NAVAL POSTGRADUATE SCHOOL THESIS

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1 NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA THESIS DIGITAL COMMUNICATIONS OVER NON-FADING AND FADING CHANNELS y Jose H. Hernandez Jr. Septemer 008 Thesis Advisor: Second Reader: Clark Roertson Frank Kragh Approved for pulic release; distriution unlimited

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3 REPORT DOCUMENTATION PAGE Form Approved OMB No Pulic reporting urden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instruction, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this urden estimate or any other aspect of this collection of information, including suggestions for reducing this urden, to Washington headquarters Services, Directorate for Information Operations and Reports, 115 Jefferson Davis Highway, Suite 104, Arlington, VA 0-430, and to the Office of Management and Budget, Paperwork Reduction Project ( ) Washington DC AGENCY USE ONLY (Leave lank). REPORT DATE Septemer TITLE AND SUBTITLE: Digital Communications Over Non-Fading and Fading Channels 3. REPORT TYPE AND DATES COVERED Master s Thesis 5. FUNDING NUMBERS 6. AUTHOR Jose H. Hernandez Jr. 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Naval Postgraduate School Monterey, CA SPONSORING /MONITORING AGENCY NAME(S) AND ADDRESS(ES) N/A 8.PERFORMING ORGANIZATION REPORT NUMBER 10. SPONSORING/MONITORING AGENCY REPORT NUMBER 11. SUPPLEMENTARY NOTES The views expressed in this thesis are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. Government. 1a. DISTRIBUTION / AVAILABILITY STATEMENT Approved for pulic release; distriution unlimited 13. ABSTRACT (maximum 00 words) 1. DISTRIBUTION CODE A The ojective of this thesis is to enhance the aility of the Improved Many-on-Many (IMOM) software package to analyze modern digital communication systems using availale intelligence. Currently, IMOM can only e used to analyze analog communication systems, ut modern systems are, increasingly digital. In this thesis, the proaility of it error expressions for many common digital modulation techniques, oth inary and non-inary, are inverted to otain expressions for the required signal-to-noise ratio as a function of proaility of channel it error. Furthermore, results are otained not only for a non-fading channel ut for channels modeled as either Rayleigh or Ricean. These equations can e implemented in IMOM to increase the accuracy of the link udget analysis when the specific modulation type eing evaluated is known. This thesis takes the approach of determining proaility of channel it error rather than information it error, which allows generic solutions independent of the specifics of the system under investigation as long as the particular modulation type is known. When even greater accuracy is desired, system specifics such as the type of error control coding must e taken into account. As an example of this, the Joint Tactical Information Distriution System (JTIDS) is considered. 14. SUBJECT TERMS coherent, noncoherent, BPSK, QPSK, MPSK, MQAM, GMSK, MFSK, JTIDS, multipath, signal fading, IMOM 15. NUMBER OF PAGES PRICE CODE 17. SECURITY CLASSIFICATION OF REPORT Unclassified 18. SECURITY CLASSIFICATION OF THIS PAGE Unclassified 19. SECURITY CLASSIFICATION OF ABSTRACT Unclassified 0. LIMITATION OF ABSTRACT NSN Standard Form 98 (Rev. -89) Prescried y ANSI Std UU i

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5 Approved for pulic release; distriution unlimited DIGITAL COMMUNICATIONS OVER NON-FADING AND FADING CHANNELS Jose H. Hernandez Jr. Captain, United States Marine Corps BS Industrial Engineering, Texas A&M University, 001 Sumitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN ELECTRICAL ENGINEERING from the NAVAL POSTGRADUATE SCHOOL Septemer 008 Author: Jose H. Hernandez Jr. Approved y: R. Clark Roertson Thesis Advisor Frank Kragh Second Reader Jeffrey B. Knorr Chairman, Department of Electrical and Computer Engineering iii

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7 ABSTRACT The ojective of this thesis is to enhance the aility of the Improved Many-on- Many (IMOM) software package to analyze modern digital communication systems using availale intelligence. Currently, IMOM can only e used to analyze analog communication systems, ut modern systems are, increasingly, digital. In this thesis, the proaility of it error expressions for many common digital modulation techniques, oth inary and non-inary, are inverted to otain expressions for the required signal-to-noise ratio as a function of proaility of channel it error. Furthermore, results are otained not only for a non-fading channel ut for channels modeled as either Rayleigh or Ricean. These equations can e implemented in IMOM to increase the accuracy of the link udget analysis when the specific modulation type eing evaluated is known. This thesis takes the approach of determining proaility of channel it error rather than information it error, which allows generic solutions independent of the specifics of the system under investigation as long as the particular modulation type is known. When even greater accuracy is desired, system specifics such as the type of error control coding must e taken into account. As an example of this, the Joint Tactical Information Distriution System (JTIDS) is considered. v

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9 TABLE OF CONTENTS I. INTRODUCTION...1 A. OVERVIEW...1 B. THESIS OBJECTIVE... C. THESIS OUTLINE... II. BACKGROUND...5 A. GENERIC, SYSTEM INDEPENDENT APPROACH...5 B. COHERENT MODULATION TECHNIQUES...7 C. NONCOHERENT, ORTHOGONAL MODULATION TECHNIQUES...9 D. MULTIPATH FADING CHANNELS...10 E. JOINT TACTICAL INFORMATION DISTRIBUTION SYSTEM (JTIDS)...11 III. NON-FADING CHANNELS...13 A. COHERENT MODULATION MPSK MQAM GMSK... B. NONCOHERENT MODULATION, M-ARY ORTHOGONAL SIGNALING...3 IV. SLOW, FLAT RAYLEIGH FADING CHANNELS...7 A. COHERENT MODULATION BPSK/QPSK...7. MPSK MQAM GMSK...36 B. NONCOHERENT MODULATION, M-ARY ORTHOGONAL SIGNALING...37 V. SLOW, FLAT RICEAN FADING CHANNELS...41 A. COHERENT MODULATION BPSK/QPSK MPSK MQAM GMSK...58 B. NONCOHERENT MODULATION, M-ARY ORTHOGONAL SIGNALING...59 VI. JOINT TACTICAL INFORMATION DISTRIBUTION SYSTEM...65 VII. CONCLUSIONS AND FUTURE WORK...69 A. CONCLUSIONS...69 B. FUTURE RESEARCH AREAS...73 vii

