the need for multicarrier code division multiple access (MC-CDMA). 7

28 1. INTRODUCTION Multicarrier code division multiple access Based on the combination of OFDM and DS-CDMA, multicarrier code division multiple access (MC-CDMA) is proposed [16]. Unlike DS-CDMA, which spreads the original data stream into the time domain, MC-CDMA spreads the original data stream into the frequency domain by initially converting the input data stream from serial to parallel then multiplying this stream by the spreading chips in different OFDM subcarriers, resulting in a MC-CDMA signal which takes on the advantages of both DS-CDMA and OFDM. The advantages of MC-CDMA are: 1. The capacity is interference limited [17] and any techniques that reduce interference are capable of increasing the capacity of MC-CDMA. 2. The signal is robust to frequency selective fading and can support high data rate transmission. 3. Bandwidth is used more efficiently as the spectra of subcarrier overlap [18]. 4. Since the received signal is combined in the frequency domain, a MC-CDMA receiver can employ all the received signal energy scattered in the frequency domain [19]. This is a significant advantage over DS-CDMA, where part of the signal energy can be lost due to insufficient number of fingers in the RAKE receiver. 5. The transmitter and receiver signals can be implemented using FFT, which does not increase the degree of complexity. However, as MC-CDMA is still a multi-carrier modulation technique, it inevitably has the same drawback as OFDM. Problems such as inter-modulation 8

29 1.2 Multiple access interference problem in MC-CDMA distortion and ICI can still be found in MC-CDMA systems. Furthermore as with other CDMA techniques, MC-CDMA suffers multiple access interference (MAI) during asynchronous transmission, which significantly degrades the performance of MC-CDMA systems. This research will focus on the analysis and reduction of MAI in MC-CDMA systems. 1.2 Multiple access interference problem in MC- CDMA In MC-CDMA systems, as with DS-CDMA systems, each user is assigned a unique orthogonal pseudo random spreading sequence which allows the receiver to distinguish individual user signals from one another. During asynchronous transmissions, however, where multiple user signals arrive at the receiver with different timing offsets, the orthogonality between users is lost [1]. This creates the first type of multiple access interference (MAI) in MC-CDMA. This type of MAI, denoted as I s, is the interference created by non-reference users transmitting information over the same subcarrier frequencies and it is commonly found in both asynchronous DS-CDMA and MC-CDMA systems. The second type of MAI, denoted as I d, is exclusive to MC-CDMA systems and it is due to the nature of the multicarrier transmission. In MC-CDMA systems the frequency separation between subcarriers is chosen as an integer multiple of 1/T s (T s denoting the symbol duration), such that subcarriers are orthogonal to each other. During asynchronous transmission, however, this orthogonality between subcarriers is also lost, which gives rise to the production of the second type of MAI. This MAI 9

30 1. INTRODUCTION Figure 1.4: MAI generation in asynchronous MC-CDMA systems term represents the interference generated by transmission over different subcarrier frequencies by the non-reference users. Fig. 1.4 illustrates the generation of the two types of MAI for a two-user system. In this figure, both reference user r and interferer k are transmitting signals through N subcarriers. It can be seen from Fig. 1.4 that in the reference user r the signal with subcarrier frequency f r,1 is suffering two types of MAI i.e. I s (1) and I d (1,j) (j = 2...N). The first type of MAI I s (1) (represented by a solid line) is generated due to the interferer signal with subcarrier frequencies f k,1, with f k,1 = f r,1. Meanwhile the signals in other interferer subcarriers f k,j (for j = 2...N) contribute to the second type of MAI I d (1,j) (represented by dash lines). The total MAI in the reference user, denoted as I, is the sum of both types of MAI in all N subcarriers, which means N N N I = I s (i) + I d (i,j). (1.1) i=1 i=1 j=1;j i The effect of MAI on the performance of asynchronous MC-CDMA was first 10

31 1.2 Multiple access interference problem in MC-CDMA Figure 1.5: Effect of MAI on the performance of asynchronous MC-CDMA system (Note: The full capacity of the system is 16 users. The level of noise is indicated by E b /N o, where E b is the bit energy and N o is the single-sided noise spectral density) 11

32 1. INTRODUCTION studied in [20] in a frequency selective fading channel. Then later in [21] and [22], similar analyses were given in the case of the Nakagami fading channel. All this research showed that during asynchronous transmission, the bit error rate (BER) performance of MC-CDMA is significantly degraded by the presence of MAI. BER is a performance measurement for telecommunication systems. It represents the percentage of bits that have errors relative to the total number of bits received in a transmission [2]. A small BER indicates that the system has a small probability of receiving bits in error, hence, the performance of the system is good. On the contrary, a large BER indicates a large probability of receiving error bits, hence, the system has a poor performance. Fig. 1.5 illustrates the effect of MAI on the BER of asynchronous MC-CDMA. This figure compares a single user system (i.e. no MAI) to a multiple user system with MC-CDMA. As seen in Fig. 1.5, MC-CDMA is capable of offering a low BER for the single user system, and this BER can be further reduced by decreasing the level of noise (indicating by increasing E b /N o ). However, when two or more users exist in the system, MAI is introduced, and the BER is increased, indicating a performance degradation. In the case when the system is fully loaded, the MAI becomes the dominant factor of the interference. As a result, further reduction of noise cannot decrease the BER, and the system has an irreducible error floor. Therefore MAI is one of the major factors that limits the performance of asynchronous MC-CDMA. 12

33 1.3 Thesis objectives and contributions 1.3 Thesis objectives and contributions The main objectives of this thesis are to analyse the multiple access interference in asynchronous MC-CDMA and to develop robust techniques to reduce the MAI effect. In order to achieve these objectives, this thesis has been separated into three parts. The first part primarily considers the statistical behavior of MAI in asynchronous MC-CDMA, and provides statistical MAI analysis and a statistical model for asynchronous MC-CDMA. The principal contributions in this part of the research are listed below 1. Define effective timing offset and derive its statistics. 2. Derive the exact expression of MAI for MC-CDMA as a function of timing offset. 3. Establish a statistical model for the PDF of the time varying MAI power using Gamma distribution. 4. Propose a computer simulation model for asynchronous MC-CDMA systems. All the above contributions led to the publications of [23] and [24]. In the second part, slow frequency hopping (SFH) as a MAI reduction technique will be introduced to MC-CDMA (SFH/MC-CDMA). SFH/MC-CDMA is further divided into two subsystems, group subcarrier frequency hopping MC- CDMA (GSFH/MC-CDMA) [25, 26] and individual subcarrier frequency hopping MC-CDMA (ISFH/MC-CDMA). Listed below are the lists of contributions presented in this part. 13

34 1. INTRODUCTION 1. Provide a MAI analysis for GSFH/MC-CDMA with asynchronous transmissions. 2. Show that the MAI power in GSFH/MC-CDMA is not affected by the length of hopping period. 3. Derive a more general SFH/MC-CDMA system based on the application of individual subcarrier frequency hopping (ISFH/MC-CDMA). 4. Derive the analytical expression of the MAI for ISFH/MC-CDMA under asynchronous conditions and compare it to that of MC-CDMA as well as GSFH/MC-CDMA. 5. Identify the relationship between the MAI power generated in ISFH/MC- CDMA and MC-CDMA. 6. Demonstrate that GSFH/MC-CDMA is a special case of ISFH/MC-CDMA. The publications that are associated with this part are [27, 28, 29]. The third part is based on developing a system which combines base station polarization diversity and MC-CDMA, referred to as Pol/MC-CDMA. The original contributions presented in this part of the thesis are as follows: 1. Introduce base station polarization diversity to MC-CDMA and derive the expression of MAI and its power for Pol/MC-CDMA. 2. Propose an optimum combining method to combine the signal in two base station antennas, called maximal signal-to-mai ratio combining (MSMAIRC) for Pol/MC-CDMA system. 14

35 1.4 Organization of this thesis 3. Derive the optimum antenna angle for Pol/MC-CDMA with MSMAIRC as well as the traditional maximal ratio combining (MRC). With this optimum antenna angle the signal-to-interference-plus-noise ratio is maximized and the BER performance of Pol/MC-CDMA is the lowest. 4. Compare the BER performance of Pol/MC-CDMA with MRC and MS- MAIRC. These contributions are documented in the publications of [30] and [31]. 1.4 Organization of this thesis After the introductory chapter, the thesis develops through seven chapters. Chapter 2 focuses on the statistical analysis of MAI for asynchronous MC- CDMA systems and a new statistical model of MAI is derived. Unlike existing statistical models where uniform distributed timing offsets and constant MAI power are assumed, in the new statistical model of MAI, different distributions of timing offset can be applied and the MAI power is found to be Gamma distributed. Utilizing the new statistical model of MAI, a computer simulation model for asynchronous MC-CDMA systems is proposed. Current models require a heavy computational load because multiple user signals must be simulated simultaneously. In the new simulation model, the computer simulation of the multiuser system is replaced by a single user system, followed by an additive noise component representing the MAI. As a result, the proposed model requires significantly less computing. The proposed model is validated by using statistical measurements such as paired t-test. 15

36 1. INTRODUCTION Chapter 3 reviews the three main techniques for MAI reduction: optimal spreading sequence design, multiuser detection (MUD) and slow frequency hopping (SFH). It provides the literature reviews and background information for Chapter 4 and Chapter 5. Chapter 4 provides a solution for reducing the MAI effect by introducing a group subcarrier frequency hopping technique to MC-CDMA (GSFH/MC- CDMA). In this chapter, the MAI performance of GSFH/MC-CDMA under asynchronous transmission is analysed. Three different detection scenarios are considered. The expression of MAI power is derived and verified using Monte- Carlo simulation results. In Chapter 5, the condition of group hopping in GSFH/MC-CDMA is dropped and a new system referred to as the individual subcarrier frequency hopping MC-CDMA (ISFH/MC-CDMA) is proposed. In this chapter, a thorough MAI analysis for asynchronous ISFH/MC-CDMA is provided. The MAI power in ISFH/MC-CDMA is then compared with MC-CDMA as well as with the previously proposed GSFH/MC-CDMA. ISFH/MC-CDMA is found to generate less MAI power than the basic MC-CDMA system during asynchronous transmission. Because of this, ISFH/MC-CDMA is shown to outperform MC-CDMA in terms of the bit error rate (BER). Moreover, in this chapter it is shown that GSFH/MC-CDMA is a special case of ISFH/MC-CDMA; in particular, when the number of available subcarrier frequencies are equal, both systems generate the same amount of MAI power. The expression of BER for ISFH/MC-CDMA is derived and all theoretical results are confirmed using Monte-Carlo simulations. Chapter 6 firstly reviews the literature for the existing research on polarization diversity. Then base station polarization is introduced as another MAI 16

37 1.4 Organization of this thesis reduction technique to asynchronous MC-CDMA. In this chapter, a new diversitycombining technique is proposed for the asynchronous MC-CDMA, with two branch base station polarization diversity (Pol/MC-CDMA). The new combining technique, called the maximal signal-to-mai ratio combing (MSMAIRC), aims to maximize the received signal-to-mai ratio (SMAIR) at the diversity combiner. Bit error rate (BER) performance of Pol/MC-CDMA using MSMAIRC is analysed and compared with maximal ratio combining (MRC). It is found that when the level of additive white Gaussian noise (AWGN) is small in terms of the E b /N 0, MSMAIRC generally outperforms MRC for most values of cross polarization discrimination (XPD). Moreover, MSMAIRC is also found to be less sensitive to the change in XPD. This indicates that MSMRC is able to guarantee a more stable performance if the transmission environment is changing. Finally, in this chapter, the optimum antenna angles for Pol/MC-CDMA with both MSMAIRC and maximum ratio combining (MRC) are also derived. By setting the antenna angles to the derived optimal values, Pol/MC-CDMA systems can obtain the lowest BER. Chapter 7 summarizes the research results and discuss the future research directions. 17

38 1. INTRODUCTION 18

39 Chapter 2 Statistical analysis of MAI in asynchronous MC-CDMA systems In this chapter, two problems with the existing statistical analysis of MAI in asynchronous MC-CDMA systems have been identified. By solving these two problems, a new statistical model of MAI is developed. This statistical model can be applied in two applications. Firstly, it can be used as a tool to analyze the performance of asynchronous MC-CDMA systems with different assumptions of MAI and its power, including different distributions of timing offsets and different statistical model of MAI power. The second application of this statistical model is computer simulations, and an efficient computer simulation model for asynchronous MC-CDMA is proposed. 19

40 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS 2.1 Problems with the existing statistical analysis of MAI in asynchronous MC-CDMA systems During the analysis of the performance in asynchronous MC-CDMA, MAI is generally approximated as a zero mean Gaussian random variable [20, 21, 22, 32]. This is mainly because the random phase of each subcarrier is assumed to be independent and identically distributed (i.i.d.) for each subcarrier and for each user. As a result, the MAI from each subcarrier of interferers is uncorrelated. According to the central limit theorem, the total MAI can be approximated as Gaussian distributed and its power has been derived as a constant in [20]. This is the generally accepted statistical property of MAI in asynchronous MC-CDMA [20, 21, 32, 33, 22]. The analysis, however, is not complete. In previous analyses of MAI [20, 21, 32, 33, 22], the timing offset, which is defined as the arrival time difference between the reference user and the interferer, is assumed to be uniformly distributed within one symbol duration. But this assumption has been contested by many other research studies such as [34, 35, 36, 37]. The main objection is that the distribution of the timing offsets between users is strongly dependent on the user distribution within a cell, and is not necessarily uniform. For example, when the user distribution is uniform, the distribution of the timing offsets is shown to be linear [34]. Hence the MAI analysis should not be restricted to the uniform distributed timing offset; a more general derivation which can be applied to any possible distribution of the timing offsets is required. 20

41 2.2 Asynchronous MC-CDMA model Another problem with the existing statistical analysis of MAI in asynchronous MC-CDMA systems relates to the assumption of a constant MAI power. Under this assumption the MAI is considered stationary in a channel that is known to be non-stationary [38]. A more realistic assumption in a non-stationary channel is that the MAI is a non-stationary process, in which case the MAI power must be time varying, indicating that the MAI power is also a random variable. The statistical properties of the MAI power are not widely documented. [39] and [40] are among the few to study the statistical behavior of the MAI power for asynchronous MC-CDMA. In [39], the probability density function (PDF) of the MAI power is derived. Then in [40] the MAI powers of MC-CDMA and DS-CDMA are compared. It is found that the MAI power in asynchronous MC- CDMA is lower than in DS-CDMA. Unfortunately in both studies only one type of MAI is considered, and the presence of the other type of MAI is ignored, which suggests that the developed results are incomplete. The aim of this chapter is to complete the analysis for MAI and its power in asynchronous MC-CDMA. By using this result, an accurate statistical model for MAI can be established. This statistical model is flexible which can be adjusted to different distributions of timing offsets according to different transmission environments, and it does not limit to a constant MAI power which is more realistic. Finally, this statistical model will lead to the proposal of an efficient model for computer simulations in asynchronous MC-CDMA systems. 21

42 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS Figure 2.1: Asynchronous MC-CDMA transmitter and receiver model 2.2 Asynchronous MC-CDMA model Fig. 2.1 shows the mathematical model for the asynchronous MC-CDMA transmitter and receiver considered in this chapter. In this model, originally proposed in [20], the transmission of binary data, b k (t) (k = 1...K) for K users, is considered. It is assumed that b k (t) are i.i.d. random variables. As shown in Fig. 2.1, the data from the k th user are first BPSK modulated, before the spreading chip sequence c k,i (i = 1...N) is applied over N orthogonal subcarriers f i (i = 1...N). The carrier frequency for the i th subcarrier, f i, is defined by f i = f 1 + i 1 T s for i = 1...N, (2.1) where T s represents symbol duration. The transmitted signal for the k th user is given by N s k (t) = 2Pbk (t)c k,i cos (2πf i t + φ k,i ), (2.2) i=1 22

