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Adaptive trellis-coded modulation for mobile communications Alamouti, Siavash Massoumi 1992

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ADAPTIVE TRELLIS-CODED MODULATIONFOR MOBILE COMMUNICATIONSbySIAVASH MASSOUMI ALAMOUTIB.A.Sc. (EE), University of British Columbia, 1989.A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF ELECTRICAL ENGINEERINGWe accept this thesis as conformingto the required standardThe UNIVERSITY OF BRITISH COLUMBIANovember 1991© Siavash M. Alamouti, 1991In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication' of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature) _Department of  giLeht;c	 /itee,1Date  /t/67/	 15 The University of British ColumbiaVancouver, CanadaDE-6 (2/88)AbstractAn adaptive scheme for trellis-coded modulation of MPSK signals called adaptivetrellis-coded multiple-phase-shift-keying (ATCMPSK) is proposed for Rayleigh fadingchannels. The adaptive scheme employs a slightly modified rate 1/2 convolutionalencoder and the corresponding Viterbi decoder to realize a family of codes of differentrates which are employed according to channel conditions. During poor channel con-ditions, trellis-coded QPSK (TCQPSK) together with repetition schemes are employed.As channel conditions improve, higher rate schemes such as trellis-coded 16PSK areused. An interleaving/deinterleaving method suitable for the adaptive scheme is proposed.Theoretical bounds for the error performance and throughput of the proposed adaptivescheme are derived, and are compared against the simulation results. Simulations havebeen performed to measure the performance of the scheme for different parameters andnonideal conditions. It is shown that ATCMPSK results in considerable improvement inbit-error-rate (BER) performance of MPSK signals. Under ideal conditions, gains in therange of 3 — 20 dB are achieved over conventional fixed rate trellis-coded schemes.Contents Abstract 	 iiList of Figures 	 viList of Tables 	 ixAcknowledgement 	Chapter 1	 Introduction 	 1Chapter 2	 Review of Trellis-Coded Modulation 	 5Chapter 3	 Pragmatic TCM 	 9Chapter 4	 Adaptive Trellis-Coded MPSK (ATCMPSK) 	 12	Section 1	 Fading Channel Model 	 12	Topic 1	 The Gaussian Metric 	 14	Section 2	 Description of the Overall System Model 	 16	Topic 1	 Interleaving/Deinterleaving 	 20Chapter 5	 Performance of Pragmatic TCM 	 23	Section 1	 Theoretical Bound on the BER in AWGN 	 23	Topic 1	 T(D) and the Error-State Diagram 	 25	Topic 2	 Simplified Error-State Diagram 	 29	Section 2	 BER Performances of Various Pragmatic TCM Schemesin AWGN 	 29	Topic 1	 Rate 1/2 TCQPSK in AWGN 	 30	Topic 2	 Rate 2/3 TC8PSK in AWGN 	 34Ill	Topic 3	 Rate 3/4 TC16PSK in AWGN 	 38	Section 3	 BER Performance of Pragmatic TCM Schemes in theRayleigh Fading Channel 	 44Chapter 6	 Performance of ATCMPSK in the Rayleigh FadingChannel 	 51	Section 1	 Bound on the BER performance of ATCMPSK in theRayleigh Fading Channel 	 51	Section 2	 Throughput of ATCMPSK 	 56Chapter 7	 Description of Computer Simulations 	 60	Section 1	 The AWGN Channel 	 60	Section 2	 The Rayleigh Fading Channel 	 61	Topic 1	 Jakes' Model of Rayleigh Fading (Land-Mobile) 	 62	Section 3	 Simulation of the codulator/decodulator 	 64	Section 4	 Performance Evaluation Results 	 66	Topic 1	 Effect of Error Thresholds on the BER Performance andThroughput 	 66	Topic 2	 Effect of Finite Interleaving 	 69	Topic 3	 Comparison of ATCMPSK with Nonadaptive TCMSchemes 	  70	Topic 4	 Effect of Vehicle Speed or BdTs Product 	 75	Topic 5	 Effect of Feedback Delay and Adaptation Interval 	 78ivChapter 8	 Conclusions and Recommendations for Future Research 	 82Bibliography 	 85Appendix A	 The Gaussian Metric 	 87Appendix B	 Comparison of Double-Gray-Coded Mapping andSectorized-Gray-Coded Mapping 	  91Section 1	 Comparison in AWGN 	 91Section 2	 Comparison in Rayleigh Fading Channel	 93Appendix C	 Confidence Interval for Simulation Results 	 96VList of FiguresFigure 1	 Trellis encoder and MPSK mapper structure. 	 6Figure 2	 Mapping by set partitioning of the 8PSK signal space. . 6Figure 3	 A rate 2/3 trellis-coded scheme and its set partitioned8PSK signal space 	 8Figure 4	 Pragmatic encoder and its trellis structure for K=3. . . .	 9Figure 5	 Double-Gray-coded mapping for 16PSK	 11Figure 6	 Comparison between the Euclidean distance and theGaussian distance. 	 15Figure 7	 Simplified block diagram of ATCMPSK	 16Figure 8	 Interleaver Structure for ATCMPSK	 21Figure 9	 The structure of the deinterleaver for ATCMPSK. 	 22Figure 10	 The general structure of the error-state diagram forpragmatic codes (K=3).	 28Figure 11	 Trellis diagram and the signal space for rate 1/2TCQPSK. 	 31Figure 12	 BER Performance of rate 1/2 TCQPSK code in AWGN. 	  34Figure 13	 Trellis structure, and the signal space for the rate 2/3pragmatic code. 	 35Figure 14	 Parallel branches in the trellis for the rate 2/3 code	 37Figure 15	 BER performance of the rate 2/3 TC8PSK code inAWGN 	 38viFigure 16	 Trellis structure and signal space for rate 3/4 pragmaticcode with double-Gray-coded mapping. 	 39Figure 17	 Parallel branches in the trellis diagram of rate 3/4pragmatic code. 	 42Figure 18	 BER performance of rate 3/4 TC16PSK code in AWGN. . 44Figure 19	 BER performance of ideally interleaved trellis-codedMPSK in Rayleigh fading. 	  49Figure 20	 Theoretical and simulation curves for the BERperformance of ATCMPSK under Rayleigh Fading. . . . . 56Figure 21	 Information throughput of ATCMPSK—simulation andtheory. 	  59Figure 22	 Block diagram of the baseband land-mobile fadingsimulator. 	  64Figure 23	 Adaptation thresholds for ATCMPSK	 66Figure 24	 BER performance of ATCMPSK with different error roofs. 	  67Figure 25	 Throughput of ATCMPSK for different error roofs. 	 68Figure 26	 Effect of finite interleaving on BER performance ofATCMPSK.     70Figure 27	 BER curves for ATCMPSK and fixed rate TCQPSK, withsuperimposed curve of the throughput performance ofATCMPSK. 	 71viiFigure 28	 BER curves for ATCMPSK and fixed rate TC8PSK, withsuperimposed curve of the throughput performance ofATCMPSK. 	 74Figure 29	 Effect of vehicle speed on the BER performance ofATCMPSK with delayless feedback. 	 76Figure 30	 Effect of vehicle speed on the BER performance ofATCMPSK, feedback delay = 1 symbol duration. 	 77Figure 31	 Effect of vehicle speed on the BER performance ofATCMPSK, feedback delay = 3 symbol durations 	 78Figure 32	 Effect of feedback delay on the BER performance ofATCMPSK. 	 79Figure 33	 Effect of nonideal adaptation on the BER performance ofATCMPSK. 	 80Figure 34	 Correlator receiver for maximum likelihood detection. . . 	  87Figure 35	 Comparison of signals on the parallel branches ofthe trellis diagram for double-Gray-coded andsectorized-Gray-coded mappings    91Figure 36	 Comparison of trellis error and probability of error due toparallel transitions for TC16PSK in AWGN with sectorized-Gray-coded and double-Gray-coded mapping. 	  93Figure 37	 BER performance of double-Gray-coded TC16PSK. . . . 	  95viiiList of Tables Table 1Table 2Table 3Table 4Error-weight profiles for the rate 2/3 pragmatic code. . . . 36Error weight profiles for rate 3/4 pragmatic code withdouble-Gray-coded mapping	 40Proportion of errors for the various schemes used inATCMPSK. Error roof = 0.01, interleaving size = 32*16,SNR=24 dB. 	 73Error weight profile for the rate 3/4 TC16PSK pragmaticcode with sectorized-Gray-coded mapping. 	  92ixAcknowledgementI would like to thank my long time companion Gaylynne Noonan for her moralsupport and constant encouragement. I send my regards to my mother Atiye Shaeri,and my father Ahmad M. Alamouti for their love, nurturing and support. I am indebtedto my supervisor, Dr. Samir Kallel, for his constant guidance and his indispensablerole in the successful completion of this thesis. I express my gratitude to Dr. A. D.Moore, and Dr. Cyril Leung for having played a major part in my decision to enterthe most fascinating field of communications. I extend my appreciation to B.C. ScienceCouncil for their financial support, to Dr. Norman Toms at MPR for his encouragementas the industrial supervisor of this project , and to MDI and MPR for their support ascooperating companies. I would also like to thank my good friends and fellow studentsin the communications group of the Electrical Engineering department of UBC for theirdedication to providing a friendly and cooperative environment for research. Finally, Iwould like to thank Dr. Takis Mathiopolous for his careful examination of this thesis.Chapter 1IntroductionThe rapid progression of mobile communication technology from analog to digitalhas made available a new variety of services. Mobile systems are no longer dedicatedsolely to speech transmission [1]. Reliable transmission of data, and integrated serviceshave gained a lot of importance in today's era of information. There is a great demandfor mobility, reliability, and speed in transmission of information. Frequency spectrum is,however, a limited resource. Therefore, in order to maximize the services offered throughmobile communications, frequency spectrum must be used efficiently. Spectrally efficientlinear modulation schemes increase the effective rate of transmission of informationwithout any added spectrum. One widely used, linear modulation scheme is multiple-phase-shift-keying or MPSK. MPSK schemes are spectrally efficient, simple to implement,and have relatively good bit-error-rate (BER) performance. But, as there is usually a tradeoff between power and spectral efficiency, MPSK schemes are less power efficient.Power efficiency is a major issue in mobile communications. To keep the size of themobile units small, the power consumption must be minimized. To minimize the requiredpower to achieve a certain BER, error control techniques can be applied. The combinationof convolutional encoding and Viterbi decoding is a widely used forward-error-correction(FEC) technique. Furthermore, considerable gains can be attained through combiningMPSK modulation and coding through trellis-coded modulation (TCM). TCM was firstexplored by Ungerboeck [2] who showed that, in the AWGN channel, gains of up to6 dB were attainable without any bandwidth expansion, and that most of the coding1gain could be obtained by doubling the size of the signal space. For example, using arate 2/3 convolutional code together with an 8PSK signalling scheme, 3-4 dB of gaincan be obtained as compared to uncoded QPSK—although both schemes have the sameinformation rate of 2 bits per symbol duration.Although conventional TCM schemes offer appreciable coding gains in AWGN, theirperformance suffers greatly in the Rayleigh fading environment. The time varying natureof the channel imposes a great constraint on the TCM scheme employed to combat theeffect of fading. MPSK schemes with more than four signals in their constellation have,for most applications, unacceptable BER performance, while lower level MPSK schemessuch as BPSK greatly sacrifice the throughput. Moreover, it has been shown that a lot canbe gained—in terms of BER and throughput performance—by variable rate transmissionof information on fading channels [3]. These facts justify the implementation of anadaptive scheme using trellis-coded MPSK signals.The proposed adaptive scheme is based on a simple pragmatic encoder/modulator[4]. The principal advantage of the scheme is that the same encoder/decoder can be usedfor a wide variety of trellis codes. The system can generate different trellis-coded signalsusing the same rate 1/2 convolutional encoder. The adaptive nature of the scheme allowsfor lowering the coding rate while keeping the same symbol transmission rate duringdeep fades. The coding/modulation (codulation) rate is changed according to channelconditions determined by the signal level detected at the receiver. If the signal leveldrops below a predetermined threshold, then the encoder is informed through a feedbackchannel, and the codulation rate is changed accordingly. If channel conditions becomeextremely bad, repetition codes together with trellis-coded QPSK (TCQPSK) are applied.2This thesis has been organized in the following manner:Chapter 2 is a brief review of trellis-coded modulation. In Chapter 3, the pragmaticapproach to trellis-coded modulation is discussed, and the fact that is suitable for use inadaptive systems is explained. In Chapter 4, we introduce the adaptive scheme and thevarious elements needed for its design. A suitable approach to interleaving/deinterleavingfor the adaptive scheme is also introduced in this Chapter. Chapter 5 has been dedicatedto a detailed analysis of the error performance of the pragmatic schemes employed in theadaptive system both in AWGN and Rayleigh fading channels. All the theoretical analysisof error performance are based on the classical transfer function approach. In Chapter 6,theoretical analysis for the bit-error probability and the throughput of ATCMPSK underideal conditions are carried out. Finally, in Chapter 7, the details of simulation programsare explained, and the results of simulations are reported. Simulations have been run forvarious parameters to take into account the effects of nonideal conditions.The contributions of this thesis to digital communications theory are as follows:• Derivation of relatively tight theoretical upper bounds for the performance of somepragmatic TCM schemes in AWGN, and Rayleigh fading channels.• Introduction of a new signal mapping scheme suitable for the adaptive scheme; and,in fact, all pragmatic trellis-coded MPSK schemes using a signal constellation with16 or more signals.