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Advanced noncoherent receivers for mobile fading channels Bouras, Dimitrios P. 1995

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ADVANCED NONCOHERENT RECEIVERS FORMOBILE FADING CHANNELSByDimitrios P. BourasDiploma (Electrical Engineering) University of Patras, Greece, 1989M. A. Sc. (Electrical Engineering) The University of British Columbia, 1991A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESELECTRICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1995© Dimitrios P. Bouras, 1995In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely available forreference and study. I further agree that permission for extensive copying of this thesis forscholarly purposes may be granted by the head ofmy department or by his or her representatives.It is understood that copying or publication of this thesis for financial gain shall not be allowedwithout my written permission.ELECTRICAL ENGINEERINGThe University of British Columbia2356 Wesbrook MallVancouver, CanadaV6T 1Z4Date:I5ABSTRACTThe purpose of this thesis, is to derive and evaluate the performance of noncoherent, maximumlikelihood receivers with improved performance, for trellis coded PSK and QAM type signals,transmitted over Rician, correlated, fast, frequency non-selective and frequency selective fadingchannels, with and without diversity.First we derive the optimal, in the maximum likelihood detection sense, receiver structurefor frequency non-selective Rician fading channels, employing diversity reception. In orderto reduce the complexity of the optimal receiver, we propose and evaluate the performance ofsuboptimal receiver structures, which show significant performance improvements as comparedto conventional techniques. Investigation of the effects on performance of the proposedalgorithms, due to imperfect statistical knowledge of the fading channel typical for a real lifeenvironment, demonstrates very small sensitivity even to large errors in estimates of channelparameters.Complementing our work in frequency non-selective fading, we derive the optimal, inthe maximum likelihood detection sense, receiver, for the correlated, fast, frequency selectiveRician fading channel. In the interest of system simplicity, we propose and evaluate reducedcomplexity versions of the decoding algorithms. The impact of simplifying assumptions inthe theoretical derivation, as well as the receiver sensitivity to non ideal channel knowledge, isinvestigated. The results show significant performance improvements over the fastest knownchannel equalization technique, accompanied by small sensitivity to imperfections.Last, we derive analytical performance bounds for simplified versions of the optimal diversity receiver, for frequency non-selective, Rician fading channels. The tightness and accuracy11of the bounds is verified, through the excellent agreement between computer simulation results, and bound calculation. Performance evaluation demonstrates significant improvements,approaching the effectiveness of coherent detection in AWGN, even with a relatively smalldiversity order, for Rician, as well as shadowed EHF fading channels.111TABLE OF CONTENTSABSTRACT iiLIST OF TABLES viiiLIST OF FIGURES ixGLOSSARY xvACKNOWLEDGEMENTS xvii1 INTRODUCTION 11.1 MODULATION SCHEMES 41.2 THE MOBILE FADING CHANNEL 71.2.1 Frequency Non-Selective Mobile Fading 91.2.2 Frequency Selective Mobile Fading 111.3 MITIGATION TECHNIQUES FOR MOBILE FADING CHANNELS 121.3.1 Coherent Detection Techniques 131.3.2 Noncoherent Detection Techniques . . . 131.3.3 Diversity Techniques 181.3.4 Equalization Techniques 201.3.5 Maximum Likelihood Sequence Estimation (MLSE) Techniques 211.4 ANALYTICAL BER PERFORMANCE BOUNDS 211.5 THESIS RESEARCH CONTRIBUTION 23iv1.6 THESIS ORGAMZATION.252 OPTIMAL SEQUENCE ESTIMATION FOR FAST CORRELATED FREQUENCYNON-SELECTIVE RICIAN FADING CHANNELS WITH DIVERSITY 262.1 TRANSMITTER AND FREQUENCY-FLATFADING DIVERSITY CHANNEL MODEL 272.2 DERIVATION OF THE STRUCTURE OF THEOPTIMAL DIVERSITY RECEIVER 342.2.1 Optimal Algorithms 352.2.2 Suboptimal Algorithms 412.3 BER PERFORMANCE EVALUATION OF THESUBOPTIMAL ALGORITHMS 432.3.1 Perfect Knowledge of Channel Parameters 442.3.2 Imperfect Knowledge of Channel Parameters 482.4 CONCLUSION . . . 533 OPTIMAL SEQUENCE ESTIMATION FOR FAST CORRELATEDFREQUENCYSELECTIVE RICIAN FADING CHANNELS 553.1 FREQUENCY SELECTIVE FADING CHANNEL MODEL 553.2 RECEIVER MODEL 583.3 DERIVATION OF THE MAXIMUM LIKELIHOODSEQUENTIAL DECODER 613.4 BER PERFORMANCE EVALUATION RESULTS AN])DISCUSSION 653.5 CONCLUSION 784 PERFORMANCE ANALYSIS OF OPTIMAL NONCOHERENT DETECTIONVFOR SLOW CORRELATED FREQUENCY NON-SELECTIVE RICIAN FADINGWITH DIVERSITY 794.1 DERIVATION OF THE MAXIMUM LIKELIHOODSEQUENTIAL DECODER FOR SLOW FADING 804.1.1 Low SNR channel 824.1.2 High SNRchannel 834.2 ANALYTICAL PERFORMANCE BOUND 844.3 BER PERFORMANCE EVALUATION RESULTS ANDDISCUSSION 884.3.1 UHF Land Mobile Fading Channel 884.3.2 EHF Mobile Satellite Channel 954.4 CONCLUSION 1005 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK 1015.1 CONCLUSIONS 1015.2 SUGGESTIONS FOR FUTURE WORK 1035.2.1 Suboptimal Receiver Algorithms of Reduced Complexity 1035.2.2 Receiver Structures for Nyquist Pulses in Frequency Selective Fading 1045.2.3 Frequency selective fading with > 1 1045.2.4 Extension of Derivation of Bounds for Multi-level Schemes 1045.2.5 Performance Bounds for Fast Fading 1055.2.6 Investigation of MDD Techniques in Spread Spectrum Systems . 1055.2.7 VLSI Implementation of Detection Algorithms Derived 1055.2.8 Neural Net Based Adaptive Receivers 1065.2.9 Digital Image Transmission in Fading Channels 106viAPPENDIX A 119A. I Derivation of Eq. (2.22) 119A.2 Derivation of Eq. (2.25) 121APPENDIX B 123B.1 Derivation of Eqs. (3.40), (3.41) and (3.42) 123B.2 Derivation of Eqs. (3.46) and (3.47) 125APPENDIX C 127C.1 Derivation of Eq. (4.59) 127C.2 Derivation of Eqs. (4.60) and (4.61) 131C.3 Derivation of Eq. (4.68) 133C.3.1 Rician Fading, Strong LOS Component and High SNR 137C.3.2 Rayleigh Fading Channels 138C.4 Derivation of Eq. (4.80) 140C.5 Derivation of Eq. (4.81) 142viiLIST OF TABLES2.1 Performance degradation due to inaccurate estimate of channel BFT at thereceiver 502.2 Performance degradation due to inaccurate estimate of channel f’ at the receiver. For the results summarized in this table, the Rician channel has beenassumed to have a K = 5 dB and BFT 0.125. The estimate error for f’ hasalso been translated to an error in the value of K which is given in dB. Theextreme case is when the receiver assumes the channel is Rayleigh (- 100% errorin f or K — —oc dB) while the channel has been assumed Rician with K = 5dB. SNR degradation is given in parenthesis under each BER performance value 522.3 Performance degradation due to inaccurate estimate of channel f’ at the receiver. In this case, the receiver assumes that the channel is Rician whereas infact it has a Rayleigh characteristic with BFT = 0.125. The value of K givenis that assumed by the receiver. SNR degradation is given in parenthesis undereach BER performance value 534.4 L-band mobile-satellite radio channel model parameters 96C.5 Ka-band satellite channel model parameters 141C.6 Ratio of LOS signal average power to multipath signal power (KAvG); comparison between L and Ka bands 141viiiLIST OF FIGURES1.1 Examples of PSK and QAM modulation formats; (a) 8-PSK (b) 16-QAM. . . 51.2 Simplified example of a propagation environment giving rise to frequencyselective mobile fading 111.3 Block diagram illustrating the multiple differential detector hardware structure.PD: phase detector 152.1 Block diagram of the communication system transmitter 272.2 The signal space of the 7r/4-shiftDQPSK transmitter; a) before spectral shaping,b) at the output of the receiver filter in the absence of interference; raised cosinefiltera=O.7 292.3 The signal space of the 7r/4-shift 8-DQAM transmitter; a) before spectral shaping, b) at the output of the receiver filter in the absence of interference; raisedcosine filter = 0.7 292.4 Convolutional coders used for ir/4-shift DQPSK, 7r/4-shift 8-DQAM and 8-DPSK, showing assignment of signal phases and amplitudes to signal numbers 302.5 Block diagram of the low-pass equivalent of the frequency selective fadingmodel 312.6 Block diagram of the diversity receiver 332.7 Block diagram of the bank of multiple differential detectors; PD: Phase Detector 392.8 Block diagram for the structure of the optimal diversity receiver decoder. . . 40ix2.9 BER performance evaluation of raised-cosine filtered (a = 0.35) rate 2/3, 8-state ‘r/4-shift 8-DQAM in a Rayleigh land-mobile fading environment withBFT=0.125 442.10 BER performance comparison of raised-cosine filtered (a = 0.35) rate 1/2,4-state ir/4-shift DQPSK in a Rayleigh land-mobile fading environment withBFT = 0.125 using the improved diversity receiver structure derived, to theperformance of a 1-symbol differential detector (DD) and Viterbi decoder. . . 452.11 BER performance comparison of raised-cosine filtered (a = 0.35) rate 2/3, 8-state 8-DPSK in a Rayleigh land-mobile fading environment with BFT = 0.125using the improved diversity receiver structure derived, to the performance ofa 1-symbol differential detector (DD) and Viterbi decoder 462.12 Effects of correlation between signals received on two diversity paths on araised-cosine filtered (a = 0.35) rate 1/2, 4-state ir/4-shift DQPSK scheme,in a Rayleigh land-mobile fading environment with BFT = 0.125, using theoptimal diversity receiver with prediction order z = 2 482.13 Effects of fading correlation and direct/diffused power ratio K on a raised-cosine filtered (a = 0.35) rate 1/2, 4-state 7r/4-shift DQPSK scheme, in aRician land-mobile fading environment with BFT = 0.125, using the optimaldiversity receiver with prediction order z = 2 and two (A = 2) diversity paths. 493.1 Block diagram of the low-pass equivalent of the frequency selective fadingmodel 573.2 BER performance of rate- 1/2 trellis coded Butterworth filtered (fB = 1 /T)ir/4-shift DQPSK scheme, using the frequency-selective maximum likelihoodreceiver, versus the equivalent frequency-flat maximum likelihood receiver, inthe presence of a 2-ray Rayleigh fading channel with BFT = 0.0625 and = 0.75. 67x3.3 BERperformance ofrate- 1/2 trellis coded Butterworth filtered ir/4-shift DQPSKscheme using the frequency-selective maximum likelihood receiver, versus theequivalent frequency-flat maximum likelihood receiver, both with z = 3, in thepresence of a 2-ray Rayleigh fading channel with BFT = 0.0625 and = 0.25 673.4 BERperformance of rate- 1/2 trellis coded Butterworth filtered ir/4-shift DQPSKscheme using the frequency-selective maximum likelihood receiver, versus theequivalent frequency-flat maximum likelihood receiver, both with z = 3, in thepresence of a 2-ray Rayleigh fading channel with BFT = 0.125 and 0.75 693.5 BER performance ofrate- 1/2 trellis coded Butterworth filtered ir/4-shift DQPSKscheme using the frequency-selective maximum likelihood receiver, versus theequivalent frequency-flat maximum likelihood receiver, both with z = 3, in thepresence of a 2-ray Rayleigh fading channel with BFT = 0.0625 and = 0.25 693.6 Frequency-selective maximum likelihood receiver (z = 2) performance degradation due to inaccurate estimate of channel delay () and BFT, in the presenceof a 2-ray Rayleigh fading channel 703.7 Frequency-selective maximum likelihood receiver (z = 3) performance degradation due to inaccurate estimate of channel delay (?) and BFT, in the presenceof a 2-ray Rayleigh fading channel 713.8 BER performance of rate- 1/2 trellis coded Butterworth filtered (fB I /T)7r/4-shift DQPSK scheme, using the frequency-selective maximum likelihoodreceiver, in a channel with triangular delay spread profile for the reflectedsignal, centered at 0.75T, with BFT = 0.0625 and DRR = 3 and 6 dB 72xi3.9 BER performance of rate- 1/2 trellis coded Nyquist filtered (a = 1.0), ir/4-shiftDQPSK scheme using the frequency- selective maximum likelihood receiver,versus the equivalent frequency-flat maximum likelihood receiver, in the presence of a 2-ray Rayleigh fading channel with BFT = 0.0625 and = 0.75 (seealso Fig. 3.2 for comparison) 743.10 BER performance of rate- 1/2 trellis coded Nyquist filtered (a = 1.0), ir/4-shiftDQPSK scheme using the frequency-selective maximum likelihood receiver,versus the equivalent frequency-flat maximum likelihood receiver, in the presence of a 2-ray Rayleigh fading channel with BFT = 0.0625, = 0.25 (seealso Fig. 3.3 for comparison) 743.11 BER performance of rate- 1/2 trellis coded Butterworth filtered (fB 1 /T),w/4-shift DQPSK scheme using the frequency-selective maximum likelihoodreceiver, versus the equivalent frequency-flat maximum likelihood receiver, inthe presence of a 2-ray Rician fading channel with BFT 0.0625, K0 = 5 dB,K1 —* —ocdB and =0.75 763.12 BERperformance of rate- 1/2 trellis coded Butterworth filtered, 7r/4-shift DQPSKscheme using the frequency-selective maximum likelihood receiver, versus theequivalent frequency-flat maximum likelihood receiver, in the presence of a2-ray Rician fading channel with BFT = 0.0625, = 0.75, K = K0 andK1 — —oc dB 763.13 BER performance comparison of the optimal receiver derived versus a receiveremploying MSRK equalization, both employing a rate- 1/2 trellis coded Butterworth filtered (fB = 1 /T) 7r/4-shift DQPSK scheme, in the presence of a 2-rayRician fading channel with BFT = 0.001 and 0.005, K0 0 dB, K1 —* —oodB and = 1.0 77xii4.1 BER performance of rate-1/2 trellis coded Nyquist filtered (a = 0.35) ir/4-shiftDQPSK scheme, in Rayleigh fading with BFT = 0.005, for diversity A = 1, 2and 3 and sequence length Z = 2, 3 and 4 894.2 BER performance of rate-l/2 trellis coded Nyquist filtered (a = 0.35) -ir/4-shiftDQPSK scheme, in Rayleigh fading with BFT = 0.01, for diversity A = 1, 2and 3 and sequence length Z = 2, 3 and 4 904.3 BER performance of rate- 1/2 trellis coded Nyquist filtered (a = 0.35) 7r/4-shiftDQPSK scheme, in Rayleigh fading with BFT = 0.05, for diversity A = 1, 2and 3 and sequence length Z = 2, 3 and 4 914.4 BER performance of rate- 1/2 trellis coded Nyquist filtered (a’ = 0.35) r/4-shiftDQPSK scheme, in Rayleigh fading with BFT = 0.1, for diversity A = 1, 2and 3 and sequence length Z = 2, 3 and 4 914.5 BER performance bound evaluation versus Monte-Carlo simulation results forrate- 1/2 trellis coded Nyquist filtered (a = 0.35) ir/4-shift DQPSK scheme, inRayleigh fading with BFT = 0.005, for diversity A = 1, 2 and 3 and sequencelength Z = 4 934.6 BER performance bound evaluation versus Monte-Carlo simulation results forrate-1/2 trellis coded Nyquist filtered (a 0.35) ir/4-shift DQPSK scheme, inRayleigh fading with BFT = 0.01, for diversity A = 1, 2 and 3 and sequencelength Z = 4 944.7 BER performance bound evaluation and Monte-Carlo simulation results forrate- 1/2 trellis coded Nyquist filtered (a = 0.35) ir/4-shift DQPSK scheme, inRician fading with BFT 0.05, for diversity A = 2 and 3, sequence lengthZ=4,andK=5andl0dB 944.8 Block diagram of the low-pass equivalent of the EHF log-normal fading model 97xlii4.9 BER performance of rate-l/2 trellis coded Nyquist filtered (a = 0.35) 7r/4-shiftDQPSK, in EHF light shadowing conditions with BFT = 0.05, for diversityA = 1, 2 and 3 and sequence length Z = 2, 3 and 4 984.10 BER performance of rate-l/2 trellis coded Nyquist filtered (a = 0.35) 7r/4-shiftDQPSK, in EHF average shadowing conditions with BFT = 0.05, for diversityA = 1, 2 and 3 and sequence length Z = 2, 3 and 4 994.11 BER performance of rate-1/2 trellis coded Nyquist filtered (a = 0.35) ir/4-shiftDQPSK, in EHF heavy shadowing conditions with BFT = 0.05, for diversityA = 1, 2 and 3 and sequence length Z = 2, 3 and 4 99xivGLOSSARYAVG: AverageAWGN: Aditive white Gaussian noiseBER: Bit error rateCCI: Co-channel interferenceCDMA: Code division multiple accessCE: Convolutional encoderDPSK: Differential phase shift keyingDQAM: Differential quadrature amplitude modulationDQPSK: Differential quadrature phase shift keyingDRR: Direct to reflected power ratioEGC: Equal gain combining diversityEHF: Extremely high frequencyETSI: European telecommunications standards instituteFDMA: Frequency division multiple accessFSK: Frequency shift keyingHF: High frequencyISI: Intersymbol interferenceMDD: Multiple differential detectorMDTS: Mobile digital telecommunication systemsMLSE: Maximum likelihood sequence estimationMMSPE: Minimum mean square prediction errorMRC: Maximal ratio combining diversityMSAT: Mobile satelliteMSK: Minimum shift keyingxvMSRK: Modified square root Kalman equalizationNEC: Nonredundant error correctionNMT: Nordic mobile telephoneNRZ: Non return to zero pulsePCS: Personal communication systemsPD: Phase detectorpdf: Probability density functionPSK: Phase shift keyingQAM: Quadrature amplitude modulationQPSK: Quadrature phase shift keyingRTMS: Radio telephone mobile systemSM: Signal mapperSNR: Signal-to-noise ratioSWC: Switched combining diversityTACS: Total access communication systemTCM: Trellis coded modulationTDMA: Time division multiple accessTIA: Telecommunications industry associationWPCS: Wireless personal communication systemsxviACKNOWLEDGEMENTSMy deepest thanks go to my parents, Tov,\ci and flczicytwric, to which this Ph.D. thesisis dedicated, for their continuous support and encouragement throughout the years it took tocomplete.I would like to express my most sincere thanks and appreciation to my thesis supervisor,Dr. P. Takis Mathiopoulos, for offering his valuable experience throughout the research effort,his moral support in trying times, as well as his encouragement and stimulation in critical pointsduring the course of the work presented in the pages to follow.Many thanks are also due to Dr. Dimitrios Makrakis, currently with the University ofOttawa, for his valuable feedback on many topics covered in this thesis, as well as his help withsome of the complicated mathematical analysis of Chapter 4.Taking this opportunity, I would also like to thank Dr. Vassilios Makios of the University ofPatras, Greece, for positively influencing my decision on coming to Canada for post-graduatestudies, and for all his help in providing an opportunity for my entrance to the M.A.Sc.programme at UBC. His later encouragement during the course of this Ph.D. thesis is alsodeeply appreciated.I would like to thank all members of my Ph.D. committee, Dr. C. Leung (committeemember), Dr. V. Leung (committee member), Dr. R. W. Donaldson (Head’s nominee) and Dr. R.K. Ward (chair, departmental examination), as well as Dr. Vasant K. Prabhu (external examiner)of the Department of Electrical Engineering, from the University of Texas at Arlington, Dr.M. J. Yedlin (university examiner), Dr. H. Chen (university examiner), and Dr. F. L. Curzon(chair, university final oral examination). Special thanks go to Dr. V. Leung who agreed to bea committee member despite the fact that he was on sabbatical leave.I would also like to thank Mr. Ian Marsland for proof-reading portions of the draft of thisthesis, and intercepting typographical mistakes in the complicated mathematical formulas ofxviiChapter 2.Last, I would also like to acknowledge the financial support provided by the NaturalSciences and Engineering Research Council of Canada (NSERC) under Grant OGP-443 12, theCentre for Integrated Computer Systems Research (CICSR), a University of British ColumbiaGraduate Fellowship (UGF), and a B.C. Advanced Systems Institute (ASI) Fellowship.xviiiCHAPTER 1INTRODUCTIONThe second half of the twentieth century has been undoubtedly marked by the rapid growthof public wireline networks, providing reliable and affordable voice, and relatively low-ratedata communications. During the last ten years there has also been a near exponential growthof specialized wire networks, focused primarily on providing high rate data communications,both on a local and a global scale [1]. These developments have changed our perception ofpossibilities in both business and scientific information processing, and in turn, have created aneed of access to these resources, unconstrained by time, place or mobility [2]. Towards thisneed for unrestricted access, the idea of using radio for providing personal communicationshas taken on a renewed meaning during this last decade, giving birth to what we broadly referto today as wireless personal communication systems (WPCS) [3]. This deep change in ourperception of communication is also reflected in the suggestion of integrating WPCS withfuture high-end networks, rather than having two separate systems [4]. Furthermore, althoughwireline phone service penetration in developed countries has reached almost 100 percent, twothirds of the world’s population still does not even have access to a public telephone [5]. WPCSare also becoming a very attractive alternative to tethered solutions for communities trying torapidly upgrade their telecommunication infrastructure, due to decreasing manufacturing costsas well as rapid and flexible deployment in both urban and rural areas [51.The current analog North American cellular phone system, known as the Advanced MobilePhone Service (AMPS) pioneered during the ‘70s by Bell Laboratories in the United States1Introduction 2[6], employs analog frequency modulation (FM) for speech transmission and frequency shiftkeying (FSK) for signaling. The same analog transmission techniques have also been used insimilar cellular networks worldwide, including Total Access Communication System (TACS)in the United Kingdom, Italy, Spain, Austria and Ireland; Nordic Mobile Telephone (NMT)in many European, African and Southeast Asian countries; C-450 in Germany and Portugal;Radiocom 2000 in France; and Radio Telephone Mobile System (RTMS) in Italy [2]. In theinterest of higher system capacity and ease of integration for information services other thanvoice, WPCS employ an all digital transmission format. Examples of possible applications forsuch WPCS include cellular phone service [7], land mobile radio phone and data networks [8],mobile satellite terminals [9] and indoor low-range personal communication systems (PCS)[10]. This thesis focuses on WPCS operated in a non frequency reuse environment, sufferingsignal distortion primarily caused by fading [11]. We will be referring to such WPCS as mobiledigital telecommunication systems (MDTS).Following the well known International Standards Organization (ISO) Open Systems Interconnection (OS 1) reference model of a network architecture [12], the three lowest layers,namely the physical layer, the data-link layer and the network layer, are of particular interest tothe design of a communication system, since they comprise each network node. The physicallayer, the lowest in the hierarchy defined in the ISO model, takes care of the raw transmissionand reception of the digital information. The data-link layer assures error-free transmissionthrough the error prone physical layer, and the network layer implements the necessary intelligence for sending information from one node to an other, via a physical route through themesh of nodes [12]. Among the most important issues in the design of a MDTS are a) theaccess technique by which users allocate network resources to themselves, b) the capacity ofthe communication system in terms of simultaneous number of users and c) the communicationquality. Multiple user access techniques fall into three categories [13]: a) Frequency divisionmultiple access (FDMA) where a transmit receive pair is allocated to communication betweenIntroduction 3two network nodes, b) time division multiple access (TDMA) where each user is allocated atime slot within a time duration of fixed length called a time slot, and c) code division multipleaccess (CDMA) where each node makes use of the same spectrum allocation as all other nodes,by using spread spectrum signaling. The type of communication service provided by the network, i.e., voice and/or low- and high-rate data, together with the type of access technique usedhas a great influence on the capacity of a MDTS. Lastly, the quality of service reflected at ahigh network layer, as for example in the voice quality of phone communications, or the speedof data communications, depends on the percentage of information received erroneously at thephysical layer. Since in digital communication the information is conveyed in bits, taking eitherone of two discrete values, 0 or 1, a commonly employed measure of quality is the averagenumber of bits in error, over a large sequence of bits received, called the bit-error-rate (BER).This thesis is concentrated on the design issues particular to the physical layer of MDTS,that is, the lowest network layer in such communication systems. Our effort is particularlyfocused on proposing and evaluating new receiver structures for improving the physical layerBER, which can also serve to increase the system capacity by accommodating systems of highercomplexity, with little or no penalty to communication quality. The proposed receiver structuresare shown to outperform other known receivers in applications where the rate of change of thecommunication channel is relatively high, i.e., when fading is relatively fast1. Such fast fadingappears in numerous real life applications, including MDTS with relatively low transmissionrates, mobile stations traveling at high velocities, employing extremely high frequency carriers.Specific examples of such systems are given in Section 1.2. The organization of this introductorychapter is as follows. In the next two sections, we summarize methods used for mobile digitalcommunication, and