Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Olutayo, Adebola
Doctor of Philosophy - PhD
University of British Columbia
Wireless communication systems have been recognized as one of the fastest-growing technologies in recent times. Wireless researchers look for viable solutions to meet higher data rates required by evolving applications. One solution is to use a dedicated ultra-high frequency (UHF) spectrum instead of the almost exhausted radio-frequency band. Emerging millimeter-wave (mmWave) band, terahertz (THz) band, femtocell, and wireless powered communication (WPC) systems are relatively new research areas, which can provide the needed solution to the rapidly growing need for higher data rates. However, there are unresolved challenges. A foremost challenge is the need for a suitable channel model to adequately characterize transmission environments. Channel modeling is a preliminary step for the successful design and deployment of any communication system; hence, channel modeling is the main focus of this thesis. This thesis presents detailed work on the newly-introduced Beaulieu-Xie (BX) distribution as a useful channel model, with extensive performance analyses. First, we analyze the BX fading model's second-order statistics for both single and diversity receptions using the probability density function (PDF) and characteristic function (CHF) methods, respectively. Our result provides insight into a communication system's dynamic performance operating on the BX fading model. Next, we carry out a robust performance analysis of correlated BX fading channels using bounds. Here, tractable analytical bit-error rate (BER) and outage probability (OP) expressions are derived using a newly-developed bounding technique that involves transforming correlated BX random variables (RVs) to correlated Gaussian RVs. Our results present insights into the primary factors affecting emerging communication systems' performance utilizing massive multiple-input and multiple-output technologies. Subsequently, we present performance analyses of the BX fading channels in the presence of interfering signals for diversity receptions of the maximal-ratio combining technique. Tractable expressions of the average BER and OP are derived and used to show the effect of interference on the communication systems' performance. The results provide insight into the performance of short-range communication systems where interference is prominent. Furthermore, we develop a new composite model, which we refer to as the shadowed Beaulieu-Xie (SBX). We derive the SBX's PDF, CHF, cumulative distribution function, and moment generating function in closed-form. Then, we carry out a detailed performance analysis of this new distribution and ultimately fit it to published experimental data obtained from fading measurements in 28 gigahertz outdoor mmWave channels. The goodness-of-fit demonstrates that our model yields far more accurate characterizations of new emerging wireless systems' communication environment than other existing unimodal and composite distributions. Finally, we apply the BX fading model to the performance analyses of WPC. We establish the validity of the BX fading model by the possible presence of multiple line-of-sight components in such communication environments. Here, we introduce two new nonlinear energy harvesters and fit them to measured data obtained from a communication system operating in a UHF band. Our fits are improved over those of existing nonlinear energy harvesters that apply the same set of measured data.
A Novel Fading Model for EmergingWireless Communication SystemsbyAdebola OLUTAYOM.Sc., Communication Networks and Signal Processing, University of Bristol, UK, 2011B.Sc., Electrical and Electronic Engineering, Obafemi Awolowo University, Nigeria, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe College of Graduate Studies(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)March 2021© Adebola OLUTAYO, 2021ii The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, the dissertation entitled: A Novel Fading Model for Emerging Wireless Communication Systems submitted by Adebola OLUTAYO in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering Examining Committee: Julian Cheng, School of Engineering, UBC Okanagan Campus Supervisor Jonathan F. Holzman, School of Engineering, UBC Okanagan Campus Co-supervisor Md. Jahangir Hossain, School of Engineering, UBC Okanagan Campus Supervisory Committee Member Thomas Johnson, School of Engineering, UBC Okanagan Campus Supervisory Committee Member Solomon Tesfamariam, School of Engineering, UBC Okanagan Campus University Examiner Jean-Yves Chouinard, Department of Electrical and Computer Engineering, Laval University External Examiner iiAbstractWireless communication systems have been recognized as one of the fastest-growing technologies inrecent times. Wireless researchers look for viable solutions to meet higher data rates required by evolvingapplications. One solution is to use a dedicated ultra-high frequency (UHF) spectrum instead of the almostexhausted radio-frequency band.Emerging millimeter-wave (mmWave) band, terahertz (THz) band, femtocell, and wireless powered com-munication (WPC) systems are relatively new research areas, which can provide the needed solution to therapidly growing need for higher data rates. However, there are unresolved challenges. A foremost challengeis the need for a suitable channel model to adequately characterize transmission environments. Channel mod-eling is a preliminary step for the successful design and deployment of any communication system; hence,channel modeling is the main focus of this thesis. This thesis presents detailed work on the newly-introducedBeaulieu-Xie (BX) distribution as a useful channel model, with extensive performance analyses.First, we analyze the BX fading model’s second-order statistics for both single and diversity receptionsusing the probability density function (PDF) and characteristic function (CHF) methods, respectively. Our re-sult provides insight into a communication system’s dynamic performance operating on the BX fading model.Next, we carry out a robust performance analysis of correlated BX fading channels using bounds. Here,tractable analytical bit-error rate (BER) and outage probability (OP) expressions are derived using a newly-developed bounding technique that involves transforming correlated BX random variables (RVs) to correlatedGaussian RVs. Our results present insights into the primary factors affecting emerging communication sys-tems’ performance utilizing massive multiple-input and multiple-output technologies.Subsequently, we present performance analyses of the BX fading channels in the presence of interferingsignals for diversity receptions of the maximal-ratio combining technique. Tractable expressions of the av-erage BER and OP are derived and used to show the effect of interference on the communication systems’performance. The results provide insight into the performance of short-range communication systems whereinterference is prominent. Furthermore, we develop a new composite model, which we refer to as the shad-owed Beaulieu-Xie (SBX). We derive the SBX’s PDF, CHF, cumulative distribution function, and momentiiiAbstractgenerating function in closed-form. Then, we carry out a detailed performance analysis of this new distribu-tion and ultimately fit it to published experimental data obtained from fading measurements in 28 gigahertzoutdoor mmWave channels. The goodness-of-fit demonstrates that our model yields far more accurate char-acterizations of new emerging wireless systems’ communication environment than other existing unimodaland composite distributions.Finally, we apply the BX fading model to the performance analyses of WPC. We establish the validity ofthe BX fading model by the possible presence of multiple line-of-sight components in such communicationenvironments. Here, we introduce two new nonlinear energy harvesters and fit them to measured data obtainedfrom a communication system operating in a UHF band. Our fits are improved over those of existing nonlinearenergy harvesters that apply the same set of measured data.ivLay SummaryThe demand for faster internet services has led to the emergence of new wireless communication systems,which require a new type of frequency resource. These communication systems promise to be efficient andeffective; however, there are unresolved challenges. One challenge is the need for a suitable channel model,which is a mathematical representation of the effects of the propagation environment on the signal.Channel models play a vital role in designing, implementing, testing, and deploying new wireless com-munication systems. This thesis aims to present a suitable channel model for these systems. Towards thisobjective, we develop and verify the proposed channel model using experimental data. We demonstrate thesuitability and versatility of the channel model through extensive analyses and simulations.vPrefaceThis thesis is based on the research conducted under the joint supervision of Drs. Julian Cheng andJonathan F. Holzman. For all the chapters, I conducted the required literature review and developed the re-search ideas. I carried out the required mathematical analyses and performed simulations. My supervisorshelped me to refine my research ideas, validate the analytical and simulation results, and improve the presen-tation of the manuscripts. All the manuscripts were also co-authored with my supervisors.Below is the list of journal and conference publications related to the chapters of the thesis. Both pub-lished, submitted, and drafted journal articles are provided in this list.Published and Submitted Journal PapersJ1. A. Olutayo, H. Ma, J. Cheng, and J. F. Holzman, “Level crossing rate and average fade duration fora new fading model,” IEEE Wireless Commun. Lett., vol. 6, no. 3, pp. 326-329, June 2017 (part ofChapter 2).J2. A. Olutayo, J. Cheng, and J. F. Holzman, “Performance bounds for diversity receptions over a newfading model with arbitrary branch correlation,” EURASIP Journal on Comm., vol. 25, no. 97, May2020 (part of Chapter 3).J3. A. Olutayo, J. Cheng, and J. F. Holzman, “Analyzing the Performance of Short-Range Random AccessChannels With Beaulieu-Xie Fading Model,” Revised for resubmission at IEEE Commun. Lett. (part ofChapter 4).J4. A. Olutayo, J. Cheng, and J. F. Holzman, “A New Statistical Channel Model for Emerging WirelessCommunication Systems,” IEEE Open Journal on Commun., vol 1, pp. 916-926, July 2020 (part ofChapter 5).J5. A. Olutayo, Y. Dong, J. Cheng, J. F. Holzman, and V. C. M. Leung, “Performance of Wireless PoweredCommunication Over Beaulieu-Xie Channels with Nonlinear Energy Harvesters,” to be submitted toviPrefaceIEEE Trans. Green Commun. (part of Chapter 6).Referred Conference PublicationsC1. A. Olutayo, J. Cheng, and J. F. Holzman, “Asymptotically tight performance bounds for selection com-bining over new fading channels with arbitrary correlation,” IEEE ICC 2017, Paris, May 21-25, 2017(part of Chapter 3).C2. A. Olutayo, J. Cheng, and J. F. Holzman, “Asymptotically tight performance bounds for equal-gaincombining over a new correlated fading channel,” IEEE CWIT 2017, Quebec, June 11-14, 2017 (partof Chapter 3).viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxivChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Paradigm shift in wireless communication . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Challenges of emerging trends in communication . . . . . . . . . . . . . . . . . . . . 51.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Multipath Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.3 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.4 Envelope Distribution of Received Signals . . . . . . . . . . . . . . . . . . . . . . . . 91.2.5 Energy Harvesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15viiiTABLE OF CONTENTS1.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Chapter 2: The Second-Order Statistics of the BX Fading Model . . . . . . . . . . . . . . . . . . 192.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Level Crossing Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Average Fade Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 Level Crossing Rate Expression for Maximal Ratio Combining . . . . . . . . . . . . . . . . . 222.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Chapter 3: Performance Bounds for Diversity Receptions over BX Fading Model with Arbi-trary Branch Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Physicality and Justification of the BX Fading Model . . . . . . . . . . . . . . . . . . . . . . 313.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4 Relationship between the Power Correlation and Gaussian Correlation Coefficients . . . . . . 353.5 Bounds to the probability density function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.6 Performance Bounds to Outage Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.6.1 Outage Probability Bounds for MRC . . . . . . . . . . . . . . . . . . . . . . . . . . 363.6.2 Outage Probability Bounds for EGC . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6.3 Outage Probability Bounds for SC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.7 Performance Bounds to Error Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.7.1 Error Rate Bounds for MRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.7.2 Error Rate Bounds for EGC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.7.3 Error Rate Bounds for SC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.8 Discussion on the Tightness of the Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.8.1 Tightness of the Outage Probability Bound . . . . . . . . . . . . . . . . . . . . . . . 483.8.2 Tightness of the Error Rate Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.9 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.10 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54ixTABLE OF CONTENTSChapter 4: On the Performance of Beaulieu-Xie Fading Channels for MRC in the Presence ofCo-channel Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2 Average Bit Error Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3 Outage Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Chapter 5: A New Statistical Channel Model for mmWave and THz Communication Systems . 685.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.2 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.3 Statistical Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.4 Effect of Shadowing on the K-factor Estimator . . . . . . . . . . . . . . . . . . . . . . . . . 745.5 Fit to Published Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.6 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.6.1 Average Bit-Error Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.6.2 Outage Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.8 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Chapter 6: Performance of Wireless Powered Communication over Beaulieu-Xie Channels WithEnergy Harvesters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.2 System Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2.2 Nonlinear Energy Harvester Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.2.3 Review of the BX Fading Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.2.4 Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.3 Performance of WPC with the MSPwL Energy Harvester . . . . . . . . . . . . . . . . . . . . 926.3.1 Expected Energy Harvesting Rate for MSPwL . . . . . . . . . . . . . . . . . . . . . 926.3.2 Transmission Outage Probability for MSPwL . . . . . . . . . . . . . . . . . . . . . . 936.4 Performance of WPC with the ArT Energy Harvester . . . . . . . . . . . . . . . . . . . . . . 94xTABLE OF CONTENTS6.4.1 Expected Energy Harvesting Rate for ArT . . . . . . . . . . . . . . . . . . . . . . . . 946.4.2 Transmission Outage Probability for ArT . . . . . . . . . . . . . . . . . . . . . . . . 956.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Chapter 7: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.2 Concluding Remarks on the Accomplished Works . . . . . . . . . . . . . . . . . . . . . . . . 1037.3 Suggested Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Appendix A: Proof that the BX PDF becomes an impulse as fading severity parameter ap-proaches infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Appendix B: The nth-Order Moment of the BX Distribution . . . . . . . . . . . . . . . . . . . . 120Appendix C: Derivation of the Diversity Order of the SBX Model . . . . . . . . . . . . . . . . . 122Appendix D: Derivation of the Moments of the PDF of the SBX Model . . . . . . . . . . . . . . 124Appendix E: Derivation of the Bound on the Truncation Error of the BER for Coherent Detection125Appendix F: Derivation of the Bound on the Truncation Error of the OP . . . . . . . . . . . . . 127Appendix G: Proof of CCDF for MSPwL Energy Harvester . . . . . . . . . . . . . . . . . . . . . 129Appendix H: Proof of CCDF for ArT energy harvester . . . . . . . . . . . . . . . . . . . . . . . 130xiList of TablesTable 5.1 Parameters of the SBX model as they appear in other fading models . . . . . . . . . . 71Table 6.1 Parameters of the fits of measured data from [32, Fig. 17d] for a load resistance of1.32 MΩ to MSPwL and ArT energy harvesters . . . . . . . . . . . . . . . . . . . . . 97Table 6.2 Parameters of the fits of measured data from [32, Fig. 17d] for a load resistance of 5.6MΩ to MSPwL and ArT energy harvesters . . . . . . . . . . . . . . . . . . . . . . . 98xiiList of FiguresFigure 1.1 Propagation phenomena in the transmission environment. . . . . . . . . . . . . . . . 2Figure 1.2 The cellular concept using hexagonal cells. . . . . . . . . . . . . . . . . . . . . . . . 8Figure 1.3 Relationships between the Rayleigh, Ricean, Nakagami-m, and BX distributions.The parameters m and λ represent the fading severity parameter and noncentralityparameter, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 1.4 The BX PDF versus r for different values of m with paramaters Ω = 5 and λ = 3. . . . 12Figure 1.5 The BX PDF versus r for different values of Ω with paramaters m = 2 and λ = 3. . . . 13Figure 1.6 The BX PDF versus r for different values of λ with paramaters m = 2 and Ω = 5. . . . 13Figure 2.1 The normalized LCR curves of the BX, Ricean and Nakagami-m fading models ver-sus crossing level, r. The BX, Ricean and Nakagami-m fading model curves havea total power of 4.88 dB. The BX and Nakagami-m curves are shown for fadingparameters of m = 1.5 and 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Figure 2.2 The normalized LCR curves of the BX fading model versus crossing level, r, with aK-factor of 0, 6.99 and 10 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Figure 2.3 The LCR curves of the BX fading model versus crossing level, r, for vehicular speedsof v = 5, 30, 80, and 100 km/h, with the parameters of m = 1.5, λ2 = 7 dB, and Ω =0 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.4 The AFD curves of the BX fading model versus crossing level, r, for vehicular speedsof v = 5, 30, 80, and 100 km/h, with the parameters of m = 1.5, λ2 = 2.6 dB, and Ω= 1 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.5 The AFD curves of the BX fading model versus crossing level, r, for fading parame-ter, m = 1, 2, 3, and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27xiiiLIST OF FIGURESFigure 2.6 The LCR curves of the Ricean, Nakagami-m and BX fading models versus crossinglevel, r, at the output of a 3-branch MRC combiner using the CHF method, with anaverage power of 7.64 dB per branch. . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 2.7 The LCR curves of BX fading versus crossing level, r, with fading parameters of m= 1.5, 2, 3, and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 2.8 The LCR curves for i.i.d. and non-i.i.d. branches have a total average power of 12.41dB and a K-factor of 10 dB per branch. . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 3.1 Outage probability curves for the asymptotic approximation, lower bound, and upperbound for MRC, SC, and EGC over correlated BX fading channels. The number ofbranches is N = 2, the fading parameter is m = 1.5, and the correlation structure isCb = [C1,02×2,02×2;02×2,C2,02×2;02×2,02×2,C3], where C1 = C2 = C3 =[1,√0.35;√0.35, 1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 3.2 Outage probability curves for the asymptotic approximation, lower bound, and upperbound for MRC over correlated and independent BX fading channels. The numberof branches is N = 2, the fading parameter is m = 1.5, and the correlation structure isCb = [C1,02×2,02×2;02×2,C2,02×2;02×2,02×2,C3], where C1 = C2 = C3 =[1,√0.35;√0.35, 1]and C1 = C2 = C3 = [1, 0; 0, 1] are for correlation andindependence, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 3.3 Outage probability curves for the asymptotic approximation, lower bound, and upperbound for SC over correlated and independent BX fading channels. The number ofbranches is N = 2, the fading parameter is m = 1.5, and the correlation structure isCb = [C1,02×2,02×2;02×2,C2,02×2;02×2,02×2,C3], where C1 = C2 = C3 =[1,√0.35;√0.35, 1]and C1 = C2 = C3 = [1, 0; 0, 1] are for the correlated andindependent cases, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 3.4 Outage probability curves for the asymptotic approximation, lower bound, and upperbound for EGC over correlated and independent BX fading channels. The numberof branches is N = 2, the fading parameter is m = 1.5, and the correlation struc-ture is Cb = [C1,02×2,02×2;02×2,C2,02×2;02×2,02×2,C3], where C1 = C2 =C3 =[1,√0.8;√0.8, 1]and C1 = C2 = C3 = [1, 0; 0, 1] are for correlated andindependent cases, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53xivLIST OF FIGURESFigure 3.5 BER curves for the asymptotic approximation, lower bound, and upper bound forMRC, SC, and EGC, over correlated BX fading channels. The number of branchesis N = 2, the fading parameter is m = 1.5, and the correlation structure is Cb =[C1, 02×2, 02×2; 02×2, C2, 02×2; 02×2, 02×2, C3], where C1 = C2 = C3 =[1,√0.4;√0.4, 1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 3.6 BER curves for the asymptotic approximation, lower bound, and upper bound forMRC, SC, and EGC, over correlated BX fading channels. The number of branchesis N = 3, the fading parameter is m = 1.5, and the correlation structure is Cb =[C1, 03×3, 03×3; 03×3, C3, 03×3; 03×3, 03×3, C3], where C1 = C2 = C3 =[1,√0.4,√0.2;√0.4, 1,√0.4;√0.2,√0.4, 1]. . . . . . . . . . . . . . . . . . . . . 54Figure 3.7 BER curves for the asymptotic approximation, lower bound, and upper bound forMRC, over correlated and independent BX fading channels. The number of branchesis N = 2, the fading parameter is m = 1.5, and the correlation structures is Cb =[C1, 02×2, 02×2; 02×2, C2, 02×2; 02×2, 02×2, C3], where C1 = C2 = C3 =[1,√0.4;√0.4, 1]and C1 = C2 = C3 = [1, 0; 0, 1] are for correlated and indepen-dent cases, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Figure 3.8 BER curves for the asymptotic approximation, lower bound, and upper bound forSC, over correlated and independent BX fading channels. The number of branchesis N = 2, the fading parameter is m = 1.5, and the correlation structures isCb =[C1, 02×2, 02×2; 02×2, C2, 02×2; 02×2, 02×2, C3], where C1 = C2 = C3 =[1,√0.4;√0.4, 1]and C1 = C2 = C3 = [1, 0; 0, 1] are for correlated and indepen-dent cases, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Figure 3.9 BER curves for the asymptotic approximation, lower bound, and upper bound forEGC, over correlated and independent BX fading channels. The number of branchesis N = 2, the fading parameter is m = 1.5, and the correlation structures is Cb =[C1, 02×2, 02×2; 02×2, C2, 02×2; 02×2, 02×2, C3], where C1 = C2 = C3 =[1,√0.4;√0.4, 1]and C1 = C2 = C3 = [1, 0; 0, 1] are for correlated and indepen-dent cases, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56xvLIST OF FIGURESFigure 4.1 Comparison of the average BER versus average SINR for coherent BFSK and BPSKdetection over BX fading channels with parameters of L = 4, ms = 1, λs = 3 dB, λI= 3 dB, mI = 0.75, and ΩI = 4 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 4.2 Comparison of average BER versus number of interferers for noncoherent BFSKdetection over BX fading channels with parameters of ms = 1, mI = 0.75, ΩI = 3dB,and SINR = 10 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Figure 4.3 Comparison of OP versus average SINR over BX fading channels with parameters ofL = 4, ms = 2, λs = 1.5 dB, λI = 1 dB, mI = 1, and ΩI = 3 dB. . . . . . . . . . . . . 66Figure 4.4 Comparison of the average BER curves versus average SINR for coherent BPSK and4-QAM detections over BX fading channels with parameters of L = 4, ms = 1,λ2s = 3 dB, λ2I = 3 dB, mI = 0.75, and ΩI = 4 dB. Here, N is the number ofbranches of the desired signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 5.1 Normalized error curves of the BXK-factor estimator of the BX fading model versusthe true K-factor of the SBX model for K˜2,4 and mX = 1.5. . . . . . . . . . . . . . . 77Figure 5.2 Normalized error curves of the K-factor estimator of the BX fading model versusthe true K-factor of the SBX model showing the combined effects of fading andshadowing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 5.3 Fits of the FTR, BX, and SBX models to empirically obtained signal amplitude [69]for vertical-vertical co-polarized LOS measurements. The Rayleigh fading modelhas 2σ2r = -4 dB, yielding KSRAYLOS = 0.1786. The Ricean fading model has theparameters of s2 = -10 dB and 2σ2 = -5.5 dB, yielding KSRICLOS = 0.1786. TheNakagami-m fading model has parameters of mn = 0.75 and Ωn = 0.5 dB, yieldingKSNAKLOS = 0.1413. The FTR fading model has parameters of m = 80, ∆ = 0.6, andKFTR = 22 dB, yielding KSFTRLOS = 0.0714. The BX fading model has parameters ofm = 0.65, Ω = -2 dB, and λ = -1 dB, yielding KSBXLOS = 0.0612. The SBX model hasparameters ofmX = 0.65, mY = 0.7, ΩX = -2 dB, and ΩY = -5 dB, yieldingKSSBXLOS= 0.0708. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79xviLIST OF FIGURESFigure 5.4 Fits of the FTR, BX, and SBX models to empirically obtained signal amplitude[69] for vertical-horizontal co-polarized NLOS measurements. The Rayleigh fad-ing model has 2σ2r = -0.8 dB, yielding KSRAYLOS = 0.1857. The Ricean fading modelhas the parameters of s2 = -0.8 dB and 2σ2 = -5.3 dB, yielding KSRICLOS = 0.1837.The Nakagami-m fading model has parameters ofmn = 0.75 and Ωn = 1.1 dB, yield-ing KSNAKLOS = 0.1082. The FTR fading model has parameters of m = 5, ∆ = 0.75,and KFTR = 12, yielding KSFTRLOS = 0.1236. The BX fading model has parametersofm = 0.7, Ω = 1.1 dB, and λ = -2 dB, yieldingKSBXNLOS = 0.0818. The SBX modelhas parameters of mX = 0.7, mY = 1.1, ΩX = 2 dB, and ΩY = -7.5 dB, yieldingKSSBXNLOS = 0.0770. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 5.5 Comparison of average BER curves versus average SNR for noncoherent FSK andcoherent BFSK detections over SBX channels with parameters of mY = 2 and ΩY =1 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Figure 5.6 Comparison of OP curves versus average SNR for noncoherent and coherent FSKdetections over SBX channels with parameters of mY = 2 and ΩY = −1 dB. . . . . . 86Figure 6.1 System model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 6.2 Fits of the harvested power for the MSPwL and ArT energy harvesters to empiricallyobtained harvested power [32, Fig. 17d] as a function of the recieved signal power fora 1.32 MΩ load resistance. The RMSE values for the 2-, 3-, and 4-segment MSPwLenergy harvesters are 1.68× 10−6, 6.90× 10−7, and 5.78× 10−7, respectively. TheRMSE value for the ArT energy harvester is 3.02× 10−6. . . . . . . . . . . . . . . . 97Figure 6.3 Fits of the harvested power for MSPwL and ArT energy harvesters to empiricallyobtained harvested power [32, Fig. 17d] as a function of the transmit power for a 5.6MΩ load resistance. The RMSE values for the 2-, 3-, and 4-segment MSPwL energyharvesters are 1.21× 10−6, 9.56× 10−8, and 5.15× 10−8, respectively. The RMSEvalue for the ArT energy harvester is 7.52× 10−7. . . . . . . . . . . . . . . . . . . . 99xviiLIST OF FIGURESFigure 6.4 Curves for the average harvested power, PRD, versus transmit power, pSR, for a3-segment MSPwL energy harvester at a distance of d1 = 10 m with a pathlossexponent of s1 = 2, unity-gain antennas, parameters values of$ = 0.5, and differentvalues of m1 and NS . The parameters an and bn have the values [0.3990 0.03895.515× 10−4] and [0 2.9136× 10−6 1.0086× 10−5], respectively. . . . . . . . . 99Figure 6.5 Curves for the average harvested power, PRD, versus TS protocol parameter, $, forthe ArT energy harvester at a distance of d1 = 10 m with a pathloss exponent ofs1 = 2, unity-gain antennas, and parameters values of m1 = 2, NS = 2. Theparameters a0 and b0 have values of 1.079× 10−5 and 7.90× 103, respectively. . . . 100Figure 6.6 The TOP curves versus transmit power, pSR, for the 3-segment MSPwL energy har-vester for parameters values of m1 = m2 = 2, ND = 2, an = [0.3990 0.03895.515× 10−4], and [0 2.9136× 10−6 1.0086× 10−5]. . . . . . . . . . . . . . . . 101Figure 6.7 The TOP curves for the ArT energy harvester versus distance, d1, for a transmitpower, pSR, of 100 mW and parameters values of m2 = 2, ND = 2, $ = % = 0.5,a0 = 1.079× 10−5, and 7.90× 103. . . . . . . . . . . . . . . . . . . . . . . . . . . 102xviiiList of AcronymsAcronyms Definitions5G Fifth Generation6G Sixth GenerationAFD Average Fade DurationAWGN Additive White Gaussian NoiseArT ArctangentBER Bit-Error RateBEP Bit-Error ProbabilityBPSK Binary Frequency-Shift KeyingBPSK Binary Phase-Shift KeyingBX Beaulieu-XieCCI Co-Channel InterferenceCCDF Complementary Cumulative Distribution FunctionCDF Cumulative Distribution FunctionCHF Characteristic FunctionCLT Central Limit TheoremCMOS Complementary Metal–Oxide–SemiconductorEGC Equal-Gain CombiningEEHR Expected Energy Harvesting RateEH Energy HarvestingES Energy SourceFTR Fluctuating Two-RayFSK Frequency-Shift KeyingFSO Free-Space OpticalGbps Giga-bits-per-secondxixList of AcronymsGHz GigahertzHz HertzIoT Internet-of-ThingsINR Interference-to-Noise-RatioKHz KilohertzLCR Level Crossing RateLOS Line-of-SightLMS Land Mobile SatelliteMC Multi-CarrierMGF Moment Generating FunctionMIMO Multiple-Input and Multiple-OutputMMSE Minimum Mean-Square-ErrorM -PSK M -ary Phase-Shift KeyingMSPwL Multi-Segment Piecewise LinearmmWave Millimeter-WaveMRC Maximal Ratio CombiningNLOS Non-Line-of-SightOP Outage ProbabilityOWC Optical Wireless CommunicationsPDF Probability Density FunctionPSK Phase-Shift KeyingQAM Quadrature Amplitude ModulationQoS Quality-of-ServiceRF Radio-FrequencyRMSE Root Mean-Square-ErrorRV Random VariableSBX Shadowed Beaulieu-XieSR Shadowed RiceanSC Selection CombiningSINR Signal-to-Interference Plus-Noise RatioxxList of AcronymsSNR Signal-to-Noise RatioSWIPT Simultaneous Wireless Information and Power TransferTbps Tera-Bits-Per-SecondTHz TerahertzTOP Transmission Outage ProbabilityTWDP Two-Wave With Diffuse PowerUAV Unmanned Aerial VehicleWPC Wireless Powered CommunicationWPCN Wireless Powered Communication NetworkWPT Wireless Power TransferxxiAcknowledgmentsI would like to express my sincere gratitude to my supervisor, Prof. Julian Cheng, for his support fromhis vast knowledge and experience throughout my Ph.D. study. I will continue to be inspired by his advice,dedication to research, and outstanding work ethics. I would also like to express my extreme gratitude to myco-supervisor, Prof. Jonathan F. Holzman, for his help both academically and non-academically. His supportcannot be matched and I am immensely grateful for his patience and guidance. I can say he made my doctoraljourney less painful. My supervisors’ guidance and outstanding insights have made this thesis a reality. It hasbeen a great privilege to have studied under their supervision.Moreover, I would like to express my deepest gratitude to Profs. Md. Jahangir Hossain and ThomasJohnson for accepting to be on my supervisory committee. I am thankful for their time, suggestions, andcritical comments before, during, and after my Ph.D. candidacy exams. My interactions with them havestirred my thoughts in the right direction, which has led to outstanding results. Also, I will like to thank Prof.Zheng Liu for serving as my examiner during my candidacy exams. His presence at my examination madethe ability to communicate my research in lay terms to be put into good use and evident to my supervisorycommittee members.I would like to thank Prof. Jean-Yves Chouinard for serving as my external examiner and Prof. SolomonTesfamarian for serving as my university examiner. I would also like to thank Prof. Norman Beaulieu for hisvaluable feedback, critical comments, and suggestions throughout my studies. I am sincerely thankful for histime and dedication despite his busy schedule.I am grateful to the University of British Columbia (UBC) for supporting my research through the univer-sity graduate funding (UGF), emergency bursary, and summer interval award.I had the opportunity to meet different people during my studies, both in the academic and non-academicspaces. I am particularly grateful for my colleagues on Profs. Cheng and Holzman’s research groups for theirvaluable insights into my analyses and research approach. I will like to thank members of the Integrated opticslaboratory for creating a very warm and relatable environment all through my studies at the UBC Okanagancampus.xxiiAcknowledgmentsMy special thanks go to my managers and colleagues at the TD Canada Trust bank for their encouragementand flexibility in scheduling to ensure I complete my studies at UBC Okanagan campus.Finally, my gratitude goes to my family members for their support both materially and immaterially. I amgrateful for my mother who left her career and life to help take care of my kids so that I can have ample timeto study. I am thankful for my father for his support throughout my studies. I am grateful for my kids fortheir sacrifices all through my studies. I am indeed grateful for my husband for his sacrifices, support, andencouragement throughout my studies. He remained a solid backbone through the rough and smooth paths. Ihave a doctoral thesis because of the support of my family members and friends.xxiiiDedicationTo Almighty GodxxivChapter 1Introduction1.1 Background and MotivationNew forms of wireless systems are emerging to meet the growing demand for mobile communications[1]. Noteworthy examples include terahertz (THz) and millimeter (mmWave) wireless systems, which in-crease capacity by pushing beyond the traditional microwave spectrum. Another example is the femtocellwireless system, which also increases capacity, but it does so via localization. Wireless powered communica-tion (WPC) system is another emerging wireless system where batteries of wireless communication devicescan be charged via wireless power transfer (WPT) technology, while information is simultaneously transmit-ted. These systems hold great promise, but their channels can be challenging to characterize and ultimatelyunderstand.The performance of any communication system depends on the communication channel, which can bewired or wireless. Wired channels arise if there exists a physical connection between the transmitter and thereceiver. A wireless channel has no physical connection; it applies the propagation of electromagnetic (EM)waves in the environment. A statistical distribution (model) 1 can best describe such EM wave propagationbecause of its random nature. The propagation environment may also differ for indoor, urban, suburban,underwater, or orbital communication systems.The propagating EM wave path can be altered (based on the environment’s nature) from direct propagationto complete obstruction by buildings, cars, trees, etc. The transmitter or receiver’s mobility can also affectthe signal’s path(s). This alteration in the signal’s path(s) can lead to phenomena like reflection, scattering,and diffraction, as shown in Fig. 1.1. Reflection occurs due to the wave being incident upon a far largersurface than the EM radiation’s wavelength. Scattering occurs due to the wave being incident upon a bodywith dimensions comparable to or smaller than the wavelength. Scattering leads to the redirection of signalenergy. Diffraction occurs due to the wave bending (according to Huygen’s principle) when it encounters anedge or a corner.1The terms ’distribution’ and ’model’ are used interchangeably throughout this thesis.11.1. Background and MotivationReflection off a moving truckMobile ReceiverReflection off a houseDiffractionReflection off a buildingScattering from a treeBase stationReflection & DiffractionDirect path (LOS)Figure 1.1: Propagation phenomena in the transmission environment.The propagating signal undergoes varying attenuation and redirection levels due to the phenomena men-tioned above. This attenuation and redirection can be as a result of large-scale propagation effects, as in thecase of pathloss and shadowing, or small-scale propagation effects as found in multipath fading.This thesis focuses on modeling fading and shadowing for wireless channels. Contemporary statisticalmodels, such as the Rayleigh, Ricean, generalized Ricean (i.e., κ − µ), and Nakagami-m fading models,have been used to describe the fading amplitude of such signals. However, they are limited in their ability toprovide the necessary flexibility and line-of-sight (LOS)/non-line-of-sight (NLOS) duality.The Rayleigh fading model has been used to characterize a wide range of communication environments.However, its fixed (two) degrees of freedom limit its flexibility in describing wireless communication environ-ments. Moreover, the Rayleigh fading model is incapable of characterizing transmission via LOS components(via direct propagation and specular reflections) because it is based upon the central chi distribution.The Ricean and generalized Ricean (i.e., κ − µ) fading models [2] are obtained from the noncentralchi distribution and can characterize transmission via LOS components (via direct propagation and specularreflections) and NLOS components. However, the Ricean fading model only has two degrees of freedom andis therefore limited in its flexibility for describing a broad range of signal fades [3]. The generalized Ricean (orκ− µ) fading model, in contrast, has 2n degrees of freedom, where n is the number of independent Gaussianrandom variables (RVs) and is thus flexible. However, it is not a meaningful fading model because it does notsatisfy the power constraint.The Nakagami-m fading model can characterize communication environments with multiple NLOS com-21.1. Background and Motivationponents via its flexible fading parameter, m. However, the Nakagami-m fading model is derived from thecentral chi distribution via normalization and cannot characterize LOS components (via direct propagationand specular reflections). Notable attempts to apply the Nakagami-m fading model to environments withboth LOS components (via direct propagation and specular reflections) and NLOS components have beenunsuccessful [4]-[6]. The Nakagami-m distribution was proposed to approximate the Ricean distribution [7];however, this approximation was flawed because the tail of the Nakagami-m distribution decays with a diver-sity order of m while the tail of the Ricean distribution decays with a diversity order of one [5]. Moreover,Yacoub showed that the Ricean fading model outperforms the Nakagami-m fading model via curve-fittingperformance comparison in a LOS propagation environment [4]. Ultimately, the Ricean fading model [6] wasshown to provide the best fit to field measurements obtained from LOS propagation environments.Given the differing successes and challenges of the above unimodal distribution, several bimodal andcomposite models were proposed to describe the random fluctuations suffered by the received signal in anenvironment. Examples include the Loo’s model [8], the shadowed Ricean (SR) fading model [9], the two-wave with diffuse power (TWDP) fading model [10], and the fluctuating two-ray (FTR) fading model [11].However, these bimodal and composite models all consider a fading scenario/environment with two specularcomponents at most.Loo’s model assumes that the LOS component undergoes shadowing according to the log-normal distribu-tion, while the multipath effect is Rayleigh distributed. This model was developed for the land-mobile-satellite(LMS) channel. Unfortunately, the probability density function (PDF) was not presented in closed form be-cause the application of the log-normal distribution for describing shadowing leads to a non-tractable result[9].Because of the complicated results presented by Loo’s model, the SR model was introduced in [9]. The SRmodel has the envelope of its LOS component be a Nakagami-m RV, making it a more versatile distributionthan the Loo’s model and the unimodal distributions. However, the SR model has only one fading parameterto describe the fluctuations in both the LOS and NLOS components.The TWDP model can describe an environment having two dominant LOS components and many NLOScomponents. However, its PDF is a definite integral, with many works approximating it to give a closed-formexpression. Because of the TWDP PDF’s challenges, the FTR was introduced in [11]. The FTR distributioncan describe two specular components and numerous NLOS components. Unlike the TWDP model, the FTRmodel’s PDF, cumulative distribution function (CDF), and moment generating function (MGF) are obtainedin closed-form; however, they are not easy to reproduce and manipulate mathematically.31.1. Background and Motivation1.1.1 Paradigm shift in wireless communicationThe radio-frequency (RF) band for wireless communication is congested, while optical wireless commu-nication (OWC) systems have safety and implementation concerns. The mmWave and THz communicationsystems and the use of femtocells and WPC systems are emerging communication systems proposed as solu-tions to these two problems; however, there are unresolved challenges within these systems.Most wireless systems are built on RF technologies, with frequencies spanning 3 kilohertz (kHz) to 300gigahertz (GHz). These systems have seen rapid growth, causing wireless communication data rate require-ments to double every 18 months. Modulation and coding techniques have been explored to accommodatehigher data rates in this band, but the capacity limitation is a significant concern. The (unlicensed) frequencieswithin the THz band, spanning 300 GHz to 10 THz, and the mmWave band, spanning 30 GHz to 300 GHz,are sufficient to support a high data rate without compromising safety.The THz band offers a larger bandwidth than the mmWave band. The THz wavelength, spanning 1millimeter (mm) to 30 micrometer (µm), and the mmWave wavelength, spanning 0.01 meter (m) to 1 mm, bothsuffer similar challenges, i.e., signals from both frequency bands exhibit the same characteristics when theypropagate in an environment. But, the THz waves are more impacted because of their smaller wavelengths.Hence, the THz band is proposed for indoor communication only.The use of femtocells in homes and offices has promised a better quality-of-service (QoS) for users. Thefemtocells are provided by mobile network operators but use the owner’s broadband internet connection. ThisQoS is only guaranteed if (i) the broadband internet connection is good, (ii) there is no interference withneighboring femtocells, and (iii) there is a seamless handoff when the user crosses from one cell to another.These reasons present a unique challenge that is not found with traditional macrocells.The use of the WPT technology has been shown to improve wireless communication devices’ uptime.WPC systems have risen as a paradigm shift in networking, improving wireless communication systems’performance by exploiting simultaneous wireless information and power transfer (SWIPT) technologies. Be-sides, WPT can reduce operational costs and sustain the rapid increase in connected devices.Certain phenomena are of little concern in the RF band but are predominant in the mmWave and THzbands, e.g., reflection loss, spreading loss, and molecular absorption loss [12]. Reflection loss occurs due tothe signal impinging on a surface that causes part of its energy to return to the source. Spreading loss occursdue to a decrease in the signal amplitude as it spreads out from the source, while molecular absorption lossoccurs due to the photons’ energy being lost to the molecules in the medium. There is a need to use highly41.1. Background and Motivationdirectional antennas and multiple-input-multiple-output (MIMO) schemes to reduce these losses’ combinedeffect. Highly directional antennas usually lead to multiple LOS and NLOS signals. Molecular absorptionplays an essential role in channel modeling because it increases with distance and frequencies.1.1.2 Challenges of emerging trends in communicationAlthough the above-mentioned emerging communication systems feature several advantages comparedto the traditional RF wireless technologies, these systems are not currently as widespread as RF wirelesstechnologies for three main reasons.The first reason relates to the transceiver architecture and its availability. Transceivers suitable for usein mmWave and THz bands are more expensive than the RF transceivers. These transceivers must exploitthe large bandwidth promised by mmWave and THz bands and provide high power, high sensitivity, and lownoise figure [12]. The following technologies have risen in response to the transceiver problem (i) silicon ger-manium (SiGe) complementary metal-oxide-semiconductor (CMOS) and BiCMOS technologies, (ii) galliumnitride, indium phosphide, and metamorphic technologies, (iii) photonic and plasma waves technologies, and(iv) graphene technologies.The second reason relates to the reliability of having mmWave and THz propagation links available allthe time. Weather-induced channel impairments degrade these communication systems’ performance andpose a significant constraint on the communication distance [12]. This performance degradation results fromspreading and absorption losses, which combine to become the propagating wave’s pathloss.The third reason relates to channel modeling. Channel models are mathematical representations of channelimpairments’ effect on the propagating signal. These channel models are essential in designing and analyzingwireless communication systems; therefore, they must adequately describe the communication environment[13]. The channel models employed in the RF frequency bands are not applicable in the mmWave and THzbands because of the weather-induced losses. Also, the channel models used with macrocells become insuf-ficient in the femtocells because the former is used for outdoor communication while the latter is used forindoor communication. Based on mmWave and THz communication systems’ potential applications, the cor-responding channels can be of three types - intra device, indoor, and outdoor [14]. The intra device channelhas not received much attention from researchers, so we will not discuss it here. The indoor propagating chan-nel relates to the indoor environments where phenomena like reflection, scattering, diffraction, and frequencydispersion are prominent [15] - [19]. Detailed experiments have been carried out in the indoor environment51.2. Literature Reviewto obtain channel measurements, and the results have been used to validate the propagation phenomena usingtheoretical models. The outdoor environment features phenomena like reflection, either from the ground orthe walls of a building, atmospheric conditions like fog and rain, causing signal attenuation, and wind, whichaffects the antennas’ directivity. Therefore, it becomes crucial to have a channel model that is versatile andmeaningful in describing the communication environment accurately.This thesis focuses on two statistical models – the Beaulieu-Xie (BX) fading model and the shadowedBeaulieu-Xie (SBX) model in support of their understanding and future application to emerging wirelesssystems. This thesis is built on the following goals:1. Investigate the dynamic performance, i.e., the level crossing rate (LCR) and average fade duration(AFD) of wireless links characterized by the BX fading model for single and diversity receptions.2. Study the performance of the BX fading model for correlated and independent channels to observe theeffect of certain channel characteristics (e.g., physical separation between fading channels).3. Study the performance of the BX fading model in the presence of multiple co-channel interference(CCI) with a focus on emerging short-range random access channels.4. Investigate the performance of the SBX model and validate it with experimental data. Such work cansupport the future application of this new model in THz and mmWave communication systems and theuse of femtocells.5. Investigate the performance of wireless powered communication networks (WPCNs) with considerationto BX channels with nonlinear energy harvesters.1.2 Literature Review1.2.1 Multipath FadingMultipath fading describes the variation of a signal’s amplitude and phase over a short period, usually froma small fraction of the wavelength. Fading is, therefore, classified based on the parameters of the transmittedsignals (i.e., signal bandwidth and symbol period) and parameters of the fading channels (i.e., delay spreadand Doppler spread).We define the signal bandwidth as the difference between the highest- and lowest-frequency signal com-ponents. The symbol period is the reciprocal of the signal bandwidth. Delay spread describes the time61.2. Literature Reviewdispersive nature of the fading channel, i.e., the time difference between the arrival of the first and last sig-nificant multipath components. The coherence bandwidth describes the frequency interval over which twofrequencies of a signal experiences correlated or comparable fading. It is expressed as the reciprocal ofthe root-mean-square (RMS) delay spread. If the signal bandwidth is larger than the coherence bandwidth,then the channel is said to be frequency-selective. In a frequency-selective fading channel, channel ampli-tude varies widely across the signal bandwidth, i.e., the frequency components experience different gains andphase shifts. However, if the signal bandwidth is smaller than the coherence bandwidth, then the channelis said to be frequency-nonselective, otherwise known to be flat fading. The channel amplitude is the sameacross the signal bandwidth for flat fading.The channel impulse response observations remain unchanged over a period called coherence time. Thecoherence time is the reciprocal of the Doppler spread, which describes the fading channel’s frequency dis-persive nature. If the symbol period is larger than the coherence time, then the channel is said to be fast fading.On the contrary, if the symbol period is smaller than the coherence time, the channel is said to be slow fadingAs described above, the delay spread and Doppler spread are two independent mechanisms that can col-lectively categorize fading channels into the fast frequency-selective fading channel, slow frequency-selectivefading channels, fast frequency-nonselective fading channel, and slow frequency-nonselective fading channel.If the channel delay spread is small compared to the reciprocal of the signal bandwidth, all the signalcomponents (i.e., LOS and multipath components) are non-resolvable. Hence, we have a narrowband fadingchannel. Otherwise, we have a wideband fading channel.1.2.2 ShadowingShadowing occurs due to the presence of obstacle(s) in the signal’s path. The receiver’s location mostlydetermines the presence of these