• Design and investigation of an adaptive system used for the purpose of communica-tions over time varying channels, specifically Rayleigh fading channels.• Derivation of a theoretical bound on the BER, and exact expressions for thethroughput performance of the proposed ATCMPSK scheme, under ideal conditions.3• Simulation of the ATCMPSK scheme, taking into account many parameters such asvehicle speed, feedback delay, adaptation interval, and adaptation thresholds.4Chapter 2Review of Trellis-Coded ModulationThe fundamental idea behind the design of any TCM scheme is combining thefunctions of coding and modulation. Before this revolutionary idea was introduced byUngerboeck [2], coding and modulation were conceived as two separate functions. Theseparation of these two functions led to an unrecoverable loss of information due tohard quantization, and nonideal code optimization. Traditionally, code optimization wascarried out independently from the signal geometry of the modulation scheme. All thatwas considered was the Hamming distance between binary sequences. Generally, theHamming distance between binary representations of two signals does not translate totheir distance in the signal space. Therefore, one can say that Hamming distance is oftennot a true measure of similarity between signal representation of two symbols. On theother hand, the geometric distance between the signals or their Euclidean distance (ED)is an exact measure.Ungerboeck [2], [5] showed that using ED as the metric in systems employingconvolutional coding and Viterbi decoding, and designing codes to achieve maximumED between all possible sequences of channel signals leads to significant coding gainswithout any sacrifice in bandwidth or the effective information rate. As opposed toconventional FEC techniques, where the redundancy is added to information bits, therequired redundancy for error correction is obtained by expanding the size of the signalspace. In fact, it was shown [2] that nearly all the possible gain in performance is achievedby doubling the size of the signal space. This led to the design of rate n/n+1 codes with5sRate n/n+1ConvolutionalEncoder•MPSKSet PartitionedMappersignal spaces of size M=2n+ 1 . Figure 1 shows the trellis encoder with an M level mapper.Figure 1 Trellis encoder and MPSK mapper structure.The optimum signal mapping is based on partitioning the signal space into subsetswith increasing minimum distances, and assigning bits to transitions to each sub-partition.The example for 8PSK is shown in Figure 2. It is interesting to note that for MPSK wecan achieve a set-partitioned signal space by consecutive binary numbering of the signalsin the signal space starting from a reference point.000	100	010	110	001	101	 011	 111Figure 2 Mapping by set partitioning of the 8PSK signal space.6The design of a trellis-coded scheme is aimed primarily at maximizing Euclideanfree distance dfree defined as:d2free = mLn [d2 (XL, XL)]L	 (2.1)= min E d2 (.„, xn)n=1where XL = (x1, x2, ..., xL) , XL =x2 , ..., 4) are two sequences of encodedsymbols of length L, and d(xn , xin ) is the ED between the two signals.The first step in designing an optimum code is to choose a suitable trellis structure.The choice of the trellis structure depends mainly on the constraint length of the code,since it determines the overall shape of the trellis and the number of states. Codes withhigher constraint length have superior performance; however, they are more costly toimplement both in terms of hardware and design effort.Having chosen the suitable trellis structure, signals from the MPSK signal spaceshould be assigned to state transitions on the trellis so that dfree is maximized. For codesof large constraint length, this is done using a computer algorithm that assigns signalto the transitions and measures dfree until it achieves the optimum configuration. Forsmaller constraint lengths, a few rules of thumb can ensure an optimum design for thechosen trellis structure.1. Having partitioned the signal space into subsets with increasing Euclidean distances,assign to parallel transitions signals in the subset of the signal space Cp (see Figure2) whose members have as many signals as the number of parallel transitions, andmaximum distance between all pairs of signals.2. To transitions originating from the same state or joining into the same state assignthe signals in the subset preceding Cp in the partition process.7S 1( 00)S2( 10)S3( 01)S4( 11)0100111001010011111101 , 53. Make sure that all states, and the signals in the signal space are on equal footing.Figure 3 shows an example of a rate 2/3 trellis-coded 8PSK scheme which has beendesigned using the above guidelines. In this example, there are two parallel branchesin each transition.Figure 3 A rate 2/3 trellis-coded scheme and its set partitioned 8PSK signal space.Note that signals on the parallel transitions have been assigned in such a way as tomaximize the distance between them which for the special case of 8PSK this correspondsto antipodal signals.One major drawback of Ungerboeck's family of trellis codes is that each signalconfiguration requires a different encoder/decoder—since optimized TCM schemes ofdifferent rates have different code structures. The use of these codes in adaptive systemswould therefore be costly. A less expensive and more pragmatic alternative is to usethe set of pragmatic trellis codes.8b) QPSK5+1 	 (S+4)/24• S/2	• S/2• (S+4)/24a) Pragmatic EncoderK=3S/2	 S •(S+4)/2 S+/ •d) 16PSKFigure 4 Pragmatic encoder and its trellis structure for K=3.c) 8PSKChapter 3Pragmatic TCMRecently, Viterbi et al. [4] proposed a pragmatic approach to trellis-coded modulationwhich is based on the realization of rate n/n+1 trellis-coded schemes using a single rate1/2 encoder/decoder, in conjunction with an MPSK signal mapper where M=2n +1 . Thename pragmatic refers to code implementation employing the widely used, best knownrate 1/2 convolutional encoder, and the corresponding Viterbi decoder.Figure 4 shows the pragmatic encoder, and the corresponding trellis diagram for the"good" 1 convolutional code of constraint length K=3.Good codes refer to codes optimized for minimum Hamming distance.9Information bits are fed serially into the encoder. One out of every n bits is sentinto the rate 1/2 encoder, and the remaining n-1 bits pass directly through without beingencoded. In this manner, trellis codes of various rate are generated. The number ofuncoded bits determine the rate of the code. Different rate codes differ only in thenumber of parallel transitions in their respective trellis diagrams. The trellis diagram forrate 1/2 TCQPSK does not have any parallel branches. However, rate 2/3 TCQPSK has2 parallel branches in each state transition. In general, rate n/n+1 TCMPSK pragmatictrellis codes have M/4 parallel branches in each state transition.The mapping scheme suggested in [4] can be called sectorized-Gray-coded mapping,and it works as follows: the n+1 bits at the output of the encoder are used to select asignal in the signal space. The two least significant bits, which are the output of the rate1/2 encoder, select one out of four signals according to Gray code. The remaining n-1bits choose a sector in the signal space lexicographically, as shown in Figure 5 for thecase of 16PSK. Note that equivalent signals in different sectors (signals with identicaltwo least significant bits) belong to parallel branches of the same state transition.In this thesis, we suggest a different mapping scheme which we choose to call double-Gray-coded mapping. As for sectorized-Gray-coded mapping, the two least significantbits corresponding to the output of the rate 1/2 encoder select one out of four signalsaccording to Gray code. The remaining n-1 bits choose a sector in the signal space againaccording to Gray code, as depicted in Figure 5 for 16PSK signalling scheme. The firsttwo bits of the encoded symbol determine the sector to which the signal belongs. Notethat both mapping schemes result in the same signal space for constellations with lessthan 16 signals.101001Sector 2100100 	 0010Sector 1	 Sector 0\:11111111 111111 001100010101\0	 0111_Sector 31111100111011010001001Sector I NSector 2 —\\I1 ---\\1101Sector 000111 11Iiiimi:111110001Sector 310001010111010011001000101011111001011	 11011010	 1100Sectorized-gray-coded1110	 1000Double-gray-codedFigure 5 Double-Gray-coded mapping for 16PSK.The obvious advantage of the double-Gray-coded mapping scheme for 16PSK isthat signals belonging to parallel branches of neighbouring sectors differ in one bit ascompared to 2 bits for sectorized-Gray-coded mapping. This will definitely improve theerror performance of the code since most of the errors happen in neighbouring sectors.The BER performance of the trellis-coded 16PSK signals using the two different mappingschemes are compared in Appendix B. The mapping scheme employed throughout thisresearch is double-Gray-coded MPSK, unless otherwise stated.It is shown in [4] that, for all practical purposes, the asymptotic coding gains (ACG)of the pragmatic codes are almost the same as Ungerboeck's codes. The amount ofsub-optimality introduced is negligible for MPSK signals. The code, however, is muchsimpler to implement, and the same decoder can be used for different codulation rates.11Chapter 4Adaptive Trellis-Coded MPSK (ATCMPSK)Although pragmatic trellis codes perform nicely in the AWGN environment, they arenot nearly as good in the fading environment. The major reason for this degradationin performance is the presence of parallel transitions in their state diagram which giverise to single signal error events. Therefore, in order to keep the BER lower than acertain level, lower rate pragmatic codes must be used—since going into higher rateswill have catastrophic results during deep fades. But channel conditions are not alwaysbad. A fixed low rate code will sacrifice the throughput even during good channelconditions. Therefore, a system is needed that can take advantage of periods of goodchannel conditions to transmit at a higher rate, and can lower its rate during deep fades.Adaptive schemes have shown to be an effective method for combating the timevarying nature of the distortion in the fading channel. A series of published papers reportconsiderable gains by adapting certain parameters in the transmitter according to channelconditions—from symbol duration and power to transmission rate [6], [7], [3].4.1 Fading Channel ModelThe model for the fading channel used in this study is that presented by Jakes in [8].It is assumed that the received field at the mobile unit is made up of a number N ofwaves arriving with different amplitudes and angles of incidence in a random fashion. Ifthe transmitted signal is vertically polarized, the received signal can be expressed as:E z = I (t) cos (21-fc t) — Q(t) sin (2r f ct)	 (4.1)12where/(t) Eo E cn cos (27r fn t + On )Q(t) = E0 E cr, sin (271-fnt + On )n=1Ar.	 (4.2)n=1are Gaussian processes. At any instant of time, the samples of these processes areuncorrelated Gaussian random variables with zero mean , and variance bo = Ed /2. Theenvelope of the faded signala = N/11/(011 2 +1104 2 	(4.3)is then Rayleigh distributed with probability density functiona	 a2f(a) = 7,-0.exp	 a > 0 ,	 (4.4)where 11'11 stands for magnitude. Throughout this thesis we shall assume that b0=0.5 sothat 4	 1. Or equivalently:f (cc) = 2aexp (—a2 )	 a > 0 .	 (4.5)The phase of the fading signal is uniformly distributed. We will assume that theeffect of phase on the received signal is fully compensated for by the receiver. Thiscan be done in practice by pilot tone insertion and calibration [9 ], [10], or tracking thephase by a phase locked loop [11]. We should keep in mind, however, that in a fadingenvironment, the exact tracking of the phase is extremely difficult. In fact, differentialdetection seems to be one of the only viable solutions for this problem [12]. Simonand Divsalar [13] have devised a differential detection scheme for MPSK signals calledmultiple-symbol differential detection, which is based on maximum-likelihood sequence13estimation. Their idea can be applied to differential detection of TCMPSK signals. Ifa more realistic model is desired, the effect of phase error can usually be modelled bya Gaussian random variable [14].Considering only the effect of fading on the amplitude of the transmitted signal isthe same as multiplying the in-phase and quadrature components of the baseband signalby the time varying fading amplitude a(t).4.1.1 The Gaussian MetricSince the transmitted signal is attenuated by the fading component a, the square ofthe Euclidean distance is no longer an accurate, or optimum measure of the similaritybetween the received signal and branch signals in the trellis diagram and hence not anoptimum metric. Figure 6 illustrates this concept geometrically.S1, 52 are signals in the signal-space, Rf is the faded signal at the receiver, and Rgwould be the received vector if fade was not present (AWGN channel). Distance d isthe ED between Rf, Si. This ED is distorted by fading. However, N, the length of theAWGN vector, and the distance between Rf and the faded transmitted signal S 1, isexactly the same as the distance between Rg , Si (the case of AWGN channel). Therefore,if the right decision is made, and N2 is chosen as the metric, one should expect thesame performance as in the AWGN channel. Of course, this is not the case since theattenuation reduces the signal energy which in turn increases the probability of error, andcauses an overall degradation in performance.In conclusion, using optimum decision rules for matched filter detection to minimizethe probability of error, one would find that the square of the distance between the faded14transmitted signal and the received signal is the optimum metric. This metric is oftencalled the Gaussian metric [15].Figure 6 Comparison between the Euclidean distance and the Gaussian distance.Assume that a sequence of signals X = (x1, s2, xL) is transmitted, signalsY = (yi, y2, ..., yL) are received, and the corresponding fading amplitudes are :A =	 (4.6)then y; = aixi ni for 1 < i < L, whereni d N(0 N° )2 (4.7)is a sample of AWGN, and	 is its power spectral density. The metric associated with15ReorganizerInformationBits .0m(Estimate) Deinterleavero-DemodulatorInformationBitsa((k - t) Ts )R b1•411111.