43 2.2 Asynchronous MC-CDMA model where P is the signal power of each subcarrier, and φ k,i is the random phase introduced by the BPSK modulator. For each subcarrier, the random phase φ k,i is assumed to be uniformly and i.i.d. over the interval [0, 2π) Channel Following the approach developed in [20], it is assumed that each subcarrier signal passes through a frequency non-selective Rayleigh fading channel, and that the fading remains constant for at least two symbol durations. For a given user, the Rayleigh fading channels for transmitting subcarriers are assumed correlated. For different users, however, the Rayleigh fading channels are assumed independent. Finally, the statistics of the Rayleigh fading for all channels are assumed to be identical. The complex impulse response of the Rayleigh fading channel of the i th subcarrier in the k th user can then be written as ν k,i (t) = β k,i e jθ k,i δ (t τ k ) (2.3) where τ k represents the timing offset of the k th user with respect to the reference user, δ ( ) is the Dirac s delta function. The random phase introduced by the channel, θ k,i, is modelled as uniformly distributed over the interval of [0, 2π), and assumed i.i.d. for each subcarrier and each user. Finally, β k,i is a Rayleigh random variable with second-order moment E[βk,i] 2 = σ 2. In previous research [20, 21, 32, 33, 39, 22], the timing offsets τ k were generally assumed to be uniformly distributed between 0 and T s. In communication system analysis, however, the assumption of uniform distribution does not always hold, and τ k can take different distributions depending on the assumptions made on 23

44 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS the user distribution within a cell [34, 35, 36, 37]. Due to the uncertainty of the distribution of the timing offset, in this chapter, timing offsets are allowed to follow any distribution. Although the results for the uniform, exponential, and Gaussian distributions will be demonstrated as examples, readers can extent the same concept to almost any possible distributions found in practical environment Receiver The signal received at the base station is written as K N r (t) = 2Pβk,i b k (t τ k )c k,i cos (2πf i t + ζ k,i ) + n (t). (2.4) k=1 i=1 In (2.4), ζ k,i = θ k,i + φ k,i 2πf i τ k represents the total phase effect for the i th subcarrier allocated to the k th user and n (t) is additive white Gaussian noise (AWGN) with double-sided power spectral density N 0 /2. At the receiver site, the signal is demodulated and despread. Equal gain combining (EGC) is then applied to the despread signals. In EGC, signals received from each subcarrier i (i = 1...N) are weighted equally by parameters µ r,i (i = 1...N) and summed. In this thesis, µ r,i = 1 (i = 1...N). The combined signal is then passed through a matched filter followed by a maximum likelihood detector. 2.3 Derivation of test statistic The test statistic at the output of the matched filter is denoted as [Z] mc, where [ ] mc represents symbols that are exclusive to MC-CDMA systems. With l de- 24

45 2.3 Derivation of test statistic fined to be an arbitrary integer, [Z] mc can be obtained as N (l+1)ts [Z] mc = r (t)µ k c r,i cos (2πf k,i t)dt i=1 lt s = [D] mc + [η] mc + [I] mc. (2.5) [Z] mc can be written as a sum of three components, as indicated above by [D] mc, [η] mc and [I] mc. [D] mc is the component which represents the desired signal, and is given by [20] [D] mc = P 2 T snb r (l) B r, (2.6) where N B r = β r,i. (2.7) i=1 It represents the sum of Rayleigh fading coefficients of the N subcarrier in the reference user. The data bit of the reference user in the current detection interval [lt s, (l + 1)T s ] is denoted by b r (l). The interference due to the AWGN is represented by [η] mc. It is a Gaussian random variable with zero mean and variance given as [20] V ar [η] mc = N 0NT s 4. (2.8) The [I] mc term represents the undesired multiple access interference. For MC- CDMA, the MAI term was shown to be composed of two independent terms referred to [I s ] mc and [I d ] mc [20]. The expressions for [I] mc, [I s ] mc and [I d ] mc can 25

46 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS be written respectively as [I] mc = [I s ] mc + [I d ] mc, (2.9) where [I s ] mc = K N I s,k (i), (2.10) k=1;k r i=1 [I d ] mc = K N N k=1;k r i=1 j=1;j i I d,k (i,j). (2.11) In (2.10), I s,k (i) represents the MAI generated by user k using subcarrier frequency f k,i, which is also used by the reference user r, that is f k,i = f r,i. Similarly, in (2.11), I d,k (i,j) represents the MAI generated by user k using subcarrier frequency f k,j, which is different from that used by reference user r, that is f k,j f r,i. The expressions of I s,k (i) and I d,k (i,j) have been developed by Gui and Ng in [20] under the assumption of uniformly distributed timing offsets. They are given respectively as P I s,k (i) = 2 µ r,ic k,i c r,i cos ζ k,i [β k,i (l 1)b k (l 1)τ k + β k,i (l)b k (l) (T s τ k )], (2.12) I d,k (i,j) = c k,j c r,i T s 2Pµ r,i [β k,j (l 1)b k (l 1) β k,j (l) b k (l)] 4π i,j [ ( sin 2π i,jτ k T s ) ] + ζ k,j sin ζ k,j, (2.13) where b k (l) and β k,i (l) are respectively as the binary data and the channel fading in the current detection interval, that is [lt s, (l + 1)T s ]. Similarly b k (l 1) and 26

47 2.3 Derivation of test statistic β k,i (l 1) represent respectively the binary data and the channel fading associated with the previous detection interval, that is [(l 1)T s, lt s ] respectively. In (2.13), i,j denotes as the spectral distance between f r,i and f k,j and is defined as i,j = i j. (2.14) The derivations of (2.12) and (2.13) are, however, based on the assumption that the timing offsets lie only within 0 and T s. When the timing offsets are not assumed to be uniformly distributed between 0 and T s, they can take any real value between and +. As a result, the expressions derived in (2.12) and (2.13) become invalid. Fig. 2.2 shows the three possible scenarios involving timing offsets. In scenario 1, the timing offset between the reference user and the interferer fall within [0, T s ). This is the scenario that was originally considered in [20]. In scenario 2, the arrival time difference, τ k, between the reference user and interferer is larger than T s, while in scenario 3, the interfering signal arrives earlier than the reference signal, resulting in a negative timing offset. What is important to understand, is that the amount of MAI affecting the reference user is not determined by the timing offset, τ k, but by the time misalignment between the reference user symbol b r (l) and the interferer symbol b k (l). This time misalignment is referred to as the effective timing offset, and denoted it as τk. e In scenario 1 of Fig. 2.2, the time misalignment between the reference symbol b r (l) and interfering symbol b k (l) is equal to the timing offset, that is τk e = τ k. 27

48 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS Figure 2.2: Effective timing offset scenarios 28

49 2.3 Derivation of test statistic In scenario 2, as illustrated in Fig. 2.2, the timing offset τ k is greater than T s. The timing offset in this case is calculated as τ k mod T s, where mod denotes the modulo operator. Finally, in scenario 3, the signal of the interfering user arrives before the reference user, and generalizes the case for τk e < 0. In this case, the effective timing offset is calculated as τk e = T s (τ k mod T s ). In summary, given a timing offset τ k between and +, the effective timing offset τ e k responsible for MAI always lies within [0, T s ), and can be expressed as τ k mod T s for τ k > 0 τk e = T s (τ k mod T s ) for τ k < 0 (2.15) Further, the probability density function (PDF) of the effective timing offset τ e k can be expressed in terms of the PDF of the timing offset τ k as: For 0 τ e k < T s prob (τ e k) = For τ e k T s prob (τ e k) = 0 l= prob (sgn (l) τ k + lt s ) for 0 τ k < T s, (2.16) where l can take any integer values from to +. As the effective timing offset is always within one symbol duration, the two MAI terms, (2.17) and (2.18) previously calculated in [20] for timing offsets, τ k, in the range [0, T s ), can be generalized to any distribution of the timing offset 29

50 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS as P I s,k (i) = 2 µ r,ic k,i c r,i cos ζ k,i [β k,i (l 1)b k (l 1)τ e k + β k,i (l) b k (l) (T s τ e k)], (2.17) I d,k (i,j) = c k,j c r,i T s 2Pµ r,i [β k,j (l 1)b k (l 1) β k,j (l) b k (l)] 4π i,j [ ( sin 2π i,jτk e T s ) ] + ζ k,j sin ζ k,j. (2.18) These results will be used in the next section for modelling the statistics of the MAI power. 2.4 Statistical modelling of MAI and its power In asynchronous MC-CDMA, due to the assumption that φ k,i and θ k,i are i.i.d. for different k and i, both types of MAI, that is I s,k (i) and I d,k (i,j) defined respectively in (2.17) and (2.18), are independent for different subcarriers and different users. Hence their corresponding summation over i and k, that is [I s ] mc and [I d ] mc defined in (2.10) and (2.11), are also uncorrelated. As a result, according to the central limit theorem, the total MAI defined in (2.9) can be modelled as Gaussian distributed [20]. Figure 2.3 to 2.5 show three examples of the PDF for the MAI in asynchronous MC-CDMA, with timing offset being uniformly, exponentially and Gaussian distributed respectively. All three examples show close fits between the PDF of the MAI and that of a zero-mean Gaussian PDF. Problems remain in the statistical modelling of MAI power. Previous research assumes that the MAI power is a constant in time. In a non-stationary environment, however, the MAI power becomes randomly time-varying. Although the 30

51 2.4 Statistical modelling of MAI and its power Figure 2.3: PDF of MAI in asynchronous MC-CDMA (with N=K=16, uniformly distributed τ k ) Figure 2.4: PDF of MAI in asynchronous MC-CDMA (with N=K=16, exponentially distributed τ k ) 31

52 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS Figure 2.5: PDF of MAI in asynchronous MC-CDMA (with N=K=16, Gaussian distributed τ k ) exact PDF of the MAI power is unknown, a good statistical model can be obtained through the fitting of its PDF to existing well-known PDFs. Because the MAI power is a positive random variable, the fitting process will be limited to positive distributions, namely Gamma, Nakagami, Rice and Weibull distributions. The statistical modelling is also performed for various distributions of the effective timing offset, and for various numbers of users and subcarriers. The Kullback-Leibier divergence (KLD) is used as a measure of closeness between the PDF of the MAI power and that of the positive distributions considered in this chapter. The expression of the KLD is given as [41] KLD = prob (V ar [I] prob (V ar [I] mc = i) log mc = i) ( ) (2.19) i Ψ prob V ar [I]mc = i where (V ar [I] mc = i) represents the i th realization of the MAI power, Ψ is the total number of realizations, prob (V ar [I] mc = i) is the true PDF of V ar [I] mc 32

53 2.4 Statistical modelling of MAI and its power Table 2.1: Distribution fitting for conditional MAI power in asynchronous MC- CDMA: Kullback-Leibier divergence values τ k Spreading factor Distribution Distribution & user numbers Gamma Nakagami Rice Weibull Uniform N=8; K= N=16; K= N=32;K= N=64;K= Gaussian N=8; K= with N=16; K= mean = 0.8T s N=32;K= variance = 0.6T s N=64;K= Exponential N=8; K= with N=16; K= mean = 0.5T s N=32;K= variance = 0.5T s N=64;K= ( ) and prob V ar [I]mc = i is the approximated PDF. The results of the statistical modelling over realizations are summarized in Table 2.1. Three different distributions of the effective timing offset have been considered, namely the uniform, exponential and Gaussian distributions. The numbers of subcarriers N and active users K have also been varied. Table 2.1 shows that the Gamma distribution consistently displays the best closeness of fit with respect to the various distributions tested, and the multiple scenarios considered. Figures 2.6 and 2.7 are two examples that confirm the choice of the Gamma distribution as a model for the PDF of the MAI power. The PDF of a Gamma distributed random variable, x, is given by prob (x) = 1 b a Γ (a) xa 1 e x/b, (2.20) where a and b are the parameters of the distribution that is aimed to be estimated. 33

54 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS Figure 2.6: PDF of MAI power for asynchronous MC-CDMA (N=K=8, uniformly distributed τ e ) Figure 2.7: PDF of MAI Power for asynchronous MC-CDMA (N=32 K=8, Gaussian distributed τ e ) 34

55 2.5 Mean and variance of MAI power with different distributions of timing offset a and b are completely determined by the knowledge of the mean and variance of x as shown below [42] a =E 2 [x]/v ar [x] (2.21) b =V ar [x] /E [x]. (2.22) It follows that the PDF of the MAI power for asynchronous MC-CDMA can be completely determine by the mean power and its variance. Therefore, the next section illustrates the derivation of the mean and variance of MAI power. 2.5 Mean and variance of MAI power with different distributions of timing offset In this section, an expression for MAI power conditional to τk e is derived. The derived expression can be used to obtain the mean and variance of MAI power for the various distribution of the effective timing offset. The expressions of the conditional MAI power are E [ I 2 s τ e k E [ I 2 d τ e k ] ] mc =E b(t),β,ζ = NPσ2 4 mc =E b(t),β,ζ [ I 2 s ] K k=1;k r [ ] I 2 d = (K 1) PT2 s σ 2 4π 2 [ (τ e k ) 2 + (T s τ e k) 2] (2.23) N N 1 i=1 j=1j i 2 i,j =E [ I 2 d]. (2.24) 35

56 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS In (2.23) and (2.24), E [ ] Is 2 τk e and E [ I 2 mc d τ ] k e mc represent the MAI power conditional to τk. e E b(t),β,ζ [ τk] e represent the expectation operation respect to b (t), β, ζ and it is conditional to τk. e It is interesting to show that in (2.24) the power of I d is independent of τk. e As a result, different distributions of the timing offset will affect the power of I s but not I d. The sum of E [ ] Is 2 τk e and E [ I 2 mc d τ ] k e asynchronous MC-CDMA, which is written as mc gives total conditional MAI power for E [ I 2 τ e k ]mc = NPσ2 4 [X + (K 1) 1 ] π 2C, (2.25) where K X = (τk) e 2 + (T s τk) e 2, (2.26) k=1;k r and the following relationship is applied [22] C = 1 N N N i=1 j=1;j i 1 i,j. (2.27) After obtaining the conditional MAI power expression, the mean and variance of the MAI power can be obtained by further taking the expectation to include τ e k. The results are shown respectively as E [ ] I 2 mc =NPσ2 [E τ [X] + (K 1) C ], (2.28) ek 4 π 2 V ar [ ( ) NPσ 2 2 [ I ]mc 2 = [ E ] τe k X 2 ( E τ e k 4 [X]) 2 ], (2.29) 36

57 2.5 Mean and variance of MAI power with different distributions of timing offset where E τ e k [X] and E τ e k [X 2 ] can be derived as K ( [ E τ e k [X] = 2 Eτ e (τ e k ) 2] 2T s E τ e [τ ) e k k] + Ts 2, (2.30) k=1;k r [ [ E ] 4E τ e τ e k X 2 = (K 1) k (τ e k ) 4] + Ts 4 4Ts 3 E [τk] e [ +8Ts 2 E τ e k (τ e k ) 2] [ 8T s E τ e k (τ e k ) 3] + (K 1) (K 2) ( 2E τ e k [ (τ e k ) 2] + T 2 s 2T s E [τ e k] ) 2, (2.31) In (2.28)-(2.31), E τ e k [ ] represents the expectation operation in respect to τk. e It is found that the key to obtaining the mean and variance of MAI power lies in the evaluation of the first four order moments of the effective timing offset. Hence the knowledge of the distribution of the effective timing offset, τk e is required. As a result, the j th moment of the effective timing offset can be easily calculated as [ E τ e k (τ e k ) j] Ts = (τk) e j prob (τk)dt e. (2.32) 0 The average MAI power for asynchronous MC-CDMA for various distributions of timing offset can finally be obtained by substituting (2.32) into (2.30) and (2.31). It follows that the mean and variance of the Gamma distribution representing the MAI power, and therefore the parameters of the Gamma distribution, are completely determined by the knowledge of the number of active users K, the symbol duration T s and the first four moments of the effective timing offset, τk. e For example, when the timing offsets are uniformly distributed between [0,T s ], the effective timing offsets are also uniformly distributed between [0,T s ] and the 37

