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Interference cancellation and macrodiversity for wideband CDMA systems employing software radio base… Nie, Hong 2003

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Interference Cancellation and Macrodiversity for Wideband CDMA Systems Employing Software Radio Base Stations By Hong Nie B. Sc. Eng., Tsinghua University, 1993 M . A. Sc., Tsinghua University, 1995 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Electrical and Computer Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 2003 © Hong Nie, 2003 Abstract This thesis focuses on the applications of Successive Interference Cancellation (SIC) and Combining Macrodiversity (CM) in software radio base station receivers to increase the reverse link capacity of wideband Code Division Multiple Access (CMDA) systems. In the software radio base station receiver, an Analog-to-Digital Converter (ADC) with very high resolution is required to digitize the wideband analog signals received by the receiver. In order to mitigate this stringent ADC resolution requirement, we have proposed a novel digitization method, Adaptive Prediction & Cancellation Digitization (APCD) method. By predicting and canceling the high power narrowband signals among the wideband analog signals, the APCD method in conjunction with Auto-Regressive (AR) or Periodic Auto-Regressive (PAR) predictor can significantly reduce the high dynamic range of the wideband analog signals, and hence, can effectively relax the steep ADC resolution requirement. Furthermore, since the high power narrowband signals can usually be modeled as cyclostationary signals, by theoretical analysis and by means of computer simulations, it is shown that the PAR predictor can achieve a much higher prediction gain than the AR predictor. Although C M and Multi-cell SIC (MSIC) have the potential to increase the reverse link capacity of cellular CDMA systems, they cannot be used in conventional base station receivers. For base station receivers employing software radio technology, we have proposed a new base station system architecture, software radio Distributed Base Station (DBS) system architecture. In this system architecture, base station receivers can simultaneously exploit the radio signals received by several DBSs to detect mobile users; hence, the applications of C M ii and MISC become possible. When either CM, Single-cell SIC (SSIC) or MSIC is employed, the receiving power of mobile users assigned to the same DBS have controlled disparity. Consequently, the existing inter-cell interference model, which is based upon the assumption of equal received power, cannot be used to evaluate the inter-cell interference of these mobiles. We have proposed a new inter-cell interference model, which can precisely evaluate the inter-cell interference of mobile users with controlled power disparity. Furthermore, it has been proven that the existing inter-cell interference model for mobile users with equal receiving power can be obtained from the inter-cell interference model for mobile users with controlled power disparity. In order to demonstrate the reverse link capacity improvement C M and MSIC offer, by applying the inter-cell interference model for mobile users with controlled power disparity, Signal-to-Interference-plus-Noise Ratio (SINR) expressions for mobile users employing the following detection techniques have been obtained by theoretical analysis: i) Single-User Detection (SUD) in conjunction with Selection Macrodiversity (SM), ii) SUD in conjunction with CM, iii) SSIC in conjunction with SM, iv) SSIC in conjunction with CM, v) MSIC in conjunction with SM, and vi) MSIC in conjunction with CM. From these SINR expressions, the corresponding reverse link capacities of cellular CDMA systems have been analyzed by evaluating the outage probability of the systems. Both the analysis results and the computer simulation results show that regardless being employed individually or jointly, C M and MSIC can significantly increase the reverse link capacity of cellular CDMA systems. iii Table of Contents Abstract i i Table of Contents iv List of Tables ix List of Figures x Glossary of Mathematical Symbols xii Acknowledgements xix Chapter 1: Introduction 1 1.1 Stacked Base Station System Architecture 5 1.2 Software Radio Technology 6 1.2.1 Generic Structure of Software Radio Base Station Receiver 7 1.2.2 Analog-to-Digital Conversion Challenge in Software Radio Base Station Receiver 8 1.3 Techniques for Capacity Improvement of Cellular CDMA Systems 11 1.3.1 Macrodiversity 12 1.3.2 Successive Interference Cancellation 14 1.4 A New Software Radio Distributed Base Station System Architecture 16 1.5 Research Contribution of the Thesis 20 1.6 Organization of the Thesis 22 iv Chapter 2: An Adaptive Prediction and Cancellation Digitization Method for Multi-Standard Software Radio Base Station Receivers .24 2.1 ADC Resolution Requirement 25 2.2 Adaptive Prediction and Cancellation Digitization (APCD) Method 29 2.2.1 System Model 29 2.2.2 Factors Influencing G p 32 2.2.3 Implement Requirements 33 2.3 Signal Prediction Algorithms 36 2.3.1 Statistical Characteristics of xs(n) 37 2.3.2 Auto-Regressive (AR) Predictor 39 2.3.3 Periodic Auto-Regressive (PAR) Predictor 44 2.4 Performance Evaluation Results and Discussion 50 2.4.1 Resolution Improvements 51 2.4.2 Effects of Ps /Pw and Rjf,( Bs / / , ) 54 2.4.3 Resolution Requirements of Q2 and DAC 56 2.4.4 Performance Comparison of the AR and the PAR Predictors 57 2.5 Summary .59 Chapter 3: Inter-cell Interference Analysis for Mobile Users with Controlled Power Disparity 61 3.1 Introduction and System Model 61 3.2 Path Loss Model 65 3.3 Inter-cell Interference Model for Controlled Power Disparity 69 3.3.1 Disparity Characteristics 69 v 3.3.2 A Novel Inter-cell Interference Model 72 3.4 Analysis of Individual Inter-cell Interference Factor 73 3.4.1 Inter-cell Interference Originated from S0 74 3.4.2 Inter-cell Interference Originated from S0 75 3.4.3 Individual Inter-cell Interference Factor 76 3.5 Numerical Evaluations and Discussions 77 3.6 Computer Simulations 81 3.6.1 Computer Simulation Methodology 81 3.6.2 Performance Evaluation Results 82 3.7 Summary 83 Chapter 4: Reverse Link Capacity Analysis for Selection and Combining Macrodiversity 84 4.1 Introduction 84 4.2 Capacity Analysis for SUD-SM 86 4.3 Power Reduction Factors for Combining Macrodiversity 89 4.3.1 SINR Expression of Mobile Users Employing SUD-CM 90 4.3.2 Theoretical Analysis for Power Reduction Factors 92 4.3.3 Numerical Evaluations and Computer Simulations 95 4.4 Capacity Analysis for SUD-SM 97 4.4.1 Performance Upper Bound for SUD-CM 97 4.4.2 Outage Probability Analysis 99 4.4.3 Numerical Results 101 4.5 Summary 103 vi Chapter 5: Reverse Link Capacity Analysis for Successive Interference Cancellation 104 5.1 Introduction • 104 5.2 SSIC-SM Capacity Analysis 106 5.2.1 SINR Expression 106 5.2.2 Outage Probability Analysis 107 5.3 MSIC-SM Capacity Analysis 110 5.3.1 A New Approach to Performing MSIC 110 5.3.2 SINR Expression I l l 5.3.3 Outage Probability Analysis I l l 5.3.4 Numerical Evaluations 113 5.4 SSIC-CM Capacity Analysis 115 5.4.1 SLNR Expression 115 5.4.2 Outage Probability Analysis 116 5.5 MSIC-CM Capacity Analysis 118 5.5.1 SLNR Expression 118 5.5.2 Outage Probability Analysis 118 5.5.3 Numerical Evaluations 120 5.6 Computer Simulation Results and Discussion 121 5.6.1 Computer Simulation Methodology 122 5.6.2 Performance Evaluation Results 125 5.7 Summary 129 Chapter 6: Conclusions and Topics for Future Research 130 6.1 Conclusions 130 6.1.1 APCD Method 130 6.1.2 Software Radio DBS System Architecture 131 6.1.3 Inter-cell Interference Model for Mobile Users with Controlled Power Disparity 131 6.1.4 Reverse Link Capacity Analysis for C M 132 6.1.5 Reverse Link Capacity Analysis for SSIC and MSIC 133 6.2 Topics for Future Research 134 Appendix A 136 Appendix B 137 Appendix C 140 References 142 List of Tables 2.1 BER Comparison for the Lower Power BPSK Signal Digitized by: (a) a Conventional ADC, and (b) the APCD method 53 2.2 Resolution Requirements of Q2 and DAC 57 A . l Relation of Clipping Probability and Full-Scale Range 136 ix List of Figures 1.1 Stacked Base Station System Architecture 5 1.2 System Model of Software Radio Base Station Receiver 8 1.3 System Model of Software Radio DBS System Architecture 17 2.1 System Model of the Digitization Domain Implemented by a Single ADC .26 2.2 System Model of the Digitization Domain Employing the APCD Method 30 2.3 Prediction Gain of AR Predictors (Analytical Results) 43 2.4 Performance Comparison between the AR and the PAR Predictors (Analytical Results). 48 2.5 Power Spectral Density Comparison between a Conventional ADC and the APCD Method 52 2.6 Influence of Ps /Pw on Prediction Gain (Computer Simulation Results) 55 2.7 Influence of Rjfs on Prediction Gain (Computer Simulation Results) 56 2.8 Performance Comparison between the AR and the PAR Predictors (Computer Simulation Results - Part I) 58 2.9 Performance Comparison between the AR and the PAR Predictors (Computer Simulation Results - Part II) ..59 3.1 Serving Area of the DBS in the Central Cell for Different Values of Nc 67 3.2 Receiving Power Matrix 72 3.3 Individual Inter-cell Interference Factors with and without the Effects of Lognormal Shadowing Fading. 78 x 3.4 Individual Inter-cell Interference Factors for Different Values of cr. 79 3.5 Inter-cell Interference Originating from S 0 vs. Total Inter-cell Interference 80 3.6 Computer Simulation Model for Individual Inter-cell Interference Factor 81 4.1 Combining Macrodiversity Region for N c =3 86 4.2 Overlapping Serving Areas for N c -4 93 4.3 Power Reduction Factors vs. Path-Loss Rank 95 4.4 Computer Simulation Model for Power Reduction Factors 96 4.5 Reverse Link Capacity of SUD-SM and SUD-CM (Numerical Results) 102 5.1 Reverse Link Capacity of SUD-SM, SSIC-SM, and MSIC-SM (Numerical Results). 114 5.2 Reverse Link Capacity of SUD-CM, SSIC-CM and MSIC-CM (Numerical Results). 121 5.3 Computer Simulations Model for the Evaluation of Outage Probability 123 5.4 Reverse Link Capacity of SUD-SM and SUD-CM (Simulation and Numerical Results). 126 5.5 Reverse Link Capacity of SSIC-SM and SSIC-CM (Simulation and Numerical Results). 127 5.6 Reverse Link Capacity of MSIC-SM and MSIC-CM (Simulation and Numerical Results) 127 5.7 Reverse Link Capacity of SUD-CM, SSIC-CM and MSIC-CM (Simulation and Additional Numerical Results) 128 xi Glossary of Mathematical Symbols Mathematical Operators T: Matrix transposition. £"[•]: Mean value <•> Arithmetic mod P tr(»): Trace of a matrix |»|: Determinant of a matrix A =: Define as Analog-to-Digital Conversion x(n): Sampled version of the wideband analog signals received by a multi-standard base station receiver x s (n): High power narrowband signals Ps: Total power of x s (n) B s : Total bandwidth of x s (n) x w (n): Difference between x(n) and x s (n) Pw: Total power of x w (n) xwk ( n ) : Weakest desired signal in x(n) Pwk: Total power of x w k (n) SNR^n: The minimum required SNR for x w k (n). A A D C : • Full-scale range of an ADC xii B Q : B i t resolution of an A D C f s : Sampling rate of an ADC P s s: Peak power of the largest spurious signal in an ADC NFR max: Maximum near-far ratio of mobile communication systems under consideration Adaptive Prediction and Cancellation Digitization y d (n) : Prediction output of the SPU y ( n ) : Analog equivalent of y d ( n ) e(n) : Residual signal, i.e. e(n) = x(n) - y(n) e d l (n): Digital equivalent of e(n) outputted by Q l x d (n): Digitial equivalent of x ( n ) , x d i n ) - y d ( n ) + e d l ( n ) B D A C : B i t resolution of a D A C A D A C : Quantization step of a D A C A D A C : Full-scale range of a D A C B Q l : B i t resolution of Ql AC1: Quantization step of Q l e q (n): Overall digitization noise of Q l SNR q : SNRofQl,i .e. SNR q = 101og10{^2(n)]/E[^2(n)]} Gp : Prediction gain, i.e. Gp = \0\ogl0{E[x2(n)]/E[e2(n)]} B: A positive integer eco (n): Cut-off noise caused by omitting the least significant bits of the prediction results to obtain y d ( n ) G™ax: Designed maximum SNR improvement of the APCD method Signal Prediction Algorithms u(n): Equivalent baseband signal of a modulated signal fc: Carrier frequency of a modulated signal. A;: Amplitude of the ith branch of received signals : Phase of the ith branch of received signals n i: Delay of the i" 1 branch of received signals J and P: Two positive integers that have no common factor larger than one <p: A constant u A ( n ) : Equivalent baseband signal of x s (n) u A R (n): An AR process with the same statistical characteristics as u A (n) r ( m ) : Autocorrelation function of u A R ( n ) rxx(n'm) '• Autocorrelation function of xs (n) s (n ) : Signal to be predicted y A R (n): Prediction result of an AR predictor L: Number of past samples used to predict the next sample W A R : Prediction coefficients of an AR predictor S ( n ) : Past samples of s(n) e A R (n ) : Prediction error of an AR predictor, i.e. e A R (n ) = s (n) - y A R (n ) W A R: Optimal prediction coefficients of an AR predictor G™^ : Highest prediction gain of an A R predictor C : A constant satisfying -1 < C < 1 v (n ) : A stationary, zero mean, memoryless Gaussian random process Q m z x . Highest prediction gain of u A R ( n ) S : Adjustment step of adaptive L M S algorithm H: Number of cycle frequencies used in a P A R predictor ypAR ( n ) : Prediction result of a P A R predictor WPAR : Prediction coefficients of a P A R predictor {JJP}: Cycle frequencies of s(n) e P A R (n): Prediction error, i.e. e P A R i n ) = s ( n ) - y P A R i n ) Wp^R : Optimal prediction coefficients of a P A R predictor G™PAR '• Highest prediction gain of a P A R predictor E b : B i t energy of a signal n A in): A W G N N A : Single-side P S D of n A i n ) Cellular CDMA System N : Number of mobile users per cell p: Number of mobile users per unit area A c : Area of a hexagonal cell i i , n ) user: z t h mobile user assigned to the n t h D B S PLinm : Path loss from the (i, n) user to the m* D B S E l " j l f : Reverse link SLNR of the (i, n ) user xv Pinm : Average receiving power of the (i, n) user at the m DBS Imm : Intra-cell interference the (i, n) user suffers at the DBS V"m : Inter-cell interference the (i, n) user suffers at the mm DBS R: Information bit rate of mobile users B w : Spreading bandwidth of mobile users yV0: Single-side PSD of thermal noise F N : Receiver noise figure a. Required SINR of mobile users pmax. Maximum transmitting power of mobile users p = l o g 10/10 Path Loss Model and Inter-cell Interference Model DAm : Distance attenuation (in dB) from a mobile user to the m"1 DBS rm: Distance from a mobile user to the DBS ju: Distance attenuation exponent D: A fixed attenuation common to all DBSs LSFm : Lognormal shadowing fading (in dB) from a mobile user to the DBS h and hm: Two independent zero-mean Gaussian distributed random variables with a standard deviation equal to a a and b: Two constants satisfying a 2 + b 2 =1 CLmn: Correlation of the lognormal shadowing fading from a mobile user to the m"1 and the nmDBSs PLm : Path loss (in dB) from a mobile user to the m" 1 DBS xvi N c : Number of DBSs among which a mobile user can minimize its path loss 0th DBS: DBS in the central cell S 0: Serving area of the 0 t h DBS S 0 : Area outside S 0 PL: Path loss of a mobile user, i.e. PL - minf.PL } 7=1 1 F(K): Cumulative probability function of the path loss of a mobile user Ks: Maximum possible path loss a mobile user can have so as to be served by the system PKS '• Ks -limited p F K s (K): K s -limited F(K) R e : Joint inter-cell interference factor [R^ (Ks)}: Individual inter-cell interference factors originating from 5 0 {R-h (KS)}: Individual inter-cell interference factors originating from 5 0 [Rf}: Overall individual inter-cell interference factors, i.e. R?(Ks) = RsaS(Ks) + R%(Ks) Reverse Link Capacity KSSUD~S: Ks of the system employing SUD-SM U SUD~S (iV): Outage probability of the system employing SUD-SM Nc-l P i n: Combined receiving power of the (/, n) user, i.e. Pin = P i n n + ^P i n m. 7=i { i i N ( K s ) } : Power reduction factors xvn PLin: Combined path loss of the (/, n) user fK (K): Ks -limited probability density function of the path loss of a mobile user q(K): Power reduction (in dB) of a mobile user whose path loss is equal to K KSSUD~C : Ks of the system employing SUD-CM U SUD~C (N): Outage probability of the system employing SUD-CM \p : Interference cancellation factor Kf'c~s: Ks of the system employing SSIC-SM U ss,c~s (AO: Outage probability of the system employing SSIC-SM KMSIC-S . K ^ o f t h e s y s t e m employing MSIC-SM U MS!C~S (N): Outage probability of the system employing MSIC-SM K s s i c - c . K^ o f t h e s y s t e m e m p i 0 y i n g SSIC-CM U ss,c~c (AO : Outage probability of the system employing SSIC-CM KMS,C-C . K ^ o f t h e s y s t e m employing MSIC-CM U MS!C~C (N): Outage probability of the system employing MSIC-CM xviii Acknowledgements Many people were involved in the preparation of this thesis. Without their help, I would not be able complete it. I am extremely grateful of my research supervisor, Professor P. Takis Mathiopoulos. During the course of my studies, his continued assistance in all aspects was essential to me. His optimism and encouragement help me pass through some difficult time of my research. I would like to thank the members of my thesis examination committee, Dr. Cyril Leung, Dr. Vikram Krishnamurthy, Dr. Victor Leung, and Dr. Mathew Yedlin, and the external examiner for their time and efforts spent reviewing this thesis. I would also like to thank Dr. Andrew Wright for his valuable suggestion during the initial stages of my research, and Dr. H. P. Bernhard who kindly provided a computer program to calculate the upper bound of prediction gain. I wish to thank my parents for their assistance in taking care of my baby daughter so that I can focus on preparing this thesis. I am forever indebted to my wife, Ning Ma, for providing me with the opportunity to complete this thesis by keeping me healthy and clean and enduring endless lonely nights. Her contribution to this thesis is as great as my own. I wish to apologize to my daughter, Melissa, for unable to stay with her since she was only six months old. I would like to dedicate this thesis to her, my lovely little angle. xix Chapter 1 Introduction Ever since lst-Generation (1G) mobile communication systems, employing analog Frequency Modulation (FM) and Frequency Division Multiple Access (FDMA) technology [1], were introduced to the public in the beginning of the 1980's, demand for mobile communication services has increased exponentially. In 1990, there were only 10 million users of mobile communication services worldwide, but by 2002, this number had increased to about 700 million [2]. Furthermore, it is expected that the number of users of mobile communication services worldwide will increase to about 1.3 billion by the end of 2003, and 1.8 billion by the end of 2007 [3]. In contrast to this increasing demand for mobile communication services, the Radio Frequency (RF) spectrum shared by all mobile communication systems is a finite natural resource. Although the RF bandwidth allocated to mobile communication systems by regulatory agencies has grown steadily in the last two decades, due to fundamental physical laws, such growth is completely unsustainable in the not-far future [4]. Therefore, achieving higher system capacity within a limited RF bandwidth has become one of the most important driving forces for the evolution of mobile communication systems. By employing digital techniques and Time Division Multiple Access (TDMA) technology [1], 2nd-Generation (2G) mobile communication systems, such as the Global System for Mobile Communication (GSM) in Europe, the Interim Standard 54 (IS-54) and the Interim Standard 136 (IS-136) in North America, and the Personal Digital Cellular (PDC) in Japan, can achieve much higher system capacity than 1G systems [5]. Thus, since becoming 1 commercially available in the early 1990's, 2G systems have quickly replaced 1G systems, and have turned into the dominant mobile communication system in the world. However, as 2G systems are designed to mainly support narrowband communications, such as voice applications, message/text applications, and transaction processing, the average throughput an individual mobile user can achieve is less than 14.4 kb/s, and even in the so-called 2.5G systems, such as the Interim Standard 95 (IS-95) [1] and the General Packet Radio Service (GPRS) [2], the average throughput of an individual mobile user is less than 40 kb/s [3]. Therefore, 2G systems cannot support wideband multimedia services, such as still image transfers, Internet/VPN access, database access, enhanced Web surfing, large file transfer, Hifi audio, interactive entertainment, video telephony, video conference, and high-quality video [3],[6]. Furthermore, although 2G systems utilizing the TDMA technology can achieve much higher system capacity than 1G systems utilizing the FDMA technology, their system capacity within a limited RF bandwidth still cannot meet the exponentially increasing demand on mobile communication services [2]-[4]. In order to further increase the capacity of mobile communication systems within a limited RF bandwidth, and to support wideband multimedia services as well, during the past few years 3rd-Generation (3G) mobile communication systems, called International Mobile Telecomunications-2000 (IMT-2000) [7], have been experiencing an increase in research and development [8]-[ll]. As compared to 2G systems, 3G systems mainly employ Code Division Multiple Access (CDMA) technology to realize the access of multiple mobile users [7]. Research and development activities over the previous decade show that CDMA technology has strong potential to achieve much higher system capacity than the TDMA and the FDMA technologies [12]-[14]. Furthermore, in addition to supporting existing narrowband 2 communications, 3G systems are designed to also support a variety of wideband multimedia services [6]. For example, as shown in [3], in 3G systems, the average throughput of an individual mobile user is in the range of 100-150 kb/s, or higher. Such a high average throughput provides adequate bandwidth for most wideband multimedia services, which 2G systems cannot support. Although 3G systems have many technological advantages over 2G systems, during the past two years, mainly for economic reasons, many mobile communication service providers have expressed serious doubts concerning the profitability of 3G systems within the present market situation [15]. Thus, a common acknowledgment has been made that 3G systems need to evolve from existing 2G systems step by step, and coexist with current 2G systems for quite a long time before they finally replace those 2G systems [16]. According to this common acknowledgement, in order to provide improved mobile communication services and to enable global and seamless roaming for mobile users, the receivers of next generation base stations must be able to support both 2G and 3G systems, such as, the GSM, the IS-136, the IS-95, and the IMT-2000. Furthermore, as shown in [4], in order to convert the potential of CDMA technology to achieve high system capacity within a limited RF bandwidth into reality, a series of signal processing techniques, such as, multi-user detection [17]-[18], macrodiversity [19]-[24], microdiversity [5], antenna arrays [25]-[26], and channel error control coding [5],[27], must be employed in conjunction with CDMA technology in the receivers of next generation base stations. Motivated by the above observations, this thesis focuses on proposing and evaluating a new system architecture for base station receivers, which satisfies the following two requirements: 3 • First, the new system architecture must enable base station receivers to support both 2G and 3G systems. In this thesis, this requirement is fulfilled by employing software radio technology [28]-[34], a technology widely considered to have the potential to solve the dilemma of simultaneously supporting multiple mobile communication standards. • Second, the new system architecture must facilitate base station receivers to employ certain signal processing techniques in conjunction with CDMA technology to improve the overall capacity of 3G systems. Since macrodiversity and Successive Interference Cancellation (SIC - a widely used multi-user detection technique) [17]-[18],[35]-[37], have strong connections with the system architecture of base station receivers, in this thesis, the performance of these two signal processing techniques in software radio base station receivers are thoroughly studied. The remainder of this introductory chapter is organized as follows. In Section 1.1, the stacked base station system architecture is discussed. Then the current state-of-the-art software radio technology is reviewed in Section 1.2. In the same section, analog-to-digital conversion is identified as one of the serious technical challenges limiting the employment of software radio technology in the base station receivers which can support both 2G and 3G systems. In Section 1.3, by reviewing the characteristics of macrodiversity and SIC techniques, the obstacles to employing Combining Macrodiversity (CM) and Multi-cell Successive Interference Cancellation (MSIC) in the stacked base station system architecture are revealed. In order to enable the applications of C M and MSIC, a novel base station receiver system architecture, referred to as software radio Distributed Base Station (DBS) system architecture, is proposed in Section 1.4, and its advantages for 3G systems are 4 thoroughly discussed. The research contributions of this thesis are summarized in Section 1.5. Finally, the organization of this thesis is given in Section 1.6. 