obstacles. Shadowing is random in nature, and it causes the received signal’spower to deviate from the mean value. Thus, shadowing is modeled together with the pathloss to capturepower loss due to signal travel and random attenuation due to signal blockage [20].The cell coverage area shows the combined effect of pathloss and shadowing, while the shadowing levelspecifies the received power’s applicable contour—the random nature of shadowing results in an amorphouspower contour with users experiencing different power levels. When shadowing is averaged, the contourbecomes a perfect circle. The perfect circle contour limits the communication system because users at anequal distance from the center may still experience different signal power levels.71.2. Literature ReviewFigure 1.2: The cellular concept using hexagonal cells.1.2.3 InterferenceThe cellular concept is a solution to the spectral congestion problem, which results from many usersaccessing the same EM spectrum [21]. Network engineers employ the cellular concept to maximize spectralefficiency (i.e., accommodate more users) while keeping the QoS acceptable. The cellular concept is an ideawhere the coverage area is divided into cells. These cells are allocated channels available to the entire systemsin a design process referred to as frequency reuse. A different set of channels is assigned to nearby cells, andthe same set of channels are reused at another cell that is sufficiently far away. The distance between cells thatuse the same set of channels is called the reuse distance. The reuse distance must be small for high-frequencyreuse efficiency but large for tolerable interference. Figure 1.2 shows the basic cellular concept where thehexagonal cell shape is employed for design purposes. In reality, the cell shape is amorphous, leading to aphenomenon called adjacent channel interference. In Fig. 1.2, the co-channel cells are labeled with the sameletter. Interference between co-channel cells is called co-channel interference. Co-channel interference (CCI)is a major source of performance degradation in microcellular systems, and it is analyzed in this thesis.81.2. Literature Review1.2.4 Envelope Distribution of Received SignalsWe consider a transmitted signal of the form [20]s (t) = <{u (t) exp [j (2pifct+ φ0)]} (1.1)where <{·} is the real value operator, u (t) is the complex envelope of the signal, fc is the carrier frequency,and φ0 is the phase. The received signal is a combination of the LOS and multipath components, which hasthe form [20]r (t) = <{N∑n=0αn (t)u (t− τn (t)) e[j2pi(fc(t−τn(t)))+φDn+φ0]}(1.2)where αn (t) is the amplitude, φDn is the Doppler shift, τn is the delay of the nth multipath component, andn = 0 corresponds to the LOS path. The varying amplitude, αn (t), due to the sum of multipath componentsthat may be received in- or out-of-phase, can best be described by statistical models.Clarke used EM field representations to develop the first model based on the statistical nature of thereceived signal [22]. The model assumes that the transmitter is stationary while the receiver is mobile. Theelectric field incident on the mobile antenna consists of N plane waves with the same mean amplitude. The zcomponent of the electric field, Ez , is expressed asEz =N∑n=1En exp (jφn) = R exp (jθ) = X + jY (1.3)where En is the amplitude and φn is the uniformly distributed phase of the nth arriving waves. When N issufficiently large, we can invoke a central limit theorem (CLT) theorem. Therefore, X and Y are GaussianRVs with mean and variance asE [X] = E [Y ] = 0; σ2x = σ2y =Ω2(1.4)where E [·] is the expectation operator and E [XY ] = 0. The envelope of R is Rayleigh distributed asfR (r) =2rΩexp(−r2Ω), r ≥ 0 (1.5)where E[R2]= Ω and the phase distribution is uniform. The Rayleigh distribution has been successful forcharacterizing urban transmission environment with no LOS components. In the case where there exists a91.2. Literature ReviewLOS component in the transmission environment, the envelope of R is Ricean distributed asfR (r) =2rΩexp(− (r2 + λ2)Ω)I0(2rλΩ)(1.6)=2r (K + 1)Ωexp(−K + (K + 1) r2Ω)I0(2r√K (K + 1)Ω), r ≥ 0where I0 (·) is the zero-order modified Bessel function of the first kind [23] and the parameter K is called theRicean K-factor, which is defined as the ratio of the power in the LOS component to the power in the NLOScomponents, i.e.,K =λ2Ω. (1.7)When the power of the LOS component is zero, then the Ricean distribution becomes the Rayleigh distribu-tion. Also, the phase of the Ricean distribution is not uniform; it takes on a more complicated form.The Nakagami-m distribution [7] has been successfully used to characterize transmission environmentwith multiple NLOS components and no LOS component. It is derived from the central chi distribution vianormalization and takes the formfR (r) =2Γ (m)(mΩ)mr2m−1 exp(−mΩr2), r ≥ 0, m ≥ 12(1.8)where E[R2]= Ω and m is the fading (severity) parameter defined as m = (E[R2])2Var[R2]. The Nakagami-m distribution includes special cases for different values of m. When m = 1, it becomes the Rayleighdistribution. When m = 0.5, it becomes the one-sided Gaussian distribution. When m→∞, the Nakagami-m PDF tends to an impluse located at√Ω.The BX distribution [24] has been introduced for flexible representation of transmission environmentswith a multitude of LOS components (via direct propagation and specular reflections) and NLOS compo-nents. This is possible because of its three parameters that can control different aspects of the PDF. The BXdistribution has also been shown to unify existing fading models like the Ricean (and generalized Ricean orκ − µ distribution) and the noncentral chi distribution. Its PDF is obtained from the noncentral chi distribu-tion via normalization in the same manner by which the Nakagami-m PDF was obtained from the central chidistribution.We consider a noncentral chi distributed RV, Z, which is related to k independent normally distributed101.2. Literature ReviewRVs, Xi, i = 1, 2, . . . , k, with means µi and variances σ2i . Then Z is expressed asZ =√√√√ k∑i=1(Xiσi)2. (1.9)The noncentrality parameter λ relates to the mean µi asλ =√√√√ k∑i=1(µiσi)2. (1.10)Thus, the PDF of Z is given asfZ (z; k, λ) =exp(− z2+λ22)zkλ(λz)k2I k2−1 (λz) (1.11)where k specifies the degree of freedom, λ is the noncentrality parameter, and In (·) is the nth-order modifiedBessel function of the first kind. To obtain the BX PDF, we normalize the RV Z by√2mΩ and make k = 2mto havefR (r;m,λ,Ω) =exp(−mΩ (r2 + λ2)) rm (2mΩ )λm−1Im−1(2mΩλr)(1.12)where λ2 is the power in the LOS components (via direct propagation and specular reflections) and Ω isthe power in the NLOS components. The K-factor for the BX distribution is defined as K = λ2Ω and isindependent of the fading parameter, m. It should be noted the noncentrality parameter, λ, in (1.11) is alsoscaled by√2mΩ , i.e., λ∗ = λ ∗√2mΩ . The PDF in (1.12) has its shape controlled by m, its spread controlledby Ω, and the location and height of the mode controlled by λ. The BX distribution includes some specialcases for different values ofm and λ. Figure 1.3 shows the relationship between the BX distribution and otherunimodal distributions.We show the effect of the BX parameters on its PDF. Figure 1.4 shows the effect of the parameter m onthe PDF. We see that as m increases, the PDF’s shape takes a slimmer form. Thus, the BX’s PDF becomesan impulse as the value of m approaches infinity. Appendix A has the proof of the impulse’s location for theBX distribution as the value of m approaches infinity. Figures 1.5 and 1.6 show the effect of the power in theLOS and NLOS components, respectively, on the BX PDF.111.2. Literature ReviewNon-central parameter, λ = 0Nakagami-mDistributionBealieu-XieDistribution m = 1RiceanDistribution Rayleigh Distributionm = 1Non-central chi-distributionCentral chi-distributionNon-central parameter, λ = 0Figure 1.3: Relationships between the Rayleigh, Ricean, Nakagami-m, and BX distributions. The parametersm and λ represent the fading severity parameter and noncentrality parameter, respectively.1 2 3 4 5 6 7 8 900.10.20.30.40.50.60.70.8Figure 1.4: The BX PDF versus r for different values of m with paramaters Ω = 5 and λ = 3.121.2. Literature Review0 1 2 3 4 5 6 7 8 900.10.20.30.40.50.60.70.8Figure 1.5: The BX PDF versus r for different values of Ω with paramaters m = 2 and λ = 3.0 1 2 3 4 5 6 7 8 900.10.20.30.40.50.6Figure 1.6: The BX PDF versus r for different values of λ with paramaters m = 2 and Ω = 5.131.2. Literature ReviewThe CDF of the BX distribution, defined as FR (r) =∫ r0 fR (t) dt, is obtained asFR (r) =∫ r0exp(−mΩ (r2 + λ2)) rm (2mΩ )λm−1Im−1(2mΩλr). (1.13)Applying the definition of the generalized Marcum Q-function defined as [39]QM (a, b) =∫ ∞bt(ta)M−1exp(− t2 + a22)IM−1 (at) dt, (1.14)we haveFR (r) = 1−QM(√2mΩλ,√2mΩr). (1.15)The nth-order moment is defined as [25]E [Rn] =∫ ∞0rnfR (r) dr. (1.16)After some manipulations, as shown in Appendix B, the nth-order moment for the BX distribution is expressedasE [Rn] =(mΩ)−n2Γ(1 +n2)L(m−1)n2(−mΩλ2)(1.17)where L(a)n (·) is the nth-order Laguerre polynomial [23].1.2.5 Energy HarvestingThe uptime of wireless communication networks has been dramatically impacted by the lifetime of thebatteries powering them. As such, energy harvesting (EH) has risen as a viable avenue to keep these networksoperational via a dedicated device scavenging energy from renewable energy sources [26]. These energysources can either be controllable (e.g., RF signals) or uncontrollable (e.g., wind and solar). For the RFsignals, a steady energy source can be provided by exploiting their far-field EM waves [27]-[29]. This meansthat RF signals can carry both information and energy simultaneously, leading to a technology called SWIPT.High-frequency signals from the mmWave and THz bands have shown potential to present a more reliableenergy source for EH [30]. This reliable energy source is due to the extensive spectrum of available frequen-141.3. Thesis Contributionscies and signal strength due to the presence of LOS signals (via direct propagation and specular reflections).However, the role of specific propagation mechanisms is not yet fully understood [31]. Since communica-tion in these high-frequency bands is successful at close range, it is intuitive that EH will be successful inthese bands. This is motivated by the large-dimensional array of beamforming antennas in mmWave and THzsystems because of their smaller wavelength size.Another challenge in EH is the energy harvester (or EH model), which earlier works considered linear.Experimental results have faulted this assumption of linearity [32]-[34]. It is shown via those experimentalresults that there exists a saturation point, beyond which the harvested energy remains unchanged. Thus, non-linear energy harvesters have been introduced to capture the EH circuit’s nonlinear characteristics. Examplesof these nonlinear EH models include the piece-wise linear function [35], the polynomial model [36], and thesigmoid function [37]-[38].There are suggested practical ways of increasing the harvested energy. One such way is to increase theefficiency by which the EH circuit converts the ambient energy from RF, mmWave, and THz wave signals toelectrical energy. Thus, many works have studied how to improve the EH circuit’s efficiency by consideringthe hardware architecture.Ultimately, we see that EH will be an effective solution for emerging technologies like the internet-of-things (IoT) where the powering of many devices is currently a major challenge.1.3 Thesis ContributionsThis thesis’s key contribution is the analysis of the new statistical models that are proposed to characterizethe transmission environments of emerging wireless communication systems. Specifically, we make the fivecontributions.− Propose a fading model for emerging wireless communication systems: This contribution presents a newcomposite distribution called the SBX model. Based on the recently proposed BX fading model, theSBX model has four key parameters that can quantify different propagation parameters. The FTR modelis currently the best in representing fading channels of mmWave communication systems; however, itsPDF is cumbersome and difficult to manipulate. In contrast, the SBX’s PDF is in terms of mathemati-cal functions that are easily reproducible. The key findings here are as follows. First, the mathematicalmanipulations reveal that the SBX model includes unimodal distributions (e.g., the Rayleigh, Ricean,and Nakagami-m models) and composite distributions (e.g., the SR and FTR models) as special cases.151.3. Thesis ContributionsSecond, the analysis of the nth-order moment shows that the BX’s K-factor estimator does not applyto the SBX fading model, even though they have the same origin. Third, the SBX model has distinctparameters that can quantify the amount of fading and shadowing in a communication environment, aproperty that is not in any existing statistical model.− Investigate the dynamic performance of the BX fading model: This contribution provides the exact clo-sed-form expression for the LCR and AFD for both single and diversity receptions. The PDF andcharacteristic function (CHF) methods are employed for single and diversity receptions, respectively.The key finding here is that the BX fading model can accurately characterize an environment withmultiple LOS components (via direct propagation and specular reflections) and NLOS components,thus, presenting a better performance result.− Investigate the performance of the BX fading model for correlated channels using bounds: This contri-bution is a closed-form expression for the error rates and outage probability (OP) for diversity tech-niques of maximal ratio combining (MRC), equal-gain combining (EGC), and selection combining(SC) for correlated BX fading channels. The key findings here are as follows. First, the closed-formexpression for the asymptotic, lower, and upper bounds for the error rates and OP are obtained usinga newly introduced technique. Second, the results obtained for the BX fading model include the re-sults of the Ricean and Nakagami-m fading models as particular cases. Third, the results obtained forcorrelated and independent channels present insights into wireless links’ performance in large antennaarrays.− Investigate the performance of BX fading model in the presence of CCI: This contribution presents theperformance of short-range random-access channels. Here, series-form expressions are derived forthe bit-error rate (BER) and OP of the BX fading model. We consider binary frequency shift-keying(BFSK), binary phase-shift keying (BPSK), and quadrature amplitude modulation (QAM) schemes withMRC diversity in the presence of multiple interferes. The key findings here are as follows. First, theBER and OP results provide insights into the possible performance of short-range communication sys-tems found in emerging mmWave and THz systems. Second, the BER and OP results for the BX fadingmodel include the results for the Nakagami-m fading as a particular case. Third, the communicationsystem’s diversity order can be derived from the results of the BER or the OP.− Investigate the performance of WPC systems with nonlinear energy harvesters: This contribution pre-161.4. Thesis Organizationsents the performance of WPC systems over the BX fading channels. The system model consists of awireless device that scavenges energy from the energy source and uses it to transmit information to thedata fusion receiver. The energy source and data fusion are assumed to have multiple antennas, whilethe wireless device has only one antenna. We introduce two new EH models/energy harvesters; themulti-segment piecewise linear (MSPwL) and the arctangent (ArT) energy harvesters. Both models arevalidated with published measured data. The key findings here are as follows. First, the fits our newnonlinear EH models are better than those of existing nonlinear EH models. Second, the simulation andanalytical results closely matched.1.4 Thesis OrganizationThis thesis is organized into seven chapters. We summarize the content of each chapter in the followingparagraphs.Chapter 1 presents a brief background on the paradigm shifts in wireless communication. Here, wedescribe the pending research challenges and proposed solutions. We also present the research objective and acomprehensive review of the statistical channel models related to this thesis. Finally, the overall contributionsof the thesis are briefly summarized.Chapter 2 presents the second-order statistics of the BX fading model with consideration to single anddiversity receptions. We employ the traditional PDF method to derive the system’s dynamic performance forsingular reception, while the CHF method is used for diversity receptions. Ultimately, we show that the BXfading model can adequately capture the presence of multiple LOS components (via direct propagation andspecular reflections) in transmission environments.In Chapter 3, we investigate the performance of the correlated and independent BX fading channels usingbounds. We carry out this analysis to see the combined effects of LOS and NLOS signals, the fading parame-ter,m, and various spatial separations between the fading channels. A newly developed bounding technique isemployed to obtain a closed-form asymptotic, lower, and upper bounds. The method used involves transform-ing correlated BX RVs to correlated Gaussian RVs. Ultimately, our performance bounds show the primaryfactors affecting wireless communication systems’ performance.Chapter 4 focuses on the performance of communication systems in the presence of interference. Weconsider the MRC diversity technique because its results are close to the optimum combining results, idealfor an interference-limited environment. Our system model assumes that both the amplitude of the desired171.4. Thesis Organizationand interfering signals are BX distributed. Our analyses show the effect of diversity, fading severity, andmodulation technique on the system’s performance.We propose a new composite statistical model for emerging wireless communication systems in Chapter 5.This model, which we refer to as the SBX model, is based on the BX fading model. The SBX’s uniqueness isits four meaningful parameters, which distinctly quantify the fading severity, shadowing severity, power of theLOS, and NLOS components. We carry out extensive performance analysis on this new model and validate itusing experimental data from the mmWave communication environment. The goodness-of-fit shows that theSBX model can describe the emerging system’s communication environment; however, more experimentalvalidation of the model is required.Chapter 6 shows the performance analysis of WPC over the BX fading channels with nonlinear energyharvesters. We employ the BX fading model to characterize an environment with multiple LOS componentsfound in close-range communication. We introduce two nonlinear EH models, MSPwL and ArT, and validateboth models using measured data.Finally, Chapter 7 concludes this thesis by outlining the major findings of this work and proposing futureresearch topics. We note here that any value expressed in dB throughout this thesis is a logarithm of a powerratio and not units of power in the logarithmic system of numbers.18Chapter 2The Second-Order Statistics of the BXFading ModelIn this chapter, we analyze the level crossing rate (LCR) and average fade duration (AFD) of the BXfading model. The PDF method is employed to derive the LCR and AFD for the singular reception, while theCHF method is used to derive the LCR for the diversity scheme using MRC. The LCR and AFD of the BXfading model present improved performance levels compared to the Ricean and Nakagami-m fading modelswhen characterizing environment with LOS components (via direct propagation and specular reflections) andNLOS components.The organization of this chapter is as follows. We present the motivation for carrying out the BX fadingmodel’s dynamic performance in Section 2.1. In Section 2.2 and 2.3, we derive the LCR and AFD for the BXfading model, respectively. The LCR expression for the diversity combining of MRC is derived in Section2.4. Section 2.5 presents the numerical results, and Section 2.6 makes some concluding remarks.2.1 MotivationThe Ricean fading model is widely accepted due to its ability to characterize wireless systems with adominant LOS and multiple NLOS components. However, the Ricean fading model has limited flexibility incharacterizing the broad range of signal fades from reflections and shadowing. In contrast, the Nakagami-m distribution has been an effective fading model because it includes a flexible fading parameter, allowingit to be adapted for varying severity levels in signal fades. However, attempts to apply the Nakagami-mfading model to wireless systems with LOS and NLOS channels have been unsuccessful. This is because theNakagami-m fading model is a normalized form of the central chi distribution, which cannot define an LOSchannel [4]-[6].Because of the benefits and challenges of the Ricean and Nakagami-m fading models mentioned above,192.2. Level Crossing Ratethese models were recently merged into a new fading model by Beaulieu and Xie [24]. The BX fading modelpossesses the benefits of the Nakagami-m fading model in that it has a flexible fading parameter m. The BXfading model also contains the benefits of the Ricean fading model in that it is derived from the noncentralchi distribution and allows for the characterization of both LOS and NLOS components.The LCR and AFD are computed to evaluate the BX fading model’s dynamic performance. The LCR andAFD are key the second-order statistics that define the quality of the received signal when the transmitter orreceiver is moving [40]. The AFD is especially critical in understanding the statistics of error bursts [41]. TheCHF method is used to analyze the LCR for diversity reception via MRC [42]. Such a technique overcomesthe analytical complexities of derivations employing the joint PDF. We conclude that the LCR and AFDobtained for the BX model present improved performance levels compared to the Ricean and Nakagami-mfading models when characterizing an environment with both LOS components (via direct propagation andspecular reflections) and NLOS components.2.2 Level Crossing RateThe LCR and AFD are two key metrics to characterize dynamic performance in mobile communicationsystems [40]. The rate of change of the received signal in mobile communication is related to the crossing(signal) level2 and velocity [43]. The LCR, NR(r), is the average number of times a fading signal crosses agiven signal level, r, within a certain period, t. Here, the envelope of the received signal, R (t), is subjectedto BX fading with the PDF in (1.12). The LCR is expressed asNR(r) =∫ ∞0r˙fR,R˙(r, r˙)dr˙ =∫ ∞0r˙fR (r) fR˙ (r˙) dr˙ r ≥ 0 (2.1)where fR,R˙ (r, r˙) is the joint pdf of R (t) and its time derivative, R˙ (t), and the second equality follows whenR(t) is assumed to be independent of R˙(t). Hereafter, we will omit t for simplicity. For single branchreception, the PDF of the time derivative of the received signal envelope, R˙ (t), is [44]fR˙ (r˙) =√12piσ˙2r˙exp{− r˙22σ˙2r˙}(2.2)2Crossing level and signal level are used interchangeably in this chapter.202.3. Average Fade Durationwhere σ˙2r˙ is the time derivative of the variance of R˙ (t). The variance of R˙ (t) for the Ricean fading model is[44]σ˙2r˙ = 2pi2f2mσ2 (2.3)where 2σ2 is the Ricean power of the NLOS component(s) and fm is the maximum Doppler frequency. Thetransformation that unifies the Ricean fading model to the BX fading model is s = σ√2mΩ λ [24], where s2 isthe Ricean LOS power, λ2 is the BX LOS power, and Ω is the BX NLOS power. Then, σ˙2r˙ for the BX fadingmodel isσ˙2r˙ =pi2f2mΩm. (2.4)We insert (2.4) into (2.2), and then into (2.1) to obtainNR(r) =pifm√Ω√2pim∗ fR (r) . (2.5)Inserting (1.12) in (2.5) yields the LCR for the BX fading model asNR(r) =pifm√Ω√2pimexp(−mΩ (r2 + λ2)) rm (2mΩ )λm−1Im−1(2mΩλr). (2.6)2.3 Average Fade DurationThe AFD, TR (r), is the average time that the received envelope, R (t), remains below a given crossing(signal) level, r, after crossing it in the negative direction [44]. The AFD isTR (r) =FR (r)NR (r)=∫ r0 fR(t)dtNR (r)(2.7)where FR (r) is the CDF of R (t). The integral in (2.7) is expressed as [24]∫ r0fR(t)dt =∫ r0exp(−mΩ (t2 + λ2)) tm (2mΩ )λm−1Im−1(2mΩλt)dt (2.8)and ultimately given asFR (r) = 1−QM(√2mΩλ,√2mΩr)(2.9)where QM (·, ·) is the generalized Marcum Q-function.212.4. Level Crossing Rate Expression for Maximal Ratio Combining2.4 Level Crossing Rate Expression for Maximal Ratio CombiningDiversity combining techniques can counteract variations in the envelope of a received signal in wirelesssystems. In [45], the LCR was calculated using a CHF method, as an alternative to the standard PDF method.Using this approach, we define the LCR as the expected number of occurrences per unit time that a stationaryrandom process R (t) crosses the signal level r. This is expressed using Rice’s formula as [45]E [Nr {R (t)}] =∫ ∞−∞|r˙|fR,R˙ (r, r˙) dr˙ (2.10)where E [·] denotes the expectation operation and |·| denotes the absolute value operator, which is obtainedfrom its joint CHF. Note that |r˙| = r˙sgn (r˙), where sgn (·) is the signum function. The joint CHF of R (t)and R˙ (t) is defined asΦR,R˙(ω1, ω2) = E[e(jω1R+jω2R˙)](2.11)where j2 = −1. We rewrite (2.11) using Fourier transforms asE [Nr {R (t)}] = 12pi∫ ∞−∞F [r˙sgn (r˙)]{F[fR,R˙ (r, r˙)]}∗dω2 (2.12)whereF [·] is the Fourier transform operator, ∗ is the complex conjugate operator, andF [sgn (r˙)] =2jω2(2.13a)F[fR,R˙ (r, r˙)]=12pi∫ ∞−∞ΦR,R˙(ω1, ω2)e−jω1rdω1. (2.13b)Substituting (2.13a) and (2.13b) into (2.12) gives [42], [46]E [Nr {R(t)}] = − 12pi2∫ ∞−∞∫ ∞−∞ΦR,R˙(ω1, ω2)− ΦR(ω1)ω22e−jω1rdω1dω2. (2.14)We define the total instantaneous signal-to-noise ratio (SNR) at the output of MRC diversity system asR =L∑l=1Rl (2.15)where L is the number of branches and Rl is the instantaneous SNR of the lth branch. Here, Rl is expressedas222.4. Level Crossing Rate Expression for Maximal Ratio CombiningRl =k∑i=1X 2l,i (2.16)assuming unity noise variance, where k = 2m signifies the degree of freedom for the diversity system andm is the fading parameter. For the lth branch, Xl,i’s are identically and independently distributed (i.i.d.)Gaussian RVs with equal mean µl and equal variance al. Thus, the joint CHF for independent branches canbe written asΦRl,R˙l(ω1, ω2) = E[exp(jω1Rl +jω2R˙l](2.17)ΦRl,R˙l(ω1, ω2) = E[exp(jω1(X2l,1 + X 2l,2 + · · ·+ X 2l,k) +jω2(2Xl,1X˙l,1 + 2Xl,2X˙l,2 + · · ·+ 2Xl,kX˙l,k))].For each l, X˙l,i’s are Gaussian RVs with zero mean and equal variance bl. Equation (2.17) can also beexpressed with the following vectors and matrices:Ml =[µl,1 0 µl,2 · · · 0]T1×2kD =ω1 0 0 · · · 0 ω20 0 0 · · · ω2 00 0 ω1 . ..0 00 0 ω2. . . 