- InformationBufferRate 112EncoderAdaptiveMPSKMapper R sInterleaver--ipChannelHistoryIdeal FadingEstimatorRayleighFadingDistortioncx(k Ts)a(ic Ts)AWGNDistortionN(t)IQTrellisDecoderRFTransmitterthe ith symbol can be written as:mi = lyi — ceixi1 2 .	 (4.8)For a mathematical proof of the optimality of Gaussian metric refer to Appendix A.4.2 Description of the Overall System ModelThe adaptive scheme presented in this thesis (ATCMPSK) is conceptually simpleto implement. We call it adaptive trellis-coded multiple -phase -shift-keying to accentuatethe combined coding/modulation (codulation) aspect of the scheme. A simplified blockdiagram of the proposed ATCMPSK scheme is shown in Figure 7.Figure 7 Simplified block diagram of ATCMPSK.Information bits enter the information buffer at rate Rb bits/sec. The buffer is neededsince, as it will be discussed shortly, the effective rate of information is a function of theconditions in the time varying channel and is hence not fixed, but the symbol transmission16rate R, = 4; is constant. In order to choose the suitable scheme for transmission we needchannel state information (CSI) which in our model is the amplitude of fading a(kT,) atthe start of the transmission and is supplied by the ideal fading estimator to the transmitterso that it can adjust its rate according to channel conditions, and to the receiver for thepurpose of decoding. As shown in the block diagram, we assume that the receiver hasideal CSI, while, in general, the transmitter receives a delayed version of CSI whichin our model we assume is an integer multiple of symbol duration, and we have hencedenoted it by a((k — 7•)Ts). The assumption that the receiver has ideal CSI without anydelay is reasonable since usually CSI is obtained by methods such as pilot tone insertionand extraction [10], [9] which determine an estimate of the channel at the receiver side.The transmitter decides what scheme should be used at the start of every transmissionaccording to a set of given thresholds that have been chosen to keep the BER below acertain level. We assume that there are five possible schemes to choose from:1. Rate 1/2 TCQPSK with 3 repetitions2. Rate 1/2 TCQPSK with 2 repetitions3. Rate 1/2 TCQPSK4. Rate 2/3 TC8PSK5. Rate 3/4 TC16PSK.Rate 1/2 TCQPSK with 3 repetitions is used during the worst, and rate 3/4 TC16PSKduring the best channel conditions. Since there are five different codes employed bythe ATCMPSK under study, we have four adaptation thresholds. Assume that theseadaptation thresholds are pi, p2, p3, p4 2. The transmitter looks at the value of the2	 The way in which these thresholds are chosen is discussed in Section 7.4.117immediate SNR defined as:SNR; = a?—No(4.9)and determines the suitable codulation scheme. If 0 < SNR; < pi rate 1/2 TCQPSKwith 3 repetitions is employed. Otherwise, if pi < SNR; < p2 rate 1/2 TCQPSK withtwo repetitions is chosen as the codulation scheme suitable for transmission, and so on.One could extend the idea to systems which employ more diverse schemes to attain largerthroughput during very good channel conditions, and superior error performance duringvery poor channel conditions at the expense of lower throughput of information. Themain trade off in such schemes is between performance and implementation cost.Going back to the block diagram, information bits are fed into a rate 1/2 convolutionalencoder, passed to the interleaver, and combined with the appropriate number ofinformation bits (according to CSI) to form the adaptively encoded channel symbol 3 .The encoded channel symbol is then mapped into a baseband signal according to double-Gray-coded mapping discussed in Chapter 2. The baseband signals are modulated by thecarrier, and sent over the channel. A more realistic model would include a digital pulseshaper to combat ISI, and pilot tone insertion to help estimate the value of fading.The only sources of distortion are assumed to be the AWGN and the envelopeattenuation due to fading. The demodulator recovers the distorted in-phase and quadrature(I/Q) components of the baseband signal which are then passed to the deinterleaver whichreestablishes the correct order of rate 1/2 encoded symbols. However, as it will bediscussed shortly, the order in which appended information bits are transmitted is notrestored until the decoding function has been performed. Finally, the decoder, which is3	 We shall shortly discuss the interleaving/deinterleaving function in detail.18a slightly modified rate 1/2 Viterbi decoder, decodes the received sequence of symbols,and delivers them to the reorganizer which restores the original order of the informationsequence. The Viterbi decoder is capable of constructing parallel branches in its trellisdiagram. Changing to a different modulation scheme is then equivalent to changing thenumber of parallel branches in each state transition. The output of the reorganizer is anestimate of the information bits produced by the source.Note that the same encoder/decoder is used for different codulation rates. Themapping scheme can either be realized by simple table-look-up methods, or, due toits regular structure, by a simple algorithm.The MPSK mapper/transmitter can accommodate repetition codes. The concept ofrepetition codes is based on repeating the information to lower the BER [16]. Duringbad channel conditions, the signal representation of the encoded symbol is transmittedR times. This has the same effect as multiplying the signal energy by R, and may alsohelp the signal recover from a relatively short fade by increasing signal's duration. Ofcourse, the cost of improvement in BER performance is a reduction in throughput.To accommodate repetition codes, the decoder should allow for multiple signals onthe same branch. That is, the metrics for the received repeated signals are added upfor the calculation of the final metric. For example, if Si was transmitted R times and= (rii, r{2,	 riR) is received, then the branch metric for the transition correspondingto branch signal Bk (1 < k < M) is:Mik = E h.; - Bk1 2	 (4.10)j=1194.2.1 Interleaving/DeinterleavingInterleaving is traditionally performed to convert channels with memory to memo-ryless channels. It has the same effect as removing the correlation between the samplesof Rayleigh fading, and hence combats the disastrous effect of deep fades on thereconstruction procedure in the trellis, during decoding [17]. Conventional interleavingand deinterleaving consists of storing encoded symbols into a matrix in successive rows,and transmitting the symbols in successive columns thus removing the correlation betweenthe symbols for the adjacent branches of the trellis. The interleaving matrix Zi has sizes x d where d is called the interleaving depth and s the interleaving span. The deinterleaverperforms the opposite operation. It stores the received symbols in successive columns,and delivers them to the decoder in successive rows, and hence restores the originalorder of the transmitted symbols.Conventional interleaving/deinterleaving, however, is not appropriate for ATCMPSK.Storing the encoded symbols in a matrix introduces an unavoidable delay; therefore, bythe time an interleaved symbol is sent for transmission, channel conditions may havechanged, and the encoded signal may no longer have the suitable scheme for transmission.We should therefore devise a new interleaving method. Figure 8 shows the structure ofthe interleaver for the proposed scheme.20AdaptiveSignalMapperInterleaved	 ChannelSymbolsChannelEstimatorInformationBits1111-111.Rate 1/2EncoderFigure 8 Interleaver Structure for ATCMPSK.Information bits are passed to the encoder, and the encoded symbols are then deliveredto the interleaver where they are stored in a serial fashion in successive rows of theinterleaving matrix Zi until it is filled. The commutator receives CSI from the channelestimator. If channel conditions allow for higher level codulation schemes (TC8PSK,TC16PSK), the commutator will redirect the information bit flow to the adaptive mapper,which will take in the appropriate number of information bit(s) (1 for 8PSK, or 2for TC16PSK) together with rate 1/2 encoded symbols which are delivered from theinterleaver in successive columns, to form the channel symbol.If channel conditions dictate the use of TCQPSK, the rate 1/2 encoded symbol in theinterleaving matrix will be mapped into a QPSK signal (no uncoded bits). Should channelconditions require R repetitions, the mapped QPSK signal is repeated R times. Thisprocedure is repeated, symbol by symbol, until the interleaved symbols are exhausted.21EstimatedInformation BitsReceivedSymbolsFigure 9 The structure of the deinterie,aver for ATCMPSK.The deinterleaving function is depicted in Figure 9. Received symbols are storedin the successive columns of the deinterleaving matrix Zd, and are then delivered insuccessive rows to the Viterbi decoder. By this time, the original order of the rate 1/2encoded symbols has been restored, but the appended bits are out of order, and cannotbe delivered until the whole sequence of symbols (of length equal to the interleavingsize) is decoded. The order of the sequence of information bits is then restored by thereorganizer which has the knowledge of the past history of received symbols. The channelhistory has been stored in an array of the same size as the interleaving/deinterleavingmatrix and is shared by the decoder and the reorganizer. The interleaving size is a designspecification that is very much application dependent. The most important factor isperhaps the allowable delay due to interleaving/deinterleaving/reorganizing of informationsymbols. Based on the maximum allowable delay one can choose the interleaving sizeto keep the delay below the maximum allowable value.In order to analyze the performance of the ATCMPSK scheme, we need to evaluatethe performance of various schemes used to construct it. In the next two chapters wewill analyze the BER performance of the pragmatic trellis codes used in ATCMPSK.22Chapter 5Performance of Pragmatic TCM5.1 Theoretical Bound on the BER in AWGNA bound for the BER performance of TCM schemes can be found by applying theidea of the generating function as in the case of linear codes [18]. We will follow thesame method as in [19] for performance evaluation of trellis codes. The steps involvedare as follows:• Determine the trellis structure, and finite state diagram of the code.• Break the finite state diagram at one state so that transitions start from a certainstate and end at that same state (commonly the all zero state.) This is called theerror-state diagram.• Using the trellis diagram of the code, and ED between symbols on the branches ofthe trellis diagram determine the branch gains of the error-state diagram.• From the error-state diagram derive the transfer function T(D, I).• From T(D, I) find the bit-error probability.Let XL =	 , X'L = (4, x 12,...,4) be two sequences of encodedsymbols with length L, and P (XL	 'CD be the pairwise error probability; that is, theprobability of choosing Xi L given XL was transmitted. The union bound on the probabilityof an error event can then be expressed as [19]:00P(e) E	 > P(XL)P(XL —+ x/L).	 (5.1)L=1 Xi XL9eXi23Assuming maximum-likelihood detection [18, p. 2471:P(XL X L` ) = Q(dm(XL,X1L)V21E	(5.2)wheredif (xL , VL) = ilfm(xL) — fm(XL)11 2 = E ilfm(sn) — fm (4)11 2 7	 (5.3)n=1where	 represents Euclidean distance, and fm•) is the nonlinear mapping functionfor the MPSK signal space normalized on E s . Applying the bound:Q(Vx	 y) 5_ Q(-‘5) exp (1)	 x > 0,y >	 (5.4)and noticing that	d 2m(xL,VL) — d2free > 0,	 (5.5)we can express the pairwise error probability as:	Es 2	 Yt — d2	d m k—L —L	 free')P (XL XL) (dfr" V2No exp (-41—To.kE,	 Es d2d(- free fa.171) exp (4No freeEJ 	 s	 )dfree NIT\i"c7) exP (Wo ufreeE Lexp	41V 	 (xn, 4)	o 	)n=1exp 4N d2m (xn , 4)).n=1	 o(5.6)Note that the first two terms in the above expression are independent of XL and X.Now, let:0o	 LT(D)= E p(xL) E ft Dd24(rn,4) ,L=1 XL	 XLOX'L n=1D = exp (— E,—4No)•	(5.7)24Referring to (5.1) and (5.6) we can then conclude that:Es DdfreeP(e) < Q	 free T(D).	 (5.8)(dfree Ai2NoA close look at (5.7) reveals that it may be possible to find a closed form expression forT(D) using a finite state diagram, as done for the case of linear convolutional codes. Infact, as it will be shown later, for pragmatic trellis codes, T(D) is a scalar transfer function.5.1.1 T(D) and the Error-State DiagramTo find T(D) for TCM schemes, we have to consider all possible paths leaving anarbitrary state p and merging into another state q in exactly L steps for all possible valuesof L, and error sequences of length L.Assume that all source symbols have the same probability 2 —n. We can then definean ./C/ x /V—Nis the number of states—error weight matrix G(ei) whose element (p,q) isa function of the distance between the output symbol xp_.q corresponding to a transitionfrom state p to q , and the same symbol added with noise pattern (es) multiplied by theprobability of that state transition (or equivalently the source symbol corresponding tothat transition). That is :1Gpq(ei) Dllfm(zp—q)—fm(xp_qseoir2 — n pt4In fact, the expression Dilfm(zP—q) — fm (zP— geet )112 is an upper bound for the probabilityof error between the symbols xp_,q , (xp,q e ei). The series in (5.9) is taken over allpossible parallel transitions from state p to state q.