1.1 Stacked Base Station System Architecture As illustrated in Figure 1.1, in 2G systems most base station receivers employ a system architecture referred to as stacked base station system architecture [28]. More specifically, the receivers employing the stacked base station system architecture are usually assembled by a stack of receiver chains, and typically, each of these receiver chains consists of an RF Processing Functionality, an Intermediate Frequency (IF) Processing Functionality and a Baseband Processing Functionality [38]. V RF Processing IF Processing Baseband Processing Mobile User 1 RF Processing IF Processing Baseband Processing Mobile User 2 A/ RF Processing IF Processing Baseband Processing Mobile User N Receiver Chains Figure 1.1 Stacked Base Station System Architecture. Since most base station receivers of 2G systems only need to support a single mobile communication standard, their receiver chains are dedicated to and optimized for that particular standard. Thus, for 2G systems, when the implementation issues of the base station receivers are considered, the performance and the cost of the base station receivers are more 5 important than their flexibility. Correspondingly, although compared with digital signal processing techniques, analog signal processing techniques are not flexible and are still widely employed in receiver chains to process RF and IF signals. For example, the RF Processing Functionality and the IF Processing Functionality are usually implemented by analog signal processing techniques [38]. With the existence of RF and IF analog signals in the receiver chains, base station receivers must be deployed close to their antennas, since otherwise expensive and complicated linear transmission systems would be required to avoid the performance degradation caused by transmitting these RF and IF analog signals over a long distance. Therefore, in the stacked base station system architecture, the radio signals received by each base station are processed by it own receivers, and it is very difficult for these receivers to exploit the radio signals received by other base stations. 1.2 Software Radio Technology In order to support both 2G and 3G systems, base station receivers must be able to simultaneously support multiple mobile communication standards. The most straightforward approach to implementing such a multi-standard base station receiver is to assemble more receiver chains into the stacked base station system architecture, illustrated in Figure 1.1, and dedicate each receiver chain to a specific standard. However, such an approach is far from flexible in the sense that most of the hardware of the receiver chains needs to be replaced whenever an old standard is dropped, a new standard is added, or the air interface of a standard is changed [16]. Furthermore, when the number of standards the base station receiver needs to support is large, or the complexity of the standards is high, the base station receiver assembled by a bank of receiver chains becomes very complicated and expensive. 6 In 1995, Mitola proposed a novel technology, which has been referred to as software radio technology, to implement multi-standard receivers [28]. The receiver employing software radio technology digitizes the received signals at the earliest possible receiving stage and implements most signal processing functionalities with software reconfigurable digital devices, such as Field Programmable Gate Arrays (FPGAs), Digital Signal Processors (DSPs), workstations and even Main Frame Computers (MFCs). Thus, with software-reconfigurable functionalities in its every architectural level, the software radio receiver can achieve a high degree of flexibility and adaptation even at the physical level [29]. Hence it is capable of adapting to different transmission parameters and supporting virtually any mobile communication standards. Therefore, software radio technology is considered as a more efficient approach toward the implementation of demanding multi-standard base station receivers [31]. 1.2.1 Generic Structure of Software Radio Base Station Receiver Since the idea of the software radio receiver was proposed, it has motivated much research and development activity from both industrial and university based research teams [28]-[34]. In summarizing the achievements of these research and development efforts, we first propose in [39] a system model for the software radio base station receivers, which is illustrated in Figure 1.2. In this system model, the main components of a software radio base station receiver include an Analog Domain, a Digitization Domain, and a Digital Domain. In the Analog Domain, received signals are usually amplified and filtered. Furthermore, because IF signals can be digitized more efficiently than RF signals, the conversion from the RF signals to the IF signals is also performed in the Analog Domain. V Configuration Unit Wideband Wideband Analog RF • Analog IF V Analog Domain Analog-to-Digital Conversion Y Digitization Domain Digital IF I Digital Baseband J Digital IF I Digital Baseband Digital IF I Digital Baseband Control Interface Channel 1 Channel 2 Channel N v Digital Domain Figure 1.2 System Model of Software Radio Base Station Receiver. The analog-to-digital conversion is implemented by the Digitization Domain, which, in the simplest case, can be a single Analog-to-Digital Converter (ADC). In other cases, however, this domain may be a complicated subsystem, composed of one or more ADCs and other signal processing devices, in order to achieve the performance requirement set by the software radio base station receivers. As the most important and complicated domain in the software radio base station receivers, the Digital Domain performs most of the IF and baseband signal processing tasks a base station receiver can have, such as de-spreading, equalization, channel selection, phase tracking, demodulation, interference cancellation/suppression, framing, de-interleaving, and forward error control [28]. Furthermore, the Digital Domain includes a Configuration Unit to provide the necessary configuration signals in accordance with the control commands arriving through the Control/Network interface. 1.2.2 Analog-to-Digital Conversion Challenge in Software Radio Base Station Receivers From the system model of the software radio base station receivers illustrated in Figure 1.2, it can be observed that unlike the receivers employing the stacked base station system architecture, such a software radio receiver requires a Digitization Domain which can effectively digitize the wideband analog signals received by the multi-standard base station receivers. Thus, the implementation of the Digitization Domain becomes an important prerequisite for the efficient and effective operation of the software radio base station receivers. In practical applications, several mobile communication systems may be neighboring each other in the same frequency band. For example, the 1900 MHz frequency band is used simultaneously by the IMT-2000, the GSM and the IS-136 [7],[40]. Thus, the Digitization Domain of the software radio base station receivers must be able to simultaneously digitize wideband signals, for example, the LMT-2000 signals with bandwidth of 5MHz, and narrowband signals, for example, the GSM and the IS-136 signals with bandwidth of 200 kHz and 30 kHz respectively [40]. Due to multi-path fading [5], lognormal shadowing fading [41], and distance attenuation [5], typically, for more than 20% of the time these signals are characterized by a significant difference in their receiving power [38]. For instance, the near-far ratio for the GSM signals could be up to 90 dB [42]. As shown in [42]-[44], if a single Analog-to-Digital Converter (ADC) is employed to implement the Digitization Domain of the software radio base station receivers, with such a high dynamic range, in order to satisfy the Signal-to-Noise Ratio (SNR) requirement for the weakest signal among the overall received signals, the resolution requirement of the ADC can be as high as 20 bits. Furthermore, because the receiving bandwidth of the software radio base station receivers must be at least 5 MHz in order to support the LMT-2000 [7], the sampling rate of the ADC must be larger than 10 Million Samples Per Second (MSPS). 9 Nevertheless, ADCs with a resolution close to 20 bits are commercially available only when the sampling rate of the ADC is less than 400 kHz, and approximately for every doubling of the sampling rate, the resolution of these commercially available ADCs reduces one bit [45]. In other words, for off-the-shelf commercially available ADCs with sampling rates higher than 10MSPS, their resolution is usually lower than 16 bits. What makes things even worse is that historically, the performance of the state-of-the-art ADC has improved only by 1 bit every 3 to 5 years [43]. Obviously, a large gap exists between the performance of commercially available ADCs and the ADC resolution requirement to support both 2G and 3G systems. Thus, it is clear that the inherent inability of the commercially available ADCs to perform well for digitizing the wideband analog signals received by the software radio base station receiver has become one of the major technical challenges which may hinder the application of software radio technology in multi-standard base station receivers. In the past, very few techniques that can effectively relax this steep ADC resolution requirement have been proposed in open technical literature. For example, the most straightforward approach is to divide the whole receiving frequency band into many sub-bands and utilize many parallel ADCs in the Digitization Domain [42]. However, if the received signals reside in more than one sub-band, this approach has serious problems for combining the digitization results obtained by several parallel ADCs to recover the original received signals. Furthermore, when the receiving bandwidth of the multi-standards base station receivers is wide, the utilization of a large number of parallel ADCs significantly increases the overall complexity and implementation cost of the software radio base station receivers. As shown in [43], another possible method is to apply analog nonlinear compression 10 techniques, such as logarithmic compression, to suppress the dynamic range of the wideband analog signals received by the multi-standard base station receivers before those signals are digitized by the ADC. Subsequently, digital inversion processing is performed on the digitization results to restore the original received signals. The main disadvantage of this method is that the nonlinear distortion produced by the analog nonlinear compression techniques cannot be completely removed by the digital inversion processing, and hence, the residual nonlinear distortion will seriously reduce the SNR of the received signals, especially those with weak receiving power. 1.3 Techniques for Capacity Improvement of Cellular CDMA Systems When software radio technology is employed, new system architectures, different from the stacked base station system architecture, can be utilized to implement base station receivers. Thus, certain signal processing techniques which cannot be used in the stacked base station system architecture now may be used in the new system architectures to increase the capacity of 3G systems. As previously discussed, in order to achieve high system capacity within a limited RF bandwidth, a series of signal processing techniques, such as multi-user detection, macrodiversity, microdiversity, antenna arrays, and channel error control coding, must be employed in the base station receivers of 3G systems. Among those signal processing techniques, the performances of macrodiversity and SIC are strongly connected with the system architecture of base station receivers. Therefore, in this thesis, our research efforts are focused on how to increase the capacity of the 3G systems by improving the performances of these two signal processing techniques in the new system architecture for base station receivers employing software radio technology. 11 1.3.1 Macrodiversity In mobile communication systems, the radio signals transmitted by a mobile user can be simultaneously received by several base stations in the vicinity of the mobile user. Consisting of distance attenuation [5] and lognormal shadowing fading [41], the path losses from the mobile user to different base stations have different values. Correspondingly, the average receiving power of the mobile user has different values at different base stations. Usually, the mobile user is assigned to the base station to which the mobile user has the minimum path loss, since the mobile user has the strongest average receiving power at that base station, and hence, in most cases, the mobile user has the highest Signal-to-Interference-plus-Noise Ratio (SINR) at that base station [21]. In this thesis, unless otherwise indicated, the path loss of a mobile user refers to the path loss from the mobile user to the base station the mobile user is assigned to, and the receiving power of a mobile user refers to the average signal power of the mobile user received by the base station the mobile user is assigned to. Generally, there are two kinds of macrodiversity techniques, Selection Macrodiversity (SM) [5], [19] and Combining Macrodiversity (CM) [20]-[24]. When SM is employed, only the radio signals received by the base station the mobile user is assigned to are used to detect the mobile user. Nevertheless, radio signals, possibly of inferior quality, received by other base stations in the vicinity of the mobile user may still be useful for improving the overall SINR of the mobile user. This is especially true when the path loss of the mobile user is large. In this case, with a high probability, the mobile user is far from the base station it is assigned to. Thus, its path losses to other base stations may have a similar value as its path loss to the base station it is assigned to, and correspondingly, at these base stations the mobile user may have a similar SINR as that at the base station it is assigned to. 12 When C M is employed, the radio signals received by several base stations in the vicinity of the mobile user are combined together, and these combined signals are exploited to detect the mobile user. Thus, to achieve the same SINR, the receiving power of the mobile user can be lower than that employing SM, especially when the path loss of the mobile user is large. For the following two reasons, such reduction on the receiving power of mobile users can significantly increase the reverse link capacity of the cellular CDMA system. First, the reduction of the receiving power of a mobile user also means the reduction of its interference to other mobile users. As shown in [12], cellular CDMA systems are interference-limited systems, that is, any reduction of the multiple access interference can lead to capacity improvement. Therefore, the systems employing C M can achieve a much higher reverse link capacity than those employing SM. Second, the reduction of the receiving power of mobile users can effectively reduce the outage probability of the systems. Usually, an outage event happens if the path loss of a mobile user is so large that even when the mobile user has reached its maximum transmitting power, its receiving power is still too low to meet the SNIR requirement. When C M is employed, with a high probability, the mobile users with a large path loss can achieve a higher reduction of their receiving power. Thus, these mobile users can have more margins to combat with their path loss, and hence, the systems employing C M have a lower outage probability than those employing SM. Correspondingly, to maintain the same outage probability, the reverse link capacity of the systems employing C M is higher than those employing SM. However, when the stacked base station system architecture illustrated in Figure 1.1 is employed, the base station receivers must be deployed close to their antennas. Thus, the radio 13 signals received by each base station are processed independently, and it is very difficult for a base station receiver to exploit the radio signals received by other base stations. From this observation, it is clear that only the SM can be utilized in the stacked base station system architecture, and new base station receiver system architectures must be developed so that C M can be employed to increase the reverse link capacity of the cellular CDMA systems. 1.3.2 Successive Interference Cancellation (SIC) In conventional cellular CDMA systems, Single-User Detection (SUD) is employed in the base station receivers to detect mobile users, that is, each mobile user is detected independently by its own matched filters without regard for other mobile users [17]. Although the implementation of SUD is simple, its ability to combat multiple access interference is weak [17]. Hence, the reverse link capacity of the systems employing SUD is seriously restricted by the multiple access interference. In order to increase the reverse link capacity, various multi-user detection techniques have been proposed to be employed in the base station receivers in order to reduce the effect of multiple access interference [17]. Among these multi-user detection techniques, SIC is considered as one of the simplest, yet most robust methods [17],[35]-[37]. For example, as shown in [17], the complexity of SIC is less than half of that of one-stage Parallel Interference Cancellation. As described in [17], when SIC is employed, mobile users assigned to the same base station are detected successively. Whenever a mobile user is detected, according to the detection results, its received signals are regenerated and cancelled from the overall received signals, and then the rest of the mobile users are detected from the modified overall received signals. Thus, the mobile users who are detected subsequently suffer less multiple access interference, and hence the reverse link capacity of the cellular CDMA systems can be 14 increased. Furthermore, because the mobile users detected subsequently suffer less interference than those detected earlier, in order to achieve the same SINR, the mobile users detected subsequently may have lower receiving power. Correspondingly, these mobile users have a larger margin to combat with a larger path loss than those detected earlier. As previously discussed, outage events mainly take place on the mobile users with a large path loss. Therefore, in order to reduce the outage probability of the systems, in most cases the mobile users assigned to the same base station are detected successively from the minimum to the maximum path loss [36]-[37]. One potential problem of SIC is that any detection error made by previous mobile users seriously reduces the SINR of subsequent mobile users [17]. Therefore, in order to achieve the performance improvement SIC offers, only the mobile users whose Bit Error Rate (BER) is lower than a required value is regenerated and cancelled from the overall received signals. In the stacked base station system architecture illustrated in Figure 1.1, the radio signals received by each base station are processed by its own receivers, and it is very difficult (if not impossible) for these receivers to exploit the radio signals received by other base stations. Furthermore, in each base station, usually, only the assigned mobile users can have a high enough SINR to meet the BER requirement set by the SIC. Thus, in the stacked base station system architecture, for each base station, only the interference produced by the mobile users assigned to it, referred to as intra-cell interference, can be cancelled. In this thesis, we refer to this type of SIC as Single-cell SIC (SSIC). As shown in [12] and [41], in addition to the intra-cell interference, the interference produced by the mobile users assigned to other base stations, referred to as inter-cell interference, also seriously limits the reverse link capacity of the cellular CDMA systems. 15 Thus, if SIC is able to cancel not only the intra-cell interference, but also the inter-cell interference, the reverse link capacity of the systems can be effectively increased. In this thesis, this type of SIC is referred to as Multi-cell SIC (MSIC). However, in order to make the utilization of MSIC possible, new base station receiver systems architecture different from the stacked base station system architecture must be developed. 1.4 A New Software Radio Distributed Base Station System Architecture In the stacked base station system architecture illustrated in Figure 1.1, analog signal processing techniques are widely used in the receiver chains. Since expensive and complicated linear analog transmission systems are required to transmit wideband analog signals over a long distance without decreasing their SNR, the receivers employing the stacked base station system architecture are usually deployed close to the physical locations of their antennas. Consequently, in the stacked base station system architecture, the radio signals received by each base station are processed by its own receivers, and it is very difficult for these receivers to exploit the radio signals received by other base stations; hence, neither CM nor MSIC can be employed. According to the system model of the software radio base station receivers illustrated in Figure 1.2, it is clear that once the technical challenge of digitizing the wideband analog signals is solved, in the software radio base station receivers no analog signal should exist beyond the Analog Domain and the Digitization Domain. As compared to analog signals, digital signals can be transmitted over long distances virtually without any error by employing simple and cost-effective transmission systems, such as digital optical fiber networks. Thus, the Digital Domain of a software radio base station receiver does not need not to be deployed 16 close to the physical location of its Analog Domain and its Digitization Domain, and in fact, can be far away from them. Meanwhile, if the Digital Domains of a cluster of software radio base station receivers are deployed in the same physical location, base station receivers can simultaneously exploit the radio signals received by several base stations to detect mobile users. Hence, both C M and MSIC can be employed in this type of base station receiver system architecture to improve the reverse link capacity of the cellular CDMA systems. Distributed Base Station (DBS) V Central Processing Station (CPS) DBS-I Analog Digitization ^ Network Domain Domain Adapter r High Speed Data Communication Network (HSDCN) Network Adapter V DBS-N Analog Digitization Network — • Domain • Domain Adapter Digital Domain For DBS-I Digital Domain For DBS-N Figure 1.3 System Model of Software Radio DBS System Architecture. Motivated by the above observations and by extending the idea of DBS given by [20] and [46], we propose for the first time in [47] a novel base station receiver system architecture, referred to as software radio DBS system architecture. As shown in Figure 1.3, in the proposed system architecture, a cluster of software radio base stations are reorganized as follows. For each software radio base station receiver, its antennas, Analog Domains and Digitization Domains, together with a network adapter form a DBS. As compared to a conventional base station, the DBS is compact, lightweight, and has a lower power 17 consumption requirement. Meanwhile, the Digital Domains of those software radio base station receivers are integrated together to form a Central Processing Station (CPS). The connection between the DBSs and the CPS is implemented by a High Speed Data Communication Network (HSDCN), which for most applications could be a digital fiber optics network. By employing this software radio DBS system architecture, the performance of the mobile communication systems, especially for 3G systems, can be significantly improved because of the following reasons. First, in order to meet the dramatically increasing demands for mobile communication services, 3G systems need to deploy many small size cells so as to increase their overall capacity. For example, as shown in the system description of cdma2000 [48], which is one of the IMT-2000 standards proposed in North America, except the cells for vehicular users, all the other three kinds of cells, the cells for pedestrian users, indoor users, and outdoor-to-indoor users, have a radius less than 500m. However, as discussed in [49], due to the shortage of space for containing equipment, high space leasing cost, and high power supply requirements, up to 50% of the desired sites are not available for installing a conventional base station. Thus, the scarcity of suitable locations for installing conventional base stations seriously restricts the deployment of small size cells for increasing the overall capacity of 3G systems. In the software radio DBS system architecture, the DBS is compact, lightweight, and has a lower power consumption requirement than conventional base stations. Therefore, it can be deployed virtually everywhere, for example, on building walls or even on street lamps [20],[46]. Consequently, more large size cells can be split into small size cells to provide mobile communication services to more users. Furthermore, in small cells, as the distance between mobile users and DBSs is shortened, the distance attenuation mobile users suffer can 18 be significantly reduced. Thus, the average transmission power of the mobile users can be reduced, and such a power reduction results in longer battery operation time for the mobile users. Moreover, the shortened distance between mobile users and DBSs can also considerably mitigate the multi-path fading that mobile users suffer [5]. Second, from the system model of the software radio DBS system architecture, illustrated in Figure 1.3, it is clear that the CPS can be installed with little consideration for the physical locations of the DBSs. Moreover, since only one CPS is required to serve a cluster of DBSs, as compared to conventional base stations, the CPS has few restrictions on its dimension, weight, and power consumption requirements. Therefore, the CPS can be implemented with general propose processors, such as a group of FPGAs, DSPs, workstations, or even MFCs. With the employment of these general propose processors, the CPS possesses the following advantages over conventional base stations: • Flexibility: The CPS can be easily reconfigured by software means to adapt to different transmission parameters and support virtually any mobile communication standard. • Scalability: Without modifying the remaining system, the CPS can be upgraded smoothly by adding more general propose processors. • Resource Reallocation: Through efficient resource allocation, the computation resources of the CPS can be dynamically shared by a cluster of DBSs to achieve a better performance. • Service Accommodation: Upon demand, new services can be accommodated quickly by installing new DBSs and connecting them to the existing CPS. • Maintenance: Every general propose processor can be tested or adjusted 19 independently, so any malfunction in the CPS can be located easily, and fixed fast. Third, in the software radio DBS system architecture, the radio signals received by a cluster of DBSs is transmitted to and processed in the same CPS. Thus, a software radio base station receiver can simultaneously exploit the radio signals received by several DBSs to detect mobile users, and hence, both C M and MSIC can be employed in the software radio DBS system architecture to increase the reverse link capacity of cellular CDMA systems. 