0 00 ω2 0 · · · ω1 0ω2 0 0 · · · 0 02k×2kCl =al 0 0 0 0 00 bl 0 0 0 00 0 al 0 0 00 0 0. . . 0 00 0 0 0 al 00 0 0 0 0 bl2k×2k232.5. Numerical Resultswhere the superscript T denotes the transpose, Ml is a mean vector, and Cl is a covariance matrix. Wecontinue by expressing (2.17) as [42], [46]ΦRl,R˙l(ω1, ω2) =exp(−MTl C−1l [I−(I−j2ClD)−1]Ml2)√det(I− j2ClD)(2.18)where I is a 2k-by-2k identity matrix and det(·) denotes the determinant. Substituting the vectors and matricesinto (2.18) and applying mathematical manipulations, we obtainΦRl,R˙l(ω1, ω2) =exp(−4µ2 (2blω22−jω1)(1+4alblω22−j2alω1))(1 + 4alblω22 − j2alω1)m(2.19)for the BX fading model. We assume that the instantaneous SNR of each branch is independent and writeΦR,R˙(ω1, ω2) =L∏l=1ΦRl,R˙l(ω1, ω2). (2.20)The LCR is then obtained by substituting (2.20) into (2.14).2.5 Numerical ResultsThe BX fading model is simulated using Matlab to show the effects of various parameters and comparethe observations to those of the existing Ricean fading model.Figure 2.1 shows the normalized LCR curves of the BX, Ricean, and Nakagami-m fading models as afunction of the crossing level, r, for different values ofm. It is seen that the normalized LCR of the BX fadingmodel approaches that of the Ricean fading model as m is reduced to 1, corresponding to increased fadingseverity at the same LOS and NLOS powers. Figure 2.2 shows the effect of the K-factor on the LCR of theBX fading model. An increase in the K-factor value corresponds to an increase in the LOS power, whichleads to lower LCR values.Figures 2.3 and 2.4 show the LCR and AFD as a function of the crossing level, r, for vehicular speeds upto 100 km/h. We observe that the LCR and AFD are dependent on the mobile device’s speed, i.e., maximumDoppler shift frequency. The LCR increases as fm becomes large while AFD decreases as the fm becomeslarge. For the LCR, we observe that an increase in the mobile device’s speed leads to the signal crossingthe specified threshold frequently, as expected. On the other hand, an increase in the mobile device’s speed242.5. Numerical Results-15 -10 -5 0 5 1010-1010-810-610-410-2100Figure 2.1: The normalized LCR curves of the BX, Ricean and Nakagami-m fading models versus crossinglevel, r. The BX, Ricean and Nakagami-m fading model curves have a total power of 4.88 dB. The BX andNakagami-m curves are shown for fading parameters of m = 1.5 and 2.-15 -10 -5 0 5 1010-1010-810-610-410-2100102Figure 2.2: The normalized LCR curves of the BX fading model versus crossing level, r, with a K-factor of0, 6.99 and 10 dB.252.5. Numerical Results-15 -10 -5 0 5 1010-1610-1410-1210-1010-810-6Figure 2.3: The LCR curves of the BX fading model versus crossing level, r, for vehicular speeds of v = 5,30, 80, and 100 km/h, with the parameters of m = 1.5, λ2 = 7 dB, and Ω = 0 dB.-10 -8 -6 -4 -2 0 2 4 6 8 1010-5100105Figure 2.4: The AFD curves of the BX fading model versus crossing level, r, for vehicular speeds of v = 5,30, 80, and 100 km/h, with the parameters of m = 1.5, λ2 = 2.6 dB, and Ω = 1 dB.262.5. Numerical Results-10 -8 -6 -4 -2 0 2 4 6 810-310-210-1100101102103Figure 2.5: The AFD curves of the BX fading model versus crossing level, r, for fading parameter, m = 1, 2,3, and 4.reduces the AFD value. This is because the mobile user spends less time below the specified threshold. Figure2.5 shows the AFD as a function of the crossing level, r, for the fading parameter, m.Figures 2.6, 2.7, and 2.8 are all plotted using the LCR expression in (2.20). Figure 2.6 shows the LCR as afunction of the crossing level, r, at the output of a three-branch MRC combiner for Ricean, Nakagami-m, andBX fading models. Figure 2.7 shows LCR curves for fading parameters of m = 1.5, 2, 3, and 4. We recordeda reduction in the LCR value as m increases. This reduction signifies that the frequency at which the signalcrosses the specified threshold reduces as fading reduces. Figure 2.8 shows the LCR curves for i.i.d. andnon-i.i.d. branches. Here, the i.i.d. branches all have Ricean-distributed envelopes. The non-i.i.d. brancheshave a BX distributed envelope in the first branch and Ricean-distributed envelopes in the second and thirdbranches. The average power per branch is 7.64 dB, and the K-factor is 10 dB for the i.i.d. case. The averagepowers are 7.64, 7.58, and 6.71 dB for the first, second, and third branches in the non-i.i.d., respectively. Anotable difference between the non-i.i.d. and i.i.d. cases is apparent here. This can be a key consideration insystem design [42].272.5. Numerical Results0 2 4 6 8 10 12 14 16 18 2010-1100Figure 2.6: The LCR curves of the Ricean, Nakagami-m and BX fading models versus crossing level, r, at theoutput of a 3-branch MRC combiner using the CHF method, with an average power of 7.64 dB per branch.5 10 15 2010-1100Figure 2.7: The LCR curves of BX fading versus crossing level, r, with fading parameters of m = 1.5, 2, 3,and 4.282.6. Chapter Summary0 2 4 6 8 10 12 14 16 18 2010-1100Figure 2.8: The LCR curves for i.i.d. and non-i.i.d. branches have a total average power of 12.41 dB and aK-factor of 10 dB per branch.2.6 Chapter SummaryThis chapter studied the LCR for the BX fading model and used the CHF method to develop an analyticalexpression for the LCR in an MRC diversity system. We analyzed the BX fading model’s performance forvarious crossing levels, LOS powers, and maximum Doppler frequencies. We compared the result for theBX fading model via simulation to those obtained for the Ricean and Nakagami-m fading models in a LOSenvironment. The simulation results show that the BX fading model presents the best characterization ofcommunication environments with multiple LOS and NLOS components. While we comment in this chapterthat the BX fading model provides a better representation of LOS plus diffuse scatter signal environment,more evidences are provided in the following chapters.29Chapter 3Performance Bounds for DiversityReceptions over BX Fading Model withArbitrary Branch CorrelationIn this chapter, we analyze the performance of the BX correlated channels using bounds. The analysesinvolve transforming the problem involving correlated BX RVs to correlated Gaussian RVs. We considerthe performance metrics of OP and error rate for MRC, EGC, and SC over arbitrarily-correlated BX fadingchannels. Closed-form expressions for upper and lower bounds to the OP and error rate are obtained andshown to be asymptotically-tight in the high SNR regime. The analytical results are verified via Monte Carlosimulations. We show that the BX fading model can be more effective than the Ricean and Nakagami-mfading models in characterizing environments with multiple LOS components (via direct propagation andspecular reflections).The organization of this chapter is as follows. The motivation for carrying out the performance analyseson BX correlated channels is in Section 3.1. In Section 3.2, we discuss the physicality (and justification) ofthe BX fading model. Section 3.3 introduces the system model for linear diversity receptions in BX fading.In Section 3.4, we show the relationship between the power correlation and Gaussian correlation coefficientswhile we derive bounds on the PDF of the channel coefficient in Section 3.5. In Sections 3.6 and 3.7, weanalyze and discuss the asymptotically-tight bounds on the OP and error rates for the three diversity schemes,respectively. Section 3.8 discusses the tightness of the bounds. Section 3.9 presents the numerical results, andSection 3.10 gives some concluding remarks.303.1. Background and Motivation3.1 Background and MotivationNumerous models have been developed to characterize wireless communication systems. The Riceanfading model has been largely employed to characterize wireless systems with LOS and NLOS signals. How-ever, the Ricean fading model has a disadvantage in its inadaptability to fading variations in the environmentand its restriction to a diversity order of one [24]. The Nakagami-m fading model, derived from the centralchi-distribution, was proposed to characterize wireless systems over different fading variations. However, theNakagami-m fading model is only suitable for systems with NLOS signals [4]-[6].The flexibility presented by the BX fading model, by way of its three parameters, makes it of interestin representations of practical fading for mmWave and THz communication systems, femtocells, and wire-less powered communication systems, where multiple LOS components (via direct propagation and specularreflections) are present [24]. Although the performance of correlated κ − µ (or generalized Ricean) fadingchannels [2] have been analyzed [49], it is not meaningful to study arbitrary correlation of such a fading modelas it does not satisfy the power constraint. In contrast, adjusting the fading parameter, m, in BX fading doesnot affect the power in the scatter component [24]. The analysis carried out in this chapter targets perfor-mance bounds to evaluate the performance—in recognition that intractable integrals or infinite series arise inthe pursuit of closed-form expressions.3.2 Physicality and Justification of the BX Fading ModelThe performance analyses put forward in this work are motivated by the BX fading model’s ability tocharacterize wireless systems with multiple LOS components (via direct propagation and specular reflec-tions) and NLOS components while satisfying the physical constraint of power conservation. In this section,we will show that the BX fading model’s ability to handle this power constraint comes about from its normal-ization—and this sets it apart from the existing κ − µ (generalized Ricean) fading model. We consider herevarious distributions for the fading models. For the κ− µ fading model, the PDF is defined for a RV of G byfG (g) =2µ (1 + κ)µ+12gˆκµ−12 exp (µκ)(ggˆ)µexp[−µ (1 + κ)(ggˆ)2]Iµ−1[2µ√κ (1 + κ)ggˆ](3.1)where µ is the positive-valued fading parameter, gˆ is the square-root of the second moment defined as gˆ =√E [G2], and κ is the ratio of total power for the LOS and NLOS components. With such definitions, d setsthe total power for the LOS components at d2 and σ sets the total power for the NLOS components at 2nσ2,313.2. Physicality and Justification of the BX Fading Modelgiven n as the number of Gaussian RVs [2]. For the generalized Ricean fading model, the PDF is defined fora RV of F by transforming (3.1) with substitutions of s2 for d2, n for µ, and√2nσ2 + s2 for dˆ to havefF (f) =fn2σ2sn−22exp(−f2 + s22σ2)× In2−1(fsσ2)(3.2)where s sets the total power for the LOS components at s2. With such definitions, K = s2/2nσ2 is theK-factor and E[F 2]= 2nσ2 + s2 is the second-order moment. Ultimately, the above definitions for κ,K, and E[F 2]all depend upon n and thus leads to two shortcomings for the κ − µ and generalized Riceandistributions: (i) the power of the NLOS components has to be adjusted to keep the total power invariant, and(ii) the fading powers of both distributions are unbounded. Nonetheless, these shortcomings can be overcomeby normalization—and this is done by the BX fading model. The BX RV, Z, can be obtained by normalizingthe generalized Ricean distribution in (3.2) such that F = Zσ√2m/Ω and s = λσ√2m/Ω. This producesthe BX’s PDF in the form offZ(z) =exp(−mΩ (z2 + λ2)) zm (2mΩ )λm−1Im−1(2mΩλz). (3.3)With such definitions, K = λ2/Ω is the K-factor given λ2 as the LOS power and Ω as the NLOS power. Forthe BX’s PDF, m controls the shape, Ω controls the spread, and λ impacts the location and height of the mode[3], [24].We observe the BX fading model’s physicality as its distribution becomes an impulse at√λ2 + Ω as thefading parameter, m, approaches infinity. To see this, we define the generalized Ricean RV asF =√√√√ n∑i=1V 2i (3.4)where Vi is a Gaussian distributed RV with mean mi and variance σ2 and n is the number of Gaussian RVs.We obtain the BX RV, Z, via normalization such thatZ =√n∑i=1V 2iσ√nΩ=√Ωσ√√√√√ n∑i=1V 2in(3.5)323.3. System Modelwhile noting that n = 2m. When m and thus n approach infinity, we are left with the BX RV’s limit ofZlim = limn→∞√Ωσ√√√√√ n∑i=1V 2in. (3.6)This limit can be simplified using the Chebyshev’s law of large numbers [25] tolimn→∞√√√√√ n∑i=1V 2in= limn→∞√√√√ 1nn∑i=1E[V 2i]. (3.7)We note here that s in (3.2) is defined as s =√n∑i=1m2i , which allows (3.7) to be cast aslimn→∞√σ2 +s2n. (3.8)We also note that the parameter λ in (3.3) is scaled by σ√2mΩ , which gives λ =sσ√2mΩ. Applying this scalingto (3.8) and inserting the result into (3.6) givesZlim = limn→∞√Ωσ√σ2 +σ2λ2n=√λ2 + Ω. (3.9)We observe from (3.9) that Zlim is a constant at√λ2 + Ω. Thus, the distribution of Z takes the form of animpulse at z =√λ2 + Ω as m approaches infinity. Moreover, the impulse’s location for the BX distributionis a bounded quantity, which corresponds to the BX distribution’s instantaneous power.3.3 System ModelWe consider linear diversity receptions with N branches operating over the BX channels. The receivedsignal isy = zx+ n (3.10)where x is the transmitted signal, n is a random vector denoting additive white Gaussian noise, and z isthe fading channel vector, i.e., the real fading channel amplitudes. In addition, z = [z1, . . . , zN ]T =[√γ¯1Z1, . . . ,√γ¯NZN ]T , where [·]T represents the transpose, γ¯n is the average received SNR of the nth333.3. System Modelbranch, Zn is the fading amplitude of the nth branch, and z is distributed according to (1.12). The respectiveoutput SNRs for MRC, EGC, and SC diversity receptions areγMRC =N∑n=1γ¯nZ2n (3.11a)γEGC =1N(N∑n=1√γ¯nZn)2(3.11b)γSC = max(γ¯1Z21 , · · · , γ¯nZ2n). (3.11c)We assume that the BX RV is associated with the nth branch asZn =√√√√ 2m∑i=1X2n,i; ∀n = 1, . . . N (3.12)where Xn,i, ∀i = 1, . . . 2m are Gaussian RVs with mean µ and variance(12m). The nth component of zcan be obtained by generating an N × 2m matrix of Gaussian RVs, B, whose entries are bn,i and whosemth column is denoted as bm, such that B = [b1, . . . ,b2m]. We obtain the vector b =[bT1 , . . . ,bT2m]T=[b1,1, b2,1, . . . , bn,2m]T . We can, therefore, express the fading amplitude over the nth branch aszn =√√√√ 2m∑i=1b2n,i; ∀n = 1, . . . N (3.13)such that b2n,i = γ¯nX2n,i. The PDF of b is expressed asfb(b) =1√(2pi)2mN |Rb|exp(−12(b− µb)TR−1b (b− µb))(3.14)where µb is the 2mN × 1 mean vector and Rb is the 2mN × 2mN covariance matrix of b. The variancesof the LOS and NLOS components are defined as σ2L,n = |µb,n|2 and σ2NL,n = E[|bn − µb,n|2], respectively,where |·| denotes magnitude. The K-factor for the BX fading channel is therefore defined as K = σ2L,nσ2NL,n.The determinant of the covariance matrix Rb is expressed in terms of the correlation matrix Cb by |Rb| =(N∏n=1γ¯2mn)|Cb|(2m)2mN[54]. This new fading model has the diversity order of mN , which is the same as that of theNakagami-m fading model. However, the BX model predicts improved performance for channels possessingLOS components [24].343.4. Relationship between the Power Correlation and Gaussian Correlation Coefficients3.4 Relationship between the Power Correlation and Gaussian CorrelationCoefficientsAs shown in [52], the correlation coefficient between RVs Xn,i and Xj,k with mean µ and variance12m is obtained by employing the Cholesky decomposition. It can be shown that Xn,i = ρ(n,i)(j,k)Xj,k +√1− ρ2(n,i)(j,k)W , where ρ(n,i)(j,k) is the correlation coefficient andW is a Gaussian RV with zero mean andvariance 12m . We note that W is independent of Xj,k. This leads to a relationship between the correlation co-efficient of the channel powers, ρz2n1z2n2 , and correlation coefficient of the Gaussian RVs, ρ(n,i)(j,k), accordingtoρz2n1z2n2=D − (2m)2T 2(2 + 2m)U − (2m)2T 2 (3.15)whereD =2m∑i=12m∑k=1(3U2ρ2(n,i)(j,k) + T2(1− ρ2(n,i)(j,k))),T = 12m + µ2, and U = 12m . A special case can be obtained when µ = 0 such thatρz2n1z2n2=12m2m∑i=12m∑k=1ρ2(n,i)(j,k) (3.16)which is the same as the one obtained for the Nakagami-m fading model in [54], as expected.3.5 Bounds to the probability density functionIn this section, we show that the PDF of b is bounded in the region bTb ≤ r2, which is a 2mN -dimensional sphere with radius r. We apply the Rayleigh quotient defined asRy(R−1b ,b)=bTR−1b bbTb(3.17)where Rb is a positive definite matrix, to the inequality expressed above. We obtain the numerical rangeof Ry as λmin ≤ bTR−1b bbTb≤ λmax, where λmin and λmax are the smallest and largest eigenvalues of R−1b ,353.6. Performance Bounds to Outage Probabilityrespectively. Thus, we consider the region bTb ≤ r2 where λmin = 0 to have0 ≤ bTR−1b b ≤ λmaxbTb. (3.18)We expand the exponential component with the knowledge that R−1bT= R−1b and obtain the expression(b− µb)TR−1b (b− µb) = bTR−1b b+ µbTR−1b µb + 2∣∣µbTR−1b b∣∣ . (3.19)We apply the 2-norm of the matrix as ‖b‖ =√bTb = r to (3.19) to have− 2 ∥∥µbTR−1b ∥∥ ‖b‖ ≤ (b− µb)TR−1b (b− µb) ≤ λmax ‖b‖+ 2 ∥∥µbTR−1b ∥∥ ‖b‖ (3.20)− 2 ∥∥µbTR−1b ∥∥ r + µbTR−1b µbT ≤ (b− µb)TR−1b (b− µb)≤ λmaxr2 + 2∥∥µbTR−1b ∥∥ r + µbTR−1b µbT . (3.21)The upper and lower bounds of fb(b) near the origin are then obtained asfb (0) exp(−12(λmaxr2 + 2∥∥µbTR−1b ∥∥ r)) ≤ fb (b) ≤ fb (0) exp (∥∥µbTR−1b ∥∥ r) . (3.22)3.6 Performance Bounds to Outage ProbabilityIn this section, we analyze the BX fading model’s performance with arbitrary channel correlation bycomputing the bounds according to [54]. Here, we set an SNR threshold, γth, below which we declareoccurrences of outage.3.6.1 Outage Probability Bounds for MRCWe express the OP for MRC with respect to a signal threshold, γth, asPMRCo (γth) = Pr {γMRC ≤ γth} . (3.23)Substituting (3.12) into (3.11a) and then substituting the result into (3.23), we have363.6. Performance Bounds to Outage ProbabilityPMRCo (γth) = Pr{N∑n=12m∑i=1b2n,i ≤ γth}=∫γMRC≤γthfb (b)db. (3.24)The integral in (3.24) is a 2mN -fold integral; however, it is expressed with just one integral sign for notationalsimplicity.Asymptotic Outage Probability Approximation for MRCThe asymptotic OP approximation is obtained by substituting fb (b) ≈ fb (0) into (3.24), wherefb(0) =1√(2pi)2mN |Rb|exp(−12µTR−1b µb)and rearranging the resulting expression to havePMRCo,∞ (γth) = fb (0)∫γMRC≤γthdb (3.25)where γMRC =N∑n=12m∑i=1b2n,i. The integral in (3.25) is the volume of a 2mN -dimensional sphere with a radiusof r =√γth. Ultimately, eq. (3.25) can be expressed asPMRCo,∞ (γth) = fb (0)(2√γth)2mN+1pimNΓ (mN + 1)(2mN + 1)!. (3.26)Lower Bound Outage Probability for MRCThe lower bound to the OP is obtained by substituting fb (0) with the lower bound of (3.22) in (3.25) to havePMRCo,LM (γth) = OLM∫γMRC≤γthdb (3.27)whereOLM = fb × (0) exp(−12(λmaxγth + 2∥∥µbTR−1b ∥∥√γth)) .Equation (3.27) can be further simplified toPMRCo,LB (γth) = OLM ×[(2√γth)(2mN+1)pimNΓ (mN + 1)(2mN + 1)!]. (3.28)373.6. Performance Bounds to Outage ProbabilityUpper Bound Outage Probability for MRCThe upper bound to the OP is obtained by substituting fb (0) with the upper bound of (3.22) in (3.25) to havePMRCo,UB (γth) = OUM∫γMRC≤γthdb (3.29)where OUM = fb (0) exp(∥∥µbTR−1b ∥∥√γth) . Equation (3.29) is further simplified toPMRCo,UB (γth) = OUM ×[(2√γth)(2mN+1)pimNΓ (mN + 1)(2mN + 1)!]. (3.30)We observe that the difference in the analytical expressions of the bounds in (3.26), (3.28), and (3.30) lie inthe exponential component. Thus, an increase to the average SNR per branch results in the lower and upperbounds of the OP converging to the asymptotic approximation.3.6.2 Outage Probability Bounds for EGCWe express the OP for EGC with respect to a signal threshold, γth, asPEGCo (γth) = Pr {γEGC ≤ γth} . (3.31)Applying (3.12) and (3.11b) to (3.31) leads to an OP expression ofPEGCo (γth) = Pr 1N N∑n=1√√√√ 2m∑i=1b2n,i2 ≤ γth =∫γEGC≤γthfb (b)db. (3.32)Asymptotic Outage Probability Approximation for EGCThe asymptotic OP approximation is obtained by substituting fb (b) ≈ fb (0) in (3.32) and rearranging theresulting expression to havePEGCo,∞ (γth) = fb (0)∫γEGC≤γthdb (3.33)383.6. Performance Bounds to Outage Probabilitywhere γEGC = 1N(N∑n=1√2m∑i=1b2n,i)2. The integral in (3.33) is obtained as [54]∫γEGC≤γthdb =(2mpimΓ (m+ 1))N ΓN (2m)(2mN)(Nγth)mN . (3.34)Substituting (3.34) into (3.33) leads to the asymptotic OP approximation ofPEGCo,∞ (γth) = fb (0)(2mpimΓ (m+ 1))N ΓN (2m)(2mN)(Nγth)mN . (3.35)Lower Bound Outage Probability for EGCWe compare the integral region of EGC to that of MRC, such thatIREGC (r)∆={1NN∑n=12m∑i=1b2n,i ≤ r2}andIRMRC (r)∆={N∑n=12m∑i=1b2n,i ≤ r2}.We note thatN∑n=12m∑i=1b2n,i ≤ Nr2 can be derived from 1NN∑n=12m∑i=1b2n,i ≤ r2, thus, the integral region IREGC (r)lies within IRMRC(√Nr). We obtain a bound on the PDF for EGC by substituting r with√Nr in (3.22)asfb (0) exp(−12(λmaxNr2 + 2∥∥µbTR−1b ∥∥√Nr)) ≤ fb (b) ≤ fb (0) exp(∥∥µbTR−1b ∥∥√Nr) (3.36)where r =√γth. Substituting the lower bound of (3.36) into (3.33), we obtain the lower bound OP for EGCasPEGCo,LB (γth) = OLE ×∫γEGC≤γthdb (3.37)whereOLE = fb (0)× exp(−12(λmaxNγth + 2∥∥µbTR−1b ∥∥√Nγth)) .Substituting (3.34) into (3.37) leads to the lower bound expression for OP as393.6. Performance Bounds to Outage ProbabilityPEGCo,LB (γth) = OLE ×(2mpimΓ (m+ 1))N ΓN (2m)(2mN)(Nγth)mN . (3.38)Upper Bound Outage Probability for EGCThe upper bound to the OP of EGC is obtained by substituting fb (0) in (3.33) with the upper bound of (3.36)to havePEGCo,UB (γth) = OUE ×∫γEGC≤γthdb (3.39)where OUE = fb (0) exp(∥∥µbTR−1b ∥∥√Nγth). Substituting (3.34) into (3.39) leads to the upper boundexpression for OP asPEGCo,UB (γth) = OUE ×(2mpimΓ (m+ 1))N ΓN (2m)(2mN)(Nγth)mN . (3.40)We observe that the difference in the analytical expressions of the bounds in (3.35), (3.38), and (3.40) lie inthe exponential component. Thus, an increase to the average SNR per branch results in the lower and upperbounds of the OP converging to the asymptotic approximation, as seen previously.3.6.3 Outage Probability Bounds for SCWe express the OP for SC with respect to a signal threshold, γth, asPSCo (γth) = Pr {γSC ≤ γth} . (3.41)Substituting (3.12) into (3.11c) givesPSCo (γth) = Pr{maxn{2m∑i=1b2n,i}≤ γth}=∫γSC≤γthfb (b)db. (3.42)Asymptotic Outage Probability Approximation for SCThe asymptotic OP approximation is obtained by substituting fb (b) ≈ fb (0) into (3.42) asPSCo,∞ (γth) = fb (0)N∏n=1∫2m∑i=1b2n,i≤γthdbn (3.43)403.6. Performance Bounds to Outage Probabilitywhere dbn = ΠNn=1dbn,i. The integral in (3.43) is the volume of 2m-dimensional sphere with a radius ofr =√γth. Ultimately, eq. (3.43) can be expressed asPSCo,∞ (γth) = fb (0)×[(2√γth)(2m+1)pimΓ (m+ 1)(2m+ 1)!]N. (3.44)Lower Bound Outage Probability for SCWe compare the integral region for SC to that of MRC, such thatIRSC (r)∆={maxn{2m∑i=1b2n,i}≤ r2}andIRMRC (r)∆={N∑n=12m∑i=1b2n,i ≤ r2}where r is the radius. We can bound the 2m-dimensional sphere in (3.43) by the hypersphere in (3.25). Inaddition, we note thatN∑n=12m∑i=1b2n,i ≤ Nr can be derived from maxn{2m∑i=1b2n,i}≤ r; therefore, IRSC (r) lieswithin IRMRC(√Nr). Therefore, we can apply the bound in (3.36) to SC. Substituting the lower bound in(3.36) into (3.43) givesPSCo,LB (γth) = OLS∫γSC≤γthdb (3.45)whereOLS = fb (0)× exp(−12(λmaxNγth + 2∥∥µbTR−1b ∥∥√Nγth)) .Equation (3.45) can be further simplified toPSCo,LB (γth) = OLS ×[(2√γth)(2m+1)pimΓ (m+ 1)(2m+ 1)!]N. (3.46)Upper Bound Outage Probability for SCThe upper bound to the OP is obtained by substituting fb (0) with the upper bound of (3.36) in (3.42) to have413.7. Performance Bounds to Error RatePSCo,UB (γth) = OUS∫γSC≤γthdb (3.47)where OUS = fb (0) exp(∥∥µbTR−1b ∥∥√Nγth). Equation (3.47) is further simplified toPSCo,UB (γth) = OUS ×[(2√γth)(2m+1)pimΓ (m+ 1)(2m+ 1)!]N. (3.48)We observe that the difference in the analytical expressions of the bounds in (3.44), (3.46), and (3.48) lie inthe exponential component. Again, an increase to the average SNR per branch results in the lower and upperbounds of the OP converging to the asymptotic approximation, as seen previously.3.7 Performance Bounds to Error Rate3.7.1 Error Rate Bounds for MRCWe can express the error rate for MRC asPMRCe = EpQ√√√√q N∑n=12m∑i=1b2n,i = p ∞∫−∞Q√√√√q N∑n=12m∑i=1b2n,ifb (b) db (3.49)where the Gaussian Q-function is defined asQ (x) =1√2pi∞∫xexp(−t22)dt.For the special case of coherent binay phase-shift keying (BPSK), we have p = 1 and q = 2.Asymptotic Error Rate Approximation for MRCThe asymptotic error rate approximation is obtained by substituting fb (b) ≈ fb (0) into (3.49) to havePMRCe,∞ = pfb (0)∞∫−∞Q√√√√q N∑n=12m∑i=1b2n,idb (3.50)where the integral in (3.50) is obtained as [54]∞∫−∞Q√√√√q N∑n=12m∑i=1b2n,idb = 2mN−1pimNqmN√piΓ(mN + 12)Γ (mN + 1). (3.51)423.7. Performance Bounds to Error RateSubstituting (3.51) into (3.50) gives the asymptotic error rate bound asPMRCe,∞ = pfb (0)2mN−1pimNqmN√piΓ(mN + 12)Γ (mN + 1). (3.52)Lower Error Rate Bound for MRCThe lower bound to the error rate is obtained by changing the integral region in (3.50) according toPMRCe = p∫RMRC≤R2thQ√√√√q N∑n=12m∑i=1b2n,i× fb (b) db (3.53)where RMRC =N∑n=12m∑i=1b2n,i. Substituting fb (b) with the lower bound in (3.22) into (3.53), noting thatr = Rth, leads toPMRCe,LB = p× ELM ×G (Rth) (3.54)whereELM = fb (0) exp(−12(λmaxRth2 + 2∥∥µbTR−1b ∥∥Rth)) (3.55)andG (Rth) =pimNΓ (mN + 1)[Rth2mNQ (√qRth) +2mN−1qmN√piγ(mN +12,q2Rth2)](3.56)with γ (α, x) =x∫0e−ttα−1dt as the incomplete gamma function [54].Upper Error Rate Bound for MRCWe obtain the upper bound to the error rate by splitting the integral region in (3.50) according toPMRCe,UB =p∫RMRC≤R2thfb (b)Q√√√√q N∑n=12m∑i=1b2n,idb+ p∫RMRC>R2thfb (b)Q√√√√q N∑n=12m∑i=1b2n,idb. (3.57)Applying the upper bound in (3.22) to (3.57), the upper bound to the error rate can be expressed as433.7. Performance Bounds to Error RatePMRCe,UB = p× EUM ×∫RMRC≤R2thQ√√√√q N∑n=12m∑i=1b2n,idb+ pfb (µb)∫RMRC>R2thQ√√√√q N∑n=12m∑i=1b2n,idb (3.58)where the first integral in (3.58) according to (3.56), fb (b) is substituted with the upper bound in (3.22),EUM = fb (0) exp(∥∥µbTR−1b ∥∥Rth), and fb(b) = 1√(2pi)2mN |Rb| is substituted with fb (µb), which is thelargest value of fb (b) in the second part of (3.57). The second integral in (3.58) is obtained asH(Rth) =∫RSC>R2thQ√√√√q N∑n=12m∑i=1b2n,i db (3.59)={pimN2mN−1Γ(mN + 12)Γ (mN + 1)√piqmN− pimNΓ (mN + 1)[Rth2mNQ (√qRth)+2mN−1√piqmNγ(mN +12,q2R2th)]}.3.7.2 Error Rate Bounds for EGCWe can express the error rate for EGC asPEGCe = EpQ√√√√√ qN N∑n=1√√√√ 2m∑i=1b2n,i2 = p∞∫−∞Q√√√√√ qN N∑n=1√√√√ 2m∑i=1b2n,i2fb (b) db.