10 LIST OF REFERENCES...75 INITIAL DISTRIBUTION LIST...77 viii

11 LIST OF FIGURES Figure 1. Information it error and channel it error relationship [4]...6 Figure. Proaility of channel it error vs. information it error [4]...7 Figure 3. BPSK performance in AWGN (comparison etween exact and approximate results)...9 Figure 4. BPSK/QPSK performance in AWGN Figure 5. 8-PSK performance for a non-fading channel Figure PSK Performance for a non-fading channel...16 Figure QAM performance for a non-fading channel Figure QAM performance for a non-fading channel Figure QAM performance for a non-fading channel Figure QAM performance for a non-fading channel....0 Figure QAM performance for a non-fading channel....1 Figure QAM performance for a non-fading channel....1 Figure 13. GMSK performance for a non-fading channel....3 Figure FSK performance for a non-fading channel....4 Figure FSK performance for a non-fading channel....5 Figure FSK performance in a non-fading channel...5 Figure 17. BPSK/QPSK performance for a Rayleigh fading channel....8 Figure PSK performance for a Rayleigh fading channel....9 Figure PSK performance for a Rayleigh fading channel Figure QAM performance for a Rayleigh fading channel Figure QAM performance for a Rayleigh fading channel....3 Figure. 56-QAM performance for a Rayleigh fading channel Figure 3. 8-QAM performance for a Rayleigh fading channel Figure 4. 3-QAM performance for a Rayleigh fading channel Figure QAM performance for a Rayleigh fading channel Figure 6. GMSK performance for a Rayleigh fading channel Figure 7. 8-FSK performance for a Rayleigh fading channel Figure FSK performance for a Rayleigh fading channel Figure 9. 3-FSK performance for a Rayleigh fading channel Figure 30. Performance of BPSK transmitted over a Ricean fading channel for, 4,10,1. Increasing denoted y reduction in required E N o for p Lines converge at approximately p Figure 31. BPSK/QPSK performance for a Ricean fading channel for Figure 3. BPSK/QPSK performance for Ricean fading channel for Figure PSK performance for a Ricean fading channel for...46 Figure PSK performance for a Ricean fading channel for Figure PSK performance in Ricean fading channel for Figure PSK performance for a Ricean fading channel for Figure QAM performance for a Ricean fading channel for...49 ix

12 Figure QAM performance for a Ricean fading channel for Figure QAM performance for a Ricean fading channel for...51 Figure QAM performance for a Ricean fading channel for Figure QAM performance for a Ricean fading channel for...5 Figure QAM performance for a Ricean fading channel for Figure QAM performance for a Ricean fading channel for...54 Figure QAM performance for a Ricean fading channel for Figure QAM performance for a Ricean fading channel for...55 Figure QAM performance for a Ricean fading channel for Figure QAM performance for a Ricean fading channel for...57 Figure QAM performance in Ricean fading channel for Figure 49. GMSK performance for a Ricean fading channel for...58 Figure 50. GMSK performance for a Ricean fading channel for Figure FSK performance for a Ricean fading channel for...60 Figure 5. 8-FSK performance for a Ricean fading channel for Figure FSK performance for a Ricean fading channel for...6 Figure FSK performance for a Ricean fading channel for Figure FSK performance in Ricean fading channel for Figure FSK performance for a Ricean fading channel for Figure 57. General overview of JTIDS applications [5] Figure 58. JTIDS proaility of info it error vs. proaility of channel symol error...66 Figure 59. JTIDS performance as a function of proaility of channel symol error x

13 LIST OF TABLES Tale 1. Modulation Dependent Constants...8 Tale. Conditional proailities of channel symol error j for JTIDS [7]...67 Tale 3. E N o for modulation techniques and a non-fading channel for p Tale 4. E N o for modulation techniques and a Rayleigh fading channel for p Tale 5. E N o for modulation techniques and a Ricean fading channel for p 10 with....7 Tale 6. E N o for modulation techniques and a Ricean fading channel for 10 with p xi

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15 EXECUTIVE SUMMARY The 453 rd Electronic Warfare Squadron is the leading authority in improving the Improved-Many-On-Many (IMOM) modeling tool for the United States Air Force. IMOM is a two-dimensional, graphical-oriented, radio frequency modeling tool used for mission and planning intelligence. In an effort to improve the product, the effects of digital modulation techniques must e included in the system. Currently, only analog modulation schemes are incorporated into the architecture. The ojective of this thesis is to enhance the aility of the Improved Many-on- Many (IMOM) software package to analyze modern digital communication systems using availale intelligence. Currently, IMOM can only e used to analyze analog communication systems, ut modern systems are, increasingly, digital. In this thesis, the proaility of it error expressions for many common digital modulation techniques, oth inary and non-inary, are inverted to otain expressions for the required signal-to-noise ratio as a function of proaility of channel it error. Furthermore, results are otained not only for a non-fading channel ut for channels modeled as either Rayleigh or Ricean. The proaility of channel it error equations for coherently detected inary phase-shift keying (BPSK), quadrature phase-shift keying (QPSK), M-ary phase-shift keying (MPSK), M-ary quadrature amplitude modulation (MQAM), and Gaussian minimumshift keying (GMSK) and noncoherently detected M-ary frequency-shift keying (MFSK) are inverted, in some cases numerically, and, in the case of fading channels, numerically using logarithmic regression techniques. These equations can e implemented in IMOM to increase the accuracy of the link udget analysis when the specific modulation type eing evaluated is known. This thesis takes the approach of determining proaility of channel it error rather than information it error, which allows generic solutions independent of the specifics of the system under investigation as long as the particular modulation type is known. When even greater accuracy is desired, system specifics such as the type of error control coding must e taken into account. As an example of this, the Joint Tactical Information Distriution System (JTIDS) is considered. xiii

