Applied Science, Faculty of
DSpace
UBCO
Mohammadnavazi, Hassan
Doctor of Philosophy - PhD
University of British Columbia
Due to its convenience, wireless communication systems have grown tremendously. Data rate requirements of various services/applications that are transmitted over wireless channels are increasing day by day. Bit interleaved coded modulation (BICM), which is a serial concatenation of a channel encoder, an interleaver and a symbol mapper, is a spectral efficient technique that is being used in many wireless systems. The performance of BICM can be significantly improved by using the iterative decoding technique at the receiver. This system is referred as BICM with iterative decoding (BICM-ID), and offers an improved performance over both fading and nonfading channels. It is well-known that BICM-ID performance is strongly dependent on the applied signal mapping. Signal mapping is the assignment of binary bits to complex symbols from a modulation alphabet. It is demanded to use a higher order modulation in BICM-ID to achieve a higher data rate and spectral efficiency. However, finding a mapping for a large modulation that offers an improved performance for BICM-ID is very complicated. This is because of the huge number of possible mappings for higher order modulations. This thesis focuses on the mapping problem for BICM-ID systems. In particular, novel mapping methods are developed for higher order modulations, including two-dimensional and multi-dimensional modulations. The proposed methods in this thesis consist of (i) heuristic methods and (ii) computer search techniques. In comparison with the previously known mappings, the proposed mappings significantly improve the BICM-ID performance over the considered channels. This is confirmed by various analytical and simulation results that are investigated in this thesis.
eIGNSL MSbbING FadTIf-INfWdLWShWV UaVWVMaVgLSfIaN iIfH IfWdSfIhWVWUaVINGbyHassan MohammadnavaziA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe College of Graduate Studies(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)April 2018c© Hassan Mohammadnavazi 2018ii The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis/dissertation entitled: Signal Mapping for Bit-Interleaved Coded Modulation with Iterative Decoding submitted by Hassan Mohammadnavazi in partial fulfillment of the requirements of the degree of Doctor of Philosophy. Dr. Md. Jahangir Hossain, School of Engineering Supervisor Prof. Julian Cheng, School of Engineering Supervisory Committee Member Dr. Thomas Johnson, School of Engineering Supervisory Committee Member Prof. Shawn Wang, Irving K. Barber School of Arts and Sciences University Examiner Dr. Nghi Tran, Electrical and Computer Engineering, University of Akron, OH, USA External Examiner iiStstrsutDue to its convenience, wireless communication systems have grown tremendously. Datarate requirements of various services/applications that are transmitted over wireless chan-nels are increasing day by day. Bit interleaved coded modulation (BICM), which is a serialconcatenation of a channel encoder, an interleaver and a symbol mapper, is a spectral effi-cient technique that is being used in many wireless systems. The performance of BICM canbe significantly improved by using the iterative decoding technique at the receiver. Thissystem is referred as BICM with iterative decoding (BICM-ID), and offers an improvedperformance over both fading and nonfading channels. It is well-known that BICM-IDperformance is strongly dependent on the applied signal mapping. Signal mapping is theassignment of binary bits to complex symbols from a modulation alphabet. It is demandedto use a higher order modulation in BICM-ID to achieve a higher data rate and spectralefficiency. However, finding a mapping for a large modulation that offers an improved per-formance for BICM-ID is very complicated. This is because of the huge number of possiblemappings for higher order modulations.This thesis focuses on the mapping problem for BICM-ID systems. In particular, novelmapping methods are developed for higher order modulations, including two-dimensionaland multi-dimensional modulations. The proposed methods in this thesis consist of (i)heuristic methods and (ii) computer search techniques. In comparison with the previouslyknown mappings, the proposed mappings significantly improve the BICM-ID performanceover the considered channels. This is confirmed by various analytical and simulation resultsthat are investigated in this thesis.iiiLsy eummsryTodayâs wireless networks should be designed to support widely varying user needs (home,office, etc.). The obvious trend is the need of very high data transmission rates to deal withthe increasing demand of multimedia communication services such as video teleconferencingand high quality media streaming. The bit interleaved coded modulation with iterativedecoding (BICM-ID) system offers excellent performance on the both wired and wirelesscommunication channels. As such, it is a great candidate to be used in future wirelesssystems. The technical challenge is to enable high data transmission rates for BICM-ID.One main approach is to increase the number of the unique signals that BICM-ID uses totransfer information. In this approach, the assignment of information to the unique signalsis a very crucial and complex problem. This problem is addressed in this thesis such that,the system performance improves without any increase in the system complexity.ivbrwxsuwThe research chapters of this thesis (Chapters 2-5) have been published in or submitted topeer reviewed journals.A version of Chapter 2 has been published as: Hassan M. Navazi and Md. J. Hossain,âA novel symbol mapping method for BICM-ID systems for higher order signal constella-tions,â IEEE CommunC aettC, Vol. 18, no. 8, pp. 1323-1326, Aug. 2014. I conducted allthe mathematical analysis and computer programming. I wrote the manuscript, and Dr.Md. J. Hossain revised and edited it to improve the presentation in the manuscript.A version of Chapter 3 has been published as: Hassan M. Navazi and Md. J. Hossain,âEfficient multi-dimensional mapping using QAM constellations for BICM-ID,â IEEEirvnsC on lireless CommunC, Vol. 16, no. 12, pp. 8067-8076, Dec. 2017. I conducted allthe mathematical analysis and computer programming. I wrote the manuscript, and Dr.Md. J. Hossain revised and edited it to improve the presentation in the manuscript.A version of Chapter 4 has been submitted to the journal IEEE Transactions onWirelessCommunications as: Hassan M. Navazi and Md. J. Hossain, âMulti-dimensional mappingof higher order QAM constellations for BICM-ID over Rayleigh fading channels,â Jan.2018. I conducted all the mathematical analysis and computer programming. I wrote themanuscript, and Dr. Md. J. Hossain helped me improve the presentation in the manuscriptby his comments.A version of Chapter 5 has been submitted to the journal IEEE Transactions onWirelessCommunications as: Hassan M. Navazi and Md. J. Hossain, âMulti-Dimensional QAMmapping for BICM-ID over AWGN channels,â Feb. 2018. I conducted all the mathematicalanalysis and computer programming. I wrote the manuscript, and Dr. Md. J. Hossainhelped me improve the presentation in the manuscript by his comments.vfstlw ox UontwntsStstrsut iiiLsy eummsry ivbrwxsuw vfstlw ox Contwnts viiiList ox fstlws xList ox Fiyurws xiiList ox Suronyms xiiiSuknowlwvymwnt xivDwviustion xvC Introvuution snv avwrviww C1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 BICM system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 BICM-ID system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Thesis objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Mapping design guidelines for BICM-ID . . . . . . . . . . . . . . . . . . . . 71.6.1 AWGN channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6.2 Rayleigh fading channel . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 Coded modulation capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.8 Literature review of mapping methods . . . . . . . . . . . . . . . . . . . . . 101.8.1 Binary switching algorithm (BSA) . . . . . . . . . . . . . . . . . . . 111.8.2 Random mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11viG45L8 BF 6BAG8AGFD Msppiny xor Hiyhwr arvwr 2-D Movulstions CD2.1 Proposed mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.1 Precoding process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Mapping process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Characteristics of Gray mapping . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Characteristics of our mapping . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Numerical results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 18E Wffiuiwnt Multi-Dimwnsionsl Msppiny ox 16-, 64-cSM xor TICM-ID DE3.1 Proposed mapping method . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.1 Method description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.2 Design considerations of iNk . . . . . . . . . . . . . . . . . . . . . . 263.1.3 Proposed vectors for i, , and i;k . . . . . . . . . . . . . . . . . . 283.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Numerical results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 303.3.1 Performance over AWGN channel . . . . . . . . . . . . . . . . . . . . 313.3.2 Performance over Rayleigh fading channels . . . . . . . . . . . . . . 32F Multi-vimwnsionsl Msppiny ox Hiyhwr arvwr cSM xor TICM-ID avwrdsylwiyh Fsviny Chsnnwls EH4.1 Proposed mapping method . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Development and optimization of cost functions . . . . . . . . . . . . . . . . 394.2.1 Optimization of cost functions . . . . . . . . . . . . . . . . . . . . . 434.3 Numerical results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 444.3.1 Resulting MD mappings of M-QAM . . . . . . . . . . . . . . . . . . 454.3.2 Performance comparison . . . . . . . . . . . . . . . . . . . . . . . . . 56G Multi-vimwnsionsl Msppiny ox M-cSM Constwllstions xor TICM-IDovwr SiGN Chsnnwls HB5.1 Optimum MD 16-QAM mapping . . . . . . . . . . . . . . . . . . . . . . . . 605.1.1 16-QAM intermediate mapping . . . . . . . . . . . . . . . . . . . . . 635.1.2 Proposed 4-D 16-QAM mapping . . . . . . . . . . . . . . . . . . . . 685.2 MD mapping of 2m-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2.1 MD mapping using 16 symbols from 2m-QAM . . . . . . . . . . . . 695.2.2 Transferring from 16-QAM to 2m-QAM . . . . . . . . . . . . . . . . 715.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73H Conulusions IIviiG45L8 BF 6BAG8AGF6.1 Work accomplished in this thesis . . . . . . . . . . . . . . . . . . . . . . . . 776.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Titlioyrsphy IKSppwnviuws 8EA Proof for Proposition 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84B Proof for Proposition 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92viiiList ox fstlwsTable 1.1 Number of possible mappings for b -QAM. . . . . . . . . . . . . . . . 5Table 1.2 CM capacity for b -QAM. . . . . . . . . . . . . . . . . . . . . . . . . 10Table 2.1 Average Euclidean distance between symbols with 3 or 4-bit Hammingdistances for 16-QAM mappings. . . . . . . . . . . . . . . . . . . . . . . . . 17Table 2.2 Evaluation parameters for different QAM mappings. . . . . . . . . . . 18Table 3.1 i in different steps of the proposed mapping method. . . . . . . . . 28Table 3.2 Conversion vector, . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Table 3.3 Different cases of i;k for 16-QAM. . . . . . . . . . . . . . . . . . . . 28Table 3.4 Different cases of i;k for 64-QAM. . . . . . . . . . . . . . . . . . . . 29Table 3.5 Our proposed 4-D 16-QAM mapping. . . . . . . . . . . . . . . . . . . 31Table 3.6 Comparison of Nmin and yË2min for different mappings. . . . . . . . . . 32Table 3.7 Comparison of Φ(P ) and Î¦Ë(P ) for different mappings. . . . . . . 33Table 4.1 Proposed xr, or, xl, and ol for 16-QAM. . . . . . . . . . . . . . . 47Table 4.2 Proposed xr, or, xl, and ol for 32-QAM. . . . . . . . . . . . . . . 47Table 4.3 Proposed xr, or, xl, and ol for 64-QAM. . . . . . . . . . . . . . . 47Table 4.4 Proposed xr, or, xl, and ol for 128-QAM. . . . . . . . . . . . . . . 48Table 4.5 Proposed xr, or, xl, and ol for 256-QAM. . . . . . . . . . . . . . . 49Table 4.6 Proposed xr and or for 512-QAM. . . . . . . . . . . . . . . . . . . . 50Table 4.7 Proposed xl and ol for 512-QAM. . . . . . . . . . . . . . . . . . . . 51Table 4.8 Proposed xr for 1024-QAM. . . . . . . . . . . . . . . . . . . . . . . . 52Table 4.9 Proposed or for 1024-QAM. . . . . . . . . . . . . . . . . . . . . . . . 53Table 4.10 Proposed xl for 1024-QAM. . . . . . . . . . . . . . . . . . . . . . . . 54Table 4.11 Proposed ol for 1024-QAM. . . . . . . . . . . . . . . . . . . . . . . . 55Table 4.12 Comparison of the harmonic mean of MSED, Î¦Ë(P ), for differentmappings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Table 4.13 Î¦Ë(P ) for proposed 4-D mapping using higher order QAMs. . . . . . 56Table 5.1 The proposed 4D mapping of 16-QAM for chosen-index equal to 1. . 68ixLIFG BF G45L8FTable 5.2 The 16 chosen symbols from 2m-QAM in the first step of the proposedmethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Table 5.3 The chart of substitution of 2m-QAM symbols for the 16-QAM symbols. 70Table 5.4 Transfer system to generate symbol coordinates for MD 2m-QAMmappings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Table 5.5 yË2min and NËmin for different mappings. . . . . . . . . . . . . . . . . . 73Table 5.6 yË2min and NËmin for MD mapping of higher order modulations. . . . . 74Table A.1 Hamming distance between adjacent symbols for proposed mappings. 87xList ox FiyurwsFigure 1.1 The block diagram of a BICM system. . . . . . . . . . . . . . . . . . 3Figure 1.2 The block diagram of a BICM-ID system. . . . . . . . . . . . . . . . 4Figure 1.3 A typical BER curve for BICM-ID. . . . . . . . . . . . . . . . . . . . 6Figure 2.1 An example of the resulting mapping for 16-QAM. . . . . . . . . . . 16Figure 2.2 BER performance of 64-QAM. . . . . . . . . . . . . . . . . . . . . . 19Figure 2.3 BER performance of 256-QAM. . . . . . . . . . . . . . . . . . . . . . 20Figure 2.4 BER performance of 1024-QAM. . . . . . . . . . . . . . . . . . . . . 21Figure 2.5 Analytical bound on the error-floor [34] of 64-QAM on Rayleighfading channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.6 Analytical bound on the error-floor [34] of 256- and 1024-QAM onRayleigh fading channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 3.1 (a) A 16-QAM constellation, (b) Four selected 16-QAM symbols(dark symbols) to be used in mapping step i = 1, i.e., 1, and (c) Eightselected 16-QAM symbols (dark symbols) to be used in mapping step i = 2,i.e., 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 3.2 BER performance over an AWGN channel. . . . . . . . . . . . . . . 32Figure 3.3 Error-floor bounds over an AWGN channel. . . . . . . . . . . . . . . 33Figure 3.4 BER performance over a Rayleigh fading channel. . . . . . . . . . . 34Figure 3.5 Error-floor bounds over a Rayleigh fading channel. . . . . . . . . . . 35Figure 4.1 Flowchart of the proposed algorithm (itCnumCr, itCnumCl, and itCnumrepresent the number of iterations for different loops). . . . . . . . . . . . . 45Figure 4.2 (a). Symbolâs arrangement in 16-QAM, and achieved 16-QAM map-pings in decimal format: (b). xr, (c). or, and (c). xl (the light symbols),ol (the dark symbols). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 4.3 BER performance of BICM-ID with 4-D and 6-D 16-QAM overRayleigh fading channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57xiLIFG BF FIGUE8FFigure 4.4 BER performance of BICM-ID with 4-D 32- and 64-QAM over Rayleighfading channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 4.5 Error-floor bounds of BER for BICM-ID with 4-D and 6-D 16-QAMover Rayleigh fading channels. . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 4.6 Error-floor bounds of BER for BICM-ID with 4-D 32- and 64-QAMover Rayleigh fading channels. . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 5.1 Symbol arrangement in a 16-QAM constellation. . . . . . . . . . . . 61Figure 5.2 Proposed intermediate 16-QAM mapping for (a). z â {1P 2}, and (b).z â {3P 4}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Figure 5.3 64-QAM constellation. The black symbols represent the 16 selectedsymbols in the first step of the proposed mapping method. . . . . . . . . . . 70Figure 5.4 BER performance of BICM-ID over the AWGN channel. . . . . . . . 75Figure 5.5 BER performance of BICM-ID over the AWGN channel. . . . . . . . 75Figure 5.6 Error-floor bounds over the AWGN channel. . . . . . . . . . . . . . . 76Figure 5.7 Error-floor bounds over the AWGN channel. . . . . . . . . . . . . . . 