1.5 Research Contribution of the Thesis The previous discussions on the stacked base station receiver system architecture, the software radio technology, the macrodiversity and SIC techniques, and the software radio DBS system architecture established the context of the research contributions of this thesis, which can be summarized as follows. • In order to mitigate the stringent ADC resolution requirement set by software radio technology, a novel digitization method, referred to as Adaptive Prediction and Cancellation Digitization (APCD), is proposed to implement the Digitization Domain of the software radio base station receiver. The APCD method can significantly reduce the high dynamic range of the wideband analog signals received by the multi-standard base station receiver, and hence, can effectively relax the steep ADC resolution requirement. This dynamic range reduction is achieved by applying appropriate signal prediction techniques to remove the strong mutual correlation presented in the adjacent samples of the received signals. By theoretical analysis, and by means of computer simulations, it is shown that when the APCD method in conjunction with Auto-Regressive (AR) [50] or Periodic Auto-Regressive (PAR) [51] prediction 20 algorithm is employed, the high ADC resolution requirement for the software radio base station receiver can be relaxed significantly. In order to obtain the SINR expressions for mobile users employing CM, SSIC or MSIC, the inter-cell interference of these mobile users suffer must be first analyzed. However, when CM, SSIC or MSIC is employed, the receiving power of the mobile users assigned to the same base station has controlled disparity. Consequently, the existing inter-cell interference model [41],[52], which is based upon the assumption of equal receiving power, cannot be used to analyze the inter-cell interference of the mobile users employing CM, SSIC or MSIC. In this thesis, a novel inter-cell interference model for mobile users with controlled power disparity is proposed. By theoretical analysis and by means of computer simulations, it will be demonstrated that by applying this inter-cell interference model, the amount and the characteristics of the inter-cell interference of the mobile users employing CM, SSIC or MSIC can be precisely evaluated. Furthermore, it is also proven that the existing inter-cell interference model for mobile users with equal receiving power can be easily obtained from the newly proposed inter-cell interference model. In order to precisely analyze the reverse link capacity improvement C M offers, a new approach to performing C M in software radio DBS system architecture is proposed. By applying this new C M approach and the inter-cell interference model for mobile users with controlled power disparity, first, a more general and more precise SLNR expression for mobile users employing C M in conjunction with SUD, denoted as SUD-CM, is obtained. Second, the receiving power reduction of these mobile users will be analyzed theoretically and verified by computer simulations. Third, the reverse 21 link capacity of the system employing SUD-CM is be evaluated by analyzing the outage probability of the system. In order to demonstrate the capacity improvement C M offers, the reverse link capacity of the system employing SM in conjunction with SUD, denoted as SUD-SM, is evaluated as well. By theoretical analysis, and by means of computer simulations, it is shown that C M can significantly increase the reverse link capacity of cellular CDMA systems. • In order to achieve an acceptable MSIC performance without significantly increasing the signal processing complexity of the system, a new approach to performing MSIC in the software radio DBS system architecture is proposed. By applying the inter-cell interference model for mobile users with controlled power disparity, the SINR expressions for mobile users respectively employing the following four combinations of detection techniques: i) SSIC in conjunction with SM, denoted as SSIC-SM, ii) SSIC in conjunction with CM, denoted as SSIC-CM, iii) MSIC in conjunction with SM, denoted as MSIC-SM, and iv) MSIC in conjunction with CM, denoted as MSIC-CM, are obtained. Furthermore, by theoretical analysis, and by means of computer simulations, it is shown that first, both SSIC and MSIC can significantly increase the reverse link capacity of cellular CDMA systems, regardless of being employed in conjunction with SM or CM. Second, MISC can achieve considerably higher reverse link capacity improvement than SSIC. 1.6 Organization of the Thesis The organization of this thesis is as follows. After this introductory chapter, the APCD method is proposed and evaluated in Chapter 2. In Chapter 3, the inter-cell interference model 22 for mobile users with controlled power disparity is proposed. The analysis for the individual inter-cell interference factors is presented in the same chapter. The reverse link capacities of the system employing SUD-SM and SUD-CM, respectively, are analyzed in Chapter 4. The analysis for the reverse link capacity improvement SSIC and MSIC offer is be given in Chapter 5. Computer simulations are performed in the same chapter to verify the validity of the reverse link capacity analysis presented in Chapters 4 and 5. Finally, contributions of this thesis and topics for future research are summarized in Chapter 6. 23 Chapter 2 An Adaptive Prediction and Cancellation Digitization Method for Multi-Standard Software Radio Base Station Receivers As shown in Figure 1.2, for the software radio base station receiver, a Digitization Domain is required to digitize the wideband analog signals received by the multi-standard base station receiver. Thus, the implementation of the Digitization Domain becomes an important prerequisite for the efficient and effective operation of the software radio base station receiver. In order to simultaneously support multiple mobile communication standards, the Digitization Domain of the software radio base station receiver must be able to digitize analog signals with a wide frequency bandwidth and a high dynamic range [42]-[44]. As shown in the same reference, with such a high dynamic range and such a wide frequency bandwidth, the Digitization Domain implemented by a commercially available ADC cannot have enough high resolution to satisfy the SNR requirement for the weakest signal among the overall received signals. Since the research for this thesis has begun, to the best of our knowledge, very few techniques for effectively relaxing this steep ADC resolution requirement have been proposed in the open technical literature. Furthermore, as it has been discussed in the previous chapter, the few existing approaches that mitigate the steep ADC resolution requirement (e.g. see [42] and [43]) have serious limitations. In this chapter, according to the characteristics of the wideband analog signals received by the software radio base station receiver, a novel digitization method, the APCD method, is proposed to build the Digitization Domain. The APCD method can significantly reduce the high dynamic range of the wideband analog 24 signals received by the software radio base station receiver, and hence, can effectively mitigate the stringent ADC resolution requirement. This dynamic range reduction is achieved by applying appropriate signal prediction techniques for removing the strong mutual correlation presented in the adjacent samples of the wideband analog signals. The organization of this chapter is as follows. In Section 2.1, the characteristics of the wideband analog signals received by the software radio base station receiver are thoroughly analyzed, and the resolution requirement for efficiently digitizing those signals with a single ADC is quantified. Then, in Section 2.2, the proposed APCD method, including its system model, the factors influencing its performance, and its implementation requirements, are discussed in detail. Since the resolution improvement the APCD method offers is mainly determined by the prediction gain achieved by the APCD method, in Section 2.3, the performance of the AR and the PAR predictors, which are employed in conjunction with the APCD method, is comprehensively analyzed. The various performance evaluation results obtained by computer simulations together with a detailed discussion are presented in Section 2.4. Finally, a summary of the chapter is contained in Section 2.5. 2.1 ADC Resolution Requirements Generally, a conventional ADC consists of an Anti-alias Filter & Sample-holding Unit and a Quantizer [53]. When the Digitization Domain of the software radio base station receiver is implemented by such an ADC, the system model of the Digitization Domain is illustrated in Figure 2.1. It is well known that in mobile communication systems, because of multi-path fading, lognormal shadowing fading, and the movement of mobile users, the average value of wideband analog signals received by the software radio base station receiver 25 varies frequently and considerably. As shown in Figure 2.1, in order to achieve a consistent digitization performance, usually, an Automatic Gain Control Unit, together with a Variable Gain Amplifier, is employed in the Digitization Domain to maintain the average value of the wideband signals at a desired level. Wideband \ Analog \ , Signal Automatic Gain Control Unit Gain Anti-alias Filter & Sample-holding Unit x(n) Quantizer (Q) , Wideband Digital Signal Amplifier Analog-to-Digital Converter (ADC) Figure 2.1 System Model of the Digitization Domain Implemented by a Single ADC. In Figure 2.1, x(n) denotes the sampled version of the wideband analog signals received by the software radio base station receiver, and hence x(n) is a discrete analog signal. As discussed in the previous chapter, because the receiving bandwidth of the software radio station receiver is usually wide, with a high probability, x(n) consists of a large number of modulated signals which have different bandwidths and originate from different sources. Due to multi-path fading, lognormal shadowing fading, and the physical location of mobile users, typically, for more than 20% of the time, these signals are characterized by a significant difference in power [38]. Therefore, the dynamic range of x(n) could be very high, and it is reasonable to consider that x(n) can be separated into the following two groups of signals: i) x s (n): one or several strong signals with a total power equal to P s; and ii) x w (n) = x(n) - x s («): the rest of the signals among x(n) with a total power equal to P w . Furthermore, these two groups of signals satisfy Ps » Pw. When x(n) is digitized by the 26 Quantizer in Figure 2.1, due to the non-linear distortion in the Quantizer, spurious signals are generated and included into the digitization output [53]. If the spurious signals happen to fall into the frequency band of the weakest desired signal in x(n), denoted as xw k (n), the power of the spurious signals determine the SNR of xw k (n). Therefore, in order to derive the ADC resolution requirement, one of the important ADC performance parameters, Spurious Free Dynamic Range (SFDR), which provides a measurement of the peak power of the spurious signals, must be considered. Usually the SFDR is defined as follows [53]: SFDR = 10iog10 A a d c ^ (2.1) P s s where A A D C is the full-scale range of the ADC, and P s s is the peak power of the largest spurious signal. If the largest spurious signal happens to fall within the frequency band of x w k (n), the SNR of x w k (n) can be expressed as follows: P 8P SNR <101og 1 0-^ = 1 0 1 o g 1 0 ^ ^ + SFDR (2.2) P s s A A D C where Pwk is the power of xwk (n). Without the loss of generality, x(n) can be assumed to follow the zero-mean Gaussian distribution. As shown in Appendix A, if a zero-mean Gaussian distributed random variable with a standard deviation equal to o s is digitized by the ADC, the clipping probability will be low enough to ensure reliable operation in practice when A A D C > 6 o s . Thus, in order to digitize x ( n ) , A A D C should approximately satisfy the following condition: 27 A A D C > 6^ (2.3) where the effect of xw (ri) is neglected since Ps » Pw. Therefore, the required SFDR is given by the following: SFDR > lOlog c A2 ^ v 8P»* J > lOlog f 9P ^ + SNR, •SNRm, n 2Pwi = 6.5 + NFR m a x +SNR m i n (2.4) where SNR m i n is the minimum required SNR for xwk(ri), and NFR m a x =10\ogl0(Ps/Pwk) is the maximum near-far ratio of the mobile communication standards under consideration. As power control techniques are employed in the IS-95 and the IMT-2000, generally, their NFR m a x is not very high. However, as pointed out in [42], for the GSM, its NFR m a x can be as high as 90 dB. Thus, since for the GSM, S N R ^ = 12 dB [42], according to Eq. 2.4, in order to digitize the wideband analog signals received by the software radio base station receiver, the SFDR of the ADC in Figure 2.1 must be greater than 108.5 dB. As shown in [45], for most commercially available ADCs, their SFDR increases approximately 6 dB when their resolution increases one bit. Therefore, the resolution of the ADC in Figure 2.1 must be at least 18 bits. In addition to these bits, one should add approximately another 1.5 bits to account for the extra error sources in the ADC [45], such as the input-referred circuit noise, the aperture jitter and comparator ambiguity. Hence, the overall ADC resolution should be at least 20 bits. Despite the rapid advancements of the ADC technology, to the best of our knowledge, no ADC with such a high resolution requirement is commercially available when 28 the sampling rate of the ADC, fs, is higher than 400 kHz. Therefore, it can be concluded that when the Digitization Domain of the software radio base station receiver is implemented by a single ADC, a large gap exists between the performance of the commercially available ADCs and the demanding resolution requirement to digitize the wideband analog signals. In order to relax this steep ADC resolution requirement, some alternative methods must be exploited to build the Digitization Domain. 2.2 Adaptive Prediction and Cancellation Digitization (APCD) Method As discussed in the previous section, in the software radio base station receiver, its Digitization Domain is required to simultaneously digitize a variety of signals with a considerable bandwidth difference. Thus, if xs(n) are narrowband signals (with a total bandwidth B s ) , for example, xs (n) are one or several GSM or IS-136 signals, it is clear that as fs » B s, xs (n) are over sampled. Consequently, a strong mutual correlation exists among the adjacent samples of xs(n). Furthermore, as Ps » Pw, the statistical characteristics of xin) are dominated by those of xs(ri), that is, a strong mutual correlation also exists among the adjacent samples of x(n). By taking advantage of this strong mutual correlation, and by employing appropriate digital signal prediction techniques, we first propose in [54] the APCD method which, by predicting and canceling xs (n), can significantly reduce the dynamic range of x(n), and hence, relax the steep ADC resolution requirement. 2.2.1 System Model When the Digitization Domain of the software radio base station receiver is implemented by the APCD method, its system model is illustrated in Figure 2.2. As compared with a 29 conventional ADC, the APCD method adds a digital function unit, referred to as a Signal Predictor Unit (SPU), to predict x s (n) based upon the past samples of x(n). Meanwhile, a Digital-to-Analog Converter (DAC) is employed to convert the prediction output of the SPU, denoted as y d (n), into a discrete analog signal, denoted as y(n). Then the residual signal, e ( n ) = x ( n ) - y ( n ) , instead of x ( n ) , is digitized by the Q l and Q2 quantizers in the APCD method. The digitized signal at the output of Q l , denoted as edl(n), is combined with yd(n) to obtain the digital equivalent of x (n) , denoted as x d (n ) . At the same time, the digitized signal at the output of Q2, denoted as ed2(n), is exploited by the SPU to predict yd (n). Wideband Analog Signal Automatic Gain Control Unit Variable Gain Amplifier Anti-alias Filter & Sample-holding Unit . x(n)= xs(n)+xw(n) y(n) e(n) Quantizer (Ql) 'eJ(»/^P Quantizer (Q2) ed2(n) D A C yd(n) Signal Predictor [Unit (SPU)| Wideband \ Digital Signal xd(n) Adaptive Prediction and Cancellation Analog-to-Digital Converter Figure 2.2 System Model of the Digitization Domain Employing the APCD Method. Since a strong mutual correlation exists among the adjacent samples of x(ri), ideally, y(n) would be approximately equal to xs (n). Consequently, e(n) features a considerably reduced dynamic range as compared to x(n). Thus, the SNR of the APCD method can be expressed as follows: 30 TOR i m E[x\n)] E[x z (n)]E[e\n)] S N R A P C D =101og10 = 101og10 = G p +SNR (2.5) E[e,(n)] E[e (n)]E[eq(n)] where £[•] denotes the mean value, and eq(n) denotes the overall digitization noise of Q l , which includes the quantization noise and other kinds of noise caused by the non-linearity of the quantizer [53]. In the above equation, SNR q = 10\ogl0{E[e2 (n)]/ E[e2(n)]} specifies the SNR that a conventional ADC with Q l as its quantizer can achieve. Furthermore, Gp = 10logw{E[x2(n)]/E[e2(n)]}, referred to as prediction gain, specifies the overall SNR improvement accomplished by the proposed new signal prediction technique. For a conventional ADC, increasing the resolution of its quantizer by one bit can decrease the digitization noise by approximately 6 dB [45],[53]. Thus, if the proposed signal prediction techniques can obtain a prediction gain, for example, equal to 24 dB, the APCD method that employs an rc-bit quantizer as Q l can achieve the same SNR as a conventional ADC that employs an (n + 4) -bit quantizer. According to the system model of the APCD method illustrated in Figure 2.1, it can be concluded that unlike the digitization methods mentioned in [42],[43], the APCD method digitizes the receiving band of the software radio base station receiver as a whole, and reduces the dynamic range of the wideband analog signals by linear signal processing techniques. Thus, it neither increases the overall complexity of software radio base station receivers, nor introduces nonlinear distortion to the digitization results. Furthermore, although XA-ADC also employs signal prediction techniques to improve its resolution [55], the approach it uses to achieve this resolution enhancement is different from that of the APCD method. As shown in [55], the ZA-ADC achieves its SNR enhancement by filtering and shaping the quantization noise within the desired frequency band, and usually the bandwidth of this desired frequency 31 band should be much smaller than the overall sampling rate. Therefore, EA-ADC is more appropriating employed in software radio mobile receivers, since these kinds of receivers need to support only one radio channel. However, according to Eq. 2.5, the APCD method improves its SNR performance by predicting and canceling xs (n) from x(ri), and hence reducing the dynamic range of e(n). Thus, the APCD method does not require that the bandwidth of x(ri) be much smaller than fs, and therefore it can be employed in both software radio mobile user receivers and software radio base station receivers. 2.2.2 Factors Influencing G p According to Eq. 2.5, the SNR improvements the APCD method offers are mainly determined by its prediction gain. Thus, in order to investigate the factors affecting the prediction gain, x s (n) and x w (n) are assumed to satisfy the following two conditions. First, because the sources of xs(n) and xw(n) are usually different, it is reasonable to assume that they are independent from each other. Second, since Pw « Ps, and thus the statistical characteristics of x(n) are dominated by those of x s (n), it is reasonable to assume that xw (n) cannot be predicted by the SPU. Thus, the prediction gain of the APDC method can be expressed as follows: Gp =10 log E[x2(n)] E[e2(n)] = 10 log P +P s w 10 E{[xs(n)-y(n)]2} + P„ w = 101og 10 E{[xs(n)-y(n)]2} + Pw ' (2.6) From the above equation, the following observations are in order. First, Gp increases as 32 PjPw increases. In fact, the maximum value that Gp can achieve is bounded by PjPw • Second, Gp decreases as Bjfs increases. This happens because if Bjfs increases, the mutual correlation among the adjacent samples of xs(n) decreases, and thus the power of prediction e r r o r , E { [ x s i n ) - y(n)]2}, increases. As it will be shown in Section 2.4, these two observations have also been verified by the performance evaluation results obtained by computer simulations. From the above two observations, it can be concluded that only when xs(n) are high power narrowband signals, can the APCD method attain improved performance as compared to a conventional ADC. However, as illustrated in Section 2.1, the high ADC resolution requirement to digitize wideband analog signals received by the software radio base station receiver is mainly caused by the high NFR m a x of narrowband signals, for example, GSM signals. Thus, the prerequisite that x s (n) must be high power narrowband signals does not limit the application of the APCD method to mitigate the steep ADC resolution requirement for the software radio base station receiver. 2.2.3 Implementation Requirements When the implementation of the APCD method is considered, as compared to a conventional ADC, the APCD method employs two quantizers (Ql and Q2) and an additional DAC to perform the necessary signal conversion. Due to this increased complexity, the following three requirements must be met so that the performance of the APCD method is satisfactory: (i) quantization steps of DAC and Q l , ii) resolution of DAC, and iii) conversion time of Q2 and DAC.) 33 Requirement I: Quantization Steps of DAC and Ql As shown in Figure 2.1, the digitization result of x(n), x d (n), is given by the following: x d ( n ) = e d l ( n ) + ^ y d ( n ) (2.7) where e r f l ( « ) is the digitization result of e{n) given by Q l , A D A C is the quantization step of the DAC, and AQl is the quantization step of Q l . In order to simplify the computation for xd(n), ADAC and AQl are usually set as follows: ADAC=2BAQi (2.8) where B is a positive integer. Requirement II: Resolution of DAC In order to let y(n) precisely track x{n), x(n) must be within the full-scale range of the DAC, A D A C , with a high probability. Since x(n) is assumed to follow a zero-mean Gaussian distribution, it is shown in Appendix A that when the following condition is satisfied, with a high probability, x(n) is within the full-scale range of the DAC: ADAC = 2B™ A D A C > 6^E[x2(n)] = 6J10°>/w E[e2(n)] (2.9) where B D A C is the bit resolution of the DAC. Meanwhile, as B D A C may be lower than the bit resolution of the SPU, the cut-off noise, denoted as eco (n), is added into e(n) when the least significant bits of the prediction results are omitted to obtain y d (n). e c o (n) may seriously degrade the performance of the signal prediction algorithm unless the power of e(n) is much stronger than that of eco(ri). Mathematically, this can be expressed as follows: 34 £[«'(.)] = 1 2 g « » ] where <?co(n) is assumed to be uniformly distributed over -AD A C/2 and AD A C/2 [53]. According to Eqs. 2.9 and 2.10, BDAC is given by the following: BDAC >-\GP +101og10 £ [ e 2 2 ( " ) ] 1 + 0.8. (2.11) 6[ " 5 l ° E [ ^ ( n ) ] j From the above equation, it can be obtained that if BDAC = (Gp/6 + 1) bits, then E[e2(n)]/E[e20(n)]<l.32; if B M C = (G p /6 + 2)bits, then E[e2 (n)]/E[e2„(n)] < 5.25 ; and if 5 D A C = (G p /6 + 3)bits, E[e2(n)]/E[e2C0(n)]< 20.9. According to Eq. 2.10, the condition that E[e2(n)]/E[e2g(n)]»1 must be satisfied so that the performance degradation caused by ecl)(n) can be omitted. Thus, if the designed maximum SNR improvement of the APCD method is G™x, it is suggested that BDAC > (G™ax/6 + 3)bits. As it will be shown in Section 2.4, computer simulation results have also confirmed that (Gj a x /6 + 3)bits can be considered as the lower bound of BR 'DAC • Requirement III: Conversion Time of Q2 and DAC If the digitized output of Q l , eAX(n), is exploited to derive yd(n), a series of signal processing procedures, including digitizing e(n -1), deriving y d (n), and converting y d in) to y(n), must be implemented serially within one sampling period to obtain e(n). Therefore, the conversion time of Q l must be much shorter than one sampling period, since it takes time to derive y d ( n ) and convert y d ( n ) to y ( n ) . However, since a quantizer approximately loses one bit of resolution for every doubling of its sampling rate [45], in order to shorten the 35 conversion time of Q l , we may have to reduce its resolution. Such a reduced resolution for Q l also means that the SNR q in Eq. 2.5 is reduced, and thus the overall SNR improvement the APCD method offers might be limited. As illustrated in Figure 2.1, this problem is solved by adding a second quantizer (Q2), which works in parallel to Q l and digitizes e(ri) with a lower resolution. Thus, the SPU can exploit the output of Q2, ed2{ri), to derive yd(n), and hence the conversion time of Q l can be up to one sampling period to achieve its highest resolution. When the bit resolution of Q2, BQ2, is larger than 3 bits, the performance degradation of the signal prediction algorithm, caused by the digitization noise produced by Q2, can be omitted. Currently, the conversion time of a commercially available 4-bit quantizer can be less than 1 nsec [45]. Thus, as long as fs < 100MSPS, it is easy to keep the conversion time of Q2 much shorter than one sampling period. Meanwhile, we note that for the same reason, the conversion time of the DAC must also be much shorter than one sampling period. However, even when G™* = 24 dB, the bit resolution of the DAC can still be as low as 7 bits. Currently, the conversion time of a commercially available 10-bit DAC can also be less than 1 nsec [56], which is much shorter than one sampling period, as long as / , <100MSPS. 2.3 Signal Prediction Algorithms As mentioned in the previous section, the SNR improvements the APCD method offers are mainly determined by the prediction gain achieved by predicting the high power narrowband signals among the wideband analog signals received by the software radio base station receiver. Therefore, in this section, appropriate signal processing algorithms which can 36 be used to predict these high power narrowband signals, xs (n), are investigated. 