(3.60)Asymptotic Error Rate Approximation of EGCThe asymptotic error rate approximation is obtained by substituting fb (b) ≈ fb (0) in (3.60) to havePEGCe,∞ = p× fb (0)×∞∫−∞Q√√√√√ qN N∑n=1√√√√ 2m∑i=1b2n,i2db. (3.61)We obtain the integral in (3.61) according to [63, eq. 61], such that (3.61) becomesPEGCe,∞ = pfb (0)× Y (Rth) (3.62)where443.7. Performance Bounds to Error RateY (Rth) =(2mpimΓ (m+ 1))N ( ΓN (2m)Γ (2mN))(NmN2mN−2√pimNqmNΓ(mN +12)). (3.63)Lower Error Rate Bound for EGCThe lower bound to the error rate is obtained by changing the integral bound in (3.61) asPEGCe = p∫REGC≤R2thQ√√√√√ qN N∑n=1√√√√ 2m∑i=1b2n,i2fb (b) db (3.64)where REGC =√√√√ qN(N∑n=1√2m∑i=1b2n,i)2. Substituting fb (b) with the lower bound in (3.36) into (3.64),noting that r = Rth, leads toPEGCe,LB = pfb (0)× ELE ×W (Rth) (3.65)whereELE = fb (0) exp[−12(λmaxNRth2 + 2∥∥µbTR−1b ∥∥√NRth)] (3.66)andW (Rth) =(2mpimΓ (m+ 1))N×(ΓN (2m)Γ (2mN))×[(2mN−2NmN√pimNqmNγ(mN +12,q2Rth2)+Q (√qRth)Rth2mNNmN2mN)]. (3.67)Upper Error Rate Bound for EGCThe upper error rate bound is obtained by splitting the integral region in (3.60) according toPEGCe,UB = p∫REGC≤R2thQ√√√√√ qN N∑n=1√√√√ 2m∑i=1b2n,i2fb (b) db+ p∫REGC>R2thQ√√√√√ qN N∑n=1√√√√ 2m∑i=1b2n,i2fb (b) db. (3.68)453.7. Performance Bounds to Error RateFor the first part of (3.68), we substitute fb (b) with the upper bound in (3.22), while for the second part, wesubstitute fb (b) with fb (µb). Thus, the expression for the upper bound of the error rate becomesPEGCe,UB = p× EUE∫REGC≤R2thQ√√√√√ qN N∑n=1√√√√ 2m∑i=1b2n,i2db+ pfb (µb)∫REGC>Rth2Q√√√√√ qN N∑n=1√√√√ 2m∑i=1b2n,i2db︸ ︷︷ ︸I(µb)(3.69)where EUE = fb (0) exp(∥∥µbTR−1b ∥∥√NRth). The first integral above is simplified according to (3.56),and the second integral is simplified asI (µb) =∞∫−∞Q√√√√√ qN N∑n=1√√√√ 2m∑i=1b2n,i2 db ∫REGC≤R2thQ√√√√√ qN N∑n=1√√√√ 2m∑i=1b2n,i2db. (3.70)After some mathematical manipulation, eq. (3.70) is further simplified to∫REGC≤R2thQ√√√√√ qN N∑n=1√√√√ 2m∑i=1b2n,i2 db ≥ ∫REGC≤R2thQ√√√√q N∑n=12m∑i=1b2n,i db, (3.71)and then=∞∫−∞Q√√√√√ qN N∑n=1√√√√ 2m∑i=1b2n,i2 db− ∫REGC≤R2thQ√√√√q N∑n=12m∑i=1b2n,idb. (3.72)Ultimately, eq. (3.70) can be expressed asZ(Rth) = Y (Rth)− pimNΓ (mN + 1)[R2mNth Q (√qRth)+2mN−1√piqmNγ(mN +12,q2R2th)]. (3.73)Substituting (3.67) and (3.73) into (3.69) gives the upper error rate bound for EGC.3.7.3 Error Rate Bounds for SCWe can express the error rate for SC as463.7. Performance Bounds to Error RatePSCe = EpQ√√√√qmaxn{2m∑i=1b2n,i} = p ∞∫−∞Q√√√√qmaxn{2m∑i=1b2n,i}fb (b) db. (3.74)Asymptotic Error Rate Approximation for SCThe asymptotic error rate approximation is obtained by substituting fb (b) ≈ fb (0) into (3.74) to havePSCe = pfb (0)∞∫−∞Q√√√√qmaxn{2m∑i=1b2n,i}db (3.75)which can be simplified to [54]PSCe,∞ = pfb (0)2mN−1pimNΓ(mN + 12)qmN√piΓN (m+ 1). (3.76)Lower Error Rate Bound for SCThe lower bound of the error rate is obtained by changing the integral region in (3.74) according toPSCe = p∫RSC≤R2thQ√√√√qmaxn{2m∑i=1b2n,i} fb (b) db (3.77)where RSC = maxn{2m∑i=1b2n,i}. Substituting fb (b) with the lower bound in (3.36) into (3.77), noting r =Rth, leads toPSCe,LB = p× ELS × S (Rth) (3.78)whereELS = fb (0) exp[−12(λmaxNRth2)+2∥∥µbTR−1b ∥∥√NRth] (3.79)andS (Rth) =pimNΓN (m+ 1)(Rth2mNQ (√qRth) +2mN−1qmN√piγ(2mN + 12,q2Rth2)). (3.80)Upper Error Rate Bound for SCThe upper error rate bound is obtained by splitting the integral region in (3.74) according to473.8. Discussion on the Tightness of the BoundsPSCe,UB = p∫RSC≤R2thQ√√√√qmaxn{2m∑i=1b2n,i} fb (b) db+ p∫RSC>R2thQ√√√√qmaxn{2m∑i=1b2n,i} fb (b) db. (3.81)Applying the upper bound in (3.36) to (3.81) leads toPSCe,UB = p× EUS∫RSC≤R2thQ√√√√qmaxn{2m∑i=1b2n,i} db+ pfb (µb)∫RSC>R2thQ√√√√qmaxn{2m∑i=1b2n,i} db (3.82)where the first integral is simplified according to (3.80) and EUB = fb (0) exp(∥∥µbTR−1b ∥∥√NRth). Thesecond integral in (3.82) is simplified asV (Rth) =∫RSC>R2thQ√√√√qmaxn{2m∑i=1b2n,i} db={pimN2mN−1qmN√piΓ(mN + 12)ΓN (m+ 1)− pimNΓN (m+ 1)[R2mNth Q (√qRth) +2mN−1qmN√piγ(mN +12,q2R2th)]}.(3.83)The upper error rate bound for SC is obtained by substituting (3.80) and (3.83) into (3.82).3.8 Discussion on the Tightness of the Bounds3.8.1 Tightness of the Outage Probability BoundWe consider our analytical results for MRC, EGC, and SC’s lower and upper bounds, which differ fromtheir corresponding asymptotic approximation in the exponential components. From these exponential com-ponents, we observe that an increase in the average SNR per branch by a factor P results in the covariancematrix Rb being multiplied by the factor P and R−1b being divided by the same factor P . Also, λmax isdivided by the same factor P , i.e., λmaxP . Thus, we can express the limit of the ratio of lower bound OP to theupper bound OP as P tends to infinity as483.8. Discussion on the Tightness of the BoundslimP→∞Po,LB (γth)Po,UB (γth)=exp(−12(λmaxγthP + 2∥∥∥∥µTbR−1bP ∥∥∥∥√γth))exp(∥∥∥∥µTbR−1bP ∥∥∥∥√γth) . (3.84)Equation (3.84) shows that the lower and upper bounds of the OP converge to the asymptotic approximationas the SNR approaches infinity. This property is highly desirable.3.8.2 Tightness of the Error Rate BoundGiven the definition of the incomplete gamma funtionΓ (s) = limx→∞ γ (s, x) (3.85)and the Chernoff bound of the Q-function [55]Q (x) ≤ e−x22 , x > 0 (3.86)we apply these definitions to G(Rth), H(Rth), W (Rth), Z(Rth), S(Rth), and V (Rth) in (3.56), (3.59),(3.67), (3.73), (3.80), and (3.83), respectively, to show the tightness of the error rate bounds, according tolimRth→∞G(Rth) =2mN−1pimNqmN√piΓ(mN + 12)Γ (mN + 1)(3.87)limRth→∞H(Rth) = 0 (3.88)limRth→∞W (Rth) = Y (Rth) (3.89)limRth→∞Z(Rth) =Y (Rth)− pimN− 12Γ (mN + 1)[2mN−1qmNΓ(mN +12)](3.90)limRth→∞S(Rth) =2mN−1pimNqmN√piΓ(mN + 12)ΓN (m+ 1)(3.91)493.9. Numerical ResultslimRth→∞V (Rth) = 0. (3.92)From the exponential components, we observe that an increase in the average SNR per branch by a factorP results in the covariance matrix Rb being multiplied by the factor P and R−1b being divided by the samefactor P . Also, λmax is divided by the same factor P . Thus, we can express the limit of the ratio of lowerbound error rate to the upper bound error rate OP as P tends to infinity aslimP→∞Pe,LBPe,UB=exp(−12(λmaxR2thP + 2∥∥∥∥µTbR−1bP ∥∥∥∥Rth))exp(∥∥∥∥µTbR−1bP ∥∥∥∥Rth) . (3.93)Expressing Rth in terms of P , i.e., Rth =√P , giveslimP→∞Pe,LBPe,UB=exp(−12(λmaxR2thP + 2∥∥∥∥µTbR−1bP ∥∥∥∥√P))exp(∥∥∥∥µTbR−1bP ∥∥∥∥√P) (3.94)which indicates that the lower and upper bounds are asymptotically tight.3.9 Numerical ResultsFigure 3.1 shows a comparison of OP curves for MRC, SC, and EGC for a 2-branch diversity systemover correlated BX fading channels with coherent BPSK modulation. A fading parameter of m = 1.5 anda correlation structure of the form Cb = [C1, 02×2, 02×2; 02×2, C2, 02×2; 02×2, 02×2, C3], where C1 =C2 = C3 =[1,√0.35;√0.35, 1], is considered. The lower and upper bounds converge to the asymptoticapproximation and simulated OPs at high SNR. It is observed that the OPs of MRC and EGC are close toeach other unlike MRC and SC. This is largely due to the factor contributed by the integral in (3.25), (3.33),and (3.43).Figures 3.2, 3.3, and 3.4 show a comparison of OP curves of MRC, SC, and EGC, respectively, for a 2-branch diversity system over correlated and independent BX fading channels with coherent BPSK modulation.We consider a fading parameter of m = 1.5 and a correlation structure of the form Cb = [C1,02×2,02×2;02×2,C2,02×2;02×2,02×2,C3], where C1 = C2 = C3 =[1,√0.35;√0.35, 1]and C1 = C2 = C3 =[1, 0; 0, 1], for correlation and independence, respectively. The independent channels outperform the corre-503.9. Numerical Results6 8 10 12 14 16 18 20 2210-610-510-410-310-210-1100Figure 3.1: Outage probability curves for the asymptotic approximation, lower bound, and upper boundfor MRC, SC, and EGC over correlated BX fading channels. The number of branches is N = 2, the fadingparameter ism = 1.5, and the correlation structure isCb = [C1,02×2,02×2;02×2,C2,02×2;02×2,02×2,C3],where C1 = C2 = C3 =[1,√0.35;√0.35, 1].lated channels, as expected. The effect of correlation is seen on the rate of convergence exhibited by the threecombining techniques.Figure 3.5 shows a comparison of BER curves for MRC, SC, and EGC for a 2-branch diversity systemover correlated BX fading channels with coherent BPSK modulation. We considered a fading parameter ofm = 1.5 and a correlation structure of the form Cb = [C1,02×2,02×2; 02×2,C2,02×2;02×2,02×2,C3],where C1 = C2 = C3 =[1,√0.4;√0.4, 1]. Again, the lower and upper bounds converge to the asymptoticapproximation at high SNR. Also, we observe that the BERs for MRC and EGC are close to each other,differing by at most 3 dB.Figure 3.6 shows a comparison of BER curves for MRC, SC, and EGC for a 3-branch diversity systemover correlated BX fading channels with coherent BPSK modulation. A fading parameter of m = 1.5 anda correlation structure of the form Cb = [C1,03×3,03×3; 03×3,C2,03×3;03×3,03×3,C3], where C1 =C2 = C3 =[1,√0.4,√0.2;√0.4, 1,√0.4;√0.2,√0.4, 1]. The bounds are seen to be tight at high SNR.We observe that the BERs for MRC and EGC are again close to each other, differing by at most 3 dB in SNR.Figures 3.7, 3.8, and 3.9 show a comparison of BER curves for MRC, SC, and EGC, for a 2-branch di-versity system over correlated and independent BX fading channels with coherent BPSK modulation. We con-513.9. Numerical Results5 10 15 20 2510-710-610-510-410-310-210-1Figure 3.2: Outage probability curves for the asymptotic approximation, lower bound, and upper bound forMRC over correlated and independent BX fading channels. The number of branches is N = 2, the fadingparameter ism = 1.5, and the correlation structure isCb = [C1,02×2,02×2;02×2,C2,02×2;02×2,02×2,C3],where C1 = C2 = C3 =[1,√0.35;√0.35, 1]and C1 = C2 = C3 = [1, 0; 0, 1] are for correlation andindependence, respectively.6 8 10 12 14 16 18 20 2210-610-410-2100Figure 3.3: Outage probability curves for the asymptotic approximation, lower bound, and upper bound for SCover correlated and independent BX fading channels. The number of branches is N = 2, the fading parameteris m = 1.5, and the correlation structure is Cb = [C1,02×2,02×2;02×2,C2,02×2;02×2,02×2,C3], whereC1 = C2 = C3 =[1,√0.35;√0.35, 1]and C1 = C2 = C3 = [1, 0; 0, 1] are for the correlated andindependent cases, respectively.523.9. Numerical Results5 10 15 20 2510-710-610-510-410-310-210-1100Figure 3.4: Outage probability curves for the asymptotic approximation, lower bound, and upper bound forEGC over correlated and independent BX fading channels. The number of branches is N = 2, the fadingparameter ism = 1.5, and the correlation structure isCb = [C1,02×2,02×2;02×2,C2,02×2;02×2,02×2,C3],where C1 = C2 = C3 =[1,√0.8;√0.8, 1]and C1 = C2 = C3 = [1, 0; 0, 1] are for correlated andindependent cases, respectively.10 12 14 16 18 20 22 24 2610-810-610-410-2100Figure 3.5: BER curves for the asymptotic approximation, lower bound, and upper bound for MRC, SC,and EGC, over correlated BX fading channels. The number of branches is N = 2, the fading parameter ism = 1.5, and the correlation structure is Cb = [C1, 02×2, 02×2; 02×2, C2, 02×2; 02×2, 02×2, C3], whereC1 = C2 = C3 =[1,√0.4;√0.4, 1].533.10. Chapter Summary8 10 12 14 16 18 20 22 24 26 2810-1410-1210-1010-810-610-410-2100Figure 3.6: BER curves for the asymptotic approximation, lower bound, and upper bound for MRC, SC,and EGC, over correlated BX fading channels. The number of branches is N = 3, the fading parameter ism = 1.5, and the correlation structure is Cb = [C1, 03×3, 03×3; 03×3, C3, 03×3; 03×3, 03×3, C3], whereC1 = C2 = C3 =[1,√0.4,√0.2;√0.4, 1,√0.4;√0.2,√0.4, 1].sider a fading parameter ofm = 1.5 and a correlation structure of the formCb = [C1,02×2,02×2;02×2,C2,02×2;02×2,02×2,C3], where [C1 = C2 = C3 =[1,√0.4;√0.4, 1]and C1 = C2 = C3 = [1, 0; 0, 1] for cor-related and independent cases. The performance improves when the channels are independent. The effect ofthe correlation matrix is also seen on the rate of convergence of the bounds, i.e., the higher the determinant ofCb (signifying greater independence between channels), the faster the rate of convergence.3.10 Chapter SummaryThis chapter derived asymptotically-tight lower and upper bounds for the OP and error rate of corre-lated BX fading channels. We observed the effects of specific channel characteristics (e.g., presence of LOScomponents, the number of diversity branches, and spatial channel separation) on the communication sys-tem’s performance. An important observation is a faster convergence rate of the lower and upper bounds toasymptotic approximation as the independence between the BX channels increases. This observation equallymeans that a positive correlation between the BX channels adversely affects the communication system’sperformance. While we analyze independence and positive correlation between the BX channels in this chap-ter, future works can analyze communication systems’ performance with a negative correlation between the543.10. Chapter Summary5 10 15 20 2510-810-610-410-2100Figure 3.7: BER curves for the asymptotic approximation, lower bound, and upper bound for MRC, overcorrelated and independent BX fading channels. The number of branches is N = 2, the fading parameter ism = 1.5, and the correlation structures is Cb = [C1, 02×2, 02×2; 02×2, C2, 02×2; 02×2, 02×2, C3], whereC1 = C2 = C3 =[1,√0.4;√0.4, 1]and C1 = C2 = C3 = [1, 0; 0, 1] are for correlated and independentcases, respectively.8 10 12 14 16 18 20 22 24 26 28 3010-1010-810-610-410-2100Figure 3.8: BER curves for the asymptotic approximation, lower bound, and upper bound for SC, over cor-related and independent BX fading channels. The number of branches is N = 2, the fading parameter is m= 1.5, and the correlation structures isCb = [C1, 02×2, 02×2; 02×2, C2, 02×2; 02×2, 02×2, C3], whereC1 = C2 = C3 =[1,√0.4;√0.4, 1]and C1 = C2 = C3 = [1, 0; 0, 1] are for correlated and independentcases, respectively.553.10. Chapter Summary10 15 20 25 30 3510-1010-810-610-410-2100Figure 3.9: BER curves for the asymptotic approximation, lower bound, and upper bound for EGC, overcorrelated and independent BX fading channels. The number of branches is N = 2, the fading parameter ism = 1.5, and the correlation structures is Cb = [C1, 02×2, 02×2; 02×2, C2, 02×2; 02×2, 02×2, C3], whereC1 = C2 = C3 =[1,√0.4;√0.4, 1]and C1 = C2 = C3 = [1, 0; 0, 1] are for correlated and independentcases, respectively.fading channels.56Chapter 4On the Performance of Beaulieu-Xie FadingChannels for MRC in the Presence ofCo-channel InterferenceA paradigm shift will unfold over the next decade by introducing fifth-generation (5G) and sixth gener-ation (6G) wireless systems. These 5G and 6G systems have high data rates, enhanced reliability, networkdensification, low energy consumption, low latency, and massive connectivity [56]. Such wireless systems canbring about new technologies, like mmWave and THz communications, femtocells, internet-of-things (IoT)services, and WPC [57].For IoT services, the use of random access channels has been proposed because of the challenges asso-ciated with efficient resource allocation [59]. Despite the extreme improvements made to the random accessprotocols, there is still a heavy presence of multiuser interference. Therefore, we must recognize that modern5G random access protocols’ success depends upon how it can constructively embrace interference [59].For the performance analysis in this chapter, we focus on CCI. In such a case, the received signal consistsof faded copies of the desired signals, interfering signals, and noise [60]. Simultaneously, received signals isimpaired by multipath fading. We consider the MRC because it presents mathematical advantages over theoptimum combiner. Besides, the MRC’s result offers a bound to those of the optimum combiner, giving agood understanding of how each system parameter affects the optimum combiner’s performance.The novelty and contributions of the analysis in this chapter lie in targeting critical challenges facingemerging wireless systems. These systems’ short-range and random-access functionality rely upon LOSsignals (via direct propagation and specular reflections). The BX fading model was developed in light ofthese challenges and can show success because of its enhanced flexibility.The organization of this chapter is as follows. We introduce the system model in Section 4.1. In Sections4.2 and 4.3, we analyze the average BER and OP for the MRC scheme, respectively. Section 4.4 presents the574.1. System Modelnumerical results, and Section 4.5 gives some concluding remarks.4.1 System ModelThis section presents a system model where we assume that the received signal is a combination of thedesired signal, interfering signal, and noise. We note that the SNR at the MRC diversity system’s output is aweighted combination of the signals of each of the diversity branches.We consider a RV, W , defined asW =Q∑q=1R2q = R21 +R22 + · · ·+R2Q (4.1)where the Rq are the i.i.d. BX RVs with a PDF given by (1.12). The RV, W , has a noncentral chi-squaredistribution with 2mQ degrees of freedom. Thus, the PDF of W is [39, p. 46]fW (w) =mΩexp[−mΩ(w + λ2)] (wλ)mQ−12 × ImQ−1(2mλΩ√w)(4.2)where λ2 =∑Qq=1 λ2q and ImQ−1 (·) is the (mQ− 1)th-order modified Bessel function of the first kind.We continue by deriving the PDF of the signal-to-interference-plus-noise ratio (SINR), γt, asγt =∑Ni=1Gi2σ2 +∑Lj=1Hj2=X1 + Y(4.3)where X =∑Ni=1Gi2/σ2 is the SNR at the output of the combiner, Gi2 is the desired signal power on theith branch, N is the number of diversity branches, Y =∑Lj=1Hj2/σ2 is the interference-to-noise (INR) atthe output of the combiner, L is the number of interfering signals, Hj2 is the interference signal power on thejth branch, σ2 is the noise power per branch, and Gi and Hj are BX distributed. The PDF of the output SNRis noncentral chi-squared distributed with a PDF given by (4.2) and is expressed asfX(x) =msΩsexp[−msΩs(x+ λ2s)]( xλ2s)msN−12ImsN−1[2msλsΩs√x], x ≥ 0 (4.4)where λ2s =∑Ni=1 λ2i given λ2i as the power in the LOS component of the ith branch, Ωs is the power in theNLOS components, and ImsN−1 (·) is the (msN − 1)th-order modified Bessel function of the first kind. We584.1. System Modelassume that there are L independent BX faded interfering signals with the PDF of the output INR followingthe noncentral chi-squared distribution asfY (y) =mIΩIexp[−mIΩI(y + λ2I)]( yλ2I)mIL−12ImIL−1[2mIλIΩI√y], y ≥ 0, (4.5)where λ2I =∑Lj=1 λ2j , λ2j is the power in the LOS component of the jth interfering signal, and ΩI is thepower in the NLOS component. We can then express the PDF of SINR, γt, asfγt (γ) =∫ ∞0(1 + y)fX ((1 + y) γ) fY (y) dy, γ ≥ 0. (4.6)Substituting (4.4) and (4.5) into (4.6) leads tofγt (γ) =msΩsmIΩIexp[−(msΩs(γ + λ2s)+mIΩIλ2I)](γλ2s)msN−12(1λ2I)mIL−12∫ ∞0(1 + y)msN+12 ymIL−12× exp[−(msΩsγ +mIΩI)y]ImIL−1(2mIλIΩI√y)ImsM−1(2msλsΩs√(1 + y) γ)dy. (4.7)Expressing the modified Bessel function of the first kind in series form using [64, 8.402], we havefγt (γ) = exp[−(msΩs(γ + λ2s)+mIΩIλ2I)]γk+msN−1∞∑k=01k!Γ (msN + k)(msΩs)2k+msNλ2ks×∞∑q=01q!1Γ (mIL+ q)(mIΩI)2q+mILλ2qI∫ ∞0(1 + y)k+msN yq+mIL−1 exp[−(msΩsγ +mIΩI)y].(4.8)Applying [60, 6] to (4.8), we obtain the PDF of the SINR asfγt (γ) = exp[−(msΩsλ2s +mIΩIλ2I)] ∞∑k=01k!Γ (k +msN)×(msΩs)2k+msNλ2ks∞∑q=01q!(mIΩI)2q+mILλ2qI× γk+msN−1 exp(−msΩsγ)Ψ(q +mIL; k + q +msN +mIL+ 1;msΩsγ +mIΩI)(4.9)where Ψ (·; ·; ·) is the confluent hypergeometric function of the second kind.594.2. Average Bit Error Rate4.2 Average Bit Error RateIn this section, the average BER is computed asPe =∫ ∞0Pe (γ) fγt (γ) dγ (4.10)where Pe (γ) is the conditional probability of error and fγt (γ) is presented in (4.9). We employ the unifiedexpression of Pe (γ) in the form of [65]Pe (γ) =Γ (b, aγ)2Γ (b)(4.11)where Γ (·, ·) is the incomplete gamma function,a =12 for frequency-shift keying (FSK)1 for phase-shift keying (PSK), and(4.12)b =12 for coherent detection1 for noncoherent/differential coherent detection.(4.13)Substituting the (4.7) and (4.11) into (4.10) leads toPe =12Γ (b)msΩsexp(−msλ2sΩs)λ1−msNsmIΩIexp(−mIλ2IΩI)λ1−mILI∫ ∞0∫ ∞0Γ (b, aγ) (1 + y)msN+12× γmsN−12 ymIL−12 exp(−msΩs(1 + y) γ)ImsN−1(2msλsΩs√(1 + y) γ)exp(−mIΩIy)× ImIL−1(2mIλIΩI√y)dydγ. (4.14)Applying [64, 8.402] to (4.14), we obtain the average BER asPe =12Γ (b)exp[−(msλsΩs+mIλIΩI)] ∞∑k=0λ2ksk!Γ (msN + k)(msΩs)2k+msN ∞∑q=0λ2qIq!Γ (q +mIL)(mIΩI)2q+mIL×∫ ∞0∫ ∞0Γ (b, aγ) γk+msN−1yq+mIL−1 (1 + y)k+msN exp(−mIΩIy)exp(−msΩs(1 + y) γ)dydγ.(4.15)604.2. Average Bit Error RateEvaluating the integral in (4.15) with respect to y leads toPe =12Γ (b)exp[−(msλsΩs+mIλIΩI)] ∞∑k=0λ2ksk!Γ (msN + k)×(msΩs)2k+msN ∞∑q=0λ2qIq!Γ (q +mIL)×(mIΩI)2q+mIL k+msN∑s=0(k +msNs)(q + s+mIL− 1)!∫ ∞0γk+msN−1Γ (b, aγ) exp(−msΩsγ)×(msΩsγ +mIΩI)−(q+s+mIL)dγ. (4.16)Applying [64, 6.455] to (4.16), we obtain the average BER asPe =ab2Γ (b)exp[−(msΩsλ2s +mIΩIλ2I)] ∞∑k=0λ2ksk!Γ (k +msN)∞∑q=0λ2qIq!Γ (q +mIL)k+msN∑s=0Γ (k +msN + 1)s!Γ (k +msN − s+ 1)×∞∑p=0(−1)p Γ (q +mIL+ s+ p)p!(mIΩI)2q+mIL−p(msΩs)2k+msN+p Γ (k +msN + p+ b)(k +msN + p)(a+ msΩs)k+msN+p+b× 2F 1(1, k +msN + p+ b; k +msN + p+ 1;msaΩs +ms)(4.17)where 2F 1 (·, ·; ·; ·) is the Gauss hypergeometric function [66].The error in truncating the infinite series in (4.17) is given byc ≤ ab2Γ (b)exp[−(msΩsλ2s +mIΩIλ2I)] ∞∑k=Kλ2ksk!Γ (k +msN)∞∑q=Qλ2qIq!Γ (q +mIL)×k+msN∑s=0Γ (k +msN + 1)s!Γ (k +msN − s+ 1)∞∑p=PΓ (q +mIL+ s+ p)p!(mIΩI)2q+mIL−p(msΩs)2k+msN+p(−1)p× Γ (k +msN + p+ b)(k +msN + p)(a+ msΩs)k+msN+p+b 2F 1(1, k +msN + p+ b; k +msN + p+ 1;msaΩs +ms).(4.18)After some mathematical manipulations, the truncation error can be shown to be bounded byc ≤ ab2Γ (b)exp[−(msλ2sΩs+mIλ2IΩI)](a+msΩs)−(b+msN)maxk,s>Bmaxq>Vmaxp>T(mIΩI)−s Γ (q + p+mIL+ s)Γ (q +mIL)︸ ︷︷ ︸=constant→1× Γ (k +msN + p+ b)Γ (k +msN) (k +msN + p)Γ (k +msN + 1)s!Γ (k +msN − s+ 1)︸ ︷︷ ︸=constant→12F 1(1, k +msN + p+ b; k +msN + p+ 1;msaΩs +ms)︸ ︷︷ ︸=constant614.2. Average Bit Error Rate× exp((msλs)2Ωs (aΩs +ms))exp(− msΩImI (aΩs +ms))exp(mIλ2IΩI)(msΩs)msN. (4.19)This expression in (4.19) approaches zero as k, q, and p approach infinity.We continue our analysis for the average BER by considering the approximate M -quadrature amplitudemodulation (QAM) constellation BER for coherent modulation, defined asPe =4log2MQ(√3γ log2MM − 1)=2log2M√piΓ(12, Aγ)(4.20)where A = 3 log 2M2(M−1) . Inserting (4.7) and (4.20) into (4.10), we havePe =2log2M√pimsΩsmIΩIexp[−msΩsλ2s −mIΩIλ2I]λ1−msNs λ1−mILI∫ ∞0∫ ∞0Γ(12, Aγ)× (1 + y)msN+12 ymIL−12 γmsN−12 exp(−msΩs(1 + y) γ)exp(−mIΩIy)× ImsN−1(2msλsΩs√1 + y)ImIL−1(2mIλIΩI√y)dydγ. (4.21)Applying [64, 6.455] to (4.21), we obtain the average BER asPe =2log2M√piexp[−msΩsλ2s −mIΩIλ2I] ∞∑k=01k!Γ (k +msN)(msΩs)2k+msNλ2ks×∞∑q=01q!Γ (q +mIL)λ2qI(mIΩI)2q+mIL ∫ ∞0∫ ∞0Γ(12, Aγ)(1 + y)k+msN× yq+mIL−1γk+msN−1 exp(−msΩs(1 + y) γ)exp(−mIΩIy)dydγ. (4.22)Applying the binomial expansion to the term (1 + y)k+msN in (4.22), with an integer value of msN , we havePe =2log2M√piexp[−msΩsλ2s −mIΩIλ2I] ∞∑k=01k!Γ (k +msN)(msΩs)2k+msNλ2ks×∞∑q=01q!Γ (q +mIL)λ2qI(mIΩI)2q+mIL k+msN∑s=0Γ (k +msN + 1)s!Γ (k +msN − s+ 1)×∫ ∞0∫ ∞0Γ(12, Aγ)yq+s+mIL−1γk+msN−1 exp(−msΩs(1 + y) γ)exp(−mIΩIy)dydγ. (4.23)624.3. Outage ProbabilityUsing [64, 3.351], we integrate (4.23) with respect to y to havePe =2log2M√piexp[−msΩsλ2s −mIΩIλ2I] ∞∑k=01k!Γ (k +msN)(msΩs)2k+msNλ2ks×∞∑q=01q!Γ (q +mIL)λ2qI(mIΩI)2q+mIL k+msN∑s=0Γ (k +msN + 1)s!Γ (k +msN − s+ 1) (q + s+mIL− 1)!×∫ ∞0Γ(12, Aγ)γk+msN−1 exp(−msΩsγ)(mIΩI+msΩsγ)−(q+s+mIL)dγ. (4.24)Expanding the term(mIΩI+ msΩs γ)−(q+s+mIL)and integrating (4.24) with respect to γ using [64, 6.455], leadsto an average BER expression ofPe =2√Alog2M√piexp[−(msΩsλ2s −mIΩIλ2I)] ∞∑k=01k!Γ (k +msN)λ2ks∞∑q=01q!Γ (q +mIL)λ2qI×k+msN∑s=0Γ (k +msN + 1)s!Γ (k +msN − s+ 1)∞∑p=0(−1)p Γ (q +mIL+ s+ p)p!(mIΩI)q−s−p(msΩs)2k+msN+p(4.25)× Γ(k +msN + p+12)(k +msN + p)(A+ msΩs)k+msN+p+ 12 × 2F 1(1, k +msN + p+12; k +msN + p+ 1;msAΩs +ms).4.3 Outage ProbabilityThe OP is another key performance metric expressed asPout = p (γt < γb) =∫ γb0fγt (γ) dγ (4.26)where γb is the defined threshold SNR below which unsatisfactory performance is recorded. To calculate theOP, we insert (4.6) into (4.26) to havePout = exp[−(msΩsλ2s +mIΩIλ2I)] ∞∑k=01k!Γ (k +msN)(msΩs)2k+msNλ2ks∞∑q=01q!Γ (q +mIL)(mIΩI)2q+mIL× λ2qI∫ ∞0(1 + y)k+msNyq+mIL+1 exp(−mIΩIy)∫ γb0γk+msN−1 exp[−msΩs(1 + y) γ]dγdy. (4.27)634.4. Numerical ResultsThe outer integral is then solved using [71] to havePout = exp[−(msΩsλ2s +mIΩIλ2I)] ∞∑k=01k!(msΩs)kλ2ks∞∑q=01q!Γ (q +mIL)(mIΩI)2q+mILλ2qI∫ ∞0yq+mIL+1× exp(−mIΩIy)×{1−(exp[−msΩs(1 + y) γb] k+msN−1∑n=01n!(msΩs(1 + y) γb)n)}dy. (4.28)After some mathematical manipulations, we obtain the OP asPout = exp[−(msΩsλ2s +mIΩIλ2I)] ∞∑k=01k!(msΩs)kλ2ks∞∑q=01q!(mIΩI)qλ2qI{1−((mIΩI)q+mILexp(−msΩsγb)×k+msM−1∑n=01n!