the impact of signal propagation on the signal quality at the receivingend. in particular, Section 1.1 describes common ways of using physical qualities of thetransmitter signal for communicating digital states, i.e., modulation schemes, while Section1A precise definition of fast fading is given in Section 1.2.Introduction 41.2 introduces the types of distortion imposed on a radio signal in propagation environmentstypically encountered in MDTS. Section 1.3 presents an overview of previously investigatedtechniques, employed by such systems for communication over mobile fading channels. Section1.4 discusses analytical methods used in the past for predicting the performance ofMDTS, whileSection 1.5 summarizes the research contributions of this thesis. Finally, Section 1.6 presentsthe thesis organization.1.1 MODULATION SCHEMESAmong the various ways of categorizing digital modulation schemes, a widely accepted oneis that in constant and non-constant envelope schemes [14]. Constant envelope schemes arethose modulation schemes which produce signals with continuous phase [15], and as a result,have a constant envelope. Perhaps one of the most extensively studied is the Gaussian-filtered minimum shift keying (GMSK) [16]. Because of its excellent spectral properties,especially in a non-linear channel, GMSK has been a very popular modulation scheme formobile communication systems, and it is the format adopted by the Groupe Special Mobile(GSM) in 19822 for the pan-European digital cellular [17].Non-constant envelope schemes, as the name implies, produce signals with variable, i.e.,non constant enve]ope. Among the various such modulation formats (with multi-amplitude!phase signals), M-ary phase shift keying (M-PSK) and M-ary quadrature amplitude modulation(M-QAM) type of modulation schemes appear to be the most popular. Typically, M= 2c,where k is the number of bits required to represent all possible transmitted symbols. PSKsignals are multi-phase, whereas QAM signals are multi-amplitude/-phase. In M-ary PSK,the transmitted digital symbols are mapped to M distinct transmitter carrier phases, uniformlydistributed in the interval [0, 2w) [14]. An example of an 8-PSK scheme is shown in Fig.2Note that currently, GSM is also referred to as the Global System for Mobile communications, of the EuropeanTelecommunications Standards Institute (ETSI) [7].Introduction 51.1 a; the arrangement of signal points on such a diagram of phase and amplitudes is calleda signal constellation. In M-ary QAM, the M digital symbols each correspond to one ofAl possible combinations of carrier signal phase and amplitude, arranged in such a way asto produce a signal constellation of rectangular form [14]. An example of a 16-QAM signalconstellation is shown in Fig. l.lb. The number of possible signals on the constellation ofQ Q(a) (b)Figure 1.1: Examples of PSK and QAM modulation formats; (a) 8-PSK(b) 16-QAM.a modulation scheme directly translates to the maximum attainable spectral efficiency whenusing that scheme. It is well known, the maximum theoretical spectral efficiency for an M-arymodulation scheme, is k bits/s/Hz [18]. For example, the 8-PSK scheme having 8 constellationpoints, can communicate with each signal 3 bits, and hence has a maximum theoretical spectralefficiency of 3 bits/s/Hz; for 16-QAM it is 4 bits/s/Hz. It is important to note that, QAMmodulation has been traditionally associated with coherent communication systems, whereasPSK with both coherent and noncoherent systems [19, 20, 18, 14].It is certain that the one modulation format common to QAM and PSK, that is, quadrature phase shift keying (QPSK), has been the most extensively investigated over the pasttwo decades. QPSK has four constellation points, associated to the signal carrier phases ofir/4, 3it-/4, 57r/4 and 77r/4, and thus a maximum theoretical spectral efficiency of 2 bits/s/Hz.Introduction 6In the 1980’s it was extensively employed in conjunction with fixed point, satellite and, to alesser extent, terrestrial microwave links [201. In the 1990’s, a modified version of QPSK,namely, ir/4-shift differential QPSK (DQPSK) was adopted as the modulation standard forthe emerging North American [21] and the Japanese [221 digital cellular systems. ir/4-shiftDQPSK uses the four differential phase shifts of Tr/4, 37r/4, 57r/4 and 77r/4, rather than theabsolute carrier phase values of QPSK, and in so doing has reduced envelope fluctuation [23].Note that QPSK can also be transmitted in another differential fashion, often referred to asdifferential QPSK (DQPSK), using differential phases of 0, Tr/2, ir and 3ir/2 with considerablymore envelope fluctuation than ir/4-shift DQPSK. When using differential phase encoding atthe transmitter, the receiver does not need to produce a coherent estimate of the transmittercarrier phase in order to detect the transmitted signal phases. Rather, each received signal phasecan be used as a reference value, from which that of the next received signal can be differentiallydetected [23], yielding the information symbol transmitted. The envelope fluctuation of Tr/4-shift DQPSK, although much improved with respect to that of DQPSK, will nevertheless resultin spectral spreading in a nonlinear channel. To reduce this spreading, amplifier linearizationtechniques have been proposed [23, 24], with excellent results. It should be noted, however,that both DQPSK and ir/4-shift DQPSK have identical performance in linear channels.There is no doubt, that 7r/4-shift DQPSK is an important modulation scheme for achievingspectral efficiencies of not more than 2 bits/s/Hz. For higher efficiencies, however, M-ary PSK(M > 4) and most importantly M-ary QAM (M > 4), have to be considered. In addition toproviding higher spectral efficiency, increasing the number of signals in the constellation canalso used in conjunction with convolutional, or otherwise called, trellis coding [14], to improvethe error rate of a communication system. The technique of combining modulation and trelliscoding, for improving the performance of digital transmission over band-limited channels, iscalled trellis coded modulation (TCM) [25]. Its main advantage is that it offers significantgains over conventional uncoded modulation, without compromising bandwidth efficiency byIntroduction 7requiring an increase in transmission rate, for accommodating the redundancy introduced bycoding [25]. Instead of increasing the rate, a higher order modulation format is used, with twiceas many signals in the constellation as compared to the uncoded case. The success of TCM liesin the special way of assigning signals to trellis code transitions, called set partitioning [25].This thesis concentrates on trellis coded M-ary PSK and QAM schemes.1.2 THE MOBILE FADING CHANNELThe most prominent from the variety of interference types imposed on a signal transmitted ina mobile-radio environment, is that oftenly referred to as fading. Fading is the time varyingsignal strength, phase and spectral-shape distortion [26], its rate of change depending on thesymbol rate of the digital transmission, appearing ttfaster” or “slower according to the relativespeed between transmitter and receiver, and following the Doppler3frequency [14]. Denotingthe maximum Doppler frequency by BF, and the inverse of the digital transmission symbolrate as T4, the product BFT gives us a measure of the average rate of change of the fadingdistortion normalized to the transmission symbol rate. As there is no universally accepted valueof BFT below which fading is characterized as slow, and above which as fast, for the purposesof work reported in this thesis, we have adopted the value of BFT = 0.001 for the point of thistransition. Values of BFT > 0.00 1 will be henceforth regarded as fast fading, while values ofBFT <0.001 as slow. In the emerging North American digital mobile cellular system, usingcarrier frequencies in the 800/900 MHz frequency range (ultra high frequency (UHF) band),and a symbol rate of about 25,000 Baud5,avehicle moving at little over 50 km/h would cause a3The Doppler frequency is defined as the ratio of the relative speed v between transmitter and receiver, over thecarrier wavelength A. For a more detailed discussion of Doppler frequency and its implications on the distortioncaused to the transmitted signal, see the sections describing the channel models in Chapter 2 and subsequentchapters.4Clearly, T is the symbol duration in seconds.5One Baud is one symbol per second; in the case of 7r/4-shift DQPSK, where 2 bits are transmitted in everysymbol, one Baud corresponds to 2 bits per second.Introduction 8BFT product of 0.0015. On the other hand, in the mobile satellite system (MSAT) developedjointly by the United States and Canada [27], with carrier frequencies of about 1600 MHz(L-band) and a transmission rate of about 3200 Baud, assuming a 45° elevation angle pointingto the satellite, traveling at a little over than 70 km/h produces a BFT of about 0.023. Assumingthe same rate, in the case of an aeronautical channel [281 where the vehicle speed is an orderof magnitude greater, or in NASA’s advanced communications technology satellite (ACTS)system [29] with carrier frequencies in the 20/30 GHz range (extremely high frequency (EHF)band), the BFT values will be well within the range of 0.1 to 1.The cause for the random signal amplitude and phase variations characteristic of fading, isthe simultaneous arrival of signal reflections, produced in an environment possessing numerousobjects capable of reflecting a considerable portion of the energy of incident radio signals,rather than absorbing them [26, 11]. As previously mentioned, well known examples of suchan environment are the UHF terrestrial radio channel [26], and the EHF mobile satellite channel[29]. Depending on the time delay spread r by which such reflections arrive at the receiver,the random amplification/attenuation and phase shift imposed on the signal by the mobilefading channel is, either of essentially constant amplitude over the received signal frequencyspectrum, or considerably varying, presenting peaks and nulls [26]. We define the normalizedto the symbol duration delay spread as = r/T. The value of can be used to characterize thetype of fading, distinguishing frequency non-selective and frequency selective cases. Similarlyto the distinction between slow and fast fading, there exists no universally accepted point inthe range of values for , where such a transition occurs. However, it is reasonable to statethat 0.1 characterizes the former case of frequency non-selective fading [261, whereas for> 0.1 we have frequency selective fading [14].In the following two sub-sections, the mathematical modeling and the different cases forboth the frequency non-selective and the frequency selective mobile fading channel will bepresented.Introduction 91.2.1 Frequency Non-Selective Mobile Fading -Depending on whether or not there exists a line-of-sight (LOS) signal path between transmitterand receiver, flat mobile fading channels can have Rayleigh or Rician characteristics6[301. Forthe Rayleigh fading channel the probability density function of the faded signal amplitude rfollows the Rayleigh distribution, i.e.,(r/2) exp (_r2/2o2) r > 0(1.1)0 elsewhereand its phase 0 is uniformly distributed in [0, 2ir). For the Rician fading channel case, inaddition to the diffused signal component (Rayleigh fading) there is also a LOS path. Thecomposite signal follows the Rice distribution given as [31]f(r) = 2r’ exp [—K — (1 + K)r2] 1o [2rVK(l + K)] (1.2)with the K-factor defined asDA =--, andindB 101og0.k (1.3)D denoting the average power of the LOS signal component, S the average power of thediffused component, and Io() the modified Bessel function of order 0 [32]. Note that K —* 0,or K — —00 dB, results in Rayleigh fading. The phase statistics of the Rician channel aredescribed by [311e \/‘RcosOexp (_Ksin2O)f(0)=——--+2/. [2—erfc(\/Kcoso)j (1.4)with erfc(.) the well known complementary error function [32]. Depending on the particularmobile fading channel environment, there exist various mathematical models adopted in the6As it will become apparent, and as a matter of fact, is well known, Rayleigh is a special case of Rician.However, for convenience in presentation, we will be using both terms.Introduction 10literature for the autocorrelation function RF(T) of the diffused signal component, with acorresponding power spectral density SF(f) [30]. The most simple one is the rectangularmodel, with SF(f) = S/(2BF) at frequencies f < and 0 elsewhere. For the aeronauticalchannel, the model adopted is Gaussian, with SF(f) S exp {_f2/B] / (\/BF). For thecase of land-mobile fading, .$p’(f) s /\/ (f2 — B) with corresponding autocorrelationfunction RF(r) = SJO(2lrBFr), J0(•) being the zero-order Bessel function of the first kind [33].In the EHF bands, the advantage of being able to accommodate much higher bandwidthsper channel does not come without a price. Since the wavelength at such frequencies is betweenI and 1.5 cm, i.e., roughly 20 to 30 times smaller than at UHF frequencies, it is very oftenthat the signal path is blocked even by relatively small objects, such as tree leaves and smallbranches. Contrary to UHF where the signal would have traveled through such obstacleswithout any significant attenuation, in EHF this signal blockage brings the receiver temporarilywithin an electromagnetic !shadow with respect to the transmitted signal. For this reason,fading in the EHF band is commonly referred to as shadowed fading, or simply shadowing.In a shadowed mobile fading channel, the direct path signal level can no longer be consideredconstant. Mathematical modeling based upon experimental data has shown that in such case, thechannel can be assumed Rician with its local mean, i.e., the LOS signal component, followinga lognormal statistical distribution [34, 35]. The probability density function (pdf) of the signalenvelope r can be expressed asr j 1 (in z — z)2 r2 + z2 ‘ rzf(r)= j- exp— 2d0 — 2b0‘ dz (1.5)where b0 is the power of the multipath signal, i and are the mean and standard deviationof the shadowing process. The pdf of the lognormally distributed LOS signal component canbe written as1 (lnz —f(z)=— 2d0(1.6)Specific values for b0, u and /ij have been determined experimentally for enabling thisIntroduction 11mathematical model to accommodate three types of shadowed fading conditions, namely, lightaverage and heavy [34].1.2.2 Frequency Selective Mobile FadingThe particular situation giving rise to frequency selective mobile fading can perhaps be explained easier using the illustration of Fig. 1.2. In the simplified propagation instance shown,Figure 1.2: Simplified example of a propagation environment giving rise to frequency selectivemobile fading.the transmitted signal reaches the mobile receiver via three discrete paths. The delay ri of thedirect signal component on path-i is the reference and hence assumed equal to 0. The delay r2of the second path is still small with respect to the symbol period T and its effect alone wouldstill be regarded as non-selective fading. The third signal ray, however, arrives at the receiverwith considerable delay. The cumulative effect of the three signals clearly causes intersymbolinterference (ISI) [14]. It is well known, that in a mobile fading environment signal reflectionsarriving with large delay spreads at the receiver, e.g., with > 0.1 cause frequency selectivefading [14]. With reference to the maximum value of , for the purpose of this thesis, weconsidered 1. The main reason for this choice is that this range is recommended in thetimeIntroduction 12model assumed by the EIA for the North American digital cellular standard [21], and is alsocommonly adopted by other researchers investigating the time dispersive frequency selectivemobile fading channel [36, and references within].In a more practical mobile radio environment than the one illustrated in Fig. 1.2, withreflectors of arbitrary size and orientation, we don’t have discrete signal rays, but rather,concentrations of signal energy arriving with different delays at the receiver. Each one of theseconcentrations can be assumed to be a signal following Rayleigh fading statistics. If a LOSsignal path exists, the concentration exhibiting the smallest time delay is by definition assumedto be the “direct” signal, for which the fading channel characteristic is Rician. In this respect,the “direct” signal will have a finite K, while all other signals, being Rayleigh, have K — —oodB.1.3 MITIGATION TECHNIQUES FOR MOBILE FADING CHANNELSThe effect of fading distortion to digital communication systems is indeed quite devastating [261.The random signal phase fluctuations of fading give rise to irreducible error rates, called errorfloors [11]. Such error floors, as the term implies, are independent of received signal strength,and in fast fading conditions can render a communication system completely unusable. Over theyears, a very large number of fading mitigation techniques have appeared in the open technicalliterature for a wide variety of fading channel conditions. Since it is virtually impossibleto present them all within the limited space of this section, we briefly summarize importantrelevant techniques which are related to the research effort reported in this thesis. The fourfollowing subsections each describe the most important qualities, advantages and disadvantagesof fading combating signal reception methods, classified by the general directions of researchcarried out in this area.Introduction 131.3.1 Coherent Detection TechniquesIn coherent transmission systems, an estimate of the carrier signal employed at the transmitteris constructed at the receiver, with the purpose of recovering the information conveyed by thereceived signal [14]. The vast majority of fading mitigation techniques for coherent systemsinvolve the transmission of pilot tones [37, 38, 39] or pilot sequences [40, 41,42,43, 44]. Themain idea behind such fading combating methods is the following. The signal impairmentis extracted by processing the received tones or sequences, and a coherent reference signal isgenerated. Assuming that the fading effects are the same on both pilot and data, this coherentreference can be used to cancel, to great extent, the effects of fading on the informationbearing signal. However, such techniques have several disadvantages. More specifically,for relatively fast fading applications, the overhead required to adequately offset the fadingdistortion is impractically high. As it has been pointed out in [45], in order to achieveacceptable performance results, the intervals between pilot symbol insertions must be lessthan I /(2BFT). in other words, the rate of pilot symbol insertion must be proportional tomore than twice the average rate of the fading distortion, rendering the redundancy for a pilotsymbol assisted scheme unacceptably high for a fast fading environment. For example, fadingwith BFT = 0.125 would require a redundancy of at least 25%. Furthermore, such pilotcalibration techniques tend to yield relatively complicated receiver structures, relying heavilyon memory to store signal samples, and introducing delay in decoding decisions as a side-effectof estimating the distortion imposed by the channel.1.3.2 Noncoherent Detection TechniquesConventional TechniquesIn contrast with coherent systems, noncoherent detection does not require an estimate of thetransmitter carrier signal at the receiver. Instead of using absolute carrier phase values, theInfroduction 14transmitter translates information symbols to carrier phase differences [14]. Hence, informationcan be recovered at the receiver by comparing the received signal with its value at some timeinstant in the past, usually at the symbol rate. Noncoherent detection can be subdivided intolimiter/discriminator detection [23, 46], and differential detection [47, 48, 49, 50, 51, 52, 53].The limiter/discriminator detection, although simple and low-cost in terms of implementation,suffers from the disadvantages of the frequency discriminator it employs. A serious problem isthe FM “click” effect [54], whereby an FM receiver will lock to a signal at an adjacent frequency,if its level is considerably higher than that of the desired signal. Differential detection, on theother hand, maintains the simplicity in implementation but does not exhibit the drawbacksassociated to the frequency discriminator. It is, in this sense, more robust, providing extremelyfast signal acquisition as compared to coherent detection techniques outlined in the previoussubsection, and a relatively good performance in additive white Gaussian noise (AWGN) andslow fading channels, equivalent to that of limiter/discriminator detection. Note, however, thatas well known, its performance in AWGN is worse than that offered by coherent detection [14].The improvement in performance provided by this classical 1-symbol7differential detector ina fading channel, can be understood by regarding its operation of subtracting adjacent symbolphases in order to recover the information bearing carrier phase change. In a slow fadingchannel, for most of the time, the phase distortion between adjacent symbols is sufficientlysmall, so that it is cancelled out to a large degree, by this phase subtraction. Performancecomparisons for these noncoherent techniques can be found in [55], and particularly for limiter/discriminator detection versus differential detection in [56]. As awareness of the benefitsinherent to differential detection in fading channels was established, the idea of expanding thesignal observation from a adjacent symbols to a multi-symbol “window” was quick to follow,as it was expected to improve the receiver performance in fading.7A I-symbol differential detector gets its name by the fact that it operates on adjacent symbols, i.e., signalsymbols with time distance of one symbol (1 T).Introduction 15Multiple Differential Detection TechniquesThe hardware structure implementing an expanded observation interval spanning more thantwo symbols, is called a multiple differential detector (MDD). A block diagram illustrating thishardware structure is shown in Fig. 1.3. At a time instance kT, the received signal sample isdenoted as Uk, and the output of a conventional differential detector as d1 (k). Assuming that theMDD has access to Z received signal inputs, the MDD hardware structure provides additionaloutputs d2(k) up to d_1, by employing delay elements of progressively increasing multiplesof T, covering all symbols within the observation interval. Clearly, the number of differentialdetectors used is z = Z — 1.Figure 1.3: Block diagram illustrating the multiple differential detector hardware structure.PD: phase detector.The MDD hardware structure was used in the past first by Chow and Ko [57] in the form of ahard decision error correction scheme referred to as nonredundant error correction (NEC). Thework in [57] has shown that a receiver employing the MDD structure and a signal observationwindow of Z symbols, employing z — 1 differential detectors, has the capability to correct upd1 (k)dz1 (k)Introduction 16to z — 2 errors. This improvement in receiver performance, as the name of this error correctionscheme implies, is achieved without introducing any redundancy in the transmission. Notethat the analysis and results reported in [571 assume only binary phase shift keying (BPSK)8inAWGN, yielding relatively small gains in overall system BER performance. In [58], Masamuraet a!. have applied the NEC technique to the minimum shift keying (MSK) constant envelopescheme. Samejima et al. in [53], have extended the work of [57] to include differential PSK(DPSK) signals, including DQPSK. The performance of 7r/4-shift DQPSK signals employingthe NEC scheme at the receiver, has been analyzed and evaluated by Wong and Mathiopoulos,for the static9 co-channel interference (CCI) channel in [591. Due to the additive nature of thestatic CCI, significant BER performance improvements were reported in [59]. However, asshown in [60], in general, the NEC technique does yield any significant performance gains formobile fading channels. Only small error floor reductions for extremely fast fading conditionshave been reported in [60].Motivated by the implementation simplicity and robustness of differential detection, Makrakis, Mathiopoulos and Bouras proposed for the first time, the use ofMDD receiver structures forthe mobile fading channel [61, 62, 63110. The general problem investigated dealt with optimaldetection of PSK and QAM signals transmitted over frequency non-selective correlated fastfading channels [63]. By using maximum likelihood detection arguments, the optimal sequenceestimator was derived, for coded digital multi-phase/multi-amplitude signals, corrupted byAWGN and multiplicative Rayleigh or Rician frequency non-selective fading. The derivationpresented in [63] does not make any assumptions on the type of fading autocorrelation function,or on how slow or fast the fading might be, although both are assumed to be known to the81n BPSK, as the name implies, there are only two points on the signal constellation, at phases of 0° and 1800,each one conveying one of two possible states, 0 or 1.9Static, in this case, implies that transmitter, receiver and interfering transmitters are all stationary, i.e., thereis no random phase/amplitude signal distortion caused by any signal reflections and movement, as in fading.