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17 ACKNOWLEDGMENTS I dedicate this work to my loving wife Janene, and three great sons Noah, Elijah and Jeremiah. Their support and dedication have een instrumental, not only to this thesis, ut to my military career. I would also like to give my sincere appreciation to Professor Clark Roertson for his support and confidence in me while at the Naval Postgraduate School. xv

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19 I. INTRODUCTION A. OVERVIEW Improved many-on-many (IMOM) is a two-dimensional, graphical-oriented, radio frequency modeling tool used for mission and planning intelligence. The tool can e used for electronic attack, electronic support, command and control, communications, computers and intelligence (C4I) and for mission planning and intelligence. Many-onmany refers to many targets versus many collectors/jammers, while improved reflects an aspiring dedication to continually advance the product. Current IMOM calculations are ased on various analog modulations. The goal of the U.S. Air Force and, specifically, the 453 rd Electronic Warfare Squadron (EWS) is to incorporate digital modulation waveforms into IMOM. In order to do so, reverse engineering of previously analyzed digital communications signals needs to e modified for easy incorporation into the link udget analysis equation [1], Equation Section 1 E M EIRP G T L L R 8.6 db dbw r dbi SdBK CdB 0 db db Hz dbw / K Hz N0 mindb (1.1) where radiated power, M db is the link margin expressed in deciels, EIRPdBW is the effective isotropic noise temperature, losses not accounted for, G rdbi is the gain of the receiving antenna, T S dbk is the system equivalent L C db is the free space channel loss, L 0dB represents miscellaneous R is the data rate of the signal, db Hz E N is the minimum 0 min db signal-to-noise ratio required to close the link, and / represents 8.6 dbw K Hz Boltzmann s constant. The term E N is the minimum allowed signal-to-noise ratio at the o mindb receiver which satisfies the proaility of it error specification of the system under consideration and accounts for the small scale fading losses of the communications link. 1

20 In order to implement Equation (1.1) directly, systems are typically evaluated to otain the proaility of channel error as a function of the signal-to-noise ratio. B. THESIS OBJECTIVE The primary ojective of this thesis is to invert proaility of it error equations for common digital communications modulation formats and determine the range of applicaility of the inverted equations. The equations are intended for implementation in the IMOM architecture. Currently, the architecture does not take into account digital modulation techniques, making it imprecise for applications where digital communications are used. The proaility of channel it error equations for coherently detected inary phase-shift keying (BPSK), quadrature phase-shift keying (QPSK), M-ary phase-shift keying (MPSK), M-ary quadrature phase-shift keying (MQAM), and Gaussian minimum-shift keying (GMSK) and noncoherently detected M-ary frequency-shift keying (MFSK) are inverted, in some cases numerically, and, in the case of Ricean fading channels, numerically using logarithmic regression techniques. To the est of the author s knowledge, this is the first time, except for the simplest cases such as noncoherent inary FSK (BFSK), that a comprehensive catalog of inverted proaility of it error equations for all the major modulation types, including the effects of small-scale fading, has een done. Furthermore, this thesis takes the novel approach of determining proaility of channel it error rather than information it error, which allows generic solutions independent of the specifics of the system under investigation as long as the particular modulation type is known. Finally, as an example of a specific system, the Joint Tactical Information Distriution System (JTIDS) is evaluated to determine E N for the first time in this thesis. 0 min db C. THESIS OUTLINE This thesis is organized into seven chapters, including the introductory material in this chapter. Insight into the ackground of the derivations otained throughout this thesis, are given in Chapter II. The proaility channel it error equations for coherent BPSK, QPSK, MPSK, MQAM and GMSK and noncoherent MFSK in additive white

21 Gaussian noise (AWGN) for nonfading, Rayleigh and Ricean fading communications channels are inverted in Chapters III, IV and V, respectively. The methodology is applied to JTIDS in Chapter VI. A culmination of the previous chapters and the results and the conclusion of the thesis are given in Chapter VII. 3

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23 II. BACKGROUND The modulation techniques evaluated in this thesis are among the most common types of digital modulation schemes availale for wireless communications and military applications. They are utilized in oth U.S. and European wireless communication standards and in aerospace applications in the Armed Forces. The approach taken in the majority of this thesis is not meant to e all inclusive ut to provide a generic methodology that can e implemented regardless of the system eing evaluated as long as the specific modulation technique is known. The coherent modulation techniques BPSK, QPSK, MPSK, MQAM and GMSK are evaluated for a non-fading, a Rayleigh fading and a Ricean fading channel. In addition a common non-coherent modulation technique, MFSK is evaluated for the same channel types. The equations used to evaluate the performance of the various modulation techniques are otained from pulished literature.equation Chapter (Next) Section A. GENERIC, SYSTEM INDEPENDENT APPROACH Forward error correction (FEC) coding significantly improves the performance of communications systems. The calculations and formulas derived for this project assume that error control coding has een implemented. The equations are intended to e generic in nature to remain flexile enough to adapt to different code rates and different code types as dictated y a specific system. The two primary types of FEC codes are lock codes and convolutional codes. Only the performance of JTIDS is calculated precisely. The intent of using JTIDS as an example, in addition to the ovious interest in this particular system, is to show the capaility that exists when specific system details are known. The generic equations can easily e tailored to specific systems if the system details, such as type of FEC coding used, are known. In order to understand the generic approach that depends only on the proaility of channel it error, it is important to understand the demodulation process. The 5