58 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS fist four moments of the effective timing offset are given as E [τ e k] = T s 2, (2.33) E [ (τ e k) 2] = T 2 s 3, (2.34) E [ (τ e k) 3] = T 3 s 4, (2.35) E [ (τ e k) 4] = T 4 s 5. (2.36) Hence E [X] = (K 1) 2 3 T s 2, (2.37) E [ X 2] [ 7 = (K 1)Ts ] + (K 2) 4. (2.38) Therefore, by substituting (2.37) and (2.38) into (2.28) and (2.29), the mean and variance of the MAI power are given by E [ I 2 ] = (K 1) NPσ 2 V ar [ I 2 ] = (K 1) ( NPT 2 s σ ( 2 3 T 2 s + C π 2 ), (2.39) ) (2.40) Applying (2.21) and (2.22), the parameters for the Gamma distributed MAI power can be obtained as a =45 (K 1) b = NPσ 2 ( C ) 2, (2.41) Ts 2 π ( C T 2 s π 2 ). (2.42) 38

59 2.6 Applications of the new developed MAI model: A tool for analyzing the effect on BER Up to this point, the problems with the existing research on statistical analysis of MAI have been corrected. The general expressions of the MAI and its power, which can accommodate different distributions of timing offset, have been developed and the MAI power has been modelled as a timing varying random variable with PDF that follows Gamma distributions. In the next section the bit error rate (BER) performance of asynchronous MC-CDMA is derived and the effect of the new developed MAI model on the BER is shown. 2.6 Applications of the new developed MAI model: A tool for analyzing the effect on BER Theoretical derivation of BER Based on the analysis for MAI, in this section the bit error rate (BER) for asynchronous MC-CDMA is derived. The mean of the test statistic is the desired signal component [D] mc, which is given in (2.6). The corresponding variance of the test statistic is the sum of the AWGN power in (2.8) and the total MAI power, which is derived in (2.28). Hence the BER for the system is given by p e B r = Q E [Z]2 mc [D] 2 mc = Q V ar [Z] mc V ar [η] mc + E [ ] I 2 mc, (2.43) where Q ( ) is the Q function. Notice that p e B r is conditional to B r which is the sum of Rayleigh fading random variables in the subcarriers of the reference user. To obtained the average BER, expectation should be taken with respect to B r 39

60 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS which gives p e = + 0 p e B r prob(b r )db r. (2.44) To evaluate the above integral, numeric methods such as Monte-Carlo integration [43] can be applied. The PDF of B r can be approximated as Nakagami-m PDF (see Appendix A) for details. Hence prob (B r ) = 2mm Br 2m 1 ( Γ (m) Ω exp m ) m Ω B2 r. (2.45) with parameters Ω =E [ Br] 2, (2.46) E [B 2 m = r] E [Br] 2 (E [B r ]) 2. (2.47) The associated first and second order moments for B r can be derived respectively as E [B r ] =Nσ π/4, (2.48) E [ ] N + n (n 1) ( ρ π Br 2 =σ 2 ) π (N n) (N + n 1) π, 4 (2.49) where n is the number of correlated subcarriers, n N and ρ is the correlation coefficient between correlated subcarriers. 40

61 2.6 Applications of the new developed MAI model: A tool for analyzing the effect on BER Figure 2.8: BER comparison between uniformly distributed τ k and exponentially distributed τ k with SNR = 10dB Simulation results of the effects on BER In this section the effect on BER for different statistical assumptions of MAI is shown through Monte Carlo simulations. An asynchronous MC-CDMA system described in Section 2.2 was simulated. The second-order moment of the Rayleigh fading channel is assumed equal to unity and the number of realizations used for the Monte-Carlo simulations is 100, 000. Further, it is assumed that all the fadings in the subcarriers are correlated with correlation coefficient ρ = 0.6, and the signal-to-noise ratio (SNR) is set to 10dB. Fig. 2.8 shows the BER comparison between uniformly distributed τ k and exponentially distributed τ k. It is found that the BER of exponentially distributed τ k decreased when the parameter λ increased. In exponential distribution, λ represents the mean as well as the variance. Fig. 2.8 also shows that when λ = 0.5T s, the resulting BER is close to the one with uniformly distributed τ k. 41

62 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS Figure 2.9: BER comparison between uniformly distributed τ k and Gaussian distributed τ k with SNR = 10dB In fact, if λ is further increased, the BER obtained will be close to the BER of uniformly distributed τ k. Similar results can be observed in Fig. 2.9 for the comparison between uniformly distributed τ k and Gaussian distributed τ k. Different sets of the parameters of the Gaussian distributed timing offset will lead to different BER. For example, when the mean and the standard deviation are 0.1T s, the PDF of the effective timing offset is significantly different from that of the uniform distribution, resulting in a BER that is different from that of the uniformly distributed τ k. However, when the mean is 0.6T s and the standard deviation is 0.7T s, the resulting BER is close to the one obtained from uniformly distribution τ k. 42

63 2.7 Application of the new developed MAI model: computer simulations Figure 2.10: Proposed asynchronous MC-CDMA simulation model 2.7 Application of the new developed MAI model: computer simulations So far in this chapter, it has been shown that the Gamma distribution provides the best fitting results for the PDF of MAI power. Furthermore the Gamma distribution can be characterized by using the expressions of the mean and variance of the MAI power, derived respectively in (2.28) and (2.29). Therefore the statistical behavior of the MAI power can now be completely described. Utilizing this result, this section proposes a new computer simulation model for asynchronous MC-CDMA systems, referred to as the meta-model. In section 2.2, the mathematical model of asynchronous MC-CDMA is described. If this mathematical model is used for computer simulations, the computation load is increased with the number of active users K. This is because in the mathematical model, MAI is generated through the simulation of K 1 interferer signals with different randomly generated timing offsets. When the number of users K is large, substantial computing resources need to be allocated to the generation of MAI. For the performance analysis of asynchronous MC-CDMA, however, only the signal from the reference user is interested and all the other 43

64 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS Figure 2.11: MAI noise generator for asynchronous MC-CDMA K 1 interferer signals are acting as interference only. With this concept in mind, a more efficient way to simulate asynchronous MC-CDMA systems has been found and a new model is proposed for computer simulations referred to as the meta-model. In the meta-model, shown in Fig. 2.10, only the transmission and reception of the reference user signal are simulated. The simulation of the K 1 interferer signals is replaced by a noise component representing MAI, which is added to the reference signal before passing into the channel. Therefore, the proposed meta-model is essentially a single-user simulation model which required significantly less computation load than a multiuser simulation. The additive MAI noise component is a MAI generator, shown in Fig It is a Gaussian random variable generator with mean and power as inputs. The mean input is set to zero and its power input is the output from a Gamma random variable generator. The Gamma random variable generator also has the mean and variance as inputs, which can be determined using the derived equation in (2.28) and (2.29). The output random variable of the MAI noise generator has 44

65 2.7 Application of the new developed MAI model: computer simulations the identical statistics of the MAI for asynchronous MC-CDMA. The performance of the proposed meta-model is validated by comparing, through Monte-Carlo simulations, the BER between the mathematical model described in Section 2.2 and the simplified model in Fig The models for two systems with randomly chosen parameters are built. In system 1, N = K = 16 is assumed and all fadings in subcarriers are correlated with the correlation coefficient ρ = 0.6. Three different distributions of timing offset are considered here. They are uniform distributed timing offset, Gaussian distributed timing offset with mean equal to 0.5T s and standard deviation equal to 0.2T s, and exponential distributed timing offset with mean and standard deviation set to 0.2T s. In system 2, N = K = 32 is assumed and the correlation coefficients for all subcarriers fading are set to 0.8. System 2 also applied three different distributions of timing offset, namely uniform distribution, Gaussian distribution with mean 0.7T s and standard deviation 0.3T s, and exponential distributed with mean and standard deviation set to 0.6T s. For both systems, the number of realizations used for the Monte Carlo simulations is 100, 000. Further, it is assumed that the second-order moment of the Rayleigh fading channel is set to unity. For each system, a paired t-test [44] is performed for the average BER samples obtained in both models. The paired t-test is a statistical test that compares the means of two groups of observations. In this case, the paired t-test determines whether the BER samples from two models differ from each other in a significant way under the assumptions that the paired differences are independent and identically normally distributed. 45

66 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS Table 2.2: Pair T-test results for BER samples obtained in the mathematical model and the proposed model (System 1) t-test for system with N = K = 16 and ρ = % confident interrval Timing Offset Eb/N 0 p for the difference Distributions in db Value Lower Upper Uniform Gaussian mean = 0.5T s variance = 0.2Ts Exponential mean = 0.2T s variance = 0.2Ts During the test, 100 samples are taken for the average BER from both models and the significant level is set to The null hypnosis of the test is that the difference between the average BER samples is a zero mean Gaussian process. Table 2.2 and 2.3 show the test results for systems 1 and system 2 respectively. It can be seen that, in all situations, the p values of the tests are larger than 0.05 and the 95% significant intervals are small and contain zero value, indicating that the tests fail to reject the null hypnosis. These test results imply that there is no significant difference between the average BER obtained from the mathemathical model and from the proposed model, therefore validating the proposed model. 46

67 2.7 Application of the new developed MAI model: computer simulations Table 2.3: Pair T-test results for BER samples obtained in the mathematical model and the proposed model (System 2) t-test for system with N = K = 32 and ρ = 0.8 Timing Offset 95% confident interrval Distributions Eb/N 0 p for the difference in db Value Lower Upper Unifrom Gaussian mean = 0.7T s variance = 0.3Ts Exponential mean = 0.6T s variance = 0.6Ts

68 2. STATISTICAL ANALYSIS OF MAI IN ASYNCHRONOUS MC-CDMA SYSTEMS 48

69 Chapter 3 Multiple access interference reduction techniques There are a number of proposed techniques that are capable of reducing multiple access interference (MAI) for DS-CDMA systems. These techniques involve mainly the application of the optimal spreading sequences, multiuser detection (MUD) and slow frequency hopping (SFH). However, not all these techniques can be applied to MC-CDMA systems. In the following sections, a review of these three MAI reduction techniques are shown and their applications on MC-CDMA systems are also discussed. 3.1 Spreading sequence As early as 1977, Pursley in [45] was among the first to derive the exact expression of MAI for asynchronous DS-CDMA systems. He argued that the amount of MAI in an asynchronous DS-CDMA system is determined by the discrete aperiodic 49

70 3. MULTIPLE ACCESS INTERFERENCE REDUCTION TECHNIQUES cross-correlation function of the spreading sequence. As a result, in order to minimize the MAI in DS-CDMA, the cross correlation of the spreading sequence is required to be low. Various spreading sequences have been developed to satisfy this requirement, for example the m-sequence [46], the Gold sequence [47] and the Kasami sequence [48]. For asynchronous MC-CDMA, the effect of spreading sequences has been studied in [49], where a selection criterion for the spreading sequence has been identified. The performances of asynchronous MC-CDMA are compared for four different spreading sequences namely, Walsh-Hadamard sequence, Gold sequence, orthogonal Gold sequence [50] and Zadoff-Chu sequence [51]. It was determined that Zadoff-Chu sequence was the optimal spreading sequence for asynchronous MC-CDMA. Based on this result, Yip and Ng derived the upper and lower bound for the bit error rate (BER) of asynchronous MC-CDMA [52]. Then in [53], the analysis was completed by considering both types of MAI. However, all the analysis discussed in [49, 52, 53] was limited to the case of additive white Gaussian noise (AWGN) channels. When a frequency selective fading channel was considered, the results in [32] show that the choice of the spreading sequence has little effect on the power of the MAI. In fact, in [20], the expression of the MAI power is shown to be independent of the spreading sequence. As frequency selective fading channels appear in most wireless communication systems, the effect of MAI for asynchronous MC-CDMA cannot be reduced by using the optimal spreading sequence. 50

71 3.2 Multiuser detection 3.2 Multiuser detection Verdu in [54] proposed the first optimal multiuser detector, referred to as the maximum-likelihood sequence (MLS) detector. He proved that, with his approach, systems with asynchronous multiuser signals can achieve a minimum BER, equivalent to that of a single-user system when the level of AWGN is low. In other words, all interferences including MAI can be eliminated. Despite this great achievement, the MLS detector has not been widely adopted in practical CDMA systems. This is mainly due to its high complexity cost. According to [55], the complexity of the MLS detector is increased exponentially with the number of active users. Since then multiuser detector (MUD) research has been focused on reducing complexity, and various types of suboptimal multiuser detectors have been developed for asynchronous CDMA systems. The two most common types of MUD are minimum mean-square error (MMSE) detector and subtractive interference cancellation Minimum mean-square error detector MMSE detectors can be implemented linearly or adaptively. Linear MMSE detectors are relying on the linear transformation of the soft outputs from the conventional single-user detector [56]. The criterion of the linear transformation is based on MMSE, which minimizes the mean-squared error between the actual data and the soft output of the conventional detector. It has been shown that linear MMSE detectors can successfully reduce MAI and other forms of interference and thus give a significant improvement for the performance of asynchronous 51

72 3. MULTIPLE ACCESS INTERFERENCE REDUCTION TECHNIQUES DS-CDMA [56]. Furthermore, unlike other linear detectors, the MMSE detectors do not enhance the background noise. However, the complexity of the system is still a problem. In order to suppress interference, linear MMSE detectors require the estimation of the phase, frequency, delays and amplitude of all users [55]. In addition, it also involves the inversion of a large-size matrix which requires high computation capacity [55]. Adaptive MMSE detector [57, 58, 59], the adaptive extension of the linear MMSE detector, can solve both the assumed knowledge and complexity problems. By making the filter parameters adaptive, adaptive MMSE detector does not require any matrix inversion; only the rough timing of the desired user is needed [57, 55]. Castoldi in [60] studied the performance of adaptive MMSE detectors for asynchronous DS-CDMA. The trained adaptive MMSE detector, a subclass of the adaptive MMSE detectors, is able to give a BER performance that is similar to the nonadaptive linear MMSE detector. Moreover the trained adaptive MMSE detector is robust to the imperfect timing recovery as well as the phase and frequency offset. The disadvantage of adaptive MMSE detector is discovered when it is applied in a frequency selective channel. In the research for the performance of DS-CDMA, the authors in [61, 62, 63] argued that it is crucial for the receiver to accurately track the channel fading parameters for all of the users signals; otherwise substantial degradation in performance [61] and capacity [62] can result. The same problem happens in MC-CDMA, where the tracking of all the users fading process is also essential for MC-CDMA [64]. Although a tracking algorithm had been developed in [64], it inevitably increased the complexity of the resulting multiuser receiver [62]. 52