76xiiList ox SuronymsAWGN Additive White Gaussian NoiseBER Bit Error RateBICM Bit Interleaved Coded ModulationBICM-ID Bit Interleaved Coded Modulation with Iterative DecodingBSA Binary Switching AlgorithmCSI Channel State InformationGA Genetic AlgorithmLLR Log-Likelihood RatioMD Multi-DimensionalMSED Minimum Squared Euclidean DistancePSK Phase Shift KeyingQAM Quadrature Amplitude ModulationQPSK Quadrature Phase Shift KeyingRTS Reactive Tabu SearchSED Squared Euclidean DistanceSNR Signal to Noise Ratio Signal constellationNmin Average Hamming distance between neighboring symbolsΦ Harmonic mean of MSEDÎ¦Ë Harmonic mean of MSED after feedbackl (l) Hamming wight of binary sequence lyH(l1P l2) Hamming distance between binary sequences l1 and l2sgn(:) Sign functionxiiiSuknowlwvymwntI would like to thank my supervisor, Dr. Jahangir Hossain, for his excellent guidance andinvaluable advice. He has always been supportive, encouraging, and patient. He is notonly a professional supervisor, but also a good friend. I cannot thank him enough for theassistance he provided at all levels of my PhD study.I also thank my thesis committee, Dr. Julian Cheng and Dr. Thomas Johnson, for theirhelpful comments and suggestions. My sincere gratitude also goes to Dr. Ha H. Nguyen,University of Saskatchewan and Dr. Robert G. Maunder, University of Southampton, fromwhom I learned priceless points regarding my research before my attending UBC. I oweparticular thanks to Lori Walter, Centre for Scholarly Communication, UBC Okanagan,for her help and support.My special thanks also go to all my family members and friends for their support.xivVwviustionTo am darents, Golnar andPanjalixvUzsptwr CIntrovuution snv avwrviwwC.C IntrovuutionIn general, data rate requirements of various services/applications that are transmittedover communication networks including wired and wireless networks are increasing day byday. Higher order modulations in conjunction with efficient channel coding will play a vitalrole to meet this high data rate requirement.Bit interleaved coded modulation (BICM) was introduced by Zehavi in [1] as an at-tractive coded modulation scheme. In BICM, channel coded bits are randomly interleavedand then the interleaved coded bits are mapped to the signals at the modulator. Due toits simplicity and design flexiblity, BICM has been standardized for contemporary wirelessand wired communication systems [2] and is a potential choice for future communicationsystems.BICM offers a good error performance over the Rayleigh fading channel. However, theemployed interleaver results in a random modulation, which degrades BICM performanceover the additive white Gaussian noise (AWGN) channel. One effective way to overcomethis problem is to use iterative decoding at the receiver. This system is known as BICM-ID, which is investigated in [3]-[5]. Because of the iterative decoding, BICM-ID offers animpressive performance over the AWGN channel as well as over the Rayleigh fading channel[6].The performance of BICM-ID is significantly dependent on the employed signal map-ping at the modulator. Signal mapping is defined as the assignment of binary digits tocomplex signals (symbols) from a modulation alphabet. Multi-dimensional (MD) mappingimproves bandwidth efficiency for BICM-ID by a reasonable increase in system complexity[7]. In MD mappings, a sequence of bits is mapped to a sequence of symbols instead of asingle symbol. Many research studies have been carried out to address the signal mappingproblem for BICM-ID, see for examples, [7]-[24].The two main methodologies to develop signal mappings for BICM-ID are as follows:(i) heuristic methods and (ii) computer search techniques. The existing heuristic methodsconstruct good 2-D/MD mappings for only smaller modulations such as 2-D 32-ary quadra-ture amplitude modulation (QAM) [22] and MD 8-ary phase shift keying (PSK) [7]. For11.2. 5I6M flfgemlarger modulations, computer search techniques are usually used. However, the achievedresults from the existing computer search techniques are suitable only for modulations withmedium sizes such as 2-D 64-QAM [13].Higher order modulations can be used to increase the data rate and improves thebandwidth efficiency of BICM-ID. However, developing efficient mappings of larger con-stellations for BICM-ID is always challenging due to the huge number of possible mappings.Even the best computer search techniques become intractable in finding good mappingsof larger constellations for BICM-ID due to the high level of complexity [16]. In [16] and[25], the authors demonstrated that random mapping can lead to efficient MD mappings.Random mapping technique searches among a set of randomly generated mappings to finda mapping that improves the performance of BICM-ID. However, as the set of randomlygenerated mappings is very large, this technique also suffers from computational complex-ity. This complexity degrades the obtained random mappingâs performance especially forlarge MD modulations. Consequently, developing efficient mappings of larger constellations(including 2-D and MD constellations) is a very important and open research question.Motivated by the above discussions, this thesis focuses on improving the error per-formance of BICM-ID via developing novel and efficient signal mappings. In particular,we investigate the mapping problem for higher order signal constellations including 2-Dand MD constellations. To design mapping functions throughout this thesis, two generalmethodologies will be followed: (i) systematic methods which generate mapping heuristi-cally and (ii) novel and computationally fast computer search techniques.In the rest of this chapter, we briefly review the BICM and BICM-ID concepts anddiscuss about the mapping problem for BICM-ID.C.D TIUM systwmA BICM system model is shown in Fig. 1.1, where the transmitter is built from serialconcatenation of an encoder, a bit interleaver Î , and a modulator. A sequence of informa-tion bits u is encoded by a convolutional encoder. Then, the coded bits x are randomlyinterleaved, and the interleaved coded bits v are grouped in blocks of mN bits, where mand N are positive integers. For notational convenience, let us denote the tth block ofinterleaved coded bits at the input of the modulator by lt = [l213t P l223t P · · · P l2ma3t ]. Themodulator maps lt to a vector of N consecutive 2m-ary signals, xt = [x213t P x223t P · · · P x2a3t ],using a mapping function  : {0P 1}ma ââ = a , where denotes the 2-D 2m-ary signalset. Mathematically, we can writext = [x213t P x223t P · · · P x2a3t ] = (lt): (1.1)21.2. 5I6M flfgem Î 1âÎ â®â®É¶u( )eL vÉ¶cu EncoderDecoderS/P MapperDemapperChannelP/Stlv txtyFigure 1.1: The block diagram of a BICM system.At the receiver, the received signal-vector corresponding to the transmitted symbol-vector xt can be expressed asyt = hTt xt + ntP (1.2)where ht = [h213t P h223t P · · · P h2a3t ] is the corresponding vector of Rayleigh fading coefficients,AT represents the transpose of A, and nt is a vector of N additive complex white Gaussiannoise samples with zero-mean and variance N0. It is important to note that if N = 1, themapping is a regular 2-D mapping; otherwise, it is referred to as an MD (2N -D) mapping.At the receiver, the demapper calculates the log-likelihood ratios (LLRs) using thereceived signal as [4]ax(lit) = logâxtâ i( P(yt|xt)âxtâ i) P(yt|xt)P (1.3)where i0 and i1 represent the subset of signals xt â whose labels have the bit value of0 and 1, respectively, in the ith bit position. P(yt|xt) is the probability density function,which is determined by the channel model. The extrinsic LLRs are then deinterleaved andfed to the decoder, where a decision is made about the transmitted bits.The idea of bit by bit interleaving in BICM improves the diversity order of the system.Diversity order is a highly influential parameter on the bit eror rate (BER) performanceof coded systems over the Rayleigh fading channel [26]. As a result, BICM offers a gooderror performance over the Rayleigh fading channel. However, as mentioned previously,the interleaver results in a random modulation, which degrades the BER performance ofBICM over the AWGN channel.31.3. 5I6M-I7 flfgemC.E TIUM-IV systwmTo improve the error performance of BICM, iterative decoding has been used at the receiver[3]-[5]. The resulting system is referred to as BICM-ID. The iterative decoding techniqueused for the BICM-ID is similar to that of a turbo code [27]. However, BICM-ID employsonly one encoder at the transmitter and only one decoder at the receiver. As such, it hasconsiderably less complexity in comparison with turbo codes [4].The BICM-ID transmitter is the same as the BICM transmitter, which has been ex-plained in section 1.2. Therefore, in what follows, we describe only the receiver of BICM-ID.Fig. 1.2 illustrates the block diagram of a BICM-ID system. It is assumed that the receiverhas the perfect channel state information (CSI). At the receiver, the demapper uses thereceived signal-vector yt and the v priori LLR of the coded bits to compute the extrinsicLLR for each of the bits in the received signal-vector as [4]ax(lit) = logâxtâ i( P(yt|xt)âmaj=1Nj Ì¸=i zâLa2ljt 3Mljtâxtâ i) P(yt|xt)âmaj=1Nj Ì¸=i zâLa2ljt 3MljtP (1.4)where at(ljt ) = log(P(ljt = 0)=P(ljt = 1)) is the v priori LLR of the coded bits. After beingpermuted by the random deinterleaver, the extrinsic LLRs are applied to the channeldecoder. The decoder then calculates the extrinsic LLR values on the coded bits. Afterbeing interleaved, these LLRs are fed back to the demapper and used as the v priori LLRsin the next iteration. Through this iterative process, BICM-ID achieves a significant codinggain and improves the error performance. Î Î 1âÎ â®â®É¶u( )eL vÉ¶cu EncoderDecoderS/P MapperDemapperChannelP/Stl( )aL vÉ¶v txtyFigure 1.2: The block diagram of a BICM-ID system.41.4. CebUlem fgTgemeagC.F brotlwm ststwmwntIt is widely known that the performance of BICM-ID strongly depends on the applied signalmapping at the modulator. Signal mapping describes how to assign binary sequences tocomplex signals from a modulation alphabet. In fact, a constellation with b signal pointshas  = b ! possible mappings, where ! denotes the factorial operation. For example,there are 8! = 40320 mappings for 8-QAM. Table 1.1 provides the number of possiblemappings for 2-D b -QAM for different values of b . It is worth noting that the numberof unique mappings for a constellation is smaller than the number of possible mappings.However, finding the unique mappings among the possible mappings for large constellationsis prohibitively complicated.Table 1.1: Number of possible mappings for b -QAM.Modulation 4-QAM 2416-QAM 2:1Ã 101=64-QAM 1:3Ã 1089256-QAM 8:6Ã 105061024-QAM 5:4Ã 1026=9As this table shows, the number of possible mappings for higher order constellationssuch as 1024-QAM approaches infinity, which makes finding the corresponding good/optimummappings difficult if not impossible. As a result, exhaustive computer search techniques arenot applicable to find suitable mappings of larger constellations. The well-known binaryswitching algorithm (BSA) [9] is the best known computer-based mapping search technique.However, the BSA becomes intractable to obtain good mappings of larger constellations forBICM-ID due to the high level of complexity [16]. In the case of MD mappings, the prob-lem is much more severe because by increasing the dimensionality, the number of possiblemappings increases exponentially. For example, as shown in Table 1.1, while the number ofpossible mappings for 2-D 64-QAM is 1:3Ã 1089, there are 3:6Ã 101=019 possible mappingsfor 4-D 64-QAM. For example, using In [16] and [25], the authors demonstrated that therandom mapping technique can lead to efficient higher dimensional mappings. Accordingto the random mapping technique, computer searching is used to obtain a good mappingfrom a large set of randomly generated mappings. Selecting a mapping randomly/blindlyfrom a large set and checking its efficiency makes the procedure complex. Moreover, itdegrades the resulting mappingsâ performance especially when the MD modulation is con-structed using a large 2-D modulation. Therefore, the problem of 2-D or MD mapping ofhigher order modulations for BICM-ID has not been solved efficiently yet.51.5. Ghefif bUjecgiieC.G fzwsis otjwutivwBER is the most common metric that is used to evaluate the performance of BICM-ID.Fig. 1.3 illustrates a typical BER curve of BICM-ID after a number of iterative decoding.In this figure, Eb denotes the transmitted energy per bit, N0 represents the power of noiseper Hertz, the threshold Eb=N0 is the point where BER starts to decrease, the turbo cliffregion is the region where the BER curve falls quickly, and the error-floor region is theregion where the BER curve is flat at very small values. Two main goals in developingsignal mappings for BICM-ID are to achieve a BER curve with (i) a lower error-floor and(ii) an earlier turbo cliff (i.e., smaller threshold Eb=N0).Eb/N0 (in dB)0 1 2 3 4BER10-810-610-410-2100Threshold Eb/N0Turbo cliffError-floorFigure 1.3: A typical BER curve for BICM-ID.The error performance of BICM-ID at low signal to noise ratios (SNRs) depends on theBER at the first iteration. This is because at low SNRs no coding gain can be achieved fromthe iterative decoding process. In this case, BICM-ID is equivalent to BICM. Therefore,the optimum mapping for BICM-ID in the low SNR region corresponds to the optimummapping for BICM. However, the mapping designed for a low SNR region usually offers apoor error-floor for BICM-ID. The mapping designed to minimize the error-floor of BICM-ID results in an extremely low BER at a very high SNR value. Hence, this mapping is notvery relevant for practical communication systems, which require a BER of 10â= to 10â6.Moreover, finding such a mapping is computationally expensive. Consequently, designingan efficient MD mapping for BICM-ID that offers good BERs at both low and high SNRvalues is very desirable.Motivated by the above discussions, the objective in this thesis is to develop efficient 2-D/MD mapping methods for BICM-ID to achieve good BER performance over AWGN and61.6. MTcciag defiga ghideliaef fbe 5I6M-I7Rayleigh fading channels in both low and high SNR regions. In what follows, we investigatemapping design guidelines for AWGN and Rayleigh fading channels. To investigate theguidelines for the low SNR region, we consider BICM-ID performance at the first iteration.C.H Msppiny vwsiyn yuivwlinws xor TIUM-IVC.H.C SiGN uzsnnwlLow eNd rwyionLet y = {y1P · · · P yp} is the set of all possible Euclidean distances between two signal pointsin , where yi Q yj if i Q j, and p depends on the constellation. For example, p takes thevalue of two and five, respectively, for 2-D and 4-D QPSK. A larger value of y1 is desiredto achieve a better BER performance of BICM over the AWGN channel [28]. Moreover, inorder to achieve a good asymptotic BER performance of BICM over AWGN channels, thevalue of Nmin should be as small as possible [28], where Nmin is defined asNmin =1mN2mamaâi=11âb=0âxâ iiN(xP i)P (1.5)where x = [x1P x2P · · · P xa ] is a 2N -D signal point, and N(xP i) is the number of signalpoints at the Euclidean distance y1 from x that are different from x in the ith bit position.Hiyh eNd rwyionThe asymptotic performance of a mapping for BICM-ID over the AWGN channel dependson Φt(P ), which is expressed as [15]Φt(P ) =0@ 1mN2mamaâi=11âb=0âxâ iiexp(ââxâ xËâ24N0)1Aâ1 P (1.6)where xË = [xË1P xË2P · · · P xËa ] is different from x only in the ith bit position and âSâ is theEuclidean norm of S. To achieve improved asymptotic performance, a greater value ofΦt(P ) is desired [15]. Let us define, yË2min as the minimum squared Euclidean distance(MSED) between two symbol-vectors with a Hamming distance of one bit in the appliedmapping, i.e., yË2min is the minimum value of âxâ xËâ2 in (1.6), and defineJ(xP xË) ={1 if âxâ xËâ2 = yË2min0 otherwise(1.7)71.6. MTcciag defiga ghideliaef fbe 5I6M-I7as an indicator function. At high SNR values, exp(â yË2min4N0)becomes the dominant termin (1.6). Therefore, (1.6) at high SNR values can be approximated as followsΦt(P ) â0@ 1mN2mamaâi=11âb=0âxâ iiJ(xP xË) exp(â yË2min4N0)1Aâ1=1NËminexp(yË2min4N0)P (1.8)where NËmin is defined asNËmin =1mN2mamaâi=11âb=0âxâ iiJ(xP xË): (1.9)In fact, NËmin is the average number of signal pairs in which the two signals are at theEuclidean distance yËmin and at the Hamming distance of one bit from each other.According to (1.8), in the high SNR region, Φt(P ) clearly depends on NËmin and yË2min,which are determined by the applied mapping. In fact, Φt(P ) increases as NËmin decreasesor as yË2min increases. However, the effect of yË2min on Φt(P ) is much more significant thanthat of NËmin due to the exponential relationship between yË2min and Φt(P ).C.H.D dsylwiyz xsviny uzsnnwlThe so called hvrmonix mevn of the MSED [28] of a mapping is a well-known parameterthat relates to the BER performance of BICM-ID on Rayleigh fading channels. In [15], theharmonic mean of the MSED is developed for 2N -D mappings and is expressed asΦ(P ) =0@ 1mN2mamaâi=11âb=0âxâ ii1âxâ xËâ21Aâ1 : (1.10)For the performance at the first iteration, xË refers to the nearest neighbor of x in ib,and Φ(P ) is referred to as the harmonic mean of the MSED before feedback. For theasymptotic performance, ib involves only one symbol-vector xË, which is different from xonly in the ith bit position. In this case, (1.10) is referred to as the harmonic mean of theMSED after feedback, which is denoted by Î¦Ë(P ).81.6. MTcciag defiga ghideliaef fbe 5I6M-I7We can rewrite Φ(P ) asΦ(P ) =(1mN2mapâi=1niy2i)â1P (1.11)where ni is defined asni =maâj=11âb=0âxâ jiIi(xP xË); i = 1P · · · P pP (1.12)where Ii(xP xË) is an indicator function that takes the value of one if the Euclidean distancebetween x and xË is equal to yi, otherwise it takes the value of zero. In what follows, wedescribe the mapping design criteria for the Rayleigh fading channel in low and high SNRregions.Low eNd rwyionTo achieve a good performance at the first iteration, a larger value of Φ(P ) is desired.In the low SNR region, xË in (1.10) is the nearest neighbour of x in ib. Since each signalis different with its nearest neighbor at least in one bit position, xË is at the Euclideandistance y1 from x for some values of i. As a result, n1 in (1.11) always has a non-zerovalue. Moreover, summation of ni over all values of i is constant, i.e.,pâi=1ni = mN2ma : (1.13)Considering (1.11) and (1.13), it is obvious that for a specific value of i, any reduction inni without increasing nj , where j Q i, yields a larger value of Φ(P ). In particular, onecan increase Φ(P ) by decreasing n1.Hiyh eNd rwyionFor the asymptotic performance of BICM-ID, xË is considered to be different from x onlyin the ith bit position. As a result, it is possible to design a mapping in which yËmin S yifor some small values of i. This gives ni = 0 for some small values of i and it yields a largervalue of Î¦Ë(P ). Numerical examples show that a significant increase in yËmin leads to aconsiderable increase in Î¦Ë(P ).In summary, it can be concluded that a mapping that offers a small value of Nminand a large value of yË2min is suitable to improve the error performance of BICM-ID over91.7. 