2.3.1 Statistical Characteristics of x s (n) It is well known that any modulated signal can be expressed as the product of an equivalent baseband signal and a carrier signal. As shown in [5], because of multi-path fading, the high power narrowband signal that only consists of one independent modulated signal can be expressed as follows: xs(n) = ^iAju(n-ni)cos(—mi + ^i) (2.12) where u(ri) is the equivalent baseband signal of the modulated signal, and A(., and ni respectively represent the amplitude, the phase and the delay of the ith branch of the received signals. Furthermore, in Eq. 2.12, J/P = 2fc/fs, where fc is the carrier frequency of the modulated signal, J and P are two positive integers that have no common factor larger than one. In a signal prediction algorithm, usually, only a number of the most recently received samples are exploited to predict the next sample. Since fs is much higher than the maximum possible Doppler shift in most mobile communication systems [5], for these adjacent samples of xs (n), their A., (ft i and ni can be assumed to be constant. Thus, xs (n) can be rewritten as follows: xs(n) = uA(n)cos{^m + (p) (2.13) where 0 is a constant and uA (n) is the equivalent baseband signal for x s (n). Furthermore, because f s » B s , for a number of adjacent samples of x s ( n ) , the value of 37 E[uA(n)uA(n + m)] varies very slowly. Thus, uA{ri) can be assumed to be a stationary signal [50] when the performance of a signal prediction algorithm is analyzed. According to Wold's Decomposition Theorem [50], any stationary signal can be decomposed into an AR process of an appropriate order and a deterministic process. Obviously, a deterministic process does not affect the performance of a signal prediction algorithm. Thus, when the performance of a signal prediction algorithm is analyzed, uA{ri) can be replaced by an AR process with the same statistical characteristics, and hence, according to Eq. 2.13, the high power narrowband signal can be expressed as follows: xs(ri) -uAR{n)cos,{^m + <l>) (2.14) where uAR{n) is an AR process with the same statistical characteristics as uA(n). From the above equation, the autocorrelation function of xs (n) can be expressed as follows: A RXX m ) = E[X! (H)XS (H + = -r(m){cos[— n(2n + m) + 20] + cos(— Tan)]} (2.15) 2 P P where r(m) = E[uAR(n)uAR(n + m)]. Obviously, rjji^ni) is a periodic function of n, hence x s (n) is a cyclostationary signal, and its cycle frequency is J/P = 2fJ f s [57]. When no prior knowledge about xs (n) is available, the AR predictor is considered as a simple but effective signal prediction algorithm. However, since the AR predictor is based upon the stationary model, that is, r^(n,m) does not vary with n [50], its performance may degrade when it is used to predict a cyclostationary signal. On the other hand, when certain parameters of xs(n), for example, its carrier frequency, are known in advance, the PAR 38 predictor, which is based upon the cyclostationary model, that is, r^in^m) is a periodic function of n, can be used to improve the overall prediction performance. It should be emphasized that, to the best of our knowledge and independent of employing either the AR predictor or the PAR predictor, no detailed analysis or evaluation on prediction gain are published by other authors in the open technical literature when such algorithms are used to predict a cyclostationary signal. Our research efforts on this subject are published in [54] and 2.3.2. Auto-Regressive (AR) Predictor A. Algorithm Description If the signal to be predicted is s(n), an Lth-order AR predictor can be expressed as follows: where yAR_L(n) is the prediction result, WAR = [wAR_l,wAR_2,...,wAR_L]T are the L prediction coefficients, T denotes transposition, and S(n) = [s(n -1), s(n - 2),..., s(n - L)]T are the L past samples of s(n). When s(n) is a cyclostationary signal with a cycle frequency equal to J/P, as shown in [57], the autocorrelation function of s(n) satisfies the following: [58]. y AR-L (n) = WATRS(n) (2.16) E[s(n)s(n + m)] = E[s(n + P)s(n + P + m)]. (2.17) From the above two equations, E[e2AR(n)] can be expressed as follows: E[e2AR(n)] = E[(s(n)- yAR_L(n))2] = E[s2(n)]-2E[s(n)ST(n)WAR +WATRE[S(n)ST (n)]WAR = E[s2 (n + P)] - 2E[s(n + P)ST (n + P)]WAR + WTARE[S(n + P)ST (n + P)]W1 = E[(s(n + P)-yAR_L(n + P))2] AR 39 = E[e2AR(n + P ) ] . (2.18) Thus, as presented in [59], based upon the Mean Square Error (MSE) criterion, to achieve the highest prediction gain, W A R should minimize J A R , given by the following: JA*=^elR^n)] = %{E[slm + WLlRkWM ~2PkTWAR} (2.19) k=0 k=0 where eAR_k(n) = eAR(nP + k), sk (n) = s(nP + k), Sk (n) = S(nP + k), Rk = E[Sk(n)STk («)], and Pk = E[sk (n)Sk (n)]. Setting dJAR /dWAR = 0, the optimal solution for WAR is given by the following: W Z = [ % R k V [ ^ P k ] (2-20) k=0 k=0 and the highest prediction gain for s(n) achieved by the Lth-order AR predictor is given by the following: ^- .max k=0 (2 21) UpAR-L ~ p-i p-1 p-i p-l • k=0 k=0 k=0 k=0 B. Algorithm Performance p-i p-i P-I I f s ( n ) = x s ( n ) , where xs(n) is given by E q . 2.14, ^ E [ s t ( n ) ] , ^ P k , and ^ R k in Eq. k=0 k=0 k=0 2.21 can be expressed as follows: f^E[s2k(n)] = ^r(0) (2.22) k=0 2 40 p-\ En k=0 P_ ~2 r(l)cos— r(2)cos P P r(L) cos LnJ AR (2.23) and p-i r(0) r(l)cos r(l)cos /Z/ r(0) n (L-1W (L-2)nJ r(L-Y) cos- — r(L-2) cos • r ( L - l ) cos r(L- 2) cos (L-IW P (L-2)nJ KP) (2.24) From Eqs. 2.20-2.24, when the signal to be predicted is given by Eq. 2.14, the optimal prediction coefficients, , based upon the MSE criterion is given by the following: W"p - R'1 P AR AR AR ' (2.25) Correspondingly, the highest prediction gain achieved by the Lth-order AR predictor can be expressed as follows: s~< max (JpAR-L r(0) r ( 0 ) - P X R P A R (2.26) In order to clearly demonstrate the performance degradation when an AR predictor is employed to predict a cyclostationary signal, in the following example, uAR{n) in Eq. 2.14 is assumed to be a lst-order AR process as follows: u A R (n ) = Cu A R (n - l ) + v (n) (2.27) where -1 < C < 1 and v(n) is a stationary, zero mean, memoryless Gaussian random process. Thus, based upon Eqs. 2.25 and 2.26, the optimal lst-order AR predictor for the 41 cyclostationary signal given by Eqs. 2.14 and 2.27 can be expressed as follows: yAR-i (n) = Ccos(nJ/P) xs (n -1) (2.28) and the highest prediction gain this optimal lst-order AR predictor can achieve is given by the following: GTR-i= T\ 7—- ( 2- 2 9) p A R l 1 -C 2 cos 2 (nJ/P) Furthermore, the optimal 2nd-order AR predictor for xs (ri) is as follows: C cos(nJ/P)[l - C2 cos(2nJ/P)] I - C2 cos2 (nJ/P) , - C 2 s i n 2 ( n J / P ) + ; ;———xJn-2) (2.30) 1 - C 2 cos2 (nJ/P) s and the highest prediction gain this optimal 2nd-order AR predictor can achieve is given by the following: G m a x = 1 - C 2 COS 2 ( 7 0 / P ) PAR-2 x_2C2 cos2(nJ/P) + 2CA cos2(7iJ/P)-CA ' By observing Eqs. 2.29 and 2.31, it is clear that when the AR predictors are employed to predict the cyclostationary signal given by Eqs. 2.14 and 2.27, their prediction gains, and G ™ K _ 2 , are periodic functions of the cycle frequency of the signal, J/P. Since the period of these functions is equal to one, for 0 < J/P < 1, the corresponding G^R_X and G ^ _ 2 are calculated and plotted in Figure 2.3. It is given in [50] that the optimal signal predictor for a 1st -order AR process is a lst-order AR predictor, and increasing the order of the AR predictor does not increase the highest 42 prediction gain for uAR{n), which is given by the following: 1 max ~ 1 - C 2 (2.32) However, as illustrated in Figure 2.3, it is clear that when J/P^O, that is, xs(n) is a cyclostationary signal, the prediction gains of the AR predictor seriously decrease, especially when J/P = 0.5. For this case, as E[xs (n)xs (n -1)] = 0, the optimal lst-order AR predictor for xs(n) is yAR_{(n) = 0, which means that the lst-order AR predictor cannot predict xs(n) at all. Furthermore, from Figure 2.3, it can be concluded that although uAR(n) is a 1st -order AR process, when J/P ^0 , increasing the order of the AR predictor can effectively increase the prediction gain of x s i n ) . T3 <3 -A- lst-Order AR Predictor -B- 2nd-Order AR Predictor "nil i nn i II i -tt ft! IIH8-1-0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 J/P=2f/fs Figure 2.3 Prediction Gain of AR Predictors (Analytical Results). 43 In Figure 2.3, when the values of and G™^_2 are calculated, C is fixed at 0.995, since, according to Eq. 2.32, this value of C lets = 20 dB. C. Algorithm Implementation One well-known adaptive algorithm for implementing the AR predictor is the Least Mean Square (LMS) algorithm, which can be mathematically described as follows [50]: yARAW = WATRA(n)S(n) (2.33) eARA (") = s(n) - y A R A (n) (2.34) WARA (n +1) = WARA (n) + 2SeARA (n)s(n) (2.35) where WARA(n) - [wARA_^(n),wARA_2(n),.. .,wARA_L(n)]T, and S , referred to as the adjustment step, is a small positive constant. In [59] it is proven that for cyclostationary signals, when 6 —> 0 and n — t h e algorithm given by Eq. 2.33-2.35 lets WA R A(n) converge to W A R , given by Eq. 2.20, in the mean, that is, it gives us the following: lim E[WARA (n)] = Rk ]"! P t ] = W% . (2.36) s->o k=0 k=0 2.3.3. Periodic Auto-Regressive (PAR) Predictor A. Algorithm Description According to [57], the PAR model is a more accurate, yet more complicated model for describing the statistical characteristics of cyclostationary signals than the AR model. In general, an LxH'h -order PAR predictor can be described as follow: y p A R - » H ( n ) = W?ARQ(n)S(n) (2.37) 44 WPAR = [WPAR-l>WPAR-2>---^PAR_L(2H+l)] (2.38) <D (n ) = [ A(l), A(cos(2^r ^  n)), A(sin(2^ y r c ) ) , . . . , A(cos(2^ ^ - n)), A(sin(2^ ^ - n ) ) f . (2.39) In the above equations, A(z) is an Lth-order diagonal matrix with z as its diagonal elements, and {jJP,ie[l,H]} are the cycle frequencies of s(n). Since s(n) is a cyclostationary signal, according to Eqs. 2.17 and 2.37, E[e2PAR(rij\ can be expressed as follows: E[e2PAR(n)] = E[(s(n) - yPAR_LxH (n))2] = E[s2 (n)] - 2E[s(n)ST (n)]Or (n)WPAR + WPTAR^(n)E[S(n)ST (n)]Or = E[s2 (n + P)] - 2E[s(n + P)ST (n + P)]O r (n + P)WPAR + WTPAR<$>(n + P)E[S(n + P)ST (n + P)]O r (n + P)WPAR = E[(s(n + P ) - y P A R _ L x H ( n + P))2] = E[e2PAR(n + P)]. (2.40) Thus, based upon the MSE criterion, to achieve the highest prediction gain, WPAR should minimize J P A R as follows: JPAR = E[^e2PAR_k(n)] k=0 = § { £ [ * > ) ] + < / I > ( £ ) / ^ (2.41) k=0 where ePAR_k(n) = ePAR(nP + k). Setting dJPAR/dWPAR = 0 , it is straightforward to obtain the following optimal solution for WPAR: 45 p-l p-l (2.42) k=0 k=0 and the following highest prediction gain for s(n): p-i *=0 ' pPAR p-l P-l P-l (2.43) £ E[s2k (n)] - [^0(k)Pk f [^(k)RkOT (k)]~l (Z <W* ] k=0 k=0 k=0 k=0 B. Algorithm Performance Since JCSin) given by Eq. 2.14 has only one cycle frequency, J/P = 2 f c / f s , when the PAR predictor described by Eqs. 2.37-2.39 is employed to predict xs(n), <&(k) is simplified to <&\ = [A(l),A(cos(2tf—£)),A(sin(27r—k))]T .Thus, ^ ®(k)Pk is given by the following: k=0 P-l X ^ P * = - k « P c o s / 2 PL/2f=-PPAR k=0 (2.44) where JTI TTJ r( l)cos(y-2^) K2)cos(2y-2^) nJ r(L)cos(L— - 2<p) (2.45) rrl 7T.1 7L.J r( l )s in(y-2^) r(2)sin(2y-20) ... r(L)sm(L—-2</)) (2.46) p-i and ^®(k)Rk&T(k) can be expressed as follows: k=0 P-l T p k=0 ^ ^cos /2 R A J 2 0 * s i „/2 0 R J 2 ' ^ ^PAR (2.47) 46 In the above equation, 0 is a L x L zero matrix, and r(O)cos(2-y-20) r(l)cos(3y-20) r(l)cos(3-y-20) r(O)cos(4-y-20) TLI TLJ r(L-l) cos[(L +1) — - 20] r(L - 2) cos[(L + 2) — - 20] /•(^-l)cos[(L + l)-y-20] r(L-2)cos[(L + 2)-y-20] r(O)cos(2L-y-20) (2.48) r(O)sin(2-y-20) r(l)sin(3y-20) r(l)sin(3-y-20) r(O)sin(4^--20) 77-/ 77"/ r(L -1) sin[(L +1) -y - 20] r(L - 2) sin[(L + 2) — - 20] r(L-l)sin[(L + l)-y-20] r(L - 2) sin[(L +2)^-20] r(O)sin(2L^--20) (2.49) Based upon Eqs. 2.42-2.49, if the signal to be predicted is given by Eq. 2.14, the optimal prediction coefficients, W°PAR, based upon the MSE criterion is given by the following: AR PAR PAR (2.50) and the highest prediction gain achieved by the PAR predictor can be expressed as follows: s . max ^pPAR-lxl r(0) r (0) PPAR RpAR PpAR (2.51) From Eq. 2.50, the optimal 1x1st -order PAR predictor for the cyclostationary signal given by Eqs. 2.14 and 2.27 can be expressed as follows: ,2nJ ypAR-M (») = C{cos(—) + sin(—)[sin(— - 2^)cos(—- n) 2nJ 2nJ - cos(—-— 2<p) sin(—— «)] }xs (n -1) , (2.52) From Eq. 2.51, the highest prediction gain this optimal 1x1st-order PAR predictor can achieve is given by the following: 47 ^ max 1 1- — [ c o s 2 ( — ) + l] 2 P (2.53) According to Eq . 2.53, for 0 < J / P < 1, G™*R_ l x l is calculated and plotted in Figure 2.4 for C = 0.995 , together with the previously obtained results for the l s t-order A R predictor. From Figure 2.4, it is clear that, as compared with the A R predictor, the P A R predictor can achieve a higher prediction gain. The maximum improvement of 3 dB is achieved at J / P = 0.5, where the l s t-order A R predictor cannot provide any prediction gain. Similar results can also be derived for higher order P A R predictors. However, because of the complexity of the mathematical analysis, performance evaluations for the higher order P A R predictors are performed by means of computer simulation and are presented in Section 2.4. 20 18 16 14 PQ 12 8 10 8 6 4 2 0 I I I I I lst-Order AR Predictor lst-Order PAR Predictor - * — 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 J/P=2f/fs Figure 2.4 Performance Comparison between the A R and the P A R Predictors (Analytical Results). In order to achieve this performance improvement, the cycle frequencies of the signal to 48 be predicted must be known in advance. Furthermore, as compared to the AR predictor, the PAR predictor requires higher computational effort. For example, in an Lth-order AR predictor, the number of prediction coefficients is L, but in an LxHth PAR predictor, that number increases to L(2H +1). C. Algorithm Implementation Contrary to the AR predictor case, the algorithmic implementation of the previously described PAR predictor is not a trivial issue, and to the best of our knowledge, this problem has not yet been addressed in the open technical literature. Perhaps the most straightforward approach to implementing the PAR predictor is to calculate WPAR by using Eq. 2.41. However, in order to derive P k , R k and R k l , which involves mean value calculation and matrix inversion, significant computational efforts are required, especially when higher order PAR predictors are considered. These computational intensive calculations make it very difficult to obtain prediction results in real time applications. As an alternative, in this thesis, an approach adaptively calculating WPAR with the LMS algorithm is proposed, and its algorithm can be described as follows: ypARA-k in) = WTPARA (nP + k)&(k)Sk in) (2.54) ePARA-k O) = sk (n) - y P A R A - k in) (2.55) WPARA inP + k + l) = WPARA inP + k) + 2SePARA_k in)<Z>ik)Sk in) (2.56) where WPARAin) = [wPARA_iin),wPARA_2in),...,wPARA_L(2H+l)in)]T. Clearly, the algorithm described by Eqs. 2.54-2.56 is not as computationally intensive as the one described by Eq. 2.42, as it only includes matrix addition and multiplication, and no mean value calculation or 49 matrix inversion is required. In Appendix B, the validity of the proposed algorithm is justified by proving that when 6 -> 0 and « t h e proposed algorithm lets WPARA{nP + k) converge to W°PAR, given by Eq. 2.42, in the mean, that is: 1 r i = K%- (2.57) p-i lim E\W, PARA (nP + k + l)] j=0 2 . 4 Performance Evaluation Results and Discussion In order to thoroughly examine the SNR improvement the proposed APCD method offers, the computer simulation approach is employed. More specifically, the APCD system, illustrated in Figure 2.2, is implemented in software in conjunction with the algorithm described by Eqs. 2.33-2.35 for the AR predictor and the algorithm described by Eqs. 2.54-2.56 for the PAR predictor. As shown in Eqs. 2.36 and 2.57, the value of 5 must be small so that the AR and the PAR algorithms can achieve high prediction gains. Thus, for both algorithms, b = 5xl0~ 4 , which is obtained by trial and error until the prediction gains of the AR and the PAR algorithms no longer increase. As for the high power narrowband signal, xs(n), besides the signal given by Eqs. 2.14 and 2.27, BPSK, QPSK and GMSK signals are also considered during the computer simulations. Furthermore, as discussed in Section 2.2, since Pw « P s, xw(n) can be reasonably assumed to be an unpredictable signal. In this sense, if xw(n) is replaced by another unpredictable signal, the total power of which is also equal to Pw, the prediction gain of x(n) does not change. Thus, for our computer simulations, unless otherwise indicated, x w { n ) is assumed to be an Additive White Gaussian Noise (AWGN). 50 For all computer simulation results, unless otherwise noted, the following parameters are been chosen: a) both the BPSK and the QPSK signals are shaped by a square root raise cosine filter with the excess bandwidth a = 0.5 , and the GMSK signal is shaped by a Gaussian filter with BT =0.3, where BT is the 3 dB-bandwidth-bit duration product of the filter; b) BQ 2 =4 bits, BD A C =7 bits and G™ax = 24dB, which are chosen according to the resolution requirements described in Section 2.2; c) L = 5, since extensive computer simulations show that for the signals investigated in this paper, when L > 5, the increase of the prediction gain is small; and d) Rj f s = 0.02, where Rs = l/Ts and Ts is the symbol duration of the BPSK, the QPSK, or the GMSK signal. Since the value of R s is proportional to that of B s (the bandwidth of x s (n)), the value of B s is modified through varying the value of R s . In the following subsections, first, the ability of the APCD method to improve the overall ADC resolution requirement is assessed. Then, in Section 2.4.2, the effects of PjPw and R s I f s ( B s I f s ) on the prediction gain are investigated. The resolution requirements of Q2 and the DAC are evaluated in Section 2.4.3. Finally, the performance of the AR and the PAR predictors are compared in Section 2.4.4. 2.4.1 Resolution Improvements In this subsection, the following signal is used as an example to demonstrate the ability of the APCD method to improve the ADC resolution requirements: x(n) = u*PSK (n) cos(nnJl/P + (j)x) + ufSK (n) cos(nnJ2/P + 02) + nA (n) (2.58) where u " P S K ( n ) c o s ( n n J j P + <j)x) and u 2 P S K ( n ) c o s ( n n J 2 / P + (/>2) are two BPSK signals, with their bit energy respectively equal to Ebl and Eb2. In the same equation, n A (n) is the 51 AWGN, and its single-side Power Spectral Density (PSD) is equal to N A . For the computer simulation results, it is assumed that E bjE b 2 - 40dB, and E b 2/NA =6.8 dB or 8.4 dB. Under these assumptions, the power of ufPSK (n)cos(nnJl/P + 0l) is much larger than the total power of the rest two signals, u 2 P S K (n)cos(n7zJ2/P + 0 2 ) and n A(n). Therefore, ufPSK(n)cos(n7ZJjP + 0,) can be considered as xs(n), and u2PSK(n)cos(n.7rJ2/P + <p2) together with nA(n) can be considered as xw{n). For the signal given by Eq. 2.58, its PSDs before and after digitized by a conventional ADC or the APCD method in conjunction with the AR predictor are obtained by computer simulations and illustrated in Figure 2.5. = / n I 0 . 4 0 . 6 F r e q u e n c y (a) Before Digitization (c) Digitized a 9-bit ADC 0 . 4 0 . 6 F r e q u e n c y (b) Digitized by a 6-bit ADC \ J J J w Witt 'HI 0 . 2 0 . 4 0 . 6 F r e q u e n c y (d) Digitized by the APCD with BQ,=6 bits Figure 2.5 Power Spectral Density Comparison between a Conventional ADC and the APCD Method. Clearly, if a 6-bit ADC is employed to digitize the signal, as shown in Figure 2.5(b), the 52 weaker BPSK signal, u 2 P S K ( n ) c o s ( n 7 l I 2 / P + <p2), is buried by the quantization noise. However, by comparing Figures. 2.5(c) and 2.5(d), it can be concluded that if the APCD method with B Q L - 6 bits is employed to digitize the signal, the PSD of the digitization results is almost the same as that when a 9-bit ADC is employed. In order words, in this example, the APCD method has relaxed the ADC resolution requirement by 3 bits. To obtain quantitative evaluation results, the following computer simulations are performed: after the signal given by Eq. 2.58 is digitized by a conventional ADC or the APCD method in conjunction with the AR predictor, the low power BPSK signal, u2PSK(n)cos,(n7iJ2/P + <f>2), is filtered by a square root raise cosine filter with the excess bandwidth a = 0.5, is coherently demodulated, and then its BER is estimated and shown in Table 1. \ . BQ 4 bits 5 bits 6 bits 7 bits 8 bits oo bits 6.8 dB 1.9 x 10"1 8.2 x 10"2 8.3 x 10"3 2.9 x 10"3 1.4 x 10"3 1.0 x 10"3 8.4 dB 1.8 x 10"1 8.3 x 10"2 4.4 x 10~3 6.2 x 10"4 2.0 x 10"4 1.0 x 10"4 (a) E b 2 / N ^ \ ^ 4 bits 5 bits 6.8 dB 2.4 x 10'3 1.5x 10"3 8.4 dB 5.7 x 10"4 2.0x 10"4 (b) Table 2.1 BER Comparison for the Lower Power B P S K Signal Digitized by: (a) A Conventional A D C , and (b) the A P C D method. 53 If a conventional ADC is employed to digitize the signal and BQ (resolution of ADC) varies from 4 to 8 bits, the estimated BERs of the low power BPSK signal are shown in Table 1(a). In this table, BQ^>°° bits means that no digitization is performed, hence, theoretically, if Eb2/NA =6.8dB, the BER of the weak power BPSK signal should be l.OxlO" 3, and if Eb2/NA =8.4dB, the BER should be l.OxlCT 4. The obtained performance results indicate that when the value of BQ is low, for example, 4 or 5 bits, the low power BPSK signal is essentially buried in the digitization noise and its obtained BERs are unacceptably high. However, when the APCD method is employed, as illustrated in Table 2.1(b), when the resolution of Q l , B Q i , is also 4 or 5 bits, the corresponding BERs of the low power BPSK signal are much lower and approximately at the same level when the conventional ADC with a BQ of 7 bits or 8 bits is employed. These simulation results clearly show that the APCD method can significantly relax the resolution requirement of the quantizer. Extensive computer simulations show that for the above two computer simulations, similar performance results can be obtained if the BPSK signals in Eq. 2.58 are replaced by other kinds of modulated signals, for example, the QPSK signals or the GMSK signals. These simulation results confirmed our observations in Section 2.2.2, that is, the prediction gain is mainly determined by the values of Ps /Pw and Rs /fs, instead of the modulation schemes of the signals to be predicted. 2.4.2 Effects of Ps/Pw and Rs/fs (Bjf,) In the computer simulations investigating the effects of Ps/Pw and Rs/fs (Bs/fs) on the prediction gain, the signal to be digitized, x(ri), is assumed to consist of xs (n) which could be either a BPSK, a QPSK or a GMSK signal, and xw(n) which is the AWGN. Under these 54 assumptions, various performance evaluation results for the prediction gain when the AR predictor is employed are obtained, and illustrated as follows: i) as a function of PjPw with Rjfs =0.02 (see Figure 2.6); and ii) as a function of Rjfs with Ps I Pw =30 dB (see Figure 2.7). 24f 18 CQ 5 14 8' - A - BPSK -+• QPSK : - Q - GMSK 15 20 25 30 Figure 2.6 Influence of P s / P w on Prediction Gain (Computer Simulation Results). These performance results clearly confirm the analytical observation presented in Section 2.2.2, that is, the prediction gain increases as PjPw increases, and decreases as Rjfs (Bs/fs) increases. It should be also mentioned that in Figure 2.6, when PjPw increases, the difference between the prediction gains of the BPSK, the QPSK and the GMSK signal increases as well. This increasing difference can be intuitively justified as follows: when PjPw is low, ed2(n), the input of the SPU (see Figure 2.2) is dominated by xw(n) instead of 55 the prediction error of xs(n), so the prediction gains of different signals are almost same. However, when PjPw is high, ed2(n) is dominated by the prediction error of xs(n), which has a strong relation with the statistical characteristics of xs (n), thus the prediction gains of the three signals exhibit an increasing difference in value. 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Figure 2.7 Influence of RJ' fs on Prediction Gain (Computer Simulation Results). 2.4.3 Resolution Requirements of Q2 and DAC In the computer simulations that explore the effects of B Q 2 and B D A C on the prediction gain, the signal to be digitized, x(n), is assumed to consist of x s (n), which is a BPSK signal, and xw(n), which is the AWGN. Furthermore, it is assumed that Ps/P„ =30dB. When BQ1 varies from 3 to 6 bits, and BD A C varies from (G p n a x/6 + 2) to (GjJUK/6 + 4) bits, the corresponding prediction gains when the AR predictor is employed are summarized in Table 56 2.2. The obtained results indicate that when BQ2 is increased from 3 to 4 bits, or BDAC is increased from (G™ax /6 + 2) to (G™** /6 + 3) bits, the increase of the prediction gain is quite large. However, when BQ2 > 4 bits and B D A C > {G™™ J6 + 3) bits, though the prediction gain still increases while BQ2 or BDAC increases, the increase of the prediction gain is quite limited. Thus, considering that the APCD method requires that Q2 and DAC have as short as possible conversion time, it can be concluded that the optimal value for B Q 2 is 4 bits and for BDAC is (5,™x + 3) bits. It should be mentioned that such observations are consistent with the resolution analysis given in Section 2.2. BDAC >V 3 bits 4 bits 5 bits 6 bits (G; a x /6 + 2) bits 14.88 dB 17.07 dB 17.46 dB 17.52 dB (G; a x /6 + 3)bits 16.77 dB 18.38 dB 18.66 dB 18.71 dB (G; a x /6 + 4)bits 17.03 dB 18.66 dB 18.93 dB 19.00 dB Table 2.2 Resolution Requirements of Q2 and DAC. 2.4.4 Performance Comparison of the AR and the PAR Predictors In this section, the following two computer simulations are performed to compare the performance of the AR and the PAR predictors. First, the signal to be digitized, x(ri), is assumed to be the signal given by Eq. 2.14, where uAR(ri) is the lst-order AR process with C = 0.995 (see Eq. 2.27). For this case, the APCD method respectively employs the following four different predictors: i) the lst-order AR predictor, ii) the 1x1st-order PAR 57 predictor, iii) the 2nd-order AR predictor, and iv) the 2 x1st-order PAR predictor. Their performance in prediction gain as a function of J/P is illustrated in Figure 2.8. By comparing the results of Figure 2.8 with Figures. 