73 3.2 Multiuser detection Subtractive interference cancellation Subtractive interference cancellation is another common type of multiuser detector. The basic principle behind these detectors relies on the regeneration of the MAI contributed from each user and then subtracting some or all of the MAI from the desired user signal [65]. The subtraction can be performed in either a serial or a parallel approach. When the serial approach is taken, the multiuser detector is referred to as the successive interference cancellation (SIC) detector [66]. Multiple stages are required for SIC detectors. In each stage, the MAI from the strongest signal is regenerated and then canceled out from the received signal, so that in the next stage a smaller amount of MAI is presented. Holtzman in [67] studied the performance of SIC detectors in DS-CDMA systems. He found that the SIC detector can offer system performance that is significantly better than that of the single-user detector. A similar conclusion was drawn in [68] for MC-CDMA systems. Parallel interference cancellation (PIC) detectors [69] take the parallel approach of interference cancellation. Similarly to SIC, PIC detectors also require multiples stages. In each stage, an estimated data bit is produced for each user and is then fed into the next stage. When all data estimations are correct, no additional stage is needed and complete MAI elimination is achieved. Numerous studies have investigated the performance of PIC detectors in DS-CDMA systems e.g. [70, 71, 72]. PIC detectors are applied to MC-CDMA systems in [73]. All this research shows that PIC detectors can improve system performance substantially. However, both SIC and PIC detectors have similar drawbacks. Some of them 53

74 3. MULTIPLE ACCESS INTERFERENCE REDUCTION TECHNIQUES are listed below [60, 65, 74]: SIC detectors require the signal to be reordered continuously if the power profile changes. Accurate amplitude estimations of all user signals are crucial for both detectors. Any estimation error can cause additional noise. Extra delay is introduced; it increases linearly with the number of stages. During high data rate transmission (e.g. 1G bps in 4G), where large delay is prohibited, the number of stages in SIC and PIC detectors must be reduced. As a result, the MAI can not be completely eliminated and the system performance is degraded. 3.3 Slow frequency hopping Due to the limitations of spreading sequence and multiuser detectors, it is necessary to investigate other possible techniques that are capable of MAI reduction for MC-CDMA. In this thesis, slow frequency hopping (SFH) technique is the focus. Frequency hopping technique involves changing or hopping carrier frequency in every predetermined time interval [2]. For slow frequency hopping, the predetermined time interval is restricted to any values that are much larger than one symbol duration. SFH has been successfully implemented in DS-CDMA systems as a way to reduce MAI during asynchronous transmission [75, 76]. SFH has also been implemented in multicarrier DS-CDMA (MC-DS-CDMA) systems and has similarly 54

75 3.3 Slow frequency hopping achieved a reduction in MAI [77, 78]. To the best of the author s knowledge, the introduction of SFH in MC-CDMA systems, as a possible solution for improving the performance of MC-CDMA systems, has been attempted in only[25] and [26]. In both [25] and [26], the total number of available subcarrier frequencies Q is divided into H groups. Each group has N subcarriers (where N is the spreading factor) and the relative position for all subcarrier frequencies in the spectrum is fixed for a given group. As a result, when hopping occurs, the whole group of N subcarriers has to hop together. In [25] and [26], this technique is referred to as frequency hopping MC-CDMA (FH/MC-CDMA). However, to clearly represents the nature of hopping and to avoid confusion, in this research, this technique is referred to as group subcarrier frequency hopping MC-CDMA (GSFH/MC-CDMA). In [25] and [26], GSFH/MC-CDMA has been implemented under synchronous conditions. It was found that GSFH/MC-CDMA can increase the capacity of the system and outperform MC-CDMA in a tap delay line correlated channel. However, in both papers, no MAI analysis has been undertaken. This research gap will be filled in Chapter 4, where a thorough MAI analysis for GSFH/MC-CDMA with asynchronous transmissions will be provided. The problem for GSFH/MC-CDMA, however, is the condition of group hopping, which is found to be unnecessary and which puts a constraint on the total bandwidth to be multiples of N subcarrier frequencies. Therefore, in Chapter 5, a new MC-CDMA system based on the use of SFH is proposed; this new system is referred to as individual subcarrier frequency hopping MC-CDMA (ISFH/MC- CDMA). In ISFH/MC-CDMA systems, each individual subcarrier can randomly hop within the available bandwidth. As a result, at every hop, N subcarrier frequencies are randomly chosen from the total available frequencies Q. The 55

76 3. MULTIPLE ACCESS INTERFERENCE REDUCTION TECHNIQUES randomness, in this case, exists for each individual subcarrier. In Chapter 5, the analytical expression of the MAI for ISFH/MC-CDMA under asynchronous conditions is derived, and comparison is made to that of MC-CDMA. It will be shown that ISFH/MC-CDMA is capable of reducing MAI power and therefore allows for better performance in terms of BER. Further, GSFH/MC-CDMA is found to be a special case of ISFH/MC-CDMA when Q = N H. 56

77 Chapter 4 Group subcarrier frequency hopping MC-CDMA Group subcarrier frequency hopping MC-CDMA (GSFH/MC-CDMA) has been previously proposed in [25] and [26], and analysis has been given under synchronous transmission. This chapter provides a MAI analysis for GSFH/MC- CDMA under asynchronous transmission by considering four different detection scenarios. These four detection scenarios represent all possible outcomes that the MAI can generate from the interferer on the reference user. The expression of MAI and its power for GSFH/MC-CDMA in all four detection scenarios will be derived and the total MAI power will be compared with that of basic asynchronous MC-CDMA systems. 57

78 4. GROUP SUBCARRIER FREQUENCY HOPPING MC-CDMA Figure 4.1: Asynchronous GSFH/MC-CDMA transmitter and receiver model 4.1 Asynchronous GSFH/MC-CDMA model As shown in Fig. 4.1, the model for asynchronous GSFH/MC-CDMA is similar to the model of asynchronous MC-CDMA described in Section 2.2. The main difference lies in the choice of subcarrier frequencies. In basic MC-CDMA, the subcarrier frequencies are fixed during transmission, while in GSFH/MC-CDMA the subcarrier frequencies randomly change every N h symbols. In the spectrum of GSFH/MC-CDMA signals, the overall bandwidth is divided into H groups of N orthogonal subcarrier frequencies. The configuration of the subcarrier frequency groups is shown in Fig At a given time, a user k is allocated group h k, where h k can take any integer value between 1 and H. The group of subcarrier frequencies allocated to user k 58

79 4.1 Asynchronous GSFH/MC-CDMA model Figure 4.2: Frequency spectrum of asynchronous GSFH/MC-CDMA signals at that time is referred to as f i (h k ), i = 1...N and k = 1...K and it is given as f i (h k ) = f i + (i 1) + N(h k 1) T s, (4.1) where h k is an integer random variable uniformly distributed between 1 and H. as The transmitted signal for the k th user in GSFH/MC-CDMA systems is given N [s k (t)] mg = 2Pbk (t)c k,i cos [2πf i (h k )t + φ k,i ], (4.2) i=1 where [ ] mg is used to denote symbols that are exclusive to GSFH/MC-CDMA systems. P is the signal power over each subcarrier. φ k,i is the random phase introduced by the BPSK modulator. It is assumed independent and identically distributed (i.i.d.) for each subcarrier and each user, and is assumed uniformly distributed over the interval [0, 2π). The channel considered here is identical to the one considered for asynchronous 59

80 4. GROUP SUBCARRIER FREQUENCY HOPPING MC-CDMA MC-CDMA, and to focus on the effect of GSFH, only uniform distributed timing offset is considered. After the channel, the received signal for GSFH/MC-CDMA can be written as K N [r(t)] mg = 2Pβk,i b k (t τ k )c k,i cos [2πf i (h k )t + ζ k,i ] + n(t), (4.3) k=1 i=1 where ζ k,i = θ k,i + φ k,i 2πf i τ k represents the total phase. At the receiver site, the signal is demodulated and despread. Equal gain combining (EGC) is then applied to the despread signals: the signals received from each subcarrier i (i = 1...N) are multiplied by the gain parameter µ r,i (i = 1...N) then added together. In this research, µ r,i = 1 (i = 1...N) is set. The resulting signal is next passed through a matched filter, then through a maximum likelihood detector. The test statistic at the output of the matched filter is then composed of the desired signal, an MAI term due to the asynchronous transmission and a noise term due to the AWGN channel. The MAI term can be further decomposed into two independent terms [I s ] mg and [I d ] mg. They represent, respectively, the MAI term due to interference within the same subcarrier frequency, and the MAI term due to interference within different subcarrier frequencies. In the next section, the expressions of [I s ] mg and [I d ] mg and their corresponding MAI power will be derived. 60

81 4.2 MAI analysis in different detection scenarios Figure 4.3: Asynchronous GSFH/MC-CDMA detection interval with length of 2N h symbols 4.2 MAI analysis in different detection scenarios Fig. 4.3 represents two hopping intervals. Each interval has length N h symbols. Interferer k is delayed by a timing offset τ k with respect to reference user r. In this figure, the symbols for the reference user are denoted as b r and the symbols for the interferer are denoted as b k. Let h r, 1 represents the subcarrier group index for the first hopping interval and h r,0 represents the subcarrier group index for the second hopping interval. The reference user is allocated subcarrier frequencies f i (h r, 1 ), i = 1...N for symbols b r ( N h + 1) to b r (0), then, after the hop, subcarrier frequencies f i (h r,0 ), i = 1...N for symbols b r (1) to b r (N h ). Similarly, the interferer uses frequencies f i (h k, 1 ), i = 1...N for symbols b k ( N h + 1) to b k (0), then frequencies f i (h k,0 ), i = 1...N for symbols b k (1) to b k (N h ). The current detection interval is assumed to be between symbols b r (1) and 61

82 4. GROUP SUBCARRIER FREQUENCY HOPPING MC-CDMA b r (N h ), and the MAI affecting these symbols will be analyzed. If b r (l) is referred to as the current symbol being detected, b r (l) is affected by the corresponding symbols in the interferer, referred to as b k (l), as well as the previous symbol, referred to as b k (l 1). More precisely, the MAI affecting each symbol in the reference user can be written as the sum of four terms, the [I s ] mg from b k (l) and b k (l 1) respectively, as well as the [I d ] mg from b k (l) and b k (l 1) respectively, where both [I s ] mg and [I d ] mg are functions of the subcarrier frequencies used. The overall MAI affecting each symbol depends on four different scenarios. These four scenarios represent all the possible resulting MAI generating on the reference user. By analyzing the MAI and its power generating in each scenario and obtaining the probability of the occurrence of each scenario, the average MAI power for GSFH/MC-CDMA can be derived. In Scenario A, the reference user chooses a subcarrier frequency group that is identical to the one used by the interferer in the current detection interval, but is different from the subcarrier frequency group used by the interferer in the previous detection interval. Hence f i (h r,0 ) =f i (h k,0 ), (4.4) f i (h r,0 ) f i (h k, 1 ). (4.5) Scenario B is the opposite case of Scenario A. In Scenario B, the reference user chooses a subcarrier frequency group that is different from the one used by the interferer in the current detection interval, but is identical to the subcarrier fre- 62

83 4.2 MAI analysis in different detection scenarios quency group used by the interferer in the previous detection interval. Hence f i (h r,0 ) f i (h k,0 ), (4.6) f i (h r,0 ) =f i (h k, 1 ). (4.7) When the system is in Scenario C, the reference user chooses a subcarrier frequency group that is different from the ones used by the interferer in both hopping intervals which means f i (h r,0 ) f i (h k,0 ), (4.8) f i (h r,0 ) f i (h k, 1 ). (4.9) The opposite of Scenario C is Scenario D, where the reference user chooses a subcarrier frequency group that is identical to the ones used by the interferer in both hopping intervals which gives f i (h r,0 ) =f i (h k,0 ), (4.10) f i (h r,0 ) =f i (h k, 1 ). (4.11) In the following, the MAI power associated with each of the detection scenarios will be analysed Scenario A Scenario A is shown in Fig In this scenario, symbols b r (2) to b r (N h ) in the reference user share the same subcarrier frequencies group with their cor- 63

84 4. GROUP SUBCARRIER FREQUENCY HOPPING MC-CDMA Figure 4.4: Asynchronous GSFH/MC-CDMA detection scenario A responding b k (l) and b k (l 1). For example, for b r (2), its corresponding b k (l) and b k (l 1) are symbols b k (2) and b k (1) in the interferer, which use the same subcarrier frequency group as b r (2). The situation here is identical to the basic MC-CDMA, where symbols in the reference user and the interferer share the same frequencies. Hence the calculation of the MAI for b r (2) is the same as the one shown in basic MC-CDMA, and two types of MAI can be found. Let [ ] Is A and [ ] I A mg d represents the two types of MAI found in b r (2). The mg 64

85 4.2 MAI analysis in different detection scenarios MAI affecting symbol b r (2) can be derived as [ I A s ]mg = K k=1;k r i=1 N (l+1)ts lt s 2Pβk,i b k (l)c k,i c r,i cos [2πf i (h k,l )t + ζ k,i ] cos [2πf i (h r,l ) t] dt K N P = 2 c k,ic r,i cos ζ k,i β k,i [b k (l 1)τ k + b k (l) (T s τ k )], (4.12) [ I A d ]mg = K k=1;k r i=1 N N k=1;k r i=1 j=1;j r (l+1)ts lt s 2Pβk,j b k (l)c k,j c r,i cos [2πf j (h k,l )t + ζ k,j ] cos [2πf i (h r,l ) t] dt K N N 2PTs β k,j c k,j c r,i = [b k (l 1) b k (l)] k=1;k r i=1 j=1;j i [ ( ) 2π i,j sin τ k T s 4π i,j ] sin (ζ k,j ), (4.13) where superscript A refers to scenario A, l can take any integer value and i,j = i j. The MAI powers associated with these two terms are given respectively as V ar [ Is A ]mg = (K 1) NPT2 s σ 2, 6 (4.14) V ar [ Id A ]mg = (K 1) PT2 s σ 2 N N 1 4π 2. (4.15) i=1 j=1;j i 2 i,j Because symbols b r (3) to b r (N h ) in the reference user are in the same situation as symbols b r (2), their MAI powers are identical to (4.14) and (4.15). For symbol b r (1), however, the MAI analysis is different. Due to subcarrier frequency hopping, the corresponding b k (l) and b k (l 1) for symbol b r (1), i.e. symbol b k (1) and b k (0) in the interferer, are on different subcarrier frequencies. As a result, the expressions of MAI generated by b k (1) and b k (0) are different 65

86 4. GROUP SUBCARRIER FREQUENCY HOPPING MC-CDMA and they need to be derived separately. The two types of MAI generated by b k (0) in the interferer can be derived as [ I A,1 s,l 1 ]mg K = k=1;k r [ I A,1 d,l 1 ]mg K = k=1;k r N lts+τk 2Pβk,i b k (l 1)c k,i c r,i i=1 lt s cos [2πf i (h k,l 1 )t + ζ k,i ] cos [2πf i (h r,l )t]dt K N 2PTs β k,i b k (l 1)c k,i c r,i = k=1;k r i=1 4πδ [ ( ) ] 2πδ sin τ k + ζ k,i sin ζ k,i, (4.16) T s N N i=1 j=1;j i lts+τ k lt s 2Pβk,j b k (l 1)c k,j c r,i cos [2πf j (h k,l 1 )t + ζ k,j ] cos [2πf i (h r,l ) t] dt K N N 2PTs β k,j b k (l 1)c k,j c r,i = k=1;k r i=1 j=1;j i 4π ( i,j + δ) [ ( ) ] 2π ( i,j + δ) sin τ k + ζ k,j sin ζ k,j, (4.17) T s where δ = N (h k, 1 h r,0 ). Meanwhile another two types of MAI are generated 66