6bded mbdhlTgiba cTcTciglboth AWGN and Rayleigh fading channels in low and high SNR regions. A small valueof Nmin implies that the average Hamming distance between neighbouring symbols, i.e.,symbols with the Euclidean distance y1, is small, and therefore, n1 is small. This eventuallyincreases Φ(P ) at the first iteration of BICM-ID in the Rayleigh fading channel. Thus,a small value of Nmin can improve the mappingâs performance at the first iteration, i.e., inlow SNR region, not only in the AWGN channel but also in the Rayleigh fading channel.On the other hand, a large value of yË2min implies that ni is equal to zero for small values of i.This increases Î¦Ë(P ) at high SNR values. As a result, the asymptotic BER performanceof BICM-ID improves over AWGN and Rayleigh fading channels as the value of yË2minincreases.C.I Uovwv movulstion uspsuityThroughout this thesis, we use the coded modulation (CM) capacity (or constrained capac-ity) [28] to indicate how far the achieved performance is from the capacity. Table 1.2 liststhe CM capacity for different modulations over the AWGN and Rayleigh fading channels.Table 1.2: CM capacity for b -QAM.Modulation AWGN (Eb=N0 in dB) Rayleigh fading (Eb=N0 in dB)16-QAM 2:09 3:9832-QAM 3:13 5:0264-QAM 4:28 6:24128-QAM 5:26 7:34256-QAM 6:48 8:65512-QAM 7:57 9:771024-QAM 8:91 11:11C.8 Litwrsturw rwviww ox msppiny mwtzovsThis section provides a brief review of the well-known methods that have been applied tofind suitable mappings for BICM-ID. To the best of our knowledge, none of the heuristicmapping methods in the literature is applicable for higher order modulations. Therefore,in this section, we review only the computer-based mapping search techniques, which canbe used for higher order modulations.101.8. LigeeTghee eeiiej bf mTcciag meghbdfC.8.C Tinsry swituziny slyoritzm (TeS)One iteration of the BSA can be described in the following steps [9]:1- A random mapping is generated for the constellation.2- A cost function is calculated for each symbol in the constellation, and then, symbols arelisted in descending cost value.3- The label of the symbol with the highest cost value is switched with the label of anothersymbol such that the total cost is reduced as much as possible.4- If a switching is done in step 3, go to step 2; otherwise, look for such a switching for thenext symbol in the list (the symbol with the next highest cost value).5- If a switching is done in step 4, go to step 2; otherwise, the algorithm ends.Usually, the BSA is applied for a number of iterations and the best achieved map-ping is selected as the BSA mapping. The BSA can be considered as the best knowncomputer search technique to find suitable mappings for BICM-ID. However, it becomesintractable for finding suitable mappings for modulations with a large alphabet size due tothe computational time constraints.C.8.D dsnvom msppinyThe random mapping technique searches among a set of randomly generated mappingsto find a mapping that minimizes a defined cost function [16], [25]. In order to achieve abetter result, the set of randomly generated mappings should be as large as possible. Thismakes the random mapping technique suffer from computational complexity and degradesthe obtained mappingâs performance.The rest of this thesis focuses on our contributions to the signal mapping problemfor BICM-ID. In particular, Chapter 2 provides our proposed mapping method for higherorder 2-D modulations. Chapter 3 presents our proposed MD mapping of 16- and 64-QAM.Chapter 4 describes our proposed MD mapping of higher order modulations for BICM-IDover Rayleigh fading channels. Chapter 5 presents our proposed MD mapping methodusing rectangular QAMs for BICM-ID over AWGN channels. Finally, Chapter 6 concludesthe thesis.11Uzsptwr DMsppiny xor Hiyzwr arvwr 2-VMovulstionsAs mentioned earlier, finding a good mapping for higher order modulations for BICM-ID is challenging. Several mapping algorithms for 2-D modulations were proposed inthe literature. The BSA was proposed in [9]. In [13], the authors used a reactive Tabusearch (RTS) method to find mappings with a minimum error-floor. A genetic algorithm(GA)-based mapping optimization was proposed in [17]. All the mentioned algorithms stillexhibited high computational complexity when finding suitable mappings for higher ordermodulations.In this chapter, we take a heuristic approach to propose a systematic mapping methodfor higher order modulations, where the computer search based methods become imprac-tical. The proposed method is a simple and explicit method and easily generates goodmappings for higher order modulations. We study the resulting mappingsâ characteristicsand compare their performance with other well-known mappings to date. Numerical resultsshow that for a target BER of 10â6 and over the Rayleigh fading channel, our resultingmapping offers a gain of 0:7 dB over the RTS mapping for 64-QAM. This gain is 2:3 dBand 4:4 dB, respectively, over the well-known BSA mappings for 256- and 1024-QAM. Forall these cases, our mappings exhibit a comparable error-floor with a gap of about 0:6 dBor less. On the AWGN channel, our achieved gains are even larger.D.C broposwv msppinyOur proposed mapping consists of a precoding process followed by an intermediate mappingas described below.D.C.C brwuoviny prouwssLet us denote the proposed precoding function by Ψ : {0P 1}m â {0P 1}m, where m isthe number of bits per symbol. According to the proposed precoding method, an arbi-trary m-bit label lt = [l1t P l2t P · · · P lmt ] is converted to a precoded m-bit label lËt = Ψ(lt) =122.1. Cebcbfed mTcciag[lË1t P lË2t P · · · P lËmt ] as follows:lËit ={l (lt) if i = chosen-indexlit âl (lt) otherwiseP(2.1)where l (x) is an indicator function that takes the value of one if the Hamming weightof x is odd, otherwise it is equal to zero, the chosen-index can take value from the set{1P 2P · · · Pm}1, and â is the modulo-2 addition.Wxsmplw D.C. huppose thvt the xhosenBinyex is equvl to one vny lvwel l1 = [l11P l21P l=1P l41] =[1P 1P 0P 1]C hinxe the Hvmming fieight of l1 is oyyA using eqC (GCF)A fie xvn firitelË11 = l (l1) =l ([1P 1P 0P 1]) = 1PlË21 = l (l1)â l21 =l ([1P 1P 0P 1])â 1 = 1â 1 = 0PlË=1 = l (l1)â l=1 =l ([1P 1P 0P 1])â 0 = 1â 0 = 1PlË41 = l (l1)â l41 =l ([1P 1P 0P 1])â 1 = 1â 1 = 0:ihusA using our proposey prexoying proxessA l1 = [1P 1P 0P 1] results in pl1 = [1P 0P 1P 0]ChimilvrlyA for l2 = [0P 1P 0P 1] the Hvmming fieight is evenP vs suxhA eqC (GCF) yielyspl2 = [0P 1P 0P 1]Cbroposition D.C. huppose thvt lk vny ln vre tfio mBwit lvwels thvt vre yifferent only in thejth wit positionC Vxxorying to the proposey prexoying in eqC (GCF)A the prexoyey lvwel of lkAiCeCA lËkA vny the prexoyey lvwel of lnA iCeCA lËnA hvve the Hvmming yistvnxe of m wits if thexhosenBinyex is equvl to jC dtherfiiseA they hvve the Hvmming yistvnxe of (mâ 1) witsCeroofC Suppose that lk = [l1kP l2kP · · · P ljkP · · · P lmk ] and ln = [l1nP l2nP · · · P ljnP · · · P lmn ], where forall i except i Ì¸= jP lik = lin. Using eq. (2.1), we can obtain the corresponding precoded labelsfor lk and ln, respectively, as lËk = [lË1kP lË2kP · · · P lËjkP · · · P lËmk ] and lËn = [lË1nP lË2nP · · · P lËjnP · · · P lËmn ].Without loss of generality, let us assume that the chosen-index is equal to q. Then, thereare two possible cases as follows.Cvse FO The chosen-index is equal to the bit position in which lk and ln differ, i.e., q = j.Using eq. (2.1), we can write lËik =l (lk) and lËin =l (ln) if i = q. Moreover, the Hammingdistance between lk and ln is equal to one bit. Therefore, we can writelËik â lËin =l (lk)âl (ln) = 1: (2.2)Now, lËik â lËin = 1 implies that lËik = ¯Ëlin, where x¯ is the 1âs complement of x.)In the next suwsextion, we yesxriwe how to xhoose the xhosenBinyex.132.1. Cebcbfed mTcciagSimilarly, using eq. (2.1), we can write lËik = likâl (lk) and lËin = linâl (ln) when i Ì¸= q,which yieldslËik â lËin = lik âl (lk)â lin âl (ln): (2.3)Using the fact that the Hamming distance of lk and ll is equal to one bit and lik â lin = 0,from eq. (3) we can write lËikâ lËin = 1, which implies that lËik = ¯Ëlin. Hence, for Case 1, lËik = ¯Ëlinfor all i, which means that lËk and lËn have a Hamming distance of m bits from each other.Cvse GO The chosen-index is equal to one of the bit positions in which lk and ln donot differ, i.e., q Ì¸= j. Similar to Case 1, using eq. (2.1), we can writelËik â lËin =l (lk)âl (ln) = 1P if i = qP (2.4)which means that lËik =¯Ëlin. However, if i Ì¸= q, using eq. (2.1) we can write lËik = lik âl (lk)and lËin = lin âl (ln). Then, we havelËik â lËin = [lik â lin]â [l (lk)âl (ln)]: (2.5)Since the Hamming distance of lk from ln is equal to one bit, and lk and ln are differentin the jth bit-position, then using eq. (2.5) we can write lËjk â lËjn = 0, which implies thatlËjk = lËjn. On the other hand, if i Ì¸= j, we can write lik â lin = 0, which yields eq. (2.5) to beequal to one. This implies that lËik =¯Ëlin. Consequently, for Case 2, for all i except i Ì¸= j,lËik =¯Ëlin, which means that lËk and lËn have a Hamming distance of (m â 1) bits from eachother.broposition D.D. ihe proposey prexoying funxtion is wijextiveA iCeCA lt = Î¨â1(Ëlt)P theoriginvl lvwel lt xvn we uniquely owtviney from the prexoyey lvwel lËtCeroofC Let us consider that the chosen-index is q, and then according to eq. (2.1), we canwritelËqt =l (lt) =ââjljt = lqt ââj Ì¸=qljt P (2.6)whereâis a modulo-2 summation. From eq. (2.6), we havelqt = lËqt ââj Ì¸=qljt : (2.7)Now, let us find the m-bit label lt = [l1t P l2t P · · · P litP · · · P lmt ] from its precoded version, i.e.,lËt = Ψ(lt) = [lË1t P lË2t P · · · P lËmt ]. There are two possible cases as follows.142.1. Cebcbfed mTcciagCvse F : i Ì¸= q. In this case, according to eq. (2.1), lËit is given bylËit = lit âl (lt)P (2.8)and using lËqt =l (lt), we can writelit = lËit âl (lt) = lËit â lËqt : (2.9)Cvse G : i = q. In this case, using eq. (2.7), lit can be obtained aslit = lqt = lËqt ââj Ì¸=qljt : (2.10)Using eq. (2.9), we simplify eq. (2.10) aslit = lËqt ââj Ì¸=q(lËjt â lËqt ) (2.11)= lËqt ââj Ì¸=qlËjt ââj Ì¸=qlËqt=ââjlËjt ââj Ì¸=qlËqt= l (Ëlt)âA(m)Ã lËqt Pwhere A(m) is an indicator function that takes the value zero if m is odd, otherwise it isequal to one. Consequently, if q denotes the chosen-index, using eqs. (2.9) and (2.11), thereverse equation to generate lt from lËt can be expressed aslit ={lËit â lËqt if i Ì¸= ql (Ëlt)âA(m)Ã lËqt if i = q:(2.12)From eq. (2.12), it is obvious that lt depends only on its precoded version lËt. In otherwords, two precoded labels, lËk and lËl yield the same original label only when lËk = lËl. Thatis, lËk = lËl leads to lk = ll. As a result, the precoding function is bijective.D.C.D Msppiny prouwssOur proposed mapping uses Gray mapping as the intermediate mapping. The reason willbe explained later in this section. In order to obtain the resulting mappings using theprecoded label lËt and Gray mapping, we map the symbols as follows. In the Gray labeledconstellation, the symbol labeled with lËt is mapped to lt. The proposed mapping is defined152.1. Cebcbfed mTcciag 0000000100110010010001010111011011001111 10111101 100110001110 10100000100100111010110001011111011010111000 01000010 111001110001 1101Gray Mapping Proposed MappingFigure 2.1: An example of the resulting mapping for 16-QAM.asΦ(lt) = G(Ëlt) = G(Ψ(lt))P (2.13)where Φ(·) is the proposed mapping and G(Â·) represents the Gray mapping.Wxsmplw D.D. For exvmpleA FigC GCF shofis the resulting 16BfVb mvpping fiith ourproposey vlgorithmC In fihvt follofisA fie yesxriwe hofi the resulting 16BfVb mvpping isowtvineyC aet us vssume thvt the xhosenBinyex is equvl to oneC jsing eqC (GCF)A theprexoyey version of lvwels l1 = [1P 1P 0P 1] vny l2 = [0P 1P 0P 1] xvn we firitten vs follofis (xCfCAExvmple GCF)Ol1 = [1P 1P 0P 1] â lË1 = [1P 0P 1P 0]Pl2 = [0P 1P 0P 1] â lË2 = [0P 1P 0P 1]:cofiA l1 = [1P 1P 0P 1] is mvppey to the symwol in the 16BfVb xonstellvtion fihose lvwelin the Grvy lvweley F6BfVb is pl1 = [1P 0P 1P 0]A vny l2 = [0P 1P 0P 1] is mvppey to the symwolfihose lvwel in the Grvy lvweley F6BfVb xonstellvtion is pl2 = [0P 1P 0P 1]C In v similvr fvshionAthe rest of the 16BfVb symwols xvn we mvppey using our proposey mvpping methoyCbroposition D.E. ihe Hvmming yistvnxe wetfieen tfio vyjvxent symwols in resulting mvpBpings is either tfioA mA or (m â 1) witsA vny the frvxtion of vyjvxent symwols fiith theHvmming yistvnxe of tfio wits xvn we vt levst (mâ1m )A fihixh tenys to we lvrger for higheroryer xonstellvtionsCeroofC See Appendix A.162.2. 6hTeTcgeeifgicf bf GeTl mTcciagD.D Uzsrsutwristius ox Grsy msppinyIn what follows, we go through three characteristics of Gray mappings that are beneficialin our proposed mapping method.â¢ Gray mapping is easy to generate. For square QAM and PSK constellations, Graylabeling can be obtained from the natural binary labeling (see [29] and [30] for de-tails). For cross QAM constellations, pseudo Gray labeling can be obtained usingthe procedure described in [31].â¢ The Hamming distance between two adjacent symbols is one bit for Gray mappingsand is at most two bits for pseudo-Gray mappings.â¢ Our study shows that among the well-known mappings of 2m-ary modulations, Graymappings have the largest average Euclidean distance between any pair of symbolswith the Hamming distance of m or (mâ 1) bits. For example, Table 2.1 shows thatamong the well-known mappings for 16-QAM (m = 4), the Gray mapping has thelargest average Euclidean distance between symbols with the Hamming distance of3 or 4 bits (please refer to [32] for the comparison of the average Euclidean distancebetween symbols with the Hamming distance of m or (m â 1) bits for other QAMconstellations).Table 2.1: Average Euclidean distance between symbols with 3 or 4-bit Hamming distancesfor 16-QAM mappings.Mapping Average Euclidean distanceSet Partitioning [33] 0.8154Modified Set Partitioning [34] 0.7946Mixed Laleling [34] 0.7520Gray [34] 0.8541MSEW-1 [11] 0.7798MSEW-2 [11] 0.7195MSEW-3 [11] 0.6584b16t [9] 0.8255b16r [9] 0.7807D.E Uzsrsutwristius ox our msppinySince our proposed mapping uses a simple precoding and Gray mapping as an intermediatemapping, it provides a simple and efficient method for generating mappings for QAM andPSK modulations of any order. Moreover, it has two interesting characteristics as follows.172.4. AhmeeicTl eefhlgf Tad difchffibaâ¢ The average Euclidean distance between the symbols with the Hamming distance ofone bit is considerably increased. This can increase yË2min and the harmonic mean ofthe MSED after feedback (denoted by Î¦Ë(P )), and as a result, it improves the BERperformance of BICM-ID in the error-floor region.