2.3 and 2.4, it can be concluded that the analytical results about the performance of the AR and the PAR predictor given in Section 2.3 are well verified by the computer simulation results. T3 -1! Ist-OrderAR Predictor - B - 2nd-Order AR Predictor lst-Order PAR Predictor -Q- 2nd-Order PAR Predictor i i i j J/P=2f/fs Figure 2.8 Performance Comparison between the A R and the PAR Predictors (Computer Simulation Results - Part /). Second, x(n) is assumed to consist of xs(n) which is a B-SPK signal, and xw(ri) which is the AWGN, with PjPw =30dB. When the 5th-order AR or the 5x1 s t PAR predictors is respectively employed by the APCD method, their performance in prediction gain as a function of J/P is illustrated in Figure. 2.9. Clearly, the obtained results indicate that the PAR predictor can achieve much better prediction gains than the AR predictor. Furthermore, 58 for x { n ) , an upper bound about its prediction gain is obtained by using the numerical method presented in [60], and is also illustrated in the same figure. By comparing the gap between the prediction gains achieved by the PAR predictor and their upper bounds, it can be concluded that a large potential still exists for improving the performance of the signal prediction algorithms. 30 28 26 T3 24 5th-order AR Predictor - B - 5th-order PAR Predictor Upper Bound of 5th-order Predictor" 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 J/P=2f/fs Figure 2.9 Performance Comparison between the A R and the PAR Predictors (Computer Simulation Results - Part II). 2.5 Summary In this chapter, by theoretical analysis and by means of computer simulation, it is illustrated that in conjunction with the AR and the PAR predictors, the APCD method can significantly improve the ADC resolution requirement for the wideband multi-standard base-59 station receivers employing the software radio technologies. Furthermore, by analyzing the performance of the AR and the PAR predictors when used to predict cyclostationary signals, it is shown that the PAR predictor is a much better algorithm for cyclostationary signals, although as the trade-off for this performance improvement, the PAR predictor requires certain prior knowledge of the signals and higher computational efforts. 60 Chapter 3 Inter-Cell Interference Analysis for Mobile Users with Controlled Power Disparity 3.1 Introduction and System Model In this thesis, a widely used cellular CDMA model (see [12] and [41]) is considered. In this model, the system consists of M identical hexagonal cells, and in the center of each cell, a DBS with an omni-directional antenna is installed to receive the radio signals transmitted by mobile users. It is also assumed that only one type of mobile users exists in the system, and these mobile users are uniformly distributed in the M cells, independent of each other. Therefore, it is reasonable to assume that there are an equal number of mobile users, denoted as N, assigned to each DBS. Correspondingly, the number of mobile users per unit area, p, is given by the following: p = N/Ac (3.1) where Ac is the area of one hexagonal cell. For the N mobile users assigned to the same DBS, it is convenient to rank and notate them according to the following rule: P L l m < P L 2 m < . . . < P L i m < . . . < P L N n n , V n e [ 0 , M - l ] (3.2) where PLim is the path loss (in dB) from the I t h mobile user assigned to the n t h DBS, denoted as the (/, n) user, to the n t h DBS. When SM is employed, only the radio signals received by the n t h DBS are used to detect the (i, n) user. Thus, as it is shown in [12], the reverse link SINR of the (i, n) user, denoted as E'b" jV" , is given by the following: 61 (3-3) (IT+O/BW + N0FN where Pim is the average receiving power of the (i, n) user at the DBS, R is the information bit rate of the mobile users, and I'"" are the intra-cell interference and the inter-cell interference the (i, n) user suffers at the DBS, respectively, BW is the spreading bandwidth of the mobile users, NQ is the single-side PSD of thermal noise, and FN is the receiver noise figure. In order to meet the BER requirement set by the system, E'b" /1'" must be higher than a required value, denoted as d. However, as shown in [5], the higher the value of E'b"/1'" is, the larger is the interference caused by the (i, n) user. Thus, in order to reduce the interference caused by the (i, n) user, a power control scheme is employed to regulate the value of Pinn so that E ? / l ? = a . Obviously, on the one hand, when ./V increases, V"" and increases as well. Consequently, Pinn must take a higher value so that the requirement that Eb /F" > a is satisfied. On the other hand, the maximum value Pinn can take is limited by the following ratio: pmax pmax _ t (n A\ inn BPL V->-^ 7 Q r11 '-inn where PtmdX is the maximum transmitting power of the mobile users, and fi = log 10/10. Thus, in order to analyze the reverse link capacity of the system, the value of FAM and V"N must be first obtained. 1 a will take different values for different type of mobile users, e.g. vehicular users, pedestrian users, outdoor-to-indoor users, and indoor users [47]. Here or is a constant, since only one type of mobile users exists in the considered system model. 62 As shown in [12], V™ can be expressed as follows: •j=hj*i The value of Pjnn is regulated by the power control scheme, hence, the value of can be easily derived from Eq. 3.5. In contrast, the derivation of V"n is not a trivial case. As shown in [41], the value of V™ is given by the following: /.'"" = E 2 j=\ m=Q,m^:n M-l j=l m=0,m^n (3.6) where Pjnm is the average receiving power of the (j, m) user at the DBS, and PLjmn is the path loss (in dB) from the (j, m) user to the DBS. Although the value of Pjmm is regulated by the power control scheme, PLjmm and PLjmn are random variables whose probability density functions (pdf) are very difficult to derive. Consequently, it is almost impossible to derive the value of I'"n directly from Eq. 3.6, and alternative inter-cell interference models must be developed so as to simplify the calculation of / . In the past, an inter-cell interference model which can simplify the calculation of I'"n has been proposed in [41]. However, this inter-cell interference model is valid only if all mobile users have equal receiving power at the DBSs they are assigned to, that is: P m n = P , Vie [ 1 , N ] and Vne [ 0 , M -1]. (3.7) As shown in [41], if the above condition is satisfied, the value of I'"" can be calculated as follows: ' * 63 jmn = N R p (3.8) where R E , referred to in this thesis as the joint inter-cell interference factor, is a constant independent of iV and P, and its value is analyzed in [41] from the following equation: When SUD-SM is employed, as shown in [12], all mobile users can be assumed to suffer the same amount of intra-cell interference and inter-cell interference, that is, and I'"" are and Vne [0,M -1], according to Eq. 3.3, P I M must satisfy Eq. 3.7, that is, all mobile users have equal receiving power at the DBS they are assigned to. Therefore, for the cellular CDMA systems employing SUD-SM, the value of I'"N can be derived by using Eq. 3.8. As discussed in Chapter 1, the reverse link capacity of cellular CDMA systems can be increased if SM is replaced with CM, or if SUD is replaced with SSIC or MSIC. In order to theoretically analyze the capacity improvement brought by these techniques, the value of I'"" must first be derived. However, when CM, SSIC or MSIC is employed, the receiving power of the mobile users assigned to the same DBS has controlled disparity. Thus, the condition set by Eq. 3.7 cannot be satisfied, and the value of I'"" can no longer be derived from the inter-cell interference model described in Eq. 3.8. To the best of our knowledge, when the mobile users assigned to the same DBS have controlled power disparity, no appropriate inter-cell interference model which can simplify the calculation of has been published in the open technical literature. According to the characteristics of the controlled power disparity when (3.9) constants and independent of i and n. Consequently, in order to let E'b" /V" = 64 CM, SSIC or MSIC is employed, we propose for the first time in [61] a novel inter-cell interference model which can effectively simplify the calculation of I'"" for mobile users with controlled power disparity. In this chapter, by applying this novel model, the inter-cell interference of the systems employing CM, SSIC or MSIC are thoroughly analyzed. The remainder of this chapter is organized as follows. After this introduction, the path loss model from mobile users to the DBSs is presented in Section 3.2. Then the characteristics of the controlled power disparity when either CM, SSIC or MSIC is employed are analyzed in Section 3.3. According to the analysis results for the characteristics of the controlled power disparity, a novel inter-cell interference model is proposed in the same section. In Section 3.4, the inter-cell interference factors for the proposed inter-cell interference model are derived through theoretical analysis. Then, in Section 3.5, numerical evaluations are employed to calculate the values of the inter-cell interference factors, and to further evaluate the characteristics of the inter-cell interference. Computer simulations are employed in Section 3.6 to verify the validity of the theoretical analysis and numerical evaluations. Finally, a summary is given in Section 3.7. 3.2 P a t h Loss Model The path loss from mobile users to the DBSs is assumed to satisfy the propagation model of IMT-2000 systems, that is, the path loss from mobile users to the DBSs only consists of distance attenuation and large scale fading (lognormal shadowing fading). According to [48], the distance attenuation (in dB), DAm, from a mobile user to the m* DBS can be expressed as follows: D A m = 1 0 M log1 0 r m +D . (3.10) 65 where rm is the distance from the mobile user to the mm DBS, ju is the distance attenuation exponent, and D is a constant defining the fixed attenuation common to all DBSs. Meanwhile, the lognormal shadowing fading (in dB), LSFm, from the mobile user to the DBS can be expressed as follows: LSFm=ah + bhm (3.11) where h and hm are two independent zero-mean Gaussian distributed random variables with a standard deviation equal to o [48]. In the same equation, a and b are two constants, and in order to let the standard deviation of LSFm be equal to o , a and b must satisfy the following: a2 + b2 =1. As presented in [41], ah represents the lognormal shadowing fading caused by the obstacles close to the mobile user, thus it is common to all DBSs, and bhm represents the lognormal shadowing fading caused by the obstacles close to the m* DBS, thus it is independent from other DBSs. From Eq. 3.11, the correlation, CLmn, of the lognormal shadowing fading from a mobile user to any two DBSs is given by the following: C L „ - (3,2) jE[LSF2]E[LSFn2] Therefore, if a - 0 and b = 1, the lognormal shadowing fading from a mobile user to different DBSs is independent from each other; also, if a - 1 and b = 0, the lognormal shadowing fading from a mobile user to different DBSs has the same value. From Eqs. 3.10 and 3.11, the path loss (in dB), PL m , from a mobile user to the m 6 1 DBS can be expressed as follows: P L m = D A m + L S F m = 10M log 1 0 r m + a h + b h m + D . (3.13) In order to reduce the inter-cell interference, according to Eqs. 3.6 and 3.13, each mobile user should be assigned to the DBS with the minimum path loss instead of to the DBS with the minimum 66 distance. However, as a mobile user may not be able to minimize its path loss over the entire DBS set, it is assumed that each mobile user can only minimize its path loss among the Nc DBSs closest to it. Consequently, for each DBS, there exists a specific area, referred to as the serving area, and only when a mobile user is located within this area, can it be assigned to the DBS. For example, as illustrated in Figure 3.1, the serving area of the DBS in the central cell, denoted as the 0 th DBS, is the shaded area, denoted as S 0 . N c = 3 N C = A Figure 3.1 Serving Area of the DBS in the Central Cell for Different Values of Nc . Obviously, when N c takes different values, the shape of S 0 is different, and the total area of S{ 67 is the function of iVc as follows: S 0 = N C A C (3.14) Since each mobile user can only minimize its path loss among the Nc DBSs closest to it, the path loss of a mobile user is given by the following: PL = min{PL } (3.15) where {my, j e [1, N c]} are the Nc nearest DBSs to the mobile user. From Eqs. 3.13 and 3.15, for a mobile user assigned to the central DBS, if its physical location is at (x, y), the probability that its path loss is less than K6B, Pr{PL < K\(x, y)}, can be expressed as follows: P r { P L < K\(x, y)} = Pr{PL0 < K; PL0 < PL, , i e [1,2,..., N c - l]|(x, y)} „ K - M n - a h , b h n + M n - M : . r, „ = Pr{/j0 < ^ ; /» , ->— 9 r L , i e [ l ,2 , . . . ,W c - l ]} b b _ h1 K-M0-ah _ hi ^ J - f i O ^ . ™ . — , ^ (3.16) where M,. = 10// log1 0 rt(x, y) + D and g(z) = f— <-——^ • Using Eqs. 3.1 and from the above equation, the cumulative probability function (cpf) of the path loss of a mobile user, F ( K ) , is given by the following: F(K) = —JJPr{PL < K\(x, y)}pdxdy h1 K-M0-ah hi = ff f—5—=— - T — T \ Q ( , L ) d h 0 d h d x d y . (3.17) 68 If it is required that for any mobile user its path loss must be less than KS dB so that it can be served by the system considered here, in order to let the number of mobile users assigned to each DBS be still equal to N, p must be revised to pKs as follows: N pK = • (3-18) H K S ACF(KS) Correspondingly, the KS -limited F(K), denoted as FK (K), can be expressed as follows: h2 K-M0-ah hj 3.3 Inter-cell Interference Model for Controlled Power Disparity In order to develop an appropriate inter-cell interference model which can simplify the calculation of V"n when the mobile users assigned to the same DBS have controlled power disparity, the characteristics of this disparity must be first investigated. 3.3.1 Disparity Characteristics A. Combining Macrodiversity (CM) When C M is employed, the radio signals received by the several DBSs in the vicinity of a mobile user are combined together so that better signal detection can be achieved. Thus, if optimal combining methods, such as Maximum Ratio Combining (MRC) [5], are applied to combine these radio signals, E'B" /F" is given by the following: pin P I Tf  M ~ l P I J? b rinn I n + y r rinm I ^ (3 20) If dT+lT)/Bw+N0FN mjt*n (ir+llnm)/Bw + "0FN 69 where I'"m and I'""1 are the intra-cell interference and the inter-cell interference the (i, n) user suffers at the m"1 DBS respectively, and 11, Radio signals received at the m* DBS are combined; Xinm ~ 1 (3-21) [O, Radio signals received at them* DBS are not combined. In the same equation, Pinm is the average receiving power of the (i, n) user at the m"1 DBS, which can be expressed as follows: p _ P(PLim-PLinm) p inm inn ' ^ ' By comparing Eqs. 3.20 and 3.8, it can be concluded that when CM is employed, in order to let E' b" IV" = a, P i m can have lower value than that when SM is employed. However, such reduction on the receiving power is not identical for every mobile user; instead, it has a strong correlation with the path loss of the mobile users. More specifically, if the value of PLim is small, according to Eq. 3.13, with high probability the (i, n) user is close to the n t h DBS. Correspondingly, with high probability, Vme [0,M -1] and m * n , PL i n m »PL i n n . Thus, according to Eqs. 3.20 and 3.22, the SINR improvement C M offers is low, and the reduction on Pim is small. On the contrary, if the value of PLim is large, according to Eq. 3.13, with high probability, the (z, n) user is far from the n t h DBS. Correspondingly, there may exist one or several PLinm which has similar value as PLinn. In that case, the SINR improvement C M offers is high, and hence the reduction on Pinn is large. From the above discussion, it can be concluded that when C M is employed, the mobile users with larger path loss have lower receiving power than those with less path loss. Therefore, according to Eq. 3.2, the receiving power of the mobile user assigned to the same DBS must satisfy that 70 P l m > P 2 m > . . . > P i n n > . . . > P N n n , V n e [ 0 , M - 1 ] . B. Single-cell Successive Interference Cancellation (SSIC) When SSIC is employed, the mobile users assigned to the same DBS are detected successively, starting from the one with the minimum path loss to the one with the maximum path loss. Whenever a mobile user is detected, its interference to the assigned DBS is cancelled. Thus, in each DBS, the intra-cell interference the mobile users suffer satisfies the following relation: jinn > jinn > . . . > / ™ > . .. > , Vn E [0, M — 1] . (3.23) From the above equation and Eq. 3.3, it can be concluded that in order to achieve the same SINR, the receiving power of the mobile user assigned to the same DBS must satisfy that P l m > P 2 n n > . . > P i n n > . . . > P N n n , V n e [ 0 , M - 1 ] . C. Multi-cell Successive Interference Cancellation (MSIC) When MSIC is employed, the mobile users assigned to the same DBS are detected successively, starting from the one with the minimum path loss to the one with the maximum path loss. Whenever a mobile user is detected, its interference to the DBSs in its vicinity is cancelled. Thus, in each DBS, the intra-cell interference the mobile users suffer also satisfies the relation given by Eq. 3.23. Furthermore, the inter-cell interference these mobile users suffer satisfies the following relation: jinn > jinn > . .. > / ™ > . .. > / Nnn ^ V« £ [0, M - 1] . (3.24) According to Eqs. 3.3, 3.23 and 3.24, in order to achieve the same SINR, the receiving power of the mobile user assigned to the same DBS must satisfy that 71 P > P > > P > > P Vn £ TO Af -11 r\nn — r2nn — •• — r i n n 1Nnn ' V 1 1 C L w > 1 Y 1 A J • 3.3.2 A Novel Inter-cell Interference Model From the above discussions on the characteristics of the controlled power disparity, it can be concluded that when C M or SIC is employed, for any mobile user, its receiving power has a strong correlation with the rank of its path loss among the path losses of all the mobile users assigned to the same DBS. Since the system considered here consists of M identical cells, it is reasonable to assume that all mobile users with the same path-loss rank have the same receiving power. In other words, for the receiving power matrix illustrated in Figure 3.2, the matrix elements within the same column have the same value, that is: P i m = P., Vie [1,N] and Vn£ [0,M -1]. (3.25) 0 DBS: P 1 0 0 P 2 0 0 "• o^o ^NOO 1th DBS: P 1 H P 2 H ••• P n i ••• P N U n* DBS: P X m P 2 n n - P i m ••• P N m (Af — 1) DBS: P\(M-\)(M-\) ^ 2 ( M - 1 ) ( M - 1 ) " • ' Pi(M-\)(M-\) "' P N -\)(M-I) Figure 3.2 Receiving Power Matrix. Based upon the above assumption, a novel inter-cell interference model which can simplify the calculation of V™ for mobile users with controlled power disparity is proposed as follows: j i n n = ^ R J » p . (3.26) where { R f , j £ [1,^]}, referred to as individual inter-cell interference factors, are a set of 72 constants independent of P.. Applying Eqs. 3.25 and 3.26 into Eq. 3.6, R]e can be expressed as follows: R jN M-l w=0,m^/i P (PL;mm PLjmn) (3.27) By comparing Eqs. 3.26 and 3.8, it is clear that for the inter-cell interference model given by Eq. 3.26, the limitation that all mobile users must have equal receiving power is removed. Thus, once the value of , j e [1,/V]} is obtained, the value of V™ when CM, or SIC is employed can be easily calculated using Eq. 3.26. Furthermore, if all mobile users have equal receiving power, P, from Eqs. 3.26 and 3.27, I'"" is given by the following: = NR.,P. M-l m = 0 , m ^ « J(PLjmm-PLjmn) = PE\ N M-l j=l m=0,m^n B(.PLJmm-PLJm„) (3.28) From the above equation, it can be concluded that, if all mobile users have equal receiving power, the inter-cell interference model presented in [41] can be easily obtained from the inter-cell interference model proposed here. In other words, the inter-cell interference model given by Eq. 3.26 can be used to calculate the value of I'"" not only for mobile users with controlled power disparity, but also for those with equal receiving power. Clearly, it is a more general inter-cell interference model than that presented in [41]. 3 . 4 Analysis of Individual Inter-cell Interference Factor Although the expression for R'eN is given by Eq. 3.27, it appears very difficult (if not 73 impossible) to obtain the value of RJN directly from this expression, and hence alternative approaches must be applied. Due to the symmetry of the system, without a loss of generality, the inter-cell interference to 0 th DBS (see Figure 3.1) is considered here. As discussed in Section 3.2, S 0 is the serving area of the 0 t h DBS, that is, the mobile users located outside S 0 , denoted as S 0 , cannot be assigned to the 0 t h DBS. Thus, the interference produced by all these mobile users is the inter-cell interference to the 0 th DBS. On the contrary, the mobile users within S 0 may be assigned to the 0 th DBS. Consequently, only the interference produced by the mobile users who are not assigned to the 0 th DBS is the inter-cell interference to the 0 th DBS. Due to this difference, the inter-cell interference originating from S 0 and that originating from S 0 must be analyzed separately, and the analysis results for these two regions are combined together at last to obtain the value of { R J N , j e [ l , N ] } . 3.4.1 Inter-cell Interference Originating from S 0 For a mobile user who is located at (x, y) e S 0 but not assigned to the 0 th DBS and whose path loss is less than K dB, using Eqs. 3.13 and 3.15, the inter-cell interference to the 0 th DBS produced by this mobile user, denoted as Isr° (x, y ) , can be mathematically expressed as follows: lr° (*, y) = XW^"^; PL„ < PLj, j e [0,Nc -1], j * m; PLm < K\(x, y)} m=l £f b b h2 K-M„-ah (hm-b/3(T2)2 t^i L<2n(j L 4lna ba ^ } b h + M m - M i TT Q( m m L ) d h j h (3.29) &i*m b a 74 where Rm - rm(x,y)/r0 (x,y)\ From the above equation, the total inter-cell interference to the 0 t h DBS produced by all the mobile users, who are within S 0 but not assigned to the 0 th BDS and whose path loss is less than K dB, is given by the following: i s N ° - K s ( K ) = l l i s r ° (x , y )p K s dxdy h2 K-Mm-ah ( h m - b f i a 2 ) 2 NJ^1 b h m + M m - M i f l Q( , J-)pKsdhmdhdxdy.(330) )=\,j*m bG Correspondingly, the total inter-cell interference to the 0 th DBS produced by all the mobile users, who are within S 0 but not assigned to the 0 t h DBS and whose path loss is equal to K dB, is given by the following: dK s 0 nrt L<2na j l n b o - b a ^z 1 K - a h - M i f l Q( L)pKdhdxdy. (3.31) j=\,j*m ° ® 3.4.2 Inter-cell Interference Originating from S 0 Similarly, for a mobile user who is located at (x, y)e S0, and whose path loss is less than K dB, again from Eqs. 3.13 and 3.15, the inter-cell interference to the 0 th DBS produced by this mobile user, denoted as 7r5° (JC, y), can be expressed as follows: 75 i f" (x , y ) = ^E{e / } ( P L "- p i - ) ;PL m < PL J t j e [ l ,N c ] , j * m;PL m < K \ (x , y ) } m=\ «=i b b h2 K-Mm-ah (h„-bPo2)2 = ^ V S ^ J - ^ = - J V - II 6 ( ^ 7 ^ ^ ^ . (3-32) m=i L < 2 n o L < 2 n o j=x,j*m bo From the above equation, the total inter-cell interference to the 0 th DBS produced by all the mobile users, who are located in SQ and whose path loss is less than K dB, is given by the following: li°-K°{K) = \jl*°(x,y)pKsdxdy So h2 K-Mm-ah (hm-bPa2)2 , , , . . N c +~ „ 2a2 b „ l a 2 Nc bk + M ~ M = ^ v j /S^IV- / V ~ n e ( 7 M j )p K s dh m dhdxdy. (3.33) Correspondingly, the total relative inter-cell interference to the 0 t h DBS produced by all the mobile users, who are located in S0, and whose path loss is equal to K dB, is given by the following: N { }~ dK h2 (K-Mm -ah-b2 Pa2)2 , w * r r & y e ' 2 ' 1 e 2bV A - K - a h - M , = «M* JJX*» J4j—£ ?=7 II G( 7 L)PKsdhdxdy. (3.34) L<2na -42nbo jX%m b o 3.4.3 Individual Inter-cell Interference Factor From Eq. 3.19, if the path loss of a mobile user is equal to K dB, the probability that this 76 mobile user has the * minimum path loss among the N mobile users assigned to the same DBS can be expressed as follows: Pr (K) = C'\F^ (K)[l - FKs (*)]"-'. (3.35) From the above equation and Eq. 3.31, the individual inter-cell interference factor for all the mobile users, who are within S0 but not assigned to the 0 th DBS, and whose path loss has the i* minimum value among that of the N mobile users assigned to the same DBS, is given by: K° (Ks)= $isN°~Ks {K)Fr*° (K)dK . (3.36) Similarly, from Eqs. 3.34 and 3.35, the individual inter-cell interference factor for all the mobile users, who are located in SQ, and whose path loss has the ^minimum value among that of the N mobile users assigned to the same DBS, is given by the following: Ks _ R-N (Ks)= $isN°~Ks (K)Pr^ (K)dK . (3.37) Thus, according to Eqs. 3.36 and 3.37, the individual inter-cell interference factor for all the mobile users, whose path loss has the z* minimum value among that of the A7 mobile users assigned to the same DBS, can be expressed as follows: RiN{Ks) = RS°(Ks) + Rf°{Ks). (3.38) 3.5 Numerical Evaluations and Discussions Since it is very difficult, if not impossible, to derive the closed-form solutions for R' e N (K s), numerical evaluations are employed to evaluate R' e N (K s). The obtained performance 77 evaluation results are presented in this section together with a related discussion on the characteristics of R'eN(Ks). In the numerical evaluations, according to the path loss model for IMT-2000 systems given by [48], the parameters of the system are set as follows: a-b- yfl/l, Nc = 3, ju = 4, N = 100 and Ks —>+°°. With this set of parameters, firstly, the values of R'eN(Ks) are evaluated when o = 0 dB and o = 10 dB respectively, and the obtained performance results are shown in Figure 3.3. These results clearly indicate that the inter-cell interference including the effects of lognormal shadowing fading, for example, o =10dB, is much larger than that excluding the effects of lognormal shadowing fading, that is, o = OdB. Therefore, in order to precisely analyze the reverse link capacity, lognormal shadowing fading must be included in the path loss model of mobile users. 