‘°Earlier work related to MDD techniques in AWGN, published by Makrakis and Mathiopoulos, can be foundin [64] for QAM and in [651 for PSK signals.Introduction 17receiver. The authors do not adopt the simplified case where the fading is assumed to be constantover several symbol intervals ([66, 52, 67]), or even during one symbol interval. Rather, boththe fading amplitude and phase distortion are assumed to change significantly, according to thefading’s statistical characteristics. Furthermore, contrary to common practice ([66,51,521), thefading correlation is not removed by using interleaving” at the transmitter and de-interleavingat the receiver. Although the advantage of interleaving is that transmission schemes derivedfor AWGN can be readily applied to fading, the disadvantage is the increased implementationcomplexity and processing requirement introduced, and their resultant impact on the practicalimplementation of small, lightweight portable units as such desirable for MDTS. Instead ofremoving the fading correlation by interleaving, the optimal sequence estimator described in[63] uses it to the receiver’s advantage. Its structure can be viewed as a combination of a squaredenvelope detector, a MDD hardware structure, and a coherent detector. The squared envelopedetector provides information on the signal amplitude, and is only needed when schemes such asQAM, which have more than one signal levels, are employed. The coherent detector is used bythe receiver only for providing an estimate of the transmitter modulator initial phase, and is onlyrequired for the case of Rician fading; it is not needed in Rayleigh fading conditions. In order toreduce the algorithmic complexity of the optimal sequence estimator derived, the authors of [63]also presented suboptimal, reduced complexity versions of these algorithms. Using computersimulation they also demonstrated that the large portion of performance gains available fromthese techniques, is obtained with relatively low complexity implementations of the suboptimalalgorithms. The performance improvements presented in [63] are rather significant, as errorfloors are shown to decrease by orders of magnitude, compared to performance achieved byconventional detection methods.“Interleaving at the transmitter is accomplished, for example, by entering symbols to be transmitted in a matrixrow by row, and then transmitting the matrix contents column by column. At the receiver, the inverse operationrestores the symbol sequence to its original order. If the row/column length of the interleavingmatrix is sufficientlylarge, the errors afflicted upon the received symbols due to the fading distortion are, for all practical purposes,uncorrelated.Introduction 18We conclude this subsection by mentioning that since 1990, i.e., after the publication of [611and [64], MDD based receivers have been investigated by various other researchers, and haveresulted in several journal and conference publications. Concurrent with [611, in [681 Wilsonet al. investigated the use of MDD for improving the performance of differential detection ofQPSK and 8-PSK in AWGN, and presented some suboptimal detection procedures. Divsalarand Simon in [69] also proposed the use of MDD techniques’2for PSK signals in AWGN. Thiswork is very similar to that of [65]. In [70], Ho and Fung have obtained analytical performancebounds for uncoded PSK signals in Rayleigh fading and AWGN. The work in [70], employsinterleaving for reducing the correlation between faded signal samples, and only considersRayleigh frequency non-selective fading and uncoded single amplitude schemes such as PSK.1.3.3 Diversity TechniquesDiversity techniques for improving the signal reception in fading channels are classified intothree general categories, namely, selection diversity combining (SC), equal-gain combining(EGC) and maximal ratio combining (MRC) [26, 11]. In SC, the least complicated of the three,the strongest signal is selected for processing at the receiver end. This method is impracticalfor use in mobile radio communication, as it requires maintaining a floating threshold level[11]. A more practical version of SC is called switched combining (SWC) and it amountsto having the receiver use the selected signal until it drops below a predetermined switchingthreshold. Following that, it switches to the strongest signal at that moment. In MRC,before combining, each signal is scaled according to an estimate of the signal strength on thatchannel. SC and its derivative SWC, both require switching signals at the receiver, whileMRC requires more complicated and hence, more costly receiver design. EGC, on the otherhand, is implemented very simply by incoherently summing up the signals available on all‘2They refer to MDD as multi-symbol differential detection.Introduction 19diversity channels. Analysis found in [11] shows that the performance of EGC is only slightlyworse than that of MRC, while providing a considerable simplification in receiver design. Asintuitively expected, the performance improvement available from all three methods decreasesas the correlation between the available diversity signals increases [11].More recent work, investigating the performance of DQPSK and r/4-shift DQPSK signals,employing all three types of signal combining in conjunction with differential detection, ispresented by Adachi eta!. in [71]. The signal is assumed to be corrupted by CCI and frequencyselective fading. An expression for the BER is derived, for the MRC and the SC case, andnumerical results show the MRC technique to be slightly superior to the SC technique, withvirtually identical BER performance for w/4-shift DQPSK and DQPSK. The work reported in[71] does not address the more general case of Rician fading, and also uses simplifications onthe nature of the fading distortion, based on the assumption of a slow fading rate, wherebythe channel transfer function is assumed approximately constant during the symbol durationT. In [72, 73], Kam and Ching investigate sequence estimation of PSK signals with diversityreception over slow, Rayleigh fading channels. In this work, the correlation present in fadedsignal samples is removed by the use of interleaving at the transmitter and de-interleaving atthe receiver, the drawbacks of which have already been mentioned in the previous section. Inaddition, reliable sequential estimation of the transmitted symbols depends heavily on fadedsignal amplitude estimation, which is the first of two stages incorporated in the proposeddecoding algorithm. More recent work involving MRC used in conjunction with a QAMmodulation scheme can be found in [74]. In that paper, experimental results from a hardwareprototype and computer simulation are provided, for frequency non-selective Rayleigh fadingand co-channel interference. Fading estimation is provided by inserting pilot symbols in theinformation sequence, and expectedly, the technique breaks down in fast fading environments.In [75], a new diversity method is introduced, called code combining (CC), for improvingthe performance of r/4-shift DQPSK, in both frequency selective and non-selective RayleighIntroduction 20fading, CCI and AWGN. The analytical performance bounds presented in this work assumethat the interleaving degree is large enough to effectively eliminate all correlation in adjacentsignal samples due to fading; also the fading BFT product is assumed approximately equalto 0, i.e., the fading is assumed to be very slow. Experimental results for SC and MRC arealso presented in [76], where the performance of 7r/4-shift DQPSK employing block codingand diversity is investigated in CCI and frequency selective Rayleigh fading. This work alsoassumes interleaving sufficient to randomize the burst errors caused by fading, and does notconsider the more general case of Rician fading.1.3.4 Equalization TechniquesFor more than two decades, several nonlinear equalization techniques, including decisionfeedback equalization and maximum likelihood sequence estimation (MLSE)13 - efficientlyimplemented by a Viterbi algorithm - have been developed for improving the performance ofdigital transmission systems, in order to provide reliable communication over time invariantor slowly varying channels, with severe inter-symbol interference (1ST) [14]. A considerablevolume of techniques and results reported on this particular subject appear in the literature; asexamples we mention equalization in the time invariant 300 Hz - 3 kHz telephone channel asfound in [77], application to slowly varying high frequency (HF) radio channels in [78, 79],and for microwave LOS communications in [801. In contrast to the wealth of techniques forequalization in static and slowly varying channel conditions, relatively little has been publishedin the open technical literature on combating both ISI and the rapidly varying characteristicof the frequency selective mobile fading channel [811. Several techniques to this end haverecently appeared, dealing with analysis and evaluation of various equalization structures[82, 831, including diversity [84, 85].‘3For a brief introduction to MLSE see Section 1.3.5.Introduction 211.3.5 Maximum Likelihood Sequence Estimation (MLSE) TechniquesInterestingly enough, the vast majority of research work on combating frequency selective fading has concentrated on equalization rather than MLSE as a means of combating the combinedeffect of ISI and fading. The frequency selective multipath Rician fading channel presentstwo major obstacles when attempting to analyze MLSE detection; the first one being the timevarying characteristic in the amplitude and phase of the fading distortion, and the second onebeing the non uniform attenuation over the signal spectrum (frequency selectivity). It is onlyrelatively recently that research work employing MLSE has been reported, using various maximum likelihood detectors for digital signals transmitted over frequency selective Rayleighfading channels [36, 86]. In [36], Alles and Pasupathy describe the derivation of optimalsymbol-by-symbol detectors for a two-ray Rayleigh fading channel. The second publication[861, by Dai and Shwedyk, presents a more general approach of sequence estimation for ageneralized frequency selective Rayleigh fading channel. More specifically, in [86] two sequence estimators, which are referred to as i) MLSE with Viterbi Algorithm (MLSE-VA) andii) Sequential Sequence Estimator (SSE), have been proposed. It has been shown that for thesame ISI channels, the SSE has much lower computational complexity while achieving a BERperformance which is almost identical to that of the MLSE-VA. However, in terms of analysisand performance evaluation, both of these publications deal exclusively with Rayleigh fading.In addition, they only consider binary modulation schemes without the use of any coding,presenting a relatively limited set of results.1.4 ANALYTICAL BER PERFORMANCE BOUNDSAlthough deriving analytical bounds for the BER performance of digital communication systems is more often than not a formidable task, it is indeed a preferred method for evaluatingsuch systems. There are two main reasons for this, both related to shortcomings of the otherIntroduction 22method commonly employed to obtain such performance evaluation results, namely, computersimulation. The first one is related to the fact that simulation results are practical only downto error rate levels of about i04, and in some rare cases, at most down to i0 [871. This isdue mainly to the large time interval required to process long signal sequences, and the numberof error events needed to acquire a good estimate of the error rate. The second reason whyanalytical bounds are attractive is related to rarely occurring phenomena which also requireextremely long signal sequences if the estimate of their effect in a computer simulation is to bestatistically significant. An interesting example of one such instance arises in the case of slowfading, where the occurrence of deep fades’4 is much less frequent than in the fast fading case,in terms of signal samples processed in a digital simulation. This can even lead to the totalabsence of such a deep fade if the number of samples is not large enough, yielding, in turn, anerroneous estimate of the system error rate.BER performance bounds for trellis coded coherent 4-PSK and 8-PSK are presented byBiglieri and McLane in [88]. In this publication, perfect channel state information is assumed,rendering the results practical only for channels exhibiting very slow fading conditions. In [891,McKay et a!. present evaluation results using analytical bounds, for the performance of trelliscoded 8-PSKin Rayleigh, Rician and EHF shadowing channels, with and without ideal channelstate information. The authors of [89] assume interleaving/de-interleaving which increases thereceiver implementation complexity, and do not examine higher spectral efficiency, multilevel modulation formats. Analytical results for the performance of receivers employing theMOD hardware structure have been presented by Ho and Fung in [701, where they havederived an expression of the pairwise error event probability of MDD for uncoded PSK signalstransmitted over a Rayleigh fading channel. Their analysis is based upon a residue theoremtechnique reported in [901, using interleaving/de-interleaving and pilot sequences for channel‘4By the term ‘deep fade” we describe the severe attenuation randomly imposed on the received signal due tofading [261.Introduction 23state estimation. In [91], Divsalar and Simon also provide analytical results for PSK systemsusing the MDD hardware structure, in AWON and slow fading channels. They also providesome computer simulated BER results for a differentially detected 16-QAM scheme in anAWGN channel, but not for a fading channel. In conclusion, it should be pointed out that noneof the aforementioned publications include the effects of diversity reception in their analyticalderivations.1.5 THESIS RESEARCH CONTRIBUTIONThe foregoing discussion on noncoherently detected, PSK and QAM type of signals, transmittedover mobile fading channels with diversity, has established the context in which the threeresearch contributions of this thesis are presented as follows.1. We derive the optimal receiver, in the maximum likelihood detection sense, for fastfrequency non-selective Rician fading channels, employing diversity reception. Thereceivers employ the MDD hardware structure, in conjunction with novel optimal detection algorithms. The proposed analysis is general enough to accommodate any type ofmodulation signal. However, for performance evaluations we considered the ir/4-shiftDQPSK, and the more spectrally efficient trellis coded 8-DPSK and r/4-shift 8-DQAMschemes, employing trellis coding in all cases. In order to reduce the overall implementation complexity, we propose suboptimal, reduced complexity versions of these algorithms, and evaluate their performance by using digital computer simulation. The resultsobtained demonstrate significant improvement over conventional schemes. Furthermore,taking into consideration the inaccuracies inherent to practical system implementations,we investigate cases where the receiver statistical knowledge of the fading channel is inconsiderable error with respect to the actual channel state. Results show relatively smallsensitivity, even to large inaccuracies in estimates of channel parameters.Introduction 242. We present the theoretical derivation of the optimal receiver, in the maximum likelihooddetection sense, for correlated fast Rician frequency selective fading channels. Theimpact of simplifying assumptions made to facilitate the theoretical derivation are investigated via digital simulation. Furthermore, we estimate the sensitivity of the receiverperformance to inaccurate knowledge of channel parameters. The results show significantperformance improvements over conventional signal detection methods, accompaniedby small sensitivity to imperfections. The performance of the derived receiver structureis also compared to the fastest known equalization technique for frequency selective fading, which employs the modified square root Kalman equalization algorithm derived in[78] for use in HF dispersive channels. Results show the optimal receiver to outperformequalization in fast fading channel conditions.3. We derive novel analytical BER performance bounds for simplified versions of the optimal diversity receiver, for frequency non-selective, Rician fading, in the most generalcase when no channel state information is available. Simplified versions of the diversityreceiver structure are derived for varying degrees of channel state information, and areshown to be optimal in the case of slow fading channel conditions. Using digital simulation and computer aided calculation of the derived bounds, we evaluate the performanceof these receivers in slow as well as fast Rician fading, employing different fading rates,and various orders of reception diversity and receiver implementation complexity. Theexcellent agreement between calculation and computer simulation results, demonstratesthe tightness and accuracy of the derived bounds. Motivated by the performance improvement gained for increasing diversity order, and given the fact that antenna diversityis more practical in higher frequencies with smaller wavelengths, we present BER evaluation results of these simplified receivers in an EHF channel, experiencing light, averageand heavy shadowing.Introduction 251.6 THESIS ORGANIZATIONAfter this introductory chapter, the three research contributions summarized in Section 1.5, arepresented in Chapters 2, 3 and 4.In Chapter 2, first, the model for the communication system employed throughout thisthesis is described in Secion 2.1. The optimal receiver structures for Rician frequency non-selective fading are derived in Section 2.2, where details regarding the decoding algorithmsare presented. Section 2.3 includes the BER performance evaluation results for the optimalreceivers derived, and results of the investigation of receiver sensitivity to errors in knowledgeof channel statistics. Section 2.4 offers the conclusions of this chapter.In Chapter 3, Section 3.1 presents the model employed for the frequency selective mobilefading channel, and Section 3.2 the implications of the channel imposed distortion to thereceived signal. In Section 3.3 we present the derivation of the optimal receiver structure anddetails on the associated decoding algorithms. BER performance evaluation and sensitivityinvestigation results are included in Section 3.4, with concluding remarks in Section 3.5.In Chapter 4, Section 4.1 includes the derivation of the simplified receiver structures for thecase of slow fading conditions. Three versions are presented, for varying degrees of channelstate information. In Section 4.2, we present the derivation of the analytical BER performancebounds for the case of no channel state information, in Rician frequency non-selective fading.Section 4.3 includes results from computer simulation and computer aided calculation of theperformance bounds derived, for Rician, as well as EHF shadowed fading channels. Section4.4 offers the conclusions of this chapter.The thesis concludes with Chapter 5, which is divided in two parts. Section 5.1 presentsthe concluding remarks of this thesis, and Section 5.2 offers suggestions on future work, asinspired during the course of this research effort.CHAPTER 2OPTIMAL SEQUENCE ESTIMATION FOR FASTCORRELATED FREQUENCY NON-SELECTIVE RICIANFADING CHANNELS WITH DIVERSITY1In this chapter, we derive and evaluate the optimal sequence estimator for digital signalsreceived over A different channels. Each of these channels corrupts the transmitted signal bya mixture of AWGN and frequency non-selective, correlated, fast Rician fading. The diversityassumed is of the equal gain combining type, and is implemented within the decoding metric.The organization of the chapter is as follows. In Section 2.1, we describe the overallsystem model, including the transmitter and channel model under study. In Section 2.2, wepresent the theoretical derivation of the optimal equal combining diversity receiver. Section2.3 summarizes various BER performance evaluation results. Finally, Section 2.4 contains theconclusions of this chapter.research reported in this chapter has been presented in part at the 1991 IEEE Pacific Rim Conference,Victoria, Canada [92], and has been published as a full paper in the IEEE Transactions on Vehicular Technology[93].26Optimal Sequence Estimation for Fast Flat Fading with Diversity 272.1 TRANSMITTER AND FREQUENCY-FLAT FADING DIVERSITYCHANNEL MODELFigure 2.1: Block diagram of the communication system transmitterThe block diagram of the model assumed for the communication system transmitter2appearsin Fig. 2.1. The transmitter consists of a Convolutional Encoder (CE), a Signal Mapper (SM),a Differential Encoder (DE), a transmit pulse-shaping filter with transfer function HT(f) and amodulator (complex multiplier). The p -bit information words = [aj, a, ..., a] of the inputdata sequence A, consist of independent and equiprobable bits taking values from the alphabet{O, l}. This input sequence is transformed by the CE to q -bit words 14 = [14 b, ..., 14]. byusing a p/q rate convolutional code. The signal mapper then converts these words to symbols= 7k exp (jl), where represents the amplitude and 2k the phase of Sk, respectively. Thesequence of &‘s is then differentially encoded, resulting in the sequence of transmitted symbolsCk = = 7k exp [j(k-1 ak)] (2.7)7k—1with 1k denoting the phase of ck and modulo 2w addition. The non-return-to-zero (NRZ)sequence of ck’S is shaped by the transmitting premodulation filter HT(f), resulting to thebaseband signalXB [(A), t] = CkhT(t — kT) (2.8)2The transmitter model illustrated in the block diagram of Fig. 2.1 is the one assumed not only for this chapter,but for all work reported throughout this thesis.x( t)Complex signal: -.Ø-Real signal:Optimal Sequence Estimation for Fast Flat Fading with Diversity 28where T is the symbol duration, Z is the number of symbols transmitted, hT(t) is the impulseresponse corresponding to HT(f), and XB {(A), t]the transmitted sequence generated by theinformation sequence A. The frequency response shapes of the filters utilized throughout thework reported in this thesis, are the raised cosine (Nyquist filtering), and the 4th orderButterworth. These are given as the transfer function [14]T 0<fI<(l—a)/2TH(f)= (2.9)[i—sin irT (j— )/aJ (1 —a)/2T < f <(i÷)/2Tfor Nyquist filtering, and as the transfer function [94]f4H(f) =JB (2.10)Ff2 1 r] 1;2 I 3ir[JB ‘ cos — J j [JB -- cos -- —for Butterworth filtering, with f denoting frequency and fB the 3 dB Butterworth filter cut-off,both in Hz. Assuming for example a rate- 1/2 trellis coded 7r/4-shift DQPSK signal, we willhave p = 1, q = 2, = {a], = [b,b], ck = Skck_1/-yk_1 and C(A) = [c1,2...,czj, with7k = 1 and 11k e {ir/4, 37r/4, 57r/4, 77r/4}, resulting in the signal-space diagram of Fig. 2.2.For a rate 2/3 trellis coded 7r/4-shift 8-DQAM signal, we will have p = 2, q = 3, a = {a, a],= [bj, b, b], ck = SkCk_1 /7k—i and C(A) = [c1, C2, ..., cz]. Such a signal will have7k e {l/3, l} and 12k é {7r/4,37r/4,57r/4,77r/4}, resulting in the signal-space diagramshown in Fig. 2.3. The rate 1/2 and rate 2/3 trellis codes used for ir/4-shift DQPSK andir/4-shift 8-DQAM respectively, are the best codes for AWGN channels found in [25]. Theconvolutional coders used as well as the signal phase and amplitude assignment to signalnumbers are depicted in Fig. 2.4.The modulator up-converts this signal to its carrier frequency f, resulting to the transmittedsignalx(t) = Re {XB [(A), t] exp[j(2irft + )]} (2.11)with Re { . } denoting real part of { . } and ii the modulator initial phase.29— I I I I•1 I—-1.00 -0.50 0.00 0.50 1.00— I I_________- L__ IOptimal Sequence Estimation for Fast Flat Fading with DiversityQ Q1.201.001.000.800.800.60 0.600.40 0.400.20 0.20-0.00 -0.00-0.20-0.20-0.40-0.40-0.60-0.60-0.80-0.80-1.00-1.00 -1.20_____________________-1.00 -0.50 0.00 0.50 1.00Figure 2.2: The signal space of the ir/4-shift DQPSK transmitter; a) before spectral shaping, b)at the output of the receiver filter in the absence of interference; raised cosine filter c = 0.7.Q Q1.20’1.001.000.800.800.600.600.400.20 0.2001 -0.00-0.20 -0.20-0.40-0.40-0.60-0.60-0.80-0.80-1.00-1.00-1.20t— .-Figure 2.3: The signal space of the 7r/4-shift 8-DQAM transmitter; a) before spectral shaping,b) at the output of the receiver filter in the absence of interference; raised cosine filter = 0.7.-1.00 -0.50 0.00 0.50 1.00 -1.00 -0.50 0.00 0.50 1.00I IOptimal Sequence Estimation for Fast Flat Fading with Diversity 302-bitinputMapping of transmitted symbolsto differential phases.Mapping of transmitted symbols todifferential phases & amplitudes.Figure 2.4: Convolutional coders used for ir/4-shift DQPSK, ir/4-shift 8-DQAM and 8-DPSK,showing assignment of signal phases and amplitudes to signal numbers.2-bittransmittedsymbol4-state convolutional encoder.(a) it/4-shitt DQPSK.I -bit__________________ ________Input2-bit Fnput8-state convolutional encoder.TQ4‘7(b) it/4-shift 8-DQAM.symbol3-bittransmitted8-state convolutional encoder.46(c) 8-DPSK.Mapping of transmitted symbolsto differential phases.Optimal Sequence Estimation for Fast Flat Fading with Diversity 31We investigate the case where x(t) is transmitted over 1 (1 1 A) in general correlatedchannels which corrupt x(t) with a mixture of multiplicative nonselective fading ft(t) andAWGNn1(t) with a double-side power spectral density 1V0/2. f1(t) is modeled as a complexsummation of two white and independent Gaussian noise processes, n’/(t) and n’(t), filteredby two identical filters Hj(f) [26], as illustrated in Fig. 2.5. Hence f1() = f’’(t) ÷jf;’(t),Figure 2.5: Block diagram of the low-pass equivalent of the frequency selective fading model.with f’(t) and f’1(t) having the same autocorrelation function, R(r), which is given byR’FQr) = F { [f’;’(t) — f1;l} [f’;’(t — r) — j77] }= E { {JQ;’(t) — j7] {fQ;’(t — T) — j7] }2= NJ H(f) e3Tdf. (2.12)In the above equation, N is the power spectral density of the white Gaussian noise processgenerating f’(t), f = F {f’;’(t)} and fQ;l = E {fQ;1(t)}, where E{.