87 4.2 MAI analysis in different detection scenarios by b k (1) in the interferer, given respectively as [ I A,1 s,l ]mg K = k=1;k r N (l+1)ts 2Pβk,i b k (l)c k,i c r,i i=1 lt s+τ k cos [2πf i (h k,l )t + ζ k,i ] cos [2πf i (h r,l ) t] dt K N P = 2 T sβ k,i b k (l) c k,i c r,i cos ζ k,i, (4.18) k=1;k r i=1 N [ I A,1 d,l ]mg K = k=1;k r N i=1 j=1;j i lts+τ k lt s 2Pβk,j b k (l)c k,j c r,i cos [2πf j (h k,l ) t + ζ k,j ] cos [2πf i (h r,l )t]dt K N N 2PTs β k,j b k (l) c k,i c r,i = k=1;k r i=1 j=1;j i [ 4π i,j ( )] 2π ( i,j + δ) sin ζ k,j sin τ k + ζ k,j. (4.19) T s Taking the summation of the MAI contributed from b k (0) and b k (1) in the interferer then [ ] I A,1 s [ I A,1 d ] mg = [ I A,1 s,l 1 mg = [ I A,1 d,l 1 ] ] + [ I A,1 mg s,l + [ I A,1 mg d,l ], (4.20) mg ]. (4.21) mg As a result the corresponding MAI powers affecting symbol b r (0) in the reference 67

88 4. GROUP SUBCARRIER FREQUENCY HOPPING MC-CDMA user are given as V ar [ Is A,1 ]mg = (K 1) NPT2 s σ 2 ( 1 2 4π 2 δ + 1 ) 2 6 =γ 1 V ar [ ] Is A, (4.22) mg V ar [ I A,1 d ]mg = (K 1) PT2 s σ 2 8π 2 =γ 2 V ar [ I A d ] N N i=1 j=1;j i 1 ( i,j + δ) i,j mg, (4.23) where γ 1 = 3 4π 2 δ , (4.24) N N 1 γ 2 = 1 i=1 ( j=1;j i i,j +δ) N N. (4.25) i=1 j=1;j i Therefore the average MAI power over N h symbols in Scenario A is given by 1 2 i,j V ar [ I A] = 1 (γ 1 V ar [ ] I mg s A N + γ 2V ar [ ] I A h mg d + N ( h 1 V ar [ ] Is A N + V ar [ ] I A h mg d ) mg ) mg. (4.26) Scenario B Fig. 4.5 demonstrates the detection Scenario B for GSFH/MC-CDMA. In this scenario, because for all symbols from b r (2) to b r (N h ) the subcarrier frequencies used by the reference user are different from the ones used by interferers, there is no [I s ] mg and only one type of MAI, [I d ] mg, presents. 68

89 4.2 MAI analysis in different detection scenarios Figure 4.5: Asynchronous GSFH/MC-CDMA dectection scenario B In Scenario B, the MAI affecting each symbols from b r (2) to b r (N h ) in the reference user, is calculated as [ I B d ]mg = K N N (l+1)ts 2Pβk,j b k (l)c k,j c r,i k=1;k r i=1 j=1 lt s cos [2πf j (h k,l )t + ζ k,j ] cos [2πf i (h r,l ) t] dt (4.27) For ease of presentation, the MAI obtained in (4.27) is decomposed into two independent terms referred to as [ ] Id,1 B and [ I B mg d,2 ]mg. [ ] I B d,1 is the result of mg (4.27) when j = i in the double summation term. In contrast, [ ] Id,2 B is the mg 69

90 4. GROUP SUBCARRIER FREQUENCY HOPPING MC-CDMA result of (4.27) when j i. They can be obtained respectively as [ ] K N I B 2Pβk,i T s c k,j c r,i d,1 = [b k (l 1) b k (l)] k=1;k r i=1 4πδ [ ( ) ] 2πδ sin τ k sin ζ k,i, (4.28) T s [ ] K N N I B 2Pβk,j T s c k,j c r,i d,2 = [b k (l 1) b k (l)] k=1;k r i=1 j=1;j i 4π ( i,j + δ) [ ( ) ] 2πδ ( i,j + δ) sin τ k sin ζ k,j. (4.29) T s and their respective power is obtained as V ar [ I B d,1 V ar [ I B d,2 ]mg = (K 1) NPT2 s σ 2 =γ 3 V ar [ I A s ]mg = (K 1) PT2 s σ 2 =γ 4 V ar [ I A d 4π 2 δ 2 ], (4.30) 8π 2 N N i=1 j=1;j i 1 ( i,j + δ) 2 ]. (4.31) where γ 3 = 3 2π 2 δ, (4.32) 2 N N γ 4 = i=1 j=1;j i N N i=1 j=1;j i 1 ( i,j +δ) i,j. (4.33) For symbol b r (1), its corresponding b k (l) and b k (l 1), i.e. b k (1) and b k (0), are on different subcarrier frequencies. In addition, the subcarrier frequencies in b k (0) are the same as the subcarrier frequencies used by the reference user in 70

91 4.2 MAI analysis in different detection scenarios the current detection interval. Hence, unlike other symbols, symbol b r (1) suffers both types of MAI and they should be derived in the same way as in Scenario A. As a result, V ar [ ] Is B,1 ] I A,1 mg s 1V ar [ ] I A mg s mg (4.34) V ar [ I B,1 ] d I A,1 ] mg d 2V ar [ ] I A mg d mg (4.35) Finally the average MAI power over N h symbols in Scenario B can be obtained as V ar [ I B] = 1 (γ 1 V ar [ ] I mg s A N + γ 2V ar [ ] ) I A h mg d mg + N ( h 1 γ 3 V ar [ ] Is A N + γ 4V ar [ ] I A h mg d mg ). (4.36) Scenario C As shown in Fig. 4.6, in Scenario C, because all symbols in the reference user and interferer use different subcarrier frequencies, only the second type of MAI, [I d ] mg can be found. The MAI power affecting all symbols including symbol b r (1) in the reference user is derived in the same way as was done in Scenario B for b r (2). Again the MAI in this scenario is divided into two individual terms. The power associated with each of these two MAI terms is given below V ar [ ] Id,1 C ] I B mg d,1 3V ar [ ] I A mg s mg (4.37) V ar [ ] Id,2 C ] I B mg d,2 4V ar [ ] I A mg d mg (4.38) 71

92 4. GROUP SUBCARRIER FREQUENCY HOPPING MC-CDMA Figure 4.6: Asynchronous GSFH/MC-CDMA detection scenario C As a result the average MAI power for Scenario C can be obtained as V ar [ I C] =V ar [ ] I C mg d,1 + V ar [ ] I C mg d,2 mg =γ 3 V ar [ ] Is A + γ 4V ar [ ] I A mg d. (4.39) mg Scenario D The last detection scenario, Scenario D, for GSFH/MC-CDMA is shown in Fig In this scenario, all symbols in the reference user and interferer share the same subcarrier frequencies. Hence, for all symbols in the reference user, their affecting MAI are identical to the ones calculated for symbol b r (2) in Scenario A. Therefore, the average MAI power in Scenario D is given as V ar [ I D] = V ar [ ] I A mg s + V ar [ ] I A mg d mg (4.40) 72

93 4.3 Average overall MAI Power Figure 4.7: Asynchronous GSFH/MC-CDMA detection scenario D 4.3 Average overall MAI Power From the above analysis, γ 1, γ 2, γ 3 and γ 4 are functions of δ = N (h k, 1 h r,0 ), determined by the distance in the spectrum between two hopping patterns in adjacent hopping intervals. Since the subcarrier frequencies group is assumed to hop following a uniform pattern, the probability density function (PDF) of δ is derived as prob ( δ ) = 2 NH ( ) NH δ H 1 for 0 δ N (H 1) 0 elsewhere, (4.41) 73

94 4. GROUP SUBCARRIER FREQUENCY HOPPING MC-CDMA where represents the absolute value operator. Because γ 1 and γ 2 can be expressed as γ 1 = 1 2 γ , (4.42) γ 2 = 1 2 γ (4.43) Only the average values of γ 3 and γ 4 are required. The results are N(H 1) γ 3 = prob ( δ ) δ=0 N(H 1) γ 4 = δ=0 ( ) 3, (4.44) 2π 2 δ 2 N N 1 i=1 ( prob ( δ ) j=1;j i i,j +δ) 2 N N. (4.45) 1 2 i,j i=1 j=1;j i After considering the four scenarios in Section 4.2, the average overall MAI power expression can now be obtained by multiplying the MAI power by its corresponding probability and then taking the summation. The resulting average MAI power is V ar [I] mg =E γ [ V ar [ I A] mg prob (A) ] + E γ [ V ar [ I B] mg prob (B) ] [ + E γ V ar [ I C] ] prob (C) + V ar [ I D] prob (D), (4.46) mg mg where E γ [ ] represents the expectation operation respect to γ 3 and γ 4. prob(a) denotes the probability of Scenario A which is given as prob(a) = H 1 H 2. (4.47) 74

95 4.4 Simulation Results It represents the probability of choosing f i (h r,0 ) = f i (h k,0 ) and f i (h r,0 ) f i (h k, 1 ). Similarly, prob(b) represent the probability of Scenario B given as prob(b) = H 1 H 2, (4.48) and it represents the probability of choosing f i (h r,0 ) = f i (h k, 1 ) and f i (h r,0 ) f i (h k,0 ). The probability of Scenario C is given as prob(c) = (H 1)2 H 2, (4.49) and it represents the probability of both f i (h r,0 ) f i (h k, 1 ) and f i (h r,0 ) f i (h k,0 ). Finally, the probability of Scenario D is given as prob(d) = 1 H 2, (4.50) Substituting (4.44), (4.45), (4.47), (4.48), (4.49) and (4.50) into (4.46), the overall MAI power can be obtained as V ar [I] mg = 1 ( V ar [ ] Is A H + V ar [ ] ) I A mg d mg + H 1 ( γ 3 V ar [ ] Is A H + γ dv ar [ I A mg s ] ). (4.51) mg It is interesting to note from (4.51) that the MAI power in asynchronous GSFH/MC-CDMA systems is independent of the hopping interval N h. 75

96 4. GROUP SUBCARRIER FREQUENCY HOPPING MC-CDMA Figure 4.8: GSFH/MC-CDMA MAI power ratio for different number of subcarrier frequency groups (N=K=16) Figure 4.9: GSFH/MC-CDMA MAI power ratio for different spreading factors 76

97 4.4 Simulation Results 4.4 Simulation Results In this section the results of Monte-Carlo simulation based on the asynchronous GSFH/MC-CDMA model described in Section 4.1 are shown. A full user system i.e. K = N was considered. The second-order moment of the Rayleigh parameter is set to unity. Walsh-Hadamard spreading sequences are applied and the number of realizations is set to be 100,000. The simulation results are shown in Fig. 4.8 and 4.9, with the theoretical results (shown as continuous lines) matching very closely to the Monte-Carlo simulation results (shown as markers). The performance of the conventional asynchronous MC-CDMA with a uniformly distributed timing offset is used as a benchmark. In order to describe the MAI power reduction effect of GSFH/MC-CDMA systems, a new term, MAI power ratio [Θ] mg is introduced. This term is defined as [Θ] mg = V ar [I] mg V ar [I] mc, (4.52) which is the ratio between the MAI power obtained from asynchronous GSFH/MC- CDMA and the MAI power obtained from conventional asynchronous MC-CDMA. To achieve better system performance, it is desirable to have the MAI power ratio as low as possible. (Note: From this chapter, for simplicity reason, V ar [I] mc is used to represent the average MAI power for MC-CDMA instead of E [ I ]mc 2 ) In Fig. 4.8, showing the MAI power ratio for asynchronous GSFH/MC-CDMA with different values of H, the MAI power ratio decreases with the increasing number of subcarrier frequency groups H. This result indicates that, by allocating extra frequencies and allowing the subcarrier to hop, GSFH can successfully reduce the amount of MAI power for asynchronous MC-CDMA systems. 77

98 4. GROUP SUBCARRIER FREQUENCY HOPPING MC-CDMA Moreover, the amount of MAI power reduction is not affected by the spreading factors. As shown in Fig. 4.9, where the MAI power ratio is plotted against different spreading factors, N, the MAI power ratio for GSFH/MC-CDMA is nearly unchanged for systems with different spreading factors. 4.5 Discussions In previous sections, it was shown that by allowing the subcarriers to hop as a group, MAI power reduction is achieved. Further, it was found that with the number of groups increased, the MAI power for GSFH/MC-CDMA reduced which suggesting better performance than the conventional MC-CDMA. However, this MAI power reduction comes with several costs. Firstly, as shown in Fig. 4.1, both of the transmitter and receiver of GSFH/MC-CDMA require the installment of frequency synthesizers to determined the group hopping pattern for subcarriers. Secondly, hopping pattern information is required to be known on both transmitter and receiver. Both of these two requirements increase the complexity of the system. Finally, in GSFH/MC-CDMA, the whole group of subcarriers has to hop together every hopping interval. As a result, the frequency spectrum for GSFH/MC- CDMA system has to be equal to the multiple of the spreading factor N and number of groups available H. In situations where this frequency spectrum requirement is not satisfied, GSFH/MC-CDMA cannot be applied. To overcome this disadvantage, a new developed system namely, individual subcarrier frequency hopping MC-CDMA (ISFH/MC-CDMA, is proposed in the next chapter. 78

99 Chapter 5 Individual subcarrier frequency hopping MC-CDMA In Chapter 4, group subcarrier frequency hopping MC-CDMA (GSFH/MC-CDMA) is introduced. Although MAI power reduction is achieved, GSFH/MC-CDMA has a limitation of inflexible frequency spectrum requirement, restricting the widely application of GSFH/MC-CDMA. In this chapter, the condition of group hopping for group subcarrier frequency hopping MC-CDMA (GSFH/MC-CDMA) is dropped and a system with a more general form of subcarrier frequency hopping is proposed. This chapter then continues with the performance analysis for the proposed system, which is referred to as the individual subcarrier frequency hopping MC-CDMA (ISFH/MC-CDMA). 79

100 5. INDIVIDUAL SUBCARRIER FREQUENCY HOPPING MC-CDMA Figure 5.1: Asynchronous ISFH/MC-CDMA transmitter and receiver Model 5.1 Asynchronous ISFH/MC-CDMA model Fig. 5.1 shows the model for the transmitter and receiver of the proposed asynchronous ISFH/MC-CDMA systems. This model is similar to the asynchronous MC-CDMA model described in Section 2.2, as well as to the GSFH/MC-CDMA model described in Section 4.1. The main difference between them lies in the transmission subcarrier frequencies. With basic MC-CDMA the transmission subcarrier frequencies remain unchanged during transmission. With GSFH/MC- CDMA, the total bandwidth is divided into H groups of subcarrier frequencies, and at every hop the transmitter randomly chooses one of the H groups as the transmission frequency group. The subcarrier frequencies are still hopping in ISFH/MC-CDMA but the frequency choice process is different. In ISFH/MC- CDMA, the total bandwidth is divided into Q orthogonal frequencies. At every 80

101 5.1 Asynchronous ISFH/MC-CDMA model hop, the ISFH/MC-CDMA transmitter randomly chooses N subcarrier frequencies from all the available Q frequencies, Q N, where N represents the spreading factor and the number of transmitting subcarriers. The expression of the randomly hopped subcarrier frequencies for the k th user is [f k,i ] mi = f i + p k,i T s for i = 1...N, (5.1) where [ ] mi is used to indicate symbols for ISFH/MC-CDMA systems, and p k,i is a random integer with values ranging between 0 and Q 1. The sequence p k = [p k,1,p k,2,...,p k,n ] is referred to as the frequency hopping pattern for the k th user. p k is the output of the frequency synthesizer and it changes every N h T s where N h is the number of symbols transmitted per hop. For slow frequency hopping, N h is restricted to taking any positive integer values that are larger than 1. For a given user, all N subcarrier frequencies must be different during the same hopping interval. The transmitted signal for the k th user is then given by N [s k (t)] mi = 2Pbk (t τ k )c k,i (t τ k ) cos [2πf k,i (t τ k ) + φ k,i ], (5.2) i=1 When comparing (5.2) to (2.2), the expressions of the transmitted signal in ISFH/MC-CDMA and MC-CDMA are similar. The only difference between them is the expression of the subcarrier frequency. The channel considered in this case is the same as the one considered in the case of asynchronous MC-CDMA. The received signal for asynchronous ISFH/MC- 81