24 proaility of channel it error is a figure of merit related to the demodulation process and is a measure of the quality of the demodulated its entering the channel decoder as shown in Figure 1. Proaility of information/data it error Proaility of channel symol error Figure 1. Information it error and channel it error relationship [4]. The proaility of channel it error, prior to the channel decoder, directly impacts the output, or information, it error. Since the specifics of some systems may e unknown, it is necessary to determine a range of proaility of channel it error that will e valid for most applications. Since wireless communications systems are often specified for a proaility of information it error of 10-5, and since all modern digital communication systems use FEC codes, a generic range was determined [6]. For example, using the performance of systems with convolutional coding and hard decision decoding for various commonly used code rates, the relationship etween information it error P and the proaility of channel it error, p, is shown in Figure. We see that the range p corresponds to P 6

25 Figure. Proaility of channel it error vs. information it error [4]. As can e seen from Figure, a proaility of channel it error p 10 is more or less in the middle of the range of p values that yield a proaility of information it error of P In the remainder of this thesis, p is used to represent the proaility of channel it error instead of p as in Figure. B. COHERENT MODULATION TECHNIQUES A precise equation for proaility of channel it error for the coherent modulation techniques considered in this thesis is given y [] a p Q q (.1) q where E No is the average energy per it ( E )-to-noise power spectral density ( N o ) ratio, and a specific modulation technique is defined y the modulation dependent parameters a, q and []. These parameters are shown in Tale 1. Note that for MQAM there is a difference for even and odd values of q. Even values for q such as 16-QAM and 64-QAM are more commonly used in applications such as in wireless communication standards. 7

26 Tale 1. Modulation Dependent Constants. Modulation q a BPSK 1 1 QPSK 1 MPSK log M sin (π/m) MQAM 4,6, 4(1- -q/ ) 3/( q 1) MQAM 3,5, 4 3/( q 1) GMSK 1 1 δ In order to use the approximate equations for the coherent modulation techniques, it was necessary to ensure that the approximate equations presented a good representation of the exact equations for the scheme eing used. In order to estalish this relationship, an example for BPSK is used. Equation (.1) can e approximated y [] a 1 1 p exp q, (.) q q min which incorporates a slightly modified version of a common approximation for the Q-function. Specifically, in the radical has een replaced with min, a constant that is determined for each of the coherent modulation techniques discussed in order to optimize the results over a range of p as previously discussed. min around The results from plotting Equations (.1) and (.) for BPSK are shown in Figure 3. The used for BPSK was optimized for the region of interest for BPSK and centered p 10. In this case min.5 db was used and resulted in a nearly identical approximation for the range of values eing considered. The other coherent modulation techniques considered are tested using the same method and yield similar results. Variations etween the actual results and approximate results differ at most y 0.5 db over the range of interest, and for BPSK differ y at most 0.3 db. These values are small enough to e considered negligile for the purpose of this project and for incorporation into IMOM. 8

27 10-1 Exact Values Approximate Values P E (db) Figure 3. BPSK performance in AWGN (comparison etween exact and approximate results). C. NONCOHERENT, ORTHOGONAL MODULATION TECHNIQUES M-ary orthogonal signaling is considered using a similar procedure as that used for coherent modulation. The proaility of channel it error for orthogonal signaling with noncoherent detection, including orthogonal MFSK, is given y [] where M q p M 1 n1 M ( 1) M 1 nq exp ( M 1) n1 n1 n n1 (.3). Equation (.3) is accurately approximated for practical applications y the first term in the summation, which corresponds to the union ound: p M q exp 4. (.4) 9

28 D. MULTIPATH FADING CHANNELS Many wireless communications channels do not have a line-of-sight (LOS) signal path. The lack of a LOS signal path entails transmitting a signal to the receiver y a phenomenon known as multipath. Multipath occurs when there are multiple paths from the transmitter to the receiver due to the reflection of the original signal off of uildings, terrain (features) or the ionosphere. The availaility of a LOS does not necessarily mean that a multipath component does not exist. A signal that goes through a multipath fading channel will arrive at the receiver multiple times with different amplitudes, phases and arrival times. Since multipath channels can vary significantly in terms of characteristics in an unpredictale manner, they must e modeled as a random process. In addition, relative motion of the transmitter and receiver makes a multipath channel a time-varying channel. Therefore, an identical signal transmitted over the same multipath channel at a different time may exhiit different characteristics. In general, the received multipath components will add oth destructively and constructively, resulting in a received signal that fluctuates in signal amplitude and phase. This is known as signal fading [3]. The most common model for a multipath fading channel is the Ricean model, where the amplitude of the received signal is modeled as a Ricean random variale. In that case, Equation (.1) must e modified to [] a 0 ( q0 ) q 0 p Q( q ) exp I 0 d 0 q, (.5) where I 0 is the modified Bessel function of the first kind and order zero, represents the non-los signal power, represents the LOS signal power and N T. The one-sided power spectral density of the AWGN is o o s N o, and T s is the symol duration [4]. Generally speaking, simple inversion techniques that are used for non-fading channels do not work for Ricean fading channels due to the complexity of Equation (.5). 10

29 E. JOINT TACTICAL INFORMATION DISTRIBUTION SYSTEM (JTIDS) JTIDS is a network radio system used y the U.S. Armed Forces and its allies to support data communications needs. JTIDS uses (31,15) Reed Solomon (RS) coding. The modulation technique used in JTIDS is cyclical code-shift keying (CCSK), which utilizes 3 phases of a 3-chip sequence and is a quasi-orthogonal modulation scheme. JTIDS operates with oth a single- and doule-pulse architecture. For this thesis, the single pulse architecture was considered. The single-pulse architecture allows for greater data throughput than the doule-pulse architecture. [5] In this chapter, expressions for the channel proaility of it error for the modulation techniques considered in this thesis were provided, and a system non-specific procedure was developed to estimate the required signal-to-noise ratio given only the type of digital modulation utilized. The fundamentals necessary to account for Rayleigh and Ricean multipath fading channels were also provided. The following chapter expands on the concepts introduced in Chapter II for non-fading channels. 11