} denotes expected value.As f’1(t) andf1(L) are independent, their cross-correlation is zero. In the case where a LOSsignal component exists, the fading follows Rician statistics. Furthermore, since the phase ofRayleigh faded component is uniformly distributed within [0, 2ir), the constant phase of theComplex signal: —Ø-Real signal:f1(t)flfQ;f)Itcf-i-Optimal Sequence Estimation for Fast Flat Fading with Diversity 32LOS signal component can be assumed to take any value within [0, 27r). Without any loss ofgenerality it will be assumed henceforth that fQ’(t) = 0, i.e., that phase of the LOS signalcomponent is equal to 0. The transfer function Hj(f) of the fading filter employed is relatedto the spectral characteristics of the fading model considered [30]. As this thesis deals withland-mobile applications, the fading model adopted throughout is that of land-mobile fading,having the autocorrelation function [26, 301R(T)—JO(27rBFT) V 1 <l<A. (2.13)For the Rician fading channel, comparisons under different fading conditions employ theK-factor, which in this particular case is given as,— 2(fI;l)K = 10 log10 2 2 dB (2.14)+ UfQ;1where and are the variances of1’(t) andf1(t), respectively.In general, the diversity receiver consists of A different branches. Diversity can be implemented either in space, frequency or time, with advantages and disadvantages in each case[11]. For relatively low carrier frequencies (i.e., with large wavelengths), for example, it mightnot be practical to employ space diversity in the form of using A different receiver antennas, especially when considering a portable unit of small physical size. In such case time orfrequency diversity would be preferable, requiring, nevertheless, additional bandwidth. OnEHF frequencies, however, where the wavelength is within the 1-2 cm range, space diversityis very easily realizable, and since it does not require additional bandwidth, it is much moreattractive solution than either one of the other two types. In terms of combining, we assumethat the average power is equal for signals on all diversity channels, which leads to EGC, aswill become apparent in the next section. The received signal at the1th branch, after corruptionby multiplicative fading and AWGN, can be mathematically expressed asx(t) = f’(t)x(t) + n’(t) 1 < 1 < A. (2.15)Optimal Sequence Estimation for Fast Flat Fading with Diversity 33For the computer simulation performance results which will be reported in Section 2.3, thegeneral case was considered where the fading interference f’(t) introduced on paths 1’ E{ 2,3, ..., A} can be correlated with that of path 1 = 1. A correlation coefficient P1,1’ is used as ameasure of correlation between the amplitudes of the fading processes on the A diversity paths.During evaluation of the proposed diversity receiver structures through computer simulation,experimental values of found in [95] were used.1 I i=kTx,(t) 1s branch_________________y’(k)..00(1)Complex signal: .0-.Real signal:Figure 2.6: Block diagram of the diversity receiver.The general block diagram of the diversity receiver is illustrated in Fig. 2.6. Each blockcontains a wideband (roofing) Band Pass Filter (BPF) which limits the Gaussian noise withoutdistorting the information bearing signal f’(t)x(t), a coherent demodulator3,apredetection filter3For purely mathematical convenience, f is assumed to be known to the receiver, so that the derivationof the optimal diversity detector can be carried out in the complex baseband domain. Notice that through thiscomplex demodulation, the rapid phase and amplitude changes caused by the fast fading, appear at the output ofthe demodulator totally uncompensated. Note that, if desirable, this can be accomplished in practice by employing.t=kTA” branch_____Optimal Sequence Estimation for Fast Flat Fading with Diversity 34H(f) matched to the pulse shaping filter HT(f) employed at the transmitter, and a receiverto be derived in Section 2.2. Observing Fig. 2.6, the signal at the output of Hj(f) can beexpressed aszy1(t) = f’(t) cke’h’Q — kT) + n’(t) — jn’(t) (2.16)where h1(t) represents the inverse Fourier transform of HT(f) Hj(f), and n(t) and n(t)represent the in-phase and quadrature baseband components of the narrowband Gaussian noiserespectively. Under the assumptions that Hr(f). Hj(f) satisfies the Nyquist I criterion andthat the BPF does not alter the fading characteristics4,the sampled signal at the output of thepredetection filter Hj(f) of the 1th branch can be expressed as= (i’+ jf;l)cke”+ I;l — Q;l (2.17)where fk’t = f”(kT), f’ = f;1(kT), n’ = n(kT) and n = n(kT). The samplesy (1 < k Z) are fed into the receiver, the optimal structure of which is derived in thefollowing section.2.2 DERIVATION OF THE STRUCTURE OF THEOPTIMAL DIVERSITY RECEIVERThe derivation of the diversity receiver structure is presented in two parts. In the first one,we derive the structure of the optimal diversity receiver and present the associated optimalalgorithms. For mathematical convenience in the derivation we assume that the fading processeson the A diversity channels are uncorrelated. In the second part, we present suboptimal butreduced complexity versions of the optimal structure and associated suboptimal algorithms.The effect of correlation between diversity signals on the A channels is also investigated viasubsystem locked infrequency, but not in phase, with the transmitter local oscillator.4This is a reasonable assumption, since the transmission rate in almost all cases will be at least an order ofmagnitude greater than the maximum Doppler frequency BF.Optimal Sequence Estimation for Fast Flat Fading with Diversity 35computer simulation. Because of the high complexity presented by the optimal algorithmsand the limitations inherent to the computer simulation, only reduced complexity suboptimalalgorithms were evaluated. The obtained BER performance evaluation results involving variousmodulation schemes can be found in Section 2.3.2.2.1 Optimal AlgorithmsThe optimal receiver will choose the data sequence A= [,, ...,] which maximizes thefollowing probability density function (pdf)C (A) , = [yyi, YL2, ...,y (A) , (2.18)1=1 k=1whereyl=[y,y,...,y] (ll<A).Let us define the following variables= fk’ + = —-—Re (yce3)=+= —--Im (yce3) (2.19)Ckwhere * denotes complex conjugate andl;l = 1 {n’;’Re (ce) + nIm (ceij]I Ck= _!- {n’Im —n1Re (ce)J. (2.20)Ick IClearly, as e’1 and e are the sum of Gaussian random variables, they are also Gaussianrandom variables with autocorrelation functionR1 [(A), (k — i)T] = R(k — i)+6(k — i) (2.21)I ckwhere ()2 = N/(2T) and 6(.) is the Kronecker 6-function [96]. From Eq. (2.21) it is evidentthat for multilevel signals, e.g., QAM, the correlation properties of e and e1 depend upon theOptimal Sequence Estimation for Fast Flat Fading with Diversity 36transmitted sequence. For PSK signals however, these correlation properties are independentof the sequence.Maximizing the pdf ç’ is shown in Appendix A. 1 to be equivalent to maximizing thefollowing pdfAZfIll1=1 k=1 27r (o [c(A)])x exp { 2(()j)2 2Re (ycej —2—[Aj ( _J1)]m=1 Ick_x exp { — 2(()])2 [2Im(yce”)2— PLrn [CM)] 21mm1 Ck_m(2.22)In the above equation, [(A)]2represents the kt1 order minimum mean square prediction error (MMSPE) of the 1th branch andp {(A)] (1 k Z; 0 m k) are thekth order prediction coefficients, calculated according to the statistics of thejth fading channel.To simplify the representation, from now on the dependence of o and P1k,m on 0(A) will bedropped. These prediction coefficients depend upon the fading model employed, the signalto-noise ratio (SNR), the ratio between the power of the fading signal and the Gaussian noiseas well as on the transmitted sequence for multilevel signals. Their values can be obtained bysolving the following set of Yule-Walker equationsPh= {]‘ D (2.23)Optimal Sequence Estimation for Fast Flat Fading with Diversity 37where R’k is a [k x k} matrix, D is a [Ic x 1] matrix and P is a [Ic x 1] matrix withR(O)+R(l) R(k — 1)R(l) R(o)+ R(k-2)R(k — 1) R(k —2) R(O)+[R’F(1),R(2),.•,R(k)]T and = [PLl,P,2, Pk,k] (2.24)where T denotes transpose. From Eq. (2.24), it is straightforward to verify that when the fadingrate is extremely slow, i.e., when R(k) c 1 for the range of values used for Ic, then the matrixR is singular, as the equations in the system of Eq. (2.23) become identical. In such a case,the maximum likelihood receiver structure is considerably simplified, if we choose the simplestsolution of setting all prediction coefficients P,k equal to 1. This subclass of receivers hasbeen shown to be optimal for noncohererit sequence estimation in the AWGN channel [63, 64J.However, the interesting implications of using this simplified structure for receivers operatingin fading channels, are further investigated in Chapter 4.In Appendix A.2 it is shown that maximization of the pdf in Eq. (2.22) is equivalent tomaximizing the following function2Re[i (l)*] Re (ckc_) +lm [Y(_j*]Im (CkC_rn) PLiiPLirn+3]k=1 n=1 1=1 ckl ck_7 k—O (÷.)2 / \2Z A I Z—kI IY k2k=1 1=1 IckI rn0 (+m)A Z Re ( “c e” Z—k I k+m+2 77Tk k Pk÷rn,mP’k+rn,j. (2.25)1=1 k=1 ck m=O j=OFrom Eq. (2.25) it is clear that the basic hardware structure of the diversity receiver derivedis the same as that in [63]. However, the difference lies in the decoding algorithm, and the wayOptimal Sequence Estimation for Fast Flat Fading with Diversity 38samples from all A diversity branches are combined to yield the overall metric. In the next twoparagraphs, we discuss the physical meaning and implications stemming from each term in Eq.(2.25), translating it to the structure of the optimal sequence estimator.The first term in the equation is that corresponding to the multiple differential detectors,the second to the envelope detector and the third to the coherent detector. As introduced inChapter 1 (see also [63]), by the term “multiple differential detectors” we refer to differentialdetectors which decode the receiver signal over a multi-symbol interval. Such a detectoremploying a delay element of niT seconds will provide Re [ (_m) ] and Em [L (i4_m)*]for the calculation of the first term of Eq. (2.25). Because of the summation whichappears in this first term, it is clear that a bank of k distinct multiple differential detectors arenecessary, each of them employing a progressively increasing by T time-delay element. Thestructure of such a detector bank is illustrated in Fig. 2.7, where for ease of notation we havedefined d(k) L (i_m).The term “envelope detector” refers to a subsystem providingthe instantaneous signal envelope Ilk!, which is then squared, to be used as yj, in the metricexpression of Eq. (2.25).Finally, the term “coherent detector” refers to the estimation of , the modulator initialphase shift5. From the Eq. (2.25), it is evident that i is required only when the receivedsignal includes a direct component, i.e., when we have Rician fading. If no direct componentis present, i.e., when we have Rayleigh fading, then f = 0 and the third term disappearsfrom Eq. (2.25). This also makes sense intuitively, as we do not expect the modulator initialphase to affect the decoding when the phase of the received signal is uniformly distributed in[0, 27r), as is the case in Rayleigh fading. It is also important to point out that extensive BERperformance evaluation via computer simulation for the Rician channel demonstrated quitesmall performance degradation, if the term including ii was omitted from the function to be51t should be noted that this “coherent detector” refers only to the estimation of the initial carrier phase andnot the random FM which is introduced by fading. The rapid phase changes caused by the fading interferencewill be compensated for by the receiver which is derived in this section.Optimal Sequence Estimation for Fast Flat Fading with Diversity 39Figure 2.7: Block diagram of the bank of multiple differential detectors; PD: Phase Detector.maximized. This seems to indicate that this term is of relatively less importance as comparedwith the other two terms, and especially with respect to the multiple differential detector term.This qualitative statement is also quantitatively expressed in the results presented in Section2.3, where the sensitivity of the proposed receiver performance to inaccurate estimates off1;, R(r) and BFT is shown to be very small. Furthermore, BER computer evaluationresults for a Rician fading channel with K taking values between 5 and 10 dB have alsoindicated that the performance improvement gained when having a LOS signal is orders ofmagnitude greater than any degradation introduced by omitting the coherent detection terminvolving i. In any case, when it is necessary, the estimation of i need only be performedonce at the beginning of the receiver operation. We conclude this interpretive discussion ofthe receiver derived, by providing the complete block diagram illustrating the structure of theoptimal diversity receiver in Fig. 2.8.For an equal combining system, the following conditions apply: fIi = f112 = f1, R =4(k)Complex signal:Optimal Sequence Estimation for Fast Flat Fading with DiversityFigure 2.8: Block diagram for the structure of the optimal diversity receiver decoder.40R = RF and(2()2 = 2 V 1 < 11,12 < A. In this case, P,m = Pm = Pk,m and2 (.1i 2 = 2 V I 11, 12 A. For such a system the function to maximize simplifiestoZ k • Z—k2 2 2Pk+j,jPk+j,rn÷j2k=1 rn=1 Ick Ck_m k=O (ok+)z Z—k 2A—(piv+m,m) \‘ i2k=I ICkI ,n0 (k+m)2 /__Yk1k=1 CkjComplex signal: .—Real signal:Re [ (_m)] Re (CkC_m) +Tm [u (yLm)*] Tm (CkC7c_m)rz—k k+m A 1I Pk+m,rn[ (k+m)2 Pk+m,jRe (yce)j=o 1=1 J(2.26)Optimal Sequence Estimation for Fast Flat Fading with Diversity 412.2.2 Suboptimal AlgorithmsAccording to Eq. (2.26), the receiver requires knowledge of the prediction coefficients withorders 1 to Z . It also requires the use of Z — 1 order6 MDD. Even for relatively shortsequences, as for example commonly encountered block sizes of Z = 256 or 512, the receiverwould require impractically high levels of computational and implementation complexity. Toresolve this problem, the following truncation approach is adopted. The number of differentialdetectors z used by the receiver, which for the optimal receiver is equal to Z — 1, is reducedto a value much smaller than Z — I (z << Z — 1). By truncating the sums in Eq. (2.26) to themaximum prediction order z, we end up with the following metric expression to be maximizedRe (ekc) Re Fd(k)] B7r [ )1] += 2 2_1 2k=1 rn=1 k Ck._rnTm (CkCm) [z1 Im [dL(k)j B [ (A)]][c(A)]k=1 ckl 11+ —L Re (yce”) z [ (A)] f’. (2.27)k=I 1=1wherez—m 1 1d(k) = yi)*B,k [ (A)] = Pz,JPz.m+?= - (p)2, z, k1 = Zpj (2.28)z j=O Z i=O jOwith q z for m. Z — z and q = Z — k for rn > Z — z. Using Eqs. (2.25) and (2.28),for PSK signals the metric expression to be maximized is simplified as= f’Re (yce) A,k [c(A)]6By Z — 1 order MDD we imply a MDD hardware structure with Z — 1 differential detectors, employingelements of time delay up to ZT seconds.Optimal Sequence Estimation for Fast Flat Fading with Diversity 42ARe [d(k)} Re (ckc,) +÷2 B,k [c (A)J .(2.29)k1 n=1 1=1Tm {d(k)] Tm (CkCm)Finally, assuming that the fading as well as the AWON processes corrupting each one of the Adiversity channels have identical statistics, the above equation is further simplified to= f’&,k [()] Re (yce371)-Re [4(k)] Re (ckc_m) ++2EBz,k [(i)] (2.30)k=1 ?fl=1Tm [dL(k)] Tm (CkC_m)whereB,k [ (fl)]=Pz,JPz,7n+Jz,k [ (fl)] = - (pj)2, (2.31)j=C)&,k[G(A)] =are now independent of the diversity channel 1.In the next section we present BER evaluation results obtained via Monte-Carlo simulation,for ir/4-shift DQPSK, 8-ary DPSK (8-DPSK) and ir/4-shift 8-DQAM modulation formats,in Rayleigh and Rician frequency non-selective fading conditions, employing the proposedreceivers with various degrees of diversity.Optimal Sequence Estimation for Fast Flat Fading with Diversity 432.3 BER PERFORMANCE EVALUATION OF THESUBOPTIMAL ALGORITHMSThe proposed diversity receivers employing suboptimal algorithms were evaluated by meansof computer simulation. As previously discussed, the fading interference assumed followsthe land-mobile correlated, fast Rayleigh and Rician channel model, with BFT = 0.125. TheBER results reported indicate average number of bits in error over long signal sequences andwere obtained via computer simulation employing Monte-Carlo error counting techniques. Thenumber of error bits counted for all simulation runs was much greater than 100, for all casesabove the error rate level of 5 x i04. For levels between iO— and 5 x l0— where the numberof errors were allowed to drop to around 50, the greater uncertainty introduced was resolved byaveraging results from more than one statistically independent simulation runs for each point.All signals in the computer simulation were represented by their baseband equivalents [14].The filtering process was also carried out in baseband [14], and more specifically in the discretefrequency domain, by employing the fast Fourier Transform (FFT) transform [94]. The SNRindicated on all figures in the following sections is equal toE3/No, where E is the signalenergy in each symbol transmitted, and No the AWGN power spectral density. This is becauseNyquist filtering is employed. In later chapters where Butterworth filtering is used, this doesnot hold; in such case the SNR at the receiver is lower than the channel E3/N0. For moredetails on the digital simulation environment the reader is referred to [62].In investigating the BER performance of the receivers employing the suboptimal algorithmsderived in the previous section, we distinguish two cases: a) that where perfect knowledge ofthe statistical and deterministic channel parameters is available to the receiver, and b) when thereceiver knowledge of those same parameters is inaccurate.Optimal Sequence Estimation for Fast Flat Fading with Diversity 44wca0)0w2.3.1 Perfect Knowledge of Channel Parameters100101010.410SNR (dB)Figure 2.9: BER performance evaluation of raised-cosine filtered (cv = 0.35) rate 2/3, 8-state7r/4-shift 8-DQAM in a Rayleigh land-mobile fading environment with BFT 0.125.The BER performance of an 8-state trellis-coded 7r/4-shift 8-DQAM scheme is depicted in Fig.2.9. The code used for the convolutional encoder is the best 8-state code found in [25], havingparity check coefficients 1L2 = 048, h’ 02, h° = 118 (see also Fig. 2.4b). Employinga second diversity path (A = 2) results in a dramatic increase in gain - approximately 22 dB- at a BER level of iO. The little over 3 dB of gain at the same BER level obtained byadding an extra diversity path, does not present a substantial performance improvement, giventhe implementation complexity of introducing a third receiver stage and associated hardware.Error floors for both the A = 2 and A = 3 cases are expected to exist below the i0 BER level.However, statistically significant results below l0 could not be obtained due to computersimulation limitations.Optimal Sequence Estimation for Fast Flat Fading with Diversity 45100101crzw-2100I-.w4-.1-410Figure 2.10: BER performance comparison of raised-cosine filtered (o = 0.35) rate 1/2,4-stateir/4-shift DQPSK in a Rayleigh land-mobile fading environment with BFT = 0.125 using theimproved diversity receiver structure derived, to the performance of a 1-symbol differentialdetector (DD) and Viterbi decoder.The performance of the proposed MDD scheme is compared against the Viterbi decodingalgorithm in Figs. 2.10 and 2.11. The Viterbi decoder processes the output from a conventional1-symbol differential detector and employs equal combining of the A Euclidean distances inthe metric calculation, i.e., the metric used during decoding isd(k) — CkC1 (2.32)k=1 1=1or equivalently, the incremental metric used by the Viterbi decoder isd(k)— CkCi2(2.33)SNR (dB)In Fig. 2.10 the modulation scheme used is a 4-state trellis-coded ir/4-shift DQPSK whileOptimal Sequence Estimation for Fast Flat Fading with Diversity 4610010:1wa:ia)2a0I-.Iw4-,-410Figure 2.11: BER performance comparison of raised-cosine filtered (c = 0.35) rate 2/3, 8-state8-DPSK in a Rayleigh land-mobile fading environment with BFT = 0.125 using the improveddiversity receiver structure derived, to the performance of a 1-symbol differential detector (DD)and Viterbi decoder.Fig. 2.11 illustrates the results for an 8-state 8-DPSK. The 4-state code used for the r/4-shiftDQPSK scheme is the best code found in [25], having parity check coefficients .b’ = 2, h° 5(see also Fig. 2.4a). The 8-state code used for the 8-DPSK is the same as the one for theTr/4-shift 8-DQAM (see Fig. 2.4c). The only thing that changes is the signal assignment,which is provided in Fig. 2.4, of Section 2.1. Both the diversity algorithms proposed andthe Viterbi decoder, use the same trellis codes, with the same number of states and length ofpath memory. It is evident by comparing the two sets of results that the differential detectorcoupled with the Viterbi decoder performs reasonably well when the number of signals inthe constellation is small, or otherwise when the distances between points in the constellationare substantial. As shown in Fig. 2.11, this is not the case for schemes of higher spectralSNR (dB)Optimal Sequence Estimation for Fast Flat Fading with Diversity 47efficiency with “crowded’ signal spaces. The proposed diversity scheme for trellis coded8-DPSK outperforms the Euclidean-metric based Viterbi decoding by dramatically reducingthe error floor. Using a prediction order of z = 2 the irreducible error rate is below i0,indicating improvement of more than two orders of magnitude. For the same 8-DPSK system,by introducing a block interleaver before the differential encoder at the transmitter, and ade-interleaver after the differential detector at the receiver as described in [70], we verifiedthe optimal interleaving span and depth experimentally determined in [70] for land mobilefading. In cases where BFT <0.03, we observed that use of this block interleaver indeed hasa beneficial effect on the BER performance, reducing the number of bits in error at least by afactor of 2. However, for the fast fading conditions considered in Fig. 2.11, use of interleavingdid not yield any BER improvement for the conventional differential detector and Viterbidecoder, both for the cases of no diversity, and for diversity A = 2.Fig. 2.12 demonstrates the effect of correlation between the faded signals received on twoantennas, i.e., diversity A = 2 is assumed. The modulation format tested is the 4-state trellis-coded r/4-shift DQPSK. The simulation results indicate that even severe correlation (p12 0.8)does not seriously affect the gain obtained by using diversity. At the io— BER level, the lossis approximately 3.7 dB as compared to the completely uncorrelated case, the performance stillaround 5.5 dB better than the no-diversity case. A moderate correlation coefficient of p12 0.5will produce a loss of no more than 1.6 dB at BER = iO. The ir/4-shift DQPSK schemewas also tested in the Rician fading channel environment. BER results for two different valuesof the K-factor, 5 dB and 10 dB, are illustrated in Fig. 2.13. As K is increased, aside fromthe expected improvement in BER performance, it is interesting to note that the degradationobserved for a given fading correlation remains essentially constant; the p12 = 0.49 case, forboth values of the K-factor shown (0 and 5 dB), is approximately 3 dB worse with respect toP12 = 0, at BER = iO.Optimal Sequence Estimation for Fast Flat Fading with Diversity 4810010Uia)-210I...0IUi4-10-410Figure 2.12: Effects of correlation between signals received on two diversity paths on araised-cosine filtered (a 0.35) rate 1/2, 4-state r/4-shift DQPSK scheme, in a Rayleighland-mobile fading environment with BFT = 0.125, using the optimal diversity receiver withprediction order z = 2.2.3.2 Imperfect Knowledge of Channel ParametersIn Section 2.2, the derivation of the optimal sequence estimator assumed knowledge of thechannel parameters characterizing the fading distortion. Such are the type of fading environment reflected in the choice of fading filter Hj(f), the BFT product value, and the Ricianfading K-factor. Using 7r/4-shift DQPSK and A = 1, we investigated via computer simulation, the sensitivity of the proposed detection algorithms, to imperfect knowledge of all theaforementioned channel parameters.The receiver sensitivity was found to be very small with respect to the fading modelemployed. For example, for receivers with z = 2, A = 1 and at BER=102observed at SNR—120SNR (dB)Optimal Sequence Estimation for Fast Flat Fading with Diversity 49a:Liici)01ILii0Figure 2.13: Effects of fading correlation and direct/diffused power ratio K on a raised-cosinefiltered (o = 0.35) rate 1/2, 4-state ir/4-shift DQPSK scheme, in a Rician land-mobile fadingenvironment with BFT = 0.125, using the optimal diversity receiver with prediction orderz = 2 and two (A = 2) diversity paths.14.5 dB, employing a rectangular fading filter for the channel, instead of the