(msγbΩs)nΨ (q +mIL; q +mIL+ n+ 1; mIΩI +msΩsγb}. (4.29)In the absence of interfering signal (i.e., a noise-limited environment), the OP reduces toPout = exp(−msλ2sΩs) ∞∑k=01k!Γ (k +msN)(msλ2sΩs)kg(k +msN,msΩsγb). (4.30)This newly-proposed result for the OP in the absence of interference is a crucial addition to the researchliterature. Ultimately, the result in (4.30) reduces monotonically in k and approaches zero as k approachesinfinity.From the results of the average BER or OP, we can obtain the system’s diversity order. At high SNR, theBER or OP scale according to SNR–η, where η is the diversity order. This scaling means that a log-log plotof the BER or OP versus SNR portray the diversity order as the slope ofη = − log (BER or OP)log (SNR). (4.31)Thus, applying (4.31) to our result of the BER, the diversity order is equivalent to msN at high SNR.4.4 Numerical ResultsThis section illustrates the series-form results obtained for the average BER and OP. For these results,the fading parameters ms and mI indicate the severity of the fading of the desired signal and interferingsignals, respectively. The parameters, λ2s and λ2I represent the power in the LOS component(s) of the desiredand interfering signals, respectively. We assume that the interfering signal experiences worse fading than the644.4. Numerical Results0 5 10 15 2010-1010-5100Figure 4.1: Comparison of the average BER versus average SINR for coherent BFSK and BPSK detectionover BX fading channels with parameters of L = 4, ms = 1, λs = 3 dB, λI = 3 dB, mI = 0.75, and ΩI = 4 dB.desired signal, as seen in practice. The effects of the number of diversity branches and fading severities areshown on the system’s performance and results are presented for BPSK and BFSK detections.Figure 4.1 shows the average BER curves for coherent BFSK and BPSK detection over BX fading chan-nels versus average SINR. The interfering signals experience more fading atm = 0.75 while the desired signalexperiences less fading atm = 1. As one might expect, increasing the number of diversity branches for the de-sired signal improves the system performance. For example, a 3 dB improvement in performance is recordedbetween N = 2 and N = 4 for BPSK detection at an average BER value of 10−4.Figure 4.2 shows the effect of the number of interfering signal, L, on average BER for noncoherent BFSKdetection. The desired signal experiences fading at ms = 2 due to the presence of LOS component(s) whilethe interfering signal experiences fading at mI = 0.75. The SINR value is 10 dB and ΩI = 3 dB. We observean approximate value of 5× 10−5 improvement to the average BER for every increase to N by two branches.As expected, the best performance is recorded when there are no interferers (i.e., L = 0).Figure 4.3 shows the curves for OP versus SINR for varying diversity order for the desired signal. Theinterfering signal experiences fading at mI = 1 while the desired signal experiences fading at ms = 2. Thepower in each LOS component for the desired and interfering signals is 1 dB.Figure 4.4 shows the average BER curves for the coherent 4-QAM detection over BX fading channelsversus SINR. The system’s interfering signal experiences more fading at mI = 0.75 while the desired signal654.4. Numerical Results1 2 3 4 5 6 7 8 9 1010-510-410-310-210-1Figure 4.2: Comparison of average BER versus number of interferers for noncoherent BFSK detection overBX fading channels with parameters of ms = 1, mI = 0.75, ΩI = 3dB, and SINR = 10 dB.0 5 10 15 2010-1510-1010-5100Figure 4.3: Comparison of OP versus average SINR over BX fading channels with parameters of L = 4, ms= 2, λs = 1.5 dB, λI = 1 dB, mI = 1, and ΩI = 3 dB.664.5. Chapter Summary0 2 4 6 8 1010-310-210-1100101Figure 4.4: Comparison of the average BER curves versus average SINR for coherent BPSK and 4-QAMdetections over BX fading channels with parameters of L = 4, ms = 1, λ2s = 3 dB, λ2I = 3 dB, mI = 0.75,and ΩI = 4 dB. Here, N is the number of branches of the desired signals.experiences less fading atms = 1. As expected, the system’s performance improves as the number of diversitybranches for the desired signal increases.4.5 Chapter SummaryIn this chapter, we analyzed the performance of short-range random access channels considering MRCin the presence of an arbitrary number of interfering signals. Our results are expressed in series form to seehow different parameters affect communication systems’ performance. Compared to integral forms, seriesexpressions are known to aid predictions and numerical optimization methods because their higher-orderderivatives can be easily obtained. We observe the effect of the number of diversity branches, interferers,and modulation schemes on the system’s performance. Notably, we recorded a worsened performance forhigher-order modulation schemes when other channel characteristics remained constant, as shown in Fig. 4.4.Ultimately, our result for the BX fading model goes beyond those obtained for existing models like the Riceanand Nakagami-m fading models, which are invalid for channels with LOS components (via direct propagationand specular reflections).67Chapter 5A New Statistical Channel Model formmWave and THz Communication SystemsThis chapter introduces a new composite model called the shadowed Beaulieu-Xie (SBX) model. Wedeveloped this new model to characterize wireless communication in an environment with an arbitrary numberof LOS components (via direct propagation and specular reflections) and NLOS components. The SBXmodel has a composite formulation with four parameters that characterize a wide range of fading conditions,unlike existing composite models. We validated the SBX model using experimental data obtained from fadingmeasurements in 28 GHz outdoor mmWave channels. We observe that the SBX model can accurately describethe mmWave and THz communication environments.The organization of this chapter is as follows. In Section 5.1, we present the motivation for developingthe SBX model. The physical model of the SBX model is presented in Section 5.2. We derive the first-order statistics of its envelope and instantaneous power in Section 5.3. In Section 5.4, we show the effectof shadowing on the K-factor estimator using moments. In Section 5.5, we present the SBX model’s fit tomeasured data. We derive closed-form expressions of the error rates and outage probability (OP) in Section5.6 while considering noncoherent and coherent detections. Section 5.7 presents and discusses the numericalresults. Concluding remarks are given in Section 5.8.5.1 MotivationThis section presents our motivation for developing the SBX model. We assume that the amplitude ofthe LOS components (via direct propagation and specular reflections) is Nakagami-m distributed3. Whileother composite models that assumed the LOS components to be fluctuating according to the Nakagami-mmodel did not justify their choice [9], [11], we will explain our assumption of Nakagami-m distributed LOS3The Nakagami-m RV is assumed for describing the fluctuating LOS components because its PDF offers analytical tractability.We have investigated the SBX model using the BX PDF, but we found it to be analytically intractable.685.2. Physical Modelcomponent as follows.We assume a varying LOS signal envelope based upon physical insight. The physical fading environmentcauses the received signal power to fluctuate in time and space due to shadowing and multipath fading [20].When the received signal combines LOS components (via direct propagation and specular reflections) andNLOS components, we have the envelope become BX distributed [24]. However, a partial or complete block-age of the LOS components can induce random variations to the received signal’s amplitude. We considershadowing only for the LOS components because they are concentrated in beams over narrow solid angles,i.e., small cross-sectional areas, making them preferentially sensitive to degradation from impinging obstruc-tions. The NLOS components are far less susceptible to degradation from impinging obstructions as they aredistributed over wide solid angles, i.e., large cross-sectional areas. The received signal envelope resultingfrom the LOS and NLOS components’ random variations presents an envelope not describable by existingunimodal, bimodal, and composite distributions. The SBX model overcomes this challenge by incorporat-ing distinct parameters to describe the fading and shadowing severities in the LOS and NLOS signals. TheSBX model’s parameters come about because the constituent Nakagami-m and BX models exhibit flexibilityvia their severity parameters. Moreover, the SBX model remains valid when there are more than two LOScomponents, a critical feature that other composite models do not incorporate.In this work, we present the SBX model’s probability functions in terms of relatively simple (hyper-geometric) functions. To justify the SBX model’s use in emerging systems, we fit its distribution to fieldmeasurements from [69].5.2 Physical ModelFor the SBX model, we assume that λ in (1.12) is fluctuating because of the dynamic nature of the channel.Hence, we use another RV, Y , to represent the fluctuating LOS amplitude, λ. The PDF of the new compositemodel can then be expressed asfR (r) = EY[2mXΩXrmXymX−1exp(−mXΩX(r2 + y2))ImX−1(2mXΩXyr)](5.1)whereEY [·] denotes the expectation with respect to Y , and ΩX is the average power of the NLOS componentsdue to multipath effects. The fading parameter, mX , characterizes the level to which both the LOS signals(via direct propagation and specular reflections) and NLOS signals are subject to fading. This parameter can695.3. Statistical Characterizationtake on values in the range of mX > 0 in the SBX model, characterizing severe fading at low mX values tolittle fading at high mX values. Note that the argument of the expectation operator in (5.1) is the PDF of theBX fading model.5.3 Statistical CharacterizationThis section obtains expressions for the first-order statistics of the SBX model in closed-form. The goal isto create an accurate statistical model with realistic and physical parameters characterizing the environmentaleffects. It then becomes possible to understand and overcome impairments in the system during the design,simulation, and testing. For this statistical model, the LOS components (via direct propagation and specularreflections) and NLOS components are assumed to experience different forms and levels of fluctuations, as isoften seen in practice.For such charaterizations, the PDF of the envelope of the SBX model isfR (r) =∫ ∞02mXΩXrmXymX−1exp(−mXΩX(r2 + y2))× ImX−1(2mXΩX yr)fY (y) dy (5.2)with fY (y) being the Nakagami-m PDFfY (y) =2mYmYΓ (mY ) ΩmYYy2mY −1 exp(−mY y2ΩY), y ≥ 0 (5.3)where Γ (·) is the gamma function, ΩY is the average power of the LOS components, and mY is the shad-owing parameter. This latter parameter characterizes the level to which the LOS components are subject toshadowing. It can take on values in the range of mY ≥ 0, depicting LOS shadowing from a limiting case ofcomplete obstruction (seen in a dense urban environment) to partial or no obstruction (seen in a suburban oropen environment). Since shadowing depends upon the transmitter and receiver’s positions and the positions(and materials) of obstacles, its impact on the LOS components would have no preferential direction in a gen-eralized wireless communication environment. Therefore, at best, one can use a statistical characterizationfor the net result of shadowing on all LOS components. Thus, shadowing severity can be indicated by thesingle parameter mY in the proposed model. Simultaneously, the SBX’s physicality can be seen by the phys-ical constraint of power conversation evident in its second-order moment. The SBX’s second-order moment,which is also its fading power, is independent of mX and mY .705.3. Statistical CharacterizationTable 5.1: Parameters of the SBX model as they appear in other fading modelsFading Model SBX ParametersRayleigh mX = 1,mY = 0,ΩX = ΩX ,ΩY = 0Ricean mX = 1,mY =∞,ΩX = ΩX ,ΩY = ΩYOne-sided Gaussian mX = 0.5,mY = 0,ΩX = ΩX ,ΩY = 0Beaulieu-Xie [24] mX = mX ,mY =∞,ΩX = ΩX ,ΩY = ΩYNakagami-m mX = 0,mY = mY ,ΩX = ΩX ,ΩY = 0SR [9] mX = 1,mY = mY ,ΩX = ΩX ,ΩY = ΩYWe obtain the PDF of the SBX’s envelope by solving (5.2), using [64, p. 706], to havefR (r) =2Γ (mX)(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mXr2mX−11F1(mY ;mX ;m2XΩY r2ΩX (mXΩY +mY ΩX))× exp(−mXr2ΩX)(5.4)where K = ΩYΩX is the K-factor given ΩY as the LOS power and ΩX as the NLOS power, and 1F1 (·; ·; ·) isthe confluent hypergeometric function defined as [64, p. 1023]1F1 (a; b;w) =∞∑n=0(a)n(b)nwnn!, (a)n =Γ (a+ n)Γ (a). (5.5)We observe that the PDF of the SBX model in (5.4) is expressed in terms of 1F1 (·; ·; ·), which can be obtainedwith standard mathematical software.From (5.4)-(5.5), the CDF of the SBX model, defined by FR (r) =r∫0fR (t)dt, can be obtained asFR (r) =2Γ (mY )(mXΩX)mX( mY ΩXmXΩY +mY ΩX)mY×∞∑n=0Γ (mY + n)n!Γ (mX + n)(m2XΩYΩX (mXΩY +mY ΩX))n×r∫0t2(mX+n)−1 exp(−mXΩXt2)dt. (5.6)After some algebraic manipulations, the CDF is ultimately obtained asFR (r) =1Γ (mY )(mXΩX)mX( mY ΩXmXΩY +mY ΩX)mY ∞∑n=0Γ (mY + n)n!Γ (mX + n)(mXΩYmXΩY +mY ΩX)n× g(mX + n,mXΩXr2)(5.7)715.3. Statistical Characterizationwhere g (v, u) is the incomplete gamma function given by [60]g (v, u) =u∫0bv−1 exp (−b) db, v > 0 = v−1uv1F1 (v; 1 + v;−u) . (5.8)The moments of the PDF in (5.4) areE[Rk]=∫ ∞−∞rkfR (r) dr=(mY ΩXmXΩY +mY ΩX)mY Γ (mX + k2)Γ (mX)(mXΩX)− k2× 2F1(mX +k2,mY ;mX ;mXΩYmXΩY +mY ΩX)(5.9)where the Gauss hypergeometric function is defined as [72]2F1 (a, b; c; z) =∞∑n=0(a)n(b)n(c)nznn!, (a)n =Γ (a+ n)Γ (a)(5.10)which converges if c is a positive value for all of |z| < 1. Note that the Gauss hypergeometric functionis also available in standard mathematical software. The CHF of the SBX model, defined by ΦR (ω) =∫∞−∞ fR (r) ejωrdr [25], can be obtained asΦR (ω) =∞∑k=0(jω)kk!E[Rk]=∞∑k=0(jω)kk!(mY ΩXmXΩY +mY ΩX)mY Γ (mX + k2)Γ (mX)× 2F1(mX +k2,mY ;mX ;mXΩYmXΩY +mY ΩX)×(mXΩX)− k2. (5.11)Equation (5.11) is a convergent infinite series representation of the CHF, which can also be computed usingthe NIntegrate function in Mathematica software directly from the definition of the CHF above.We define the instantaneous power as S (t) = R2 (t) [42], whose PDF is obtained by making the substi-tution fS (γ) = 12rfr (r) to havefS (γ) =1Γ (mX)(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mXγmX−1×1F1(mY ;mX ;m2XΩY γΩX (mXΩY +mY ΩX))× exp(−mXΩXγ). (5.12)725.3. Statistical CharacterizationThe CDF of the instantaneous power plays a vital role in obtaining the OP. We obtain the CDF of the instan-taneous power, defined by FS (γ) =γ∫0fS (t)dt, asFS (γ) =(mY ΩXmXΩY +mY ΩX)mY ∞∑n=0Γ (mY + n)n!Γ (mX + n) Γ (mY )×(mXΩYmXΩY +mY ΩX)n× g(mX + n,mXΩXγ). (5.13)The MGF of the instantaneous power, defined as Mγ (s) = E [exp (−γs)], is obtained asMS (γ) =(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mX ∞∑n=0Γ (mY + n)n!Γ (mY )(m2XΩYΩX (mXΩY +mY ΩX))n×(γ +mXΩX)−(mX+n). (5.14)The expressions for the first-order statistics shown above are relatively simple, allowing for easy mathematicalmanipulation and verification via simulations with standard mathematical software. The CDF and MGF ofthe instantaneous power shown in (5.13) and (5.14) also allow for straightforward calculation of the OP andaverage BER, respectively.The derived expression for the PDF of the SBX model’s envelope and the MGF of its SNR can presentinsight into its diversity order. We obtain the diversity order by observing its PDF’s behavior near the originor the asymptotic behavior of the MGF of the SNR as the SNR approaches infinity [70]. Equally, the slopeof the log-log plot of the BER or OP can define the diversity order of a system. Thus, we obtain the diversityorder of the SBX model as the fading parameter mX , which is associated with the degree of freedom of theSBX model. Appendix C shows the derivation for the diversity order of the SBX from the expression of itsBER.As a final point on statistical characterization, we identify the relationships between the SBX model’sparameters and other fading models. Table 5.1 summarizes the connections between these models. As avital illustration, we justify the SBX model by establishing its relevant relationship with the Ricean model’sparameter. The SBX model’s parameter, (mX ,ΩX ,mY ,ΩY ) are linked to the the Ricean model’s parameter,(1, b0,∞,Ω), where b0 sets the power of the NLOS components, and Ω is the power of the LOS components.In fitting the measured data of [69], we found that the SBX model’s parameters, mX and mY , can bettermatch the given PDF and provide a better overall fit than the Ricean fading model.735.4. Effect of Shadowing on the K-factor Estimator5.4 Effect of Shadowing on the K-factor EstimatorThis section shows that the BX fading model’s estimator cannot be applied to the SBX model, even thoughthe LOS and NLOS components’ power in both models have the same origin. The comparison between thesemodels shows LOS shadowing’s effect on the K-factor of the SBX model. This comparison is an essentialstep in our analysis because the underlying BX fading model is a newly-prosed fading model with no priorvalidations between analytical results and field measurementsFollowing the identity [71, p. 559]2F1 (a, b; c; z) = (1− z)−b 2F1(c− a, b; c; zz − 1); z < 1, (5.15)we can rewrite the moment of the SBX model in (5.9) asµk = E[Rk]=Γ(mX +k2)Γ (mX)(mXΩX)− k22F1(−k2,mY ;mX ;−mXΩYmY ΩX). (5.16)We consider the expression of the confluent hypergeometric function, which can be applied to the case wherethere is no shadowing, i.e., mY =∞, as [60]1F1 (a; c; z) = limb→∞2F1(a, b; c; b−1z). (5.17)Applying (5.17) to (5.16), we obtain the moment of the SBX model for mY =∞ asµk = E[Rk]=Γ(mX +k2)Γ (mX)(mXΩX)− k22F1(−k2;mX ;−mXΩYΩX)(5.18)which is equivalent to the moment of the BX fading model. Given the relationship that exists between theconfluent hypergeometric function and the Laguerre polynomial as [71, p.775]Lαn (x) =(α+ 1)nn!1F1 (−n;α;x) , (5.19)the moment of the BX fading model in (5.18) can be written exactly as [24]µk = E[Rk]= Γ(k2+ 1)(ΩXmX)− k2LmX+1k2(−mXΩYΩX). (5.20)745.4. Effect of Shadowing on the K-factor EstimatorWe continue by defining a moment-ratio function as [76]fi,j (K,mY ,mX) =µjiµij; i 6= j (5.21)where µi and µj are the ith and the jth moment of R. According to our derivation in Appendix D, we showthe first-order, second-order, and fourth-order moments of R areµ1 =Γ(mX +12)Γ (mX)(mXΩX)− 12× 2F1(−12,mY ;mX ;−mXΩYmY ΩX), (5.22)µ2 = ΩX [1 +K] , (5.23)µ4 = Ω2X[1 + 2K +mX (mY + 1)mY (mX + 1)K2], (5.24)respectively. We obtain the moment-ratio functions defined in (5.21) asf1,2 (K,mX ,mY ) =µ21µ12={Γ(mX +12)Γ (mX)}22F21(−12 ,mY ;mX ;−mXKmY)(1 +K)(5.25)andf2,4 (K,mX ,mY ) =µ42µ24= (K + 1)2(1 + 2K + mX(mY +1)mY (mX+1))K22 . (5.26)In the absence of shadowing, i.e., mY =∞, f1,2 (K,mX ,∞) and f2,4 (K,mX ,∞) are given byf1,2 (K,mX ,∞) ={Γ(mX +12)Γ (mX)}21F1(12 ;mX ;−mXK)(1 +K)(5.27)andf2,4 (K,mX ,∞) = (1 +K)2(1 + 2K +K2 + K2mX )2. (5.28)To show the effect of shadowing, we employ the performance criterion of normalized error as [74]εi,j (K,mX ,mY ) =∣∣∣∣∣K − K˜K∣∣∣∣∣ (5.29)755.5. Fit to Published Measurementswhere i = 1, j = 2 or i = 2, j = 4, K is the true K-factor of the SBX model, and K˜ is the estimated K-factorof the BX fading model, in the absence of shadowing. Thus, K˜ is computed by solving the expressionfi,j (K,mX ,mY ) = fi,j(K˜,mX ,∞). (5.30)To ensure that we obtain a closed-form expression from (5.30) , we employ a combination of the moment-ratiofunctions K2,4 in (5.26) and K∞2,4 in (5.28) only. This is because a combination of K1,2 in (5.25) and K∞1,2in (5.27) will lead to an expression whose result can only be obtained numerically. Therefore, we continueto show the effect of shadowing on the estimated K-factor for the SBX model (with shadowing present) incomparison to that of the BX fading model (with shadowing absent) by solving the expression in (5.30) forthe BX estimator analytically, K˜, to haveK˜ =mX(µ22 − µ4)(mXµ22 + µ2 −mXµ4)+√(mX(µ22 − µ4))2 −mX (µ22 − µ4) (mXµ22 + µ22 −mXµ4)(mXµ22 + µ2 −mXµ4) . (5.31)Thus, the normalized error is obtained by inserting (5.31) into (5.29). The following figures show comparisonsof the normalized error curves for different fading and shadowing severities.Figure 5.1 shows the normalized error curves for the BXK-factor estimator (in the absence of shadowing)as a function of the SBX K-factor (as the true value of the K-factor given the presence of shadowing) fordifferent values of mY . We observe that the normalized error improves as the value of mY increases for thesame value of mX . Also, the normalized error is seen to increase as the true value of the K-factor increases,as expected. Ultimately, the resulting normalized error trend shows that the BX K-factor estimator based on(5.28) does not provide the correct K-factor estimates in the presence of shadowing.Figure 5.2 shows the normalized error for the BX K-factor estimator as a function of the SBX K-factorfor different values ofmX andmY . We observe that the BXK-factor estimator performs better, with reducednormalized error, in the presence of severe fading and less shadowing. This reduced normalized error suggeststhat the BX K-factor estimator based on (5.28) is best applied to fading scenarios only.5.5 Fit to Published MeasurementsTo validate the proposed SBX model, we compare its fit to published data for a mmWave communi-cation environment with small-scale fading in the 28 GHz band [69], against those of the Rayleigh, Ricean,765.5. Fit to Published Measurements1 2 3 4 5 6 7 8 9 100.650.70.750.80.850.90.951Figure 5.1: Normalized error curves of the BX K-factor estimator of the BX fading model versus the trueK-factor of the SBX model for K˜2,4 and mX = 1.5.1 2 3 4 5 6 7 8 9 100.70.750.80.850.90.951Figure 5.2: Normalized error curves of theK-factor estimator of the BX fading model versus the trueK-factorof the SBX model showing the combined effects of fading and shadowing.775.5. Fit to Published MeasurementsNakagami-m, FTR, and BX fading models. The Rayleigh model has the parameter, σr, to represent the powerof the NLOS components as 2σ2r . The Ricean model has the parameters, s2 and 2σ2, to quantify the power ofthe LOS and NLOS components, respectively. The Nakagami-m fading model has the parameters, mn andΩn, to quantify the fading severity and power of the NLOS components, respectively. The fittings carried outin this section make a fair comparison between the BX model, SBX model, and other existing models thathave characterized (the same) experimental data [11] [69]. At the same time, such characterizations provideinsight into the SBX model’s differing functionality and those of its parent BX model. In doing this, we havefound that the combined effects of the parameters mX and mY can better encompass (the more challenging)characteristics of emerging systems, leading to better overall fitting.To carry out the fitting, we compare the analytical CDF of the envelope of our model, FR (r), with theempirical CDF of the published data, FˆR (r). We obtain the parameter values for minimized error betweenthese CDFs and quantify the analytical CDF’s deviation from the empirical CDF using the Kolmogorov-Smirnov (KS) test [67] in two ways. We obtained the SBX model’s parameters by establishing their relevantlink to the Ricean model’s parameter obtained via the fit in [69]. We obtained other SBX model’s parametersthat were not based upon existing measurements via a grid search that minimizes the KS test value. The KStest is given asKS∆= max∣∣∣FˆR (r)− FR (r)∣∣∣ . (5.32)Based upon (5.32), the smaller the value of KS obtained, the better the fit of the analytical CDF to theempirical CDF.Figure 5.3 compares the empirical CDF obtained from the vertical-vertical co-polarized LOS measure-ments in [69] with the FTR, the BX, and the SBX models’ analytical CDFs. The analytical CDFs for thesefading models fit the measured data closely. However, the BX fading model provides the best fit. This isbecause the measurement environment had no shadowing or fluctuations to the LOS component(s). We con-firmed the absence of shadowing via the SBX model’s PDF, which remained unchanged despite changes tothe value ofmY (quantifying the shadowing severity). The goodness-of-fit for the models are calculated using(5.32), which gives KSRAYLOS = 0.1786, KSRICLOS = 0.1786, KSNAKLOS = 0.1071, KSFTRLOS = 0.0714, KSBXLOS =0.0612, andKSSBXLOS = 0.0708, respectively. Ultimately, the BX fading model shows its advantage in fitting anenvironment with no shadowing. Besides, the results in Fig. 5.3 confirm that the SBX model indeed reducesto the BX fading model when mY →∞.Figure 5.4 compares between the empirical CDF for vertical-horizontal co-polarized NLOS measurements785.5. Fit to Published Measurements-25 -20 -15 -10 -5 0 510-510-410-310-210-1100Figure 5.3: Fits of the FTR, BX, and SBX models to empirically obtained signal amplitude [69] for vertical-vertical co-polarized LOS measurements. The Rayleigh fading model has 2σ2r = -4 dB, yielding KSRAYLOS =0.1786. The Ricean fading model has the parameters of s2 = -10 dB and 2σ2 = -5.5 dB, yielding KSRICLOS =0.1786. The Nakagami-m fading model has parameters of mn = 0.75 and Ωn = 0.5 dB, yielding KSNAKLOS =0.1413. The FTR fading model has parameters of m = 80, ∆ = 0.6, and KFTR = 22 dB, yielding KSFTRLOS= 0.0714. The BX fading model has parameters of m = 0.65, Ω = -2 dB, and λ = -1 dB, yielding KSBXLOS =0.0612. The SBX model has parameters of mX = 0.65, mY = 0.7, ΩX = -2 dB, and ΩY = -5 dB, yieldingKSSBXLOS = 0.0708.795.5. Fit to Published Measurements-20 -15 -10 -5 0 510-410-310-210-1100Figure 5.4: Fits of the FTR, BX, and SBX models to empirically obtained signal amplitude [69] for vertical-horizontal co-polarized NLOS measurements. The Rayleigh fading model has 2σ2r = -0.8 dB, yieldingKSRAYLOS = 0.1857. The Ricean fading model has the parameters of s2 = -0.8 dB and 2σ2 = -5.3 dB, yieldingKSRICLOS = 0.1837. The Nakagami-m fading model has parameters of mn = 0.75 and Ωn = 1.1 dB, yieldingKSNAKLOS = 0.1082. The FTR fading model has parameters of m = 5, ∆ = 0.75, and KFTR = 12, yieldingKSFTRLOS = 0.1236. The BX fading model has parameters of m = 0.7, Ω = 1.1 dB, and λ = -2 dB, yieldingKSBXNLOS = 0.0818. The SBX model has parameters of mX = 0.7, mY = 1.1, ΩX = 2 dB, and ΩY = -7.5 dB,yielding KSSBXNLOS = 0.0770.in [69] with the FTR, BX, and SBX models’ analytical CDFs. The goodness-of-fit is obtained as KSRAYLOS =0.1857 for the Rayleigh model, KSRICLOS = 0.1837 for the Ricean model, KSNAKLOS = 0.1082 for the Nakagami-m model, KSFTRNLOS = 0.1236 for the FTR model, KSBXNLOS = 0.0818 for the BX model, and KSSBXNLOS =0.0770 for the SBX model. Here, the SBX model provides the best fit because the LOS components experiencefluctuations due to varying channel characteristics.We note that the value of each data point in the measured CDF depends on (i) the individual multipathcomponent in the bin over the local area and (ii) the amplitude of the previous data point, considering thatthe distribution is cumulative. Thus, the SBX model’s performance when characterizing the measured datain the low r regime is based on the two points. The amplitude fluctuations over the local area that producesthe values below –15 dB are high in Fig. 5.3, indicating significant fading compared to the lower amplitudefluctuations that lead to higher values above –15 dB. In Fig. 3, the CDF of the measured data deviates far lessfrom the analytical CDF because the measured data was obtained for an environment with prevalent NLOSpropagation. Thus, the local areas are rich in multipath signal components, leading to higher amplitude values805.6. Performance Analysisand better overall fits in the figure.5.6 Performance AnalysisThis section presents performance results for the average BER and OP. Because the SBX model has simpleexpressions for its first-order statistics and those related to the SNR, the BER and OP results are obtained withminimal mathematical manipulations.5.6.1 Average Bit-Error RateIn this section, we calculate the average BER, PE , which is defined as the conditional error probability,Pe (γ), averaged over the output SNR in an additive white Gaussian noise channel,PE =∞∫0Pe (γ) fS (γ) dγ (5.33)where fS (γ) is the PDF of the SNR obtained in (5.12). We continue in the following subsections by derivingthe expressions for noncoherent and coherent detections.Noncoherent DetectionThe average BER is defined for noncoherent frequency-shift keying (FSK) or differential phase-shiftkeying (PSK) detections as [60]Pe (γ) =12exp (−$γ) , (5.34)where$ = 1, for Binary PSK12 , for Binary FSK.