0 10 20 30 40 50 60 70 80 90 100 Mobile User Index i Figure 3.3 Individual Inter-cell Interference Factors With and Without the Effects of Lognormal Shadowing Fading. 78 Second, when o =4dB, o =8dB, and o =12dB, respectively, the values of R'eN(Ks) are evaluated and the obtained performance results are shown in Figure 3.4. From both Figures 3.3 and 3.4, it can be concluded that the mobile users with a larger path loss produce much more inter-cell inference than those with less path loss do. Therefore, because when either CM, SIC or SIIC is employed, the mobile users with larger path loss have lower receiving power than those with less path loss do, the total inter-cell interference can be reduced significantly, and hence the reverse link capacity of the cellular CDMA system can be increased considerably. This is especially true when the value of cris larger. As illustrated in Figure 3.4, when the value of a increases, the inter-cell interference produced by the mobile users with a larger path loss increases more significantly than that produced by the mobile users with less path loss. Q- O o » u sd 5 <u u s <D I-V 4 E 8 3 I •— c I—I 3 -a c 1 -- o=4dB • CR=8 dB * <*=12dB * + + + • 0 10 20 30 40 50 60 70 80 90 100 Mobile User Index i Figure 3.4 Individual Inter-cell Interference Factors for Different Values of a. 79 Finally, in order to compare the inter-cell interference originated from S 0 with total inter-cell interference, as a function of a , the following ratios are obtained by numerical evaluations: ^ = E ^ o ( + ~ ) / £ * r ( + ~ ) (3.40) 1=1 From the obtained numerical results depicted in Figure 3.5 it can be concluded that although there only are ( N c - l)N mobile users who are within S 0 , but not assigned to the 0 th DBS, they produce a large portion of the inter-cell interference to the 0 th DBS, especially when the value of a is small. Therefore, by canceling the intra-cell interference and the inter-cell interference originating from S 0 , SEC can considerably increase the reverse link capacity of the system without significantly increasing its signal processing complexity. 0.91 c K 0) 0.8 •S 0.7 Pi 0.6 5 0.5 U 3 0.4 0.3 0.2 2 3 4 5 6 7 8 9 10 11 12 Standard Deviation of Lognormal Shadowing Fading cr(dB) Figure 3.5 Inter-cell Interference Originating from 5 0 vs. Total Inter-cell Interference. 80 3.6 Computer Simulations In order to verify the validity of the theoretical analysis and numerical evaluations for R'eN ( K s ) presented in the previous section, computer simulations are performed to evaluate the value of R'eN(Ks). 3.6.1 Computer Simulation Methodology Following the cellular CDMA model presented in Section 3.1, a cellular system up to five tiers and 61 identical hexagonal cells is implemented in the software. For this system, the individual inter-cell interference factor to the central DBS (the 0 t h DBS) is evaluated as follows by employing Monte Carlo simulation techniques. As illustrated in Figure 3.6, first, mobile users following the uniform distribution are added into the system one by one. If the minimum path loss from the mobile user to the Nc nearest DBSs is less than K s dB, the mobile user is assigned to the DBS with the minimum path loss; otherwise, the mobile user is discarded. Add Mobile Users into Each DBS Sort N Mobile Users Assigned to Each DBS by Path Loss Calculate Individual Inter-cell Interference Factor Figure 3.6 Computer Simulation Model for Individual Inter-cell Interference Factor. Second, if at least N mobile users are assigned to each of the (M -1) DBSs around the 0 th DBS, the N mobile users assigned to each of the (M -1) DBSs are sorted and notated from the one with the minimum path loss to the one with the maximum path loss. Third, for i e [1, N], the individual inter-cell interference factors, R'eN ( K s ) , are calculated as follows: 81 M-\ R^(K s ) = f j e / } i P L ^ P L ^ . (3.39) By performing the above three-step a sufficient number of times, the mean value of R[N{KS) can be obtained. 3.6.2 Performance Evaluation Results In computer simulations, the same parameters as those in the numerical evaluations are taken, that is, a - b = V2/2, N c = 3 , ju = 4, N = 100 and K s — > +0=. Furthermore, as it is found by our extensive computer simulations, the inter-cell interference to the 0 th DBS is mainly produced by the mobile users who are assigned to the first three tiers of DBSs around the 0 th DBS. In these three tiers, there are a total of 36 DBSs, so by adding the 0 t h DBS, M = 37 is selected. With the above set of parameters, the values of R'eN ( K s ) when o = 0 dB and o = 10 dB are evaluated by computer simulations. In order to compare the numerical results obtained in Section 3.4.4, the obtained simulation results are also depicted in Figure 3.3. Clearly, as it is illustrated in Figure 3.3, the simulation results obtained here perfectly confirm the numerical results obtained in Section 3.4.4. Furthermore, as it is found by our extensive numerical evaluations and computer simulations, independent of the value of a , b , N c , p . , N and K s , the simulation results always match the numerical results perfectly. Therefore, the validity of the theoretical analysis and numerical evaluations for R'eN(Ks) presented in the previous section is verified. 82 3.7 Summary When either CM, SSIC, or MSIC is employed, the mobile users assigned to the same DBS have controlled power disparity. Consequently, the inter-cell interference model presented in [41] for mobile users with equal receiving power is no longer valid. According to the characteristics of this controlled power disparity, a new model that can effectively simplify the calculation of the inter-cell interference is proposed in this chapter. Both theoretical analysis and computer simulations show that the new inter-cell interference model can precisely illustrate the characteristics of the inter-cell interference for mobile users with controlled power disparity. 83 Chapter 4 Reverse Link Capacity Analysis for Selection and Combining Macrodiversity 4.1 Introduction In the stacked base station system architecture, the radio signals received by every base station are processed independently by its own receivers, and it is very difficult for a receiver to exploit the radio signals received by other base stations. Consequently, although CM has the potential to achieve a much higher reverse link capacity than SM, it cannot be employed in the stacked base station system architecture. However, in the software radio DBS system architecture, the radio signals received by a cluster of DBSs are processed in the same CPS, so the software radio receivers can simultaneously exploit the radio signals received by several DBSs within the same cluster. Thus, in the software radio DBS system architecture, the obstacles to employing C M are removed, and C M becomes a promising technique for improving the reverse link capacity of cellular CDMA systems. In the open technical literature, for cellular CDMA systems employing CM, a few papers investigating their reverse link capacity have been published (e.g. see [22]-[24]). However, in the past, no appropriate model that can evaluate the value of inter-cell interference for mobile users with controlled power disparity is available. Consequently, all those papers assume that when C M is employed, the radio signals received by all the base stations in the system are combined to detect mobile users. As shown in [22], the advantage of this assumption is that if the radio signals received by all the base stations in the system are combined to detect mobile 84 users, the reverse link capacity can be derived without the necessity of obtaining the value of I'"". Nevertheless, the disadvantage of the assumption is that it has a large discrepancy with a practicable CM. In practice, when C M is employed, for any mobile user, only the radio signals received by a small number of base stations in its vicinity are combined to detect that mobile user. Thus, the analysis results represented in [22]-[24] are for the best-case scenarios, and can only be considered as upper bounds of the reverse link capacity. In the previous chapter, a new inter-cell interference model is proposed to simplify the calculation of V"n for mobile users with controlled power disparity. Thus, it is no longer necessary to avoid the calculation of I'™ by assuming that for any mobile user, its radio signals received by all the base stations in the system are combined to detect that mobile user. In order to precisely analyze the reverse link capacity of the system employing CM, we propose for the first time in [62] a new approach for performing CM. In this approach, for any mobile user, only the radio signals received by the N c DBSs closest to it are combined to detect that mobile user. For example, as shown in Figure 4.1, when N c - 3 , for the mobile users within the shaded area S, only the radio signals received by DBSO, DBS1, and DBS2 are combined to detect those mobile users. According to this model, by applying the inter-cell interference model for mobile users with controlled power disparity proposed in the previous chapter, a more precise and more general SINR expression for mobile users employing C M is derived in this chapter. Furthermore, the reverse link capacity improvement C M offers is thoroughly analyzed by comparing the outage probability of the system employing C M with that of the system employing SM. 85 Figure 4.1 Combining Macrodiversity Region for N c = 3. The remainder of this chapter is organized as follows. In Section 4.2, the reverse link capacity of the system employing SUD-SM is obtained by analyzing the outage probability of the system. In Section 4.3, SINR expression and power reduction factors of mobile users employing SUD-CM is thoroughly analyzed. In Section 4.4, the reverse link capacity of the system when the performance of SUD-CM achieves its upper bound is discussed, and the outage probability of the system employing SUD-CM is evaluated theoretically. Finally, in Section 4.5, the conclusions of the chapter are given. 4.2 Capacity Analysis for SUD-SM When SM is employed, for the (i, n) user, only the radio signals received by the n t h DBS are used for detection. Thus, as presented in Section 3.1, the reverse link SINR of the (z, n) user, E'b" I/,'" , is given by the following: i : ( r r + i ' D / B w + N 0 F N -If SUD is employed, the intra-cell interference the (i, n) user suffers at the n t h DBS, V™, is given by 86 the following: jinn __ Y P a 7 J jnn ' (4.2) Furthermore, according to the inter-cell interference model proposed in Chapter 3, the inter-cell interference the (/, n) user suffers at the n t h DBS, I'"", is given by: jnn (4.3) Applying Eqs. 4.2 and 4.3 into Eq. 4.1, when SUD-SM is employed, the reverse link SINR of the (/, n) user, E b /V" , can be rewritten as follows: ( Z P j n n + £ Rf ) P J n n ) /V, + ^ (4.4) In order to maintain a required BER on the reverse link, Vi G [1, N], E'b" /i'" must be higher than the required value, a . Thus, if power control is assumed to be perfect, that is, E b /V" = a , Vr G [1, N], from Eq. 4.4, the following set of linear equations on {P im, i G [1, N]} are obtained: A S U D ~ S P = B (4.5) where , SUD-S Gs-aRl"(Ks) -a[l + R™(Ks)] - a [ l + R l e " ( K s ) ] G s - a R l e N ( K s ) \ -a[l + R2N(Ks)] •a[l + RleN(Ks)] -a[l + R2N(Ks) •a[l + R(eN-i)N(Ks)] -tfl + R™(Ks)] -a[l + R(eN-l)N(Ks)] i G s - a R [ N - X ) N ( K s ) -a[l + R f ( K s ) ] - a[l + R(eN~1)N (Ks)] Gs - aR™ (Ks) _ P n n = k p \nn Inn P P T 1 (JV-l)nn 1 Nnn J ' 87 B = [aN0FBw aN0FBw ••• aN0FBw aN0FBwJ, GS=BW/R. Using linear algebra methods, it is easy to show that ASUD'S is a full rank matrix. Thus, by solving the linear equation set given by Eq. 4.5, the values of { P i n n , i £ IhN]} can be obtained as follows: P = „ 0 " = f» S £ / °-\ Vie [!,#]• (4-6) aNnFB,., A j Af ' G s - a C £ R * ( K s ) + N-l] From the above equation, it can be concluded that the N mobile users assigned to the same DBS have the same required receiving power, p^UD~s . in other words, the inter-cell interference model proposed in Chapter 3 can also be used to calculate the value of Fem for mobile users with equal receiving power. If the maximum transmission power of the mobile users in the system is P r m a x , in order to satisfy the condition that Ebn/I't" >a, Vie [1,N], from Eq. 4.6, the path loss of the Af mobile users assigned to the n"1 DBS, {PLinn, i e [1, N]} must satisfy the following condition: PL,,„<101og 1 0 (P ( m aV^ D" 5)- (4-7) Therefore, for any mobile user, in order to be served by the system, its path loss must be less than 101og 1 0(P,m a x/P^D- s),thatits, KSSUD~S = 10 log 1 0 (P™ x / ^ £ / D _ S ) - Consequently, Kssu°-5 must satisfy the follow equation: ( G s -o[£/ff ( K S S U D ' S ) + N - l ] } P t m m KSUD-S = L F J 1 £=, ( 4 G ) cxN0FBw 88 Since it is very difficult (if not impossible) to derive closed-form solutions for KSSUD 5 , the numerical solutions of K^UD~S are calculated recursively from Eq. 4.8. Generally, the outage probability of a cellular CDMA system must be small, for example, less than 5%, and hence, the value of KSSUD~S must be large. Thus, in the recursive calculations, the initial value of gsuo-s - g s e t a g a y e r y i a r g e v a i u e In Section 3.1, the cpf of the path loss of a mobile user is given as follows: h1 K-M0-ah hi ™ - i J f f e 1 fen«^^^^. (4.9, Thus, according to Eqs. 4.7 and 4.9, when SUD-SM is employed, the outage probability of the system, u S U D ~ s ( N ) , as a function of N is given by the following: t U S U D - S ( N ) = l _ F ( K S U D - S y ( 4 1 0 ) 4.3 Power Reduction Factors for Combining Macrodiversity As argued in Section 3.3.1, in order to achieve the same SINR requirement, the mobile users employing C M have a lower receiving power than those employing SM do. In Section 4.3.1, the expression of this power reduction is obtained by analyzing the SINR of the mobile users employing CM. Then, in Section 4.3.2, the values of power reduction factors are analyzed theoretically. In Section 4.3.3, numerical evaluations and computer simulations are employed to evaluate the characteristics of the power reduction factors and to verify the validity of the theoretically analysis. 4 . 3 . 1 SINR Expression of Mobile Users Employing SUD-CM When C M is employed, for any mobile user, the radio signals received by the Nc D B S S closest to it are combined to detect the mobile user. Thus, as it is presented in Section 3.3, if optimal combining methods, such as for example the MRC [5], are applied to combine these radio signals, for the (i, n) user, its E'b" /V" is given by the following: K" = PJR W * ( 4 1 1 ) /;'" ( C + 01 B w + N 0 F N f t {i:m> + I ' p ) / B w + N 0 F N where {n,ml,m2,...,mNc_l} are the yVc DBSs located closest to the (i, n) user. In the same equation, l a n m ' and I'""1' are the intra-cell and the inter-cell interferences the (i, n) user suffers at the m* DBS, respectively. When SUD is employed, according to Eqs. 4.2 and 4.3, the total interference the (/, n) user suffers at the n t h DBS, (7™ + l ' e m ) , is given by the following: C + C = t P t ™ + S R ? ) p m n • ( 4 - 1 2 ) /=i,/*; ;=1 Furthermore, by comparing V™' with l™'m', it is easy to obtain the following: jbm, = I l m , m i p = f p + p . (4.13) l = l , M Similarly, by comparing l'"m' with j'^"1'; it is easy to obtain the following: C = C'MI - Pinmj = £ RlN (Ks)Plmjmj - Pmmj. (4.14) 1=1 Consequently, the total interference the (i, n) user suffers at the mf DBS, {V™1 + l'"mj), is given 90 by the following: c + o = + t R , * ( K s ) % m j . + v > • ( 4 i 5 ) l=l,M 1=1 Obviously, for N»l, ( + S ^ C ^ s ) ^ ) » - ^ ) - Consequently, /=i,/*i /=i the effect of (Pimm. -Pinm.) on (/""'•'+/J""') can be omitted, and hence approximately (/"m> + ) can be expressed as follows: i'r +/r - ix,m, + £ * f ( * s ) ^ , m j . (4.16) Furthermore, as discussed in Section 3.3, when the C M is employed, it is reasonable to assume that: Pul=Pn V / e [ 0 , M - l ] . (4.17) According to the above assumption, (I'™' + l'"mj) presented in Eq. 4.16 can be expressed as follows: iP+ir=%Plm+ZR>eN(Ks)Plnn. (4.18) l=\,M 1=1 Applying Eqs. 4.12 and 4.18 into Eq. 4.11, when SUD-CM is employed, the Ef/f," of the (i, n) user is given by the following: K _ PJR N N 1 ^Pjnn+lR?(Ks)Pjm j=\,j*i j=\ BW + N0FN (4.19) where 91 Nr-l P. =P. +\P. . . (4.20) m inn / i mm j v ' Comparing Eq. 4.19 with Eq. 4.4, it is clear that when C M is employed, in order to achieve the same SINR, the average power reduction ratio of the (i, n) user can be expressed as follows: AV-l E[PjPim}-l+Y,E[Pinm]lPim] = 1+ X £ { e x p [ / ? ( P L , , n - P L i n m j ) ] } t l + T i N ( K s ) (4.21) where {tiN(Ks),ie[l,N]} are referred to as power reduction factors. From the above equation, Vie [l,N], Pim can be expressed as follows: P = 5«_ . (4.22) Consequently, the E'b" /l't" of the (i, n) user presented in Eq. 4.19 and 4.28 can be rewritten as follows: E: _ PJR ir N p N P , 4 , l + T ; W ( ^ ) ft 1 + TJ N(KS) (4.23) fBw+N0F 4.3.2 Theoretical Analysis for Power Reduction Factors Although the expression for z i N (K s ) is given by Eq. 4.21, the value of z i N (K s ) cannot be directly calculated from this expression, and hence alternative approaches must be applied. Without the loss of generality, the power reduction factors of the mobile users assigned to the 0 t h DBS are considered here. For a mobile user who is assigned to the 0 t h DBS and whose path 92 loss is less than K, if its physical location is at (x, y), its power reduction ratio, denoted as Qr(x,y), can be mathematically expressed as follows: Q r ( * , y ) = ^ E i e ^ - ^ ; P L Q < P L . , j e [1,yVc - 1 ] ; P L 0 < K \ ( x , y ) } . (4.24) m=\ Since the system considered here consists of a cluster of identical hexagonal cells, if any two DBSs have an overlapping serving area, they must be symmetrical to each other in this area. For example, when N c - 4, the dotted area in Figure 4.2(a) is the overlapping serving area of the 0 t h DBS and the 1 th DBS, and obviously, these two DSBs are symmetrical to each other in the dotted area. Similarly, the dotted area in Figure 4.2(b) is the overlapping serving area of the 0 t h DBS and the 2 t h DBS, and they are symmetrical to each other in this area. (a) (b) Figure 4.2 Overlapping Serving Areas for N c = 4. Therefore, in the overlapping serving area of the 0 t h DBS and the m t h DBS, for any mobile user located at (x, y) and assigned to the 0 t h DBS, there always exists a symmetrical mobile user located at (xm,ym) and assigned to the mth DBS. For these two mobile users, the following relation must be satisfied: 93 E { e ^ - ^ . p L 0 < p L j J e [ l N C _ 1 ] ; P L 0 < K \ ( x , y ) } = E { e ^ - P ^ • PLm < PL}, j 6 [0, Nc -1], j * m; PLm < K\(xm , ym)}. (4.25) Using Eqs. 4.24 and 4.25, Q r ( x , y) can be rewritten as follows: Q r (*, y) = ^ E i e ^ - ^ ; P L m < PLj,; G [0, Nc -1], j * m\PLm < K\(xm, y m ) } . (4.26) m=\ Thus, the total power reduction ratio produced by all the mobile users, who are assigned to the 0 t h DBS and whose path losses are less than K, is given by the following: Qs'(K) = jJQr(x,y)pKsdxdy So = Z j J E { e ^ - p ^ ; P L m < PLj, j e [ 0 , N c - l ] , j * m;PLm < K \ ( x m , y m ) } p K s d x m d y m . m=l 5 0 (4.27) Comparing the above equation with IsN°~Ks (K) presented in Section 3.4.1 (see Eqs. 3.29 and 3.30), it is clear that Q*s (K) = IsN"~Ks (K). From Eq. 4.27, the total power reduction ratio produced by all the mobile users, who are assigned to the 0 t h DBS and whose path losses are equal to K, is given by the following: K { K ) = dQ£(K) = dIsN°-K° (K) = .s Ks w dK dK N Therefore, the power reduction factor of the (i, 0) user, i i N ( K s ) , can be expressed as follows: ^ ( K s ) = j q K N s ( K ) J > r , K N * ( K ) d K = \ i f ? - K ° ( K ) V r * ' ( K ) d K = R % ( K S ) (4.29) where Pr^ s (K) is the probability that if the path loss of a mobile user is equal to K, this 94 mobile user has the * minimum path loss among the N mobile users assigned to the same DBS (see Eq. 3.35). 4.3.3 Numerical Evaluations and Computer Simulations A. Numerical Evaluations In order to clearly illustrate the relation between the power reduction factors and the path-loss rank of mobile users assigned to the same DBS, numerical evaluations are employed to evaluate the values of t iN(Ks) as a function of i. In the numerical evaluations, according to the path loss model of IMT-2000 [48], the parameters of the system are set as follows: a = b = V 2 / 2 , N c = 3, p = 4, N = 100 , o = lOdB and K s - » + ° ° . + o P-l c .0 o 3 <u Pi o a 7 6 -Th ^ oretical Results nulation Results 5 A • Sh A 3 1 y Mobile User Index i Figure 4.3 Power Reduction Factors vs. Path-Loss Rank. As illustrated in Figure 4.3, the results obtained from the numerical evaluations clearly indicate that the mobile users with larger path loss can achieve much higher power reduction 95 than those with less path loss can. Thus, when CM is employed, both the outage probability and the inter-cell interference of the system can effectively be reduced, and hence the reverse link capacity can be significantly increased. B. Computer Simulations In order to verify the validity of the analysis for the power reduction factors, following the cellular CDMA model presented in Section 3.1, a cellular system up to five tiers and 61 identical hexagonal cells is implemented in the software. For this system, the power reduction factors of the mobile users assigned to the central DBS (the 0 th DBS) are evaluated as follows by employing Monte Carlo simulation techniques. Add Mobile Users into the 0th DBS Sort N Mobile Users Assigned to the 0>h DBS Calculate Power Reduction Factors ^ Figure 4.4 Computer Simulation Model for Power Reduction Factors. As illustrated in Figure 4.4, first, mobile users following the uniform distribution are added into the system one by one. If the mobile user is not assigned to the 0 t h DBS, or if its path loss is larger than K s dB, the mobile user is discarded. Second, if N mobile users are assigned to the 0 t h DBS, these mobile users are sorted and notated from the one with the minimum path loss to the one with the maximum path loss. Third, for i e [1, /V], the power reduction factors, i iN (Ks), are calculated as follows: TiN(Ks) = N^PL--pL-) . (4.30) By performing the above three-step simulations a sufficient number of times, the mean value o f z i N ( K s ) can be obtained. 96 In the computer simulations, the same parameters as those in the numerical evaluations are applied, that is, a = Z? = V 2 / 2 , N c =3, ju = 4, N = 100, o =10dB and K s ->+«=. With this set of parameters, the value of z jN(Ks) are evaluated by performing the computer simulations described above. In order to compare these simulation results with the numerical results of t i N(K s), the simulation results obtained here are also depicted in Figure 4.3. Clearly, the simulation results have perfectly confirmed numerical results. 4.4 Capacity Analysis for SUD-CM In this section, the capacity of the system when the performance of C M achieves its upper bound is first discussed. Then in Section 4.4.2, the outage probability of the system employing SUD-CM is analyzed theoretically. Finally, numerical evaluations are employed in Section 4.4.3 to demonstrate the reverse link capacity improvement SUD-CM offers. 