land-mobilecharacteristic assumed by the receiver, resulted in a performance of 1.5 x 10—2 (degradationequivalent to less than 0.7 dB). Using a Gaussian fading filter for the channel, the obtainedBER was approximately 2 x 10—2 (degradation equivalent to less than 2 dB). The differencein degradation is explained by the fact that the impulse response of the Gaussian filter issignificantly different from that of the land-mobile filter. On the other hand, the impulseresponses of the rectangular and land-mobile filters are more similar. Nevertheless, in bothcases the degradation is insignificant as compared to the overall gains obtained by the proposedreceivers. Similarly, the performance was found to be very insensitive to variations of theSNR (dB)Optimal Sequence Estimation for Fast Flat Fading with Diversity 50channel BFT product, with the receiver assuming a fixed value. Again we assumed the land-mobile fading filter, at SNR 14.5 dB, z = 2, A = 1. In Table 2.1 the “IDEAL RX” isdesigned for the actual BFT present in the channel, whereas “FIXED RX” is designed forBFT = 0.125. By observing the simulation results summarized in the table, it is clear that evenextreme variations of ±50% on the BFT product value used by the receiver result in relativelyinsignificant performance degradation, equivalent to not more than 2 dB of power loss.Table 2.1: Performance degradation due to inaccurate estimate of channel BFT at the receiver.BFT FTXED RX ttIDEALRX___________(BET 0125) (B7T)0.125 1.07E-2 1.07E-20.0625 (-50%) 1.564E-2 1.020E-20.100 (-20%) 1.27 lE-2 1.136E-20.150 (+20%) l.326E-2 l.283E-20.1875 (+50%) 2.322E-2 1.680E-2The performance of the proposed receivers was also tested against the third fading parameterof interest, namely the value of the K-factor, as it is reflected in the value of fI’7 . For thepurpose of this test only one channel was employed so = f’. A value of 0 indicatesRayleigh fading while anything above that implies a direct signal path and consequently aRician fading channel. Two series of tests were performed in order to assess the impact of aninaccurate estimation of f in the receiver. In both cases, the fading channel BFT product wasassumed equal to 0.125, z = 2, A = 1. The results of the first series of tests are shown in Table2.2, where the BER performance is given for three values of the channel SNR, around a BER71t is evident from Eq. (2.14) that the K-factor is linearly related to the value of (JITT)2.Optimal Sequence Estimation for Fast Flat Fading with Diversity 51level of l0— (7.5, 10 and 12.5 dB) and various degrees of error in the estimation of f’. TheBER performance with accurate channel information is given in the column under “IDEALRX”. The channel is assumed to be Rician having a K 5 dB. The receiver, on the other hand,uses values of f’ which deviate from the correct value by some percentage. An extreme caseoccurs when the estimate of f is equal to 0 (i.e., -100% error or assumed K —* — dB), inother words, the receiver assumes that the channel is Rayleigh while in fact it is Rician withK = 5 dB. The table summarizes the results for other percentage errors and gives performancedegradation8in dB for the three SNR values. The error percentages have also been translatedin terms of K and are given in dB. For example, a -90% error in f’ (i.e., the receiver estimateis equal to 0.1 of the actual value) corresponds to -20 dB of error in K , which in this casemeans that the receiver has assumed K = -15 dB while the channel presents K = 5 dB. Fromthe results summarized in Table 2.2 it is clear that a maximum degradation of about 1 dB willoccur, around the io— BER level, if the receiver estimate of the K -factor for the Rician fadingchannel is within ± 15 dB off the actual value. It should also be noted that for errors in f above-70% (error in K above -10.5 dB) and below +180% (error in K below 9dB) the performancedegradation is negligible.The results from the second series of tests regarding the receiver performance sensitivitywith respect to f’ are shown in Table 2.3. In this case the receiver has assumed a Ricianfading characteristic for the channel while in fact the channel is Rayleigh. BER performanceand corresponding degradation in dB is given for three values of K (-5, 0 and 5 dB) and fivevalues of the channel SNR (10, 12.5, 15, 17.5 and 20 dB). The BER performance of the idealreceiver, i.e., one which has knowledge of the fading in the channel being Rayleigh, is alsogiven in the first row and is marked as “Ideal”. As expected, the performance degrades rapidlyas the assumed K-factor is increased, since the channel in this case exhibits a K —* — dB.8By degradation of x dB here, we imply that the IDEAL RX will require x dB less signal power in order toprovide the same BER performance.Optimal Sequence Estimation for Fast Flat Fading with Diversity 52Table 2.2: Performance degradation due to inaccurate estimate of channel f’ at the receiver.For the results summarized in this table, the Rician channel has been assumed to have a K = 5dB and BFT = 0.125. The estimate error for f’ has also been translated to an error in thevalue of K which is given indB. The extreme case is when the receiver assumes the channelis Rayleigh (-100% error in f1 or K —* —c’c dB) while the channel has been assumed Ricianwith K = 5 dB. SNR degradation is given in parenthesis under each BER performance value.Deviation of receiver’s estimate of f’ from that corresponding to K=5dB This translates to an error in the estimationof J which is given in parenthesis.+200% +400%IDEAL RX -100% -90% error error errorerror (-20 dB (+9 5 dB (+14 dB(Rayleigh) error) error) error)7.5 7.OOE-3 l.84E-2 l.16E-2 7.76E-3 1.03E-2(1.7 dB) (0.7 dB) (0.2 dB) (0.7 dB)SNR 10 1.56E-3 7.09E-3 3.06E-3 1.58E-3 2.57E-3[dB] (2.5 dB) (1.0 dB) (0.2 dB) (0.8 dB)12.5 3.30E-4 2.69E-3 8.OOE-4 4.OOE-4 6.OOE-4(3.3 dB) (1.3 dB) (0.3 dB) (0.9 dB)Note, however, that the performance degradation would be quite small (approximately 1 dB)at a BER level around io, if the receiver had assumed a K-factor of -10 dB. The test resultssummarized in Tables 2.2 and 2.3 suggest that a practical approach towards system designwould be: first to establish a most probable 30 dB range for the K-factor, i.e., within ±15 dB ofa mean value and, second, to use this mean value to calculate f in the receiver. This guaranteesthat as long as the K-factor lies within the 30 dB range selected, the receiver SNR performancewill be at most 1 dB worse as compared to the theoretically attainable. Degradation resultspresented in Table 2.3 suggest that the term of the metric to be maximized involving f’ (orin more general terms fI;l) should not be dropped (effectively leading to a Rayleigh fadingreceiver design), unless the K -factor is guaranteed to be below -10 dB most of the time.Optimal Sequence Estimation for Fast Flat Fading with Diversity 53Table 2.3: Performance degradation due to inaccurate estimate of channel f’ at the receiver.In this case, the receiver assumes that the channel is Rician whereas in fact it has a Rayleighcharacteristic with BFT = 0.125. The value of K given is that assumed by the receiver. SNRdegradation is given in parenthesis under each BER performance value.10 12.5 15 17.5 206.26E-2 2.57E-2 9.23E-3 3.67E-3 l.77E-3(Ideal (Ideal (Ideal (Ideal (IdealReceiver) Receiver) Receiver) Receiver) Receiver)0 7.99E-2 3.92E-2 1.74E-2 8.86E-3 6.23E-3(1.1 dB) (1.1 dB) (1.4dB) (2.2dB) (3.7dB)2.4 CONCLUSIONWe have derived the optimal sequence estimator using diversity reception for digital signalstransmitted over A frequency flat, correlated fast Rician fading channels. The analysis presentedhas considered the more general case where both phase and amplitude distortion due to fadingvary significantly over the symbol duration. The decoding algorithm does not require the useof interleaving for removing the correlation between adjacent signal samples, a quality whichgreatly simplifies receiver design. Various reduced complexity versions of the optimal decoderstructure, which consists of a combination of envelope, multiple differential and coherentdetectors, have been presented for differentially encoded PSK and QAM signal constellations.Performance evaluation results obtained by means of computer simulation, have shown themto outperform conventional Viterbi decoding using differential detection, especially for morebandwidth efficient modulation schemes, i.e., more dense signal constellations. For example,______ _______SNR[dBJ_____________-5 7.28E-2 3.2lE-2 l.30E-2 5.23E-3 3.12E-2_______(0.5 dB) (0.5 dB) (0.7 dB) (1.1 dB) (1.8 dB)K[dB]5 9.97E-2(1.8 dB)5.1 1E-2(1.8 dB)2.78E-2(2.5 dB)1.61E-2(3.4 dB)1 .22E-2(5.4 dB)Optimal Sequence Estimation for Fast Flat Fading with Diversity 54for a trellis coded 8-DPSK scheme using two diversity channels (A = 2) and two differentialdetectors in a fast Rayleigh fading channel (BFT = 0.125) there was no error floor observedat BER = i04. Under the same channel conditions, the conventional Viterbi decoder usinga 1-symbol differential detector and two diversity channels, exhibits an irreducible error rateof about 4 x 10_2. BER performance tests also demonstrated that the proposed sequenceestimation receivers are quite insensitive to significantly varying channel conditions includingthe fading model (e.g., fading autoconelation function, BFT and the K-factor), thus makingthem particularly well suited for practical system implementation.CHAPTER 3OPTIMAL SEQUENCE ESTIMATION FOR FASTCORRELATED FREQUENCY SELECTIVE RICIANFADING CHANNELS’In this chapter, following our previous work on frequency flat (i.e., frequency non-selective)fading channels [63], we derive the optimal, in the maximum likelihood detection sense,receiver for coded digital signals transmitted over channels corrupted by correlated fast Ricianfrequency selective fading and AWGN.The organization of the chapter is as follows. Section 3.1 describes the channel modelassumed while Section 3.3 presents the derivation of the optimal receiver for the frequencyselective fading channel. The performance evaluation results obtained by means of computersimulation for r/4-shift DQPSK signals are presented in Section 3.4, while in Section 3.5 weoffer some concluding remarks.3.1 FREQUENCY SELECTIVE FADING CHANNEL MODELFor the frequency selective fading channel model we assume the existence of two independentpropagation paths, the “direct” and the “reflected”, both corrupted by frequency non-selective‘The research reported in this chapter has been presented in part at the 1993 IEEE Pacific Rim Conference,Victoria, Canada [97], and is to be published as a paper in the IEEE Transactions on Vehicular Technology.55Optimal Sequence Estimation for Fast Frequency Selective Rician Fading 56Rician fading. As previously mentioned, this two-ray model has been recommended by theTelecommunications Industry Association (TIA) Standards Committee to evaluate the toleranceof delay spread in the new North-American digital cellular system [21]. The delay by whichthe reflected signal is assumed to arrive at the receiver, normalized to the symbol duration T, isdenoted by 3. Upon arrival, it is superimposed on the direct signal. The multiplicative fadingprocesses on the direct path (denoted as path 0) and the reflected path (denoted as path 1), f°(t)and f’ (t) respectively, are modeled as a complex summation of two independent Gaussian noiseprocesses. In other words, f°(t) f”°(t) +jf°”(t) and f’(t) = f”(t) +jf’(t). Consideringthe same transmitter as that of Section 2.1 (see Eq. (2.8)), and that the faded signal is corruptedby AWGN n(t), the signal arriving at the receiver can then be expressed asr(t) = f°(t)x(t) + f’(t)x(t — T) + n(t). (3.34)Adopting the land-mobile fading model for both f°(t) and f’ (t) fading processes, their auto-correlation function can be expressed identically to Eq. (2.13) asR3) = Jo(2TrBFB) m € {0, l}. (3.35)Furthermore, and similarly to Eq. (2.12), the real and imaginary parts of both ftm (t), m E {0, 1 },have the same autocorrelation function, given byR(8) = E { [fIm(t) — flm] {fJrn(t — 3) — J’] } == E { [fQm(t) fQ’rn] [fQm(t ) - fQmj }- rrn L H(f)12e’2df. (3.36)In Eq. (3.36), f1m = E {fIm(t)} fQ,m = E {IQ’rn(t)} is, without any loss of generality,assumed to be equal to zero, m E {0, 1 } and N is the power spectral density of the whiteGaussian noise processesnm(t) fllm(t) + jQm(t) generating the fading processesfm(t) =fIm(t) + jfQm(t). As flIm(t) and n”(t) are assumed independent, the cross-correlationOptimal Sequence Estimation for Fast Frequency Selective Rician Fading 57between fIm(t) and fQm(t) is zero. Considering the same fading filters and power spectraldensity for both paths, we have H(f) HF(f), and N = NF for m E {0, 1}. A blockdiagram of the low-pass equivalent of the channel model is depicted in Fig. 3.1. Similar to Eq.Figure 3.1: Block diagram of the low-pass equivalent of the frequency selective fading model.(2.14), the K-factor for the m-th path, Km, is given in dB asF 1Km = 10 log10[im + OfQmj(3.37)n”°(t)HF(J)n (t)s(t)itt)Complex signal:Real signal:n(t)n”1(t)HF(J)n“ (t)s(t)where and cT2JQm are the variances of fJ:m(t) and fQm(t), respectively.Optimal Sequence Estimation for Fast Frequency Selective Rician Fading 583.2 RECEIVER MODELFor simplicity in the theoretical derivation of the optimal sequential decoder we assume thatthe impulse response hT(t) of the premodulation shaping filter is very closely approximatedby a Dirac 6-function. In other words, the signal up-converted by the modulator very closelyresembles a non-return-to-zero (NRZ) [14] pulse-sequence, i.e., we consider an unfilteredtransmitted sequence. It should be pointed out that including the effects of signal filtering inthe analysis presented in the next section is straightforward but mathematically very tedioustask. Because of this, the filtering effects, i.e., assessing the impact of both small and largedeviations on the pulse shape from that of an NRZ signal, were included only in the performanceevaluation results obtained via computer simulation (see Section 3.4).For mathematical convenience we assume that the receiver front-end down-converts theincoming signal by making use of a local oscillator in frequency lock with the signal carrier2.By using an integrate-and-dump predetection filter and sampling its output every t = kT, thesignal provided as input to the detector can be expressed asy(kT) = ckf°(kT) + ck_lf(kT) + ckf(kT) + n(kT) (3.38)withfAO(t) = f°(a)da1 t—T÷TfBl(t)= —J f1(a)daT-TfGl(t) = J f’(a)da. (3.39)T t-T+rTSince f°(t) and f’(t) are complex Gaussian random processes, fAo(t), fBl(t) and f’(t) arealso complex Gaussian random processes. Using well known results for the statistics of linear2Similarly to Section 2.1, this complex demodulation only serves as a mathematical tool for downconvertingthe signals to the complex baseband domain. Rapid phase and amplitude fluctuations caused by fading appear atthe output of the demodulator totally uncompensated.Optimal Sequence Estimation for Fast Frequency Selective Rician Fading 59system response to random process input, it shown in Appendix B. 1 that their autocorrelationfunctions areRA(kT) = f(1 — a)Jo [2BFT(k — a)] daRB(kT) = J. (i — jo [27rBFT(k — a)] da(1—) / a( \R0(kT) = (1 — r) J - l — — ) Jo [2rBT(k — a)] da (3.40)(1 T) 1 Twhile the cross-correlations between fB1 () and fC’ () are given byRBC(kT)= { f’ (a + 1) j0 [2BFT(k — a)] daJo [2rBT(k — a)] daf—a”1 (—-) Jo [2irBp’T(k — a)] da—1 \ T IRCB(kT) = —RBC(kT) (3.41)forO < <0.5,anda+1RBc(kT) (1 — ) j L ( ) Jo [2rBiT(k — a)] dai-—I+ J Jo [27rBFT(k — a)] da+ f () Jo [2BFT(k — a)] da}RGB(kT) = —RBC(kT) (3.42)for 0.5 3 < 1. Using the notation yk = y(kT), f(ABC)m(kT)= fLABc)m and n(kT) =it is convenient to define the following random variable (r.v.)e(k)= ykcke= + + ‘Icck1 + nkce (3.43)Ck CkOptimal Sequence Estimation for Fast Frequency Selective Rician Fading 60The real and imaginary parts of e(k), e’(k) ande0(k) respectively, are given bye’(k) = jI;AO ++ Re{cc_1} — Im{cc_i }f2;B1 + n(k)Icke(k) = f;AO ÷f;CI ÷Im{cck_1}jJ;B1 — Re{cc_1}f;B1 +n(k)Ck Ck(3.44)with the noise terms n(k) and n(k) given similarly to Eq. (2.20) asn(k)= njRe {ce3?1} — nIm {ce}Ckn(k)= nIm {ce} + nRe {ce”} (345)I Ckwith n and n denoting the real and imaginary parts of n(kT), respectively. From Eqs. (3.44)and (3.45) it is not difficult to see that both e1 (k) and e (k) are Gaussian r.v.’ s. In Appendix B.2their autoconelation and cross-correlation functions are derived, and are given by the followingfunctionsR11(k, 1) = £ {e’(k)c’(k — 1)]= RA(l) +R0(l)+ Re{cck1ck_lc_l_lRB(l)Icki kk_1IRe {cck_1 } Re{c_lck_1_I }+ 2 RCB(l) + 2 RBC(l)ckj6K (1)CkRQQ(JC, 1) = £ [eQ(k)eQ(k — 1)] = E [e’(k)e’(k — 1)] =R11(k, 1) (3.46)with6K(l) denoting the Kronecker 6-function, andRJQ(k, 1) = £ [e’(k)eQ(k — 1)]— Tm {ckc_ic_ick_1_1 } Tm {c_lck_1_1 }2 2 RB(l)+ 2 RBC(l)CkI Ck_lI CkIIOptimal Sequence Estimation for Fast Frequency Selective Rician Fading 61Im{cck_l}— 2 RCB(l)RQJ(k, 1) = E [eQ(k)c’(k — 1)]= —RIQ(k.l). (347)Note that since E [e’(k)eQ(k)] = 0, and since e’(k), e(k) are Gaussian, they are also independent.3.3 DERIVATION OF THE MAXIMUM LIKELIHOODSEQUENTIAL DECODERThe maximum likelihood sequential detector must choose that sequence which maximizes theconditional pdf ( [yi, Y2, ..., Yz [(A), ii]. Since e(k) is a scaled version of Ilk maximizationcan be carried out on the joint pdf resulting from the sequence of e(k)’s rather than from thereceived signal samples. It is mathematically convenient to further rearrange the Z complexvalued e(k)’s into a vector of real numbers, associating the real parts with the even vectorindexes and the imaginary with the odd, i.e.,{ Eo, El, E2, E2Z_2, E2Z_l}E2k = e’(k), E2k+1 e’(k) V 0 < k < Z — 1. (3.48)From Eq. (3.44), the expected values for EZk and 62k+1, denoted Z and EZk+I respectively, aregiven by= fIo+(l_)fJ÷e{k_l}fI,lE2k+1Im{ccl}-f (3.49)(ek IUsing the expression for the conditional probability density function of a Gaussian r.v. givena sample vector of Gaussian r.v.’s, [98), and a mathematical approach equivalent to the one inOptimal Sequence Estimation for Fast Frequency Selective Rician Fading 62Chapter 2 (for details see Appendix A. 1), it is obtained that the maximum likelihood detectormust choose the sequence C(A) which maximizes the following pdff[[c4)j = ij1 1/27ra [C(A), kj21 —xexp — —— 2k ——_____2 [c(A), k]— PE,i {c(A), k] (E2k1 — 2k-1-i)[() k]1 (E2k÷1 — E2k+1)2xexp — 2k2o [(A), k]—kj . (E2k— — E2k_)(3.50)In the above equation, similarly to Eq. (2.22), o- {(4), k] and o- {C(A). k] represent the kthorder MMSPE’s and pE(k) {C(A)],p0,(k) [C(A)] (1 k Z) are the kth order predictioncoefficients [631. For representation simplicity, in the equations henceforth, the dependenceof 02k, 2k+1 and PE,i, po, on G(A) will be omitted. Similar to Section 2.2.1, the predictioncoefficient values are found by solving the two sets of Yule-Walker equationsPEk = []1ffPO,k DQ,I, (3.51)and they depend upon the type of fading model employed, the ratio of powers between thefading signal and the Gaussian noise, as well as, in general, on the transmitted sequence. Thedifference in the two linear systems of Eq. (3.51) as compared the one in Eq. (2.23), stemsfrom the fact that in the case of frequency non-selective fading in Chapter 2, samples from theI and Q channels are uncorrelated. In frequency selective fading, however, the 1ST caused byOptima] Sequence Estimation for Fast Frequency Selective Rician Fading 63the delay spread combined with the signal distortion caused by fading, introduces correlationbetween the samples from the I and Q channels, as evident in Eq. (3.47). Due to this, predictionfor each I and Q channels, is based not only on the k signal samples from that channel, buton the 2k samples from both channels combined. In Eq. (3.51), Rk is a [2k x 2k] matrix (ora [k x k] matrix of [2 x 2] sub-matrixes M(m) - see Eq. (3.53) below), D,k and Do,k are[2k x 1] matrixes and P,k and Fo,k are [2k x 1] matrixes withM(k—1,O) M(k—2,—1) ... M(1,2—k) M(O,1—k)M(k—1,l) M(k—2,O) . M(1,3—k) M(O,2—k)Rk=M(k—1,k—2) M(k—2,k—3) M(1,O) M(O,1)M(k —1, k—i) M(k —2, k —2) M(1, 1) M(O,O)Rjj(k,m) RQJ(k,m)M(k,rn) =RJQ(k,m) RQQ(k,rn)(3.52)= [R11(k, 1), RJQ(k, 1),. . . ,R11(k, k), RIQ(k, k)]T= [R1(k, 1), RQQ(k, 1),. . ., RQJ(k, k), RQQ(k, k)]TPE,k = [PE,1,pE,2,. . ,I’E,2k—1,PE,2k]PQ,k = [p0,1,730,2,. ..,p0,2k—1,PO,2k]where T denotes transpose. The kth order MMSPE’s are given byk koj(k) =R11(k,O) —ZpE,2j_l(k)RII(k — j, —j) —pE,2(k)RQI(k — j, —j)j=1 j=1k k= RQQ(k,O) —pQ,2_l(k)RJQ(k—j, —j) — po,23(k)RQQ(k — j, —j). (3.53)j=1 j=1Optimal Sequence Estimation for Fast Frequency Selective ELi cian Fading 64Maximizing the product in the pdf of Eq. (3.50) is equivalent to minimizing the sum oflogarithms in the following metric function21 (62k — )—2k + in [27r2 (k)]k=O PE,(k) . (_ — 62k_i)z—i (62k+1—62k÷1—2+ 2k____(3.54)k=O 0 po,j(k) . (62k_i — 6zk_)i=1According to Eq. (3.54), in order to implement a sequential detection algorithm using thismetric expression, we need to compute the 2 x 2Z prediction coefficients and also needstorage for all Z complex received signal samples. The formidable computational load ofsolving the two linear systems to yield the prediction coefficients and then computing themetric expression, even for moderately large values of Z, can be a serious limiting factoras far as practical system implementation is concerned. However, similarly to the methodfollowed in Chapter 2, taking advantage of the statistical properties of the fading channel, i.e.,knowing that the autocorrelation function RF(/3)values become quite small as we move awayfrom 3 = 0, we can truncate both sums in Eq. (3.54) up to a maximum prediction order z[63]. This decreases dramatically the computational load imposed by the algorithm withoutsignificantly compromising the attainable gain. A practical system would typically employvalues of z ranging between 2 and 4. Substituting the expressions for 62k and 62k+1 in Eq.(3.54) and truncating the sums beyond the maximum prediction order z yields the function tobe maximized given in the following equation2IckE pE,2i(k) — 62k_2i+1) — +ln [2iro(k)j +k=O E( ) i=1z /‘ yk_Ic_lePE,2_1(k) (Im 2 —62k—2ii=1 I.. Ck_1I )Optimal Sequence Estimation for Fast Frequency Selective Hician Fading 65(3.55)( *—• 2I ykcke ‘ I_____Im 2 E2k+11 )po.2(k) (Re {Yk-1_i3”} — — + in [2ir(k)]z (Ep0,2_i(k) j Tm 2 —\ L. ck_1I )3.4 BER PERFORMANCE EVALUATION RESULTS ANDDISCUSSIONThe BER performance of the maximum likelihood decoder derived, was evaluated via computersimulation, using Monte-Carlo error counting techniques. The modulation format used was4-state, rate-l/2 trellis-coded 7r/4-shift DQPSK. For comparison purposes, as in Chapter 2,the code used is the best 4-state code listed in [25]. Fading channel frequency selectivity wassimulated by one strong signal reflection arriving at the receiver at = 0.25 and = 0.75,or by a triangular delay spread profile. Both Rayleigh and Rician channel characteristicswere investigated. The ir/4-shift DQPSK NRZ sequences were filtered with square-root, 4-pole Butterworth or Nyquist filters. The Butterworth filters using a 3 dB cut-off frequencyfB = 1 /T, and the Nyquist filters an excess bandwidth of = 1.0. The sensitivity of thereceiving algorithms was also tested, by recording the BER degradation as a function ofthe error in the receiver’s estimate of and BFT product. For comparison purposes, usingcomputer simulation, we also evaluated the BER performance of a receiver employing themodified square-root Kalman (MSRK) equalizer for fading dispersive channels found in [78].We chose this equalization method to compare against, as it is the most well suited to the rapidsignal phase and amplitude fluctuation caused by fading [14], exhibiting orders of magnitudefaster convergence [78] as compared to conventional decision feedback equalization methodsfound in [14]. The performance of this receiver is compared to that of the maximum likelihoodOptimal Sequence Estirnatioii for Fast Frequency Selective Rician Fading 66decoder derived, in identical channel conditions, for increasing values of the BFT product.In the following paragraphs, we will be presenting the detailed BER performance evaluationresults.Fig. 3.2 illustrates a comparison between the performance of the optimal sequence estimation receiver for frequency selective fading, versus the equivalent receiver for flat fading foundin 163j. Results for complexities of z = 2 and 3 are shown, as well as two levels of the powerratio between direct-diffused and reflected-diffused paths, referred to as direct-to-reflected ratio(DRR). Both paths are Rayleigh faded, the reflected one arriving at = 0.75 of the symbolduration T, while the BFT product is 0.0625, corresponding for example, to a transmission rateof 1200 Baud on 850 MHz while the vehicle is moving at a speed of 95.3 km/h. In this figure,it is interesting to note the very small effect the DRR has on the frequency-selective receiver,and also the gain in performance when increasing the receiver complexity from z 2 to 3.As expected, the frequency-flat receiver performance worsens when its receiver complexityis increased. This is because an increase in z implies an increase in the number of termscontributing to the decoding metric used by the frequency-flat receiver, thus increasing its inaccuracy as compared to the metric used by the frequency-selective receiver. The improvement inperformance for both the frequency-flat and frequency-selective receivers when the reflectiondelay is decreased to = 0.25, is shown in Fig. 3.3. The complexity for both receivers is setto z = 3, and DRR takes values of 3 and 6 dB. Comparing the results illustrated in Figs. 3.2and 3.3, it can be seen that, as intuitively expected, the frequency-flat receiver performanceapproaches that achieved by the frequency-selective receiver, as —÷ 0. Note also the contrastin performance variation between the two types of receivers with the change in DRR. As isintuitively expected, the frequency-flat receiver performance is more sensitive to the value ofDRR, as compared to the the frequency-selective receiver, where the reflected ray signal isaccounted for by the decoding algorithm. The slight performance improvement observed forthe frequency-flat receiver in medium values of SNR, is attributed to the randomizing effect ofCDCDCD.CDCD •_CDCD--CL’CCD—C0II CDC-)‘_)-,tJ-,—C-—1)-_CIIoCD0CD.