102 5. INDIVIDUAL SUBCARRIER FREQUENCY HOPPING MC-CDMA CDMA is K [r(t)] mi = [r k (t)] mi, (5.3) k=1 where [r k (t)] mi is the received signal for the k th user, and is given as N [r k (t)] mi = 2Pβk,i b k (t τ k )c k,i (t τ k ) cos(2π [f k,i ] mi t + ζ k,i ) + n(t). (5.4) i=1 At the receiver site, the signal is first dehopped according to the reference user s hopping pattern, p r, and then passed through a MC-CDMA receiver. The signal at the output of the matched filter also has three components: the desired signal, a MAI term and an AWGN noise term. In the next section, the expressions of the MAI term for the asynchronous ISFH/MC-CDMA are derived. 5.2 MAI analysis for ISFH/MC-CDMA Unlike MC-CDMA, where all users share the same N subcarrier frequencies, in ISFH/MC-CDMA the reference user and a given interferer might only share q subcarrier frequencies with q N, where q is random and is referred to as the number of hits. Because the MAI received in the q subcarriers have different characteristics from the MAI received in the remaining N q subcarriers, for the purpose of MAI analysis, the subcarriers in the reference user are divided into two groups. The first group consists of the q subcarriers that have frequencies shared with the interferer. The MAI calculations for this group of subcarriers are similar to those of the MAI in MC-CDMA. The expressions of the two types of MAI affecting this groups are similar to those of (2.10) and (2.11), except that the 82

103 5.2 MAI analysis for ISFH/MC-CDMA number of subcarrier is q instead of N. Hence [I s ] mi = K q I s,k (i) (5.5) k=1;k r i=1 [I d,1 ] mi = K q N k=1;k r i=1 j=1;j i I k,d (i,j), (5.6) where [I s ] mi represents the MAI generated by the subcarriers with the same frequencies in the interferer, and [I d,1 ] mi represents the MAI generated by the subcarriers with different frequencies in the interferer. In (5.5) and (5.6), I s,k (i) and I d,k (i,j) are given in (2.12) and (2.13) respectively. In the second group of subcarriers, no common frequencies are shared between the reference user and interferers. In the case of q hits, there will be N q subcarriers that can be found in this group. Unlike the first group, only one type of MAI can be found here and it is referred to [I d,2 ] mi. The source of [I d,2 ] mi in ISFH/MC-CDMA is the same as [I d ] mc in MC-CDMA, i.e. it is due to the existing of subcarriers with different frequencies in interferers. Hence the calculation of [I d,2 ] mi can be based on the calculation of [I d ] mc stated in (2.11), which leads to [I d,2 ] mi = K N N I k,d (i,j) (5.7) k=1;k r i=q+1 j=1 where I d,k (i,j) is given in (2.13). As stated in Section 2.2, I s,k (i) and I d,k (i,j) are uncorrelated for different i and k. Hence [I s,k ] mi, [I d,k,1 ] mi and [I d,k,2 ] mi are also uncorrelated. As a result, [I s,k ] mi, [I d,k,1 ] mi and [I d,k,2 ] mi can be approximated as Gaussian random variables. By taking the expectation respect to β k,i, b k and ζ k,i, the mean of [I s,k ] mi, [I d,k,1 ] mi 83

104 5. INDIVIDUAL SUBCARRIER FREQUENCY HOPPING MC-CDMA and [I d,k,2 ] mi are found to be zero and their powers are derived respectively as V ar [I s ] mi = (K 1) qpt2 s σ 2 6 V ar [I d,1 ] mi = (K 1) PT2 s σ 2 q 4π 2 V ar [I d,2 ] mi = (K 1) PT2 s σ 2 4π 2 N 1 i=1 j=1;j i 2 i,j N N 1 i=q+1 j=1 2 i,j (5.8) (5.9). (5.10) Therefore the total power of the MAI for asynchronous ISFH/MC-CDMA is the sum of (5.8), (5.9) and (5.10), which is given as V ar [I q, i,j ] mi = V ar [I s ] mi + V ar [I d,1 ] mi + V ar [I d,2 ] mi. (5.11) In (5.11) above, V ar [I q, i,j ] mi is conditional to the number of hits q and the spectral separation between subcarriers. In order to obtain the average MAI power the probability density functions (PDF) of q and are required. The probability of choosing N out of Q available subcarrier frequencies and having q hits follows the hypergeometric distribution [79] and is given by prob(q) = N q Q N N q Q N. (5.12) 84

105 5.2 MAI analysis for ISFH/MC-CDMA The average value of q is given by [79] N E[q] = q prob(q) = N2 q=q min Q (5.13) where q min = 2N Q is the minimum number of hits for a given Q and N. In MC-CDMA, the spectral distance i,j is deterministic, whereas in ISFH/MC- CDMA, i,j is a random variable with its absolute value ranging between 1 and (Q 1). The general form of the probability density function (PDF) for i,j can be found as prob ( i,j ) = 2(Q i,j ) Q 2 Q for (1) (Q 1) 0 elsewhere (5.14) In (5.14) represents the absolute value. After obtaining the PDF of q and i,j,the average power of the total MAI can be calculated by taking the expectation of (5.11) further with respect to q and i,j. The result is derived as (Proof can be found in Appendix B) V ar [I] mi = (K 1) Pσ2 Ts 2 N Q 1 Q 1 Q 6Q π 2 Q i 1. (5.15) 2 i i=1 i=1 Meanwhile, the average MAI power for MC-CDMA, derived in (2.28), can be rewritten as V ar [I] mc = (K 1) Pσ2 T 2 s N 6 [ π 2 N ( N 1 i=1 N 1 N i 2 i=1 )] 1, (5.16) i 85

106 5. INDIVIDUAL SUBCARRIER FREQUENCY HOPPING MC-CDMA where the following relation is applied N N i=1 j=1;j i ( 1 N 1 = 2 N 2 i=1 N 1 i 2 i=1 ) 1. (5.17) i 5.3 Special case: GSFH/MC-CDMA In the GSFH/MC-CDMA system stated in Chapter 4, one out of H groups of subcarriers is chosen for each user. Because each group has N subcarriers, the total number of subcarriers used in GSFH/MC-CDMA is N H. In ISFH/MC- CDMA, N subcarriers are randomly chosen from the pool of Q available subcarriers for each user, provided Q > N. Since there is no restriction on the selection of subcarrier frequencies in ISFH/MC-CDMA, when the pool of subcarriers in ISFH/MC-CDMA is equal to the total number of subcarriers required for GSFH/MC-CDMA, i.e. Q = N H, the spectrum of the GSFH/MC-CDMA signal is equivalent to one of the possible outcomes resulting from the random selection of subcarriers for ISFH/MC-CDMA. Hence GSFH/MC-CDMA can be viewed as a special case of ISFH/MC-CDMA. As a result, when Q = N H, the MAI power for asynchronous GSFH/MC-CDMA is the same as the MAI power for asynchronous ISFH/MC-CDMA, i.e. V ar [I] mg = V ar [I] mi when Q = N H (5.18) where V ar [I] mg refers to the average MAI power for asynchronous GSFH/MC- CDMA. The proof of this equation can be easily obtained by substituting Q = N H into (5.15) and comparing the result with the expression of V ar [I] mg given 86

107 5.4 MAI power comparison between asynchronous MC-CDMA and asynchronous ISFH/MC-CDMA in (4.51). Based on this result, it can be further concluded that the configuration of subcarrier groups in GSFH/MC-CDMA does not affect the amount of MAI power as long as full user systems (i.e. K = N) are considered. This is because any configuration of subcarrier groups for GSFH/MC-CDMA is just another example of a possible outcome from the random subcarrier selection in ISFH/MC-CDMA and its MAI power is again equivalent to the MAI power in ISFH/MC-CDMA for Q = N H. 5.4 MAI power comparison between asynchronous MC-CDMA and asynchronous ISFH/MC- CDMA In order to describe the MAI power reduction effect of ISFH/MC-CDMA system, the MAI power of the basic asynchronous MC-CDMA system is used as a benchmark and MAI power ratio, [Θ] mi, is introduced, which is defined as [Θ] mi = V ar [I] mi V ar [I] mc = N Q [ [ π 2 Q π 2 N ( Q 1 i=1 ( N 1 i=1 Q Q 1 i 2 i=1 N N 1 i 2 i=1 1 i 1 i )] )] (5.19) where V ar [I] mc is defined in (5.16) and V ar [I] mi is defined in (5.15). In (5.19), when Q = N, ISFH/MC-CDMA generates the same amount of MAI power as MC-CDMA systems. For a given N, the MAI power ratio decreases when Q 87

108 5. INDIVIDUAL SUBCARRIER FREQUENCY HOPPING MC-CDMA increases. When N is large, [Θ] mi approaches to a constant value [Θ] mi N Q. (5.20) The proof of (5.20) can be found in Appendix C. 5.5 Bit error rate analysis In this section, the MAI power derived in the previous sections will be used to obtain the bit error rate (BER) for ISFH/MC-CDMA systems. To derive the BER expression, the mean and variance of the test statistic are required. The mean of the test statistic E [Z] is the desired signal component D. The expression of D in ISFH/MC-CDMA are identical to the one in M-CDMA and it is given in (2.6). Hence for both MC-CDMA and ISFH/MC-CDMA E [Z] = D = P 2 T sb r (t)b r, (5.21) where N B r = β r,i. (5.22) i=1 The corresponding variance of Z is the summation of AWGN variance and MAI power. The AWGN variance is given in (2.8) and it has equal value for both MC-CDMA and ISFH/MC-CDMA. Hence V ar [η] mc = V ar [η] mi. (5.23) 88

109 5.6 Simulation results The MAI power for ISFH/MC-CDMA is given in (5.15). Hence the variance of Z can be written as V ar [Z] mi = V ar [η] mi + V ar [I] mi. (5.24) The general form of bit error rate (BER) conditional to B r is given by p e B r =Q E2 [Z] V ar [Z] = (K 1) σ2 N 2 3Q [ B 2 r π 2 Q ( Q 1 i=1 Q Q 1 i 2 i=1 1 i )], (5.25) where Q( ) represents the Q function. This BER, however, is conditional to B r which represents the sum of N Rayleigh fading coefficients in the reference user. To find the average BER, expectation is taken in respect to B r and it is calculated as E [p e ] = + p e B r prob(b r )db r, (5.26) where prob(b r ) represents the PDF for B r and is given in (2.45). It has been derived in Appendix A. Numerical methods such as Monte Carlo integration [43] can be applied to evaluate the integral in (5.26). 5.6 Simulation results Monte-Carlo simulations based on the asynchronous ISFH/MC-CDMA model described in Section 5.1 were used to verify the theory developed in this chapter. 89

110 5. INDIVIDUAL SUBCARRIER FREQUENCY HOPPING MC-CDMA Figure 5.2: MAI power ratios (as a percentage) for asynchronous ISFH/MC- CDMA systems with different values of Q and N Full user systems are considered i.e. N = K. The second order moments of the Rayleigh fading coefficients for all channels are set to unity. 100, 000 realizations were used for the Monte-Carlo simulations. The simulation results of the MAI power ratio are shown in Fig. 5.2 and Fig In all figures, the theoretical values are shown as continuous lines and the Monte-Carlo simulation results are shown as markers MAI power ratio The MAI power ratio for asynchronous ISFH/MC-CDMA is shown in Fig In this figure, a fixed number of extra subcarrier frequencies (i.e. Q N) is allocated (e.g. 2, 5, 10 and 20). As it shows, when no extra subcarrier frequencies is allocated, i.e. Q = N, the MAI power ratios (shown as a percentage) are remaining as in 100% for all values of N and K, indicating that no MAI power reduction occurs, and introducing slow frequency hopping in this case does not reduce the MAI power. When two extra subcarrier frequencies, i.e. Q = N +2, are allocated 90

111 5.6 Simulation results Figure 5.3: MAI power ratio for asynchronous GSFH/MC-CDMA and ISFH/MC- CDMA systems for N=K=8 to enable hopping, the MAI power ratios start to drop, which suggests that the MAI power in ISFH/MC-CDMA is now lower than the basic MC-CDMA. Then when more extra subcarrier frequencies are allocated for hopping, the resulting MAI power ratios are lower. For example, when N = K = 32 and an extra 20 subcarrier frequecies are allocated, i.e. Q = N +20, the MAI power in ISFH/MC- CDMA is only 60% of the MAI power in the basic MC-CDMA. Therefore, for a given N and K, the MAI power ratio decreases as the number of available subcarriers Q increases; however, this is at the expense of bandwidth. It can also be seen that the MAI power reduction effect of using subcarrier frequency hopping is more significant for small spreading factors (a small number of users). On the other hand, where large spreading factors and a large number of users are applied, the MAI power ratio tends to approach its limit, defined in (5.20). The MAI power ratios for asynchronous GSFH/MC-CDMA and ISFH/MC- CDMA are shown in Fig It can be seen that at the left end, where the number of available subcarrier Q equals the spreading factor N, both GSFH/MC- 91

112 5. INDIVIDUAL SUBCARRIER FREQUENCY HOPPING MC-CDMA CDMA and ISFH/MC-CDMA are equivalent to the basic MC-CDMA, hence the MAI power ratio is 100%. When Q increases, the MAI power ratios for GSFH/MC-CDMA and ISFH/MC-CDMA decrease. It should be noted that at the point where Q = N H, the amounts of MAI power generated from asynchronous GSFH/MC-CDMA and ISFH/MC-CDMA are identical, which verifies the validity of (5.18) Bit error rate Fig. 5.4 shows the bit error rate performance for the asynchronous ISFH/MC- CDMA system with N = K = 16 and the correlation coefficient for subcarrier fading is set to 0.7. The BER for basic asynchronous MC-CDMA is also shown in Fig. 5.4 for comparison purposes. It can be seen that the introduction of the SFH technique to MC-CDMA can improve the BER performance due to the reduction of MAI power. Furthermore, with an increase for Q, the BER of ISFH/MC-CDMA can be further reduced. 5.7 Discussion In Sections 5.2 to 5.6, it has been shown that the introduction of SFH to MC- CDMA can improve the performance of the system by generating less MAI power. This improvement, however, is at the expense of spectral widening because extra subcarrier frequencies must be allocated to the system to enable hopping. In the situation when a fixed bandwidth is given and when the MAI is so large that the system performance (e.g. BER) cannot meet the minimum quality requirement, the capacity of the system must be reduced, i.e. the maximum number of users 92

113 5.7 Discussion Figure 5.4: Bit error rate performance for asynchronous ISFH/MC-CDMA with (N = K = 16 and the correlation coefficient for subcarrier fading is 0.7) must be less than the number of available subcarriers. In this case, the use of ISFH/MC-CDMA will allow for a higher capacity than that achieved by the standard MC-CDMA system. 93

114 5. INDIVIDUAL SUBCARRIER FREQUENCY HOPPING MC-CDMA 94

115 Chapter 6 Asynchronous MC-CDMA with base station polarization diversity This chapter firstly provides the literature review for polarization diversity. Comparisons between polarization diversity and other diversity techniques are given. Then the two factors affecting polarization diversity are discussed, followed by the review of the existing diversity combining technique. After the literature review, this chapter proposes a new system which combines the MC-CDMA system with a base station polarization diversity scheme. The new system will be referred to as Pol/MC-CDMA. The performance of Pol/MC-CDMA in asynchronous transmission is analyzed and, in particular, an expression for the multiple access interference (MAI) is derived. Furthermore, this chapter also proposes a new diversity combining technique, namely maximal signal-to-mai ratio combining (MS- MAIRC), to combine the signals in two diversity antennas for Pol/MC-CDMA. Finally, in this chapter the effects of antenna angles are studied and the optimal antenna angles for both MRC and MSMAIRC are derived. 95