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31 III. NON-FADING CHANNELS Non-fading channels require the minimal amount of signal power for a given modulation type and, therefore, represent a channel with only a line-of-sight signal component. While this is overly optimistic for many of today s digital communications systems, it serves as a reference when incorporating fading channels. As mentioned in the last chapter, E N0, and is the parameter to e determined. Equation Chapter (Next) Section 3 where Solving Equation (.) for, we get q 10log ln q 4 optimized q a min p (3.1), (3.) min q and optimized is otained using trial-and-error to optimize Equation (3.1) for the range of p of interest. A. COHERENT MODULATION For BPSK and QPSK, the modulation dependent parameters listed in Tale 1 have the same ratio and products and, therefore, exhiit identical performance. In the case of BPSK and QPSK,.5 db, or min Sustituting parameters for BPSK/QPSK from Tale 1 into Equation (3.1), we get 10 log ln (410.5 and the appropriate min 0.5 p. (3.3) 13

32 A plot of Equation (3.3) is shown in Figure 4. Each graph in this thesis is marked according to the proaility of channel it error p 10 and the associated signal-tonoise ratio. As can e seen from Figure 4, for BPSK and QPSK 4.3 db is required for p BPSK/QPSK 6 4 E p Figure 4. BPSK/QPSK performance in AWGN. 1. MPSK With regard to MPSK, 8-PSK and 16-PSK are the most commonly used modulation schemes. MPSK with M 3 is rarely used due to the excessive power required [4]. 8-PSK is used in the IEEE 80.11g wireless standard and is widely used in satellite communications [6]. As with BPSK,.5 db optimizes Equation (3.1) for 8-PSK and 16-PSK. As previously mentioned, since the ojective is to evaluate unknown systems, the range of proaility of channel it error of interest is Sustituting 14 min p 0.5 min 10 and the appropriate parameters from Tale 1 into Equation (3.1), we otain for 8-PSK

33 and, similarly, for 16-PSK log ln p, (3.4) log ln p, (3.5) with the units in deciels or db. Plots of Equations (3.4) and (3.5) are shown in Figures 5 and 6, respectively. As can e seen, 8-PSK requires 9.7 db while 16-PSK requires 15.5 db to meet the desired proaility of channel it error p PSK 1 10 E p Figure 5. 8-PSK performance for a non-fading channel. 15

34 0 16PSK E p Figure PSK Performance for a non-fading channel.. MQAM The analysis of MQAM is more complex than the previously discussed MPSK schemes. However, MQAM can e thought of as a generalization of MPSK where the signal constellation is not restricted to eing circular. MQAM also differs in that different values for a, as shown in Tale 1, are required for even and odd values of q. Even constellations are more common in practical applications. 16-QAM and 64-QAM are used in wireless communication standards 80.11a/g [6]. Sustituting the appropriate parameters from Tale 1 for 16-QAM into Equation (3.1), we get 10 log.5ln Similarly, for 64-QAM 10 log 6.9 ln p (3.6) 0.5 p, (3.7)

35 and for 56-QAM 10 log 64.1ln p (3.8) where.5 db is used to optimize Equations (3.6), (3.7) and (3.8) for the range of min interest. Equations (3.6), (3.7) and (3.8) are plotted in Figures 7, 8 and 9, respectively. In order to achieve p 10, signal-to-noise ratios of 7.8 db, 11.8 db and 16. db are required for 16-QAM, 64-QAM and 56-QAM, respectively QAM 8 6 E p Figure QAM performance for a non-fading channel. 17

36 15 64QAM 10 E p Figure QAM performance for a non-fading channel QAM E p Figure QAM performance for a non-fading channel. 18

37 The equations developed for MQAM are only valid for a proaility of channel it error less than 0.065, since p values greater than this yield inaccurate results. This does not present any concerns since this limit is well within the desired ranges for Due to its large power consumption, 56-QAM is rarely used in wireless communications [6]. Sustituting the appropriate parameters from Tale 1 into Equation (3.1), for 8- QAM we get Similarly, for 3-QAM 10 log 1.6 ln p. 0.5 p. (3.9) 10 log 4.1ln p, (3.10) and for 18-QAM, 10 log 1.1ln p. (3.11) Equations (3.9), (3.10) and (3.11) are plotted in Figures 10, 11 and 1, respectively. As can e seen, signal-to-noise ratios of 6.7 db, 10 db and 14.1 db are required for 8-QAM, 3-QAM and 18-QAM, respectively, to achieve a proaility of channel it error of

38 10 9 8QAM E p Figure QAM performance for a non-fading channel. 0

39 14 1 3QAM 10 8 E p Figure QAM performance for a non-fading channel QAM 14 1 E p Figure QAM performance for a non-fading channel. 1

40 3. GMSK GMSK is used for the European wireless standard Gloal System for Moile (GSM) [6]. As the military ecomes joint in nature, it is also expected that increasing coordination with other nations will follow. Thus, it is important to explore standards and implementations other than our own. The wireless communication industry is always exploring ways to incorporate gloal standards. This is evident, as seen in the transition from dual and cellular phones to the quad-and phones currently availale. Even with the advent of these technologies there is much research that continues in an attempt to merge wireless standards. For GMSK, the modulation constant varies, as shown in Tale 1, with respect to. The value for varies due to the andwidth-it duration product BT, ut there is only aout a one-db variation in the signal-to-noise ratio required to otain a specified proaility of it error [], so for this thesis 0.7 is used, which corresponds approximately to GSM. 0.5 Sustituting min 10 and the appropriate parameters from Tale 1, we get 10 log 1.4 ln p (3.1) for GMSK. Equation (3.1) is plotted in Figure 13. A of 5.7 db is required to otain a proaility of channel it error of 10.