-‘.CDCD-f0CDoCD-‘t’J• CDl)0PCDCI)CDJ.)c, 0CDCD0rIISD—II C0‘CDZ CCD9> II Coc,oak)CDCDCD-.C’cnCDCDZrjI—CD-o—CD 0CDt’JC’I•,—I’CDI—CDc’ -ZCD CD-CD C-)CD Cl) I CD CD CD CDBitErrorRate(BER)000BitErrorRate(BER)0,0Optimal Sequence Estimation for Fast Frequency Selective Rician Fading 68the AWGN, which cancels a portion of the error inherent to the frequency-flat receiver metricwhen used in such frequency-selective channel conditions.Figs. 3.4 and 3.5 illustrate results for conditions same to those of Figs. 3.2 and 3.3, withthe exception of a higher BFT product, set to 0.125. As expected, the performance for bothreceiver types is worse as compared to that for BFT = 0.0625. Results in both figures arepresented for receiver complexity z = 3. Indeed the receiver BER shown in Fig. 3.4 is quitehigh. While the illustration is provided for the purpose of comparison to the frequency-flatreceiver case, it should be noted that the performance, even at such a high fading rate, could beenhanced, if the receiver complexity was increased to z 4 or 5. However, due to the fact thatthe time required for computer simulation at such receiver complexity was impractically high,we will not be presenting evaluation results for z > 3.The sensitivity of the optimal receiver was investigated by obtaining the BER performancefor various degrees of inaccuracy in the receiver’s estimate of the channel delay and thefading BFT product. The results are depicted in Figs. 3.6 and 3.7. Since we are interestedin both positive and negative errors in the receiver’s estimate of the delay spread, reflectedray delay is set to = 0.5; the receiver’s estimate of BFT is set to 0.0625. The value ofassumed by the receiver is allowed to take values between 0 and 1, while the channel’sBFT product varies within ±50%. Note that this variation in BFT product could directlycorrespond to a ±50% change in vehicle speed, i.e., in our case, BFT = 0.03 125 —* 0.09375.The BER performance with both receiver estimates perfect, using a complexity of z = 2, isslightly over l0, while using z 3, below iO, corresponding to the indentation on thetwo three-dimensional surfaces, at BFT 0% error and = 0.5. It is interesting to note that amoderate error in BFT of ±20% causes relatively little degradation. A decrease in the fadingrate, although it translates to inaccuracy in the detection algorithms employed, yields betterperformance due to the fact that, the slower signal phase changes are cancelled out to a greaterdegree by the operation of the differential detectors. On the other hand, when the receiver’s0BitErrorRate(BER)0,--011rriiiiTTITITI10•CCD CDCDCDrTl0CDCDCD)-..C)HogII CDI—’C)Ct’ZrJI-’.k)•;;I—.oC) CD(Jo•oo,CDZIICD (t’CD•••‘I-*I-tCD o‘CDCDOCDE. CDClBitErrorRate(BER)00C”‘-—. —•.——•TIrnrri—TIrrml1FHfI1IEi11iii0 C” 0.CDCDCDrC)Li)p•0p-44N—t1t[ti1Ho00...III0k)•p-‘“CD.-i>90IICD•CD0..CD___II___I___ICDED..C-.CDCDCD o CDCDC-.oCD‘-‘•‘‘J<,o-CDCF0 Cl)Cb CtC C) (b C-.0 Cl Ct CD C-)CD 0—,,-1,- 0—,D.—-—0-DA—-I-’—1):.)voa-—><><P?P?——1—-—-—i‘CIC”-—1 :C”---——---*-Optimal Sequence Fstimaion for Fast. &equencv Selective Rician Fading0070Figure 3.6: Frequency-selective maximum likelihood receiver (z = 2) performance degradationdue to inaccurate estimate of channel delay () and BFT, in the presence of a 2-ray Rayleighfading channel.!oglO(BER)—50Oitiinal Sequence .Estiniation for Fast Prequezicv Selective Rician Fading 71ogi O(BEP)—1—2—3-—4,-—550Figure 3.7: Frequency-selective maximum likelihood receiver (z = 3) performance degradationdue to inaccurate estimate of channel delay () and BFT, in the presence of a 2-ray Rayleighfading channel.BT % error—50 1—50T’F I F I F F I F: = = z.. DRR = S dB, single ray •— —— Z=3 DRR = 6 dB, delay spread p. — — Z=2, DRR 6 dB, single ray— — —-, Z2DRR=6dS.delayspread——--——Z3, DPR = 3 dB, single ray •— — — — J Z=3, DRR 3 dB delay spread D= = Z=2 DPR 3 dB, single ray .z — z Z=2, DRR 3 dB, delay spread ——-o--——— — — —1.. U I I U U U U U U U‘EEEEEE E—‘---—-- .c — — = ——‘‘b10 15 20 25 30 35 40 45 50SNR (dB)Figure 3.8: BER performance of rate- 1/2 trellis coded Butterworth filtered (fB 1 /T) Tr/4-shiftDQPSK scheme, using the frequency-selective maximum likelihood receiver, in a channel withtriangular delay spread profile for the reflected signal, centered at 0.75T, with BFT = 0.0625andDRR=3and6dB.The effect of having a delay spread profile rather than a single reflected signal path, isillustrated in the BER results presented in Fig. 3.8. The channel conditions assumed in thesimulation are the same as in Fig. 3.2; the results depicted in Fig. 3.2 are duplicated here forease of comparison. The triangular delay profile used, simulated in discrete time as 5 signalrays centered around 0.75T and spaced T/l6 apart. Total signal power for the single ray isthe same as that for the delay profile, while powers of reflections before and after the centralOptimal Sequence Estimation for Fast Frequency Selective Rician Fading 72estimate of the reflected ray delay is decreased beyond the actual channel value, the performancedeteriorates quite quickly. Interestingly enough, this is not the case when the receiver estimateof is greater than that corresponding to the actual channel delay. The trend implies that apractical system should implement a channel delay estimate which would be more likely tohave a positive rather than a negative error. As intuitively expected, the higher complexityscheme of z = 3 is more sensitive to high inaccuracies in the channel delay estimate, than whenemploying z = 2.100__________________________________________10ccci,10CuJ1010Optima] Sequence Estimation for Fast Frequency Selective Rician Fading 73reflection roll-off at 3 x I 6/T dB/s. In other words, assuming 0 dB for the central reflection,the ones adjacent to it are 3 dB down, and the two outer ones 6 dB down. Such an exaggeratedreflected energy profile was chosen in order to investigate the sensitivity of the receiver to thedispersion of the delay profile; in a practical multipath environment, and for the transmissionrates of interest, the signal power around a peak caused by a strong reflection will present amuch steeper roll-off, resulting in much less degradation [99}. As seen in Fig. 3.8, the actualperformance loss for = 3 and DRR 6 dB, is around 5 dB at a BER level of 2 x i0,bringing the error floor previously below i0, at about 2 x iO—. For DRR 3 dB, the errorfloor is slightly higher, as intuitively expected, and around 3 x i0. For the z = 2 case, theirreducible error floor is higher and the observable loss in performance less. It is interesting tonote that even in such an extreme case of high signal energy around the central reflection, theperformance loss is within reasonable limits, given the fact the the receiver is designed for asingle reflection.Although the derivation of the investigated maximum likelihood receivers has assumed NRZpulse signaling, the results presented thus far are for 4-pole Butterworth filtered NRZ pulses.Comparison to results generated for unfiltered NRZ signals has revealed that the receiverstructures derived suffer no measurable performance degradation when 4-pole Butterworthfiltering is employed. The results depicted in Figs. 3.9 and 3.10 demonstrate the relativelysmall sensitivity of the algorithm to the pulse shape used. Both figures present BER evaluationresults for Nyquist filtered signals, using excess bandwidth c = 1.0, under channel conditionsidentical to those in Figs. 3.2 and 3.5. It is interesting to note the great similarity between the twosets of results, keeping in mind the vast differences between the Butterworth and Nyquist pulseshapes. The slight degradation — rather than the expected enhancement — of performance whendecreasing DRR in the Nyquist filter case, is attributed to the degradation introduced by themetric inaccuracies, due to pulses interfering from both sides of the current symbol rather thanjust the past symbol as in the Butterworth pulse case. The fact that the performance degradationBitErrorRate(8ER)0•4.0•BitErrorRate(BER)0•CDP’L,)CDCDH-CCCD-iUCDCD CCCD-‘-.IN.)“••‘-I,—‘0 QCDCDg3 CDCDoC•-÷oCDCOozCD-E.CD-•0i-tCD-CDCCDS.-.•CD-I-.CDCD0‘.CCDII- CC-)CCD9)CCDII UiCDCD0CDCDCDCDz••-‘ 4-’.0 CC•CDLi)CDo1..•)4-iCD‘—-Co o00CD004—i0i—ICDCo4-’•CDIICD CD4-CCD CD—•0 jD (b (1:4 (-:4 I) -4.0CO z :xi 0.Optima] Sequence Estimation for Fast Frequency Selective Hician Fading 75is small, is a strong indication that this error could be rectified by modifying the derivation toinclude symbols on both sides of a sampling instant, something that would only moderatelyincrease the metric, and hence the receiver complexity. However, decreasing values of theroll-off factor & has the well known effect of increasing the amplitude of pulse side-lobes [14].This, in turn, would increase the inaccuracies in the metric computation and have a detrimentaleffect on performance. If low values of are to be accommodated successfully, the signalexpression of Eq. (3.38) has to be modified to include more terms from previous and futuresymbols.The BER performance of the optimal sequence estimation receiver in a Rician frequencyselective fading environment is depicted in Figs. 3.11 and 3.12. In both cases the delay forthe Rayleigh (K1 —* —dB) faded reflection is set to = 0.75. The first figure shows acomparison between receivers with complexities of z = 2 and 3, both for the frequency-flat andfrequency-selective cases, at a Rician channel Js.’4 = 5 dB. As expected for this type of channel,the performance improvement gained by increasing the receiver complexity from z = 2 to 3 isquite smaller than for the Rayleigh fading case. Decreasing the K-factor from 5 dB down to 0dB. as illustrated in Fig. 3.12, has quite an impact on the frequency-flat receiver by raising itsBER performance error floor a little over an order of magnitude. The optimal receiver, on theother hand, suffers a loss of slightly less than 6 dB at a BER level of i0, but still presents noobservable error floor above i0”.The performance of the derived optimal sequence estimation receiver is also compared toa receiver employing an decision feedback equalization method using the MSRK algorithmreported in [78]. As mentioned previously, this algorithm was chosen for its ultra fast convergence properties, as compared to other conventional weight-updating algorithms - a vital trait inthe time-varying fading channel [14]. Since the communication format employs differentiallyencoded signals, we have made the simplifying assumption that the equalizer operates on theTr/4-shift DQPSK signal space, rather than the space of differential phases (which is the QPSKr1CDZCDII c-rrIk)>CDIIoCDCDCD-4-0CD_ 0I o4-CDoCD0C•0oCDQCDo-tCDCD‘-.--•CDCDoC’,zC’)C’,.-‘•ZCDCDCd—•---•-.QCD.QC CDCriiCDZCDCDHoIICC-•‘CoC‘ CD’o- CDII—C CD-0-00’)CD0-CDICD.04-0O.CD0-I1CDoCDC’).-.zIIc.•CStjC)0CDoCD‘-I‘)1•‘II0•H0 cc)CD CD CD Cl)0•CBitErrorRate(BER)0•-4-HF-H—-+1—--I-14I-I-I4—--I-HH4---——--4-44I4I4—--4-HI-H‘1CI,—,•1t/BitErrorRate(BER)——,r,r11?IrrrTlrPJ-rTTr1D—-++4H*—-Q--+F1i--11Et1llll1-1EIIIll1E[Rll1—u-+I4’,4+—--F-1-414-I-fl—-M-H-f10!IIiiIIiC,) z 0 EU1I1_mOCl)l00ID-.(DO,--?<?<=:flEEaffiEID—--l-IiIl-ll—1N-I-I-I-I-1—-—-—--l-H11141—-N-l4I41——-:E 1HE—-——-:t’m—-H41-I-ll—-1-14141-----fFffIl——-rTrIirrnii—-1’fH—RH—-“fl’IUNi‘1Ill:HI[BHI111Et0)IIIIUOptimal Sequence Estimation for Fast Frequency Selective Rician Fading 7710010’LUma)(at 100LU10100 5 10 15 20 25 30 35 40 45 50 55 60SNR (dB)Figure 3.13: BER performance comparison of the optimal receiver derived versus a receiveremploying MSRK equalization, both employing a rate- 1/2 trellis coded Butterworth filtered(fB = 1 /T) ir/4-shift DQPSK scheme, in the presence of a 2-ray Rician fading channel withBFT = 0.001 and 0.005, K0 = 0 dB, K1 .‘ — dB and = 1.0.constellation). This assumption, implying knowledge of the transmitter initial phase at thereceiver, shall under no circumstance make the MSRK equalizer-based receiver performanceworse than it would be, if a suitable equalization method was found to operate on the outputof a single-symbol differential detector. Such a modified equalizer is beyond the scope ofthe performance comparison sought here, since employing a differential detector before theequalizer input introduces nonlinearity in the channel [100], invalidating the assumptions onwhich linear-channel equalization is based. The MSRK equalizer output is processed by adifferential detector, and the information-bearing phase shifts are then fed to a Viterbi decoder.Experimentation with this type of receiver, in slow and fast frequency selective Rician fadingchannels, while optimizing the equalizer parameters [78] by trial and error, demonstrated thatreasonable operation is possible in fading channels with BFT products < 0.001. As the fadingspeed approaches BFT = 0.00 1, and beyond that value, the irreducible error floor observed isOptimal Sequence Estimation for Fast Frequency Selective Rician Fading 78unacceptably high. Fig. 3.13 illustrates BER performance results comparing the optimal receiver derived to the MSRK equalizer-based receiver. Clearly the optimal sequence estimationreceiver has superior performance for BFT values greater than 0.001, even though large valuesof training overhead (50% for BFT = 0.001, and 100% for BFT = 0.005) are required forsecuring the MSRK performance illustrated in Fig. 3.13.3.5 CONCLUSIONWe have presented the derivation of an optimal, in the maximum likelihood detection sense,sequence estimation receiver for coded digital signals, transmitted over frequency selectiveRician fading and additive white Gaussian noise (AWGN) channels. Computer simulationresults of several reduced complexity versions of the optimal detection algorithm, for ir/4-shiftDQPSK signals under varying channel conditions, have demonstrated the merit of using the derived receiver structures. Sensitivity evaluation employing various degrees of implementationcomplexity has also shown the optimal receiver derived to suffer relatively small performancedegradation, while enduring appreciable errors in channel condition estimates, an importantquality when considering practical system implementation. Comparison of performance withlinear-channel equalization-based receiver structures specifically tailored to the fading dispersive channel, has demonstrated the merit of using the MLSE receiver derived in cases wherethe fading BFT product is greater than or equal to 0.001.CHAPTER 4PERFORMANCE ANALYSIS OF OPTIMALNONCOHERENT DETECTION FOR SLOWCORRELATED FREQUENCY NON-SELECTIVE RICIANFADING WITH DIVERSITY1In Chapter 2 we presented and evaluated the optimal multiple differential detector using diversity reception for fast Rician frequency non-selective fading. When fading is slow, i.e.,assuming a transmission rate of two to three orders of magnitude higher than that assumedin Chapter 2, with vehicle speed and operating frequency remaining the same, considerablesimplification to the receiver structure is possible. In such case it is straight-forward to verifythat the values of the fading autocorrelation function RF(kT) for symbols close to the samplinginstant, i.e., for small values of k, are approximately equal to 1. Then, the prediction coefficients Pk,rn can be all set equal to 1. The resulting simplified receiver structure alsoarises from the derivation of the maximum likelihood receiver under slow fading conditionspresented in this chapter. Both the optimal receiver decoding metric for varying degrees ofchannel state information, and a performance bound for the case of no channel state information are presented. Although designed with slow fading in mind, computer simulation andanalytical performance bound evaluation demonstrate the merits of using diversity reception,‘The research methodology for obtaining the analytical bounds reported in this chapter, has been published inpart, as a full paper in the IEEE Journal on Selected Areas in Communications [1011.79Performance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 80and especially how its use offsets the inaccuracies inherent to using such receivers in moderateand even fast fading conditions. The BER performance is also investigated in shadowing EHFchannel environments, where the small wavelength makes a higher order of diversity muchmore practical than in UHF frequencies. The rest of this chapter is organized as follows. Section 4.1 presents the decoder metric derivation, and Section 4.2 the the analytical performancebound for the case of no channel state information. Monte-Carlo simulation results togetherwith computer-aided evaluation of the performance bound are presented in Section 4.3, withthe conclusions appearing in Section 4.4.4.1 DERIVATION OF THE MAXIMUM LIKELIHOODSEQUENTIAL DECODER FOR SLOW FADINGIn Chapter 2 we pointed out that the modulator initial phase ii has been proven of relativelysmall importance as far as the performance of the derived MDD receiver is concerned. At leasttheoretically, however, such a receiver must estimate . In this chapter, in order to furthersimplify the receiver structure, we assume that the receiver has no knowledge of ii’. This isequivalent to assuming that i follows a uniform distribution in the interval [0, 2ir). For thepurpose of the analysis presented here, we consider the signal received on the lth diversitychannel to be as in Eq. (2.15), which is repeated here for the reader’s convenience.x(t) = f(t)x(t) + nQ) 1 i < A (4.56)whereft(t) = f’(t) +jfQ’(t) is the frequency non-selective Rician fading distortion and n(t)the AWGN with two-sided power spectral densityN0/2. Using Eq. (2.8), the above equationcan be rewritten asx(t) e(t)e’x(t) +nt(i)= (t) exp [j(t)] exp [j(2ft)1 ckhT(t — kT) + n(t) (4.57)Performance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 81where -fQ;z(t)(t) = (t) = tan1 [fI(t)] (1) = (t) + (4.58)with denoting the modulator initial phase, and Z the number of symbols in the transmittedsequence of ck’S. Clearly, )(t) represents the interference in the envelope, and /4(t) theinterference in the phase of the received signal. b(t) denotes the combined effect of the twophases, ib(t) and ,j. Since we consider a “totally noncoherent receiver, ‘/‘(t) is unknown tothe receiver, and hence assumed to be uniformly distributed in [0, 27r). We also make the usualassumption of uncorrelated diversity, i.e., j’(t), ib(t) are uncorrelated, 1 < i, 1 < A, i 1.We shall consider the following three cases as representative of the receiver’s knowledgeof the channel state1. The receiver has knowledge of both the envelope (t) and phase J4(t) of the fadingprocess V i, I i A.2. The receiver has knowledge only of the envelope }t).3. The receiver has no knowledge of either the envelope }(t) or the phase b(t).For the first and second case, it is assumed that the fading is slow enough, i.e., not changingconsiderably over the transmission period, so that a suitable subsystem can be used to track thephase and/or amplitude variation with a reasonable degree of success.For the first case, it is shown in Appendix C. 1 that the optimal maximum likelihood sequenceestimator must maximize the following metric expression‘0where is the estimate of the signal strength on theth diversity channel, and y is the outputof a Nyquist filter with input x(t) after it has been downconverted by exp [—j(2irft)].Performance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 82For the second and third case, the optimal decoding metric derivation which is almostidentical to that used for the first case, is outlined in Appendix C.2. The resulting decodingmetric for the second case isA Z_likT i*fl1 [ k=0 YkCk (4.60)and for the third case it is given by. (4.61)There are implications for the implementation of the receiver structure as expressed by Eqs.(4.59), (4.60) and (4.61), which bring about the necessity of performing simplifications. Theproblem is the complexity involved in implementing the Bessel function Io(). However, byusing an approximation for Jo(S), and also distinguishing low SNR and high SNR channel operating conditions, we are able to arrive to the following two, reduced complexity asymptoticallyoptimal2forms of the decoding metrics.4.1.1 Low SNR channelFor low SNR (e.g., <5 dB), the Bessel function Io(x) can be approximated as follows [33]Io(x) = i + + + 224262 = 2(i!)(4.62)Use of Eq. (4.62) with Eq. (4.60), elimination of all terms which are independent from theconsidered sequence 0(A) and truncation of all the terms where 1/N0 appears with powerhigher than 2, leads to the following metric expressionA z—i 2= ((kT)) uici(4.63)2For the high SNR case, “asymptotically optimal” implies that as the noise power after the receiver filterapproaches zero, the performance of the simplified metric form becomes identical to that of the original metricbefore simplification. For low SNR, this convergence is achieved by using higher accuracy approximations ofIo() for increasing noise power.Performance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 83where y = [yb, y,• —J. Following a procedure very similar to that used in AppendixA.2, eliminating terms which are independent from the considered sequence C(A), leads to thefollowing metric expression[ (kTdf(k) cos [9i(k)1 + ((kT))2d?(k) sin [Zei(k)J] (4.64)k=1 11 i=1 i=Iwhered1,(k) = y(y_j)*, df(k) = Re{di,(k)}, d?(k) = Im{di,(k)}, z81(k) = arg(4.65)The termd1,(k) is the product between the signal sample yk and the complex conjugate ofthe Yk—1 sample, that is, a signal sample received iT seconds before. This is the outputof a differential detector with delay element equal to iT. From Eq. (4.64) it is apparentthat the maximum likelihood sequence estimation receiver processes the output of a multipledifferential detector structure with a total of Z — 1 differential detectors. In this respect, muchlike in the case for fast fading in Chapter 2, this receiver employs a MDD hardware structure forprocessing the signal samples within an observation window of Z symbols in length. However,in contrast to the algorithm for fast fading, we do not have the added complexity of calculatingprediction coefficients, or estimating the initial phase i of the transmitter modulator. This greatreduction in algorithmic complexity does not come without a price, as will become evident inthe performance evaluation results presented in Section 4.3.4.1.2 High SNR channelFor high SNR, the following approximation for the 1(x) will be used [33)Io(x exp(jxj) x>3. (4.66)Performance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 84In Appendix C.2 it is shown that use of the above approximation in the metric expression ofEq. (4.60) leads to the following asymptotically optimal decoding metricA Z-1‘NB(Y’,Y,. . .= kT)yc . (4.67)i=1 k=OEqs. (4.63) and (4.67) indicate that a square envelope structure is asymptotically optimal forlow SNR, while an envelope structure for high SNR.4.2 ANALYTICAL PERFORMANCE BOUNDFrom the cases of channel state information examined in the previous section, undoubtedly themost simple and, at the same time, the most desirable in a practical system implementation isthat whereby no knowledge of channel state information is available at the receiver. Becauseof this, our effort was concentrated on deriving an analytical BER performance bound for thisparticular case. For mathematical convenience, the type of modulation format assumed in thederivation is single level, accommodating, for example, ir/4-shift DQPSK, or 8-DPSK.The bound calculation used in Section 4.3 employs averaging of the union bound [141 forall possible signal sequences of length Z. In calculating this union bound for every possiblesequence, we need an expression of the pairwise error event probability, for any two sequencesfrom the total number of sequences having length Z symbols. The derivation of this pairwiseerror event probability PA ((A) — G(A’’)) for a single level modulation format in a Ricianfading channel with no channel state information available at the receiver is presented inAppendix C.3, the final result of which is the followingFor ak,,(] > 0,PA(4o) () = Qf ±) (4.68)o_fl o•fl—‘ {——-a[,(]——-/3[v,]] expPerformance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 85(1 +V)A(1 —D) A-i (2\ —1(1 +V)’(1 —V)’ -x221_1 A+l Jj ++exp (— 2c jXm=0A+m—l A+m—1 / 2A — I 1{ [_1 1(1 +D1 —V)’I22 1 A+l—m.) j1 (1 + V)A+m_l(1 — D)A_m A+m-1 ( 2A — 1 + D)’(l — V)1] }— [1_ 2A-1 1=0 A+1—m jx [1 —6(A— 1)].In the above equation, 6(o) is the Kronecker 6-function, Qf(x, y) is Marcum’s Q function [32,p. 585] and { c[v,C] A . 2 2 — 2Re { (E) * }+( E + E1 — 17vd2— ‘/2+ 2(4.69)V1-7d jjwith2vYv( = (v +vvv — vv( (4.70)2’+ v)2 —4Performance of Optimal Non coherent Detection in Slow Flat Fading with Diversity 86and— 2 _, 2 _, * _, _, 2 — 2E —2Re{(E) Ey} EVI — \/1 — 17vd2 4712- 2Re { (_v)* _v}_2 - (• )1 — — 1 — I7vdFor = 0,PA (() CMv)) exp [_ci] [‘ 2-’(2A_1 )]. (4.72)Denoting byFZ (v) = y(c)* (4.73)the argument of I(.) in Eq. (4.61), with Ck’S taking values according to the informationsequence AX, then , is the conditional expected value of (, C(Av)) given that At’was the transmitted sequence, and , is the conditional expected value of (w, .