(5.9)25Now, assume that we have an error sequence EL = (el, e2, eL) and calculate thematrix:LG( ) = fl G(en).	 (5.10)n=1The element (p, q) of this matrix is the probability of a transition from state p to qwith an error sequence EL in exactly L steps. Now if we add all the elements of thismatrix, and multiply by the probability of state p ( lifor equiprobable states), we find theprobability of error for all possible state transitions corresponding to the error sequenceEL. Finally, if we sum all the probabilities for all possible error sequences of any length,we will have:00T(D) = Tr [lj i .* (E E H G(en)) [1j* .„	 (5.11)L=1 E./40 n=1where [1]„, x n represents an m x n matrix all of whose elements are one. The term insidethe brackets is called the matrix transfer function of the code and is denoted by G.Moreover, the error vectors in the sequence EL = (el, e2, eL) cannot be treated asindependent. In order to take into account their dependence, we can use a state diagramcalled the error-state diagram to describe the connection between the error vectors. Ithas been proven in [19, p. 107] that the error-state diagram is the exact copy of thestate diagram of the code except for its labels. We have to label the error-state diagramwith the matrix G(ei) where ei is the error pattern for the corresponding transition onthe error-state diagram.Furthermore, in order to find the bit-error probability, the labels of the error-statediagram can be modified to account for the number of erroneous bits in any transition.This can be accomplished by multiplying the elements Gpq(ei) by Ik where k indicates26the number of bits in error for that transition. Note that if there are parallel transitionsin the trellis diagram—that is, Gpq(ei) is a series—every element in that series has to bemultiplied by the corresponding power of / which are not necessarily equal. Now, if thenew function T(D, I) is differentiated with respect to I, and I is set to 1, we will thenhave the symbol error probability multiplied by n, the number of information bits perchannel symbol. Therefore, in order to find the bit-error probability, we must divide theexpression by n. In other words:(.1 ( 4 	E	 8T(1), 3 D ree	D = expPb1 VIree'127V-0-(5.12)As an example, the error-state diagrams of the pragmatic codes of constraint length3 have the general structure shown in Figure 10. The labels of the error-state diagramare obviously distinct for different rate pragmatic codes, but the general structure is thesame—since they differ only in the number of parallel transitions in their trellis diagram.The transfer function of the above error-state diagram is found to be [19]:1	 91397T(D, I) — 1	.— 94951[ (1 — 9495)( 1929— 96) — 929395 gigod(5.13)27g 1S 1•	S27	S 1 Figure 10 The general structure of the error-state diagram for pragmatic codes (K=3).Note that (5.12) does not take into account the probability of error associated withthe single signal error events due to parallel transitions. The transfer function approachwill not account for the probability of error due to parallel transition since the transferfunction of the error-state diagram takes into account only those transitions going froma certain state to that same state through other states. However, it is possible to findthe exact expression for probability of bit-error due to parallel transitions which will bedenoted by P11. We will discuss how to find P11 later on in this Chapter.The overall probability of bit-error is the expression found using the transfer functionapproach plus the probability of error due to parallel transitions. That is:1	 E,․ ) ,d2 OT(D Ifree 	 , )	1ai 	D = exp (—	 ) (5.14)Pb < 111 	ufree V 2N0 ij 	4NOwhere dfr, is the free distance of the code found ignoring the parallel transitions. dfr„,together with the minimum distance in parallel transitions, is the most important factor in28the determination of the performance of the code, and can be calculated in many ways .For small constraint lengths, it can be determined by inspections. In fact, if there existsa closed form expression for T(D, I), die is the smallest exponent of D in the powerseries expansion of T(D, I). For more complex codes, computational algorithms such asthe one explained in [20], [21] can be used. We can also employ a technique which isbased on the error-state diagram and its corresponding transfer function. This techniquehas been explained in sufficient detail in [19, p. 125].5.1.2 Simplified Error-State DiagramIf all the states are on equal footing, we would expect that the sum of the elementsin any row (or column) of the matrix transfer function should be independent of therow (or column) itself. In that case, we can consider only those error events leaving orcoming into a certain state without any loss of generality. That is, we can replace matrixlabels of the error-state diagram G(el) with the sum of the elements in any of its rows(or columns). All calculations can then be carried out in terms of scalars. These scalarlabels are called error-weight profiles and will be referred to by W(e1). The algebraicconditions for symmetry are explained in great detail in [22],[19, p. 106]. The set ofpragmatic trellis codes satisfy the symmetry condition to the extent that transitions fromor to any state can be considered without any loss of generality.5.2 BER Performances of Various PragmaticTCM Schemes in AWGNIn this section, we derive bounds for BER performance of three pragmatic schemesemployed in ATCMPSK; that is:1. Rate 1/2 TCQPSK292. Rate 2/3 TC8PSK3. Rate 3/4 TC16PSK.All these schemes have the same mother code, and differ in their trellis structure onlyby the number of parallel branches in their state transitions. The same conventional"good" rate 1/2 encoder is used in all three schemes. For our purposes, and to beconsistent throughout this research, we will consider the good rate 1/2 convolutionalcode of constraint length 3 as the mother code.The decoder is a slightly modified rate 1/2 Viterbi decoder which first determines theclosest symbol to the received symbol among the symbols in every parallel transition,and then operates in the conventional manner.The method of analysis is that pursued in [19] which can be extended to codeswith longer constraint lengths, but theoretical analysis become more complicated withincreasing number of states.5.2.1 Rate 1/2 TCQPSK in AWGNFigure 11, shows the trellis structure, and the signal space for this scheme. From thetrellis diagram we can find the state transition matrix :S =00 11 0	 00 0 01 1011 00 0	 00	 0 10 01(5.15)The element Spq indicates the output symbol corresponding to a transition from statep to state q. For example, to find Sib see the trellis diagram in Figure 11, and find theoutput symbol corresponding to the transition from state one (Si) to itself which in this30case is 00. As another example, the null element (0) in S13 indicates that there is nopossible transition from state Si to S3.Figure 11 Trellis diagram and the signal space for rate 1/2 TCQPSK.The next step is to find the error-weight profiles of the code. We can, as discussedbefore, use the symmetry property and simply consider one state. Assuming that thesignal energy is normalized to one, and considering the transitions from state Si (the firstrow of the transition matrix 5') the general expression for the error-weight profile W(e)of the rate 1/2 scheme is:1W(e) = 2 Dilf4(Sii)—.14(SnSe)ll 2 	DIlf4(512)—f4(S12$01 2 } , (5.16)or equivalently:31W(00) = 1W(01) –2[D11/4(oo_h(oov1)11 2 	Dllf4(11)–ho1e01l–12[D2 + D2] = D2147 (10) = {Dllh(oo–h(oo$ 10 )11 2 	Dllf4(n)–foielo)11 2 ] (5.17)= {D2 + D2] =D2W(11) 1–2[D11/4(oo)–f4(ooe11)112 + D11/4(n)–f4(11.911)11211= 2_ ED4 +D4 ] =D4.INote again that all distances have been normalized to have signal energy equal to one,and the actual energy has been accounted for in the indeterminate D.Finally, inspecting the trellis diagram, and finding the number of bits in error—powerof /—for the transitions on the error-state diagram, we find the labels for the error-statediagram. For example, in order to find g j, we find the input symbol for the transition fromSi to S2 which is 1 in this case (see Figure 11). Since there is one information bit in error,we multiply the error-weight profile corresponding to that transition, W(11), by I, andthat forms the label for the transition in the error-state diagram. Similarly, we find that:= W(11)/ = D41	 g5 = W(00)/ = I92 = W(10)I = D2I	 g6 = W(01)I = D2 Ig3 = W(10) = D 2 	g7 = W(11) = D4= W(01) = D2 .(5.18)We can then substitute these labels into (5.13), and perform some algebraic simplifications,32to obtain:Dio	= D101- + 2D 12 12 + ••• •T(D, I) = 	 (1 — 2D 2I) (5.19)Therefore,aT(D I)D1° (5.20)— 2D2 1)2 •Furthermore, from the closed form expression for T(D, I), we find that:df2 ree = 10.	 (5.21)That is, the smallest exponent of D. Therefore, applying (5.14)--noticing that there areno parallel transitions—we can conclude that:Pb Q 61.5Z1  exp ( 5E8 ) OT (D, I) INo	 2No	 al I I= 1,D=exP (—k)Q — 2exp (-150 2 •(5.22)The theoretical upper bound and the simulation results are shown in Figure 12.33-7100 3	 4	 5Nso (dB)Figure 12 BER Performance of rate 1t2 TCQPSK code in AWGN.5.2.2 Rate 2/3 TC8PSK in AWGNIn order to find a bound for the BER performance of this scheme, we will followthe same method used for rate 1/2 TCQPSK. The only difference is that now there areparallel branches due to the one extra uncoded bit.-11 0Pb -210-310-410-510-610i	2 6	7	8The trellis diagram and the signal space for this scheme are shown in Figure 13.3401010000/ 000;10/10000/011; 10/11101 /011 ;11 /11101 / 000; 11/10000/001; 10/10100 / 010;10/11001/010;11/11001 /001; 11/101W(e) = – E411 E Dilf4(si2)-f4(si2e012 .	 (5.24)parallel	 parallelDi i .f4(S11)—f4(S11ee)ii 2 +Figure 13 Trellis structure, and the signal space for the rate 2/3 pragmatic code.From the trellis diagram, we find the state transition matrix:	000;100 011;111	 0	 00	 0	 001;101 010;110	011;111 000;100	 0	 00	 0	 01 0;110 00 1 ;101S = (5.23)Since there are two parallel transitions in the trellis diagram, every element of the statetransition matrix is comprised of two symbols—corresponding to the output symbolsof the parallel transitions. The general expression for the error-weight profiles of thisscheme is therefore:The error-weight profiles for this scheme, calculated according to (5.24), are given inTable 1.35Error Pattern (el) Error-Weight Profile W(er000 1001 d2 	d2	 d2	 d2D i+D i+D i+D 1 = D0.5864010 Dd3.+Dd3+Dcg+Dd1 _ D3.414	 0 5864011 Dd3-1-Dd3+Dd3+Dd3 = D24100 Dq+Dcq+Dd:+pd:	 D4=4101 Ddg+Ddi+Dd3+Dd3 = D3.4144110 Dcqi-D4+Dd?+Dd?	 D0.586=4111 Ddi-f-Dd3+Dq+Dq	 D2=4Table 1 Error-weight profiles for the rate 213 pragmatic code.From the trellis diagram, the labels for the error-state diagram are found to be:	= W(011)I W(111)I 2	g5 = W(000)1 + W(100)12	g2 = W(010)I W(110)I 2	g6 = W(001)I W(101)I 2	93 = W(010) + W(110)I	 g7 = W(O11) + W(111)I94 = W(001) + W(101)I.Substituting these labels in (5.13) we obtain T(D, I).By either inspection or the methods described in Section 5.1, we find that :d2	 = 4.59free — • •(5.25)(5.26)To find the bit-error probability due to parallel transitions PH, note that all states andsignals on the branches emanating from those states are on equal footing. Also recall36that the probability of error between two signals separated by a distance d is equal toQ (767) . Now, looking at any branch of the trellis diagram such as the one depictedin Figure 14, we can conclude that:1P11 = —2P(100 was received1000 was transmited)d(100,000) 	1 Q (  2 \ars-) 1 ( 	2	 ON° ) 2 -ON()	 2 Q	2E.,No) •(5.27)Figure 14 Parallel branches in the trellis for the rate 2/3 code.Note that the factor 1/2 in the expression for PH stems from the fact that only onebit out of the two information bits would be in error.Finally, the overall expression for the probability of error can be expressed as:Pb < 2 V 2N01  4 . 5 8 6 E, 	 ) ex_ (  4No4.586E3) 	aT(81); I) 1 1=1; D_exp (-1190+	 (. /2E5 )2‘ 	 No(5.28)375 8 10 11 1210Pb-21010454f,...410-s10-6107	 8	 9E sR-0 (dB)-71 04The expression for the probability of error is complicated, and would not add anyinsight to the performance of the scheme. Instead, the theoretical curve 4 for the probabilityof error, and the corresponding simulation results are shown in Figure 15.Figure 15 BER performance of the rate 2/3 TC8PSK code in AWGN.5.2.3 Rate 3/4 TC16PSK in AWGNThe trellis structure and the signal space for this scheme are shown in Figure 16. Thesame method as for the other two schemes can be applied to find the distance profilesand the labels for the error-state diagram of this scheme.4	 The theoretical curves were obtained by carrying out the operations in (5.28) using the symbolic computation program Maple.381110 10000100 00100101011101101100110111110011000100001010101110010/0; 2/4; 4/8; 6/12.411111111111111111111110/3; 217; 4/11; 6/154401001/3; 3/7; 5/11; 7/151/0; 3/4; 518; 7/121111*ifr	 0/1; 2./5; 4/9; 6/130/2; 2/6; 4/10; 6114..dgiiiigg/IIIIIIIIIIIIII112; 3/6; 5110; 7/141/1; 3/5; 5/9; 7/13Figure 16 Trellis structure and signal space for rate 3/4 pragmatic code with double-Gray-coded mapping.Let di stand for the distance between two signals separated by i signals. In general,for symmetric MPSKit= 2 sin (ri.).Therefore, for the 16PSK signalling scheme:	4 = 0.152	 4 2.765	  0.586	 4 3.414	 1.235	 4 3.848	4 2.000	 4 4.000 .