41 10 GMSK 8 6 E p Figure 13. GMSK performance for a non-fading channel. B. NONCOHERENT MODULATION, M-ARY ORTHOGONAL SIGNALING M-ary orthogonal signaling such as MFSK is a modulation used for oth commercial and military communications. MFSK is used in Milstar, a U.S. Military satellite communications system. In addition, M-ary orthogonal signaling is used for the reverse channel of the IS-95 wireless standard [3]. Although 8-FSK, 16-FSK and 3-FSK are shown in this work, it is important to note that noncoherent, M-ary orthogonal modulation techniques have identical performance. Evaluating the performance of MFSK is rather trivial when compared to the coherent techniques discussed earlier. By inverting Equation (.4), repeated here for convenience, where M q, we get p M exp q, (3.13) 4 3

42 4 10 log ln p q M. (3.14) From Equation (3.14), we get (3.15) for 8-FSK Similarly, for 16-FSK and for 3-FSK, 10 log 0.67 ln 0.5 p. (3.15) 10 log 0.5ln 0.5 p, (3.16) 10 log 0.4 ln 0.15 p. (3.17) Equations (3.15), (3.16) and (3.17) are plotted in Figures 14, 15 and 16, respectively. As can e seen, a of 5.5 db, 4.7 db and 4.3 db is required for for 8-FSK, 16-FSK and 3-FSK, respectively. p FSK 8 6 E p Figure FSK performance for a non-fading channel. 4

43 FSK E p Figure FSK performance for a non-fading channel FSK E p Figure FSK performance in a non-fading channel. 5

44 In contrast to the coherent modulation techniques evaluated, noncoherent MFSK differs in that, as M increases, the required signal-to-noise ratio decreases [3]. Expressions for the signal-to-noise ratio as functions of the proaility of channel it error were developed in this chapter. The next chapter extends the expressions to encompass the effects of Rayleigh fading channels for the various modulation techniques eing considered. 6

45 IV. SLOW, FLAT RAYLEIGH FADING CHANNELS A slow, flat Rayleigh fading channel is a model of a communications channel that is used when there is no line-of-sight etween the transmitter and the receiver, and all of the received signal power is due to multipath. In contrast to no fading, this model represents the opposite end of the spectrum. For coherent modulation techniques, the proaility of it error for a slow, flat Rayleigh fading channel is otained from Equation (.5) with 0, which results in [] Equation Chapter (Next) Section 4 where a 0 s 0 p Q( s) exp d 0 s q (4.1) E s. (4.) N0 q For Rayleigh fading channels, Equation (4.1) can e evaluated, and for each coherent modulation technique considered in this thesis, the analytical expression otained y evaluating Equation (4.1) can e algeraically inverted. Logarithmic regression is applied to MFSK since algeraic inversion is impractical in this case. The same modulation schemes examined for non-fading channels are evaluated for Rayleigh fading channels. A. COHERENT MODULATION 1. BPSK/QPSK For BPSK and QPSK and a Rayleigh fading channel, Equation (4.1) can e evaluated to otain p Equation (4.3) can e algeraically inverted to otain. (4.3) (1 p ) 10log 4 4 p p (4.4) where the result is expressed in db and plotted in Figure 17. 7

46 5 BPSK/QPSK 0 15 E p Figure 17. BPSK/QPSK performance for a Rayleigh fading channel. The performance for BPSK/QPSK for a Rayleigh fading channel exhiits the expected performance degradation when compared to the non-fading channel performance. In order to otain a proaility of channel it error of p 10, 14 db of signal power is required. This is significantly more than the 4.3 db required for a nonfading environment.. MPSK Evaluating Equation (4.1) for MPSK and Rayleigh fading, we get [] p 1 q sin ( / ) 1 M q 1 q sin ( / M) (4.5) 8

47 In order to meet demands for current applications, oth 8-PSK and 16-PSK were evaluated with satisfactory results. After inverting Equation (4.5) for 8-PSK and 16-PSK, we get for 8-PSK and 1 (1 8 p ) 10log 8sin ( 8) p p 1 (116 p ) 10log 16sin ( 16) 3 56 p p (4.6) (4.7) for 16-PSK. Equations (4.6) and (4.7) are plotted in Figures 18 and 19, respectively. 5 8PSK 0 15 E p Figure PSK performance for a Rayleigh fading channel. 9

48 30 16PSK 5 0 E p Figure PSK performance for a Rayleigh fading channel. For 8-PSK and 16-PSK, for a channel p of 10, E o N of 15.5 db and 18.6 db, respectively, are required. As expected, the E No required increases when no line-ofsight exists. This is an increase from the values of 9.7 db and 15.5 db, respectively, for 8-PSK and 16-PSK in a non-fading channel. 3. MQAM As descried for a non-fading channel model, for MQAM it is necessary to examine MQAM for two variations of the modulation-type parameters, one when q is even and one when q is odd as seen in Tale 1. Although oth modulation-type parameters provide good results, the q -even modulation-type parameters produce slightly more accurate results [4]. Analyzing 16-QAM and 64-QAM are important ecause they are used in the IEEE 80.11g wireless standard [6]. 30