4.4.1 Performance Upper Bound for SUD-CM Obviously, the performance of SUD-CM improves if the value of N c increases. Thus, the performance of SUD-CM achieves its upper bound when N c = M , that is, the radio signals received by all M DBSs in the system are combined to detect mobile users. In that case, the serving area of each DBS covers the serving area of the whole system, and hence R%(Ks) = 0 and R'eN(Ks) = (Ks). Furthermore, according to Eq. 4.29, TiN (Ks) = R % ( K s ) . Therefore, under the condition that N C = M , from Eq. 4.19, E b n / / /" can be rewritten as follows: 97 Clearly, if N » l , the effect of ^ P j n can be omitted, and thereafter, N 1 + J#(*s) in can be approximated as follows: (4.32) f , P J n / B w + N 0 F j = l . j * ' The similar SINR expressions as the above equation are also presented in [22]-[24]. In another words, when N c = M , the SINR expression given by Eq. 4.19 is equivalent to those given by [22]-[24]. However, it should be noted that that the SINR expressions given by [22]-[24] are valid only when N c - M , while Eq. 4.19 is valid \/Nc E [ 1 , M ] . Therefore, the SINR expression given by Eq. 4.19 is much more general other expressions. As shown in [12], for a cellular CDMA system consisting of an isolated cell, the SINR of the I t h mobile user among the TV mobile users served by the isolated cell is given by the following: Comparing the above two equations, it can be concluded that when the performance of SUD-C M achieves its upper bound, that is, N C = M , nearly all the impact of the inter-cell interference on the SINR of the mobile users can be compensated by CM,.and thus, the cell capacity of the system is approximately equal to that of an isolated cell. (4.33) 98 4.4.2 Outage Probability Analysis In order to maintain a required BER on the reverse link, Vi 6 [1, N], the condition that Eb jV" > a must be satisfied. Under the assumption of a perfect power control, that is, E'b/r"=a, \/ie[l,N], from Eq. 4.19, the following set of linear equations {Pinn,ie [l,N]} is obtained: on ASUD-CP =B (4.34) where G s -• a [ l + R™(Ks)] aRlN(Ks) l + r . N ( K s ) l + r w ( K s ) • a [ l + RlN(Ks)] l + ^ v ( K S ) , aRlN(Ks) ' S 1 + T 2 „ ( K , ) •a[l + R ™ ( K s ) ] l + r 2 N ( K s ) l + r 2 N ( K s ) - a [ l + R(eN-l)N(Ks)] 1 + - a [ l + RlN-i)N(Ks)] 1 + ^ W v ( ^ ) • a [ l + R™(Ks)] \ + Tnn(KS) G< -• (JV-l)Jl aR{"-1)N(Ks) - a [ l + R™(Ks)] •a[l + R 1 " ( K s ) ] -a[l + R 2 e N ( K s ) } -a[l + R ( e N - l ) N ( K s ) } 1 + T ( N , ) N ( K s ) l + rNN(KS) aR™(Ks) 1 + Tnn(KS) Pn=[Pln P2n ••• P<N-l)n PN„], and B has been given by Eq. 4.5. Using linear algebra method, it is easy to show that ASUD c is a full rank matrix. Thus, by solving the linear equation set given by Eq. 4.34, the value of {Pin,i£ [hN]} is given by the following: aN0FBw P„ = • [l + tiN(Ks)]{Gs[l + zlN(Ks)] + a} [l + rlN(Ks)]{Gs[l + TiN(Ks)] + a} ccR\N (Ks) ^ [1 + Rf (Ks )]{GS [1 + TiN(Ks)] + a} Gs- [l + tlN(Ks)] % [l + rlN(Ks)]{Gs[l + TjN(Ks)] + a} =p.r-c(Ks). (4.35) 99 When CM is employed, the combined path loss of the (i, n) user, PLin, is given by the following: PLin = 10 l°g 10 Pf" I Pin p i n = 101og10 p . + Y P inn mnij p i n 2-exp(/#L,,n) ficxp(f3PLinmj) = -l0logl0[cxP([3PLim) + XexpC/iPL,,^ )] (4.36) ;=i where Ptm is the average transmitting power of the (i, n) user. According to the path loss model presented in Section 3.2, PL i m < PL i n m j, V/e [1,iVc -1]. Consequently, it is clear that the value of PL i n is mainly determined by that of PL i m. Thus, since PL i n n has the i' t h minimum value among {PLJnn, j e [1, A7]}, it is reasonable to assume that PLin also has the ith minimum value among { P L j n , je[l ,N]}. In order to let E' b"/P" >a, V/e [ 1 , N ] , from Eq. 4.35, the Nmobile users assigned to the nth DBS must satisfy the following condition: PLin <101og 1 0 [P , m a x /P , f ° - c (^ ) ] . (4.37) Since PLNn has the maximum value among {PLjn,je [l,N]} and from the above equation PLNn <101og I 0[P ( m a x/P^D" c(A: s)], it can be concluded that for any mobile user, in order to be served by the system its combined path loss must satisfy the following condition: 100 PLin <[mogw(Pr/P^ (KS)) = PLZD-C(KS). (4.38) From the K s -limited cpf of the path loss of a mobile user, F K (AT), which is given in Section 3.2, the Ks -limited pdf of the path loss of a mobile user can be mathematically expressed as follows: dFKs(K) dK h2 (K-M0-ah)2 =_^rrr«T e ^ NtJQf-ah~M^y• (4.39) AcF (Ksyli,^[llia Jzitbo LtJt ba From the above equation and Eq. 4.27, for a mobile user whose path loss is equal to K, its power reduction (in dB) is given by the following: <7(/O = 101og10 (4.40) Therefore, when SUD-CM is employed, for any mobile user, in order to be served by the system its maximum path loss, K^UD'C , must satisfy the following recursive equation: KSUD-C = pjJUD-C (KSUD-C ) + q { K ^ D - C ) ( 4 4 1 ) By setting a very large initial value for KSSUD~C, the numerical solution of KSSUD~C is calculated recursively from the above equation. Once the value of KSSUD'C is obtained, the outage probability of the system employing SUD-CM is given by the following: USUD-C(N) = 1-F(KSSUD-C). (4.42) 4.4.3 Numerical Results In order to demonstrate the capacity improvement C M offers, pedestrian mobile users in 101 cdma2000 [48] are considered here as an example. For these pedestrian mobile users, according to the cdma2000 proposal presented in [48], the following parameters are selected: a = b o=10dB, N c =3, ju=4, R = 9.6kbps, G s =384, and a = 5dB. Under the above conditions, by varying the value of N, numerical evaluations are employed to evaluate the values of U SUD~S (N) from Eq. 4.10 and the values of U SUD'C (AO from Eq. 4.42. 1 0 " 3 10'2 10"1 10° Outage Probability Figure 4.5 Reverse Link Capacity of SUD-SM and SUD-CM (Numerical Results). As illustrated in Figure 4.5, the obtained results on USUD'S(N) and USUD'C(N) clearly indicate that the system employing C M can achieve a much higher reverse link capacity than that employing SM. For example, as shown in Figure 4.5, when the outage probability is 1%, SM can only support 17 mobile users per cell, while CM can support 51 mobile users per cell; when the outage probability is relaxed to 5%, SM can only support 57 mobile users per cell, while CM can support 81 mobile users per cell. 102 4.5 Summary In this chapter, a new approach to performing C M is proposed. By applying this approach, a more accurate and more general SNIR expression for mobile users employing SUD-CM is obtained, and the power reduction factors of mobile users employing C M are evaluated by theoretical analysis and by means of computer simulations. Furthermore, the reverse link capacity of the system employing SUD-CM is evaluated by analyzing the outage probability of the system. In order to demonstrate the capacity improvement C M offers, the reverse link capacity of the system employing SUD-SM is analyzed as well. The analytical results demonstrate that C M can significantly increase the reverse link capacity of cellular CDMA systems. 103 Chapter 5 Reverse Link Capacity Analysis for Successive Interference Cancellation 5.1 Introduction As compared to SUD, SIC has a much stronger ability to combat the multiple access interference, and therefore, can significantly increase the reverse link capacity of cellular CDMA systems. However, one of the potential problems of SIC is that any detection errors in previous mobile users seriously reduce the SINR of subsequent users [17]. Thus, in order to achieve the performance improvement SIC offers, in each base station, only the interference produced by the mobile users whose BERs are lower than a required value are cancelled by SIC. Since in the stacked base station system architecture the radio signals received by every base station are processed independently by its own receivers, it is very difficult for these receivers to exploit the radio signals received by other base stations. Consequently, in each base station, only the mobile users assigned to it can have a high enough SINR to achieve the BER requirement set by the SIC. Therefore, only the interference produced by these mobile users, that is, the intra-cell interference, can be cancelled. This type of SIC is referred to in this thesis as SSIC. On the contrary, in the software radio DBS system architecture, the radio signals received by a cluster of DBSs are processed in the same CPS, and hence the base station receivers can simultaneously exploit the radio signals received by. several DBSs within the same cluster. Therefore, a new type of SIC, referred to as MSIC, is proposed in this thesis to be employed 104 in the software radio DBS system architecture to cancel not only the intra-cell interference but also the interference produced by the mobile users assigned to other DBSs, that is, the inter-cell interference. Consequently, the systems employing MSIC can achieve a higher reverse link capacity as compared to equivalent systems employing SSIC. In the open technical literature, for cellular CDMA systems employing SSIC, only a few papers investigating their reverse link capacity have been published (e.g. [35]-[37]). However, since no appropriate model simplifies the analysis of inter-cell interference for mobile users with controlled power disparity is available, the results presented in these papers are valid only for certain special cases. More specifically, the system considered in [35] is a single cell, and [36],[37] simplify their inter-cell interference analysis by excluding the effects of lognormal shadowing fading from the reverse link path loss model, that is, only the case where o = OdB is considered. As demonstrated in Section 3.4, when the effects of lognormal shadowing fading are included in the path loss model, for example, o = 10 dB, the inter-cell interference is much larger than that when the effects of lognormal shadowing fading are excluded. Therefore, the reverse link capacity of most cellular CDMA systems cannot be precisely evaluated by the results presented in [36]-[37]. Furthermore, to the best of our knowledge, no paper investigating the reverse link capacity of the systems employing MSIC has been published in the open technical literature. A new inter-cell interference model simplifying the calculation of I'"" for mobile users with controlled power disparity is proposed in Chapter 3. By applying this inter-cell interference model, a more precise SINR expression for mobile users employing SSIC, which was published for the first time in [63], is presented, and the reverse link capacity of the systems employing SSIC is analyzed in this chapter. Furthermore, in order to analyze the 105 reverse link capacity of the systems employing MSIC, a new approach to performing MSIC in the software radio DBS system architecture is proposed. By applying this novel approach, the SINR expression of mobile users employing SSIC is presented, and the reverse link capacity improvement MSIC offers is evaluated by theoretical analysis and by means of computer simulations. The remainder of this chapter is organized as follows. In Section 5.2, when SSIC-SM is employed, the SINR expression of mobile users and the reverse link capacity of the system are analyzed. In Section 5.3, a new approach to performing MSIC is proposed. By applying this new model, the SINR expression of mobile users and the reverse link capacity of the system when MSIC-SM is employed are analyzed in the same section. When SSIC-SM and MSIC-CM are employed respectively, the SINR expressions of mobile users and the reverse link capacities of the system are analyzed in Sections 5.4 and 5.5. Computer simulations are reported in Section 5.6 that verify the validity of the reverse link capacity analysis presented in the previous chapter and this chapter. Finally, a summary of this chapter is given in Section 5.7. 5.2 SSIC-SM Capacity Analysis 5.2.1 SINR Expression When SSIC is employed, the mobile users assigned to the same base station are detected successively, starting from the mobile user with the minimum path loss to that with the maximum path loss. Whenever a mobile user is detected, its interference to the DBS it is assigned to is cancelled. Thus, for the (i, n) user, the intra-cell interference it suffers at the rt DBS, V™, is given by the following: 106 1-1 ;=i j=i+i where y/ is an interference cancellation factor indicating the portion of the interference still left after the cancellation. Hence, yj e [0,1). Since SSIC can only cancel the intra-cell interference, according to the inter-cell interference model for mobile users with controlled power disparity, the inter-cell interference the (/, n) user suffers at the DBS, I'"", can be expressed as follows: rr=tvN(Ks)Pjnri. (5.2) ;=i As demonstrated in Section 3.1, when SM is employed, the reverse link SINR for the (i, n) user is given by the following: jin [ r r + I i n n ^ B w + N o F -When SSIC-SM is employed, by applying Eqs. 5.1 and 5.2 into Eq. 5.3, the SINR expression for the (i, n) user, E'b /1'" , is given by the following: E: _ PJR f,VPJnn+tPJnn+lRtN(Ks)Pjnn j=\ j=M ;=1 (5.4) 'BW+N0F 5.2.2 Outage Probability Analysis The outage probability analysis for the system employing SSIC-SM, which is performed in the following, is similar to the SUD-SM case presented in Section 4.2. In order to maintain a certain BER on the reverse link, Vi e [ 1 , N ] , Eb /1'" must be higher than a certain value, a Under 107 ^ SSIC-S the assumption of perfect power control, that is, Ebin /if = a , V i e [1, AT] , from Eq. 5.4, the following set of linear equations on {P im, i e [1, N]} can be obtained: Ass,c-sPnn=B (5.5) where " Gs-aRle"(Ks) ~a[\ + aR2eN{Ks)] - - a [ l + R(eN'l)N(Ks)] - a [ l +R™(Ks)] -a[v + RlN(Ks)] Gs-aR2N(Ks) \ I i i -aty+R™ {Ks)] ••. -a[l + ^ N - 1 ) A , ( / i: s )] ; i ; - Gs-aRf^N(Ks) - a [ l + R™(Ks)] -a[iff + aR]N{Ks)} - a [ y / + R2eN{Ks)} ••• -aty + R(eN-,)N (K s)] Gs-aR™{Ks) P =\p p ... p p y 1 nn \f \nn 1 Inn 1 (N-\)nn Nnn J ' B = [aNaFBw aN0FBw ••• aN(jFBw cN0FBj. Using linear algebra methods, it is easy to show that ASSIC~S is a full rank matrix. Thus, by solving the linear equation set given by Eq. 5.5, the value for {Pim, i e [1, N]} is given by the following: PIM =Y'-1 = P,SNS'C-S(KS) (5-6) Gs -aRlN(Ks)-a^yj-l[l + RiN(Ks)] 7=2 where r =G * + a r (5.7) Gs +a Since 0 < yj < 1, from the above equation, it is clear that 0 < y < 1, and hence, from Eq. 5.6, {PtsNs,c-s(Ks),ie[l,N]} must satisfy P,f c " s (K s ) > P^'5(Ks) > ... > P^c~s(Ks). This result confirms our discussions in Section 3.3.1, that is, when SSIC is employed, the receiving power of the mobile users assigned to the same DBS has controlled disparity, and the ones with a larger path loss require lower receiving power than those with less path loss. 108 In order to let E' b"/i'" >a, V i e [ l ,N] , from Eq . 5.6, the path loss of the N mobile users assigned to the n"1 cell must satisfy the following: P L ( , „ < 1 0 1 o g 1 0 [ P , m a 7 ^ f C " 5 ( ^ 5 ) ] > V i e [ 1 , ( 5 - 8 ) Since PLNrm takes the largest value among {PLim,ie [1,7V]} and from the above equation, PLNm <lOlogw[P™x/P*SN!C~S (Ks)], it is clear that when SSIC is employed, for any mobile user, in order to be served by the system, its path loss must be less than 10 l o g ] 0 [P ( m a x /P^ c~ s (K s)], ft that is, K s s s , c~ s = 10\og l 0[P t m M /P™ c~ s (Kj*' c)]. Consequently, K s s s' c~ s must satisfy the following equation: { G s - a R { e N ( K S S S ! C - S ) - a f j y J ~ l [ 1 + R J N ( t f f c " s )]}P,max ATf c~ s = 101og10 A / l 2 p (5-9) Since it is very difficult (if not impossible) to derive closed-form solutions for K^SIC~S, numerical solution of Kfc~s is calculated recursively from Eq . 5.9. Generally, the outage probability of a cellular C D M A system must be small, for example, less than 5%, and hence, the value of K s s s , c~ s must be large. Thus, in the recursive calculations, the initial value of Kfc~s is set as a very large value. Once the values of K^S!C~S are obtained, the outage probability of the system employing S S I C - S M , TJSS!C~S (TV), as a function of TV is given by the following: U 5 S , C - S ( N ) = 1 - F ( K S S 5 I C ~ S ) (5.10) where F(K) is the cpf of the path loss of a mobile user given by Eq . 3.17. 109 5.3 MSIC-SM Capacity Analysis 5.3.1 A New Approach to Performing MSIC When MSIC is employed, the mobile users assigned to the same base station are detected successively, starting from the one with the minimum path loss to the one with the maximum path loss. Whenever a mobile user is detected, not only is its interference to its assigned DBS, but also its interference to a certain number of DBSs in its vicinity is cancelled. Obviously, for each mobile user, the larger the number of DBSs canceling its interference, the better the performance of MSIC is. However, by increasing this number, the signal processing complexity for performing MSIC increases as well. Therefore, in order to achieve an acceptable MSIC performance without significantly increasing the signal processing complexity of the system, the number of DBSs canceling the interference of each mobile user must be properly chosen. As shown in Section 3.5, for any DBS, although there are a total of (M - l)N mobile users producing inter-cell interference to it, a large portion of the inter-cell interference is produced by the (Nc -l)N mobile users, who are within the serving area of the DBS, but not assigned to it, especially when the value of o is small. Motivated by the above, a new approach to performing MSIC is proposed here. In this new approach, for each DBS, MSIC only cancels the intra-cell interference and the inter-cell interference originating from the serving area of the DBS. Correspondingly, for each mobile user, its interference to the Nc DBSs closest to it is cancelled by MSIC. Obviously, the signal processing complexity for performing this type of MSIC is Nc times as large as performing SSIC. Therefore, when the value of Nc is small, for example, Nc - 3 , without significantly increasing the signal processing complexity of the system, both intra-cell interference and inter-cell interference originating from the serving area of a DBS can 110 be canceled; hence, the reverse link capacity of the cellular CDMA system can be considerably increased. 5.3.2 SINR Expression When the above proposed MSIC is employed, the intra-cell interference the (i, n) user suffers at the nth DBS can still be expressed as in Eq. 5.1. After the inter-cell interference originating from the serving area of each DBS is canceled, the inter-cell interference the (/, n) user suffers at the nth DBS can be expressed as follows: C = f J R ^ ( K s ) P j m + t R e N ( K s ) P J m (5-11) j=l pi where R'eN+ (Ks) is the revised individual inter-cell interference factor of the (/, n) user given by the following: R ™ ( K s ) = yR% ( K s ) + 4° ( K s ) . (5.12) Thus, when MSIC-SM is employed, by applying Eqs. 5.1 and 5.11 into Eq. 5.3, the SINR of the (i, ri) user, E'b" /V tn , is given by the following: K _ P J R TM | i - l N 11 (5.13) ^ [ V + R ! " ( K s ) ] P j n n + R ' e N { K s ) P m n + £ [ 1 + Bw + N0F 5.3.3 Outage Probability Analysis The outage probability analysis for the system employing MSIC-SM, which is performed in the following, is similar to the SSIC-SM case presented in the previous section. In order to maintain a certain BER on the reverse link, Vi e [1, N], E'b" /1'" must be higher than a certain value, a. Under the assumption of perfect power control, that is, E b /V" = a , Vi e [1, N], from Eq. 5.13, i l l the following set of linear equations on {Pim, i e [1, N]} is obtained: A M S , C ~ S Pnn = B (5.14) where ~Gs-aRlN(Ks) -afl + flf - -a[l +R(eN~1)N (K s)] -a[l + R™(Ks)] ^ MSIC-S -a[^ + RlUKs)l Gs-aR™(Ks) -a[¥ + RlN(Ks)] '•. -a[l + R^N(Ks)] ! i . - Gs-aRlN~m(Ks) -a[l + R™{Ks)] -a[v + R™(Ks)] -a[yr + R™(Ks)] - -a[^ +R(e^1)N (Ks)] Gs-aR™(Ks) Using linear algebra methods, it is easy to show that A M S I C ~ S is a full rank matrix. Thus, by solving the linear equation set given by Eq. 5.14, the value for {Pjnn, i e [1, N]} is given by the following: p,m = —firApir-'&s) (5-i5) j=2 1=1 where, for i e [1, N -1], Gs+ays + a l R ^ i K ^ - R i f i K s ) ] Gs +a Gs + ay/-aR%(Ks)(l-y/) Gs +a (5.16) In order to let Eb jlf >a, Vie [1,N], from Eq. 5.15, the path loss of the N mobile users assigned to the cell must satisfy the following: PLim <101og10[P(maV/irC"' (Ks)], Vie [l,N]. (5.17) From the above equation, it is clear that that when MSIC-SM is employed, for any mobile user, in order to be served by the system, its path loss must be less than 10 log1 0 [P,max /P^'c' s (Ks)], that 112 is, K^SIC~S =101og10[P,max/^vTC"S (Ks)]. Consequently, Kf'c~s must satisfy the following equation: {Gs -aRlN(Kf'c~s) - a|) [1 + R!N(Kf'c~s)]f\Yl(K^,C'SWr Kf'C-S = 101og10 '± . aN0FBwHYj(Kf'c-s) 7=1 (5.18) By setting a very large initial value for K$ S1C~S, the numerical solution of K^s,c~s is calculated recursively from the above equation. Once the values of K^s,c~s are obtained, the outage probability of the system employing MSIC/SM, UMS,C~S (N), as a function of TV is given by the following: U M S , c - s ( N ) = l-F(K^s'c-s). (5.19) Comparing Eq. 5.16 with Eq. 5.7, it is clear that Vi e [1, N -1], ( £ S M 5 / C _ S ) < y. Thus, from Eqs. 5.15 and 5.6, it is clear that for the same N, Kf c~ s>K s s S I C' s, and hence JJ MSIC-S^ < u s s , c ~ s ( N ) . In other words, if the outage probability requirement is the same, the system employing MSIC can support more mobile users than that employing SSIC can. 5.3.4 Numerical Evaluations Numerical evaluations are applied to demonstrate the capacity improvement SSIC and MSIC offer. In order to compare with the numerical results of U SUD~S (N) obtained in Section 4.4.3, the same type of mobile users as those in Section 4.4.3 are considered here, that is, a = b = 4ill, N c =3, ji = 4, 0=lOdB, fl = 9.6kbps, G s =384, and a=5dB. Furthermore, according to the analysis presented in [35], the interference cancellation factor, 113 yj, is assumed to satisfy y/ = l/a . With this set of parameters, numerical evaluations are applied to evaluate the values of U S S I C ' S ( N ) from Eq. 5.10, and the values of U M S , C ~ S ( N ) from Eq. 5.19. The obtained numerical results are illustrated in Figure 5.1, together with the numerical results of J J S U D ' S (N) obtained in Section 4.4.3. 120 IOO U a, I-l 1 S3 o 10 10 10"' 10" Outage Probability Figure 5.1 Reverse Link Capacity of SUD-SM, SSIC-SM, and MSIC-SM (Numerical Results). The numerical results shown in Figure 5.1 clearly indicate that when SSIC-SM or MSIC-SM is employed, the reverse link capacity of the system can be significantly increased. For example, if the outage probability is equal to 1%, SUD-SM can only support up to 17 mobile users per cell, but SSIC-SM and MSIC-SM can respectively support up to 26 and to 30 mobile users per cell. If the outage probability is relaxed to 5%, the number that SUD-SM can support increases to 57 mobile users per cell, but the numbers that SSIC-SM and MSIC-SM can support increase to 80 and to 89 mobile users per cell, respectively. 114 In the previous two sections, the performance of SSIC-SM and MSIC-SM are thoroughly analyzed. As discussed in Chapter 4, compared with SM, CM can significantly improve the reverse link capacity of cellular CDMA systems. Thus, in the following two sections, the performance of SSIC-CM and MSIC-CM are evaluated. 5.4 SSIC-CM Capacity Analysis 5.4.1 SINR Expression As discussed in Section 4.3, when SUD-CM is employed, the reverse link SINR of the (i, n) user is given by the following: PJR if (rr + f;n)/Bw + N0F N p N p y —j-—+yRf,(Ks)—^— >B..,+N0F (5.20) If SUD is replaced by SSIC, from Eq. 5.1, if is given by the following: i-l N i-l y,p N P iim=yii/p. +yp. = y — ^ — + y — j - — a ^ jnn jnn -i , (K \ Z-l 1 , _ / v \ (5.21) Thus, when the SSIC-CM is employed, using Eqs. 5.20 and 5.21, the SIRN of the (i, n) user, Ef j If , can be mathematically expressed as follows: PJR f , f Pjn {fR Pjn | ^ JN (Ks )PJn (5.22) IB,.,+N0F 115 5.4.2 Outage Probability Analysis The outage probability analysis for the system employing SSIC-CM, which is performed in the following, is similar to the SUD-CM case presented in the Section 4.4. In order to maintain a certain BER on the reverse link, V/ e [1, N], E'b" /V" must be higher than a certain value, a. Under the assumption of perfect power control, that is, E'b" /1'" = a , Vi 6 [1, N], from Eq. 5.22, the following set of linear equations on {PIN, i 6 [1, N]} is obtained: ^SSlC-Cp = £ (5.23) where pn = k 2n (N-l). p T n 1 Nn J ' ' SSIC-C 3 aR\N{Ks) J s l + TlN(Ks) -a[¥ + RlN(Ks)] l + *i*(*s) -a[l + R™(Ks)} l + z2N(Ks) „ aR™{Ks) J s l + T2N(Ks) -aty + R2eN{Ks)] l + r2N(Ks) -a[y/ + RlN(Ks)] -afor + R™ (Ks)] l + T2N(Ks) - a[l + R("-1)N (Ks)] - a[l + R™ (Ks)] 1 + T -a[l + R(eN-»N(Ks)] 1 + T I N - Q N ( . K S ) aR^iKs) -a[l + R™(Ks)] l + r, (N-l)N (^ s) a[v + RlN-i)N(Ks)] (/V-l)/V ("s <xR™(Ks) l + tNN(Ks) By using linear algebra methods, it is easy to show that KfSIC 5 is a full rank matrix. Thus, by solving the linear equation set given by Eq. 5.23, the value for {PIN, / e [1, N]} is given by the following: p. = •=p, ssic-c iN (5-24) where, for i e [1, N -1], 116 = {Gs[l + i i N ( K s ) ] + ay} { G s [ l + r m ) N ( K s ) ] + «}' In order to let Ef / I't" > a , VZ'G [l,N], fromEq. 5.24, the combined path loss of the N mobile users assigned to the n* cell must satisfy the following: PLin <101og10[F,ma7P,.fc-c(^ s)], Vie[l,N]. (5.26) As shown in Section 4.4.2, it is reasonable to assume that PLNn has the largest value among {PLin,ie[l,N]}. Furthermore, from the above equation, it is clear that PLNn <10\ogw[P,™K/P™c~c (Ks)]. Therefore, when SSIC/CM is employed, for any mobile user, in order to be served by the system its combined path loss must satisfy the following: PLin <101og 1 0[P,m a x/P^ / c- c (KS)] = PLZC~C(KS). (5.27) As shown in Section 4.4.2, if the path loss of a mobile user is equal to K, when CM is employed, the power reduction of the mobile user is given by: 4(/O = 101ogl (5.28) Thus, from Eqs. 5.27 and 5.28, for any mobile user, in order to be served by the system its maximum path loss, Kf'c~c , must satisfy the following recursive equation: K s s I C - c = p L s * c - c ( K s s , c - c ) + q ( K ™ c - c ) ( 5 2 9 ) By setting a very large initial value for KSSSIC'C, the numerical solution of Kls,c~c is calculated recursively from the above equation. Once the values of K^SIC~C are obtained, the outage probability of the system employing SSIC-CM, USSIC~C(N), as a function of N is 117 given by the following: U s s , c ~ c ( N ) = l - F ( K S S S , C ' C ) . (5.30) 5.5 MSIC-CM Capacity Analysis 5.5.1 SINR Expression When MSIC-CM is employed, for the (i, n) user, 7™ is given by Eq. 5.21, and from Eq. 5.11, V™ is given by the following: i-i IT = E RZ + E ) P J n * j=i • H _^RJN+{Ks)P]n "RJN{Ks)P]n Thus, from Eqs. 5.20, 5.21 and 5. 31, the SINR expression of the (i, n) user, E* jlf , is given by the following: K _ PJR 1 + T j N ( K s ) l + TIN(KS) fit, l + rJN(KS) (5.32) IBW + N0F 5.5.2 Outage Probability Analysis The outage probability analysis for the system employing MSIC-CM, which is performed in the following, is similar to the SSIC-CM case presented in the previous section. In order to maintain a certain BER on the reverse link, Vi e [1, iV], E'b" /F" must be higher than a certain value, a. Under the assumption of perfect power control, that is, E'b" /1'" - a , Vz e [1, N] , from Eq. 5.32, 118 the following set of linear equations on {Pin, i £ [1, N]} is obtained: (5.33) where oRleN(Ks) -a[l + R2N(Ks)] l + TIN(Ks) l + r2N(Ks) 1 + *IN(KS) g aR2N(Ks) S l + T2N(Ks) -a[y + RlN(Ks)] 1 + T 2 N ( K S ) -aty + Rl1{Ks)] -a[v + RlN(Ks)} l + rlN(Ks) l + r2N(Ks) - a[l + RlN~»N (Ks)] - a[l + R?N (Ks)] -a[l + RlN-l)N(Ks)] . , aRlN-1)N(Ks) -a[l + R™(Ks)] -aW + R£-l)N(Ks)} 1 + T(N-DN(KS) l + Tm(Ks) Using linear algebra methods, it is easy to show that AMSIC c is a full rank matrix. Thus, by solving the linear equation set given by Eq. 5.33, the value for {Pin, i £ [1, N]} is given by the following: [l + TlN(Ks)]%* s l + Tw(Ks) £2[l + TlN (Ks)] if :P™IC-C(KS) (5.34) where, for i £ [1, N -1], {Gs [1 + TiN (Ks )] + ay + a[R';+ (Ks) - R? (Ks)]} {Gs[l + r,i+1)N(Ks)] + a} _{Gs[l + riN (Ks )] + ay/~ aR% (Ks )(1 - y) {Gs[l + T(M)N(Ks)] + a} (5.35) In order to let E* /V" > a, V/ £ [1, N], from Eq. 5.33, the combined path loss of the N mobile users assigned to the cell must satisfy the following: 119 PLin <101og10[/ima7F,.rC"C(^)], Vie [1,N]. (5.36) Since PLNn has the largest value among [PLin,ie[l,N]} and from the above equation, PLNn <101og 1 0 [P ( m a x / /^ s / c _ c (^ s )] , it is clear that when MSIC-CM is employed, for any mobile user, in order to be served by the system, its combined path loss must satisfy the following: PLm < 10log10[Pr /PN7C~C (Ks)] = ^ C / C _ C ( K s ) - (5-37) Thus, from the above equation and Eq. 5.28, for any mobile user, in order to be served by the system its maximum path loss, KfSIC~c , must satisfy the following recursive equation: jrMSlC-C Tij MSIC-C / j?