â¢ According to proposition 2.3, the Hamming distance between most of the adjacentsymbols in our proposed mappings is two bits 2. This decreases Nmin and increasesthe harmonic mean of MSED before feedback (denoted by Φ(P )), and therefore, itimproves the BER performance of BICM-ID in the turbo cliff region.D.F Numwriusl rwsults snv visuussionIn this section, we compare the performance of our proposed mappings and the best previ-ously known mappings for BICM-ID over AWGN and Rayleigh fading channels. Table 2.2compares various evaluation parameters for different QAM mappings. This table showsthat the proposed mappings improve Nmin and Φ, while offering significantly large valuesof yË2min and Î¦Ë. As a consequence, it is expected that the proposed mappings offer a bettererror performance for BICM-ID in the low SNR region.Table 2.2: Evaluation parameters for different QAM mappings.Modulation Mapping Nmin yË2min Φ Î¦Ë64-QAMRTS-based mapping [13] 2.7500 1.6190 0.1048 2.8742TV mapping [14] 5.0000 1.2381 0.0952 2.2784Proposed mapping 2.2143 1.2381 0.1127 2.4986256-QAMBSA mapping [9] 3.6104 1.1765 0.0257 2.8741TV mapping [14] 7.0000 0.5882 0.02353 1.7876eProposed mapping 2.1667 1.0588 0.0333 2.53011024-QAMBSA mapping [9] 4.4899 0.5865 0.0064 2.7515Proposed mapping 2.1129 0.9208 0.0100 2.5617Simulation results for the BER performance of the BICM-ID system using differentmappings are shown in Fig. 2.2-2.6. We use a rate-12 convolutional code with the genera-tor polynomial of (13P 15)8. The length of the used interleaver is about 10000 bits. All gainsare quantified at a BER of 10â6, which is the target BER of many practical data commu-nications. All BER curves are presented with seven iterations, and BER performance forvarious iterations can be found in [32].2le vre thvnkful to the vnonymous geviewer of the originvl vrtixle vs his/her xomment motivvtey usto investigvte why our mvppings perform wetter in the turwo xliff region vny we founy this xhvrvxteristix.182.4. AhmeeicTl eefhlgf Tad difchffibaEb/N0 (in dB)4 5 6 7 8 9 10 11 12 13BER10-810-610-410-2100TV (Rayleigh)RTS(Rayleigh)Proposed (Rayleigh)TV (AWGN)RTS(AWGN)Proposed (AWGN)CM Capacity (Rayleigh)CM Capacity (AWGN)Eb/N0 =4.28 dBEb/N0 =6.24 dBFigure 2.2: BER performance of 64-QAM.Fig. 2.2 plots the simulation results for the BER performance of BICM-ID using ourresulting 64-QAM mapping and the previously known 64-QAM mappings, i.e., the RTS-based mapping, which is optimal in the error-floor region [13], and TV mapping=. In thisfigure, we also plot the corresponding analytical bound on the error-floor using the Gauss-Chebyshev method [34]. From this figure, we can observe that our proposed mappingoutperforms the RTS-based mapping in the turbo cliff region with a certain degradationof the error-floor. This is expected as our proposed mapping has a larger value of Φ whileit has a comparable value of Î¦Ë, as listed in Table 2.2. As shown in Fig. 2.2, in comparisonwith the TV mapping, our proposed mapping significantly improves the error performanceof BICM-ID in both the turbo cliff and error-floor regions. This is because the values of Φand Î¦Ë for our proposed mapping are large (see Table 2.2). On Rayleigh fading channels,the proposed mapping outperforms the RTS-based mapping by 0:7 dB and TV mappingby 2:7 dB. The gain on AWGN channels is even larger.In Fig. 2.3, we compare the BER performance of BICM-ID using our proposed mapping,TV mapping, and the BSA mapping for 256-QAM. We have used the BSA mapping thatis optimized for the error-floor region. By optimizing the generalized weighted BSA costfunction [9] via exhaustive search, the performance of BSA mapping in the turbo cliffregion can be improved at the expense of the error-floor. Moreover, the time complexity of+[or the xonstellvtions presentey in this sextion, we hvve usey the svme ik mvppings thvt vre reporteyin pFIr.192.4. AhmeeicTl eefhlgf Tad difchffibaEb/N0 (in dB)6 8 10 12 14 16 18BER10-610-510-410-310-210-1100TV (Rayleigh)BSA(Rayleigh)Proposed (Rayleigh)TV (AWGN)BSA(AWGN)Proposed (AWGN)CM Capacity (Rayleigh)CM Capacity (AWGN)Eb/N0 =8.65 dBEb/N0 =6.48 dBFigure 2.3: BER performance of 256-QAM.this exhaustive search can prohibit finding a BSA mapping for higher order constellations.Fig. 2.3 shows that our mapping outperforms the BSA and TV mappings in the turbocliff region and it achieves a gain of about 2:3 dB over the BSA mapping. In Fig. 2.4, wecompare the BER performance of our proposed mapping with that of the BSA mappingfor 1024-QAM. On Rayleigh fading channels, our mapping for 1024-QAM offers a gain ofabout 4:4 dB. This is expected as our proposed mapping has a larger value for Φ thanthat of the BSA mapping (see Table 2.2). The obtained gains on AWGN channels areeven larger. Finally, in Fig. 2.5 and Fig. 2.6, we compare the error-floor of BICM-IDusing 64-QAM and using 256- and 1024-QAM mappings, receptively on Rayleigh fadingchannels. These figures show that our mappings offer comparable error-floors (the gap is0:6 dB or less) to those of the best known mappings.It is worth noting that it takes more than a day for the BSA to complete only one roundof the search algorithm for 1024-QAM. However, our method requires only a fraction of asecond to obtain the corresponding proposed mapping. Moreover, our mappings improvethe system performance compared to the BSA mappings. This indicates our methodâsefficiency compared to the BSA.202.4. AhmeeicTl eefhlgf Tad difchffibaEb/N0 (in dB)8 10 12 14 16 18 20BER10-610-510-410-310-210-1100BSA(Rayleigh)Proposed (Rayleigh)BSA(AWGN)Proposed (AWGN)CM Capacity (Rayleigh)CM Capacity (AWGN)Eb/N0 =8.91 dBEb/N0 =11.11 dBFigure 2.4: BER performance of 1024-QAM.Eb/N0 (in dB)5 6 7 8 9 10 11 12 13BER10-1110-1010-910-810-710-610-5TVRTSProposedFigure 2.5: Analytical bound on the error-floor [34] of 64-QAM on Rayleigh fading channels.212.4. AhmeeicTl eefhlgf Tad difchffibaEb/N0 in (dB)10 11 12 13 14 15 16 17 18 19 20BER10-1610-1410-1210-1010-8TVProposedBSA1024-QAM256-QAMFigure 2.6: Analytical bound on the error-floor [34] of 256- and 1024-QAM on Rayleighfading channels.22Uzsptwr EWffiuiwnt Multi-VimwnsionslMsppiny ox 16-, 64-cSM xorTIUM-IVAssigning a sequence of binary digits to a vector of symbols rather than a single symbolis referred to as a MD mapping. MD mapping improves system bandwidth efficiency andalso offers more flexibility in generating good mappings for BICM-ID [15]. However, ittremendously increases the number of possible mappings and makes it difficult to findgood/optimum mappings for MD modulations. This problem is more severe when MDmodulations are constructed using higher order 2-D modulations. Suitable MD mappingsare obtained by computer search techniques except for MD modulations that use a smallermodulation, e.g., BPSK, QPSK and 8-PSK, as a basic modulation.The BSA can be considered as the best known computer search method for findinggood mappings. However, to obtain suitable mappings for larger modulations such asMD modulations, the BSA becomes intractable due to its complexity [16]. In [16] and[25], the authors demonstrated that random mapping can lead to efficient MD mappings.According to the random mapping technique, computer search is used to obtain a goodmapping from a large set of randomly generated mappings, which makes the procedurecomplex. Moreover, it degrades the resulting mappingsâ performance especially for largerMD modulations.In this chapter, we propose a more efficient mapping method for MD modulationsthat use 2m-QAM (m = 4P 6) as the basic modulation. Our goal is to obtain mappingsthat improve the error performance of BICM-ID at low SNR values as well as at highSNR values. A similar objective is considered in [11] where the authors used a dopingtechnique (combining two mappings) to obtain mappings for 2-D modulations. However,instead of combining mappings, we develop a single mapping for a given MD modulation,in order to achieve a lower error rate for BICM-ID in both low and high SNR regions.Furthermore, our proposed method yields mappings for MD modulations rather than 2-D ones. The proposed method is a heuristic-based technique and does not employ any233.1. Cebcbfed mTcciag meghbdcomputer searching. The presented numerical results show that our approach not onlyefficiently generates mappings but also improves the BER performance of BICM-ID overboth AWGN and Rayleigh fading channels. For example, our method can save about 3 dBof transmit signal energy for a target BER of 10â6 compared to the mappings found bythe BSA and random mappings.E.C broposwv msppiny mwtzovAs it is discussed in section 1.6, a mapping that offers a small value of Nmin while itgives a large value of yËmin is suitable to achieve good error performance of BICM-ID overboth AWGN and Rayleigh fading channels in low and high SNR regions. Based on thisdiscussion, we take a heuristic approach to construct a mapping that improves the BERperformance of BICM-ID in low and high SNR regions over AWGN and Rayleigh fadingchannels. In particular, we apply two key techniques as follows. First, to generate amapping with a large value of yËmin, we map binary labels with a Hamming distance of onebit to the symbol-vectors with a large Euclidean distance. This leads to improved error-floors over AWGN and Rayleigh fading channels. Second, most of the nearest neighbouringsymbol-vectors are mapped to the binary labels with a Hamming distance of two bits. Thisresults in a small value of Nmin and yields good BER performance in a low SNR regionover AWGN and Rayleigh fading channels.E.C.C Mwtzov vwsuriptionThe proposed MD mapping using 2m-QAM symbols is constructed progressively in (mâ1)steps. The mappings in step i (1 â¤ i â¤ (m â 2)) are intermediate mappings whereas themapping in step i = (mâ1) is the final mapping. In the ith step, 2i51 symbols from 2m-aryconstellation are selected to be used in the mapping process. In what follows, we describeour mapping method in details.eymtols swlwutionLet hj represent the symbol with position-index j in a square QAM constellation, j =1P · · · P 2m. We assume that j increases from left to right and from top to bottom in theconstellation. The general principles in choosing 2i51 symbols from a 2m-QAM constella-tion in the ith step are as follows: (i) by moving the set of selected symbols one can coverall symbols of the constellation such that each symbol is covered only one time. In otherwords, the squareb -QAM constellations can be partitioned into a number of subsets wherethe structures/shapes formed by these subsets are congruent with one another. Thus, by243.1. Cebcbfed mTcciag meghbdmoving one of the subsets and superimposing it on the remaining subsets, one can cover allsymbols in the constellation such that each symbol is covered only once. (ii) The MSEDbetween the chosen symbols is as large as possible. Without loss of generality, assumethat i denotes the set of 2i51 chosen symbols in step i and i = [ 213i P 223i P · · · P 22i+)3i ]indicates the position-indexes of symbols in i. The set of used symbols in step (i + 1)contains all the used symbols in step i, i.e., i â i51 and i â i51. As such in step(mâ 1), all symbols in the constellation will be used to construct the MD mapping.Msppiny prouwssSuppose that l = [l213P l223P · · · P l2ma3] is an mN -bit binary label, and in step i, vi =[a213i P a223i P · · · P a22i513a3i ] denotes the (i + 1)N least significant bits of l where a2k3i is givenby:a2k3i = l2maâ2i513a5k3P k = 1P 2P · · · P (i+ 1)N: (3.1)Assume that vi is mapped to symbol-vector xi = [x213i P · · · P x2a3i ], where x2k3i â i. Thecorresponding position-index vector for xi is denoted by ji = [j213i P · · · P j2a3i ], where j2k3i â i refers to the position-index of symbol x2k3i in the constellation. Now, we describe thesteps of the mapping process.First step: In step i = 1, the selected symbol set 1 is equivalent to QPSK symbols interms of intersymbol Euclidean distances. Therefore, in order to achieve a good mapping,we use the optimum MD QPSK mapping method introduced in [12]. In particular, a 2N -bit label v1 is mapped to N consecutive QPSK symbols using the method proposed in[12]. Then, we use a conversion vector, denoted by = [213P · · · P 243], to convert eachsymbol in the achieved MD QPSK mapping to one of the symbols in 1. Without loss ofgenerality, we assume the QPSK symbols are expressed as:ek = zj r2 ; k = 1P · · · P 4; j2 = â1P (3.2)where k is the symbol position-index in the QPSK constellation. A particular QPSKsymbol, ek, is converted to one of the symbols in 1, as given below:ek â hz; z = 2k3P (3.3)where hz is the symbol with position-index z in 2m-QAM constellation. It is important tonote that converts each QPSK symbol to the corresponding symbol in the 4-ary constel-lation created using the four chosen 2m-QAM symbols. As a consequence, all properties ofthe developed MD QPSK mapping in [12] are conserved for our MD mapping using fourselected 2m-QAM symbols.253.1. Cebcbfed mTcciag meghbdhuwsequent steps: In step i (i = 2P 3 · · · Pm â 1), we use the intermediate mapping inthe previous step to map label vi to a vector of N symbols from i. Assume that in stepi, bi = [w213i P w223i P · · · P w2a3i ] denotes the N most significant bits of vi, i.e., w2k3i = a2k3i fork = 1P · · · P N . Each symbol in xiâ1 is transformed to obtain the symbol-vector in step i,xi. The transformation rule is defined by iNk, i.e.,x2k3iâ1 i;râââ x2k3i P x2k3iâ1 â iâ1P x2k3i â iP (3.4)where iNk is a 2i-tuple vector and it is determined based on the Hamming weight of biand the bit value of w2k3i .hymwol trvnsformvtion using iNkO For given vectors jiâ1 and iâ1 and for a particularvalue of k (k = 1P · · · P N), there exists a q â {1P · · · P 2i} such that j2k3iâ1 = 2q3iâ1: Then theposition-index of the kth symbol in xi, i.e., j2k3i , is given by:j2k3i = 2q3iNk P (3.5)where 2q3iNk is the qth element of the corresponding vector iNk. The values of j2k3i determinethe symbols in xi.E.C.D Vwsiyn uonsivwrstions ox iNkThere are two key ideas in designing iNk (i S 1) as follows. As discussed in Section 1.6, alarge value of yË2min is desired to achieve a good error performance at high SNRs over AWGNand Rayleigh fading channels. Let yË2minNi be the MSED between two symbol-vectors with aHamming distance of one bit in the ith step of our proposed mapping process. As mentionedearlier, the intermediate MD mapping in the first step is equivalent to the optimum MDQPSK mapping, which is developed in [12]. Therefore, it yields the largest possible valueof yË2minN1 for the selected four symbols from 2m-QAM. To achieve a large value of yË2min, iNk should be designed such that yË2minNi â¥ yË2minN1 for i = 2P 3P · · · Pm â 1. To develop amapping with a small value of Nmin, iNk is designed such that the most of the symbol-vectors with the Euclidean distance yminNi in step i are mapped by binary labels with aHamming distance of two bits, where yminNi is the minimum Euclidean distance betweenthe symbols in i. Based on the above discussion, we design iNk as discussed below.Let vi = [biPviâ1] be a given label in step i where bi and viâ1 are two binary sequencesof lengths N and iN bits, respectively. Assume that vËi is a binary sequence of (i + 1)Nbits and it is different from vi only in the kth bit position. Then, there are two possible263.1. Cebcbfed mTcciag meghbdcases for vËi as given below:vËi ={[bËiPviâ1] if k â¤ NP[biP vËiâ1] if k S NP(3.6)where bËi and vËiâ1 are two binary sequences of lengths N and iN , respectively that havea Hamming distance of one bit from bi and viâ1, respectively. Our first goal is to map viand vËi to the symbol-vectors xi and xËi, respectively such that âxi â xËiâ2 â¥ yË2minN1.Let vËi be a sequence of (i + 1)N bits. Also assume that vËi is different from vi in thejth and the kth bit positions where j Q k â¤ (i + 1)N . Then, vËi can be defined as one ofthe three following possible cases:vËi =8><>:[bËiPviâ1] if j Q NP k â¤ NP[bËiP vËiâ1] if j â¤ NP k S NP[biP vËiâ1] if j S NP k S NP(3.7)where bËi is an N -bit sequence with a Hamming distance of two bits from bi and vËiâ1 is aniN -bit sequence with a Hamming distance of two bits from viâ1. Suppose that in step i, xËiis an element of i, i. e., xËi â i, where i denotes the set of the nearest symbol-vectorsto xi in i = ai . Our second goal is to map most of the symbol-vectors in i by one ofthe possible cases of vËi in (3.7). In this way, one can have:yH(xiP xËi) = 2 (3.8)for most cases of xËi, where yH(aP w) denotes the Hamming distance between a and w.The two mentioned goals are achieved via a systematic symbol transformation fromstep (iâ 1) to step i using iNk. Specifically, iNk depends on the Hamming weight of bi aswell as on the bit value w2k3i . We consider four different cases for iNk as follows: iNk =8>>>><>>>>: E0 if wH(bi) â EP w2k3i = 0P E1 if wH(bi) â EP w2k3i = 1P b0 if wH(bi) â OP w2k3i = 0P b1 if wH(bi) â OP w2k3i = 1P(3.9)where E0, E1, b0, and b1 are row vectors with 2i elements and each of them is asubset of i.273.2. 