(5.29)(5.30)The error weight profiles of the scheme are given in Table 2.39Error Pattern (et) fileW(ei)0000 10001 D400100011 D40100 D40101 D 42	 423+D 520110 d2 	d2	 d2	 d2D 1+D 3+D 5+D 740111 D 422 +D 42621000 D41001  d2D d23+D 521010 d2 0 1+D 423+D42	 425+0 741011  42D 422+0 621100 Dd:1101 D41110 d2	 42D 5+D 721111 D4Table 2 Error weight profiles for rate 3/4 pragmatic code with double-Gray-coded mapping.The labels for the error-state diagram—following the steps explained in the previoussection—are found to be:40gi = W(0011)/ + {W(0111) + W(1011)}/ 2 + W(1111)/ 3g2 = W(0010)/ + {W(0110) + W(1010)}/2 + W(1110)/3g3 = W(0010) + {W(0110) + W(1010)}/ + W(1110)/ 2= W(0001) + {W(0101) + W(1001)}/ + wo101)r2 	(5.31)g5 = W(0000)/ + fW(0100) + W(1000)}/2 + W(1100)/3g6 W(0001)/ + {W(0101) + W(1001)}/2 + W(1101)/3= W(0011) + {W(0111) + W(1011)}I + W(1111)1 2Having determined that:df2 ree -- 1.32 ,	 (5.32)the bound on BER can be expressed as:exp	 4/1ro ) x 3 	ai Ii=i ;D.exp(1.32Es	1 OT(D,	(5.33)1.32E3 Pb 111 + Q (1,1 2N0To find F11, we can once again take advantage of the fact that all transitions are onequal footing and simply look at one transition—the transition from state S1 to itself asshown in Figure 17.41000/ 0000 0100	 01100W. 0110 / 1100Figure 17 Parallel branches in the trellis diagram of rate 3/4 pragmatic code.Now, assuming that symbol 0000 was transmitted, we can see that := -}"P(0100 rx i ed10000 tx'ed)•3P(1000 rs'ed10000 tx'ed)+2—P(1100 rxi ed10000 txi ed)3(5.34)Let R=(ri, r2) be the received signal , X=(x/, x2) the transmitted signal, and N=(ni,n2) the samples of AWGN with zero mean and variance N(/2.P(0100 rx i ed10000 tx'ed) = P(ri < 0, r2 > 01.51 = 82 ==	 + < 0, + n2 > 0) = P(ni < — 2 )P(n2 > —	 (5.35)=Q •Equivalently, we have:P(1000 rec'ed10000 tx'ed) = P(0100 reci ed10000 tx'ed)= (A) - (M)	 (5.36)42andP(1100 rec l ed10000 tx 1 ed) = P (ri < 0, r2 < Oisi = 82 = 2)= PGd	 d+ ni < 0, -2- +n2 < 0) = P(ni < --id) P (n2 < --2-d)2	 2d E,)= Q (  7  ) = Q (No(5.37)Now let Q (	 = p. We can then express the probability of bit-error due to paralleltransitions by:1	 2 „'ii = 3p(1 — p) + 3p2 1+ p(J- — p)2= e.(5.38)Finally, we can conclude that:32E,	 1.32E5	 1 OT (D, I) tPb _._ 3p + Q(\11. 2N0 ) e 	4N0	 ) x 3 at 1 /=1;D=e p (- -9-)* (5.39)Figure 18 shows theoretical and simulation results for BER performance of the scheme.4310 11 14 15 16 17-110min...n•••	 •Ellnn 	 aPb-2s n11,- A".=.1=0M F11/n11.1n111.111 11kIW	 ‘4' TO1 0-310-41010-512	 13Nso (dB)-8109Figure 18 BER performance of rate 3/4 TC16PSK code in AWGN.5.3 BER Performance of Pragmatic TCM Schemesin the Rayleigh Fading ChannelIn order to analyze the error performance of the pragmatic scheme under Rayleighfading conditions, we will assume the channel model defined in Chapter 3. To keep theanalysis as simple as possible the following assumptions are made:• The channel is frequency nonselective, and slowly fading; therefore, the amplitudeof fade is assumed constant for one symbol period or longer.• The effect of phase is fully compensated for by the receiver.Ideal pulse shaping, and hence no ISI.44• Ideal channel state information (CSI) is available at the receiver side. CSI is theamplitude of the fade which is assumed constant over at least one symbol duration.• Encoded symbols are ideally interleaved, so that the effect of deep fading isspread over time. This assumption allows us to assume that samples of fading areuncorrelated and independent, and that the channel is memoryless.The first step is to apply the union bound as done for the case of AWGN:00a2, a„,) E E P(XL) E P(XL -4 XL a2, .••,aL)•L=1 XL	 Xj$XLTherefore:[P(e) 5_E EEp(xL) > P(XL -4 ziai, a2, ..., aL)L=1 XL	 XI,OXi,00.EEp(xL) E E[p(xL nal, a2, aL)] •L=1 Xi,	 XI,OXLCO(5.40)(5.41)Now, since, as discussed in the last section, the effect of the fading is equivalent toattenuating the signal energy, we have:P(XL	 = QLE, E avm (s„,x1n )n=1 (5.42)2N0I45Furthermore, we can use the bound (5.4) to obtains :I —Es ( a!d2m(sn, xn)) \1 	 n=1 2 exP	4No\	 I—Es (CiVm (Xn 7 S nt )) 9	= 1 A exp	 .4N0'' n=1Moreover,E[P (X L -+ nal, a2, ..., aL)]= I...	 P (XL --0 XL lai, ..., aL)fai,...,at (al, ..., aL)dai—daL,al	 ar, Q/ L	 \Es E aid2m (x„, xn )n=1\ 2N0 (5.43)(5.44)and since we have assumed ideal interleaving ai's are independent, and identicallydistributed Rayleigh random variables. That is:fai,...,aL(cri, ..., aL) = fai (al)... fa L (aL).Hence,L—a2nEsd2m(s„, s'nE[P (XL --+ X'L lai , a2..., a LA = H f exp ( ) 4N0n= lan )f(5.45)clan . (5.46)Using the Rayleigh probability density function, we find that:cof exp ( —a2 .Es d2m (xn , x'n)) f (a)da = I exp ( —a2Esclif(xn, xn)) 2cte-'2da4No	 4Noa	 01= 	 i1 + +4 o df(X n , X n )(5.47)5 Using the Chernoff bound as done in [191 leads to a bound twice as large.46Substituting the above expression in (5.41) we obtain:oo1	 1 P(e) 9 EEP(xL) E 1L=1 XL	 XiOXL n=1 + if 1 dif(Xn, Xin) (5.48)Now, let:1 T () = E p(xL) E TTNo	 + tkci2m(xn,x1n)	(5.49)L=1 XL	X1OXL n=1TO can be found using the error-state diagram. The labels of the error state diagrammust, however, be changed to reflect the effect of fading. In fact, comparing (5.7), thetransfer function for the error-state diagram in AWGN, with (5.49), we can concludethat in order to obtain T (ft) all one must do is to modify the labels of the error-statediagram—elements of matrices G(e1) — according to the following transformation:Dem(xn,4) 	1 (5J0)1 + Atd2m (sn ,xin )The corresponding transfer function is no longer an explicit function of D. The labels ofthe transitions in the error-state diagram in Rayleigh fading are in fact the probability oferror for those transitions averaged over the Rayleigh distribution.Finally, to find the bit-error probability, we have to multiply every transition labelby the power of 1 corresponding to the number of bit errors in that transition. We canthen express the overall probability of error by:1 	Pll + 2n	 ar	 11=1 • (5.51)47=11	 2 E8exp (—y ITT;) a exp (—a2)dydaNig (5.55)o av*-1(1 \s/ Es /No = -4- 	1 + Es /NoThe expressions for the probability of error due to parallel transitions PH in AWGNcan be used to derive the error probability for the Rayleigh fading channel. In general, ifPHIAWGN} = g Nothen for the Rayleigh fading channel with ideal CSI we have :E \Pill f adingla = ail = g (as2 7V-S70)Averaging out the expression over the Rayleigh distribution, we get:PII E [pH { fading la = ai}] = E[g 	 k )}.= g(a2Ivo ) 2a exp (—a2)da.oTherefore, using (5.27), we can conclude that for rate 2/3 TC8PSK:Pit = 1	 \/ 2E,--,17—) 2a exp (—a2)da.o	( 	No00 0000(5.52)(5.53)(5.54)and for rate 3/4 TC16PSK, using (5.38), we find that:PII = 2 3 Q a Tifo-E )a exp (—a2)da.0I— 3	2 + E(—	 SN()Es' NO	)(5.56)48-6103 3113 19 21 23 25 27 297 11 15 175 9Rayleigh fadingIdeal interleavingIdeal CSIK=3-210-310-401.7.7110-410S (dB)N010PbThe upper bound for overall probability of error for TC8PSK can therefore beexpressed as:Pb 171	 + E i(1 \ I  E3 iNON0 )and for TC16PSK as:1 aT( 0 ,i)+ 2n	 sat	Ii= i (5.57)(5.58)aT(41,I3	) Pb < 1 (1 	 E, /No	 —2 + E,IN0	 2n	 al	 1-1Performance curves for different rate TCMPSK schemes are shown in Figure 19.Figure 19 BER performance of ideally interleaved trellis-coded MPSK in Rayleigh fading.The horizontal axis of Figure 19 is the average SNR which is defined as:E, „ E,— =No	 No0(5.59)49As shown in Figure 19, there is a 2 dB asymptotic difference between the simu-lation6and theoretical BER curves for rate 1/2 TCQPSK. However, for the other twoschemes, at high SNR, simulations and theory lead to identical results. This is becausethe theoretical expressions for PH are exact. Moreover, the overall probability of erroris dominated by PH. Therefore, at high SNR, the theoretical upper bound on the BERperformance of TC8PSK and TC16PSK are almost exact.6 	For details regarding the simulations refer to Chapter 7.50Chapter 6Performance of ATCMPSK in theRayleigh Fading Channel6.1 Bound on the BER performance of ATCMPSKin the Rayleigh Fading ChannelFor the purpose of theoretical analysis, the assumptions for the nonadaptive case willremain as such. The ATCMPSK scheme under study is the one discussed in Section 4.2.We assume symbol by symbol adaptation. That is, the codulator adjusts its rate accordingto ideal CSI obtained from the receiver at the start of every symbol transmission interval,and the decoder adjust its parameters at every decoding instant. The metric at everydecoding instant is calculated according to the CSI, and is thus a function of the fadingamplitude.The union bound for the probability of error is still valid for the adaptive scheme:00p(e)<EEp(x,) E	 aL)i•	 (6.1)L=1 XL,	 Xi OXLBut the conditional probability of an error event of length L is different from thenonadaptive case. The scheme employed at each signalling interval depends on theamplitude of fading in that interval. Let Rn be the number of repetitions for the intervaln. The BER performance of a scheme with Rn repetitions and energy Es is the same asits performance without any repetitions and energy RnEs. In fact, the effect of repetitionis equivalent to increasing the signal energy, and hence a system which adjusts thesignal energy according to channel conditions would theoretically have the same BER51performance. Moreover, it is assumed that all R„ symbols are equally attenuated; that is,fading is constant over R,, symbols. This assumption is made only for signalling intervalswhere repetition schemes are employed. For all other schemes, fading is assumed constantover one symbol period only. Therefore: LE, E Rn aict2m(n) (sn ,xinn=1\P (XL -4 XL lai, ..., aL) = Q (6.2)2 No 7/Note that M(n) can take on 3 different values: 4 for distances in QPSK, and 8, 16for 8PSK, and 16PSK, respectively. Applying the bound (5.4), we obtain:Q/	 LE, E R„cq,d2m(n) (x.,Xin)n=1 2N0\Li ex_ H p (:-.E3 	 2 2Rnandm( (xn)kt'zi 4) . (6.3))< 2 n=1 1- 4NoTherefore,LE[P (X L --4 'CIL I ai , a2..., a L)} = fi f expn=1„n (---Es D 2 2 	,	 , \4N0 n"andM(n)(Xn7 x 'n)) f (an)dan.(6.4)The adaptation thresholds are pi, p2, p3, p4. The thresholds have been obtainedusing the set of BER performance curves of all the different codes in AWGN 7 so thatBER is kept below a certain error roof E. To determine the adaptation thresholds in terms7	 The way in which these thresholds are chosen is discussed in great detail Section 7.4.152vl	 v2exp 4No23andisn , xn)) j (an Man + exp (No4 2a2 ai	 Xn )) I(an )a„ann—Es 	„	 Es	 „ of the fading amplitude note that:Es< an  <	 Ai n	 — Ai+i	 < a <	Pi -1-1 Es /No — n 	Es I No • (6.5)The adaptation thresholds for the fading amplitude are then determined by therelationship	 = vi. Therefore, the integral in (6.4) becomes:o0f• eXpV200▪J exp--ana4 kX11, Xnj)4NO—Es 2 j2—Es 2 ,./2 (a "16	 xin)4No nf (cen)dan + f exp	o(-4N ce;,di(xn, 4)) f (an)dan—E,1/3f(an )dan ,V4where f(an ) is the Rayleigh density function. Furthermore,viJ	—Ria2Esd;2t(sn, xin- ) 	exp	 ) exp (—a2 )da4Novi-iexp (—vim ign,iEs/No) — exp (— 4,3n,iEs I No) fin,i Es I No	exp	— exp ( -10n,i) fin,iE s I Nowhere1 	+ Rid2n (s n s in) fin = /No 4Substituting (6.7) into (6.1) we obtain:(6.6)(6.7)(6.8)5300P(e)	P(XL)	 HL=1 XL	 XiOXL n=11 1 — exP (—piN,i) + exP ( —µ1/3n,2) exP ( — A2/8-n,2) +fin,lEs/NO	 /3n,2Es/NOexP (-11218n,3) — exP (-443,8n,3) .4.18n,3 Es / NoexP ( -143/3n,4) — exP ( -414/3n,4)	 exP (- 1140n,5))18n,4E8 /NO	 18n,5Es/N0 •(6.9)Call the term inside the brackets in (6.9) h(xn , 4). We can then express the upperbound on the error-event probability by:1 "P(e) 5_ -EEP(X.L) > H h(xn ,L=1 XL	 XiOXL 11=1(6.10)The right hand side of (6.10) is the generating function T (40 of the error-statediagram of the adaptive scheme. Comparing it with (5.7), we can conclude that in orderto find T (), the labels of the error-state diagram for AWGN have to be modifiedaccording to the following transformation:Dem(x-r in)	 h(xn, x in) •	 (6.11)Finally, to obtain T	 I) we must multiply each term in h(xn , 4) by Ik, where k isthe number of bits in error corresponding to that term. The upper bound on BER canthen be expressed by:1 aT(k,i)Pb < Pll 27-2 	 aI	 11=1 (6.12)54where TT is the average number of information bits per adaptation interval, and is equalto:Ti = P(TCQPSK) + 2P(TC8PSK) + 3P(TC16PSK)103	 V4	 00= f 2a exp (-a2) da + 2 f 2a exp (--a2) da + 3 f 2a exp (-a2) da1/3 I/4= 1 + exp (_a_)+exp(+Eis_Lirvo).(6.13)To find PH we should note that parallel transitions contribute to the probability of erroronly when channel conditions dictate the use of TC8PSK and TC16PSK. Therefore:V4	 COPII = f 2 Q	 2E\/--1)2ae-a2 da + I -2 Q (a\ I --±E )2ae-'2 da.No	 3	 No	 (6.14)1/3 V4Applying (5.4) and integrating the expression above, we find:1 e -/23 (1+ ES2No	 e-P4 (1 + Es2NoPll 4	 E,1+1 e-it4(1+ Es/1 N 3	 E,1+ N.0Finally, we can express the overall probability of bit-error by:12Tz	 ai	 Ii=i •2N0 (6.15)(6.16)The BER performance curves obtained by simulations and theory are given in Figure20.5518 20 22	 24 261 	Error Roof = 0.1Symbol adaptationDelayless FeedbackPb-21041010-5100	 2	 4	 6	 8	 10	 12	 14	 16-ffs37-0 (dB)For the purpose of simulations, a finite interleaving depth of 64 x 32 was chosen.This interleaving size was found to be almost optimum (see Section 7.4.2.) Furthermore,symbols during repetition periods have been attenuated according to fading at the start oftheir transmission while for theory we have assumed ideal interleaving of non-repeatedsymbols and equal attenuation during repetition periods. As expected, the upper boundis fairly accurate at high SNR where it is 2 dB away from the simulation results.Figure 20 Theoretical and simulation curves for the BER performance of ATCMPSK under Rayleigh Fading.6.2 Throughput of ATCMPSKDue to its adaptive nature, the throughput of the ATCMPSK scheme will vary as afunction of the immediate SNR. The throughput can be defined as the average numberof information bits transmitted per symbol duration Ts.5631	 1	 (	 P1	 1	 . 	