116 6. ASYNCHRONOUS MC-CDMA WITH BASE STATION POLARIZATION DIVERSITY 6.1 Polarization diversity versus other diversity techniques During wireless communication transmission, the envelope amplitudes of the received signals are subjected to fading due to multipath propagation. To reduce the fading effect, various diversity techniques have been developed. The most commonly used techniques are 1) time diversity, 2) frequency diversity, 3) space diversity and 4) polarization diversity. The simplest form of diversity is time diversity, which involves transmitting information repeatedly at a time spacing that exceeds the coherence time of the channel [1]. The time-spacing set-up is to make sure that the signals carrying repeated information are subjected to independent fading in different transmission time. Despite its simplicity, this form of diversity is wasting important time communication resources [80]. Frequency diversity involves transmitting the same information on two or more carrier frequencies. If these frequencies are separated by more than the coherence bandwidth of the channel, the signals on different transmitted frequencies are approximately uncorrelated [81]. However, frequency diversity is expensive in terms of transmitters and required bandwidth. It has hardly been used because frequency bandwidth is one of the most scarce resources in wireless communications [1]. Space diversity is shown to be more economical in terms of communication resources, than both time and frequency. With space diversity, the same information is transmitted in two antennas that are separated in space [80]. However, space diversity has spacing issues in the base station (BS) because BS is nor- 96

117 6.2 The mechanism of polarization diversity mally elevated above most local reflectors, signals arrive through narrow angles and hence need wide separation in space to keep the correlation coefficient between signals below 0.7 [82], the threshold for an effective diversity scheme [80]. Experimental results show that space diversity at a BS requires antenna spacing of up to about 20 to 30 wavelengths for the broadside case and more for the in-line case [83, 82]. Giving the limited size of BS, space diversity is impractical. Polarization diversity technique has overcome all the disadvantages of the previous three techniques. It involves transmitting and receiving the same information with two antennas that are in vertical and horizontal polarization states [82]. Unlike time and frequency diversity, polarization diversity does not require either extra time or extra frequency bandwidths. The two polarized antennas can be co-located, which solves the spacing problem for space diversity. Since the signals in both polarizations are utilized, polarization diversity has the benefit of recovering the energy residing in the polarization orthogonal to the polarization of the receive antenna [84]. Therefore, due to the above advantages over other forms of diversity, polarization diversity has been chosen to be the focus of this research. Its capability of reducing the effect of MAI will be explored. 6.2 The mechanism of polarization diversity During the propagation between mobile and base stations, signals in both vertical polarization (Vpol) and horizontal polarization (Hpol) undertake multiple reflections due to the presence of obstacles (such as buildings and vehicles) in the propagation path. These multiple reflections can cause two effects which make polarization diversity possible. 97

118 6. ASYNCHRONOUS MC-CDMA WITH BASE STATION POLARIZATION DIVERSITY First, because signals in different polarization states generally have different reflection coefficients; when reflection occurs, both Vpol and Hpol signals undergo different phase shifts [85]. Such phase shifts can decorrelate the signals in Vpol and Hpol, resulting in independent signals in two polarizations [86]. Secondly, because reflections occur in three dimensions, the signal power of one polarization can be cross coupled into its orthogonal polarization [85]. For example, assume that the signals are transmitted in Vpol. After a sufficient number of random reflections, some of the transmitted signal power in Vpol can be decoupled into Hpol. As a result, the receiver can have independent signals carrying identical information available in both Vpol and Hpol, indicating that a receiving polarization diversity system can be established. 6.3 Two factors affecting polarization diversity Diversity systems are built to counter the fading effect, but not all diversity systems can achieve this goal. For a diversity system to be effective, signals in all branches of the diversity must be independent (or weakly correlated) to each other and they must have comparable mean signal levels [87]. Consequently two factors affect polarization diversity: 1) the correlation coefficient, which is the measure of the degree of independency between signals in two polarizations, and 2) the cross polarization discrimination (XPD), which is the measure for the mean signal levels in Vpol and Hpol. 98

119 6.3 Two factors affecting polarization diversity Correlation coefficient For a diversity system to be effective, the correlation coefficients between branches are required to be below 0.7 [80]. Many experiments have been conducted to measure the correlation coefficients for polarization diversity in different transmission environments and in different transmission frequencies. For example, as far back as 1953, through experimenting on ionospherically propagated radio signal, and by showing the joint distribution of the signal amplitudes received by vertical and horizontal antennas, Glaser and Faber [88] declared that the fading of signals received on vertically and horizontally polarized antennas is approximately independent, but no specific value of correlation coefficient was claimed at that time. In 1972, Lee et al. [89] conducted their experiment in a suburban environment at 836MHz and formally proposed the polarization diversity. They showed that the correlation between two polarized waves is low, with value mostly below 0.2. Then in [85], an experiment, conducted in 463MHz for both suburban and urban areas, showed that the correlation coefficient between the signal in Vpol and Hpol is for the suburban area and for the urban area. In [87], Turkmani et al. conducted the experiments in the 1800MHz frequency band, with five different environments chosen for measurements, including urban and suburban areas. They reported at least 95% of the collected data for the correlation coefficient to be less than 0.7 in all environments. Lempiainen and Laiho-Steffens [90] conducted another experiment in the 1800MHz frequency band, focusing on small cell system only. All experiment results in [90] were collected in semiurban area and the analyses were separated by two situations: 1) with a line of sight (LOS) path and 2) without a line of sight (NLOS) path. They found that in LOS 99

120 6. ASYNCHRONOUS MC-CDMA WITH BASE STATION POLARIZATION DIVERSITY and NLOS there are no considerable differences between correlation coefficients. Moreover, the average correlation coefficient is less than 0.2. A more detailed experiment was conducted by Dietrich et al. in [91]. The frequency used by the experiment was 2.05GHz. Eight different locations were considered including urban, suburban, rural, indoor and outdoor-to-indoor areas. The correlation coefficients were reported to be between 0.02 for the indoor area and 0.32 for the outdoor area. Therefore in almost all experiments conducted so far, the signals in Vpol and Hpol have a correlation coefficient much less than 0.7. In some cases, such as the experiment conducted in [85], the two signals in Vpol and Hpol can be considered as independent Cross polarization discrimination (XPD) Another important factor for polarization diversity is the cross polarization discrimination (XPD) factor. XPD is defined as the ratio between the signal power in Vpol and the signal power in Hpol. In polarization diversity the mean signal difference between the signals in Vpol and Hpol is measured by XPD. For systems to achieve effective polarization diversity, XPD is desired to have values that are close to 0dB. Empirical studies [85, 87, 90, 91] found that XPD is largely dependent on the transmission environment. For example, in the experiment conducted in [85], it was reported that XPD is 7dB in urban area and 12dB in suburban area. In [87], XPD was found to be in the range between 9.5dB for urban area and 12.6dB for Rural area. Lempiainen et al. in [90] tried to separate the analyses of the cases 100

121 6.4 Diversity combining technique LOS and NLOS. They reported that for a pure LOS connection the XPD can be as high as 14dB and for a completely NLOS environment XPD can still be 7.5dB. The more comprehensive study in [91] showed that XPD is in the range between 6 and 7dB for an outdoor area but that for an indoor area XPD can be as low as 2.3dB. In general XPD is relatively low in urban and indoor areas. This is because in urban and especially in indoor area, the large number of obstacles in the propagation path causes sufficient reflections and refractions; these in term lead to effective polarization cross coupling [91]. On the other hand, in suburban and rural area, the number of obstacles is relatively small. Hence reflection and refraction are insufficient. As a result a larger XPD is obtained. XPD is the major factor that limits the wide implementation of polarization diversity. Having a large XPD means the receiver can received the signals in one branch only and thus the advantage of introducing polarization diversity scheme is lost. 6.4 Diversity combining technique Three types of diversity combining techniques can be used to combine the outputs in polarization diversity, namely selection combining, equal gain combing (EGC) and maximum ratio combing (MRC). Discussion of each of these techniques follows. 101

122 6. ASYNCHRONOUS MC-CDMA WITH BASE STATION POLARIZATION DIVERSITY Selection combining With selection combining, the system simply selects the diversity branch output with the largest signal-to-noise ratio (SNR) and discards the output into other branches [80]. Such branch selection can increase the average SNR of the system and thus offers a better performance [1]. The advantage of selection combining is its simple implementation, as it requires only a side monitoring station and an antenna switch at the receiver [1]. However, this technique is not optimal because it does not fully utilize the signals available in all possible diversity branches [1, 3] Maximal ratio combining Maximal ratio combining (MRC) overcomes the limitations of selection combining: it combines the input signals in all diversity branches. MRC has been considered as the optimal combining technique in the presence of additive white Gaussian noise (AWGN) due to its ability to maximize the instantaneous output SNR [1]. This is demonstrated as below. Assume a system with N d diversity branches, the instantaneous output SNR is given by [3] SNR = N d ( ) Eb i=1 N N 0 d µ i β i e jθ i i=1 2 µ i 2 (6.1) where E b is bit energy; N 0 is noise spectral density, µ i is the combining weight and β i and θ i are the magnitude and phase of the received signal respectively. 102

123 6.4 Diversity combining technique To obtain the maximum instantaneous output SNR, Cauchy-Schwarz inequality is applied, giving the maximum value as [3] SNR N d ( ) Eb i=1 N 0 µ i 2 N d N d i=1 2 β i e jθ i i=1 = µ i 2 ( ) Nd Eb The only condition to reach this maximum value is to set [3] N 0 i=1 N d βi 2 = SNR i (6.2) i=1 µ i = cβ i e jθ i for i = 1...N d (6.3) where c is some arbitrary complex constant. Therefore, according to (6.3), in MRC, the magnitude of the combining weight is proportional to the magnitude of the received signal, and the phase of the combining weight is the negative value of the phase of the received signal. The maximum SNR in (6.2) also suggests that MRC can produce an output SNR equal to the sum of the individual SNRs in each diversity branch. It follows that MRC can offer the advantage of producing an acceptable output SNR even when none of the SNR in individual branches is acceptable [1] Equal gain combining Although MRC has the ability to maximize the instantaneous output SNR, in certain cases it is difficult to track the magnitude and phase of the received signal in order to produce a time varying combining weight [3, 1]. Equal gain combining (EGC) provides a more convenient solution whereby the combining weights in all branches are equal to a constant [80]. 103

124 6. ASYNCHRONOUS MC-CDMA WITH BASE STATION POLARIZATION DIVERSITY With constant gain, EGC can not maximize the instantaneous output SNR. Hence it is not optimal and its performance is inferior to MRC. However, EGC still has the ability to exploit the signals in all diversity branches, allowing it to give better performance than selection combining [1]. 6.5 Base station polarization diversity reception model and its applications To overcome the XPD limitations and to make polarization diversity more feasible, the Kozono model was proposed in [83], as a base station reception model using polarization diversity. In the Kozono model, the MS transmits signals in Vpol. The receiver in the BS is composed of two antennas elements which have ±α inclination angles from the vertical axises. This BS antenna configuration is aimed to equalize the received mean signal level in two antenna elements. However, this equalization is at the expense of raised correlation coefficients [85]. As reported in [83], by letting α = 45 degree, the value of the mean signal level differences between two antenna elements can be lower than 2.5dB but the correlation coefficient will increase to a value not larger than 0.6. The increased correlation coefficient, however, is still lower than the 0.7 threshold for effective diversity scheme. Hence the Kozono model is able to unlock the potential advantages of polarization diversity, providing a solution for countering multipath fading in the BS. Despite its significance and simplicity, very few studies have been done regarding the application of polarization diversity using the Kozono configuration with 104

125 6.5 Base station polarization diversity reception model and its applications CDMA systems. To the best of the author s knowledge, the analysis of DS-CDMA system using a polarization diversity scheme at the BS has been attempted only in [92] and [93]. In both papers, a new BS receiver architecture which combines multistage interference cancellation and polarization diversity using the Kozono configuration is proposed. The performance of uplink DS-CDMA using this receiver was analyzed and it was claimed that the proposed receiver can achieve significant performance gains over the conventional DS-CDMA receiver. However, the application of the Kozono model to MC-CDMA has never been studied before. One of the aims of this research is to fill this gap. In the following section, a new system which combines the Kozono model with asynchronous MC- CDMA is proposed. The new system is referred to as Pol/MC-CDMA and its performance during asynchronous transmission is analyzed. Further, in this chapter an optimum combining method referred to as maximal signal-to-mai ratio combining (MSMAIRC) is proposed for the Pol/MC-CDMA system. Unlike MRC, which is aimed to maximize the instantaneous SNR and to provide optimal bit error rate (BER) in the presence of additive white Gaussian noise (AWGN), MSIRC is aimed to maximize the instantaneous signal-to-mai ratio (SMAIR) and provide optimal BER in the presence of MAI. Hence in asynchronous MC-CDMA systems, where performance is typically limited by the MAI, applying MSMAIRC is more appropriate than applying MRC. However, in reality the performance of asynchronous MC-CDMA is not subjected to only AWGN or MAI but both of them. Hence the overall BER performance for asynchronous MC-CDMA is determined by signal-to-mai-plus-noise ratio (SMAINR) instead of SMAIR and SNR only. In this chapter, it will be shown that when MSIRC and MRC are applied to Pol/MC-CDMA, the SMAINR 105

126 6. ASYNCHRONOUS MC-CDMA WITH BASE STATION POLARIZATION DIVERSITY of the system can be affected by the antenna angle α, and then the optimum antenna angles for both combining methods are derived. 6.6 System model for Pol/MC-CDMA system The transmitter structure for the Pol/MC-CDMA system model is identical to the one for asynchronous MC-CDMA systems, described in Section 2.2, but since this chapter is dealing with base station polarization diversity, it needs to be emphasized that each user signal in the Pol/MC-CDMA system model is transmitted through a single vertically polarized antenna. The channel considered in Pol/MC-CDMA is the same as the one considered in the case for asynchronous MC-CDMA. However, to focus on the effect of polarization diversity, only uniform distributed timing offsets are considered in this analysis. The main differences between the asynchronous Pol/MC-CDMA and MC- CDMA models lie in the structures of the base station receiver. As illustrated in Fig. 6.1, the base station receiver architecture is based on the two-branch polarization diversity configuration proposed by Kozono et al. The two base station antennas, V 1 and V 2, are co-located and are inclined at angles ±α relative to the vertical axis. Azimuthal dependence of user k is introduced with δ k ; it is assumed as a random variable uniformly distributed between π/2 and π/2. Polarization diversity is achieved through the reception of signals that have undergone independent fading in both polarizations. Even though the mobile subscriber (MS) is transmitting in a principally vertical polarization, the presence of obstacles acting as reflection, scattering sources in the mobile environment causes signal power cross coupling between two polarizations. As a consequence, some 106

127 6.6 System model for Pol/MC-CDMA system Figure 6.1: Two-branch receiver model for base station of the power in the vertical polarization (Vpol) is decoupled into the horizontal polarization (Hpol) [85]. At the receiver the resulting signal is modelled as a signal that contains both a vertically polarized component r k,v and a horizontally polarized component r k,h. The expressions of r k,v and r k,h are given respectively as N r k,v = 2Pbk β k,v,i c k,i cos [2πf i (t τ k ) + φ k,i + θ k,v,i ] (6.4) i=1 N r k,h = 2Pbk β k,h,i c k,i cos [2πf i (t τ k ) + φ k,i + θ k,h,i ]. (6.5) i=1 The random phases introduced by the vertically and horizontally polarized channels are modelled with θ v,k,i and θ h,k,i, respectively. It is assumed that θ v,k,i and θ h,k,i are independent and are uniformly distributed over the interval [0, 2π). The 107