8kTmclefE.C.E broposwv vwutors xor i, , snv i;kTable 3.1 shows the position-index vector of the selected symbols in step i, i. e., i, ofthe proposed mapping method. The proposed conversion vector, , for 16-QAM and 64-QAM is also indicated in Table 3.2. Furthermore, Tables 3.3 and 3.4 provide the proposedvectors for E0, E1, b0, and b1 for different steps in the MD mapping using 16-QAMand 64-QAM, respectively.Table 3.1: i in different steps of the proposed mapping method.Basic Modulation i i16-QAM1 [1, 3, 9, 11]2 [1, 2, 3, 4, 9, 10, 11, 12]64-QAM1 [1, 5, 33, 37]2 [1, 3, 5, 7, 33, 35, 37, 39]3 [1, 3, 5, 7, 17, 19, 21, 23, 33, 35, 37, 39, 49, 51, 53, 55]4[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33,35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63]Table 3.2: Conversion vector, .Basic Modulation 16-QAM [11, 3, 1, 9]64-QAM [37, 5, 1, 33]Table 3.3: Different cases of i;k for 16-QAM. i;k i = 2 i = 3 E0 [1, 3, 9, 11] [1, 2, 3, 4, 9, 10, 11, 12] E1 [2, 4, 10, 12] [5, 6, 7, 8, 13, 14, 15, 16] b0 [11, 9, 3, 1] [11, 12, 9, 10, 3, 4, 1, 2] b1 [12, 10, 4, 2] [15, 16, 13, 14, 7, 8, 5, 6]E.D WxsmplwsIn what follows, we provide a number of examples to illustrate how to use Tables 3.1-3.4to construct the MD mapping using 16-QAM.Wxsmplw E.C. Fig. 3.1(a) indicates the symbols in a 16-QAM constellation. According toTable 3.1, 1 equals [1P 3P 9P 11], which means the set of selected symbols to be used in step 1is 1 = {h1P h=P h9P h11}. Similarly, using this table we have 2 = {h1P h2P h=P h4P h9P h10P h11P h12}.The symbols in 1 and 2 are shown in black colour in Fig. 3.1(b) and Fig. 3.1(c), respec-tively.283.2. 8kTmclef ( )a1S6S7S10S11S5S2S9S13S14S3S4S8S 12S 16S15S( )c1S6S7S10S11S5S2S9S13S14S3S4S 8S 12S 16S15S( )b1S6S7S10S11S5S2S9S13S14S3S4S8S 12S 16S15SFigure 3.1: (a) A 16-QAM constellation, (b) Four selected 16-QAM symbols (dark symbols)to be used in mapping step i = 1, i.e., 1, and (c) Eight selected 16-QAM symbols (darksymbols) to be used in mapping step i = 2, i.e., 2.Table 3.4: Different cases of i;k for 64-QAM. i;k i Index-vector E02 [1, 5, 33, 37]3 [1, 3, 5, 7, 33, 35, 37, 39]4 [1, 3, 5, 7, 17, 19, 21, 23, 33, 35, 37, 39, 49, 51, 53, 55]5[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33,35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63] E12 [3, 7, 35, 39]3 [17, 19, 21, 23, 49, 51, 53, 55]4 [9, 11, 13, 15, 25, 27, 29, 31, 41, 43, 45, 47, 57, 59, 61, 63]5[2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34,36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64] b02 [37, 33, 5, 1]3 [37, 39, 33, 35, 5, 7, 1, 3]4 [37, 39, 33, 35, 53, 55, 49, 51, 5, 7, 1, 3, 21, 23, 17, 19]5[37, 39, 33, 35, 45, 47, 41, 43, 53, 55, 49, 51, 61, 63, 57,59, 5, 7, 1, 3, 13, 15, 9, 11, 21, 23, 17, 19, 29, 31, 25, 27] b12 [39, 35, 7, 3]3 [53, 55, 49, 51, 21, 23, 17, 19]4 [45, 47, 41, 43, 61, 63, 57, 59, 13, 15, 9, 11, 29, 31, 25, 27]5[38, 40, 34, 36, 46, 48, 42, 44, 54, 56, 50, 52, 62, 64, 58,60, 6, 8, 2, 4, 14, 16, 10, 12, 22, 24, 18, 20, 30, 32, 26, 28]Wxsmplw E.D. In the proposed MD mapping method, let us set m = 4 (16-QAM), N = 2,and l = [1P 1P 1P 0P 0P 1P 1P 1]. For this example, v1 is made of the four least significantbits of l, i.e., v1 = [0P 1P 1P 1]. In step i = 1, v1 is mapped to a vector of two QPSKsymbols following the proposed method in [12], which results in the QPSK symbol-vectore = [e4P e2]. Then, e is converted to the 16-QAM symbol-vector x1 using . By applying293.3. AhmeeicTl eefhlgf Tad difchffiba(3.3) and setting = [11P 3P 1P 9] (c.f., Table 3.2 for 16-QAM), x1 is obtained as:x1 = [h(4) P h(2) ] = [h9P h=]: (3.10)The vector of the position-indexes corresponding to the symbol-vector x1 is j1 = [9P 3].Wxsmplw E.E. Using example 3.2, and for step i = 2, v2 is equal to [1P 0P 0P 1P 1P 1] and thenb2 equals [1P 0]. Let us consider that v2 is mapped to symbol-vector x2 = [x2132 P x2232 ] in thisstep. Since b2 has an odd Hamming weight and w2132 = 1, according to (3.9) 2N1 equals b1. From Table 3.1, 1 = [1P 3P 9P 11] and from Table 3.3, b1 = [12P 10P 4P 2]. Since j1is equal to [9P 3] (see Example 3.2), then j2131 = 2q31 when q = 3. Using (3.5) and settingq = 3, the result is j2132 = 2=32N1 = 4. Similarly, since w22 = 0, then 2N2 equals b0, where b0 = [11P 9P 3P 1] (c.f., Table III for i = 2). Moreover, when q = 2, j2231 = 2q31 . Using (3.5)and setting q = 2, the result is j2232 = 2232N2 = 9. As a consequence, j2 equals [4P 9], whichmeans that x1 will be transformed to x2 = [h4P h9]. In other words, in step i = 2, v2 ismapped to x2 = [h4P h9].In step i = 3, v= equals [1P 1P 1P 0P 0P 1P 1P 1], and then, b= is equal to [1P 1]. The Hammingweight of b= is even and both elements of b= are equal to 1. As a result, in order todetermine the elements of j= = [j213= P j223= ], we set =N1 = E1 and =N2 = E1. From Table3.3 for i = 3, E1 equals [5P 6P 7P 8P 13P 14P 15P 16] and from Table 3.1 for 16-QAM, 2 isequal to [1P 2P 3P 4P 9P 10P 11P 12]. Furthermore, in step i = 2, j2 equals [4P 9]. As a result,j2132 = 2q32 when q = 4, and j2232 = 2q32 when q = 5. By applying (3.5), one can obtainj213= = 243E1 = 8 and j223= = 253E1 = 13. Consequently, j= equals [8P 13], which means that x2will be transformed to x= = [h8P h1=]. In other words, l is finally mapped to symbol-vectorx= = [h8P h1=].Wxsmplw E.F. Table 3.5 illustrates the proposed 4-D mapping using 16-QAM symbols.In this table, the decimal label in the (j + 1P k + 1)th entry is mapped to symbol-vectorx = [hj P hk]. For example, the decimal label 231, which corresponds to binary labell = [1P 1P 1P 0P 0P 1P 1P 1], is the (9P 14)th entry of Table 3.5 and is mapped to symbol-vectorx = [h8P h1=].E.E Numwriusl rwsults snv visuussionIn this section, we provide a selection of select numerical examples to demonstrate theperformance and advantages of our proposed MD mapping for BICM-ID systems. Wecompare our mappings with random mappings and also with the mappings that are opti-mized using well-known BSA. However, the BSA becomes intractable for MD modulations303.3. AhmeeicTl eefhlgf Tad difchffibaTable 3.5: Our proposed 4-D 16-QAM mapping.h1 h2 h= h4 h5 h6 h7 h8 h9 h10 h11 h12 h1= h14 h15 h16h1 0 17 14 31 65 80 79 94 3 18 13 28 66 83 76 93h2 33 48 47 62 96 113 110 127 34 51 44 61 99 114 109 124h= 5 20 11 26 68 85 74 91 6 23 8 25 71 86 73 88h4 36 53 42 59 101 116 107 122 39 54 41 56 102 119 104 121h5 129 144 143 158 192 209 206 223 130 147 140 157 195 210 205 220h6 160 177 174 191 225 240 239 254 163 178 173 188 226 243 236 253h7 132 149 138 155 197 212 203 218 135 150 137 152 198 215 200 217h8 165 180 171 186 228 245 234 251 166 183 168 185 231 246 233 248h9 9 24 7 22 72 89 70 87 10 27 4 21 75 90 69 84h10 40 57 38 55 105 120 103 118 43 58 37 52 106 123 100 117h11 12 29 2 19 77 92 67 82 15 30 1 16 78 95 64 81h12 45 60 35 50 108 125 98 115 46 63 32 49 111 126 97 112h1= 136 153 134 151 201 216 199 214 139 154 133 148 202 219 196 213h14 169 184 167 182 232 249 230 247 170 187 164 181 235 250 229 244h15 141 156 131 146 204 221 194 211 142 159 128 145 207 222 193 208h16 172 189 162 179 237 252 227 242 175 190 161 176 238 255 224 241with a large alphabet size, e.g., 6-D 64-QAM, due to its computational time constraints.The random mappings for AWGN and Rayleigh fading channels are obtained by selectingthe best mappings from a large number of randomly generated mappings. We considera rate-1=2 convolutional code with the generator polynomial of (13P 15)8. The length ofinterleaver used is 10008 bits. All BER curves are presented with seven iterations, and allreported gains are measured at a BER of 10â6. Also, throughout this section, SNR refersto the SNR per bit, i.e., Eia( .E.E.C bwrxormsnuw ovwr SiGN uzsnnwlAs previously discussed, there are two important parameters for a mapping, i.e., Nmin andyË2min, that are relevant to BER performance of BICM-ID systems over AWGN channel. InTable 3.6, we compare Nmin and yË2min values of our MD mappings using 16-QAM and 64-QAM with those of the well-known BSA mappings that are optimized for AWGN channelsand random mappings. In this table, BSA MD 64-QAMmapping for higher dimension, e.g.,N = 3, is not reported as it could not be obtained due to the computational complexity.Table 3.6 clearly shows that our mappings offer smaller values of Nmin compared to theircounterparts. So, the proposed MD mappings will improve the BER performance of BICM-ID at low SNR values over AWGN channels. This is confirmed in Fig. 3.2 which plotsthe BER of various mappings over an AWGN channel. As it can be observed from thisfigure, the proposed mappings outperform the BSA mappings by 1:5 dB, 2:5 dB, and 3:5dB for 4-D 16-QAM, 6-D 16-QAM, and 4-D 64-QAM, respectively in the low SNR region.The error rate improvement with the proposed mappings over random mappings is even313.3. AhmeeicTl eefhlgf Tad difchffibaEb/N0 (in dB)0 2 4 6 8 10 12 14BER10-610-510-410-310-210-1100Random 4D 16QAMBSA 4D 16QAMProposed 4D 16QAMRandom 6D 16QAMBSA 6D 16QAMProposed 6D 16QAMRandom 4D 64QAMBSA 4D 64QAMProposed 4D 64QAMCM capacity 16QAMCM capacity 64QAMEb/N0 =2.09 dBEb/N0 =4.28 dBFigure 3.2: BER performance over an AWGN channel.larger. Table 3.6 also shows that our mappings yield larger values of yË2min compared tothe BSA and random mappings. Thus, the proposed mappings result in improved BERperformance in the high SNR region over AWGN channels. This can be observed fromthe plotted error-floor bounds in Fig. 3.3. It is worth noting that the gap between Nminfor different mappings in Table 3.6, can explain the gap between the corresponding BERcurves in Fig. 3.2. This is because for large constellations such as MD constellations, theimpact of Nmin on BER is more significant than that of yË2min.Table 3.6: Comparison of Nmin and yË2min for different mappings.MappingN = 2 N = 3Nmin yË2min Nmin yË2minRandom MD 16-QAM 4.0412 0.2069 6.0137 0.1333BSA MD 16-QAM 3.7305 1.2069 5.8254 1.3333Proposed MD 16-QAM 2.2500 2.4000 2.2778 2.6667Random MD 64-QAM 6.0114 0.0476 8.9979 0.0317BSA MD 64-QAM 5.8240 1.1905 - -Proposed MD 64-QAM 2.3214 2.2857 2.3571 2.5397E.E.D bwrxormsnuw ovwr dsylwiyz xsviny uzsnnwlsFor Rayleigh fading channels, Φ(P ) and Î¦Ë(P ) are two important mapping parametersto compare the BER performance of BICM-ID systems. We compare the values of Φ(P )323.3. AhmeeicTl eefhlgf Tad difchffibaEb/N0 (in dB)0 1 2 3 4 5 6 7 8 9 10BER10-5010-4010-3010-2010-10100Random 4D 16QAMBSA 4D 16QAMProposed 4D 16QAMRandom 6D 16QAMBSA 6D 16QAMProposed 6D 16QAMRandom 4D 64QAMBSA 4D 64QAMProposed 4D 64QAMFigure 3.3: Error-floor bounds over an AWGN channel.and Î¦Ë(P ) for the mappings already considered in the previous section in Table 3.7.The proposed mappings offer larger values of Φ(P ) in comparison with the BSA andrandom mappings. This results in better BER performance in the low SNR region withour mappings. The BER plots in Fig. 3.4 show that the proposed mappings offer gains of1:5 dB, 1:6 dB, and 3 dB for 4-D 16-QAM, 6-D 16-QAM, and 4-D 64-QAM, respectively,compared to the BSA mappings that are optimized for Rayleigh fading channels. Theperformance gain with respect to the random mappings is larger. Similar to the AWGNchannel, the gap between BER curves in Fig. 3.4 can be explained by the gap betweenthe corresponding Φ(P ) in Table 3.7. From the listed values of Î¦Ë(P ) in Table 3.7, weobserve that the proposed mappings also increase the values of Î¦Ë(P ). Therefore, ourmappings offer improved error-floor bounds as illustrated in Fig. 3.5.Table 3.7: Comparison of Φ(P ) and Î¦Ë(P ) for different mappings.MappingN = 2 N = 3Φ(P ) Î¦Ë(P ) Φ(P ) Î¦Ë(P )Random MD 16-QAM 0.2012 1.4350 0.1335 1.4934BSA MD 16-QAM 0.2026 2.5814 0.1342 2.8047Proposed MD 16-QAM 0.2151 2.8491 0.1446 2.9741Random MD 64-QAM 0.0478 1.1688 0.0318 1.4370BSA MD 64-QAM 0.0481 2.6899 - -Proposed MD 64-QAM 0.0579 2.8166 0.0392 2.9040333.3. AhmeeicTl eefhlgf Tad difchffibaPlease note that the BSA requires a very long time to finish only one round of the searchalgorithm for 6-D 16-QAM. But, our proposed method is a heuristic method and generatesthe proposed mappings instantaneously. Moreover, our proposed mappings improve thesystem error performance compared to the BSA mappings. This shows the efficiency ofour proposed method compared to the BSA.Eb/N0 (in dB)0 2 4 6 8 10 12 14BER10-610-510-410-310-210-1100Random 4D 16QAMBSA 4D 16QAMProposed 4D 16QAMRandom 6D 16QAMBSA 6D 16QAMProposed 6D 16QAMRandom 4D 64QAMBSA 4D 64QAMProposed 4D 64QAMCM capacity 16QAMCM capacity 64QAMEb/N0 =6.24 dBEb/N0 =3.98 dBFigure 3.4: BER performance over a Rayleigh fading channel.343.3. AhmeeicTl eefhlgf Tad difchffibaEb/N0 (in dB)0 5 10 15BER10-1410-1210-1010-810-610-410-2100Random 4D 16QAMBSA 4D 16QAMProposed 4D 16QAMRandom 6D 16QAMBSA 6D 16QAMProposed 6D 16QAMRandom 4D 64QAMBSA 4D 64QAMProposed 4D 64QAMFigure 3.5: Error-floor bounds over a Rayleigh fading channel.35Uzsptwr FMulti-vimwnsionsl Msppiny oxHiyzwr arvwr cSM xor TIUM-IVavwr dsylwiyz Fsviny UzsnnwlsThe number of possible mappings for an MD modulation increases exponentially as theorder of the employed 2-D basic modulation increases. For example, there are 4! = 24distinct mappings for a 2-D QPSK while for a 4-D QPSK modulation, there are 16! =2:1 Ã 101= possible mappings. For the higher order modulations such as 1024-QAM, thenumber of possible MD mappings approaches infinity. This makes it complicated to findgood mappings for larger constellations. Indeed, the large number of possible mappingsis the main pitfall of all the proposed computer search based methods. For instance, thewell-known mapping search method known as the BSA [9], becomes intractable if theorder of the modulation increases [7]. As a result, the BSA is not applicable to directlysearch for good MD mappings for higher order modulations. Analytical mapping searchmethods are investigated in [13] and [16] for higher order modulations. However, due to ahigh computational complexity, in these works, the authors have not reported results forconstellations larger than 64-QAM.Motivated by the above discussions, the present study focuses on finding good MDmappings for a wide range of modulations including higher order modulations. The ob-jective is to improve the error performance of the BICM-ID system over Rayleigh fadingchannels. First, we introduce a MD mapping method employing four 2-D mappings. Then,a lower bound for the harmonic mean of the MSED [6] is derived for this MD mapping.We next develop mutual cost functions and minimize them over the 2-D mappings in orderto achieve a MD mapping with a large value for the harmonic mean of the MSED. Theproposed method is a low complexity approach and can be easily applied to find good MDmappings for higher order modulations such as 512- and 1024-QAM. The reported numer-ical results confirm the efficiency of the achieved mappings. In Chapter 2, we developedMD mappings for two specific modulations, i.e., 16- and 64-QAM. In the current chapter,we develop a generalized mapping method to construct MD mappings for a vast range of364.1. Cebcbfed mTcciag meghbdmodulations that includes 2m-QAM (m = 4P · · · P 10). Moreover, the proposed method inthis chapter, is not limited to QAM and it is applicable to any kind of constellation.F.C broposwv msppiny mwtzovAs mentioned earlier, for a 2m-ary signal constellation there are 2m! possible mappings. Infact, a comprehensive computer search to find good mappings becomes intractable quicklyas the modulation order increases. The well-known BSA mapping search method cannotbe used directly to obtain good mappings. Therefore, we propose an efficient technique tofind good MD mappings for BICM-ID systems over Rayleigh fading channels.We consider the asymptotic performance of BICM-ID in which the ideal v priori infor-mation about the decoded bits is available at the demodulator. In this case, to detect aparticular bit carried by the received signal, the demodulator has perfect information aboutthe remaining (mN â 1)-bits carried by the signal. Thus, the modulation is converted to abinary modulation and Euclidean distances can easily be increased by employing a carefullydesigned mapping. As it is discussed in 1.6.2, the asymptotic performance of BICM-ID overRayleigh fading channels is influenced by the harmonic mean of the MSED after feedback,which is calculated for a given mapping function, , applied to signal set . Accordingto 1.10, for a 2N -D mapping of a 2m-ary constellation, the harmonic mean of the MSEDafter feedback is given byÎ¦Ë(P ) =0@ 1mN2mamaâi=11âb=0âxâ ii1âxâ xËâ21Aâ1 P (4.1)where x = [x1P x2P · · · P xa ] is a 2N -D signal point and ib is the subset of whose labelstake value w at the ith bit position, and xË = [xË1P xË2P · · · P xËa ] is a signal point in ib whoselabel is different with that of x only in the ith bit position. For BICM-ID with a particularcode, a larger value of Î¦Ë(P ) offers a lower error floor [34]. However, maximizing Î¦Ë(P )is a complicated problem even for 2-D modulations such as 64-QAM [13]. Therefore, wepropose an innovative approach to generate MD mappings using 2-D mappings. Next, wedevelop cost functions that are optimized over the employed 2-D mappings to achieve ahigh value of Î¦Ë(P ) for the MD mapping. Our cost functions are simple and give excellentresults, even for higher order modulations such as 1024-QAM.Let l = [l1P l2P · · · P lma ] be an mN -bit binary label. Equivalently l = [l1P l2P · · · P la ]where li is an m-bit binary label and is given byli = [l2iâ13m51P · · · P lim]; i = 1P :::P N: (4.2)374.1. Cebcbfed mTcciag meghbdSuppose that L denotes the set of all mN -bit binary labels and Lx and Lo representthe subset of all l â L with even and odd Hamming weights, respectively. The MDmapping problem can be broken into four mappings in 2-D signal space as described below.According to the proposed MD mapping function, i.e., , label l is mapped to the 2N -Dsignal point x = [x1P · · · P xa ] as given belowxi =8>>>><>>>>:xl(li) if i = 1 , l â LxPol(li) if i = 1 , l â LoPxr(li) if i â¥ 2 , l â LxPor(li) if i â¥ 2 , l â LoP; i = 1P :::P NP (4.3)where xl, ol, xr, and or are 2-D mapping functions, which will be discussed later inthis section. In the applied mapping, let x and o represent the subset of signal points in whose labels belong to Lx and Lo, respectively. Without loss of generality, assume thatx â x and xË â o where xË = [xË1P xË2P · · · P xËa ] is a signal point whose label is differentwith that of x only in one bit position. We partition the 2-D signal constellation intotwo separate subsets with equal cardinalities and denote them as xl and ol. Then, welimit the first element in x and xË, i.e., x1 and xË1, to belong to xl and ol, respectively. In(4.3), xl(:) and ol(:), each map an m-bit label to a 2-D signal point chosen from xl and ol, respectively. However, xl and ol involve only 2mâ1 signal points while there are 2mdistinct m-bit labels. As a consequence, each signal point in xl and ol should be mappedby two m-bit labels simultaneously. In order to obtain a one-to-one MD mapping function,we restrict the two labels that are mapped to a particular signal point in either xl or olto be different in an odd number of bit positions. For simplicity, we assume that these twolabels are different just in the first bit position. On the other hand, there is no constrainton xr(:) and or(:) except they need to be bijective.broposition F.C. In the proposey bD mvpping funxtionA A there is v oneBtoBone xorreBsponyenxe wetfieen bD signvl points vny winvry lvwelsCeroofC It is obvious that the Hamming distance between two labels from a particular label-set, i.e., Lx or Lo, is even. However, two labels one from Lx and the other from Lo havean odd Hamming distance from each other. Since there is no common 2-D signal pointbetween xl and ol, there is no common 2N -D signal point between x and o. As a result,none of the 2N -D signal points will be mapped simultaneously by a label from Lx and alabel from Lo. Therefore, it is sufficient to prove that there is a one-to-one correspondencebetween labels from Lx and signal points from x and similarly between labels in Lo andsignal points in o. In what follows, we prove this for the even subsets, i.e., for labels in384.2. 7eielbcmeag Tad bcgimimTgiba bf cbfg fhacgibafLx and signal points in x.Assume that l = [l1P l2P · · · P lma ] and lË = [lË1P lË2P · · · P lËma ] are two labels in Lx andare mapped to x = [x1P · · · P xa ] and xË = [xË1P · · · P xËa ], respectively where both x and xËare in x. Let us define li and lËi as the ith m-tuple bits of l and lË, respectively. Thenl = [l1P l2P · · · P lma ] and lË = [Ël1P lË2P · · · P lËma ]. Based on the relation between li and lËi fordifferent values of i, there are two possible cases as follows:Cvse FO There exists a value of i (i â¥ 2) such that li Ì¸= lËi. Let j â¥ 2, then accordingto (4.3) the same one-to-one mapping function, i.e., xr(:), is used to map lj to xj and lËjto xËj . Therefore, because lj Ì¸= lËj , we havexj Ì¸= xËj â x Ì¸= xË: (4.4)Cvse GO li = lËi for all i â¥ 2. In this case, li Ì¸= lËi only when i = 1, and as a result,the Hamming distance between l and lË is equal to the Hamming distance between l1 andlË1. Since l and lË belong to Lx, they have an even Hamming distance from each other.Consequently, the Hamming distance between l1 and lË1 is even as well. However, the twolabels that are mapped to each symbol in xl have an odd Hamming distance from eachother. Therefore, because xl(l1) Ì¸= xl(Ël1), we havex1 Ì¸= xË1 â x Ì¸= xË: (4.5)From (4.4) and (4.5), it is concluded that in the proposed mapping function, different labelsfrom Lx are mapped to different signal points in x. In a similar way, it can be proven thatthe different labels from Lo are mapped to the different signal points in o. As a result,the proposed MD mapping function is bijective.F.D Vwvwlopmwnt snv optimizstion ox uost xunutionsAs it is mentioned previously, maximizing Î¦Ë(P ) is a complicated task. In what follows,we break Î¦Ë(P )â1 into two separated parts. Next, we derive a cost function for each partto maximize Î¦Ë(P ). Both cost functions operate in 2-D signal space rather than MDsignal space. As such, they have much lower complexity. We use (4.1) to write394.2. 7eielbcmeag Tad bcgimimTgiba bf cbfg fhacgibafÎ¦Ë(P )â1 =1mN2mamaâi=11âb=00BBB@âxâ iixâ l1âxâ xËâ2 +âxâ iixâ o1âxâ xËâ21CCCA P (4.6)where x and o are the same size. Moreover, when a given x is in x then the corre-sponding xË belongs to o and vice versa. Therefore, the two parts in (4.6) are equivalent,and as a result, we haveÎ¦Ë(P )â1 = 2â¦(P )P (4.7)where â¦(P ) is given byâ¦(P ) =1mN2mamaâi=11âb=0âxâ iixâ l1âxâ xËâ2 : (4.8)Since âxâ xËâ2 =âaj=1 |xj â xËj |2, then (4.8) can be rewritten asâ¦(P ) =1mN2mamaâi=11âb=0âxâ iixâ l1âaj=1 |xj â xËj |2: (4.9)broposition F.D. aet y = [y1P y2P · · · P ya ] is v vextor of positive revl numwersC ihen fiehvve1âai=1 yi6 1N2aâj=11yj: (4.10)eroofC As f(y) = 1y is convex on R5, sof(aâi=1iyi) â¤aâi=1if(yi)P (4.11)whereâai=1 i = 1. Let i =1a . Thenf(1Naâi=1yi) â¤aâi=11Nf(yi)P (4.12)i.e.,404.2. 7eielbcmeag Tad bcgimimTgiba bf cbfg fhacgibaf11aâai=1 yiâ¤aâi=11N1yi(4.13)â Nâai=1 yiâ¤ 1Naâi=11yiâ 1âai=1 yiâ¤ 1N2aâi=11yi:Applying (4.10) in (4.9), we can writeâ¦(P ) 6 KΨ(P )P (4.14)where K is a constant value and Ψ(P ) is defined asΨ(P ) =maâi=11âb=0âxâ iixâ laâj=11|xj â xËj |2 :We can decompose Ψ(P ) asΨ(P ) = Ψl(P ) + Ψr(P )P (4.15)where Ψl(P ) and Ψr(P ) are given byΨl(P ) =maâi=11âb=0âxâ iixâ l1|x1 â xË1|2 (4.16)andΨr(P ) =maâi=11âb=0âxâ iixâ laâj=21|xj â xËj |2 : (4.17)Let l = [l1P l2P · · · P lma ] and lË = [lË1P lË2P · · · P lËma ] are twomN -bit labels, which are differ-ent only in the ith bit position, and are mapped to x = [x1P · · · P xa ] and xË = [xË1P · · · P xËa ],respectively. We define li and lËi respectively as the ith m-tuple bits of l and lË, and rewritel = [l1P l2P · · · P lma ] and lË = [Ël1P lË2P · · · P lËma ]. Then, (4.16) is equivalent to414.2. 7eielbcmeag Tad bcgimimTgiba bf cbfg fhacgibafÎ¨â²l(xlP olPL) =maâi=11âb=0âlâLiilâLl1|xl(l1)â ol(Ël1)|2P (4.18)where Lib â L is the subset of labels with value w in their ith bit position. For a given m-bitsequence li, lËi can take (m+ 1) distinct m-bit sequences, where each one is the same as lior different from li only in one bit position. For example, if m = 4 and li = [0P 0P 0P 0], lË cantake either of the 5 labels in {[0P 0P 0P 0]P [0P 0P 0P 1]P [0P 0P 1P 0]P [0P 1P 0P 0]P [1P 0P 0P 0]}. Let =[ 1P · · · P m] and = [ 1P · · · P m] are two binary sequences, where has the Hammingdistance of either zero or one from . The set of (m+ 1) possibilities for is denoted byB. Assume that for a given i, li = and lËi = . Then (4.18) is equivalent to l(xlP olP xlP ) =â â âBa2l3 N |xl( )â ol( )|2 (4.19)where a2l3 N is given bya2l3 N =maâi=11âb=0âlâLiilâLl[l1 = P lË1 = ]P (4.20)where [x] is an indicator function and is defined as[x] ={1 if x is true0 otherwise:(4.21)Similarly, (4.17) is equivalent toÎ¨â²r(xrP orPL) =maâi=11âb=0âlâLiilâLlaâj=21|xr(lj)â or (Ëlj)|2: (4.22)The m-bit elements li in l = [l1P l2P · · · P la ] are independent from one another for all i.Then (4.22) is equivalent to r(xrP orP ) =maâi=11âb=0âlâLiilâLlN â 1|xr(l2)â or (Ël2)|2P (4.23)which can be rewritten as424.2. 7eielbcmeag Tad bcgimimTgiba bf cbfg fhacgibaf r(xrP orP ) =â â âBa2r3 N (N â 1)|xr( )â or( )|2 P (4.24)where a2r3 N is computed asa2r3 N =maâi=11âb=0âlâLiilâLl[l2 = P lË2 = ]: (4.25)Using (4.7), (4.14), and (4.15), a lower bound of Î¦Ë(P ) can be derived as followsÎ¦Ëâ1(P ) 6 2K (Ψl(P ) + Ψr(P )) (4.26)â â 6 Î¦Ë(P )Pwhere â is given byâ =K â²Î¨l(P ) + Ψr(P ): (4.27)Because Ψl(P ) â¡ l(xlP olP xlP ) and Ψr(P ) â¡ r(xrP orP ), we rewrite â asâ =K â² l(xlP olP xlP ) + r(xrP orP ): (4.28)Note that (4.28) operates in 2-D signal space rather than MD signal space, and as a result,optimization is much simpler.Our objective is to maximize â and then to calculate the corresponding Î¦Ë(P ). Since l(xlP olP xlP ) and r(xrP orP ) are independent from each other, then the maximumvalue of â, âmtx is given byâmtx = Kâ²[minllNolN ll l(xlP olP xlP ) + minlyNoy r(xrP orP )]â1: (4.29)F.D.C aptimizstion ox uost xunutionsThe minimization of l and r in (4.29) can be done using the BSA [9]. To minimize r, tworandommappings are initially considered as xr and or. Then, the BSA is used to minimizethe cost function introduced in (4.24) for xr. In fact, this is a mutual cost function wherethe cost value for a given symbol in xr, is computed by using (m+1) corresponding symbols434.3. AhmeeicTl eefhlgf Tad difchffibafrom or. Our approach is to minimize this cost function by alternatingly modifying eachof xr and xr. In other words, the BSA is used to modify xr to reduce the cost function.After a given number of iterations, xr and or are exchanged. Again, the BSA is used todecrease the cost value. After a certain number of iterations, xr and or are exchangedagain and the BSA modifies xr. This procedure is repeated up to a given number ofiterations.In addition to xl and ol, xl is another effective argument in computing l. Asthere is no constraint on xl, it is a complex process to optimize l. To simplify theoptimization process, xl is constrained to involve only the symbols whose labels in theresulting xr take binary value w in a given bit position. In this chapter, we assume that xlinvolves the symbols whose labels in xr take the value zero in the first bit-position. Thefunctions l and r are computed by considering the similar Euclidean distances betweentwo dimensional symbols. As a result, there is a potential advantage in applying the abovementioned constraint on xl because it will be easier to find a suitable xl correspondingto a given ol. After determining xl and ol, two random mappings are generated as xland ol. As mentioned previously, two assigned labels to a given symbol by either of xland ol are different only in the first bit position. Similar to the previous step, the BSAis applied to minimize l by modifying xl. Then, xl is exchanged by ol and the BSAminimizes l by modifying the new xl. This procedure is repeated up to a given numberof iterations.By executing the proposed algorithm for a certain number of iterations, a local maxi-mum value is calculated using (4.29). The search algorithm is executed several times andeach time the corresponding value for Î¦Ë(P ) is calculated. Finally, the modulations cor-responding to the maximum obtained Î¦Ë(P ) are chosen. Fig. 4.1 illustrates the flowchartof the proposed algorithm.Numerical results confirm that the proposed algorithm generates mappings with sig-nificantly large values of Î¦Ë(P ). As a result, the obtained mappings would improve theerror performance of BICM-ID systems over Rayleigh fading channels.F.E Numwriusl rwsults snv visuussionIn this section, we provide our resulting mappings and selected numerical results to illus-trate the performance and advantage of our proposed MD mappings for BICM-ID.444.3. AhmeeicTl eefhlgf Tad difchffiba â¢ Choose erλ and orλ randomly â¢ 0j = Use BSA to minimize rÏ â¢ Exchange erλ with orλ â¢ 1j j= + . .j it num r<Calculate â maxâ > â â¢ maxâ = â â¢ 1i i= + â¢ Calculate Î¦Ë â¢ Choose elλ and olλ randomly â¢ 0k = Use BSA to minimize lÏ â¢ Exchange elλ with olλ â¢ 1k k= + . .k it num l< .i it num< End Start Yes No Yes No No Yes Yes No Determine elÏ and olÏ max 0â = , maxË 0Φ = , 0i = maxË ËÎ¦ > Î¦â¢ maxË ËÎ¦ = Φ â¢ Save erλ , orλ , elλ , and olλ No Yes Figure 4.1: Flowchart of the proposed algorithm (itCnumCr, itCnumCl, and itCnum representthe number of iterations for different loops).F.E.C dwsultiny MV msppinys ox M-cSMOur proposed algorithm is used to obtain MD mappings of 2m-QAM for m = 4P 5P · · · P 10.Tables 4.1-4.11 show the resulting 2-D mappings such as xr, or, xl, and ol in decimalformat. In these tables, the resulting 2-D mappings for higher order constellations are454.3. AhmeeicTl eefhlgf Tad difchffibaindicated in multiple rows. For example in Table 4.3, xr for 64-QAM is indicated intwo rows where the first element in the second row is the label of the 33rw symbol in theconstellation. For square QAMs, it is assumed that the symbol order starts from the topleft corner in the constellation and increases from top to bottom and from left to right(see Fig. 4.2.(a) as an example for 16-QAM). For cross QAM constellations such as 32-QAM, we consider the symbol order used in [19]. In Tables 4.1-4.11, two labels in theith parentheses in xl and ol belong to the ith symbol in xl and ol, respectively. Forexample, Fig. 4.2(b), (c), and (d) illustrate the 16-QAM mappings reported in Table 4.1. 3,11( )d2,105,134,127,156,141,90,80,84,121,95,132,106,147,153,1112( )c854139101011314152673( )b215117614100412158913( )a1S6S7S10S11S5S2S9S 13S14S3S4S 8S 12S 16S15SFigure 4.2: (a). Symbolâs arrangement in 16-QAM, and achieved 16-QAM mappings indecimal format: (b). xr, (c). or, and (c). xl (the light symbols), ol (the dark symbols).As mentioned previously, xl involves the symbols whose binary labels in xr takethe value zero at the first bit position. Equivalently, xl is constructed by the sym-bols whose decimal label in xr is smaller thanM2 . As a result, for 16-QAM, xl ={h1P h2P h5P h6P h9P h10P h1=P h14}, where xl is indicated by light symbols in Fig. 4.2(d).464.3. AhmeeicTl eefhlgf Tad difchffibaTable 4.1: Proposed xr, or, xl, and ol for 16-QAM.xr [3 2 15 11 7 6 14 10 0 4 12 13 1 5 8 9]or [12 8 5 4 13 9 1 0 10 11 3 7 14 15 2 6]xl [(3, 11) (2, 10) (7, 15) (6, 14) (0, 8) (4, 12) (1, 9) (5, 13)]ol [(5, 13) (4, 12) (1, 9) (0, 8) (3, 11) (7, 15) (2, 10) (6, 14)]Table 4.2: Proposed xr, or, xl, and ol for 32-QAM.xr[24 17 13 8 25 29 1 9 21 28 0 5 16 20 4 12,30 22 6 2 23 18 14 7 26 19 15 11 27 31 3 10]or[7 2 30 22 6 14 18 23 10 15 19 26 3 11 27 31,13 9 25 24 8 1 29 21 4 0 28 20 5 12 17 16]xl[(13 29) (8 24) (1 17) (9 25) (0 16) (5 21) (4 20) (12 28),(6 22) (2 18) (14 30) (7 23) (15 31) (11 27) (3 19) (10 26)]ol[(7 23) (2 18) (6 22) (14 30) (10 26) (15 31) (3 19) (11 27),(13 29), (9 25) (8 24) (1 17) (4 20) (0 16) (5 21) (12 28)]The remaining 16-QAM symbols belong to ol, which are shaded in Fig. 4.2(d). Example4.1 clarifies how to use xl, ol, xr, and or to construct the proposed MD of 16-QAM.Wxsmplw F.C. In the proposed MD mapping method, let us set m = 4 (16-QAM), N = 3and l = [0P 1P 1P 0P 1P 1P 1P 1P 0P 1P 1P 1]. l is considered as a sequence of three 4-bit labels, i.e.,l = [l1P l2P l=], where l1 = [0P 1P 1P 0], l2 = [1P 1P 1P 1], and l= = [0P 1P 1P 1]. The mappingrule in (4.3) is used to map l = [l1P l2P l=] to signal point x = [x1P x2P x=], as follows. TheHamming weight of l is odd, i.e., l â Lo, thus x1 = ol(l1), x2 = or(l2), and x= = or(l=).The decimal format of l1, l2, and l= are 6, 15, and 7, respectively. In Fig. 4.2(d), it can beobserved that among the shaded symbols that ol operates on, symbol h16 is mapped bydecimal label 6. As a result, x1 = h16. Considering the mapping function or indicated inFig. 