(5.35)Substituting (5.12) and (5.34) into (5.33), we havePE =12Γ (mX)(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mX ∞∫0γmX−11F1(mY ;mX ;m2XΩY γΩX (mXΩY +mY ΩX))× exp[−(mXΩX+$)γ](5.36)815.6. Performance Analysiswhere 1F1 (·; ·; ·) is defined in (5.5). Using [64, p. 822], we obtain the average BER for noncoherent detectionasPE =12(mY ΩXmXΩY +mY ΩX)mY2F1(mY ,mX ;mX ;m2XΩY(mX +$ΩX) (mXΩY +mY ΩX))×(mXmX +$ΩX)mX(5.37)where 2F1 (·, ·; ·; ·) is defined in (5.10).Coherent DetectionFor coherent detection, Pe (γ) is defined as [60]Pe (γ) =12erfc (√$γ) (5.38)where $ is defined in (5.35) and erfc is the complementary error function defined in terms of the upperincomplete gamma function aserfc (x) =Γ(12 , x2)2√pi. (5.39)Inserting (5.12) and (5.39) into (5.33) givesPE =12√piΓ (mX)(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mX×∞∑n=0Γ (mY + n)n!Γ (mY )Γ (mX)Γ (mX + n)×(m2XΩYΩX (mXΩY +mY ΩX))n ∞∫0γmX+n−1 exp(−mXΩXγ)Γ(12, $γ)dγ. (5.40)Using [64, p. 657], we obtain the average BER for coherent detection asPE =√$2√pi(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mX ∞∑n=0Γ (mY + n)n!Γ (mY )×(m2XΩYmXΩY +mY ΩX)n× Γ(mX + n+12)Γ (mX + n+ 1)×(ΩX$ΩX +mX)mX+n+ 12×2F1(1,mX + n+12;mX + n+ 1;mX$ΩX +mX)(5.41)where 2F1 (·, ·; ·; ·) is defined in (5.10). We observe from our results of the average BER with coherentdetection that the relative error drops from 1 percent to 0.5 percent when the number of terms that are appliedin the summations is increased from 50 to 100. In Appendix E, we show that the result in (5.41) approaches825.6. Performance Analysiszero as n approaches infinity. The error in truncating the infinite series in (5.41) is given byC =√$2√pi(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mX ∞∑n=NΓ (mY + n)n!Γ (mY )(m2XΩYmXΩY +mY ΩX)n× Γ(mX + n+12)Γ (mX + n+ 1)×(ΩX$ΩX +mX)mX+n+ 12×2F1(1,mX + n+12;mX + n+ 1;mX$ΩX +mX).(5.42)After some algebraic manipulation, the truncation error expressed in (5.42) can be bounded byC ≤√$2√piΓ (mY )(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mX ($ +mXΩX)−(mX+N+ 12)( m2XΩYmXΩY +mY ΩX)N×2 F1(1,mX +N +12;mX +N + 1;mX$ΩX +mX)Γ (mY +N − 1) Γ(mX +N +12)Γ (mX +N)×2F2(mY +N,mX +N +12;N + 1,mX +N + 1;m2XΩYmXΩY +mY ΩX)(5.43)where 2F2 (·, ·; ·, ·; ·) is the generalized hpergeometric function, pFq (a1, · · · , ap; b1, · · · , bq; z), with p = 2and q = 2 [64].5.6.2 Outage ProbabilityThe OP is defined as the probability that the SNR falls below a given threshold level, γb [20] [60]. It isgiven byPout = p (γ < γb) =γb∫0fS (γ) dγ, (5.44)whereγb = −1$ ln (2P∗e ) , for noncoherent detection1$erfc−1 (2P ∗e ) , coherent detection.(5.45)Inserting (5.12) into (5.44), we havePout =1Γ (mX)(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mX×γb∫0γmX−1 exp(−mXΩXγ)×1F1(mY ;mX ;m2XΩY γΩX (mXΩY +mY ΩX))dγ. (5.46)835.7. Numerical ResultsFrom (5.5)-(5.46), the OP becomesPout =1Γ (mX)(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mX ∞∑n=0Γ (mY + n) Γ (mX)n!Γ (mX + n) Γ (mY )(m2XΩYΩX (mXΩY +mY ΩX))n×γb∫0γmX+n−1 exp(−mXΩXγ)dγ. (5.47)Using [64, p. 346], we obtain the OP asPout =(mY ΩXmXΩY +mY ΩX)mY ∞∑n=0Γ (mY + n)n!Γ (mX + n) Γ (mY )(mXΩYmXΩY +mY ΩX)ng(mX + n,mXΩXγb).(5.48)We show in Appendix D that the result in (5.48) approaches zero as n approaches infinity. The error intruncating the infinite series in (5.48) is given byOP =(mY ΩXmXΩY +mY ΩX)mY ∞∑n=NΓ (mY + n)n!Γ (mX + n) Γ (mY )(mXΩYmXΩY +mY ΩX)ng(mX + n,mXΩXγb).(5.49)After some algebraic manipulation, the truncation error can be bounded byOP ≤(mY ΩXmXΩY +mY ΩX)mYg(mX +N,mXΩXγb)1Γ (mY )Γ (mY +N)Γ (mX +N)(mXΩYmXΩY +mY ΩX)N× 2F2(1,my +N ;N + 1;mX +N ;mXΩYmXΩY +mY ΩX). (5.50)The expression in (5.50) approaches zero as n approaches infinity.5.7 Numerical ResultsThis section presents simulation results of the average BER and the OP for the SBX model. We ob-tained the figures using the closed-form expressions in (5.37) for the noncoherent BER detection, in (5.41)for the coherent BER detection, and in (5.48) for the OP. In comparison with numerical integration, theseclosed-form expressions have the advantage of providing insight into the effects of parameters in wirelesscommunication systems. These insights enable better evaluations of system performance. It is reasonable toassume that the NLOS components experience worse fading in a typical communication environment than the845.8. Chapter Summary0 2 4 6 8 10 12 14 16 18 2010-610-410-2100Figure 5.5: Comparison of average BER curves versus average SNR for noncoherent FSK and coherent BFSKdetections over SBX channels with parameters of mY = 2 and ΩY = 1 dB.LOS components. Therefore, we allow the fading parameters to take values in the range mX ≤ mY ≤ 0.5.Figure 5.5 shows a comparison of the BER curves for the noncoherent FSK and coherent binary FSKdetections as a function of the average SNR for different values of mX . We observe that the coherent binaryFSK detection have improved performance over the noncoherent detection, with both having improved systemperformance as the value of mX increases. For example, the coherent detection shows approximately a 5 dBimprovement for mX = 4 over mX = 2 at a BER value of 10−3 with mY = 2. For noncoherent detection,an approximate of 5 dB improvement is seen for mX = 4 over mX = 2.Figure 5.6 shows a comparison of the OP curves as a function of the average SNR for different values ofmX for noncoherent and coherent FSK detections. Improved performance is observed for coherent detectionover noncoherent detection while keeping mX constant. For example, a value of mX = 3 and an OP of 10−2yields a 3.5 dB improvement for coherent FSK detection over noncoherent FSK detection.5.8 Chapter SummaryThis chapter proposed a new composite model referred to as the SBX model. The model is based upona recently proposed BX fading model for generalized LOS components (via direct propagation and specu-lar reflections) and NLOS components. We assume that the LOS component’s amplitude is Nakagami-m855.8. Chapter Summary-5 0 5 10 1510-410-310-210-1100Figure 5.6: Comparison of OP curves versus average SNR for noncoherent and coherent FSK detections overSBX channels with parameters of mY = 2 and ΩY = −1 dB.distributed for easy mathematical manipulation. We obtained the SBX model’s probability functions of theamplitude and power, average BER, and OP in closed-form. While we concluded that further experimentalstudies are required, we could see both the BX and SBX models’ potential. Ultimately, the BX and SBXmodels could lay the groundwork for characterizing emerging mmWave and THz wireless communicationsystems.86Chapter 6Performance of Wireless PoweredCommunication over Beaulieu-Xie ChannelsWith Energy HarvestersThis chapter investigates the performance of WPC systems over the BX fading model for two new non-linear energy harvesters - multi-segment piecewise linear (MSPwL) and arctangent (ArT). The WPC systemconsiders a wireless device that scavenges energy from the energy source in the downlink and uses the har-vested energy to communicate to a data fusion receiver in the uplink. To validate these new energy harvesters,we fit them to measured data obtained from far-field RF environment and quantify their goodness-of-fit viaroot mean-square-error (RMSE). We employ performance metrics of average harvested energy and trans-mission OP for the newly-proposed nonlinear EH models, assuming BX fading in the downlink and uplinkchannels. The analytical results are verified via simulations.This chapter is organized as follows. Section 6.1 presents the motivation for carrying out the analysis inthis chapter. Section 6.2 presents the system overview for this work. Section 6.3 has the performance analysisof the MSPwL EH model. Section 6.4 presents the performance analysis of the ArT EH model. We presentand discuss the numerical results in Section 6.5 and give some concluding remarks in Section 6.6.6.1 MotivationEnergy harvesting (EH) has received attention from researchers because of the seamless and sustainablepower supply from energy sources like communication signals, solar and wind power, vibration, and manymore. In many studies, RF signals’ ability to carry information and energy at the same time has been ex-ploited with the introduction of wireless powered communication networks (WPCNs) [79]. To this end, pro-tocols have been introduced to allow for the harvest of energy over a fraction of a time slot and information876.1. Motivationtransmission in the remaining fraction of that time slot [80]. The importance of EH has become even morepronounced with the emergence of IoT technology, which consists of many low-power sensor nodes (eitherin the human body or in physical structures like buildings and bridges) that demand power [81]-[82]. Thesenodes are usually powered with batteries and have a limited life span. Thus, EH has become an attractivesolution to keep the nodes active and the network operational.A key challenge to implementing effective EH has been its circuitry’s characterization. It was consideredas linear by early works [80]-[86]. However, these circuits’ nonlinear characteristics are evident in the experi-mental results because of diodes and rectifiers [87]-[88]. A significant performance degradation was recordedfor resource allocation schemes based on linear EH models compared to nonlinear EH models [38]. Thelinear EH models’ deficiencies led to the introduction of nonlinear EH models. Examples of those nonlinearEH model include the piece-wise linear function [35], the polynomial model [36], and the sigmoid function[30]-[37].A second challenge to implementing effective EH has emerged through the development of high-efficiencyantennas and rectifiers for mmWave and THz systems [89]. These wireless communication systems havereceived tremendous interest because of their ability to increase capacity beyond that of the traditional RFspectrum [90]. However, their high carrier frequencies lead to more significant energy usage and an overallreliance on wireless communication through multiple LOS components (via direct propagation and specularreflections) and NLOS components. Such systems are mostly employed for short-range communications insmall cells due to high propagation loss. Thus, mmWave and THz systems’ communication channels mustbe characterized by a fading model that can incorporate multiple LOS (via direct propagation and specularreflections) and NLOS components.The Nakagami-m fading model was introduced as a suitable model for characterizing the channels incommunication hops of EH systems [35], [91]. However, the Nakagami-m model’s deficiency in characteriz-ing multiple LOS components (via direct propagation and specular reflections) makes it less adequate for EHuse. Thus, we employ the BX fading model for this section’s analysis.We introduce two nonlinear EH models, which we refer to as MSPwL and ArT. These EH models areshown to present an excellent fit to the measured data from [32]. The fitting is done via a best-fit search, withthe goodness-of-fit quantified using RMSE. To our best knowledge, the analysis in this chapter is the first toemploy the BX fading model in performance analyses of WPC networks.886.2. System OverviewFigure 6.1: System model.6.2 System Overview6.2.1 System ModelWe consider a point-to-point WPC system where a single-antenna wireless device (R) scavenges energyfrom an NS-antenna energy source (S) and communicates with an ND-antenna data fusion receiver (D) asshown in Fig. 6.1. The distance between the S and R is d1 in meters while the distance between R and Dis d2 in meters. After obtaining the channel state information, hSR, between the energy source, S and, thewireless device, R, with hSR ∈ CNS , where CNS is the set of all column vectors of size NS with complex-value entries, the energy source performs maximal-ratio transmission using the beamforming vector hSR‖hSR‖and transmit power pSR. Therefore, the received signal at the wireless device, R, is denoted byyS = ‖hSR‖√pSRxSR + zS (6.1)where xSR is the transmitted signal, and zS is additive white Gaussian noise (AWGN) with zero mean and avariance of σ2S . The wireless device, R, harvests energy from the RF signal, and its usable power is given byPRD = $PφEH(pSR ‖hSR‖2)(6.2)where φ represents the energy harvester model, and $ is a parameter that depends on the switching proto-col, such as the time-switching (TS) protocol [35] and the antenna-switching (AS) protocol [96]. After the896.2. System Overviewmaximal-ratio combining via hRD‖hRD‖ , the received signal at the data fusion receiver, D, is denoted byyD = ‖hRD‖2√PRDxRD +hHRD‖hRD‖zD (6.3)where hRD ∈ CND is the complex channel coefficients, and zD ∈ CND is AWGN with zero mean and acovariance of σ2DIND , where IND is a unit matrix. Therefore, the received SNR at the data fusion receiver, DisγD =PRD ‖hRD‖2σ2D(6.4)where σ2D is the noise power.6.2.2 Nonlinear Energy Harvester ModelsIn this work, we consider two EH models: the MSPwL and the ArT energy harvesters.Multi-Segment Piecewise Linear Energy HarvesterThe MSPwL energy harvester approximates its input-output relation with linear segments. We consider anMSPwL energy harvester with N segments, for which the input-output relation isPMSPwLEH(pSR ‖hSR‖2)= anpSR ‖hSR‖2 + bn, cn−1 ≤ pSR ‖hSR‖2 ≤ cn (6.5)where an ≥ 0 and bn are the slope and intercept of the nth segment, respectively. Specifically, bn =∑n−1k=1 (ak − ak+1) ck + b1, with b1 = 0, n = 2, 3, . . . , N , and 0 = c0 < c1 < . . . < cN =∞.Arctangent Energy HarvesterThe ArT energy harvester approximates its input-output relation by an arctangent function asPArTEH(pSR ‖hSR‖2)= a0 tanh(b0pSR ‖hSR‖2)(6.6)where a0 ≥ 0 and b0 ≥ 0 are the two key parameters.906.2. System Overview6.2.3 Review of the BX Fading ModelThe key features of the BX fading model are reviewed here. We consider a BX RV, Y , which is derivedfrom the noncentral chi RV via normalization [24], [90]. The distribution of X = |Y |2 is noncentral chi-squared, i.e., χ2(2m, 2mΩ λ2), according tofX (x) =mΩexp[−mΩ(x+ λ2)] ( xλ2)m−12Im−1(2mλΩ√x). (6.7)The complementary cumulative distribution function (CCDF) of the RV, X , isF cX (x) = QM(λ√2mΩ,√2mxΩ)(6.8)where QM (·, ·) is the Marcum Q-function [97]. Hence, the distributions of ‖hSR‖2 and ‖hRD‖2 are χ2(2m1NS ,2m1ΩSRλ2SR)and χ2(2m2ND,2m2ΩRDλ2RD), respectively, where m1, λ2SR, and ΩSR are the fadingseverity parameter, power of the LOS component(s), and power of the NLOS components of S-R link, re-spectively, while m2, λ2RD, and ΩRD are the fading severity parameter, power of the LOS component(s), andpower of the NLOS components of R-D link, respectively. It is important to note that λ2SR and λ2RD can beobtained via the relation λ2SR =∑NSl=1 λ2l and λ2RD =∑NDj=1 λ2j , respectively, where λ2l and λ2j are the powerof the LOS component per branch in the S-R and R-D links, respectively.6.2.4 Performance MetricsWe consider the performance metrics of expected EH rate (EEHR) and transmission outage probability(TOP). The EEHR is defined asPRD = E [PRD] . (6.9)Based on (6.3), the information rate between the wireless device, R, and the data fusion receiver, D, is denotedbyRD = %W log(1 +γD%)(6.10)where W is the system bandwidth, and % is specified by the switching protocol under study. Ultimately, theparameters $ and % are factors that denote the EH architecture. The case with $ = 1 and % = 1 corresponds916.3. Performance of WPC with the MSPwL Energy Harvesterto the AS protocol, while the case with $ = ρ and % = 1− ρ corresponds to the TS protocol with ρ ∈ [0, 1].Thus, the TOP, which is defined as the probability that the transmission rate falls below a predefined threshold,can be expressed asTOP = Pr {RD ≤ Rreq} = Pr{‖hRD‖2 ≤ %σ2DPRD[exp(Rreq%W)− 1]}(6.11)where Rreq is a predefined threshold.6.3 Performance of WPC with the MSPwL Energy Harvester6.3.1 Expected Energy Harvesting Rate for MSPwLIn this section, we analyze the EEHR for the MSPwL energy harvester. We express the EEHR asPRD = E [PRD] = Pr {PRD > x} =∫ ∞0F cRD (x) dx (6.12)where the CCDF of PRD (as derived in Appendix G) isF cPRD = QNSm1(λSR√2m1ΩSR,√2m1aipSRΩSR( x$− bi)), x ∈ [$di−1, $di] . (6.13)Inserting (6.13) into (6.12) leads to an EEHR expression ofPRD =∫ $di$di−1QNSm1(λSR√2m1ΩSR,√2m1aipSRΩSR( x$− bi))dx. (6.14)Applying the series expression of the Marcum Q-function [97]QM (α, β) = exp(−(α2 + β2)2) ∞∑q=1−M(αβ)qIq (αβ) (6.15)to (6.14) leads toPRD = exp(−m1λ2SRΩSR) ∞∑q=1−m1NS(λSR√aipSR)q∫ $di$di−1exp(− m1ΩSRaipSR( x$− bi))×( x$− bi)− q2Iq(2m1λSRΩSR√aipSR√x$− bi)dx. (6.16)926.3. Performance of WPC with the MSPwL Energy HarvesterApplying an identity to the modified Bessel function of the first kind according to [64, p. 919]Iv (z) =∞∑w=01w!Γ (w + v + 1)(z2)2w+v(6.17)where Γ (·) is the gamma function, eq. (6.16) becomesPRD = exp(−m1λ2SRΩSR) ∞∑q=1−m1NS∞∑w=01w!Γ (w + q + 1)λ2q+2wSR (aipSR)−w(m1ΩSR)2w+q×∫ $di$di−1( x$− bi)wexp(− m1ΩSRaipSR( x$− bi))dx. (6.18)After rearranging (6.18), we obtain the expression for the EEHR asPRD = exp[− m1ΩSR(λ2SR −biaipSR)] ∞∑q=1−m1NS∞∑w=01w!Γ (w + q + 1)(m1ΩSR)2w+q× λ2q+2wSR (ai$pSR)−w∫ $di$di−1(x−$bi)w exp(− m1ΩSR(xai$pSR))dx. (6.19)6.3.2 Transmission Outage Probability for MSPwLHere, we analyze the TOP for the MSPwL energy harvester. As expressed in (6.11), the TOP is defined asTOP = Pr {RD ≤ Rreq} = Pr{PRD ≤ ε‖hRD‖2}= 1−∫ Pmax01εf 1‖hRD‖2(xε)F cpRD (x) dx (6.20)where ε , σ2D%[exp(Rreq%W)− 1]. The CDF of 1‖hRD‖2 isF 1‖hRD‖2(p) = Pr{1‖hRD‖2≤ p}= Pr{‖hRD‖2 ≥ 1p}= Qm2ND(λRD√2m2ΩRD,√2m2ΩRDp). (6.21)The PDF of 1‖hRD‖2 is obtained by taking the first-order derivative of (6.21) as [97]f 1‖hRD‖2(p) =m2ΩRDλ1−NDm2RD(1p)NDm2+32exp(− m2ΩRD(λ2RD +1p))INDm2−1(2m2λRDΩRD√p). (6.22)936.4. Performance of WPC with the ArT Energy HarvesterThe TOP is obtained by substituting (6.13) and (6.22) into (6.20) to haveTOP = 1−N−1∑i=1m2ΩRDεNDm2+12 λ1−NDm2RD exp(− m2ΩRDλ2RD)∫ $di$di−1(1x)NDm2+32exp(− m2εΩRDx)× INDm2−1(2m2λRD√εΩRD√x)QNSm1(λSR√2m1ΩSR,√2m1aipSRΩSR( x$− bi))dx. (6.23)Applying (6.15) to (6.23) leads toTOP = 1−N−1∑i=1m2ΩRDεNDm2+12 λ1−NDm2RD exp(− m1ΩSRλ2SR −m2ΩRDλ2RD) ∞∑q=1−m1NS(λSR√aipSR)q×∫ $di$di−1(1x)NDm2+32 ( x$− bi)− q2exp(− m1aiΩSRpSR( x$− bi)− m2εΩRDx)× Im2ND−1(2m2λRD√εΩRD√x)Iq(2m1λSRΩSR√aipSR√( x$− bi))dx. (6.24)Applying (6.17) to (6.24), we obtain the expression for the TOP asTOP = 1−N−1∑i=1exp(− m1ΩSRλ2SR −m2ΩRDλ2RD) ∞∑q=1−m1NS∞∑w=01w!Γ (w +m2ND)×∞∑s=01s!Γ (s+ q + 1)εw+NDm2(m1ΩSR)2s+q (λ2SR)s+q(aipSR)−s λ2wRD(m2ΩRD)2w+m2ND×∫ $di$di−1(1x)w+NDm2+1 ( x$− bi)sexp(− m1aiΩSRpSR( x$− bi)− m2εΩRDx)dx. (6.25)PRD and TOP both contain one integral, which can be solved using standard mathematical software likeMatlab or Mathematica.6.4 Performance of WPC with the ArT Energy Harvester6.4.1 Expected Energy Harvesting Rate for ArTThe EEHR is defined for the ArT energy harvester asPRD = E [PRD] =∫ ∞0Pr {PRD > x} =∫ ∞0F cRD (x) dx (6.26)946.4. Performance of WPC with the ArT Energy Harvesterwhere the CCDF of PRD (as derived in Appendix H) isF cPRD (x) = Qm1NS(λSR√2m1ΩSR,√m1b0ΩSRpSRln(a0$ + xa0$ − x)). (6.27)We insert (6.27) into (6.26) to havePRD =∫ ∞0Qm1NS(λSR√2m1ΩSR,√m1b0pSRΩSRln(a0$ + xa0$ − x))dx. (6.28)Applying (6.15) to (6.28) leads toPRD =∫ ∞0exp[− m1ΩSR(λ2SR +1b0pSRln(a0$ + xa0$ − x))] ∞∑q=1−m1NS√√√√ 2b0pSRλ2SRln(a0$+xa0$−x)q× Iq(m1λSRΩSR√2b0pSRln(a0$ + xa0$ − x))dx. (6.29)Applying (6.17) to (6.29), we obtain the expression for the EEHR asPRD = exp(−m1λ2SRΩSR) ∞∑q=1−m1NS∞∑w=01w!Γ (w + q + 1)(2b0pSR)w (λ2SR)w+q ( m1ΩSR)2w+q×∫ ∞0[ln(a0$ + xa0$ − x)]wexp[− m1b0pSRΩSRln(a0$ + xa0$ − x)]dx. (6.30)6.4.2 Transmission Outage Probability for ArTHere, we analyze the TOP for the ArT energy harvester. The TOP is defined for the ArT model asTOP = 1−∫ pia0$01εf 1‖hRD‖2(xε)F cpRD (x) dx. (6.31)The TOP is obtained by substituting (6.22) and (6.27) into (6.31) to haveTOP = 1− m2ΩRDεNDm2+12 λ1−NDm2RD exp(− m2ΩRDλ2RD)∫ pia0$0(1x)NDm2+32exp(− m2εΩRDx)× INDm2−1(2m2λRD√εΩRD√x)QNSm1(λSR√2m1ΩSR,√m1b0pSRΩSRln(a0$ + xa0$ − x))dx. (6.32)956.5. Numerical ResultsApplying (6.15) to (6.32) leads toTOP = 1−N−1∑i=1m2ΩRDεNDm2+12 λ1−NDm2RD exp(− m1ΩSRλ2SR −m2ΩRDλ2RD) ∞∑q=1−m1NS(λSR√2b0pSR)q×∫ a0$0(1x)NDm2+32(ln(a0$ + xa0$ − x))− q2exp(− m12b0ΩSRpSRln(a0$ + xa0$ − x)− m2εΩRDx)× INDm2(2m2λRD√εΩRD√x)Iq(m1√2λSRΩSR√b0pSR√(a0$ + xa0$ − x))dx. (6.33)Applying (6.17) to (6.33), we obtain the expression for the TOP asTOP = 1− exp(− m1ΩSRλ2SR −m2ΩRDλ2RD) ∞∑q=1−m1NS∞∑w=01w!Γ (w +m2ND)∞∑s=01s!