128 6. ASYNCHRONOUS MC-CDMA WITH BASE STATION POLARIZATION DIVERSITY Rayleigh fadings associated with the Vpol and Hpol are introduced with the β k,v,i and β k,h,i parameters respectively. From the experimental results published in [85, 87, 89], it is reasonable to assume that β k,v,i and β k,h,i are uncorrelated. The power of β k,v,i and β k,h,i is related by cross polarization discrimination (XPD), which is defined as the ratio between the available power in the Vpol and the available power in the Hpol. χ = E [ ] βk,v,i 2 E [ ] = σ2 v. (6.6) βk,h,i 2 σh 2 The signal received in the diversity antennas is the summation of the resultant projections of r k,v and r k,h onto V 1 and V 2 [83]. V 1 and V 2 are calculated as K V 1 = r k,h (t) sin α cosδ k + r k,v (t) cosα, (6.7) V 2 = k=1 K k=1 r k,h (t) sin α cos δ k + r k,v (t) cosα. (6.8) The newly proposed receiver and combiner are an extension to a single user MC- CDMA receiver. Each of the separate diversity branches as realized by V 1 and V 2 is fed into a separate instance of the single user MC-CDMA receiver, as shown in Fig The signals received in the two branches are detected independently and combined together before being input through the decision device to determine the transmitted bit. Notice that diversity combining of the V 1 and V 2 branches occurs after the matched filter stage. This type of combining is called postdetection combining, and signal combination is achieved when the output test statistics Z 1 and Z 2 from the correlation receivers are first weighted with a branch gain ε q (q = 1, 2) and then summed to give the final test statistic [Z] mp where 108

129 6.7 Derivation of the test statistic for Pol/MC-CDMA Figure 6.2: MC-CDMA receiver with polarization diversity [ ] mp represents symbols that are exclusive for Pol/MC-CDMA systems. 6.7 Derivation of the test statistic for Pol/MC- CDMA The final test statistic [Z] mp is a weighted sum of the individual test statistics output from each of the diversity branches and it is calculated as 2 (l+1)ts 2 N [Z] mp = Z q ε q = V q c r,i cos (2πf i t)µ q,i ε q dt q=1 lt s q=1 i=1 = [D] mp + [η] mp + [I] mp. (6.9) The l parameter can be any arbitrary integer, and is defined to select the symbol of interest. µ q,i are the combining gain parameters used to weight each of the 109

130 6. ASYNCHRONOUS MC-CDMA WITH BASE STATION POLARIZATION DIVERSITY individual subcarriers prior to the matched filter. At this stage, equal gain combining (EGC) is assumed, and, for the sake of simplicity, these parameters have all been set to unity, such that µ q,i = µ i = 1 for q = 1, 2 and i = 1...N. The resulting calculation for the final test statistic [Z] mp in (6.9) is most easily represented as a sum of three components, the desired signal [D] mp, the additive white Gaussian noise [η] mp, and the multiple access interference component [I] mp. The desired signal component [D] mp is calculated as [D] mp = P 2 T sb r (l) [sinαcosδ r (ε 1 ε 2 )B h + cos α (ε 1 + ε 2 )B v ], (6.10) where N B v = β r,v,i (6.11) i=1 N B h = β r,h,i. (6.12) i=1 The data bit of the reference used in the current detection interval [lt s, (l + 1)T s ] is denoted by b r (l). The interference due to noise introduced by the AWGN channel is represented by [η] mp. It is a Gaussian random variable with zero mean and variance given as V ar [η] mp = N 0T s N 4 ( ε ε 2 2 ). (6.13) The MAI component, as a result of the loss in orthogonality between user codes for the other K 1 active users, is represented by [I] mp. In asynchronous MC-CDMA, the MAI is analyzed by decomposing [I] mp into two independent 110

131 6.7 Derivation of the test statistic for Pol/MC-CDMA terms, [I s ] mp and [I d ] mp [20]. [I s ] mp represents the MAI generated by other users using the same subcarrier frequencies, while [I d ] mp represents the MAI generated by other users using different subcarrier frequencies. The expressions for [I s ] mp and [I d ] mp can be written respectively as P K [I s ] mp = 2 T s [b k (l 1)τ k + b k (l) (T s τ k )] k=1;k r N β k,h,i cos ζ k,h,i sin α cos δ k (ε 1 ε 2 ) c k,i c r,i µ i i=1 +β k,v,i cos ζ k,v,i cos α (ε 1 + ε 2 ), (6.14) [I d ] mp = 2P K k=1;k r N µ i N i=1 j=1;j i [b k (l 1) b k (l)] c k,j c r,i T s 2 i,j β k,h,j sin α cos δ k (ε 1 ε 2 ) [sin ( i,j τ k + ζ k,h,j ) sin ζ k,h,j ] +β k,v,j cos α (ε 1 + ε 2 ) [sin ( i,j τ k + ζ k,v,j ) sin ζ k,v,j ]. (6.15) where b k (l) is denoted as the binary data for the k th user in the current detection interval i.e. [lt s, (l + 1)T s ]. Similarly, b k (l 1) represents the binary data associated with the previous detection interval i.e. [(l 1)T s, lt s ]. The ζ parameters has been introduced to represent the total combined phase, where ζ k,h,i = θ k,h,i + φ k,i + 2πf i τ k and ζ k,v,i = θ k,v,i + φ k,i + 2πf i τ k. In (6.15), denotes the spectral distance between subcarrier i of the reference user and subcarrier j of the interfering user k. The spectral distance is defined as = i j. Due to the assumption that φ k,i, θ k,v,i and θ k,h,i are i.i.d. for different users and different subcarriers, [I s ] mp and [I d ] mp are also independent for different sub- 111

132 6. ASYNCHRONOUS MC-CDMA WITH BASE STATION POLARIZATION DIVERSITY carriers and different users. Because of this, the central limit theorem allows the approximation of [I s ] mp and [I d ] mp as Gaussian random variables [20]. When the expectation is taken with respect to β k,i, b k, ζ k,v,i and ζ k,h,i, the average values of both [I s ] mp and [I d ] mp are found to be zero and their variance given respectively as V ar [I s ] mp = (K 1)PT2 s σ 2 vn 6 [ sin 2 ] α 2χ (ε 1 ε 2 ) 2 + cos 2 α (ε 1 + ε 2 ) 2 (6.16) V ar [I d ] mp = (K 1)PT2 s σ 2 vnc 4π 2 [ sin 2 ] α 2χ (ε 1 ε 2 ) 2 + cos 2 α (ε 1 + ε 2 ) 2 (6.17) where C = 1 N N N 1 i=1 j=1;j i 2 i,j. (6.18) The total MAI power V ar [I] mp is calculated as the summation of (6.16) and (6.17), which gives V ar [I] mp =V ar [I s ] mp + V ar [I d ] mp = (K 1)PT 2 s σ 2 vn [ C ] 4π 2 [ sin 2 ] α 2χ (ε 1 ε 2 ) 2 + cos 2 α (ε 1 + ε 2 ) 2. (6.19) The three main combining techniques employed in practice are selection combining, equal gain combining and maximal ratio combining. When selection combining is used, the diversity branch with the highest instantaneous baseband SNR is chosen and fed into the receiver. In equal gain combining, ε 1 is set equal to 112

133 6.8 Maximal signal-to-mai ratio combining ε 2, where they are typically chosen to equal one. If MRC is applied, each of the diversity branches is weighted by their respective SNRs, cophased and then summed. In the absence of MAI, the MRC gives the best statistical reduction to fading, and the resultant SNR at the baseband is maximum [1]. Nevertheless, where Pol/MC-CDMA is considered, when the contribution of interference from MAI is significant, MRC can not guarantee the best combining performance because the combining gains do not account for the interference due to MAI. The objective of this paper is to derive the optimal combining gain parameters, ε 1 and ε 2, for Pol/MC-CDMA, in the presence of MAI. 6.8 Maximal signal-to-mai ratio combining During asynchronous MC-CDMA transmission, MAI is the major limiting factor of system performance. Better performance is expected when the combining gains are designed to maximize the signal-to-mai ratio (SMAIR). In this section, the combining gains that will maximize the instantaneous SMAIR is derived. This new combining scheme will be referred to as the maximal signal-to-mai ratio combining (MSMAIRC). The instantaneous SMAIR of an asynchronous MC-CDMA signal is derived by taking the ratio between the power of the desired signal and the power of the MAI SMAIR = [sinαcos δ r (ε 1 ε 2 )B h + cos α (ε 1 + ε 2 )B v ] 2 2 (K 1)σvN ( ) [ C sin 2 α (ε 6 4π 2 2χ 1 ε 2 ) 2 + cos 2 α (ε 1 + ε 2 ) 2]. (6.20) 113

134 6. ASYNCHRONOUS MC-CDMA WITH BASE STATION POLARIZATION DIVERSITY To further simplify the instantaneous SMAIR expression, let Υ s = (ε 1 + ε 2 ), and Υ d = (ε 1 ε 2 ). The numerator and the denominator are then divided by Υ 2 d. SMAIR = ( sin α cosδr B h + cosαb v Υ s Υ d ) 2 2 (K 1)σvN ( ) [ C sin 2 α + 6 4π 2 2χ cos2 α ( ) ] 2. (6.21) Υ s Υ d The new combining gains to yield the maximum instantaneous SMAIR are derived by taking the gradient of (6.21) relative to Υ s /Υ d, and setting the result to zero. This equation is then solved for Υ s /Υ d. Υ s Υ d = sin αb v 2χ cos α cos δ r B h. (6.22) The interpretation of this result is that, as long as the combining gains ε 1 and ε 2 satisfy the relationship stated in (6.22), the maximum SMAIR is achieved. One such example of this is to set the combining gains as ε 1 = sinαb v + 2χ cos α cos δ r B h, (6.23) ε 2 = sinαb v 2χ cos α cos δ r B h. (6.24) Using these combining gains, the maximum obtainable instantaneous SMAIR can be calculated by substituting (6.22) into (6.21). SMAIR max = 2χ cos2 δ r Bh 2 + Bv 2 (K 1)σvN ( ). (6.25) C 6 4π 2 From this analysis, one of the important details to note when using MSMAIRC in the absence of AWGN is that the result of the maximum obtainable instantaneous 114

135 6.8 Maximal signal-to-mai ratio combining SMAIR in (6.25) is independent of the antenna angle α. It should be also note that if the expectation of the SMAIR max is taken with respect to Bv 2 and Bh, 2 the average SMAIR becomes independent of the XPD. The process of calculation is shown below. E [SMAIR max ] = 2χ cos2 δ r E [Bh] 2 + E [Bv] 2 2 (K 1)σvN ( ). (6.26) C 6 4π 2 The second-order moments E [B 2 v] and E [B 2 h] are required to evaluate the expression for the average SMR. The details of the calculations are found in Appendix D. E [ B 2 v ] [ ( =Nσ 2 v 1 + (N 1) ρ v π 4 ρ v + π )] 4 (6.27) E [ B 2 h ] [ ( =Nσ 2 h 1 + (N 1) =N σ2 v χ ρ h π 4 ρ h + π 4 [ ( 1 + (N 1) ρ h π 4 ρ h + π )], (6.28) 4 )] where ρ v and ρ h are the correlation coefficients for the fading between two subcarriers in Vpol and Hpol, respectively. By substituting (6.27) and (6.28) into (6.26), this research shows that the average of SMR is independent of the XPD. E [SMAIR max ] = [ ( 2 cos 2 δ r 1 + (N 1) ρh πρ )] 4 h + π 4 + [ 1 + (N 1) ( ρ v πρ )] 4 v + π 4 2 (K 1) ( 1 + ). (6.29) C 6 4π 2 115

136 6. ASYNCHRONOUS MC-CDMA WITH BASE STATION POLARIZATION DIVERSITY The notable characteristics of using the new MSMAIRC for Pol/MC-CDMA in the absence of AWGN are: 1)the instantaneous SMAIR is maximized; 2)the instantaneous SMAIR is invariant of the antenna angle; 3)the average SMAIR is independent of XPD Performance analysis of Pol/MC-CDMA in the presence of both AWGN and MAI Even though the contribution to the source of interference for MC-CDMA is dominated by the MAI, in practice the interference as a result of the AWGN cannot be ignored. In this section, both the MAI and the AWGN are considered as sources of interference, and the performance analysis of Pol/MC-CDMA is presented by evaluating the instantaneous signal-to-mai-plus-noise ratio (SMAINR) and the bit error rate (BER). The general formula for the instantaneous SMAINR is given as [ P 2 T 2 s b 2 r (l) ] (sinαcos δ r Υ d B h + cos αυ s B v ) 2 SMAINR = [ (K 1)PT 2 s σvn ( )] ( ) C 6 4π sin 2 α 2 2χ Υ2 d + cos 2 αυ 2 s. (6.30) + ( ) N 0 T sn 8 (Υ 2 d + Υ 2 s) The instantaneous BER can then be calculated with the application of the Q function to the obtained SMAINR [33], which gives, p e B v, B h = Q ( SMAINR ), (6.31) where Q ( ) is the Q function. The average BER is obtained by taking the 116

137 6.8 Maximal signal-to-mai ratio combining expectation with respect to B v and B h, which is presented here by E [p e ] = + + p e B v,b h prob (B v,b h )db v db h. (6.32) Previously, in section 6.6, it was assumed that the fadings of the Vpol and Hpol are independent, hence the joint distribution of B v and B h, prob (B v,b h ) can be written as the product of their respective distributions, prob (B v ) and prob (B h ). As a result, the average BER can be written as E [p e ] = + + p e B v,b h prob (B v ) prob (B h )db v db h. (6.33) The evaluation of the average BER requires knowledge of the PDF for B h and B v i.e. prob (B v ) and prob (B h ). Because the Rayleigh random variable is a special case of the Nakagami-m random variable (with m = 1), the PDF for the sum of N independent Rayleigh random variables can be approximated by a Nakagami-m distribution [94]. In the case where the Rayleigh random variables are correlated, the exact expression of the PDF for B has been derived in [95]. However, the evaluation of this PDF requires the summation of an infinite series, which is impractical especially when the number of random variable N is large. Through statistical (Shown in Appendix A), this research found that the PDF for the sum of N correlated Rayleigh random variables can also be approximated by the Nakagami-m distribution. prob (B v ) and prob (B h ) are approximated by prob (B v ) = 2m v mv Bv 2mv 1 Γ (m v ) Ω mv v prob (B h ) = 2m h m h B 2m h 1 h Γ (m h ) Ω m h h ( exp m v Bv 2 Ω v ( exp m ) h Bh 2 Ω h ) (6.34), (6.35) 117

138 6. ASYNCHRONOUS MC-CDMA WITH BASE STATION POLARIZATION DIVERSITY Figure 6.3: BER comparison between Pol/MC-CDMA with MRC and MC- CDMA (with α = π/4; δ r = 0 and N = K = 16) with parameters of the Nakagami-m distribution given as ] Ω v = E [ Bv 2 Ω h = E [ ] Bh 2 m v m h (6.36) (6.37) E [B 2 v] 5 [ E [B 2 v] (E [B v ]) 2] (6.38) E [B 2 h] 5 [ E [B 2 h ] (E [B h]) 2]. (6.39) The associated first order moments for B v and B h are E [B v ] = Nσ v π/4 and E [B h ] = Nσ h π/4 respectively. The expressions of E [Bv] 2 and E [Bh] 2 have been given earlier in (6.27) and (6.28) respectively. Finally, the double integral for the average BER is evaluated using the Monte Carlo integration technique [43]. 118

139 6.8 Maximal signal-to-mai ratio combining Figure 6.4: BER comparison between Pol/MC-CDMA with MSMAIRC and MC- CDMA (with α = π/4; δ r = 0 and N = K = 16) Figure 6.5: BER comparison between Pol/MC-CDMA with MSMAIRC and MRC (with α = π/4; δ r = 0 and N = K = 16) 119