4.2(c), we also have x2 = or((15)2) = h14 and x= = or((7)2) = h12. Consequently, lis mapped to x = [h16P h14P h12].Table 4.3: Proposed xr, or, xl, and ol for 64-QAM.xr[63 55 61 53 48 56 50 58 62 54 46 38 35 43 51 59,60 52 36 37 32 33 49 57 47 39 44 45 40 41 34 42,21 6 4 5 0 1 3 16 14 29 12 13 8 9 24 11,22 30 28 20 17 25 27 19 23 31 15 7 2 10 26 18]or[0 8 17 2 7 20 13 5 1 9 25 10 15 28 12 4,16 24 27 26 31 30 29 21 3 11 19 18 23 22 14 6,49 57 59 58 63 62 60 52 34 42 51 50 55 54 47 39,41 33 56 48 53 61 36 44 40 32 43 35 38 46 37 45]xl[(14 46) (6 38) (4 36) (5 37) (0 32) (1 33) (3 35) (11 43),(30 62) (21 53) (12 44) (13 45) (8 40) (9 41) (16 48) (27 59),(22 54) (29 61) (20 52) (7 39) (2 34) (17 49) (24 56) (19 51),(23 55) (31 63) (28 60) (15 47) (10 42) (25 57) (26 58) (18 50)]ol[(0 32) (8 40) (17 49) (2 34) (7 39) (20 52) (13 45) (5 37),(1 33) (16 48) (25 57) (10 42) (15 47) (28 60) (21 53) (4 36),(9 41) (24 56) (27 59) (26 58) (31 63) (22 54) (29 61) (12 44),(3 35) (11 43) (19 51) (18 50) (23 55) (30 62) (14 46) (6 38)]474.3. AhmeeicTl eefhlgf Tad difchffibaTable 4.4: Proposed xr, or, xl, and ol for 128-QAM.xr[100 103 37 86 105 122 32 115 116 118 36 70 104 107 106 113,102 110 126 111 127 96 114 98 119 101 108 124 109 82 66 99,117 94 78 125 64 67 112 97 87 76 79 95 80 72 83 74,68 71 92 77 88 65 75 90 61 84 69 93 73 89 91 120,60 31 13 85 24 8 81 121 14 15 12 29 25 11 16 3,30 5 21 28 9 27 17 19 52 23 7 20 57 56 26 1,39 4 22 59 43 10 49 51 38 54 6 41 58 40 42 33,53 46 47 63 18 2 50 48 62 55 44 45 0 123 34 35]or[17 24 121 0 44 61 85 15 16 8 81 3 60 31 13 14,25 9 27 56 20 21 28 29 11 26 40 59 57 4 5 12,1 10 49 43 41 22 7 30 19 48 58 42 6 52 53 23,18 35 51 33 54 39 55 47 123 2 50 34 38 46 62 37,107 122 32 98 102 118 36 63 104 106 115 114 126 119 70 86,113 97 99 96 110 100 103 116 83 112 82 66 111 101 78 117,72 67 64 109 127 108 94 76 65 88 80 125 124 95 79 77,74 89 90 120 68 71 69 87 75 73 91 105 45 84 93 92]xl[(37 101) (32 96) (36 100) (61 125) (60 124) (31 95) (13 77) (24 88),(8 72) (14 78) (15 79) (12 76) (29 93) (25 89) (11 75) (16 80),(3 67) (30 94) (5 69) (21 85) (28 92) (9 73) (27 91) (17 81),(19 83) (52 116) (23 87) (7 71) (20 84) (57 121) (56 120) (26 90),(1 65) (39 103) (4 68) (22 86) (59 123) (43 107) (10 74) (49 113),(51 115) (38 102) (54 118) (6 70) (41 105) (58 122) (40 104) (42 106),(33 97) (53 117) (46 110) (47 111) (63 127) (18 82) (2 66) (50 114),(48 112) (62 126) (55 119) (44 108) (45 109) (0 64) (34 98 (35 99)]ol[(17 81) (24 88) (0 64) (45 109) (60 124) (15 79) (16 80) (8 72),(3 67) (63 127) (31 95) (13 77) (14 78) (25 89) (11 75) (9 73),(27 91) (57 121) (21 85) (28 92) (29 93) (56 120) (10 74) (40 104),(59 123) (20 84) (4 68) (7 71) (12 76) (26 90) (1 65) (49 113),(43 107) (41 105) (22 86) (23 87) (5 69) (19 83) (48 112) (51 115),(58 122) (6 70) (39 103) (52 116) (30 94) (18 82) (50 114) (35 99),(42 106) (54 118) (55 119) (62 126) (53 117) (2 66) (34 98) (33 97),(38 102) (46 110) (37 101) (47 111) (32 96) (36 100) (44 108) (61 125)]484.3. AhmeeicTl eefhlgf Tad difchffibaTable 4.5: Proposed xr, or, xl, and ol for 256-QAM.xr[206 205 222 237 236 220 252 211 227 195 194 135 215 204 231 199,207 254 221 141 253 200 140 243 167 226 151 210 247 83 230 198,238 142 201 88 217 157 216 156 131 209 242 166 193 134 213 197,239 255 89 158 173 232 172 188 163 130 183 225 208 150 246 214,223 218 174 233 249 189 248 136 162 147 241 129 165 81 245 229,202 219 159 191 190 137 153 152 179 145 146 182 224 149 192 133,203 234 250 138 154 169 185 168 177 161 240 144 181 80 244 212,143 175 251 170 155 187 186 184 178 176 160 128 164 148 132 228,91 235 139 60 171 25 24 56 49 17 51 180 35 3 115 99,92 124 44 28 120 40 57 59 48 16 50 34 19 2 39 67,12 72 29 61 121 8 41 58 32 33 113 18 55 114 98 7,108 125 73 104 62 9 27 26 52 112 1 54 97 38 23 66,93 109 13 45 105 63 10 42 43 0 53 96 22 65 82 103,77 94 90 30 46 122 123 11 36 20 37 21 5 6 119 196,76 95 126 14 127 31 106 47 116 4 117 64 118 102 85 71,78 79 110 111 74 75 15 107 84 101 100 68 69 86 70 87]or[49 48 16 32 112 0 52 36 43 42 27 26 58 59 57 56,17 33 1 53 21 117 20 116 47 11 123 10 63 41 40 24,50 113 54 96 37 64 84 4 107 106 46 122 105 62 9 25,51 18 97 65 22 5 101 100 15 127 31 90 30 104 121 8,34 55 114 38 6 118 69 68 75 74 14 13 73 45 61 120,19 2 23 82 119 102 86 111 110 95 126 109 125 72 29 60,35 3 98 7 66 103 85 70 79 77 94 108 12 124 44 28,180 115 39 99 67 196 71 87 78 76 93 92 91 235 139 171,80 148 244 228 212 229 197 198 206 207 239 203 143 234 175 170,164 149 132 133 246 214 231 199 205 238 223 142 202 219 251 138,128 224 192 245 213 230 215 204 222 237 221 255 218 159 250 155,181 81 150 193 134 210 247 135 220 236 254 141 174 89 191 154,240 165 225 208 151 83 194 195 211 200 253 201 158 233 249 187,160 144 129 183 166 209 226 227 252 140 217 173 88 190 137 169,176 161 182 241 130 242 167 131 243 216 157 232 189 153 185 186,178 177 179 146 145 162 147 163 156 188 172 248 136 152 168 184]xl[(83 211) (88 216) (89 217) (81 209) (80 208) (91 219) (60 188) (25 153),(24 152) (56 184) (49 177) (17 145) (51 179) (35 163) (3 131) (115 243),(99 227) (92 220) (124 252) (44 172) (28 156) (120 248) (40 168) (57 185) ,(59 187) (48 176) (16 144) (50 178) (34 162) (19 147) (2 130) (39 167),(67 195) (12 140) (72 200) (29 157) (61 189) (121 249) (8 136) (41 169) ,(58 186) (32 160) (33 161) (113 241) (18 146) (55 183) (114 242) (98 226),(7 135) (108 236) (125 253) (73 201) (104 232) (62 190) (9 137) (27 155),(26 154) (52 180) (112 240) (1 129) (54 182) (97 225) (38 166) (23 151),(66 194) (93 221) (109 237) (13 141) (45 173) (105 233) (63 191) (10 138),(42 170) (43 171) (0 128) (53 181) (96 224) (22 150) (65 193) (82 210),(103 231) (77 205) (94 222) (90 218) (30 158) (46 174) (122 250) (123 251),(11 139) (36 164) (20 148) (37 165) (21 149) (5 133) (6 134) (119 247),(76 204) (95 223) (126 254) (14 142) (127 255) (31 159) (106 234) (47 175),(116 244) (4 132) (117 245) (64 192) (118 246) (102 230) (85 213) (71 199),(78 206) (79 207) (110 238) (111 239) (74 202) (75 203) (15 143) (107 235),(84 212) (101 229) (100 228) (68 196) (69 197) (86 214) (70 198) (87 215)]ol[(49 177) (48 176) (32 160) (112 240) (0 128) (53 181) (43 171) (52 180),(11 139) (10 138) (42 170) (26 154) (27 155) (58 186) (59 187) (56 184),(17 145) (33 161) (1 129) (54 182) (96 224) (21 149) (20 148) (36 164),(116 244) (122 250) (123 251) (63 191) (9 137) (41 169) (57 185) (16 144),(113 241) (97 225) (37 165) (117 245) (84 212) (4 132) (47 175) (31 159),(46 174) (30 158) (105 233) (62 190) (8 136) (24 152) (50 178) (18 146),(22 150) (5 133) (64 192) (101 229) (107 235) (106 234) (75 203) (127 255),(13 141) (45 173) (104 232) (121 249) (25 153) (51 179) (55 183) (38 166),(65 193) (6 134) (118 246) (68 196) (100 228) (15 143) (74 202) (14 142),(90 218) (73 201) (61 189) (40 168) (34 162) (2 130) (114 242) (23 151),(119 247) (102 230) (86 214) (69 197) (111 239) (110 238) (126 254) (109 237),(125 253) (72 200) (29 157) (120 248) (19 147) (115 243) (98 226) (82 210),(66 194) (103 231) (85 213) (70 198) (79 207) (95 223) (77 205) (94 222),(12 140) (44 172) (28 156) (35 163) (3 131) (39 167) (99 227) (7 135),(83 211) (71 199) (87 215) (78 206) (76 204) (93 221) (108 236) (92 220),(91 219) (124 252) (60 188) (80 208) (81 209) (67 195) (89 217) (88 216)]494.3. AhmeeicTl eefhlgf Tad difchffibaTable 4.6: Proposed xr and or for 512-QAM.xr[397 493 479 462 477 456 335 269 162 165 455 502 468 453 439 503,411 334 509 494 461 472 488 285 374 342 372 402 485 501 434 385,172 440 408 510 495 504 491 473 262 451 498 466 452 406 386 326,475 174 409 511 492 507 271 366 164 481 449 486 471 390 340 260,463 415 399 446 398 447 410 430 420 422 464 438 388 405 389 484,414 412 396 474 458 442 428 394 423 421 448 384 487 436 437 404,413 445 459 431 270 395 429 427 416 418 480 432 401 400 496 407,444 506 443 393 489 286 424 426 417 483 261 358 433 403 465 391,392 350 284 457 380 382 300 425 419 263 327 309 277 499 482 497,490 505 268 173 318 301 282 264 259 257 256 258 308 356 435 387,287 348 302 170 319 365 280 266 295 359 294 167 279 467 166 276,364 317 316 281 299 330 314 298 275 288 304 273 160 292 310 324,441 332 267 346 315 313 312 296 291 289 320 305 306 272 293 325,303 383 265 362 378 10 361 297 355 290 352 370 336 353 311 274,283 171 328 360 169 329 381 376 371 307 354 321 3 163 369 368,367 379 347 8 331 344 345 377 35 339 323 375 337 0 322 373,168 363 58 11 42 333 40 41 19 39 32 338 17 48 161 2,74 26 27 45 43 24 56 57 50 34 33 38 96 16 49 1,126 44 123 59 122 9 25 120 115 99 113 103 97 102 53 240,62 90 46 106 107 104 105 121 114 51 67 65 112 64 71 227,201 91 47 124 75 72 125 73 83 98 18 81 36 80 52 7,12 28 109 29 61 60 127 89 82 66 119 37 23 20 211 21,219 251 31 92 249 110 111 88 117 118 55 100 243 6 241 209,203 153 185 94 76 108 77 93 101 87 69 54 70 68 179 147,136 253 235 349 217 13 248 95 85 116 22 225 193 226 4 129,222 351 15 184 78 232 200 79 86 242 84 146 195 178 215 181,191 220 236 255 152 239 252 216 194 210 245 244 246 198 148 130,143 159 223 204 206 238 254 207 213 197 229 212 196 214 149 133,234 137 218 139 508 157 237 14 177 208 183 128 470 199 230 5,187 205 189 478 476 154 186 30 231 341 192 151 150 180 228 145,155 156 140 142 141 202 250 63 224 176 144 454 500 134 131 247,221 188 190 158 460 138 175 233 357 278 450 135 469 182 132 343]or[48 37 96 33 306 64 5 308 186 60 90 441 41 123 45 233,225 112 113 339 161 129 224 240 250 26 10 42 40 107 61 127,179 103 39 32 17 241 341 69 126 74 58 347 169 59 11 235,343 16 0 1 65 81 80 21 63 124 122 43 345 171 106 44,35 49 51 99 98 34 114 50 89 25 73 9 105 120 121 57,163 3 115 53 83 66 2 18 72 185 88 249 91 75 104 56,97 67 117 119 55 116 82 54 93 13 77 232 111 8 125 109,19 227 131 102 226 38 118 52 153 248 95 29 137 24 108 27,101 100 36 195 178 87 242 6 22 217 79 201 184 76 47 349,243 68 85 23 84 70 194 86 216 152 200 136 15 219 187 251,7 211 245 4 247 20 130 146 221 205 155 253 189 92 139 110,193 147 229 183 230 215 210 246 157 223 236 239 255 237 12 203,71 231 181 199 244 198 150 182 204 207 252 141 78 28 31 351,176 144 135 197 213 151 134 214 159 220 238 222 254 94 202 234,145 128 192 228 149 196 212 148 206 143 140 191 188 218 14 46,310 209 342 208 133 166 132 180 156 158 142 174 154 30 138 62,374 450 278 438 471 406 164 470 478 479 476 477 190 172 397 472,262 402 466 502 390 454 468 404 462 415 412 460 413 461 408 456,434 386 326 340 407 500 469 452 414 350 463 399 396 492 175 392,279 439 276 422 486 455 405 464 398 510 494 511 508 495 411 509,358 167 503 436 391 453 388 484 446 430 474 444 428 445 348 332,387 260 418 501 437 400 448 389 286 410 447 284 493 287 475 443,165 467 324 485 487 423 384 420 394 458 431 334 395 268 424 303,419 356 435 403 277 496 421 401 270 382 490 429 459 366 507 267,357 481 272 327 449 432 465 318 426 442 506 380 170 491 383 425,293 295 160 336 263 385 480 416 316 302 319 300 427 301 296 365,353 288 256 359 325 497 433 261 282 266 346 317 330 315 312 297,305 289 352 321 257 273 320 417 298 314 378 362 299 376 379 377,451 292 373 371 354 309 375 372 457 173 367 281 265 344 440 271,499 368 307 369 322 258 311 482 285 488 393 328 329 331 280 364,483 304 291 323 290 338 274 294 335 504 168 505 363 360 264 489,275 259 337 355 370 177 162 498 409 473 269 333 361 313 381 283]504.3. AhmeeicTl eefhlgf Tad difchffibaTable 4.7: Proposed xl and ol for 512-QAM.xl[(172 428) (168 424) (162 418) (165 421) (175 431) (173 429) (164 420) (170 426),(160 416) (167 423) (161 417) (42 298) (43 299) (169 425) (41 297) (57 313),(35 291) (33 289) (32 288) (1 257) (64 320) (177 433) (10 266) (58 314),(171 427) (122 378) (59 315) (123 379) (40 296) (121 377) (51 307) (49 305),(163 419) (96 352) (65 321) (17 273) (241 497) (80 336) (90 346) (74 330),(106 362) (45 301) (107 363) (56 312) (105 361) (120 376) (34 290 99 355),(97 353) (113 369) (37 293) (39 295) (103 359) (69 325) (124 380) (235 491),(61 317) (11 267) (109 365) (75 331) (104 360) (9 265) (50 306) (98 354),(115 371) (19 275) (48 304) (112 368) (81 337) (7 263) (63 319) (44 300),(127 383) (233 489) (125 381) (91 347) (73 329) (25 281) (89 345) (114 370),(3 259) (67 323) (225 481) (0 256) (16 272) (100 356) (139 395) (110 366),(47 303) (27 283) (108 364) (8 264) (88 344) (72 328) (66 322) (83 339),(53 309) (227 483) (101 357) (243 499) (179 435) (71 327) (12 268) (92 348),(251 507) (24 280) (111 367) (249 505) (185 441) (2 258) (117 373) (116 372),(119 375) (195 451) (131 387) (68 324) (211 467) (189 445) (219 475) (76 332),(187 443) (137 393) (232 488) (77 333) (93 349) (18 274) (82 338) (55 311),(226 482) (102 358) (85 341) (4 260) (229 485) (255 511) (253 509) (15 271),(201 457) (29 285) (95 351) (13 269) (54 310) (242 498) (118 374) (38 294),(87 343) (23 279) (36 292) (247 503) (230 486) (252 508) (239 495) (155 411),(136 392) (184 440) (79 335) (248 504) (153 409) (52 308) (6 262) (178 434),(70 326) (84 340) (20 276) (244 500) (198 454) (238 494) (207 463) (141 397),(205 461) (152 408) (200 456) (216 472) (217 473) (86 342) (22 278) (194 450),(130 386) (210 466) (215 471) (150 406) (212 468) (156 412) (159 415) (206 462),(220 476) (236 492) (204 460) (223 479) (221 477) (157 413) (146 402) (246 502),(182 438) (134 390) (214 470) (148 404) (180 436) (203 459) (94 350) (154 410),(142 398) (174 430) (14 270) (46 302) (234 490) (21 277) (176 432) (144 400),(208 464) (133 389) (149 405) (147 403) (193 449) (31 287) (78 334) (188 444),(158 414) (218 474) (138 394) (250 506) (126 382) (129 385) (128 384) (228 484),(166 422) (197 453) (231 487) (28 284) (254 510) (222 478) (140 396) (190 446),(202 458) (186 442) (26 282) (5 261) (240 496) (209 465) (135 391) (196 452),(213 469) (199 455) (245 501) (237 493) (191 447) (143 399) (30 286) (62 318),(60 316) (224 480) (145 401) (192 448) (132 388) (151 407) (183 439) (181 437)]ol[(225 481) (19 275) (163 419) (33 289) (64 320) (177 433) (60 316) (58 314),(42 298) (57 313) (40 296) (56 312) (125 381) (179 435) (48 304) (113 369),(49 305) (161 417) (17 273) (241 497) (224 480) (126 382) (10 266) (90 346),(43 299) (41 297) (123 379) (45 301) (103 359) (112 368) (37 293) (97 353),(1 257) (65 321) (80 336) (129 385) (234 490) (26 282) (74 330) (171 427),(169 425) (107 363) (11 267) (127 383) (69 325) (0 256) (39 295) (32 288),(96 352) (81 337) (63 319) (124 380) (106 362) (122 378) (59 315) (61 317),(235 491) (44 300) (35 291) (34 290) (51 307) (98 354) (114 370) (66 322),(89 345) (50 306) (185 441) (72 328) (25 281) (73 329) (9 265) (120 376),(121 377) (99 355) (3 259) (227 483) (53 309) (83 339) (55 311) (82 338),(18 274) (2 258) (249 505) (88 344) (111 367) (8 264) (91 347) (104 360),(105 361) (115 371) (67 323) (117 373) (116 372) (38 294) (242 498) (118 374),(13 269) (93 349) (77 333) (232 488) (137 393) (24 280) (27 283) (109 365),(75 331) (101 357) (119 375) (226 482) (102 358) (178 434) (52 308) (153 409),(54 310) (248 504) (29 285) (95 351) (184 440) (108 364) (47 303) (233 489),(243 499) (131 387) (195 451) (23 279) (70 326) (87 343) (6 262) (194 450),(217 473) (79 335) (201 457) (15 271) (76 332) (187 443) (16 272) (36 292),(85 341) (211 467) (84 340) (130 386) (146 402) (86 342) (22 278) (200 456),(216 472) (136 392) (219 475) (12 268) (92 348) (251 507) (100 356) (71 327),(68 324) (4 260) (247 503) (20 276) (210 466) (157 413) (221 477) (152 408),(155 411) (253 509) (189 445) (237 493) (139 395) (110 366) (7 263) (193 449),(229 485) (245 501) (244 500) (215 471) (134 390) (246 502) (205 461) (236 492),(141 397) (239 495) (255 511) (28 284) (31 287) (203 459) (5 261) (21 277),(181 437) (199 455) (230 486) (198 454) (150 406) (182 438) (223 479) (220 476),(207 463) (252 508) (254 510) (78 334) (94 350) (250 506) (240 496) (147 403),(231 487) (197 453) (183 439) (151 407) (212 468) (214 470) (204 460) (206 462),(238 494) (191 447) (188 444) (218 474) (14 270) (46 302) (176 432) (128 384),(208 464) (149 405) (213 469) (196 452) (132 388) (148 404) (159 415) (156 412),(222 478) (142 398) (190 446) (154 410) (202 458) (186 442) (209 465) (144 400),(228 484) (135 391) (133 389) (166 422) (164 420) (180 436) (143 399) (140 396),(158 414) (174 430) (30 286) (138 394) (62 318) (145 401) (192 448) (167 423),(172 428) (175 431) (165 421) (173 429) (170 426) (160 416) (162 418) (168 424)]514.3. AhmeeicTl eefhlgf Tad difchffibaTable 4.8: Proposed xr for 1024-QAM.xr[674 195 226 193 197 229 67 225 129 163 161 165 97 69 203 65,182 54 252 222 220 180 118 176 86 151 148 150 247 246 214 212,194 199 227 231 131 133 200 99 234 202 37 101 1 33 235 201,158 156 254 22 52 470 23 20 116 183 144 146 215 242 244 213,192 228 224 162 66 64 3 5 35 451 483 139 485 481 449 233,406 190 404 407 502 223 48 50 468 112 178 87 240 245 210 208,196 230 130 167 205 237 450 487 453 389 385 77 72 171 137 169,438 188 94 436 400 159 191 500 16 255 114 179 181 84 243 694,198 128 98 68 232 39 448 141 138 419 355 421 74 417 75 73,126 30 124 432 434 402 92 184 496 18 248 119 82 147 149 662,134 135 103 96 239 0 136 173 104 109 13 106 323 353 107 105,62 278 28 342 374 127 95 186 471 498 503 55 221 80 145 209,132 160 71 2 455 482 170 423 359 322 45 10 11 325 357 321,60 310 304 476 31 368 370 439 403 435 152 464 253 19 115 211,166 164 7 32 480 387 168 324 320 8 456 257 461 491 9 41,412 510 306 308 274 276 372 343 336 340 189 157 51 250 177 241,70 100 4 34 452 418 327 356 495 490 42 261 493 293 43 457,446 508 478 272 279 307 63 56 375 371 120 154 466 218 216 83,102 204 207 484 386 76 354 260 40 291 488 458 289 459 393 489,414 444 447 408 479 511 275 472 58 338 122 405 499 469 21 117,238 36 143 175 384 79 111 47 259 295 397 429 427 363 361 425,284 286 415 440 442 410 311 504 506 24 125 401 155 467 53 85,6 486 454 420 12 15 352 292 258 392 426 394 395 301 331 329,318 380 314 348 445 413 509 305 61 26 433 437 187 501 49 113,236 140 391 416 78 326 263 256 463 431 424 365 269 328 265 297,316 280 312 376 443 411 309 477 474 273 29 93 