49 For MQAM with even q, Equation (4.1) can e evaluated to otain p q 3q 1 1 q q ( 1) 3q. (4.8) After inverting Equation (4.8), we otain the result for 16-QAM as (1 8 7 p ) p 5 10log p which is plotted in Figure 0. As can e seen, E N 10dB is required for o p 10. (4.9) 5 16QAM 0 15 E p Figure QAM performance for a Rayleigh fading channel. The same method was applied to 64-QAM. The result is given y (1 4 7 p ) 48 7 p log 7 p which is plotted in Figure 1. As can e seen, E N 16.4 db is required for o (4.10) p

50 5 64QAM 0 15 E p Figure QAM performance for a Rayleigh fading channel. Despite its limited use in many practical applications, 56-QAM was also evaluated. However, it does provide an insight into the power that is required when this modulation technique is used. Using the same methodology as previously, we invert Equation (4.8) with M 56 to otain ( p ) p log p which is plotted in Figure. As can e seen, E N 0 db is required for o (4.11) p 10. 3

51 5 56QAM 0 15 E p Figure. 56-QAM performance for a Rayleigh fading channel. MQAM was also evaluated for odd values of q. For MQAM with q as an odd value that is greater than or equal to 3, Equation (4.1) can e evaluated to otain [] p 3q 1 q q ( 1) 3q. (4.1) Utilizing the same concept as for even values of q, we otained inverted equations for 8-QAM, 3-QAM and 18-QAM. Inverting Equation (4.1) for 8-QAM, we get p 14 (1 3 ) 10log p p which is plotted in Figure 3. As can e seen, E N 17 db is required for o (4.13) p

52 30 8QAM 5 0 E p Figure 3. 8-QAM performance for a Rayleigh fading channel. Similarly, for 3-QAM we get 6 (1 5 p ) 10log p p (4.14) which is plotted in Figure 4. As can e seen, E N 19dB is required for o p

53 30 3QAM 5 0 E p Figure 4. 3-QAM performance for a Rayleigh fading channel. Again as efore, for 18-QAM, we get 54 (1 7 p ) 10log p p, (4.15) which is plotted in Figure 5. As can e seen, E N 1dB is required for o p 10. As q increases, as expected, the signal-to-noise ratio required for a given channel it error performance increases [3]. 35

54 6 4 18QAM 0 E p Figure QAM performance for a Rayleigh fading channel. 4. GMSK The same algeraic inversion technique was used to invert the GMSK expression for a Rayleigh fading channel. Just as in the non-fading case,, where 0.7, was used. Evaluating Equation (4.1) for GMSK, we get [] p Equation (4.16) can e inverted to otain. (4.16) 1 (1 p ) 10log (4.17) 4p 4p which is plotted in Figure 6. As can e seen, E N 15.4dB is required for 10. o p 36

55 30 GMSK 5 0 E p Figure 6. GMSK performance for a Rayleigh fading channel. B. NONCOHERENT MODULATION, M-ARY ORTHOGONAL SIGNALING MFSK, and M-ary orthogonal signaling in general, are the noncoherent modulation techniques evaluated in this thesis. We will evaluate 8-FSK, 16-FSK and 3- FSK ecause they are the most commonly used in practice [4]. Since the expression for MFSK and Rayleigh fading cannot e inverted analytically, a logarithmic regression technique is used to get equations for in terms of p. This logarithmic regression technique was used in cases where algeraic inversion is impractical. The proaility of channel it error for MFSK transmitted over a Rayleigh fading channel is given y [] p M 1 n1 M ( 1) M 1. (4.18) ( M 1) (1 n nq ) n n1 The exact expression is used since the union ound is inaccurate for Rayleigh fading channels [4]. 37

56 Using the logarithmic regression technique on Equation (4.18) when M 8, M 16 and M 3, we get for 8-FSK, 3.9 ln( p) 9.1 p (4.19) for 16-FSK and 4.1ln( p) 7.1 p 1.5, (4.0) 4ln( p) 6.7 p 1.7 (4.1) for 3-FSK. Equations (4.19), (4.0) and (4.1) are plotted in Figures 7, 8 and 9, respectively. As can e seen, E N 16.7 db, E N 16.3dB and E N 15.8dB are o o o required to achieve p 10 for M = 8, 16 and 3, respectively. Similarly, if necessary, can e otained for M 64. As expected, as M increases, the required decreases. 30 8FSK 5 0 E p Figure 7. 8-FSK performance for a Rayleigh fading channel. 38

57 30 16FSK 5 0 E p Figure FSK performance for a Rayleigh fading channel. 30 3FSK 5 0 E p Figure 9. 3-FSK performance for a Rayleigh fading channel. 39

58 Expressions for the signal-to-noise ratio as functions of the proaility of channel it error for Rayleigh fading channels were developed in this chapter. The next chapter repeats the process for Ricean fading channels. 40

59 V. SLOW, FLAT RICEAN FADING CHANNELS The slow, flat Ricean model is used where there is a line-of-sight etween transmitter and receiver ut a sustantial portion of the received signal power is also due to multipath. The most noticeale difference in the equations for a Ricean channel is the direct-to-diffuse signal power ratio. As increases, the model tends to approach a non-fading model. In contrast, as approaches 0, the model approaches that of a Rayleigh fading model. Equation Chapter (Next) Section 5 The direct-to diffuse signal power ratio is defined y [], (5.1) where represents the received line-of-sight signal power, and is the non-line-ofsight signal power. For coherent demodulation techniques, the equations for Ricean fading channels are derived from [] a 0 ( s0 ) s 0 p Q( ) exp 0 s I 0 d s q, (5.) where s q. Implementing Equation (5.) for BPSK and for varying values of the parameter, we otain the results shown in Figure 30. In order to invert the equations for Ricean fading channels, the same logarithmic regression technique used for MFSK and Rayleigh fading is used. 41