(:4())given that was the transmitted sequence. V[t’(] and Vf[] are the conditional variancesfor (, (At’)) and (vi, G(A’)) respectively, while vjj is their conditional crosscovariance. Detailed expressions for all above moments are given in Appendix C.3.In the derivation of the pairwise error event probability in Appendix C.3, in order to providesimplified expressions of Eqs. (4.71) and (4.72), we have distinguished two cases of commonlyencountered fading channel conditions. These are• Rician fading with strong direct signal component at high SNR, and• Rayleigh fading, i.e., no LOS signal component.In the following two paragraphs, we present the final results for the pairwise error eventprobability for these two special cases.Performance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 87In a Rician fading channel with strong direct signal component, i.e., when K 5 dB, theexpression of Eq. (4.68) can be tightly upper bounded by the following expression(+P1 ({M) — (Av)}) exp— 2 )(4.74)For > 0,Z,A 1=_________________x { i — exp 2A-i A + 1F,v,<] (1 + D)A(l — D)A_i f (2A — i ) (1 + D)’(l —aLvJ[v(A—iIv,C][,(1+exp[ 2n Jm=0 \I (l+D)’(1 —V)’I[1 — (1+ V)A_m(1 — V)A+m_I A+m-1 2A — 12211=0 ( A + 1 — m ) J— I(l÷V)1(l_D)1i[l(l+V)A+m_l(1 _V)A_+m1 2A—l ]}22A—1 1=0 (A+l_m)x {l_s(A_l)]}. (4.75)For a[,(] = 0, the expression is identical to the one in Eq. (4.72).For a Rayleigh fading channel, i.e., K —* —co, the pairwise error probability is reduced to({(() (V)}) (1— V)A(l — V) { (2A — (1 +V)(1 — D)1}.221 1=0 A+l )(4.76)Performance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 884.3 BER PERFORMANCE EVALUATION RESULTS ANDDISCUSSIONThis section offers BER evaluation results for the multiple differential decoder structure developed in Section 4.1, in the case when no channel state information is available to the receiver.There is no doubt that this is the most attractive case in terms of practical implementation, sinceno special subsystem is needed for providing an estimate of the amplitude and/or the phase ofthe faded signal. Furthermore, since it is our intention to investigate the receiver behaviour infast fading channels, although the metric design is based on slow fading, assuming availabilityof an accurate signal amplitude and/or phase estimate is not an option. In Section 4.3.1 wepresent results for Rayleigh and Rician fading channels, at various fading rates, degrees ofreceiver structure complexity and diversity order. BER performance obtained via Monte-Carlodigital simulation is compared to the analytical results using the bound described in Section4.2. In Section 4.3.2, the multiple-differential detector structure is also evaluated in an EHFKa-band channel, exhibiting a LOS signal component assumed to be a random variable following a log-normal distribution. Motivation for providing these results was provided by thefact that the small wavelength for carrier frequencies around 20 GHz (\ 1.5 cm), makesimplementation of antenna diversity much easier than at UHF frequencies.4.3.1 UHF Land Mobile Fading ChannelComputer simulation BER evaluationUsing the decoder metric derived for the case where no channel state information is availableat the receiver (see Eq. (4.61)), Monte-Carlo error counting techniques were used to evaluatethe BER performance in a Rayleigh fading channel, using sequence lengths Z of 2, 3 and 4,without diversity (A 1), and with diversity A 2 and 3. Note that Z = 2 corresponds toPerformance of Optimal Non coherent Detection in Slow Flat Fading with Diversity 8910010’‘IuJa)t 100w10_a10Figure 4.1: BER performance of rate- 1/2 trellis coded Nyquist filtered (c = 0.35) Tr/4-shiftDQPSK scheme, in Rayleigh fading with BFT = 0.005, for diversity A 1, 2 and 3 andsequence length Z = 2, 3 and 4.conventional differential detection. The fading BFT product was varied from a relatively slowfading rate of 0.005, to the very fast fading value of 0.1. Results for BFT = 0.005 are depictedin Fig. 4.1. Evident in this figure is the merit of employing diversity, as opposed to using largevalues of Z for reducing the BER at a specific SNR. The gain when going from no diversityto diversity A = 2, employing a sequence length Z = 4 at a BER level of i03, is almost 13dB. Increasing the order of diversity by one to A = 3 yields an additional 5 dB. Increasing thefading rate by a factor of two, to BFT = 0.01 has negligible effect on the trends observed inFig. 4.1. The results presented in Fig. 4.2 reveal approximately the same amount of gain whenintroducing diversity of A = 2 as compared to the no diversity case, at the same BER level of10. The gain available from increasing A to 3 is again approximately equal to 5 dB. In boththe BFT 0.005 case of Fig. 4.1 and the BFT = 0.01 case of Fig. 4.2, when no diversity isemployed, it is interesting to note that the incremental gain obtained when increasing Z from2 (simple differential detector) to 3, is considerably more than that obtained by increasing theSNR (dB)Performance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 90sequence length further, to Z = 4.10010LU-2a 100UJ-31010SNR (dB)Figure 4.2: BER performance of rate-l/2 trellis coded Nyquist filtered (a = 0.35) ir/4-shiftDQPSK scheme, in Rayleigh fading with BFT = 0.01, for diversity A = 1, 2 and 3 andsequence length Z = 2, 3 and 4.The problems caused by inaccuracies inherent to employing this type of decoder metricin channels with high fading rate are depicted in Figs. 4.3 and 4.4. In Fig. 4.3, in the caseof no diversity, i.e., A = 1, and for a fading BFT = 0.05 we observe an intuitively expectedperformance penalty when the sequence length used is increased to Z 4. The very interesting point to note in this figure is that, employing diversity not only continues to secure theperformance gain observed in the previous two cases, but also seems to offset the effect of theinaccuracies inherent to using the metric of Eq. (4.61) in such a high fading rate. Quantitatively,the inaccuracy introduced can be appreciated by comparing the solution of the linear system ofEq. (2.23), to setting all prediction coefficients equal to 1, as implied by Eq. (4.61) — the errorincreasing proportionally to an increase in the fading BFT product. Diversity A = 2 seems tocure part of the problem, with a BER performance curve cross-over for Z =3 and 4 still takingplace around an SNR of 12.5 dB, while employing A = 3 seems to offset the losses due to0BitErrorRate(BER)0•0,BitErrorRate(BER)Cd)CDtcCl)CD tTICDii-‘CDt1-‘.,-t CDHc’IICDcDCl)iXC’)CDN4-II 3CDtr1. .— . Jc - c2_Cci.CD) -1::tHCD — -IICd)pr,_0 0-•CDg o-Z‘.1<CI)Cl)ZZ13‘-a-a>- -II a t’3,-0-IlpuzCl) CDCDCd)CI)I ciPerformance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 92inaccuracy, at least down to the minimum observable BER level of iO—, with the Monte-Carloerror counting techniques used. Fig. 4.4, however, clearly illustrates how the decoder breaksdown, as the fading rate is increased further to the very fast fading value of BFT = 0.1. Asexpected, the greatest performance penalty occurs with Z = 4. Note, that as in the case ofBFT = 0.05, using diversity of A = 3 appears to offset the loss caused by the mismatch betweenthe fading statistics assumed by the receiver and those actually present in the channel, at leastfor Z = 3. The trend observed in Figs. 4.3 and 4.4 leads us to the conclusion that, if a high orderof diversity is practically feasible, there is much more merit in using large values of A and asimple differential detector for fast fading environments, than using no diversity or a diversityof 2, and a MDD based maximum likelihood receiver designed for slow fading, with Z > 3. If,on the other hand, diversity higher that A = 2 is not practical, the alternative for improving theperformance would be to employ the receiver structures derived for fast fading, as presentedin Chapter 2. Comparison of the performance in Fig. 2.10 with z = 2 and A = 2, to that ofFig. 4.4 with Z = 3 and A = 2 clearly ifiustrates this point. Both receivers use two differentialdetectors, and the one in Fig. 4.4 has the advantage of operating at 20% lower fading rate thanthat in Fig. 2.10. Nevertheless, the receiver designed for fast fading, performs about 3 dBbetter. This difference in performance will increase even further if z is increased to 3, and theBFT product for Fig. 4.4 is increased to 0.125.Analytical BER evaluationThe performance of the MDD receivers using the metric of Eq. (4.61) when no channel stateinformation is available was also investigated using the performance bound developed inSection 4.2. The computer aided calculation of this bound greatly reduces the time requiredto evaluate the receiver structures, as it is between 3 and 4 orders of magnitude faster thanthat required for the Monte-Carlo digital simulation. Furthermore, it enables us to predict thePerformance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 93LUa)cr0LUFigure 4.5: BER performance bound evaluation versus Monte-Carlo simulation results forrate- 1/2 trellis coded Nyquist filtered (o = 0.35) ir/4-shift DQPSK scheme, in Rayleigh fadingwith BFT = 0.005, for diversity A 1, 2 and 3 and sequence length Z 4.receiver performance at BER levels well below the practical limit of iO— in a typical computersimulation. Figs. 4.5 and 4.6 illustrate results obtained by averaging the union bound over allpossible sequences generated by the rate- 1/2 trellis-code used and associated error paths withinZ symbols. The probability for each error event is calculated by evaluating the expressionof Eq. (4.68) and (4.72). Results illustrated for Rayleigh fading with BFT 0.005 in Fig.4.5 and BFT = 0.01, in Fig. 4.6, together with Monte-Carlo simulation results, demonstratethe tightness of the analytical bounds derived. As intuitively expected, the irreducible errorrate (error-floor) due to the random FM is reduced by increasing the diversity order, while itincreases in a faster fading environment. Results for a Rician fading channel with BFT = 0.05are presented in Fig. 4.7, for a K-factor of 5 and 10 dB, using diversity A 2 and 3, and asequence length Z 3. The Monte-Carlo simulation results for each case illustrate once morethe tightness of the analytical bound. At a BER level of iO and K = 5 dB there is gain ofapproximately 4.5 dB when using diversity A = 3 as compared to A = 2. Increasing the valueSNR (dB)Performance of Optimal Non coherent Detection in Slow Flat Fading with Diversity 94LU10-aa)0-6LU 10-210 ;;;;;;— -—-— =‘- -S —Simulation DI A=310 Simulation 0A=2__Simulation 0— Al-0 I100 10 20 30Figure 4.6: BER performance bound evaluation versus Monte-Carlo simulation results forrate- 1/2 trellis coded Nyquist filtered (a = 0.35) ir/4-shift DQPSK scheme, in Rayleigh fadingwith BFT = 0.01, for diversity A 1, 2 and 3 and sequence length Z = 4.40SNR (dB)LUa>ca0LU10_7SNR (dB)Figure 4.7: BER performance bound evaluation and Monte-Carlo simulation results for rate- 1/2trellis coded Nyquist filtered (a = 0.35) ir/4-shift DQPSK scheme, in Rician fading withBFT = 0.05, for diversity A = 2 and 3, sequence length Z =4, and K = 5 and 10 dB.Performance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 95of K to 10 dB improves the BER performance for both diversity cases, moving the calculatederror-floor below 10— and, as expected, decreases the incremental gain between A = 2 and 3.It is interesting to note that the BER receiver performance for the case of A = 3 and K = 10dB is only slightly worse that of the rate- 1/2 4-state trellis coded QPSK in AWGN.4.3.2 EHF Mobile Satellite ChannelPropagation measurement campaigns have shown [34, 102] that the UHF (800-1000 MHz) andL-band (1.5 0Hz) mobile-satellite radio channel can be modeled as Rician fading with its localmean, the LOS component, following a lognormal statistical distribution. The fading appearingin the ii” diversity channel can be expressed asf(t) = p(t) exp {j(t)]= fs(t)exP [jg(t)] + .IFD(t)exp [iDtj (4.77)where p(t), L4(t) are the amplitude and phase of the fading process, respectively. qs(t) andcFD(t) are uniformly distributed over [0, 2ir). f() is a lognormally distributed stochasticprocess which represents the amplitude of the LOS signal, while (t) is a Rayleigh distributedstochastic process which represents the envelope of the multipath component. The pdf of thereceived fading envelope p(t) can be mathematically represented as [34]r 1 (in z — jj’)2 r2 + z2 “rz’\=b0/Joexp— 2d0 — 2b0‘° j—) dz (4.78)where b0 is the power of the multipath signal, and and are the mean and standarddeviation of the shadowing process. The pdf of the lognormally distributed LOS component isp(z)=exp[Onz_2](4.79)The parameters for the L-band mobile satellite channel [34, 351 are shown in Table 4.4. Thewavelength at the EHF Ka-band (20 0Hz) is approximately 1.5 cm. As this is comparablePerformance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 96Table 4.4: L-band mobile-satellite radio channel model parameters.L-band channellight shadowing 0.158 0.115 0.115average shadowing 0.126 -0.115 0.161heavy shadowing 0.0631 -3.91 0.806to even small obstructions in the signal path, such as tree leaves and small branches, wewould expect higher losses from shadowing in such frequencies. Since, to the best of ourknowledge, experimental data for the Ka-band is not presently available, we attempt to estimatean appropriate scaling factor for the model parameters, in order to account for this higher signaldegradation. Using a scaling technique described in detail in Appendix C.4, we modify p andd0 according to the following equationsx 0.45F°284x 0.45F°84 (4.80)with F in GHz, using b0 and i from Table 4.4. The model of the fading channel used inthe Monte-Carlo simulation is depicted in Fig. 4.8. The independent white noise processesn’(t), n(t) and n(t) are filtered by identical filters having the land-mobile fading characteristic. The bandwidth B for the real and imaginary part of the complex process for the diffusedcomponent is equal to the maximum Doppler frequency BF, while for the lognormally distributed LOS component, the bandwidth BD is about 1/20 of BF in UHF frequencies [26, 34].For comparison with the Rician fading case, the ratio of the LOS signal average power to themultipath power is derived in Appendix C.5 asexp(2,u+2do)kAyo = 10 log10 dB. (4.81)u0BER simulation results for an EHF channel with carrier frequency around 21 GHz, vehiclePerformance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 97Figure 4.8: Block diagram of the low-pass equivalent of the EHF log-normal fading model.speed of about 65 km/h and antenna pointing elevation of 2003 are depicted in Figs. 4.9,4.10and 4.11. The parameter scaling factor is approximately equal to 2.1, the resulting maximumDoppler frequency is about 1200 Hz, and the bandwidth of the lognormally distributed LOScomponent approximately equal to 43 Hz4. The BFT product for the diffused signal componentis then 0.05, given a data rate of 24 kBaud. The results for light shadowing conditions withKAyO 7.34 dB, sequence lengths of Z 2,3 and 4, and diversity A = 1, 2 and 3, areillustrated in Fig. 4.9. Similarly to our previous observation for the UHF fading channel, it isinteresting to note the large gain in performance by using diversity A = 2 as compared to theno diversity case (A 1). At a BER level of l0— and using sequence length Z = 4 this gain3M elevation angle of 200 makes the effective speed with respect to the satellite equal to 65km/h x cos(2 10)which is approximately 61 km/h.4The reason why Bs is more than 20 x B in this EHF band example is that, the two phenomena of LOSshadowing and Doppler, are independent. The LOS shadowing rate is obtained by scaling the UHF band valuesaccording to the increase in frequency, while the Doppler depends on the carrier frequency as well as the vehiclespeed.n’(t) HF&)Bn(t)s(t)n (t)Complex signal:Real signal: r(t)Performance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 98is approximately 10 dB, while at a BER level of iO— it increases to over 13 dB. When usingdiversity order A = 3 besides the additional gain available, we also note a greater incrementalgain when going from using Z = 2 to Z = 3 and 4, as compared to the case with A = 2. This isattributed to the fact that most of the distortion imposed by fading has been offset by the highdiversity order, which in turn decreases the penalty in performance inherent to employing thistype of receivers in fast fading channels. It is interesting to note that the receiver performancewith Z = 4 and A = 3 is only approximately 1 dB worse than the rate-1/2 trellis-coded QPSKperformance in AWGN.1001 0_iwci)-2100wSNR (dB)Figure 4.9: BER performance of rate-1/2 trellis coded Nyquist filtered (a = 0.35) 7r/4-shiftDQPSK, in EHF light shadowing conditions with BFT = 0.05, for diversity A = 1, 2 and 3and sequence length Z = 2, 3 and 4.Figs. 4.10 and 4.11 illustrate the results for average and heavy fading conditions, having‘(AVG = 4.36 dB, and -50.48 dB, respectively. It is interesting to note that even for heavyshadowing, the receivers perform reasonably well, if sufficient order of diversity is employed.The inaccuracies due to fast fading are all the more visible as we go from light, to averageand then to heavy shadowing. They are manifested as performance degradation for increasingCDI—0 CDtljCDrltTI<CD0-tCDk)CDI—-0—0vJjCDII’E.Ui_0‘--“CDII>2 Ui.0 CDrTItjT1—CDIIQCD000CD-.0c1-tCDII<o•CDCD.II>9“Ui(I, z 0 w0,BitErrorRate(BER)0,BitErrorRate(BER)0•Cl) z 0 wIPerformance of Optimal Noncoherent Detection in Slow Flat Fading with Diversity 100values of Z, and are more pronounced in higher SNR values, where the randomization effectof AWGN is negligible.4.4 CONCLUSIONWe have derived the optimal sequence estimator for slow frequency non-selective fadingchannels, and varying degrees of channel state information. The algorithmic structure of theestimator is a simplified version of that derived for frequency non-selective fading in Chapter2. For the case when no such channel state information is available, we have derived analyticalBER performance bounds for Rician and Rayleigh frequency non-selective fading channels.The very close agreement of Monte-Carlo computer simulation and computer aided boundcalculation have demonstrated the accuracy and tightness of these analytical bounds. Theresults obtained for Tr/4-shiftDQPSK demonstrate great merit in using diversity in conjunctionwith these simplified receiver structures, even in fast fading environments where considerableinaccuracies would normally prohibit their use. Furthermore, with practical systems in mind,we have used computer simulation to evaluate the performance offered by these receivers in theEHF shadowing channel. In this we were motivated by the fact that due to the very small wavelength in the 20 GHz and 30 GHz EHF bands, large orders of diversity are considerably easierto implement than in the UHF band. The results obtained demonstrate a great improvement inperformance, as we increase the diversity order from 2 to 3. In the case of light shadowing, theattainable BER approaches that of trellis coded coherent QPSK, employing the same code.CHAPTER 5CONCLUSIONS AND SUGGESTIONS FOR FUTUREWORK5.1 CONCLUSIONSIn this thesis we have addressed the general problem of noncoherent detection in fast, correlated, Rician fading channels. In our derivation of improved receivers, for this most prominentof distortions among those present in mobile communication environments, we have distinguished cases of frequency non-selective and frequency selective fading. For all cases we haveconsidered trellis coded PSK and QAM type signals, with particular emphasis on the emergingNorth American digital cellular standard ir/4-shift DQPSK, and the more spectrally efficient8-DPSK and ir/4-shift 8-DQAM modulation formats.For the frequency non-selective fading channel we have derived the optimal receiver, in themaximum likelihood detection sense, based on the MDD hardware structure, employing novelmaximum likelihood sequence estimation algorithms, using diversity reception. In the interestof system simplicity, we have proposed and evaluated reduced complexity versions of theoptimal receiver structures. Computer simulation results have demonstrated significant gains inperformance, as compared to conventional signal detection schemes. Taking into considerationthe inaccuracies inherent to practical system implementations, we have investigated cases wherethe receiver statistical knowledge of the fading channel is in considerable error with respect tothe actual channel state. Results show relatively small sensitivity, even to large inaccuracies in101Conclusions and Suggestions for Future Work 102estimates of channel parameters.For the frequency selective fading channel, we have derived the optimal receiver, in themaximum likelihood detection sense. Although the effect of signal filtering was omitted forthe purpose of reducing the mathematical complexity in the derivation, using digital simulationwe have investigated the effects of such filtering on the receiver performance. Furthermore,we have also presented computer simulation results, illustrating the relatively small sensitivityof the proposed algorithms to inaccuracies in the receiver estimates of channel parameters.The performance of the derived receiver structure has been compared to the fastest knownequalization technique for frequency selective fading, which employs the modified square rootKalman equalization algorithm, developed for use in HF dispersive channels. Results haveshown the proposed receiver to significantly outperform equalization in fast fading channelconditions.Finally, we have derived novel analytical BER performance bounds for simplified versionsof the optimal diversity receiver, for frequency non-selective, Rician fading. These simplifiedreceiver structures were derived for varying degrees of channel state information, and wereshown to be optimal in the case of slow fading channel conditions. The bound derivation,however, assumed the most general case, where no channel state information is available atthe receiver. Using digital simulation and computer aided calculation of the derived bounds,we have evaluated the performance of these receivers in slow as well as fast Rician fading,employing different fading rates, and various orders of reception diversity and receiver implementation complexity. The tightness and accuracy of the derived bounds was demonstrated bythe excellent agreement between calculated and computer simulation results. BER evaluationhas also indicated that, with increased diversity order, the performance gains are so great, thatin most cases they offset completely the inaccuracies inherent in using this type of receiversin fast fading conditions. Motivated by this result, and with practical implementation in mind,we have also evaluated these receiver structures in the Ka-band EHF shadowing channel.Conclusions and Suggestions for Future Work 103Computer simulation results, for both Rician and EHF shadowing conditions, have shown thereceiver performance approach the effectiveness of coherent detection in AWGN, even whenemploying a relatively small diversity order.5.2 SUGGESTIONS FOR FUTURE WORKIn the course of the work reported throughout this thesis, there have been instances in whichinteresting questions were raised; questions, which in our opinion, warrant further research. Inthe following subsections, we briefly present the most interesting of these topics.5.2.1 Suboptimal Receiver Algorithms of Reduced ComplexityBoth in Chapters 2 and 3, we pointed out the complexity involved in calculating the metric of theoptimal receiver structures derived. This complexity is a limiting factor, both when attemptingto evaluate such structures using computer simulation, and, perhaps more importantly, whenimplementing them in practical communication systems. Although the truncation method usedin both aforementioned chapters resulted in considerable receiver simplification, the remainingcomplexity is still a problem, especially when trying to accommodate signal constellations withlarge numbers of signals, and large orders of diversity. The problem is definitely even morepronounced in the case of frequency selective fading, than for frequency flat fading, due to thecorrelation between I and Q channels introduced by the delay spread. In light of all this, it isworthwhile investigating the possibility of further simplifying the receiver structure, hopefullyyielding forms of much reduced complexity, with only small sacrifice in receiver performance.Conclusions and Suggestions for Future 14’Tork 1045.2.2 Receiver Structures for Nyquist Pulses in Frequency Selective FadingIn Chapter 3 we pointed out that, although straightforward, it is nevertheless tedious to includethe effects of signal filtering in the derivation of the optimal receiver structure. For conveniencein the mathematical derivation, we adopted the use of unfiltered NRZ signals. Althoughthis approach provided considerable simplification, as mentioned in Section 3.4, the receiverperformance is expected to deteriorate as the pulse shape employed departs considerably fromthat of the assumed NRZ pulse. This is especially the case for Nyquist filtering employing smallvalues of the roll-off factor . Although modifying the derivation to include