Γ (s+ q + 1)×(m1ΩSR)2s+qλ(2s+2q)SR (2b0pSR)−s(m2ΩRD)2w+NDm2λ2wRD εw+NDm2∫ a0$0(1x)w+NDm2+1×(ln(a0$ + xa0$ − x))sexp(− m2εΩRDx− m12b0ΩSRpSRln(a0$ + xa0$ − x))dx.Again, PRD and TOP both contain one integral, which can be solved using standard mathematical softwarelike Matlab or Mathematica.6.5 Numerical ResultsIn this section, we first fit the MSPwL and ArT energy harvesters to measured data [32]. We then presentperformance analyses for the average harvested energy and TOP for both energy harvesters. The measureddata used for the EH models’ fitting was obtained from a wirelessly transmitted high-intensity RF signal. Asensor harvests energy from the signal and converts it to DC voltage [32].Figure 6.2 shows the fits of the harvested power versus received signal power for the MSPwL and ArTenergy harvesters to measured data from [32, Fig. 17d] for a load resistance of 1.32 MΩ. For the MSPwLenergy harvesters, we consider 2, 3, and 4 segments. We obtain the parameters’ values via a best-fit searchto the measured data with the goodness-of-fit quantified by RMSE values. Table 1 shows the values for eachof the parameters for 2-, 3-, 4-segment MSPwL, and ArT energy harvesters. We observe that the 4-segmentMSPwL energy harvester gives a better fit than the other energy harvesters.Figure 6.3 shows fits of the harvested power for the MSPwL and ArT energy harvesters to measured datafrom [32, Fig. 17d] versus transmit power for a load resistance of 5.6 MΩ. For the MSPwL energy harvester,966.5. Numerical ResultsTable 6.1: Parameters of the fits of measured data from [32, Fig. 17d] for a load resistance of 1.32 MΩ toMSPwL and ArT energy harvestersType of energy har-vesteran bn RMSE value2-segment MSPwL 0.1677, 0.0224 0, 2.2961× 10−5 1.68×10−63-segment MSPwL 0.1551, 0.0328, 0.0124 0, 1.9312 × 10−5, 3.3781 ×10−56.90×10−74-segment MSPwL 0.1755, 0.1383, 0.0422,0.01590, 1.5611 × 10−6, 1.6747 ×10−5, 2.9789× 10−55.78×10−7ArT 4.9232×10−5 2.250×103 3.02×10−60 0.2 0.4 0.6 0.8 1 1.2 1.410-30123456 10-5Figure 6.2: Fits of the harvested power for the MSPwL and ArT energy harvesters to empirically obtainedharvested power [32, Fig. 17d] as a function of the recieved signal power for a 1.32 MΩ load resistance. TheRMSE values for the 2-, 3-, and 4-segment MSPwL energy harvesters are 1.68 × 10−6, 6.90 × 10−7, and5.78× 10−7, respectively. The RMSE value for the ArT energy harvester is 3.02× 10−6.976.5. Numerical ResultsTable 6.2: Parameters of the fits of measured data from [32, Fig. 17d] for a load resistance of 5.6 MΩ toMSPwL and ArT energy harvestersType of energy har-vesteran bn RMSE value2-segment MSPwL 0.1317, 0.0042 0, 7.3721× 10−6 1.21×10−63-segment MSPwL 0.3990, 0.0389, 5.515 ×10−40, 2.9136 × 10−6, 1.0086 ×10−59.56×10−84-segment MSPwL 0.3990, 0.0427, 0.0018, 2.34×10−40, 2.6964 × 10−6, 9.6167 ×10−6, 1.0366× 10−55.15×10−8ArT 1.079×10−5 7.90×103 7.52×10−7we consider 2, 3, and 4 segments. Again, we obtain the parameters’ values via a best-fit search to the measureddata with the goodness-of-fit quantified by RMSE values. Table 2 shows the values for each of the parametersfor 2-, 3-, 4-segment MSPwL, and ArT energy harvesters. Again, we observe that the 4-segment MSPwLenergy harvester provides a better fit than the other energy harvesters. However, we see that both of ourenergy harvesters present a better fit in comparison to the EH models in [91, Fig. 1a], which are fit to thesame measured data as used here.Figure 6.4 shows curves of the average harvested power, PRD, for a 3-segment MSPwL energy harvester.Here, we show the plots of the PRD versus transmit power, pSR, of the energy source, S, at a distance, d1 = 10m with a pathloss exponent of s1 = 2 and unity-gain antennas. The simulation is done for parameter valuesof $ = 0.5 and different values of m1 and NS . The energy harvester has the parameters an and bn take onvalues of[0.3990 0.0389 5.515× 10−4] and [0 2.9136× 10−6 1.0086× 10−5], respectively. We observethat all the curves start to roll-off from a linear to nonlinear for higher transmit power values, pSR. When thefading parameter m1 increases from 1 to 2, the harvested power increases. However, the increase in harvestedpower becomes less significant as it approaches the EH circuit’s saturation level, even for an increased valueof NS . Overall, we observe that less severe fading in the first-hop yields more harvested power. Moreover,increasing the number of energy beamforming antennas, NS , does not increase the harvested power above theEH circuit’s saturation level.In Fig. 6.5, we show curves of the average harvested power, PRD, for the ArT energy harvester versusthe TS protocol parameter, $, for different transmit power, pSR. Here, we consider an energy source, S,with a set number of unity-gain energy beamforming antennas, NS = 2, and a fading severity of m1 = 2.The energy source, S, is separated from the wireless device, R, at a distance of d1 = 10 m with a pathlossexponent of 2. The ArT energy harvester has the parameters a0 and b0 take on values of 1.079 × 10−5 and986.5. Numerical Results0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110-300.20.40.60.811.2 10-5Figure 6.3: Fits of the harvested power for MSPwL and ArT energy harvesters to empirically obtained har-vested power [32, Fig. 17d] as a function of the transmit power for a 5.6 MΩ load resistance. The RMSEvalues for the 2-, 3-, and 4-segment MSPwL energy harvesters are 1.21×10−6, 9.56×10−8, and 5.15×10−8,respectively. The RMSE value for the ArT energy harvester is 7.52× 10−7.0 0.05 0.1 0.150123456 10-6Figure 6.4: Curves for the average harvested power, PRD, versus transmit power, pSR, for a 3-segmentMSPwL energy harvester at a distance of d1 = 10 m with a pathloss exponent of s1 = 2, unity-gain antennas,parameters values of $ = 0.5, and different values of m1 and NS . The parameters an and bn have the values[0.3990 0.0389 5.515× 10−4] and [0 2.9136× 10−6 1.0086× 10−5], respectively.996.5. Numerical Results0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.901234567 10-6Figure 6.5: Curves for the average harvested power, PRD, versus TS protocol parameter, $, for the ArTenergy harvester at a distance of d1 = 10 m with a pathloss exponent of s1 = 2, unity-gain antennas, andparameters values ofm1 = 2,NS = 2. The parameters a0 and b0 have values of 1.079×10−5 and 7.90×103,respectively.7.90 × 103, respectively. We observe from the curves that PRD increases as $ increases. This trend showsthat the wireless device, R, has enough time to harvest energy. Also, an increase in PRD is recorded as pSRincreases, as expected. However, we observe that the increase in PRD reduces as it approaches the saturationlevel of the EH circuit.Figure 6.6 shows TOP curves for a 3-segment MSPwL energy harvester over the BX fading channels ver-sus transmit power, pSR. We consider the TS protocol for different values of ρ and different numbers of theenergy beamforming antennas, NS . The number of data fusion receiver antennas, ND, is 2, and the fading pa-rametersm1 andm2 both take on values of 2. The parameter an has values of[0.3990 0.0389 5.515× 10−4],while the parameter bn has the values of[0 2.9136× 10−6 1.0086× 10−5]. The value of the expressionRreq%W is 0.2, while the distances d1 and d2 are both 10 m with a pathloss exponent of 2. We observe that theTOP curves exhibit an outage floor as the transmit power, pSR, increases. This outage floor is due to thesaturation level of the EH circuit. In particular, we observe that as the number of the energy beamforming an-tennas, NS increases, the outage floor is attained at a lower transmit power, pSR. We observe an improvementto the TOP values as ρ increases from 0.5 to 0.7. However, the significance of the improvement reduces asNS ,increases due to the saturation level of the EH circuit. For example, the TOP value reduces by approximately1006.5. Numerical Results0 0.02 0.04 0.06 0.08 0.1 0.1210-1100Figure 6.6: The TOP curves versus transmit power, pSR, for the 3-segment MSPwL energy harvesterfor parameters values of m1 = m2 = 2, ND = 2, an = [0.3990 0.0389 5.515× 10−4], and[0 2.9136× 10−6 1.0086× 10−5].0.15 as the value of NS increases from 2 to 3 for a transmit power of 80 mW and ρ = 0.7. However, for anincrease in the value of NS from 3 to 4, only a reduction of approximately 0.035 is recorded.Figure 6.7 shows TOP curves for the ArT energy harvester versus the distance, d1, with a pathloss ex-ponent of 2 and a transmit power, pSR, 100 mW. We consider different numbers of energy beamformingantennas, NS , and differing fading parameter values, m1. The parameters a0 and b0 are 1.079 × 10−5 and7.90× 103, respectively. We note that the expression Rreq%W takes on a value of 0.1. It is also important to em-phasize that the distance between the wireless device, R, and the data fusion receiver, D, remains unchangedat 8 m. From the curves, we observe that the TOP improves when the fading parameter m1 increases from 1to 2, as expected. This improvement indicates that less severe fading between the energy source and wirelessdevice leads to better wireless device coverage. Also, increasing the number of diversity branches leads toimproved outage performance. Ultimately, we see that the WPC system attains better performance when thewireless device is close to the energy source, as one would expect.1016.6. Chapter Summary1 2 3 4 5 6 7 8 9 1010-210-1100Figure 6.7: The TOP curves for the ArT energy harvester versus distance, d1, for a transmit power, pSR, of100 mW and parameters values of m2 = 2, ND = 2, $ = % = 0.5, a0 = 1.079× 10−5, and 7.90× 103.6.6 Chapter SummaryThis chapter analyzed the EEHR and TOP of the WPC systems while considering two newly proposednonlinear EH models. The fading was considered to be BX distributed because of the (likely) presence ofmany LOS components (via direct propagation and specular reflections) and NLOS components. Our resultsshowed that our models’ EH circuitry exhibits saturation levels as the transmit power increases. We alsoshow the effect of increasing the number of energy beamforming antennas, fading parameter, distance, andharvesting time (as defined by the TS protocol parameters) on the average harvested energy and TOP. Weobserved that increasing the number of energy beamforming antennas reduces the transmit power required foran outage floor to be recorded. Also, increasing energy harvesting time leads to an improved outage value.However, the improvement becomes less significant as the number of energy beamforming antennas increasesbecause of the EH circuit’s saturation level. Overall, our new EH models showed good fits to newly-publishedmeasured data, supporting their use in designing future WPC systems.102Chapter 7ConclusionsIn this chapter, we summarize the thesis contributions and provide concluding remarks on the accom-plished works. We also discuss some future research topics related to the BX and SBX distributions.7.1 SummaryCommunication in mmWave and THz systems often manifests through LOS signals (either via directpropagation or specular reflections). Unlike the RF systems, the signals in these higher-frequency bands aresusceptible to atmospheric attenuation (i.e., gases in the atmosphere absorb them). Thus, they are suitable forshort-range communication, as seen in femtocells and WPC systems. While the Nakagami-m fading modelhas gained popularity in describing the RF systems’ communication environment, it becomes inadequate withthese emerging wireless systems. The BX and SBX models are two models that can be applied to these newsystems.7.2 Concluding Remarks on the Accomplished WorksWe conclude this thesis with the following remarks.− In Chapter 2, we presented the second-order statistics of the BX fading model by analyzing its LCR andAFD for singular reception and diversity scheme of MRC. The traditional PDF method was employed toobtain the LCR and AFD for singular reception, while the CHF method was used for the MRC diversityscheme. We investigated the effects of various parameters via simulation and compared the results tothose obtained for the Ricean and Nakagami-m fading models. We observed that an increase in themobile device’s speed (i.e., dependent on the maximum Doppler shift frequency) increases the LCRand decreases the AFD. We also observed that less fading severity (i.e., a higher value of m) resultsin reduced LCR. We observed from the results of the LCR for the MRC diversity scheme that the BXfading model presents a better result compared to the Ricean and the Nakagami-m fading models in the1037.2. Concluding Remarks on the Accomplished Workspresence of LOS signals. We concluded our analysis by considering i.i.d. and non-i.i.d branches, wherethe i.i.d. branches all have Ricean-distributed envelopes while the non-i.i.d. branches have the firstbranch be BX-distributed and the rest be Ricean distributed. We observed a notable difference betweenthese two cases, in favor of the non-i.i.d. branches.− In Chapter 3, we analyzed the BX fading model’s performance with arbitrary branch correlation usingbounds. The use of bounds became necessary to avoid the intractable or infinite series when derivingperformance results over correlated fading channels. This intractability manifests from the joint PDFfor the branch amplitude of the diversity channels – given the presence of a Bessel function in thePDF of the BX fading model. This analysis employed a newly bounding technique where a probleminvolving correlated BX RVs is transformed into a new problem involving correlated Gaussian RVs. Weconsidered the performance metric of OP and BER for MRC, SC, and EGC diversity techniques. Ourresults, verified via Monte Carlo simulation, were shown to outperform those of the Nakagami-mmodelin a LOS environment. These results showed the combined effects of the LOS and NLOS signals, thefading parameter, m, and various correlation structures. Ultimately, the derived performance boundscan show the primary factors affecting the performance of emerging systems (with large antenna arrays).− In Chapter 4, we analyzed the performance of BX fading channels for MRC in the presence of co-channel interference. The MRC techniques offer a mathematically tractable result compared to theoptimum combining technique, which is more suited for an interference-limited environment. We notethat the MRC technique can provide equivalent performance to optimum combining in the presenceof interference with perfect or imperfect knowledge of channel coefficients [78]. From our results, weobtain the system’s diversity order, which is the slope of the log-log plot of the BER or OP versus SNR.Ultimately, this analysis presents an insight into short-range communication systems’ performance, asfound in the mmWave, THz, femtocells, and WPC systems.− Chapter 5 introduced a new composite model to accommodate the fluctuating amplitude of the LOScomponents (either via direct propagation or specular reflections). This fluctuation is due to the chang-ing position of objects in their paths or varying atmospheric conditions. This composite model, calledthe SBX model, assumes that the LOS components’ amplitude is Nakagami-m distributed. Having in-vestigated the BX model, we employed the Nakagami-m model because it offers analytical intractabil-ity. Unlike other composite models, the SBX model can describe an environment with an arbitrarynumber of LOS components and a wide range of fading and shadowing conditions. To validate the BX1047.3. Suggested Future Workand SBX models, we fit both models to experimental data obtained from fading measurement in the28 GHz outdoor mmWave channels. The results provide the groundwork for characterizing emergingmmWave and THz wireless communication systems.− Chapter 6 investigated the WPC system’s performance over the BX channels with nonlinear energyharvesters. The system consists of a wireless device, which scavenges energy from the energy sourcein the downlink and uses the harvested energy to communicate to a data fusion receiver in the uplink.We introduced two new nonlinear energy harvesters in this chapter - the MSPwL and ArT energyharvesters. We employed the performance metric of EEHR and TOP energy harvesters to analyze theenergy harvesters’ performance. We verified our analytical results via simulation, and our simulationresults showed the combined effect of the fading severity, the number of diversity branches, propagatingdistance, transmit power, and harvesting time. Ultimately, our two nonlinear EH models presented anexcellent fit to measured data obtained from a far-field RF environment.7.3 Suggested Future Work− Phase distribution of the BX fading modelThis task is of utmost importance because understanding the phase characteristics is particularly im-portant for high-frequency operation in mmWave, THz, and WPC systems. This is because spatial per-turbations within the environment can lead to relatively large changes in the transmitted signal phasewhen the wavelength is small. In [4], Nakagami-m model’s phase was derived and shown to agree withthe phases of the Ricean and Hoyt models. However, this result is arguable because the Ricean andNakagami models have different origins [24]. Therefore, the phase of the Nakagami-m fading modelis not yet fully known. The research goal is to derive the phase of the BX fading model and obtainthe phase of the Nakagami-m fading model as its particular case in the absence of the noncentralityparameter. This analysis would also show the Ricean fading model phase as a specific case of the BXfading model when the fading parameter is equal to one.− Full performance analysis of diversity systems operating according to the BX fading modelIt is crucial to analyze the performance of diversity systems operating according to the BX fadingmodel. The main objective is to form a stronger foundation for future work on the BX fading model.The analysis should apply the four main performance metrics of OP, error rate, LCR, and AFD. The OP1057.3. Suggested Future Workand error rate will quantify the effect of the various modulation techniques on the generalized wirelesslink performance, while LCR and AFD will quantify the wireless links’ dynamic performance. Becauseof the Bessel function in the BX PDF, the results for the EGC will be intractable. To circumvent thisissue, we propose the CHF method and Hermite numerical integration technique to derive the PDF forEGC. Hermite numerical integration converts the integral into a summation with all terms having theBX RVs taken as a function of the same RV. Ultimately, the CHF should present a more straightforwardapproach when dealing with Bessel functions.− The sum of the BX random variablesThe sum of RVs frequently occurs in the study of wireless communication systems’ performance. Thissum’s particular use is found in the performance analysis involving the EGC technique. Many workshave shown the accurate closed-form expression of the Ricean sum distribution by modifying the non-central chi-square distribution [98]. Other works employed the Nakagami-m distribution to approx-imate the Ricean distribution [99]-[101]. However, these approximations involving the Nakagami-mdistribution are inaccurate because the Ricean and the Nakagami-m fading models have different ori-gins. Therefore, we propose the method in [98] as an option to obtain the BX sum distribution.− The second-order statistics of the SBX modelThe dynamic performance of the SBX model is yet to be analyzed. The SR’s second-order statisticswere analyzed for land-mobile-satellite (LMS) channels in [9]. Unlike the SR model, the SBX modelhas distinct fading and shadowing parameters, thus increasing the SBX model’s flexibility. 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Technol., vol. 49, pp.173-180, Jan. 2000.116Appendices117Appendix AProof that the BX PDF becomes an impulseas fading severity parameter approachesinfinityIn this appendix, we show the proof that the BX PDF becomes an impulsive function at r =√λ2 + Ω asm approaches infinity. Let Vi be independent Gaussian RVs with mean µi and equal variance σ2. We have Zdefined asZ =√√√√ n∑i=1V 2i (A.1)to be generalized Ricean distributed. Since the BX fading model is a normalized version of the generalizedRicean distribution by applying the scaling factor σ√nΩ , where n = 2m, then we can have a new RV R asR =√∑ni=1 V2iσ√nΩ=√Ωσ√∑ni=1 V2in. (A.2)When m approached infinity, then we can have the limit asRlim = limn→∞R = limn→∞√Ωσ√∑ni=1 V2in′(A.3)Applying Tchebycheff’s condition for large numbers leads to [25]limn→∞√∑ni=1 V2in= limn→∞√√√√ 1nn∑i=1E[V 2i]. (A.4)118Appendix A. Proof that the BX PDF becomes an impulse as fading severity parameter approaches infinityThe non-centrality parameter in the generalized Ricean distribution is the same with that of the non-centralchi distribution as s =√∑ni=1 µ2i , the expression in (A.4) can be rewritten aslimn→∞√1n(s2 + nσ2) = limn→∞√1n(s2n+ σ2). (A.5)As earlier mention that λ be a scale version of s as λ = sσ√2mΩ, then (A.5) becomeslimn→∞√∑ni=1 V2in= limn→∞√σ2λ2Ω+ σ2. (A.6)Combining (A.3) and (A.6), we haveRlim =√λ2 + Ω. (A.7)In the absence of LOS component, then Rlim =√Ω, which is the same for Nakagami-m distribution.119Appendix BThe nth-Order Moment of the BXDistributionIn this appendix, we show the proof for the nth order moment for the BX distribution obtained in (1.16).Earlier on, we have shown the BX distribution to be a normalized version of the generalized Ricean distribu-tion. Thus, the kth order moment for the generalized Ricean distribution is obtained as [39]E[Xk]=(2σ2) k2 exp(− s22σ2) Γ(n+kn2)1F1(n+K2 ;n2 ;s22σ2) (B.1)where 1F1 (·; ·;x) is the confluent hypergeometric (Kummers) function. We can modify (B.1) to have themoment of the BX distribution by applying the scaling factor σ√2mΩ to the generalized Ricean RV andsetting n = 2m asE [Rn] =(mΩ)n2exp(−mΩλ2) Γ (2m+n2 )Γ (m)1F1(2m+ n2;m;mΩλ2). (B.2)Applying the Kummers transformation to (B.2) which states that1F1 (a; b; z) = ez1F1 (b− a; b;−z) , (B.3)then (B.2) can be rewritten asE [Rn] =(mΩ)n2 Γ(2m+n2)Γ (m)1F1(−n2;m;−mΩλ2). (B.4)Noting the relationship between the Kummers function and the Laguerre polynomial asL(a)n (r) =(n+ an)1F1 (−n; a+ 1; r) (B.5)120Appendix B. The nth-Order Moment of the BX Distributionand applying to (B.4), then the nth moment of the BX distribution is obtained as (1.16).121Appendix CDerivation of the Diversity Order of theSBX ModelIn this section, we derive the diversity order of the shadowed BX model by considering the derived ex-pression for the BER for noncoherent detection asPE =12(mY ΩXmXΩY +mY ΩX)mY ( mXmX +$ΩX)mX× 2F1(mY ,mX ;mX ;m2XΩY(mX +$ΩX) (mXΩY +mY ΩX))(C.1)where 2F1 (·, ·; ·; ·) is defined as [73]2F 1 (a, b; c; z) = 1 +abcz +a (1 + a) b (1 + b)c (1 + c)z2 + · · · ;(z → 0) . (C.2)At high SNR, the BER or OP scale according to SNR–N , where N is the diversity order, meaning that alog-log plot of the BER or OP versus SNR will portray the diversity order as a slope ofN = − log (BER or OP)log (SNR)(C.3)Since the ΩX and ΩY are the power of the LOS and NLOS components, respectively, we can assume ΩX =ΩY and let B =(mXmX+$ΩX). Applying (C.2) to (C.1) givesPE =12(mYmX +mY)mYBmX×[1 +mXmYmX +mYB +mY (1 +mY )2(BmXmX +mY)2+ · · ·]. (C.4)122Appendix C. Derivation of the Diversity Order of the SBX ModelWe see that as the SNR approaches infinity, and ΩX → ∞, we get B → 0. Since B → 0, PE is dominatedby first term of the square brackets in (C.4). Therefore,limB→0PE =12(mYmX +mY)mYBmX=12(mYmX +mY)mY ( mXmX +$ΩX)mX. (C.5)Since ΩX →∞, we have mX +$ΩX ≈ $ΩX . Therefore, the BER becomeslimΩX→∞PE =12(mYmX +mY)mY ( mX$ΩX)mX. (C.6)This equally meansPE ∝(1ΩX)mX. (C.7)Applying (C.3) to (C.7) leads to a diversity order of mX for the shadowed BX model. This is expectedbecause the diversity order is set by the small scale fading parameter.123Appendix DDerivation of the Moments of the PDF ofthe SBX ModelIn this section, we derive the 1st-order, 2nd-order, and 4th-order moments using (5.9). First, we define theGauss hypergeometric function as [64]2F1 (a, b; c; z) = 1 +ab1!cz +a (a+ 1) b (b+ 1)2!c (c+ 1)z2 + . . .=∞∑n=0(a)n (b)n(c)nznn!. (D.1)Based on the above definition, the moments are obtained asµ1 = E [R] (D.2)=Γ(mX +12)Γ (mX)(mXΩX)− 122F1(−12,mY ;mX ;−mXΩYmY ΩX),µ2 = E[R2]=Γ (mX + 1)Γ (mX)(ΩXmX)2F1(−1,mY ;mX ;−mXΩYmY ΩX)=Γ (mX + 1)Γ (mX)(ΩXmX)[1 +(−1)1 (mY )11! (mX)1(−mYmXK)]= ΩX [1 +K] (D.3)µ4 = E[R4]=Γ (mX + 2)Γ (mX)(ΩXmX)2× 2F1(−2,mY ;mX ;−mXΩYmY ΩX)=Γ (mX + 2)Γ (mX)(ΩXmX)2 [1 +(−2)1(mY )11!(mX)1(−mYmXK)+(−2)2(mY )22!(mX)2(−mYmXK)2]= Ω2X[1 + 2K +mX (mY + 1)mY (mX + 1)K2]. (D.4)124Appendix EDerivation of the Bound on the TruncationError of the BER for Coherent DetectionIn this section, we derive the bound to the truncation error in (5.42) as given byC =√$2√pi(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mX ∞∑n=NΓ (mY + n)n!Γ (mY )(m2XΩYmXΩY +mY ΩX)n× Γ(mX + n+12)Γ (mX + n+ 1)×(ΩX$ΩX +mX)mX+n+ 12× 2F1(1,mX + n+12;mX + n+ 1;mX$ΩX +mX).(E.1)It is seen that 2F1(1,mX +N +12 ;mX +N + 1;mX$ΩX+mX)×($ + mXΩX)−(mX+N+ 12) decreases over n,therefore the error in the truncating series can be bounded byC ≤√$2√piΓ (mY )(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mX2F1(1,mX +N +12;mX +N + 1;mX$ΩX +mX)×($ +mXΩX)−(mX+N+ 12) ∞∑n=NΓ (mY + n) Γ(mX + n+12)n! (mX + n) Γ (mX + n)(m2XΩYmXΩY +mY ΩX)n. (E.2)By a change of variable, eq. (E.2) becomesC ≤√$2√piΓ (mY )(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mX2F1(1,mX +N +12;mX +N + 1;mX$ΩX +mX)×($ +mXΩX)−(mX+N+ 12) ∞∑j=0Γ (mY +N + j) Γ(mX +N + j +12)(N + j)! (mX +N + j) Γ (mX +N + j)(m2XΩYmXΩY +mY ΩX)N+j.(E.3)Applying the definition (a)k = a (a+ 1) · · · (a+ k − 1) to (E.3) givesC ≤√$2√piΓ (mY )(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mX ($ +mXΩX)−(mX+N+ 12)( m2XΩYmXΩY +mY ΩY)N125Appendix E. Derivation of the Bound on the Truncation Error of the BER for Coherent Detection× 2F1(1,mX +N +12;mX +N + 1;mX$ΩX +mX)Γ (mY +N − 1) Γ(mX +N +12)Γ (mX +N)×∞∑j=0(mY +N)j(mX +N +12)jN ! (N + 1)j (mX +N + 1)j(m2XΩYmXΩY +mY ΩY)j(E.4)which is equivalent toC ≤√$2√piΓ (mY )(mY ΩXmXΩY +mY ΩX)mY (mXΩX)mX ($ +mXΩX)−(mX+N+ 12)( m2XΩYmXΩY +mY ΩY)N× Γ (mY +N − 1) Γ(mX +N +12)Γ (mX +N)× 2F1(1,mX +N +12;mX +N + 1;mX$ΩX +mX)×∞∑n=0(mY +N)n(mX +N +12)nN ! (N + 1)n (mX +N + 1)n(m2XΩYmXΩY +mY ΩY)n. (E.5)This last result agrees with (5.43).126Appendix FDerivation of the Bound on the TruncationError of the OPIn this section, we derive the bound on the truncation in (5.49) as given byOP =(mY ΩXmXΩY +mY ΩX)mY ∞∑n=NΓ (mY + n)n!Γ (mX + n) Γ (mY )(mXΩYmXΩY +mY ΩX)ng(mX + n,mXΩXγb).(F.1)It is seen that g(mX + n,mXΩXγb)approaches 1 as n approaches infinity. Therefore, the error in the truncationseries can be bounded byOP ≤(mY ΩXmXΩY +mY ΩX)mYg(mX +N,mXΩXγb) ∞∑n=NΓ (mY + n)n!Γ (mX + n) Γ (mY )×(mXΩYmXΩY +mY ΩX)n.(F.2)By change of variable, (F.2) becomesOP ≤(mY ΩXmXΩY +mY ΩX)mYg(mX +N,mXΩXγb)1Γ (mY )∞∑j=0Γ (mY +N + j)(N + j)!Γ (mX +N + j)×(mXΩYmXΩY +mY ΩX)N+j. (F.3)Applying the definition (a)k = a (a+ 1) · · · (a+ k − 1) to (F.3) givesOP ≤(mY ΩXmXΩY +mY ΩX)mYg(mX +N,mXΩXγb)1Γ (mY )Γ (mY +N)Γ (mX +N)(mXΩYmXΩY +mY ΩX)N×∞∑j=0(mY +N)j(1)jN ! (mX +N)j (N + 1)j(mXΩYmXΩY +mY ΩX)j(F.4)127Appendix F. Derivation of the Bound on the Truncation Error of the OPwhich is equivalent toOP ≤(mY ΩXmXΩY +mY ΩX)mYg(mX +N,mXΩXγb)1Γ (mY )Γ (mY +N)Γ (mX +N)(mXΩYmXΩY +mY ΩX)N×∞∑n=0(mY +N)n(1)nN ! (mX +N)n (N + 1)n(mXΩYmXΩY +mY ΩX)n. (F.5)The result in (F.5) agrees with (5.50).128Appendix GProof of CCDF for MSPwL EnergyHarvesterThe CCDF of the transmission power of sensor PRD is obtained asF cPRD (x) = Pr {PRD ≥ x} = Pr{$PMSPwLEH(pSR ‖hSR‖2 ≥ x)}.First, we define an interval [di−1, di] with d0 = 0, dN = dN−1, and di = aici + bi, i = 1, . . . , N − 1.Since PRD is a RV, the probability of incidence that x = $dN−1 is equal to zero. When the value x satisfiesx$ ∈ [di−1, di] with i = 1, 2, . . . , N − 1, thus, eq. (G.1) can be simplified based on (5) toF cPRD (x) = Pr {PRD ≥ x}= Pr{1ai( x$)≤ pSR∥∥h2SR∥∥ < ci}+N∑n=i+1Pr{anpSR∥∥h2SR∥∥+ bn ≥ x$, cn−1 ≤ pSR ∥∥h2SR ≤ cn∥∥}= Pr{‖hSR‖2 ≥ 1aipSR( x$− bi)}. (G.1)Noting that ‖hSR‖2 is noncentral chi-squared distributed, the CCDF of PRD isF cPRD (p) = Qm1NS(λSR√2m1ΩSR,√2m1aipSRΩSR( x$− bi)), x ∈ [$di−1, $di] . (G.2)129Appendix HProof of CCDF for ArT energy harvesterThe CCDF of the transmission power for the sensor, PRD, is obtained asF cPRD (x) = Pr {PRD ≥ x} = Pr{$PArTEH(pSR ‖hSR‖2)≥ x}= Pr{$a0 tanh(b0pSR ‖hSR‖2)≥ x}= Pr{‖hSR‖2 ≥ 1b0pSRtanh−1(x$a0)}. (H.1)We then express tanh−1 (·) in terms of natural logarithm, ln (·), in the formtanh−1 (v) =12ln(1 + v1− v)(H.2)while noting that ‖hSR‖2 is noncentral chi-squared distributed, to give a CCDF of PRD in the form ofF cPRD (x) = Qm1NS(λSR√2m1ΩSR,√m1b0ΩSRpSRln(a0$ + xa0$ − x)).130
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A novel fading model for emerging wireless communication systems
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