60 p E /N 0 Figure 30. Performance of BPSK transmitted over a Ricean fading channel for, 4,10,1. Increasing denoted y reduction in required E N o for p Lines converge at approximately p For values of greater than 1, the numerical evaluation of Equation (5.) is impractical. Generally, if the line-of-sight signal strength is more than 1 times that of the non-line-of-sight signal strength, the LOS component dominates the channel, and the effect of fading is not as important. The modulation techniques eing considered were evaluated for Ricean fading channels with and 10 in order to give an idea of the effect of. 0 0 s Sustituting 1 s (5.3) 0 where N T, into Equation (5.), we can evaluate performance in terms of the directto-diffuse ratio parameter []. For Ricean fading channels [], p is a good approximation where c a (1 ) q exp q c (1 ) q (1 q) (5.4) 4

61 A. COHERENT MODULATION 1. BPSK/QPSK 1, For BPSK and QPSK, for a Ricean fading channel, from Equation (5.4) and Tale p 1 1 exp. (5.5) c 1 1 Equation (5.5) and logarithmic regression is used to otain and 4ln( p) p 7 (5.6) 1.5ln( p) 8 p for and 10, respectively. (5.7) Equations (5.6) and (5.7) are plotted in Figures 31 and 3, respectively. As can e seen, E N 11dB and E N 5.8 db are required to achieve o o p 10 when and 10, respectively. As expected, increasing results in a decrease in required signal-to-noise ratio. 43

62 5 BPSK/QPSK 0 15 E p Figure 31. BPSK/QPSK performance for a Ricean fading channel for. 1 BPSK/QPSK 10 8 E p Figure 3. BPSK/QPSK performance for Ricean fading channel for

63 . MPSK For MPSK, for a Ricean fading channel from Equation (5.4) and Tale 1, p sin q (1 ) M exp q c(1 qsin 1 sin. (5.8) q M M Equation (5.8) and logarithmic regression are used to otain for 8-PSK and and.4 ln( p) 30 p 5.5 (5.9) 1.3ln( p) 0 p 4.7 for 8-PSK and 10, respectively. (5.10) Equations (5.9) and (5.10) are plotted for and 10 in Figures 33 and 34, respectively. As can e seen, E N 13.3dB and E N 8.8 db are required to o o achieve p 10 for and 10, respectively. 45

64 5 8PSK 0 15 E p Figure PSK performance for a Ricean fading channel for. 14 8PSK 1 10 E Figure PSK performance for a Ricean fading channel for 10. p 46

65 Similarly, the same method was used for 16-PSK. For 16-PSK and a Ricean fading channel, when and when 10..5ln( p) 33 p ln( p) 36 p 11 (5.11) (5.1) Equations (5.11) and (5.1) are plotted in Figures 35 and 36, respectively. As can e seen, E N o 16.8 db and E N o 1 db are required to achieve p 10 when and 10, respectively. 5 16PSK 0 15 E Figure PSK performance in Ricean fading channel for. p 47

66 PSK 14 1 E Figure PSK performance for a Ricean fading channel for 10. p The introduction of the parameter has a ig impact on the required signal-tonoise ratio. Based on the value of M, there is a wide range of proaility of channel it error that can e introduced into any system as a result of fluctuations in. 3. MQAM MQAM for a Ricean fading channel, as previously, is roken down into even and odd values of q. For MQAM for a Ricean fading channel and even values of q, p whereas for odd values of q, 1 4 q 1 ( 1)( 1) q 3q exp 3 ( q 1)( 1) 3 ( q q c q q 1)( 1), (5.13) 48

67 p 3q q 4( 1)( 1) exp 3 ( q 1)( 1) 3 ( q q c q q 1)( 1). (5.14) Equations (5.13) and (5.14) were evaluated for and 10 for 16-QAM. Inverting Equation (5.13) for even values of q results in.3ln( p) 4 p 5.7 for 16-QAM and and (5.15) for 16-QAM and ln( p) 11 p 1.1 (5.16) Equations (5.15) and (5.16) are plotted in Figures 37 and 38, respectively. As can e seen, E N o 13 db and E N o 6 db are required to achieve p 10 when and 10, respectively. 0 16QAM E Figure QAM performance for a Ricean fading channel for. p 49

68 QAM E p Figure QAM performance for a Ricean fading channel for 10. For 64-QAM and, for a Ricean fading channel, we get and 0.8ln( p) 66 p 7.5, (5.17) for 64-QAM and ln( p) 51 p 0 (5.18) Equations (5.17) and (5.18) are plotted in Figures 39 and 40, respectively. As can e seen, E N 17 db and E N 13.5dB are required to achieve when and 10, respectively. o o p 10 50

69 QAM E p Figure QAM performance for a Ricean fading channel for QAM 14 1 E p Figure QAM performance for a Ricean fading channel for

70 and For 56-QAM and, for a Ricean fading channel, we get 0.37 ln( p) 78 p 3 0.3ln( p) 6 p 3 (5.19) (5.0) for 56-QAM and 10. Equations (5.19) and (5.0) are plotted in Figures 41 and 4, respectively. As can e seen, E N db and E N 18dB are required to achieve and 10, respectively. o o p 10 when 30 56QAM 5 0 E p Figure QAM performance for a Ricean fading channel for. 5

71 4 56QAM E Figure QAM performance for a Ricean fading channel for 10. p Odd numered values of q were evaluated in the same manner y inverting Equation (5.14) for and 10. For 8-QAM and, for a Ricean fading channel, we get and 3.7 ln( p) 10.7 p 0.6, (5.1) for 8-QAM and ln( p) 16 p 4.8 (5.) Equations (5.1) and (5.) are plotted in Figures 43 and 44, respectively. As can e seen, E N 14dB and E N 8.4 db are required to achieve and 10, respectively. o o p 10 when 53

72 4 8QAM E p Figure QAM performance for a Ricean fading channel for. 14 8QAM 1 10 E Figure QAM performance for a Ricean fading channel for 10. p 54