such effects is notexpected to change the general structure of the receiver, it will certainly be of great importance,considering the fact that it will eliminate the performance penalty otherwise observed in systemsusing this type of filtering.5.2.3 Frequency selective fading with > 1The normalized delay spread, in both the derivation and the evaluation in Chapter 3, wasassumed to be at most equal to 1. This assumption was based on the fact that such range ofvalues is typical for the digital cellular applications considered. However, since values ofgreater than 1 are possible in other digital communication applications, there certainly is meritin extending the derivation to include such cases.5.2.4 Extension of Derivation of Bounds for Multi-level SchemesAs mentioned in Chapter 4, the analytical bound derivation for the performance of the MDDreceivers in slow fading, assumed single level modulation formats such as 2r/4-shift DQPSK and8-DPSK. Since higher order modulation schemes employing more than single level signals aredesirable for their increased spectral efficiency, it would be very useful to extend the derivationin Chapter 4 to include multi-level signals. The results of such an undertaking would be usefulConclusions and Suggestions for Future Work 105in predicting the performance of the receivers presented in Chapter 4, when employing suchsignals in Rician frequency non-selective fading channels, without the limitations inherent toMonte-Carlo computer simulations.5.2.5 Performance Bounds for Fast FadingThe BER performance bounds derived in Chapter 4 are based on the simplified MDD receiversin the case of slow frequency non-selective fading. The derivation of such analytical boundsfor the case of fast frequency non-selective and also frequency selective fading is not at allobvious. However, derivation of equations bounding the performance of the derived receiverstructures in such channel conditions is quite desirable. Using the results of such a derivation, itwould be possible to predict their performance at BER levels well below the iO— level, whereMonte-Carlo computer simulation becomes impractical.5.2.6 Investigation of MDD Techniques in Spread Spectrum SystemsA preliminary investigation in using the MDD hardware structure with systems employingspread-spectrum transmission, has yielded receiver structures similar to those derived fordiversity reception. Nevertheless, considerably more research is warranted in this area, towardsevaluating such receiver structures in fading conditions typical to spread-spectrum systems, andalso deriving analytical bounds for their performance.5.2.7 VLSI Implementation of Detection Algorithms DerivedIn order to apply the findings of this thesis to receivers in practical systems, the algorithmsderived for optimal detection in frequency flat as well as frequency selective fading, with orwithout diversity, must be implemented in hardware. The type of hardware implementationwill certainly be affected by the processing speed requirement of the particular application.Conclusions and Suggestions for Future Work 106However, based on techno-economical criteria, the modem integrated circuit implementationin the form of a very large scale integration (VLSI) chip” is perhaps the most preferred methodof hardware realization. There are certainly many interesting issues in designing such a VLSIchip, ranging from techniques for parallelizing portions of the receiving algorithm to achievehigh processing speed, to assessing the impact of signal value quantization, and implementingtrade-offs between speed and chip physical area, to name just a few. It is without any doubt,though, that application of the receiving techniques presented throughout this thesis to a widerange of digital transmission systems, depend to a very large extent on successful hardwareimplementation of the associated algorithms.5.2.8 Neural Net Based Adaptive ReceiversAlthough the receiver structures derived and presented in this thesis were shown to sufferrelatively small performance degradation due to inaccuracies in estimates of channel parameters,there exist cases where it is desirable to obtain the best possible performance at any given time.To this end, it should be possible to derive neural net based receiver structures, employingthe detection algorithms derived, with the additional advantage of being able to track varyingchannel conditions. 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Benedicto, and M. Sforza, “Mobile satellite systems at Ku and Ka-band,”in the Proceedings of the Second European Conference on Satellite Communications,Liege, France, pp. 55—62, Oct. 1991.APPENDIX AA.1 Derivation of Eq. (2.22)The derivation of the joint pdf of Eq. (2.18) is based on the following probabilistic theorem[981.Theorem: For the zero mean Gaussian random variables Xm, Xm_l, Xm_2, ..., Xo with covariance R_ = E {xx}, the conditional pdf of Xm given Xm_1, Xm_2, ..., xo is given by:1 (x)2((XmIXin_I,Xrn_2,,XO) exp (A.l.l);/Tp2m 2owhere (.) represents the prediction of (.). is the mth order minimummean square predictionerror (MMSPE), with based on Xm_l, ..., x and given by:= Pin,jXm_j (A.1.2)with Pm,j being the linear prediction coefficients and m the order of prediction. The MMSPEcan be calculated from:= — Prn,jRj. (A.1.3)Since in Eq. (2.18) Yk is a complex random variable with independent Re{.} and Im{.}parts, this equation can be rewritten asACRe [Re(y)y1,y2,y,C(A),?1]C ,71] jjj1=1 k=1[Re yC (A) ](A.l.4)119APPENDIX A. 120where cRe and denote the pdf of [Re (L) yLi, yL2 . . , y,C (A) , ] and[Im (Yi) L15 Yk—2, , y, (A) , ,], respectively. But from Eq. (2.19), (Re is a scaledversion of the pdf of e’ and (Tm a scaled version of the pdf ofe1. Hence, instead of the pdf.of YL1,yL2, , (A) , 71] we can use the pdf of [e e_1,eL2, . . . , e, C (A) , 77].Note here, that in the general case e1 is not a zero mean Gaussian random variable sincef1; 0 in which case we use the random variable e’1 — fl;l instead. Using the above theoremand the fact that e11 and e are Gaussian random variables, maximization of the pdf in Eq.(2.18) is equivalent to maximizing the following pdfAZ 1( [ (A)]) 2 ( [ (A)])which by substituting the expressions for e, e of Eq. (2.19), and= Pk,m [ (A)] [ 1 2Re (y-m-m”) _i]m=1 Ck_m= Pk,rn [ (A)] [ 21m (A.l.6)m1 Ck_mI(A.l.5)yields the pdf in Eq. (2.22).APPENDIX A. 121A.2 Derivation of Eq. (2.25)Because of the exponential nature of Eq. (2.22), maximization of this pdf is equivalent tomaximizing the following function:k Re {yLmc_me} kPk,rn 2[ m=O Ck_m m=O1ZA2(A.2.1)2 k=1 1=1 (at)k Im {y_mc_me”}l22 I +[ m—O Ck_nI J -where we have defined PLo = —l V rn,1, 0 m Z, 0 1 A. The dependence of2and P,m on the transmitted sequence (A) will be dropped for the rest of the derivationfor clarity purposes. By expanding the squared terms inside the brackets of the above equationand dropping the 1/2 scaling factor, Eq. (A.2.1) becomesRe {y_m6_rn32?}(pm2÷2fp, 2\ m=O / n0 m=O I2k 2+(Pk,m) 2m=O ICk_miZA2. (A.2.2)k=1 1=1 (Uj)‘e{y_ (yL_m)*}{Ck_n_m}+k k im{y_ (y_m)*}Im{ck_n_m}+PflPm 2 2n=0m0 Ck_nI ICk_mim’nAPPENDIX A. 122Further simplification is possible if Eq. (A.2.2) is split into the following four distinct terms (T1,T2,T3 andT4), the sum of which must be maximizedRe {yLm(i)*}Re {CkmC_n} +Z A k k Tm {v_m(j_)*}Tm {Ck_mC_n}T1/ 2 Pk,mPk,n 2 2k=1 1=1 (oL) m=On=O Ckm Ck_n\ / \..__.,___dmn2T2 = 2 ()2Yk-m (A.2.3)1=1 (Uk) m=OCkrnZ A 1 k k Ref’ c_j?lT3 = 2f’ PLmk-n k—nk1 1=1 (uj) m’O n=O kk—nZA k2T4 = 2 PLmk=1 l1 (°-O mOThe term T4 is independent of the transmitted sequence and hence can be removed from themaximization process. By writing out in more detail the sum in term T1, it is not difficult tosee that the individual terms can be grouped together in a different way, namely, with respectto combinations of the form [yi (yLm)*]•In the same fashion, we group the terms in T2 withrespect to combinations of the form / I c. and in T3 with respect to combinations of theform Re {yce3} / ICkI. The resulting three terms, are the three terms in the sum of Eq.(2.25).APPENDIX BB.1 Derivation of Eqs. (3.40), (3.41) and (3.42)The three fading functions in Eq. (3.39) can be regarded as the outputs from three correspondinglinear filters with the following impulse responsesI 1/T if O<t<ThA(t) = —I. 0 otherwiseI l/T if (l—)T<t<ThB(t) —( 0 otherwiseI l/T if 0<t<(l—)Thc(t) = — (B.l.l)( 0 otherwise.The input to hA(t) is f°(), and the input to hB(t) and hc(t) is f’(t). The autocorrelationfunctions of the output random processes can then be calculated byRAQi) = E [fA(t)fa(t + \)j = R°Q) 0 hAQ\) ® hA(X)= E[fB(t)fB(t+ \)] = R’(\) 0 hB(A) ®= E [fc(t)fc(t + \)] = R1 ()) 0 h(\) 0 h(—\)(B.l.2)where 0 denotes convolution. By substituting the expressions for hA(t), hB(t) and hc(t) in theabove equations and using the change of variable a = b/T where b is the integration variablefor the convolution integrals, Eqs. (3.40) are easily obtained.123APPENDIX B. 124For the cross-correlations between fB(t) and fc(t) we use= E [fB(t)fc(t + ))] = R’(\) 0 hB(—\) 0f0(A)RCB(A) = E [fc(t)fB(t + ))j = R)) 0h0(—)) 0 (B.l.3)Substituting the impulse responses, distinguishing cases for (0 < 0.5) and (0.5 <1.0), and performing the same change of variable as in the case of the calculation for theautocorrelations, one can arrive easily at Eq. (3.41) and (3.42).APPENDIX B.125B.2 Derivation of Eqs.(3.46) and (3.47)Using the functions forthe real and imaginary partsof e(k) as they are given in Eqs. (3.44)and (3.45), making useof their auto- and cross-correlation functions in Eqs. (3.40),(3.41) and(3.42), and eliminating non-zero terms, yieldsE [e(k)e’(k — 1)] =E [fi;AofJ;Ao + fIC1fC1 +Ck.1Re{cck..l } J;B,I J;C,I Re{cck_l }Re{c_ckJl } I;B,1 I;B,12 fk fk_1+ 2 2fk fk_1CkIcki Ck_1IIm{cck_1 }Im{c_lckI1 } Q;B,1 Q;B,12 2fk fk_l+CkI Ck_lIE [ri(k)n(k — 1)]= RA(l) + Rc(l)+ Re{cck1ck_1c_l_RB(l)Ck Ck_1IRe {cck_1 } Re{c_1ck_l_l I+ 2RCB(l) + 2RBC(l)+E [n(k)n(k — 1)](B.2,l)with E [ne’(k)ne’(k — 1)] calculated asE [n(k)n(k 1)] =Enn Re{c&”}Re{c_ } + . (R.2.2)I CJCj CklISince E [nLni_i] = E= c76K(l) with SK(•) denoting the Kronecker 6-function, withthe aid of the mathematical identity Re(A)Re(B)+Im(A)Im(B) = Re(A*B) the above equationbecomesRe {ckc_i}E [nknk — = o 6K(l) E 2 2kkI ICk..41= —6K(l).(B.2.3)APPENDIX B.126Inserting Eq. (B.2.3) in Eq.(B.2.l), and following an identical procedure for derivingE [eQ(k)eQ(k —1)], we obtain the expressions forR11(k, 1) andRQQ(k, 1) givenin Eq. (3.46).The non-zero terms of the cross-correlation function E [e’(k)eQ(k — 1)] yieldE [e’(k)eQ(k —1)] = E[Im{_1ck2_1_1}fk1;C fk11+Ck_l IRe{cckl } Im{c_lck_l_1 } I;B,1 I;B,122 fk .Ik_1 +ICkI Ck..41Im{cck_1} Q;B,1 rQ;C,1—2 JkJk—1+Ck IRe{c_1ck_j_l } Im{cc } Q;B,1 Q;B,1—22 fkfk_1+Ck_II ICkIE [(k)n(k —1)]Tm {ckc_i c_ick_l1}Tm {c_ick_l_1 }2 2B +2BCI Ck ICk_1IICklITm {cZck_1 }—2RCB(l)÷E [fle’(k)(k— 1)]with E [nkn€Qk — 1)] calculated asE[n’(k)n(k—l)] =E[ _1Re}Tm _1i}2 . (BCkI ICklIFurther, by using the mathematical identity Re(A)Im(B) — Tm(A)Re(B)= Tm(A*B) the akz———__.equation becomesTm {ckcj}E [(k)n(k —1)] = o6K(l)E Ick2 Ick_1I=(lEEInserting Eq.(B.2.6) in Eq.(B.2.4), and following an identical procedure forE {eQ(k)e’(k —1)], we obtain the expressions for RJQ(k, 1) andRQJ(k, 1) givenin Eq.. c.APPENDIX CC.1 Derivation of Eq. (4.59)Assuming the receiver has knowledge of (t) and /-(t), V i, 1 < i < A, the maximumlikelihood sequence estimation receiver must maximize the pdf of x(t) x(t) conditionedOfl XB [(A),t], , . . ,, i.e., the functionf (x(t),. . . , x(t) XB [() ] ,. . . , , (C.l.l)= KCII1 exp LU 4(t) — xB (M, t] exp {i(2ft + + (t)+ )}2dt}.In the above equation, K is a normalizing constant and tL, tu represent the limits of theintegration. Their values depend on the spreading of the signal XB [(A), t] in the timedomain. Considering a bandlimited channel, in which case hT(t) extends from —00 to cc, wehave tL —* —cc and t —* cc. Furthermore, since i has been assumed uniformly distributedover [0, 27r), the above equation can be rewritten asf (x(i),. . . , x(t) XB [0(A), t] , ,. . . , ,. . . ,=jf (x(t) ,x(t)xB [(A),t] di 1 °° 2= fl Kexp [—- j 4(t) di] (C.l.2)x exp {-- L (,(t))2 XB [‘cci), t]2di]ri ‘COexp L J [x(t)(t)x (0(A), t] exp (—j2irfi — j(t) - +2iV0 -127APPENDIX C. 128[xt)]*tXB tj exp (i2ft + j(t) + i) j dt] } d.Note that, as the first two exponentials in the integral are independent of ,j and the third onecan be transformed by using the identity [32]1 r2irIo(2IE) — I exp [Eexp(_jl4)+E*exp(jb)j d’i/’ (C.1.3)27r JOwhere is an arbitrary complex number, Eq. (C. 1.2) is simplified tof (xt x(t) jXB [(A), tj , . , . . . ,1 A= Kexp [-—J x(t)2d]2N0 —A ((t)) 2xexp2Nf XB{C(A),t] dt (C.1.4)Ax fl1 J t)x(t)x [(A), t] exp (—j2irft — j(t)) dt . (C.1.5)i=1 —cx’In the above equation, the first exponential term is common for all G(A) and hence can beremoved from the maximization process. Therefore, the expression to be maximized furthersimplifies toexp— )2j XB [(A), t]2dt] (C.1.6)xio {ñ [ J (t)x(t)x [(A),t] exp (-j2ft - j(t)) dt] } .(C.1.7)The integral inside the above exponential term can be expanded by using Eq. (2.8) as00 2 00z-1J XB [C), t] d = f ckhT(t — kT) cTh — iT)—00 —cx k=0 1=0z-1z-1 00= CkC1 J hT(t — kT)h(t — 1T)dtk=0 1=0APPENDIX C. 129z-1 z—1= CCi hi_kk=0 1=0z-1z_1 z—1= +C1j_ (h_k)*j + h0 Ck2 (C.l.8)k=1 i=k k=0where= h(1T) hT(1T) 0 h(—lT)= f hT(t)h(t — 1T)dt. (C.l.9)Note that the function h() represents the total impulse response of the concatenation of hT(t)with its matched version h(—t), which is the receiver filter assumed for every diversity channel.When HT(f) is a square root Nyquist I filter, i.e., H1(f) = /Hcj where H(f) is a Nyquistfilter with impulse response h(t), thenh(t) = hT(t) 0 h(t) = h(t) (C.l.10)and thereforeh0>0 for 1=0= h(1T) = (C. 1.11)0 elsewhere.For this Nyquist channel, since there is no ISI, the first sum of Eq. (C.l.8) is 0. For singlelevel schemes (e.g., PSK type signals), the second sum of Eq. (C.l.8) is independent of thetransmitted sequence C(A), and hence can be removed from the metric maximization process.Now returning back to Eq. (C.l.7), the integral inside the Io{} term can be rewritten asL x(t)(t)x [C), t] exp (—j2irft — j(t)) dE00Z1= J x(t)(i) exp (_j(t)) ch(t — kT) exp (—j2Trft) dt—00 k=0z_1 fc’O= c J x(t)(t) exp (—j(t)) x(t) exp (—j27rf) h(t — kT)dt.k=O —00Given the fact the the fading process f)(t) occupies a considerably smaller bandwidth asAPPENDIX C. 130compared to the receiver matched filter h(—t), the above equation is transformed to --00cR(kT) exp (_j4(kT)) j x(t) exp (—j2irf0t)h(t — kT)dt= c(kT)exp (_j(kT)) y (C.l.13)with= y(kT)y’(t)= f x(r)exp(—j2f0)h(r — kT)drz-1= cj J (t) exp (j(t) + i) hT(T — lT)h(r — t)dr +nt(t)—00= e c1(iT) exp (j(lT)) h(t — iT) + nh(t)and= j n) exp (—j2irf0r)h(r — kT)drn = n(kT). (C.1.14)Note that the samples y(t) can be derived, for example, by downconverting x.(t) using a localoscillator locked in frequency with the incoming signal’ , and then passing it through a thereceiver filter filter HR(f), which is matched to HT(f). Finally, by straightforward substitutionof Eqs. (C.1.12), (C.l.13) and (C.1.14) into Eq. (C.1.7), the metric to be maximized, as givenin Eq. (4.59), is easily obtained.‘By “locked in frequency” we imply a local oscillator generating the waveform exp(—j2irfi). During thisdownconversion, the rapid phase and amplitude changes caused by fading, appear at the output of the demodulatortotally uncompensated.APPENDIX C. 131C.2 Derivation of Eqs. (4.60) and (4.61)When estimates ?t) for the )t) of the the signal strength are available, the maximum likelihood sequence estimation receiver must maximize the pdf f (x(t),.. . , x(t) XB [(A), t],This function can be calculated from Eq. (C. 1.1), by integrating with respectto all J(t) = + /(t). This integration is over the interval [0, 2ir) where i(t) is uniformlydistributed. Also, since J(t), 4(t) (i ‘ 1) have also been assumed to be independent randomprocesses, then t’(t), k(t) will also be independent. This yieldsf (x(t),. . . ,x(t) jXB [(A), t] , . . . (C.2.1)= ñ 1 f2 (x(t),. . . , x(t) XB [M) t] , b,.. . , (t)) d(t)Using a procedure identical to the one followed in Appendix C. 1, eliminating all the terms inEq. (C.2.l) which are independent from the transmitted sequence C(A), results to Eq. (4.60).In the case where no estimates for the fading interference envelope (t) or the total randomphase shift J’(t) is available at the receiver, the maximum likelihood sequence estimationreceiver must maximize the pdf f (x(t),. . . , x(t) XB ((A), t]). Since we have assumedto be uniformly distributed within [0, 27r), the pdf to be maximized can be rewritten asf (x(t). . . . ,x) XB [(A), t].) (C.2.2)A 1 2ir= fl- j f (x(t),. . . , x(t) XB [(), t] , ) di7= { A K exp [- if dt] (C.2.3)x exp if XB [CA,tj dt]x exp {-- f {x)x {(A), t] exp (—j2irft — ji) +[x(i)j*xB [(A),t] exp(j27rfi+j11)] d] d,7.APPENDIX C. 132Using a procedure identical to the one followed in Appendix C. 1, eliminating all the terms inEq. (C.2.2) which are independent from the transmitted sequence 0(A), results to Eq. (4.61).APPENDIX C. 133C.3 Derivation of Eq. (4.68)Let us assume that A” is the transmitted information sequence, and that the signal sequenceC(A”) corresponding to A”, is compared to the signal sequence C(A), corresponding tobeing one of the remaining possible information sequences of length Z. Since wehave assumed no knowledge of channel state information, the maximum likelihood sequenceestimation receiver will use the metric function of Eq. (4.61), and will base its decision onwhich one of the two sequences G(A”) and (A’) is more likely to have been transmitted onthe following comparisonFV (i(v))2>< (Vi,A)2 (C.3.1)with(i,()) = y(c = ef(kT) c (c)* + n (c)*. (C.3.2)In the above equation, 1 <i <A, and c represents the value the symbol ck would have, if thesequence A” was the one transmitted. C(A”)), C(A)) can be expressed as(, C(A”)) = + .N + .IVJ (C.3.3)where ,\ can be either ii, i.e., ç (\ E {v, ç}) and=1()*(C.3.4)= e (f)’c (c)*, (C.3.5)= n(c (C.3.6)with (ft)’ = — f”. Assuming that the total signal power on each diversity channel isnormalized to 1, i.e.,()2 + ()2= 1=1’(C.3.7)APPENDIX C. 134and that the Rician K-factor is the same in all A diversity channels. Furthermore, assumingthat the operating SNR is the same for all diversity paths, we have consequently that= E, V = and .A1[,\] = dj V i. (C.3.8)Because of the zero mean complex Gaussian nature of f and %, F (, C(A>)) is acomplex Gaussian random variable with average and variance= F—= E{A1+ArG2}= R’ c2c2+2 Re (C_k)*()*Ct_k]+o.= i [z+2p cos [ze(l) —(C.3.9)where R’ = RF(kT), and p’ = R’/R’. The cross covariance between FL (,C(Av)) and(,c(A)) equals=Z1+‘{2 (Ck)* Ck+Cjk(cfl*+cT (Ck) Ckk=O[ (v)*F {(Ck)*Ck+ (cfl*C}]+cr (c)* c. (C.3.lO)APPENDIX C. 135We define the normalized covariance and crosscovariance as1=SNR ‘ Z-1 Z-1= K+1 k=l l=k(C.3.l1)—SNR Z-i Z—I= K+1(c)*c+pI k=O k=1 l=kZ- 1+ (v) (C.3.12)k=0where we have usedR’1 1 SNR (C.3.13)— u(K + 1) K + 1Using the material in Appendix 4.B of[14, pp.223-228], substituting A = 1, B = —1 andC = 0 in Eq. (4B.1) of [14], we find the following expression for the PA({(A) — (Av)}).For > 0,/1 1PA (C) i— = Q (—c,—/3) (C.3.14)‘o+I L 2 ]ij exp I—(1 + V)A(1 — A-I / 2A — 1(1 + V)1(1 —2A—11=0 \ A+l )F a[,J(] ++exp [ 2o jA—i 11rn(a[v,(/3v, ]){rn [i(1 +V)A_m(1 V)A÷rn_l A+m-l ( 2A — 1 ) (1 +V)’(l —22 1=0 A+l—mAPPENDIX C. 1361 F1 — (1 +V)A+m_l(1 — 73)A-rn A+m-1 / 2A — 1 ) (1 + D)’(1 — D)1] }2A-1 l- \A+l—mx [i —6(A— 1)].In the above equation, Qf(x, y)is Marcum’s Q function [32, P. 585] andJ 1[v,(1 } = F A2+ E2— 2Re { (ii)*}+ vv ( 1 — 21—1/2+ 2 1 I (C.3.15)I]where2v‘Yv( = (v + v()’— VL( (C.3.16)+ v()2 — 4 V2’and+— 2Re{(_..J)*_v _2 —+i 2i/i21— I7V —(C.3.17)—v 2EV+— 2Re {(EL)*y} E2—2 — 2Fora 0, Eq. (C.3.14) simplifies to[ Lv]1 1 (2A — 1 (C.3.18)PA ((A) — C( )) = exp [ 2o j 221 i=o A + 1For the case where diversity is not used, i.e., for A = 1, the pairwise error probability canbe expressed asp ({(Ac) (Av)}) 11 1/21—Ii +(n L [(v +v2 — 4v2j jAPPENDIX C. 137x i exp (C.3.19)It is clear that the expression provided by Eq. (C.3.14) is quite general, providing us withthe means of evaluating the performance of maximum likelihood sequence estimation receiversusing the MDD hardware structure, and operating under any SNR value and Rician fading K-factor. However, for some cases commonly encountered in practical communication systems,the expression provided by Eq. (C.3. 14) can be simplified further. More specifically, we shallexamine the cases of• Rician fading with strong direct signal component at high SNR, and• Rayleigh fading, i.e., no LOS signal component.C.3.1 Rician Fading, Strong LOS Component and High SNRUnder these fading channel conditions, the value of K/ [i+ AK1)} is quite large. Thismakes the the terms (—a[,(]), (-/3[j) quite large, and permits the use of the followingapproximations for the Qf(x, y) and 1m(Z) functions [33]exp(_y2/2) forx=OQj(x,y) (C.3.20)(1/\/2)exp(_h1) forx >0,17fl(z) exP(lzI)G() (C.3.2l)whereI form=0Gm(z) = ¶j (4rn2 — (2i — 1)2) (C.3.22)1 + (__l)ht1(l)!(8z)’for rn 0.APPENDIX C. 138Substituting these approximations in Eqs. (C.3.14), (C.3.18), yields the following expression for PA({(A)÷_ (j4L)})/ + 3,,(] \ Z,APA — = exp— 2o ) (C.3.23)In the above equation, for > 0,1pZA[v]=____________1 — exp 2 j 2A1 1=0 ( A +A—i {A_1 2faiii]1 (1 + D)A(l — D) A — 1 ) (1 +V)1(1 —+exp 2 21 A-I (Iv:C]/3[v,i)L n irn=1+rn—i A+m—i 2A — 1 1x {m [ — + — J(1 + V)’(l — V)-’ I1=0 (A+l_mJ j221rn-i / 2A—1 ii1 U + vA+m_1U— (1 + v)-’ —Ym 2A-1 1=0 A+l—rn) Hx [i — 6(A — (C.3.24)For = 0, the expression for PA({(A) — (A’)}) is identical to the one in Eq. (C.3.18).C.3.2 Rayleigh Fading ChannelsFor a Rayleigh fading channel, K —÷ —oo dB, i.e., K —* 0, which leads to K/ (i + —* 0.“ SNRUnder such channel conditions, from Eq. (C.3.4) we have that E = 0, and from Eq. (C.3.15)APPENDIX C. 139it is easy to verify that = = 0. Using the identities Qj(O,O) = 1, Io(O) landJm(0) = 0 for in > 0 [33], in Eq.(C.3.14) yieldsPA ({M) (A)}) =(1 +V)A(l_ D) { (2A — i ) (1 ÷V)’(l — 7)1}.i=o A+l(C.3.25)For very slow Rayleigh fading, no diversity (A = 1), a BPSK signal and for Z = 2, which corresponds to the conventional differential detector, the pairwise error event probability simplifiestoP1 ({(A)‘+ 1(C.3.26)which is identical to the result given in [14, Eq. 7.3.10].APPENDIX C. 140C.4 Derivation of Eq. (4.80)The channel model parameters for a mobile-satellite radio channel as found in [34, 35] arebased on data collected by propagation experiments in the UHF and L (1.5 0Hz) frequencybands. Since experimental data for the Ka-band is not presently available, we use a parameterscaling technique based on the modified exponential decay (MED) model developed from staticpropagation measurements through deciduous trees [1051. The attenuation coefficient in dB/mcan then be computed by the following equation [106]c 0.45F°284 for 0 <dT < 141.33F°284(dT)412 for 14 dT <400 (C.4.1)where F is the operating frequency in 0Hz and dT the depth of trees in meters (m) interceptedby the LOS signal. Since in the case of satellite communications having a depth of trees ofmore than 14 m would be a rather rare occurrence, we use the first of the equations abovefor attenuation scaling. According to the log-normal model [34], the attenuation in signalamplitude at UHF and L-band frequencies isA exp(x) (C.4.2)with x a Gaussian random variable with mean t and variance d0 given by Table 4.4. Theattenuation scaling factor of Eq. (C.4. 1) can be used to compute the new values of and d0 asfollows. The signal power at a frequency F can be written as4 [dB] = F x A2 [dBj (C.4.3)and taking the exponent of base 10 for both sides in the above equation,4 = (A2)’AF = AF = (exp(x)) = exp(cxFx). (C.4.4)APPENDIX C. 141Consequently, calculation of the new mean i and standard deviation involves multiplyingwith the scaling factor For the frequency of approximately 21 GHz used for the simulationresults presented in Chapter 4, the scaling factor is approximately equal to 2.1, and the resultingparameters are shown in Table C.5. Using Eq. (4.81), the ratio of LOS signal average powerTable C.5: Ka-band satellite channel model parameters.L-band channellight shadowing 0.158 0.2415 0.2415average shadowing 0.126 -0.2415 0.3381heavy shadowing 0.0631 -8.211 1.6926to multipath signal power can be calculated for both the L-band and the EHF Ka-band. Resultsof this calculation are presented for comparison in Table C.6. As intuitively expected, in theTable C.6: Ratio of LOS signal average power to multipath signal power (KAyo); comparisonbetween L and Ka bands.Channel Band dBlight shadowing L 6.12Ka 7.34average shadowing L 5.21Ka 4.36heavy shadowing L -19.33Ka -50.48case of heavy shadowing, the results in the above table show a considerable decrease of about30 dB in KAVG at the 20 GHz Ka-band, as compared to that for the 1.5 GHz L-band.APPENDIX C. 142C.5 Derivation of Eq. (4.81)Calculation of the total average power of the LOS signal component amounts to calculatingE{y}, where, y = exp(2x), and x is a Gaussian random variable with variance d0 and mean. Using the well known theorem for calculating the expected value of a function of a randomvariable with known pdf [98], the average E{y} can be written asE{y} = J exp(2x)exp(—2d2)dx. (C.5.1)Rearranging the argument of exp() inside the above integral, E{y} becomes(C.5.2)Finally, making use of the following integral expression [33]exp [_(ax2+ bx+c)] dx = exp[b2 — 4ac](C53)- 2a 4aand by substituting a = 1/(2d0), b = —( +2d0)/ and c =jt2/(2d0),Eq. (4.81) results.

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