P2  )= — —6 exp —	(exp Es/No+ exp E,AI3No ) exp/14 E, /No ).(6.20)But the number of information bits per channel symbol is 1/3 for QPSK with threerepetitions, 1/2 for QPSK with two repetitions, one for TCQPSK, two for TC8PSK, andthree for TC16PSK. Therefore:1	 1= 3—P(0 < a < ) + 2—P(vi < a < v2 ) + P(v2 < < v3)—  + 2P(v3 < a < v4) + 3P(v4 < a < oo).(6.17)Assuming that the there is no correlation between adjacent transmission periods withregards to the choice of throughput, and using the fact that samples of fading are Rayleighdistributed, we have that:P(vi < a < vi+ i) = exp (—vn — exp (-4+1 ).	 (6.18)Recall that111  Pi = Es IlVo' (6.19)where pi's are the adaptation thresholds found from the performance of the differentschemes in AWGN. The throughput of the proposed ATCMPSK can hence be expressedas:The above expression has been derived by treating every symbol transmission periodas independent from its adjacent neighbours. This is true when there are no repetitionsused. However, when repetition schemes are employed the adjacent repeated symbols57are not independent; however, since we have slow fading, this does not effect the overallexpression for throughput, and the approximation is hence very accurate.Note that, for very small SNR, the above expression approaches 1/3 (the throughputof the triple repetition TCQPSK scheme), and for very high SNR it approaches 3 (thethroughput of the TC16PSK scheme).In general, for a system employing up to R repetitions and 2" 1 PSK, the throughputcan be expressed as:R-2 exp	n-2p PR-I-i =	 (R — i)(R — i — 1) E ex 	 ES No) .1=0	 i=0 (6.21)Throughput curves obtained by simulations and theory are given in Figure 21. The curvesoverlap since theoretical results are almost exact—there is a negligible discrepancy whichis attributed to the correlation between the adjacent symbols during repetition periodswhich has been ignored in theory.583210 i	 1	 I	 i_ Error Roof = 0.1- Symbol adaptation- Delayless Feedback0	 2	 4	 6	 8	 10	 12	 14	 16	 18	 20	 22	 24	 26.5N0Figure 21 Information throughput of ATCMPSK—simulation and theory.59Chapter 7Description of Computer Simulations7.1 The AWGN ChannelAll simulations have been carried out in baseband, and the overall transfer functionof the transmit and received filters was assumed to satisfy Nyquist's first criterion forpulse transmission; that is, there is no ISI at the sampling instant. Therefore, the effectof the AWGN on the transmitted signal can be modelled in the following manner.Assume that (sib sit) is the baseband equivalent of the transmitted signal. Thebaseband representation of the received signal is hence:(ril , ri2)	 (811 + nil, si2	 nit)	 (7.1)where nu, nit are two independent Gaussian random variables with zero mean andvariance N0/2.In our simulations, to generate a sequence of AWGN samples, the Gaussian randomgenerator routine in [23, p. 217] together with the whitening procedure in [23, p. 207]were used. The first routine generates an array of independent Gaussian noise samples.In all the simulations performed for the purpose of this thesis, the length of this arraywas chosen as 10000. Whenever the simulations run out of noise samples, a whole newarray is generated. The seed used for the random number generator is the //sec field ofthe time function in the C library.The simulated signal space has unit energy (E3=1). The variance of Gaussian noiseto operate at a certain SNR is hence determined as follows:60No	 Es	 1•Q2 = — SNR = =2	 No No Thereforecr 2 —	 1 2SNR •7.2 The Rayleigh Fading ChannelThe following assumptions were made in the simulation of the Rayleigh fadingchannel:1. The effect of fading on the phase of the transmitted signal is not included in themodel. It has been assumed, as in the derivation of the theoretical bounds, that theeffect of phase has been fully compensated for.2. Unless indicated otherwise, the carrier frequency fc has been chosen as 864 MHz,and the Baud rate as 10000 symbols/sec or equivalently the symbol durationTS = 0.1 msec. Therefore,Bd -v-c f = 0.8v,	(7.4)BdT, = 8 x 10 -5 vwhere Bd is the maximum Doppler shift, v is the speed of the vehicle, and c ispropagation speed assumed equal to the speed of light— 108 x 10 7 km/hr.3. It is assumed that fading is sufficiently slow so that it can be considered constant atleast during one symbol interval. That is, the symbol duration Ts is much smaller(7.2)(7.3)61	than the coherence time of the channel defined as (At), =	 or equivalently:8 x 10-5v « 1 or(7.5)v << 104 km/hr.4. The attenuation during one symbol period is the value of fading at the start of eachtransmission.5. Unlike the theory, the signals transmitted during a repetition period are fadedaccording to channel conditions at the start of each symbol transmission.According to these assumptions, the received signal can be modelled by:rig) = (aisii nil, aisi2 ni2), (7.6)where the samples of fading ai's are generated according to Jakes' model of Rayleighfading channel.7.2.1 Jakes' Model of Rayleigh Fading (Land-Mobile)Jakes [8] showed that the in-phase and quadrature components of the envelope ofthe fading signal can be expressed by:x,(t) = 2 E cos (7) cos (27rf,,t) + cos (27rfrnt),n=1goXs (t) = 2 E sin ( '7rn ) cos (27rf,t) + cos (21- fmt).n=1	 NOfm is the maximum Doppler frequency, andfn = frn cos ,2rn , n = 1, 2, —, go.1V1Cr = 4A.to + 2.(7.7)(7.8)62N0 8 is the number of low frequency oscillators needed to effectively simulate the multipatheffect. It has been shown in [8, p. 69] that 8 oscillators are sufficient for this purpose.A sample of fading at time kT, can be expressed by:a(kT) = Jx2(kT) xi(kT).	 (7.9)We have [8, p. 72): E [4(0] = No +1,E[4(t)] = go,(7.10)or equivalently:E [x,2 (t) 4(t)] = 21Cr0 + 1. (7.11)Therefore, in order to normalize the fading energy to one, we must have:4(107) . -1- xi(kT) a(kT) =2N0 + 1(7.12)The block diagram of the baseband fading simulator is shown in Figure 22. The modelwas simulated using the C programming language.In order to simulate ideal interleaving, one can simply choose x,(t), xs(t) to be twoindependent Gaussian random numbers with zero mean and variance 0.5, hence removingthe correlation between the symbols transmitted over the channel. The Gaussian randomgenerator explained in the last section was used to generate all the Gaussian numbersneeded for this purpose.8	 Note that in this section No does not represent the AWGN spectral density.632sin( 3 1)cos(2n fit) 2cost Oi )cos(2 irfd t)a(t)••••••41,0cos(2 itfAot)2sin(13A0) 2cos(fi AFigure 22 Block diagram of the baseband land-mobile fading simulator.7.3 Simulation of the codulator/decodulatorWe have chosen an encoder of constraint length three, and a decoding depth of 18(six times the constraint length of code [18].) It was verified, by means of computersimulations, that for all practical purposes the coding loss due to the finite decodingdepth is negligible for a decoding depth of 18.64Information bits are generated using the C library function lrand48 which returns apositive long integer in the interval [0, 2 31 ]. The 18th least significant bit of the returnedinteger value is chosen as the random bit. The codulation scheme to be used is determinedaccording to the value of the immediate SNR. It is assumed that CSI is available to boththe encoder and the decoder. The encoder is a rate 1/2 convolutional encoder with extrainputs for uncoded bits. Changing the coding rate is equivalent to varying the number ofuncoded bits per symbol. Encoded symbols are then mapped into the MPSK signal spacewith unit energy. If channel conditions indicate the use of repetitions, the transmitterrepeats the codulated symbol R times. Note that in the theoretical analysis we assumedthat repeated symbols are faded equally. For the purpose of simulations, however, a morerealistic model is used, and the repeated signals are attenuated at each signalling periodby the fading component in that period.The decoder has ideal CSI, and is aware of the codulation scheme employed for anyreceived signal since the request has been made at the receiver side. The decoder makesits first decision on parallel branches of the trellis by choosing the closest signal to thereceived signal. The metric used is Gaussian metric which, for any state, and at eachdecoding instant is normalized with respect to the smallest metric for that state. Thevalues of the state metrics are not quantized. The decoder has to store past informationabout state transitions, and input symbols leading to those transmissions in a historyarray whose size is determined by the decoding depth. Finally, decoded symbols andcorresponding information symbols are compared, and the BER performance of thescheme is determined.65-410-3	-1 1-1, 1 1	 g 3	 5	 7 gE, (dB)No157.4 Performance Evaluation Results7.4.1 Effect of Error Thresholds on the BERPerformance and ThroughputAs it was discussed before, the adaptation thresholds were chosen in a way as tokeep the BER below an error roof. Performance curves of the scheme in AWGN wereused to determine the thresholds for changing from one scheme to another. Figure 23depicts this concept.-110Pb-210-310-510Figure 23 Adaptation thresholds for ATCMPSK.662618 22 24-410-61010	 12	 14ES (dB)Tio-a10 	0 4 16As the error roof is lowered further, the thresholds for all scheme move towards theright. That is, schemes with worse BER performance and higher throughput are employedat higher SNR. This will result in a better overall BER performance but lower throughput.For high SNR, however, the difference in throughput between two schemes with differenterror roofs becomes very small while the BER performance remains significantly different.Moreover, as the error roof is lowered, slopes of the performance curves approachinfinity, and, therefore, further lowering of the roof will not have any effect on thethresholds. Figure 24 shows the BER performance of ATCMPSK with different errorroofs.-1Rayleigh fadingNo interleavingSpeed=1D0 km/hrIdeal CSIK=3-------- ME1111111111111MMINMI10Pb10-310Figure 24 BER performance of ATCMPSK with different error roofs.6760 2 4 22 24 2618 20I Rayleigh fadingNo interleavingIdeal CSIK=3Tl321010	 12	 14ES (dB)17 016It can be seen that BER performance of the scheme using an error roof of 0.1 andthose using lower roofs are significantly different. However, as the roof is lowered, thedifference becomes smaller and smaller—up to the point where the thresholds overlapand the performance remains constant.The throughput curves for schemes using different error roofs are shown in Figure 25.Figure 25 Throughput of ATCMPSK for different error roofs.As expected, the throughput of the schemes using larger error roofs is significantlybetter than those with smaller ones. Therefore, the question: "what are the optimumthresholds?" is not easy to answer. In fact, because of the trade-off between error68performance and throughput, it is impossible to quantify the difference between twoschemes using different error roofs. One can observe the BER and throughput curves ata certain SNR and decide which one is more suitable for a certain application. However,at high SNR the difference between throughputs becomes small while the differencebetween BER performances is still significant. Therefore, if the system is to operate athigh SNR, it is better to use a relatively small error roof.Introducing error roofs is not the only possible way of choosing the thresholds. Infact, we can obtain an infinite family of performance and throughput curves within theconstraints that bound the error performance and the throughput. We can shape theperformance curves in any way we wish—within those constraints.7.4.2 Effect of Finite InterleavingAs discussed before, the effect of interleaving is the same as removing the correlationbetween adjacent symbols on the trellis diagram. As the interleaving depth is increased,the correlation between adjacent symbols is reduced until any further increase wouldnot result in any more improvement. BER performance of ATCMPSK with an errorroof of 0.01 and different interleaving depths are given in Figure 26. It is evident thatinterleaving results in considerable improvement in performance. The interleaving depthof 32 proved to be nearly optimum for the specific example shown above. It was verifiedthat an interleaving depth of 64 does not result in any further improvement.It is important to note that the improvement resulting from interleaving is moresignificant at high values of SNR. This is because, at low SNR, the distortion due tothe AWGN is more prevalent while at high SNR it becomes almost negligible. At high69Pb-2101 0-410SNR, therefore, the major reason for distortion is the attenuation due to fading whosecontribution to BER can be reduced by interleaving.	*4EMI-on-interleaved	 Error Roof = 0.1Delay less FeedbackFinite interleaving =Speed=100 km/hrSymbol Adaptation	NENTOOTTE NE 111111112• 2* 16	 16 * 8-5100	2	 4	 8	 8	 10	 12	 14	 18	18	 20	 22	 24	 28Ns (dB)0Figure 26 Effect of finite interleaving on BER performance of ATCMPSK.7.4.3 Comparison of ATCMPSK with Nonadaptive TCM SchemesBER performance curves for ATCMPSK with an error roof of 0.01, and the fixedrate 1/2 TCQPSK are given in Figure 27. Included in the same graph is the throughputcurve for the adaptive schemes 9 .9	 Confidence intervals for the least accurate points of the BER performance curves for ATCMPSK are given in Appendix C.70TCQPSKinterleavednterleavedl	TCMPSK32 * 16nn'Nom	 milmhz•-n 	 mirtlan1111101111111•	 rimall ==.1PralTCQPSKnon-interleavedWMEWI.