- MSIC—C \ . /-r^MSIC—C\ / r o n \ K s = rLNN (Ks ) + q{Ks ). By setting a very large initial value for K^ s , c'c, the numerical solution of K^S I C~C is calculated recursively from the above equation. Once the values of K^SIC'C are obtained, the outage probability of the system employing MSIC-CM, UM51C~C(N), as a function of N is given by the following: JJMSIC-C (#) = 1 _ F { K f , c ' c ) . (5.39) 5.5.3 Numerical Evaluations In order to demonstrate the performance of SSIC-CM and MSIC-CM, the same numerical method presented in Section 5.3.4 is applied to evaluate the values of USSIC~C (N) using Eq. 5.30, and the values of UMS,C~C(N) using Eq. 5.39. The obtained numerical results are shown in Figure 5.2 together with the numerical results of USUD~C(N) obtained in Section 4.4.3. 120 140 Outage Probability Figure 5.2 Reverse Link Capacity of SUD-CM, SSIC-CM and MSIC-CM (Numerical Results). These performance evaluation results clearly indicate that systems employing SSIC-CM and MSIC-CM can achieve a much larger reverse link capacity than those employing SUD-CM. For example, if the outage probability is equal to 1%, SUD-CM can only support up to 51 mobile users per cell, but SSIC-CM and MSIC-CM can respectively support up to 76 and to 86 mobile users per cell. If the outage probability is relaxed to 5%, the capacity of SUD-CM increases to 81 mobile users per cell, but the capacities of SSIC-CM and MSIC-CM, respectively, increase to 114 and to 125 mobile users per cell. 5.6 Computer Simulation Results and Discussion Up to now, the reverse link capacities of cellular CDMA systems respectively employing the following six types of detection techniques: i) SUD-SM, ii) SUD-CM, iii) SSIC-SM, iv) 121 SSIC-CM, v) MSIC-SM, and vi) MSIC-CM, are thoroughly analyzed by theoretical analysis and numerical evaluations. In order to verify the accuracy of the performance results obtained from the theoretical analysis and numerical evaluations, computer simulations are also performed to evaluate the reverse link capacity of the system employing these six types of detection techniques. 5.6.1 Computer Simulation Methodology In Chapter 4 and the previous sections of this chapter, cdma2000 systems are selected as examples in numerical evaluations to demonstrate the reverse link capacity improvement CM, SSIC and MSIC offer. In order to compare with the performance evaluation results obtained from the numerical evaluations, the same cdma2000 system is considered in computer simulations. Following the cellular CDMA model presented in Section 3.1, a cdma2000 system [48] with up to five tiers and 61 identical hexagonal cells is implemented in the software. For this cdma2000 system, the outage probability of the DBS of the central cell (the 0 th DBS) has been evaluated as follows by employing Monte Carlo simulation techniques. As illustrated in Figure 5.3, in Step I, the initial value of N is set to be zero. In Step II, a new mobile user whose position follows the uniform distribution is created in the serving area of the 0 t h DBS, S 0 , and its path losses to the N c nearest DBS are randomly produced according to the path loss model presented in Section 3.2. If the mobile user is not assigned to the 0 t h DBS, it is discarded, and another new mobile user is created. This procedure is repeated until the newly created mobile user is assigned to the 0 t h DBS, and then N = N + l. 122 N =0 Add a new mobile user into the 0th DBS, and N = N + l For ie [l,N], compute Ef/ / ; ' 0 ; adjust P/0 so that a<Eb°/lj0 <a + e Record the value of N. Step I Step II Step III Step IV Step V Figure 5.3 Computer Simulations Model for the Evaluation of Outage Probability. In Step III, first, the SINR of the N mobile user assigned to 0 t h DBS is calculated as follows: for SUD-SM, "b _ rlO j00 7=1.7*' 'BW+N0F (5.40) for SUD-CM, j?i0 Nr-l TlO 2-1 fBw + N0F (5.41) for SSIC-SM, -(0 £^oo+i ;^oo+£f l r (+°°) f l J yOO 7=1 j=M 7=1 'BW+N0F (5.42) for MSIC-SM, pio /v c-i 7-, m = f l ^ o J f l E ^ 0 0 + E P700 + E ^ ( + ° ° ) P 7 0 0 " ^ 7=1 7=1 'BW+N0F (5.43) for SSIC-CM, 123 £[^ + /^(+°°)]P,oo +KN(+°°)Pm + S[l + ^W(+-)]^oo /BW + N0F Pm/R ; (5.44) for MSIC-CM, Nc-\ X -m=0 5 > + ^ (+~)]^oo + Stl + * f (+°°)]^oo - ^  A. + #0*" (5.45) Second, the transmission powers of the N mobile users assigned to the 0 DBS, {Pt'°,i e [1, N]} , are adjusted iteratively until the following condition is satisfied: where e is a small positive constant, for example, O.OldB. In Step IV, the values of {Pti0,ie [l,N]} are checked. If all of these are less than P, m a x , the simulation goes back to Step II to add another new mobile user to the 0 th DBS. On the contrary, if any of these are larger than P, m a x , it means that even when the mobile user has reached its maximum transmission power, P ( m a x , its SINR still cannot meet the system requirement. Thus, an outage event occurs, and the simulation goes forward to Step V. In Step V, the value of N is recorded, and this round of simulation is implemented. By performing the above five-step simulations a sufficient number of times, the pdf for outage events, denoted as uF(N), can be obtained. This pdf depicts the relation between N and the probability that among the N mobile users assigned to the 0 t h cell, at least one mobile user fails to meet the SNIR requirement. Hence, since these mobile users are independent from each other, the outage probability of the cdma2000 system is given by the following: a<E'°/i;° <a + e Vie [1,/V] (5.46) 124 1/(A0 = 1 - « | I - | > F ( 0 . (5-47) 5.6.2 Performance Evaluation Results In order to compare the simulation results for the outage probability with the numerical results obtained in previous sections, the same parameters as those in the numerical evaluations are applied in computer simulations, that is, a = & = V2/2, N c =3, o =10dB, p . = 4 , R = 9.6kbps, G s =384, a=5dB and y j - l / a . Furthermore, in the computer simulations, £ is set to O.OldB, which is small enough to represent an almost perfect power control. With this set of parameters, when the previously mentioned six types of detection techniques are employed, the outage probabilities of the cdma2000 system are evaluated by computer simulations. The computer simulation results for SUD-SM and SUD-CM are depicted in Figure 5.4, for SSIC-SM and SSIC-CM are in Figure 5.5, and for MSIC-CM and MSIC-CM are in Figure 5.6. As illustrated in Figures 5.4, 5.5 and 5.6, when SUD-SM, SSIC-SM, and MSIC-SM are employed, the obtained computer simulation results confirm the accuracy of the theoretical analysis represented in Sections 4.2, 5.2 and 5.4. However, when SUD-CM, SSIC-CM, and MSIC-CM are employed, small differences exist between the computer simulation and the numerical performance results. These small differences can be justified as follows. As discussed in Section 4.3.1, in order to simplify the theoretical analysis, for j e [ l ,N c -1], the effect of (Pimjmj - P m m j ) on (l'°aj +1?"1') is omitted, that is: (C1 + C) = (C + C ) + (P,m,m j - P i 0 m j ) - ( C ° + C) • (5-48) However, in the computer simulations, from Eqs. 5.41,5.43 and 5.45, it is clear that the actual value of 125 {Vp +Iienmi) is appHed to calculate Ej^/lj0 . Thus, the Ej?/7,'° of the theoretical analysis is expected to take higher values than those values obtained by means of computer simulations. Correspondingly, lower transmission power is required, and hence with the same value of N the outage probability obtained by the computer simulations are slightly higher than the values obtained from the numerical evaluations. 10'* 10"1 Outage Probability Figure 5.4 Reverse Link Capacity of SUD-SM and SUD-CM(Simulation and Numerical Results). 126 Outage Probability Figure 5.5 Reverse Link Capacity of SSIC-SM and SSIC-CM(Simulation and Numerical Results). 140 10"3 10"2 10"1 10° Outage Probability Figure 5.6 Reverse Link Capacity of MSIC-SM and M S I C - C M (Simulation and Numerical Results). 127 In order to verify the validity of the above justification, additional numerical evaluations are undertaken to evaluate the outage probabilities of the systems employing SUD-CM, SSIC-C M and MSIC-CM. For these numerical evaluations, with all the other parameters kept the same, a is increased by a small value, for example, from 5.0dB to 5.2dB, so as to compensate for the small SINR differences caused by omitting the effects of -P i 0 m . ) on (l'°m' + l'e0mj). The obtained numerical results are illustrated in Figure 5.7, together with the computer simulation results obtained previously. It is clear that for the system employing CM, once the small SINR differences are compensated for, the obtained numerical results match very well with the computer simulation results. 1401 1 1 I I I MM 1 1 II I I i 11 1 1 I I I I I II Outage Probability Figure 5.7 Reverse Link Capacity of S U D - C M , SSIC-CM and M S I C - C M (Simulation and Additional Numerical Results). 128 5.7 Summary In this chapter, when SSIC-SM and SSIC-CM are employed, the SINR expression of the mobile users and the reverse link capacity of the systems are thoroughly analyzed. Furthermore, a new approach to performing MSIC in software radio DBS system architecture is proposed. By applying this approach, when MSIC-SM and MSIC-CM are employed, the SINR expression of mobile users and the reverse link capacity of the systems are also analyzed. The results obtained from the numerical evaluations show that regardless of being employed in conjunction with S M or C M , SSIC and MSIC can significantly increase the reverse link capacity of the cellular C D M A system. Finally, by means of computer simulations, the validity of the reverse link capacity analysis presented in the previous chapter and in this chapter is verified. 129 Chapter 6 Conclusions and Topics for Future Research 6.1 Conclusion In this thesis, the applications of CM, SSIC and MSIC in wideband cellular CDMA systems employing software radio base station receivers have been investigated. The major contributions of the thesis are summarized as follows. 6.1.1 APCD Method If a single ADC is employed to digitize the wideband analog signals received by multi-standard base station receivers, due to the high dynamic range of these signals, no off-the-shelf commercially available ADC has a high enough resolution to satisfy the SNR requirement for the weakest signal among the overall received signals. We have proposed a novel digitization method, the APCD method, to mitigate the stringent ADC resolution requirement for digitizing the signals received by multi-standard base station receivers. The APCD method can significantly reduce the high dynamic range of the received signals, and hence, can effectively relax the steep ADC resolution requirement. This dynamic range reduction is achieved by applying appropriate signal prediction techniques to. predict and cancel high power narrowband signals among the received signals. Both theoretical analysis and computer simulations demonstrate that when the APCD method is employed in conjunction with the AR or the PAR prediction algorithm, the steep ADC resolution requirement to digitize the wideband analog signals received by multi-standard base station receivers can be reduced significantly. 130 Furthermore, since the high power narrowband signals among the overall received signals are usually cyclostationary signals, the performance of the AR and the PAR predictors when used to predict cyclostationary signals has been thoroughly evaluated by theoretical analysis and by means of computer simulations. The obtained performance evaluation results show that when used to predict cyclostationary signals, the PAR predictor can achieve a much higher prediction gain than the AR predictor can. As the trade-off for this performance improvement, the PAR predictor requires certain prior knowledge of the cyclostationary signals and higher computational efforts. 6.1.2 Software Radio DBS System Architecture In the stacked base station system architecture, the radio signals received by every base station are processed independently by its own receivers; therefore, base station receivers cannot exploit the radio signals received by other base stations. Consequently, although C M and MSIC have the potential to achieve a much higher reverse link capacity than SM and SSIC, it cannot be employed in the stacked base station system architecture. In order to enable the applications of C M and MSIC, we have proposed a novel system architecture, software radio DBS system architecture, to implement the multi-standard base station receivers. In this system architecture, radio signals received by a cluster of DBSs are transmitted to and processed in the same CPS. Thus, base station receivers can simultaneously exploit the radio signals received by several DBSs to detect mobile users, and hence both C M and MSIC can be employed to increase the reverse link capacity of cellular CDMA systems. 6.1.3 Inter-cell Interference Model for Mobile Users with Controlled Power Disparity In order to obtain the SINR expression for mobile users employing CM, SSIC or MSIC, the inter-cell interference these mobile users suffer must first be analyzed. When either CM, 131 SSIC or MSIC is employed, the receiving power of the mobile users assigned to the same DBS has controlled disparity. Consequently, the existing inter-cell interference model, which is based upon the assumption of equal receiving power, cannot be used to analyze the inter-cell interference of mobile users employing CM, SSIC or MSIC. We have proposed a new inter-cell interference model for mobile users with controlled power disparity. In this inter-cell interference model, the mobile users assigned to each DBS are ranked starting from the mobile user with the minimum path loss to that with the maximum path loss; it is assumed that the mobile users with the same path-loss rank have the same receiving power. Therefore, in this inter-cell interference model, the receiving power of the mobile users assigned to the same DBS can have controlled disparity. Both theoretical analysis and computer simulations demonstrate that by applying the inter-cell interference model for mobile users with controlled power disparity, the inter-cell interference of the mobile users employing CM, SSIC or MSIC can be precisely evaluated. Furthermore, it is also proven that the existing inter-cell interference model for mobile users with equal receiving power can be easily obtained from the newly proposed inter-cell interference model. 6.1.4 Reverse Link Capacity Analysis for CM In order to precisely analyze the reverse link capacity improvement CM offers, we have proposed a new approach to performing CM in the software radio DBS system architecture. In this approach, for any mobile user, only the radio signals received by the several DBSs in its vicinity are combined to detect that mobile user. By applying this new CM approach and the inter-cell interference model for mobile users with controlled power disparity, a more accurate and more general SNIR expression for mobile users employing SUD-CM has been obtained, 132 and the power reduction factors of mobile users employing C M have been evaluated by theoretical analysis and computer simulations. Furthermore, the reverse link capacity of the system employing SUD-CM has been evaluated by analyzing the outage probability of the system. In order to demonstrate the capacity improvement C M offers, the reverse link capacity of the system employing SUD-SM has been analyzed as well. The performance evaluation results obtained from both analytical analysis and computer simulations demonstrate that C M can significantly increase the reverse link capacity of cellular CDMA systems. 6.1.5 Reverse Link Capacity Analysis for SSIC and MSIC In order to achieve an acceptable MSIC performance without significantly increasing the signal processing complexity of the system, we have proposed a new approach to performing MSIC in the software radio DBS system architecture. In this approach, whenever a mobile user is detected, not only is its interference to its assigned DBS cancelled, but also its interference to several DBSs in its vicinity. By applying the inter-cell interference model for mobile users with controlled power disparity, the SINR expressions for mobile users employing the following four combinations of detection techniques have been obtained: i) SSIC-SM, ii) SSIC-CM, iii) MSIC-SM, and iv) MSIC-CM. Furthermore, the reverse link capacity improvement offered by the above four combinations of detection techniques have been thoroughly evaluated by theoretical analysis and by means of computer simulations. The results obtained from both theoretical analysis and computer simulations show that SSIC and MSIC can significantly increase the reverse link capacity of cellular CDMA systems, regardless of being employed in conjunction with SM or CM. Moreover, the obtained performance evaluation results show that MISC can achieve considerably higher reverse link capacity improvement than SSIC. 133 6.2 Topics for Future Research From the contributions of this thesis summarized in the perversion section, it is clear that both the APCD method and the applications of CM, SSIC and MSIC in the software radio DBS system architecture are very promising ideas, so there are a multitude of avenues for additional research in these areas. In the following, some topics of future research are suggested. • When the prediction gains of the high power narrowband signal among the wideband analog signals received by software radio base station receivers are evaluated, it is assumed that the high power narrowband signal only consists of one independent modulated signal. In practice, however, the high power narrowband signal may consist of multiple independent modulated signals with similar receiving power. Therefore, it is a worthwhile task to investigate the prediction gains of the combinations of multiple independent modulated signals. • Although the PAR predictor can achieve a much better prediction gain than the AR predictor when used to predict the high power narrowband signal, the prediction gain achieved by the PAR predictor is still far less than the upper bound of the prediction gain of the high power narrowband signal. Therefore, some improved prediction algorithms should be developed to increase the prediction gain of the high power narrowband signal. • According to the inter-cell interference model for mobile users with controlled power disparity, the mean values of the individual inter-cell interference factors are evaluated by theoretical analysis and by means of computer simulations in this thesis. In order to extensively investigate the statistical characteristics of the inter-cell interference of the 134 mobile users with controlled power disparity, it is a worthwhile task to also evaluate the variances of the individual inter-cell interference factors. When the inter-cell interference of the mobile users with controlled power disparity is evaluated, it is assumed that there are an equal number of mobile users assigned to each DBS. Nevertheless, a more accurate, but more complicated assumption should be that the number of mobile users assigned to each DBS follows the Poisson distribution. If this improved assumption is employed, both the mean values and the variances of the individual inter-cell interference factors need to be re-analyzed. When the reverse link capacity of cellular CDMA systems is analyzed, the power control for each mobile user is assumed to be perfect. In practice, however, there always exists a certain amount of power control error. Therefore, it is a worthwhile task to investigate the effect of the power control error on the reverse link capacity of cellular CDMA systems. According to the individual inter-cell interference factors, mobile users with larger path loss produce much more inter-cell interference than those with less path loss do. Consequently, if mobile users with larger path loss have lower receiving power that those with less path loss do, the overall inter-cell interference of cellular CDMA systems can be significantly reduced. Currently existing PIC techniques treat all mobile users equally, and hence, all mobile users have equal receiving power. It is a worthwhile task to develop a new PIC technique, which let mobile users with larger path loss have lower receiving power that those with less path loss do. With the same system complexity, the new PIC technique should have better performance than currently existing PIC techniques. 135 Appendix A The purpose of this appendix is to illustrate the relation between the clipping probability and the full-scale range of an ADC when a zero-mean Gaussian distributed random variable with its standard deviation equal to o s is digitized by the ADC. For a zero-mean Gaussian distributed random variable with its standard deviation equal to os, its probability density function can be expressed as follows: /G(^ ) = ^ i - e x p ( - ^ ) . (A.l) - 4 2 n a s 2o~s Thus, if the full-scale range of the ADC is equal to A A D C , the clipping probability, P c , is given by: pc=j~AADc/2fG(t)dz+r fG(t)dz=2r jG^. ^ J — •>-AADC/2 J-AADC/2 According to Eqs. A . l and A.2, the relation between the clipping probability and the full-scale range of the ADC can be obtained as follows: Full-Scale Range ' Clipping Probability ^ADC ~ 2° s 31.7% A'ADC = % •" •* 4.55'; A ADC = 60 s 0.27% Table A. 1 Relation of Clipping Probability and Full-Scale Range. 136 Appendix B The purpose of this appendix B is to justify the validity of the algorithm described by Eqs. 2.54-2.56, that is, to prove that when b — > 0 and n —the algorithm lets WPARA (nP + k) converge to W P A R , given by Eq. 2.42, in the mean. By substituting Eq. 2.54 into Eq. 2.55, and then into Eq. 2.56, the following stochastic difference equation can be obtained: WPARA(nP + k + l) = [I - 2d$(k)Sk (n)STk (n)O r (k)]WPARA(jiP + k) + 26<$>(k)sk(n)Sk{n) (B.l) where I is a L{2H +1) x L(2H +1) unit matrix and k e [0, P -1]. Taking the mean value on both sides of the above equation, the above equation can be rewritten as: E[WPARA (nP + k+1)] = £{[/- 2$P(k)Sk (n)STk (n)<S>r (k)WPARA (nP + k)} + 2&P(k)Pk .(B.2) According to the direct-average method [50], the solution of Eq. B.2, operating under the assumption of a small S , is close to the solution of the following stochastic difference equation: EWPARA (nP + k +1)] = [/ - 28®(k)Rk<S>T (k)]E[WPARA (nP + k)] + 25®{k)Pk. (B.3) From the above equation, E [ W P A R A ( n P + k + 1 ) ] can be retrospectively expressed with E [ W P A R A ( ( n - l ) P + k +1)], as follows: E\WPAKA(nP + k+l)] = n aj V J=° J E[WPARA((n-l)P + k + l)] 137 p-x / I ,'=1 \^ m = \ (B.4) where <•> indicates arithmetic mod P and ak =l-2S®(k)Rk<S>T(k), bk = 26®(k)Pk. (B.5) (B.6) Furthermore, based upon Eq. B.4, E[WPARA(nP + k +1)] can be retrospectively expressed with E[WPARA(k +1)], as follows: f p-x Y E[WPARA(nP + k + l)] = n a, E[WPARA(k + l)] v;=o , 7 -+ (p-x Y n P-l ( i </fc+l-m> 1 1 - I \ aj v J = 0 / (B.7) As it is proven in Appendix C, when 8 < p-x , where tr(») is the trace of the (H +l)maxtr(Ri) j=o 1 matrix, the absolute value of p-\ j=0 is less than one, where |«| is the determinant of the matrix. Thus, matrix f p-x V V j = 0 J becomes a zero matrix for the limiting case where n —> °o; and hence Eq. B.7 can be rewritten as follows: UmE[WPARA(nP + k + l)} = f P-X Y 1 7=0 P-1 1=1 ^ m=l l<k+l-m> b<k-i> +bk (B.8) 138 Finally, by considering another limiting case where 6 —> 0, from Eq. B.5, expressed as follows: p-i. can be lim P-l '-11; p - l l - l + 2 S ^ ( j ) R j ^ T U ) y=0 = 2Sjj<S>(j)Rj<S>T(j) j=0 (B.9) and from Eqs. B.5 and B.6, f < x ria<*-m> i=l \^ m=l y can be expressed as follows: lim /i->0 P-l / i <k-m> b<k-i>+K = 2S p - i (B.10) According to Eqs. B.8-B.10, when 6 —» 0 and n—E[WP A R A(nP + k +1)] is given by the following: \\mE[WPARA(nP + k + V)] = y=o - i r p - i ;=o r y P A P ' (B.ll) Therefore, it is clear that when —> 0 and n —> °°, the algorithm described by Eqs. 2.54-2.56 lets £1^^ (nP + k +1)] converge to W°PAR , given by Eq. 2.42, in the mean. 139 Appendix C The purpose of this appendix is to prove that for S < ^ , -1 < (H + l)maxtr(Rk) k=0 k=0 <1. Following [50], by performing the unitary similarity transformation, the matrix <£>(k)Rk<£>T (k) can be expressed as follows: <S>{k)Rk<5>T{k) = GkAkGTk (C.l) where Gk is an L(2H +1) x L(2H +1) matrix which has as its columns an orthogonal set of eigenvectors associated with {Ak [l,L(2H + 1)]},, the eigenvalues of the matrix <3>(k)Rk<3>T (k), and Ak is an L(2H +1)x L(2H +1) diagonal matrix and has {Ak j , j e [1,L(2H +1)]} as its diagonal elements. Thus, a k described by B.5 can be expressed as follows: a k = I - 2&3>{k)Rk<$>T(k) = Gk(I- 2SAk)GTk . (C.2) As it is shown in [50], Gk is a unitary matrix, that is, \Gk \ = Gl = 1. Therefore, from Eq. C.2, the determinant of a k can be expressed as follows: U2H+1) h I = lG* |(7 - 2SA> p l I = k7 - 2 S A * )| = FI d - 28Ki) • (c.3) Let A™ denote the maximum value of {AkJ,; £ [1, L{2H +1)]}. When 0 < 6 < 1/ , it is easy to obtain the following: -\<l-2SAk} <1. Thus, from Eq. C.3, it is clear that if 140 0 < S < l / / l m a x , -\<ak < 1. Furthermore, if 0 < 5 < maxAr , the following is guaranteed: k=0 •1< n k=0 a, p-i =nhi<i-*=0 As the value of /1™X cannot be easily calculated, the value of 6 can be determined by exploiting the following property of eigenvalues [50]: U2H+1) "£AkJ=tr[<l>(k)Rk<l>T(k)] (CA) 7,(2/7+1) By substituting Eq. 2.39 into <&(k)Rk<E>r (k), ^AkJ can be rewritten as follows: 7=1 H2H+1) = tr[$(k)Rk*T (*)] = (H+ l)tr(Rk). (C.5) Since Rk = E[Sk(n)STk (n)], as shown in [50], <I>(fc)/ct<I>r(&) is an almost always positive definite matrix, that is, Ak . > 0. 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