=111•Nn10101nWw4111•1111	 11n110.11r,.........._mmin ,w.....m...mi.n 	ni	 n..--.....,um -n 	 nommilow	 mownMENIMPAIMI	 'NUM	 11n7MMIIIAMM , -,‘r......,-.....,... 	 ._-.....n1/411n•nn••••n•nnn•1	 M....'"••, • IIIHMONNIIMn11=11WEMPrAIIIIMIIIMMIIIIIIIIIIIIIIMMI	 EMIli nPal	 IIIII.	 .16M.41MMIIIIMIIIMIMEMMI••n11Id1111. 	mi...nn.:41nnnn•••=•n.-Iinnnnnnnn•11111n4•11111•nnnWAMUMEWORMIII. WOWIATCMPSKnoninterleaved32—1-110Pb -210-310-410-510-8102410	 12	 14	 16Es. (dB)N0Figure 27 BER curves for ATCMPSK and fixed rate TCQPSK, withsuperimposed curve of the throughput performance of ATCMPSK.18	20	22As shown in Figure 27, there is a lot to be gained through interleaving in eitherATCMPSK or TCQPSK. It is also evident that interleaved ATCMPSK is significantlybetter in terms of BER performance. To quantify the gain, we can look at the pointwhere the BER curves of interleaved ATCMPSK and TCQPSK meet. This occurs at10.5 dB, where the throughput of ATCMPSK is almost 1.8. That is, for the sameBER the information throughput has almost doubled. We can look at the gain in adifferent way. The throughput of the ATCMPSK increases monotonically with SNR, and71-7100 — 026at SNR 5.2 dB it becomes equal to one which is the throughput of TCQPSK. At thatSNR, the BER of the adaptive scheme is around le while TCQPSK has a BER of 10-2 .That is, we have a BER improvement of almost one order of magnitude for the samethroughput. Yet another way of comparing the performances is to find the SNR at whichthe performance of TCQPSK is the same as that of ATCMPSK with a throughput of one.We find that this happens at about 8.5 dB. Hence, we can conclude that there is morethan 3 dB gain for the special case of interleaved ATCMPSK with an error roof of 0.01.Now, if we compare the noninterleaved versions of both schemes, we see that theBER curves are almost parallel, with the adaptive scheme 8 – 10 dB better than the fixedrate code. At the first glance, this may seem unusual and contrary to common sense.One could—wrongly—suggest that since at high SNR most of the channel uses in theadaptive scheme are TC16PSK, then a better scheme such as TCQPSK should eventuallybecome better in terms of its BER performance. However, this suggestion is not true.The reason for this is a phenomenon that can be called error leakage. It is obviousthat occurrence of errors at high SNR are due to deep fades. It is also true that at highSNR the prevailing scheme with regards to the number of channel usage is the highestrate code TC16PSK. However, most errors for fixed rate TCQPSK happen at very lowimmediate SNR values at which ATCMPSK uses the least prevalent schemes; that is,TCQPSK with repetitions, which are superior to single transmission of TCQPSK signals.To clarify this point, the details of the error events that have contributed to the BER ofthe noninterleaved ATCMPSK—shown in Figure 27—are given in Table 3.72Scheme Number ofChannel usesNumber ofbits in errorPercentage ofTotal usagePercentage oftotal errorsTCQPSK 3repetitions 23569 65 0.12% 6.7%TCQPSK 2repetitions 37374 21 0.19% 2.2%TCQPSK norepetitions 227752 49 1.1% 5.3%TCQPSK 919350 244 4.6% 25%TC16PSK 18791955 596 94% 60.8%Table 3 Proportion of errors for the various schemes used in ATCMPSK.Error roof = 0.01, interleaving size = 32*16, SNR=24 dB.The results given in Table 3 are for ATCMPSK with no interleaving and an error roofof 0.01 with SNR = 24 dB. There are two important points illustrated by this table. Firstof all, 0.3% of total transmissions are carried out using schemes better than TCQPSK.This may appear as a small number, but if those transmission had been performed usingTCQPSK, and if they had all been in error, then the probability of error would haveincreased by 0.003. Of course, not all these symbols are in error when TCQPSK isemployed. This—and the fact that higher rate codes with worse BER are used—is whythe difference between the two schemes at that SNR is about 0.00025 (see Figure 27).One other important point is that although the low rate repetition schemes are usedseldom, their contribution to BER is significant. To explain this we must realize thatthese schemes occur during very deep fades and that is why they contribute so muchto the overall BER.The comparison made between ATCMPSK and TCQPSK is only valid at relativelylow values of SNR where the throughput of the adaptive scheme is comparable to that of73the fixed rate code. At high SNR, where the throughput of ATCMPSK is considerablylarger than one we can compare the scheme with the rate 2/3 pragmatic TC8PSK scheme,as given in Figure 28.I...„	 i TC8PSKAMMIM interleaved•nn 	 nnnnnnn••MIONIMMININIMINYnnn	nnnnn 	 W• nnnn•n•	 -..••n•••n••n••••••••=1AZUMMINNINIMOMM=11.011M-MraIMM===1111r 	• 111111p	ATCMPSKa noninterleaved• 11111101111/41111a110•01	 nn=1INIIIMI11105111111M	MMEEN	 Mib."51.	_par 	Throughput ofATCMPSK 	n411 ATCMPSK	 interleaved-110Pb -210-310-410-510-6103ll— 2n• •::.V1111111111111-7100 2 8	 10	 12	 14	 16 18 20 22 24 — 026N(dB)N0Figure 28 BER curves for ATCMPSK and fixed rate TC8PSK, withsuperimposed curve of the throughput performance of ATCMPSK.The throughput of the ATCMPSK scheme is 2 which is the throughput of TC8PSKscheme at SNR = 12dB. The BER for ATCMPSK at 12 dB is 2x 10 -4 while for TC8PSK itis 10 -2 which is 50 times as large. Therefore we get an improvement of almost two orders74of magnitude. In terms of dB gain, we can observe—by interpolation—that TC8PSK hasthe same BER (as ATCMPSK at 12 dB) at SNR=28 dB which translates to 16 dB gain.Moreover, to achieve a throughput of 3 using a fixed rate code we need to use ascheme such as TC16PSK. The gain obtained by applying the adaptive scheme is evengreater in this case (see Figure 19). We can therefore conclude that a substantial gainon the order of 3 — 20 dB is obtained using ATCMPSK with the error roof of 0.01.Using different thresholds, we can further improve the BER performance at the expenseof a reduction in throughput.7.4.4 Effect of Vehicle Speed or BdTs ProductSince we are considering only the effect of fading amplitude on the transmittedsignal, we should expect that increasing the speed of the vehicle—and hence the BdTsproduct—would only correspond to less correlated adjacent signals. That is, as the speedof the vehicle increases, the fading process becomes faster, and attenuated signals becomeless and less correlated. Therefore, we would expect that the performance of the systemwill improve with increasing speed. In fact, the limiting case for this improvement isthe case of ideally interleaved scheme in which there is no correlation between adjacentsignals. Figure 29 shows the performance curves for the adaptive scheme for differentvehicle speeds.One must be aware, however, that, in most practical applications using noncoherentand differential detection schemes, the BER performance becomes progressively worsewith increasing speed. The degradation in performance is reflected through error floorswhich occur at higher error probabilities for higher speeds.75-21 0-31 0'4INn 	 111n111=1n11111111•I'n%:%-s.n311111111.	 I 4 	 s1n111 1111.11 nLL.M111 11No interleavingError Roof = 0.1Symbol AdaptationDelayless FeedbackmirrammiwnoinountanINIMMOMMIII.Rum%OMMINI,.1•1••nnnnnVnnnn=111•nn•nnn• • •aing/kM •••/11n71. •nnnnnnNIIIIIIIII A•nn•nn•11:1•nn•nn11orm0111111/IMIFMO WI-41 0    100 Km/hr-510 	0	2	 4	 6	 8	 10	 12	 14	16	18	 20	22	24Es. (dB)NoInterleaved64 x 32,26Figure 29 Effect of vehicle speed on the BER performance of ATCMPSK with delayless feedback.Furthermore, if there is delay in the feedback channel, increasing the speed willmake CSI less accurate to the point that eventually increasing the speed will result indegradation of performance. The effect of speed on the ATCMPSK scheme with afeedback delay equivalent to one symbol duration (0.1 msec for a Baud rate of 10000)is shown in Figure 30.761010No interleavingError Roof = 0.1Symbol AdaptationFeedback Delay = Ts10WAnnn.•1n111111•MIE111111111111M11111n∎n111/41111111► 31gW∎n111.".442	 4	 6	 8	 10	 12	 14	 16	18	20	22	24	26ES (dB)N0Figure 30 Effect of vehicle speed on the BER performance of ATCMPSK, feedback delay = 1 symbol duration.The BER performance of a faster moving mobile is still better, but the improvementin performance due to less correlation in adjacent symbols has become less significant.If the feedback delay is increased to three symbol durations, then the picture changesgreatly. The performance gets progressively worse as the speed of the mobile increases.Figure 31 depicts this phenomenon.-177-310No interleavingError Roof = 0.1Symbol AdaptationFeedback Delay = 3 Ts-110-2100	 2	 6	 8	 10	 12	 14	 16	 18	 20	 22	 24	 26ES (dB)N0Figure 31 Effect of vehicle speed on the BER performance of ATCMPSK, feedback delay = 3 symbol durations.As the speed of the vehicle increases, CSI becomes less and less accurate; that is,the delay in the feedback channel translates into a greater change in channel conditions.Therefore, the BER performance degrades with increasing speed.7.4.5 Effect of Feedback Delay and Adaptation IntervalIn practice, it is very difficult to achieve delayless feedback. In order to take intoaccount the effect of delay, we will assume that delay r is constant, and therefore thereceiver is aware of the behaviour of the transmitter at any signalling instant. That is,the receiver knows exactly what codulation scheme has been used by the transmitter.78In order to observe the effect of feedback delay, a vehicle speed of 100 km/hr andfeedback delays equivalent to up to five symbol duration are considered; that is the rangeBdr 0.01...0.04. Figure 32 shows how the performance of the scheme deteriorateswith increasing feedback delay.-110No interleavingError Roof = 0.1Speed=100 km/hrSymbol Adaptation-210-310le4.4.3a15?ek.0- 2	 4	 6	 8	 10	 12	 14	 16	 18ES (dB)NoFigure 32 Effect of feedback delay on the BER performance of ATCMPSK.A small feedback delay—much smaller than the maximum duration of fades—doesnot amount to a considerable change in performance. However, when the delay becomeslarge—as for the case of 3 symbol durations in Figure 32—the degradation in performance-410 	0 20	 22	 24	 2679becomes very significant. This is due to the fact that CSI becomes more outdated as thedelay in the feedback channel increases.All the simulations so far have assumed ideal (symbol-by-symbol) adaptation. Theeffect of adapting the codulation rate to channel conditions at fixed intervals greater than1 symbol duration is given in Figure 33.-1No interleavingError Roof = 0.1Speed=100 km/hrDelayless Feedback-410 	0 2	 4	 6	 8	 10	 12	 14	 16	 18	 205 (dB)N0Figure 33 Effect of nonideal adaptation on the BER performance of ATCMPSK.As expected, the BER performance degrades progressively as the duration of theadaptation interval is increased. This is again due to the fact that channel conditions atthe start of an adaptation interval may change significantly before the next interval has22	 24	 2680been started. Moreover, the effect of the adaptation interval is of course a function of thefading process. For fast fading, a few symbol durations may result in considerable changein channel conditions, and therefore an adaptation interval of such length will degrade theperformance significantly. However, for very slow fading we may be able to get awaywith a few symbol duration between two adaptation instants without paying a big pricein performance. Symbol-by-symbol adaptation is of course ideal for every application.But, it may not be practically feasible in which case the length of the adaptation intervalshould be chosen to keep the BER at a tolerable level.81Chapter 8Conclusions and Recommendationsfor Future ResearchAlthough fixed rate trellis codes are comparably good in terms of error performancein Rayleigh fading channels, they fail to explore the time varying nature of these channels,and are often designed conservatively to combat very poor channel conditions. The fixedrate of transmission hence results in sacrificing the effective information rate duringgood channel conditions.We have proposed an adaptive trellis-coded scheme called ATCMPSK which exploresthe time varying nature of Rayleigh fading channels, and is extremely effective incombating fading. The ATCMPSK scheme is pragmatic in terms of its design andimplementation. The most complicated component of our system (regarding the codingand the modulation aspects of the design) is a rate 1/2 convolutional encoder, and aViterbi decoder that is capable of constructing parallel branches in its state transitions,uses the Gaussian metric for decoding, and treats repeated symbols on a single transitionin the trellis diagram. We have suggested a mapping scheme called double-Gray-codedmapping which results in better performance for MPSK signals with M=16 or larger, ascompared to sectorized-Gray-coded mapping.Since the conventional deterministic approach to interleaving/deinterleaving does notsuit the adaptive scheme, an alternative method to remove the memory in the channelhas been introduced. It has been shown that there is lot to be gained from interleavingspecially at high SNR values where the high rate schemes are prevalent.82The ATCMPSK scheme results in 3-20 dB gain in BER performance, as comparedto fixed rate codes. These gains have been obtained using a simple encoder of constraintlength three, under ideal conditions. We have investigated the loss in performance dueto nonideal conditions, and shown that even under nonideal conditions one achieves avery substantial gain using the adaptive scheme.We have suggested that one way of determining the adaptation intervals is to usethe BER performance curves (in the AWGN channel) of the individual schemes usedin ATCMPSK, and set the threshold in such a way as to keep BER below a certainlevel. However, there are many other ways in which the threshold can be chosen, andwe can in fact shape the BER curves, within the performance constraints of the scheme,by choosing different sets of thresholds.We have used the conventional transfer function approach to find an upper bound forthe BER performance, and used the statistics of the Rayleigh fading channel to formulatean almost exact expression for the throughput. Our theoretical results were consistentin terms of their agreement with simulations. By way of simulations, we showed theeffects of some of the nonideal conditions, such as the delay in the feedback channel,and adaptation intervals larger than one symbol duration.For future research, we suggest the following:• Study a more realistic and practical model which takes into account the effect ofmultipath fading on the phase of the transmitted signals, and addresses the carrierphase tracking issue for the adaptive system. Implement the proposed scheme takinginto account all the practical implications and issues involved in the design of thescheme.83• For the more realistic model, consider a system for which CSI consists of both theattenuation and the phase of the fading process.• Consider prediction-adaptation techniques to reduce the transmission rate of CSI inthe feedback channel.• Study the implementation of the system with differential detection schemes such asmultiple-symbol differential detection suggested in [13].• For practical channels, there are many considerations such as geography, ambient andclimate which can change the characteristics of the channel. Study the applicationof learning systems and fuzzy logic to the adaptation process to optimize theperformance of the scheme for specific channel considerations.• Study effective methods of monitoring the channel conditions, and providing CSI tothe transmitter in the most efficient manner.84Bibliography[1] Guest Editor: W. Y. C. Lee, "Invited special issue papers on digital cellulartechnologies," IEEE Transactions on Vehicular Technology, vol. 40, pp. 291-375,May 1991.[2] G. Ungerboeck, "Channel coding with multilevel/phase signals," IEEE Transactionson Communications, vol. IT-28, pp. 55-67, January 1982.[3] J. K. Cavers, "Variable-rate transmission for Rayleigh fading channels," IEEETransations on Communications, vol. COM-20, pp. 15-22, Feb. 1972.[4] A. J. Viterbi, J. K. Wolf, E. Zehavi, and R. Padovani, "A pragmatic approach totrellis-coded modulation," IEEE Communications Magazine, pp. 11-19, July 1989.[5] G. Ungerboeck, "Trellis-coded modulation with redundant signal sets. Part I:Introduction; Part II: State of the art," IEEE communications Magazine, vol. 25,pp. 5-21, February 1987.[6] J. F. Hayes, "Adaptive feedback communications," IEEE Transations on Communi-cations, vol. COM-16, pp. 29-34, Feb. 1968.[7] V. 0. Hentinen, "Error performance for adaptive transmission on fading channels,"IEEE Transations on Communications, vol. COM-22, pp. 1331-1337, Sept. 1974.[8] W. Jakes, Microwave mobile communications, ch. 1. New York: Wiley, 1974.[9] M. K. Simon, "Dual pilot tone calibration technique (DTCT)," IEEE Transaction onVehicular Technology, vol. VT-35, pp. 63-70, May 1986.[10]F. Davarian, "Mobile digital communications via tone calibration," IEEE Transac-tions on Vehicular Technology, vol. VT-36, pp. 55-62, May 1987.[11]F. M. Gardner, Phaselock techniques, ch. 9. Wiley-interscience, 2 ed., 1979.[12]D. Makrakis and P. T. Mathiopoulous, "Optimal decoding in fading channels:a combined envelope, multiple differential and coherent detection approach,"Proceedings of GLOBECOM '89, Dallas, Texas, pp. 1551-1557, November 1989.85[13] D. Divsalar and M. K. Simon, "Multiple symbol differential detection of MPSK,"IEEE Transactions on Communications, vol. 38, pp. 300-308, March 1990.[14] R. E. Blahut, Digital transmission of information, ch. 11, pp. 111 115. AddisonWesley, 1990.[15] J. Hagenauer, "Viterbi decoding of convolutional codes for fading—And burstchannels," 1980 Int. Zurich Seminar on Digital Communications, pp. G2.1—G2.7,March 1980.[16] S. Kallel, "Analysis of a type II, hybrid, ARQ Scheme with code combining," IEEETransactions on Communications, vol. 38, pp. 1133-1137, August 1990.[17] S. Lin and J. D. J. Costello, Error control coding: fundamentals and applications,ch. 14. Prentice-Hall, 1983.[18] A. J. Viterbi and J. K. Omura, Principles of digital communication and coding,ch. 3. McGraw-Hill,New York, 1979.[19] E. Bigfieri and D. Divsalar and P. J. McLane and M. K. Simon, Introduction totrellis-coded modulation with applications, ch. 4. MacMillan, 1991.[20] M. M. Mulligan and S. G. Wilson, "An improved algorithm for evaluating trellisphase codes," IEEE Transactions on Information Theory, vol. IT-30, pp. 846-851,November 1984.[21] R. C. P. Saxena, "Optimum encoding in finite state coded modulation," Tech. Rep.TR83-2, Department of Electrical, Computer and System Engineering, RensselaerPolytechnic Institute, Troy, N.Y., 1983.[22] E. Zehavi and J. K. Wolf, "On the performance evaluation of trellis codes," IEEETransactions on Information Theory, vol. IT-33, pp. 196-201, March 1987.[23] W. H. Press, B. P. Flannery, A. Teukolsky, and W. T. Vetterling, Numerical recipesin C, ch. 7. Cambridge, 1988.86correlatorcorrelatora(t)s(t) + n(t)Appendix A The Gaussian MetricTo prove that the Gaussian metric is a maximum likelihood metric for Rayleighfading channels with ideal CSI, we can use a model as shown in Figure 34. A matchedfilter receiver decides what signal has been transmitted.(t)tFigure 34 Correlator receiver for maximum likelihood detection.The transmitted signal s(t) is attenuated by the time varying amplitude of fading a(t),and corrupted by AWGN n(t). col(t) and (p2(t) are the following orthonormal functions:—2 cos (2itict) •(P2 (t) =	 72 sin (27rfct).	(A.1)The signals in the signal space can then be expressed as:si(t) = sivpi(t) siz,o2(t).	(A.2)R87The output of the correlator is the received signal:Ri = f r(t)0i(t) dt0= f (a(t)s(t) + n(t)) dt.0(A.3)The probability density function of the received signal given signal si was transmittedis:NiRi I si = f a(t)s1(t)15 1 (t) dt + f n(t)0 1 (t) dt0	 0= f a(t)(si 1 01 (t) + sivp 2 (t)) dt + N1 .0(A.4)Assuming that a(t) is constant for the duration of symbol [0,71 we have:R1 I si = a f sii401(t)(1)1(t) dt I si2Coi(t)4, 2(t) dt + N10	 0	 (A.5)=where the second integral is zero since coi(t) and yo2(t) are orthonormal functions. Wecan conclude that:E[Ri I sil = asil + E[Nii(A.6)The same argument holds for R2 I Si.88The variance of the random variable:var[Ri I si] = E [{(Ri I si) — E(R2 I si)} 2 1= E [{(asii + N1) — asii} 2 1	(A.7)= E[M] = NoIn conclusion:Ri I si ti N(asq, No/2),	(A.8)or equivalently1fR, 13, = virNo exp — (ri — asii)2 /No) • (A.9)To prove that R1 , R2 are uncorrelated, note that:ERR ]. — E[Ri ]) {R2 — E[R2]}] = ERasi i + N1 — «so_ Hasi2 + N2 — asi211= EININ21 = 0.	(A.10)Now, since R1 , R2 are independent and uncorrelated, we can state that:fRIss(ri , r2 ) = fR i ls i (ri)fR2 1, ; (r2)1= 	 — asii) 200) exp	 — asi (—(r2	 2)2/No) (A.11)(N/rN0)2 exp (—(ri1= r/v,o exp (— {(ri — asii) 2 + (r2 — cesi2 ) 2] /NO) •The optimum decision rule for maximum likelihood is to choose signal sk iff:fRlak (rl, r2 ) > fga,(ri , r2 )	 Vi	 k,	 (A.12)89or equivalently:[(ri — ask i ) 2 (r2 — ask2) 2] < [(ri — ashi) 2 + (r2 — as12) 2 1	 k. (A.13)For a sequence of length L of transmitted signals, the optimum decision is therefore:ChooseLSk iff :LE (ri —{	 ajskl)2 -I- (r2 — aisk2, 	,	 	 )2] < E [fri — aisio2 + (r2 — aisi2)2]i=i j=iQED.	 Vi	 k(A.14)90Appendix B Comparison of Double-Gray-CodedMapping and Sectorized-Gray-Coded Mapping Sectorized-Gray-coded signal space for an MPSK signalling scheme has M/4 sectors.The first log2 (M) — 2 bits in the encoded symbol choose one of those sectorslexicographically. In double-Gray-coded scheme, however, the sectors are also chosenaccording to Gray code. This concept is shown in Figure 5.B.1 Comparison in AWGNThe effect of Gray coding of the sectors on the signals in the parallel branches ofthe signal space is shown in Figure 35.Sectorized-Gray-coded	 Double-Gray-codedFigure 35 Comparison of signals on the parallel branches of the trellisdiagram for double-Gray-coded and sectorized-Gray-coded mappings.The two neighbouring signal differ in one bit , as compared to two for sectorized-Gray-coded mapping. For TC16PSK with sectorized-Gray-coded mapping, the probabil-ity of error due to parallel transitions is:1	 1	 2=	— p) + 5/t + i-p(1 -p)P=P - 5	 Tr;')2 2	 Es '(BA)91This is approximately 2/3 of the same probability using sectorized-Gray-coded schemeindicated in 5.38. The expression for the overall probability of error is therefore:	 1.324E,Pb ^	Q /1.324E3) e 4N0	 x 1 OT (D , I),.V 2N0	 3 aI 1 /=1;D=exp (—AO'	 (B 2)The error weight profile of the code is given in Table 4. Note that the first 7 elementsin the Table are identical to those for the sectorized-Gray-coded mapping scheme. Thesecond term in B.2—lets call it the trellis error—was found using the entries in Table 4,and the error-state diagram's set of branch labels given in (5.31).Error Pattern (ed Error Weight Profile W(ed0000 10001 Ddg0010	d2 	 d2	D° 1	 30011 DA0100 DA0101 D d2	 d231D 50110 Dd21+D d23+D d25+D a2740111 Dd3 +D ail21 000 Ddg1001 Dd?1010 Ddg1011 Ddg1100 Ddg1101 ci2 	d2D 3+D 521110 d2	 d2	 d2D 1+	 3+D 741111 a26Table 4 Error weight profile for the rate 3/4 TC16PSK pragmatic code with sectorized-Gray-coded mapping.92-110Trellis error probabilit  of double-Gray-coded 	mapping schemePb1010-710Parallel error probabilityof double-Gray-codedmapping schemeParallel error probabilityof sectorized mapping-91013	 14	 15Es (dB)N0-1310 12 18	 17	 18Note that dfree is the same as the code with double-Gray-coded mapping. However,the new mapping results in a better code that has a slightly better BER performance asdepicted in Figure 36.Figure 36 Comparison of trellis error and probability of error due to parallel transitions forTC16PSK in AWGN with sectorized-Gray-coded and double-Gray-coded mapping.B.2 Comparison in Rayleigh Fading ChannelIt was observed in the last section that the new mapping scheme does not result in aconsiderable improvement in BER performance in AWGN. However, in the Rayleighfading channel, the probability of bit-error due to parallel transitions becomes the93dominating factor. Parallel branches do not have the opportunity to recover from deepfades. Therefore, one would anticipate that an improvement in the structure of the signalsin parallel branches would have a relatively considerable effect on the BER performance.Integrating B.1 over the Rayleigh fading density function, PH for the double-Gray-coded TC16PSK scheme if found to be:p,	11/ Es/NO II = 3 2 + Es /No (B.3)We can observe, that the first term is 2/3 of that of the scheme using sectorized Gray-coded mapping. The performance of rate 3/4 TC16PSK for both mapping schemes aregiven in Figure 37.The asymptotic gain of the new mapping scheme for TC16PSK in Rayleigh fadingis therefore around 1 dB, and it is as easy to implement as the sectorized-Gray-codedmapping. For larger constellations, the gain obtained from the new mapping is evenmore significant. For example, it is simple to show that for 32PSK the asymptoticBER with double-Gray-coded mapping is approximately half that of the scheme usingsectorized-Gray-coded mapping.94-3101013Rayleigh fading 	Ideal interleavingIdeal CSIK=3..1	 =Mi.SIMI 0.11Emei......... 	 Theory: 16PSK withi ouble-gray mappingn—ale%:1=16 -6 NIWPP. i	 1	 1-o	 ' . Theory: 16PSK withnsectorized mappingtip.Simulation: 16PSK withsectorized mapping' s.. .,Angliallw.....1-71.y...m..Simulation: 16PSK withdouble-gray mappingr mi,-..	 -0..4,..,15	17	19	21	23	25	27	29	31E! (dB)NFigure 37 BER performance of double-Gray-coded TCI6PSK.-110Pb-21095Appendix C Confidence Intervalfor Simulation ResultsThe minimum number of the error events counted for the simulation results presentedin this thesis was 200. However, for the case of ATCMPSK with ideal interleaving, andan error roof of 0.01 (see Figure 27) it would have taken about a week of computing timeto obtain so many error events for SNR=24 dB (using a SPARC station 2). The actualnumber of error events was 58 in this case. To give an indication of how the numberof error events reflect the accuracy of the simulation results, that particular point in theperformance curve was run 5 times and the 95% confidence interval was calculated tobe:.P,(24 dB) = 1.39 x 10-6 ± 0.25 x 10-6. (C.1)The worst point for the case of noninterleaved ATCMPSK with an error roof of 0.1, atSNR = 24 dB was run for more than 1500 error events. The 95% confidence intervalfor this point is:Pe (24 dB) = 3.03 x 10 -4 ± 0.07 x 10-4. (C.2)The simulation points for lower values of SNR correspond to greater numbers of errorevents. Therefore, the confidence interval for these points is smaller, and they are hencemore accurate.96BIOGRAPHICAL INFORMATION NAME:	 Cla-r	 s-soc,(1/1 2	 (atttip4 1MAILING ADDRESS:	/5- &yea /2 ft, 4-	 ,C .	vs- y	 6,PLACE AND DATE OF BIRTH: /	 -7-0e#4 4\-1	/1- 1,77? C	 /G, (5EDUCATION (Colleges and Universities attended, dates, and degrees):Oeti- e. Aro cat le e5 	 e	 oor s/9cs--/-)i-e	 f -- /1/POSITIONS HELD:PUBLICATIONS (if necessary, use a second sheet):AWARDS:e ,c	 e-€4, Cot-A	 f 4.(142.4-ciComplete one biographical form for each copy of a thesis presentedto the Special Collections Division, University Library.OE-5

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