90 88 251 17,38 206 388 108 358 294 460 290 288 399 268 367 271 360 299 267,287 350 383 315 282 378 441 475 277 507 27 339 123 497 219 81,673 142 172 422 14 44 494 492 428 270 364 332 303 264 266 330,382 319 317 283 313 379 346 409 59 373 341 369 185 153 249 217,581 174 390 110 46 262 396 398 430 462 302 300 333 362 298 296,351 285 381 281 347 344 377 473 505 25 337 121 91 465 702 566,609 745 961 681 585 617 833 553 521 969 335 873 366 777 334 809,829 831 349 828 830 345 798 924 57 572 574 542 89 638 918 534,577 713 993 649 837 869 865 555 805 1001 939 843 937 875 813 811,825 827 792 826 796 892 862 956 958 926 822 790 636 700 670 764,677 747 997 619 587 835 523 971 1003 905 845 841 781 815 779 810,793 797 824 799 794 894 952 927 816 818 1022 540 944 950 668 766,579 513 545 683 933 525 522 773 1005 941 907 877 842 776 778 808,893 891 795 895 888 860 959 954 1020 820 990 543 606 916 1014 732,741 715 651 586 929 621 557 968 973 769 801 840 879 783 812 780,953 859 856 863 957 890 920 784 788 786 791 854 912 948 564 734,611 613 549 584 897 589 970 554 807 1000 909 906 872 874 876 814,921 889 858 955 923 922 823 991 575 988 882 886 919 671 560 688,641 547 963 995 901 871 867 520 1007 803 943 938 904 936 847 782,861 1021 821 989 925 984 1016 819 787 880 639 914 946 735 982 630,705 714 965 650 931 618 834 1002 552 771 975 844 768 911 940 878,857 987 789 817 573 986 1018 570 568 884 604 1008 703 980 532 656,737 578 517 999 685 616 836 832 866 559 772 770 775 1004 908 846,910 1017 1019 785 536 1023 887 852 855 915 607 983 1012 562 535 660,675 746 515 962 749 653 935 899 839 868 804 972 802 800 974 942,985 569 539 538 541 883 850 848 947 951 696 1010 767 624 628 663,709 643 712 717 744 682 960 996 623 591 864 527 838 1006 556 774,885 537 571 851 945 637 634 632 669 664 698 528 567 631 598 690,739 707 645 551 514 994 648 680 687 898 588 870 524 806 526 558,849 853 881 605 913 600 917 976 701 733 765 530 596 599 658 692,679 704 610 576 615 751 967 512 930 964 655 896 903 590 620 633,601 603 635 602 949 667 666 978 760 1015 531 762 626 752 727 695,706 642 740 580 583 546 612 516 992 719 966 932 928 900 902 622,977 697 665 1009 699 979 981 728 730 563 592 659 720 722 754 759,738 736 743 676 708 608 644 519 544 548 716 998 934 652 654 684,761 729 1013 731 529 1011 565 533 627 594 657 661 691 724 756 726,678 672 742 710 711 646 640 647 582 614 550 518 718 748 686 750,593 625 763 561 597 629 595 721 689 753 723 725 755 757 693 758]524.3. AhmeeicTl eefhlgf Tad difchffibaTable 4.9: Proposed or for 1024-QAM.or[829 825 827 797 793 893 891 889 856 859 921 861 987 857 1017 985,910 846 942 908 940 878 782 876 814 783 812 778 808 810 811 809,831 792 824 795 953 957 955 858 1021 821 789 1019 571 569 537 849,774 556 1006 974 802 911 1004 844 847 936 874 780 776 815 779 334,349 826 799 895 863 923 817 573 989 541 785 539 885 881 853 633,558 526 806 524 972 800 775 772 768 904 938 872 879 781 875 813,828 796 794 890 925 1016 984 986 538 536 945 851 605 635 603 697,601 590 620 870 838 527 864 804 770 975 771 943 840 842 843 777,345 888 860 922 823 1023 570 887 883 637 913 600 602 1009 665 902,622 900 928 903 896 588 623 866 868 559 554 803 906 941 939 366,830 894 920 991 819 1018 848 850 632 917 949 667 699 731 1013 977,654 684 934 932 655 898 591 899 839 832 552 807 1000 909 907 937,892 959 954 575 787 568 855 634 947 976 666 979 1011 561 729 761,686 652 966 719 964 687 930 680 935 836 1002 1007 970 801 845 877,798 784 788 880 884 882 852 701 669 978 1015 728 981 565 529 593,718 998 548 544 992 996 648 960 653 931 616 834 520 968 769 841,927 1020 791 543 639 607 915 951 664 765 733 730 533 595 763 625,748 518 716 516 751 967 512 682 685 901 618 871 867 557 1005 873,952 820 786 604 914 696 1008 1010 980 760 531 762 563 627 629 597,750 519 612 615 546 994 514 962 749 650 897 525 522 773 973 905,862 818 988 886 912 703 698 528 530 567 596 659 592 594 657 721,550 614 582 608 583 551 744 717 999 965 584 589 621 835 971 805,816 1022 990 854 946 671 983 1012 767 631 626 599 661 720 753 689,644 647 640 676 580 610 712 515 517 549 995 586 929 865 523 1003,924 822 540 636 919 948 735 560 532 624 752 658 691 722 725 723,646 710 711 708 645 576 746 714 547 513 963 651 933 837 555 1001,956 572 542 944 916 564 982 732 562 628 598 663 727 724 755 693,742 743 740 642 707 578 705 611 613 997 545 683 587 619 521 969,958 926 638 606 950 668 1014 764 734 535 656 660 754 756 759 757,736 672 679 704 709 643 641 579 747 715 993 961 649 585 869 553,57 790 89 700 918 702 670 566 766 534 630 690 692 695 758 726,678 738 706 739 675 737 677 673 581 609 713 745 681 617 833 398,25 574 121 91 465 249 219 217 81 113 688 241 662 694 208 213,194 674 196 230 198 166 741 6 38 577 142 390 172 110 262 430,505 337 369 185 153 497 49 251 17 85 117 83 209 210 242 212,195 192 199 228 134 132 70 102 236 238 140 174 388 14 46 396,473 341 93 123 90 467 501 53 218 115 177 145 211 245 247 214,227 226 231 224 128 135 164 7 204 486 206 454 422 108 44 428,373 27 339 437 187 88 469 21 19 216 80 149 243 240 215 244,193 197 131 162 130 98 160 4 207 36 143 391 420 78 294 492,59 507 433 371 401 155 499 51 250 82 181 147 146 144 148 246,225 229 133 64 167 68 71 100 32 452 175 384 416 358 494 462,409 277 29 125 122 405 466 253 248 221 179 84 183 178 176 151,129 67 99 163 66 103 96 239 34 484 386 418 12 326 460 288,475 305 273 26 120 154 157 464 55 119 114 87 116 118 86 150,182 161 165 200 205 232 2 455 0 480 76 111 15 292 290 270,377 477 474 24 338 435 189 498 503 16 255 468 112 220 22 180,65 97 234 202 5 237 39 448 482 168 79 354 352 263 463 302,346 309 61 506 375 340 186 152 471 18 50 23 20 52 222 54,69 1 101 37 35 3 487 136 170 423 356 260 495 256 399 335,379 441 509 504 58 336 439 403 184 496 500 223 48 156 254 252,235 203 33 483 453 450 138 141 387 327 320 40 259 258 431 364,313 378 411 472 307 56 372 127 92 402 191 159 470 404 158 406,201 449 485 451 385 389 173 355 104 324 47 291 295 392 332 300,347 344 443 511 311 343 276 95 370 434 400 407 436 502 188 190,233 481 139 171 421 109 419 106 359 322 490 488 426 367 268 303,281 283 445 413 442 410 275 63 274 368 31 28 94 432 438 73,169 137 75 72 417 77 13 10 8 456 42 397 424 365 271 333,381 315 282 376 415 479 440 408 279 476 374 342 310 124 30 126,105 107 323 353 74 45 257 261 461 289 429 394 269 360 264 362,285 317 383 312 314 348 380 447 272 508 308 304 306 60 278 62,321 357 9 325 11 491 293 493 458 427 395 328 301 299 266 298,351 382 319 287 350 280 318 316 284 286 478 444 510 414 446 412,457 41 489 43 459 425 393 361 363 331 265 329 297 267 330 296]534.3. AhmeeicTl eefhlgf Tad difchffibaTable 4.10: Proposed xl for 1024-QAM.xl[(195 707)) ((226 738) (193 705) (197 709) (229 741) (67 579) (225 737) (129 641),(163 675) (161 673) (165 677) (97 609) (69 581) (203 715) (65 577) (182 694),(54 566) (252 764) (222 734) (220 732) (180 692) (118 630) (176 688) (86 598),(151 663) (148 660) (150 662) (247 759) (246 758) (214 726) (212 724) (194 706),(199 711) (227 739) (231 743) (131 643) (133 645) (200 712) (99 611) (234 746),(202 714) (37 549) (101 613) (1 513) (33 545) (235 747) (201 713) (158 670),(156 668) (254 766) (22 534) (52 564) (470 982) (23 535) (20 532) (116 628),(183 695) (144 656) (146 658) (215 727) (242 754) (244 756) (213 725) (192 704),(228 740) (224 736) (162 674) (66 578) (64 576) (3 515) (5 517) (35 547),(451 963) (483 995) (139 651) (485 997) (481 993) (449 961) (233 745) (406 918),(190 702) (404 916) (407 919) (502 1014) (223 735) (48 560) (50 562) (468 980),(112 624) (178 690) (87 599) (240 752) (245 757) (210 722) (208 720) (196 708),(230 742) (130 642) (167 679) (205 717) (237 749) (450 962) (487 999) (453 965),(389 901) (385 897) (77 589) (72 584) (171 683) (137 649) (169 681) (438 950),(188 700) (94 606) (436 948) (400 912) (159 671) (191 703) (500 1012) (16 528),(255 767) (114 626) (179 691) (181 693) (84 596) (243 755) (198 710) (128 640),(98 610) (68 580) (232 744) (39 551) (448 960) (141 653) (138 650) (419 931),(355 867) (421 933) (74 586) (417 929) (75 587) (73 585) (126 638) (30 542),(124 636) (432 944) (434 946) (402 914) (92 604) (184 696) (496 1008) (18 530),(248 760) (119 631) (82 594) (147 659) (149 661) (134 646) (135 647) (103 615),(96 608) (239 751) (0 512) (136 648) (173 685) (104 616) (109 621) (13 525),(106 618) (323 835) (353 865) (107 619) (105 617) (62 574) (278 790) (28 540),(342 854) (374 886) (127 639) (95 607) (186 698) (471 983) (498 1010) (503 1015),(55 567) (221 733) (80 592) (145 657) (209 721) (132 644) (160 672) (71 583),(2 514) (455 967) (482 994) (170 682) (423 935) (359 871) (322 834) (45 557),(10 522) (11 523) (325 837) (357 869) (321 833) (60 572) (310 822) (304 816),(476 988) (31 543) (368 880) (370 882) (439 951) (403 915) (435 947) (152 664),(464 976) (253 765) (19 531) (115 627) (211 723) (166 678) (164 676) (7 519),(32 544) (480 992) (387 899) (168 680) (324 836) (320 832) (8 520) (456 968),(257 769) (461 973) (491 1003) (9 521) (41 553) (412 924) (510 1022) (306 818),(308 820) (274 786) (276 788) (372 884) (343 855) (336 848) (340 852) (189 701),(157 669) (51 563) (250 762) (177 689) (241 753) (70 582) (100 612) (4 516),(34 546) (452 964) (418 930) (327 839) (356 868) (495 1007) (490 1002) (42 554),(261 773) (493 1005) (293 805) (43 555) (457 969) (446 958) (508 1020) (478 990),(272 784) (279 791) (307 819) (63 575) (56 568) (375 887) (371 883) (120 632),(154 666) (466 978) (218 730) (216 728) (83 595) (102 614) (204 716) (207 719),(484 996) (386 898) (76 588) (354 866) (260 772) (40 552) (291 803) (488 1000),(458 970) (289 801) (459 971) (393 905) (489 1001) (414 926) (444 956) (447 959),(408 920) (479 991) (511 1023) (275 787) (472 984) (58 570) (338 850) (122 634),(405 917) (499 1011) (469 981) (21 533) (117 629) (238 750) (36 548) (143 655),(175 687) (384 896) (79 591) (111 623) (47 559) (259 771) (295 807) (397 909),(429 941) (427 939) (363 875) (361 873) (425 937) (284 796) (286 798) (415 927),(440 952) (442 954) (410 922) (311 823) (504 1016) (506 1018) (24 536) (125 637),(401 913) (155 667) (467 979) (53 565) (85 597) (6 518) (486 998) (454 966),(420 932) (12 524) (15 527) (352 864) (292 804) (258 770) (392 904) (426 938),(394 906) (395 907) (301 813) (331 843) (329 841) (318 830) (380 892) (314 826),(348 860) (445 957) (413 925) (509 1021) (305 817) (61 573) (26 538) (433 945),(437 949) (187 699) (501 1013) (49 561) (113 625) (236 748) (140 652) (391 903),(416 928) (78 590) (326 838) (263 775) (256 768) (463 975) (431 943) (424 936),(365 877) (269 781) (328 840) (265 777) (297 809) (316 828) (280 792) (312 824),(376 888) (443 955) (411 923) (309 821) (477 989) (474 986) (273 785) (29 541),(93 605) (90 602) (88 600) (251 763) (17 529) (38 550) (206 718) (388 900),(108 620) (358 870) (294 806) (460 972) (290 802) (288 800) (399 911) (268 780),(367 879) (271 783) (360 872) (299 811) (267 779) (287 799) (350 862) (383 895),(315 827) (282 794) (378 890) (441 953) (475 987) (277 789) (507 1019) (27 539),(339 851) (123 635) (497 1009) (219 731) (81 593) (142 654) (172 684) (422 934),(14 526) (44 556) (494 1006) (492 1004) (428 940) (270 782) (364 876) (332 844),(303 815) (264 776) (266 778) (330 842) (382 894) (319 831) (317 829) (283 795),(313 825) (379 891) (346 858) (409 921) (59 571) (373 885) (341 853) (369 881),(185 697) (153 665) (249 761) (217 729) (174 686) (390 902) (110 622) (46 558),(262 774) (396 908) (398 910) (430 942) (462 974) (302 814) (300 812) (333 845),(362 874) (298 810) (296 808) (351 863) (285 797) (381 893) (281 793) (347 859),(344 856) (377 889) (473 985) (505 1017) (25 537) (337 849) (121 633) (91 603),(465 977) (335 847) (366 878) (334 846) (349 861) (345 857) (57 569) (89 601)]544.3. AhmeeicTl eefhlgf Tad difchffibaTable 4.11: Proposed ol for 1024-QAM.ol[(349 861) (334 846) (366 878) (345 857) (473 985) (398 910) (335 847) (505 1017),(57 569) (121 633) (91 603) (89 601) (465 977) (249 761) (217 729) (81 593),(113 625) (209 721) (208 720) (212 724) (195 707) (196 708) (198 710) (134 646),(102 614) (38 550) (142 654) (390 902) (422 934) (46 558) (396 908) (430 942),(377 889) (25 537) (337 849) (185 697) (153 665) (497 1009) (219 731) (49 561),(17 529) (85 597) (83 595) (211 723) (243 755) (210 722) (213 725) (214 726),(194 706) (199 711) (230 742) (135 647) (132 644) (166 678) (70 582) (6 518),(236 748) (140 652) (174 686) (110 622) (14 526) (262 774) (294 806) (302 814),(409 921) (341 853) (369 881) (123 635) (88 600) (501 1013) (53 565) (251 763),(117 629) (177 689) (145 657) (241 753) (245 757) (247 759) (242 754) (246 758),(226 738) (192 704) (228 740) (128 640) (160 672) (71 583) (164 676) (204 716),(238 750) (206 718) (388 900) (172 684) (108 620) (44 556) (494 1006) (428 940),(346 858) (373 885) (339 851) (93 605) (187 699) (499 1011) (467 979) (218 730),(216 728) (115 627) (181 693) (149 661) (240 752) (215 727) (244 756) (193 705),(227 739) (231 743) (224 736) (130 642) (162 674) (68 580) (100 612) (7 519),(36 548) (486 998) (454 966) (420 932) (78 590) (460 972) (492 1004) (462 974),(475 987) (59 571) (27 539) (433 945) (90 602) (155 667) (51 563) (21 533),(19 531) (82 594) (80 592) (84 596) (147 659) (146 658) (148 660) (150 662),(225 737) (197 709) (229 741) (131 643) (167 679) (103 615) (4 516) (32 544),(207 719) (484 996) (391 903) (416 928) (12 524) (326 838) (290 802) (270 782),(347 859) (507 1019) (29 541) (437 949) (401 913) (469 981) (466 978) (253 765),(250 762) (221 733) (179 691) (114 626) (144 656) (151 663) (86 598) (118 630),(67 579) (129 641) (133 645) (66 578) (98 610) (96 608) (239 751) (34 546),(455 967) (143 655) (175 687) (384 896) (358 870) (292 804) (288 800) (268 780),(379 891) (277 789) (273 785) (125 637) (122 634) (405 917) (157 669) (464 976),(55 567) (119 631) (112 624) (87 599) (183 695) (176 688) (180 692) (182 694),(161 673) (163 675) (99 611) (64 576) (3 515) (205 717) (232 744) (2 514),(452 964) (386 898) (418 930) (76 588) (15 527) (263 775) (258 770) (399 911),(344 856) (309 821) (61 573) (26 538) (371 883) (120 632) (154 666) (503 1015),(255 767) (248 760) (16 528) (468 980) (178 690) (116 628) (220 732) (22 534),(65 577) (165 677) (200 712) (234 746) (5 517) (237 749) (39 551) (0 512),(480 992) (168 680) (327 839) (79 591) (354 866) (256 768) (463 975) (364 876),(411 923) (441 953) (305 817) (506 1018) (24 536) (338 850) (189 701) (152 664),(496 1008) (498 1010) (18 530) (23 535) (20 532) (52 564) (222 734) (203 715),(97 609) (69 581) (101 613) (202 714) (487 999) (450 962) (448 960) (482 994),(170 682) (423 935) (111 623) (260 772) (352 864) (47 559) (431 943) (300 812),(313 825) (443 955) (477 989) (474 986) (58 570) (375 887) (340 852) (435 947),(471 983) (184 696) (50 562) (223 735) (502 1014) (254 766) (252 764) (54 566),(201 713) (1 513) (37 549) (483 995) (35 547) (138 650) (141 653) (136 648),(387 899) (324 836) (356 868) (40 552) (259 771) (392 904) (424 936) (332 844),(283 795) (376 888) (410 922) (504 1016) (472 984) (56 568) (336 848) (403 915),(439 951) (186 698) (500 1012) (191 703) (48 560) (470 982) (158 670) (156 668),(235 747) (33 545) (485 997) (451 963) (453 965) (389 901) (173 685) (104 616),(359 871) (320 832) (495 1007) (490 1002) (291 803) (426 938) (367 879) (271 783),(381 893) (378 890) (413 925) (509 1021) (275 787) (343 855) (372 884) (370 882),(127 639) (92 604) (402 914) (159 671) (407 919) (400 912) (404 916) (233 745),(481 993) (406 918) (190 702) (139 651) (385 897) (419 931) (109 621) (355 867),(322 834) (8 520) (456 968) (295 807) (488 1000) (365 877) (269 781) (333 845),(317 829) (281 793) (282 794) (445 957) (311 823) (307 819) (63 575) (276 788),(95 607) (368 880) (432 944) (434 946) (94 606) (124 636) (436 948) (449 961),(137 649) (438 950) (171 683) (417 929) (72 584) (421 933) (106 618) (13 525),(45 557) (261 773) (42 554) (397 909) (394 906) (301 813) (303 815) (264 776),(383 895) (315 827) (314 826) (348 860) (442 954) (408 920) (511 1023) (274 786),(308 820) (31 543) (374 886) (342 854) (28 540) (30 542) (73 585) (188 700),(169 681) (75 587) (107 619) (74 586) (323 835) (77 589) (10 522) (257 769),(493 1005) (461 973) (429 941) (427 939) (395 907) (265 777) (360 872) (298 810),(285 797) (319 831) (280 792) (312 824) (286 798) (440 952) (479 991) (279 791),(272 784) (304 816) (476 988) (306 818) (310 822) (60 572) (105 617) (62 574),(278 790) (126 638) (353 865) (357 869) (11 523) (325 837) (491 1003) (293 805),(289 801) (458 970) (393 905) (331 843) (328 840) (299 811) (362 874) (296 808),(351 863) (382 894) (287 799) (350 862) (380 892) (318 830) (284 796) (415 927),(447 959) (478 990) (444 956) (508 1020) (457 969) (510 1022) (41 553) (9 521),(321 833) (412 924) (489 1001) (446 958) (414 926) (43 555) (425 937) (361 873),(459 971) (363 875) (329 841) (297 809) (316 828) (267 779) (266 778) (330 842)]554.3. AhmeeicTl eefhlgf Tad difchffibaF.E.D bwrxormsnuw uompsrisonThe proposed mappings are compared to the MD mappings that are optimized for Rayleighfading channels using the BSA. Typically, the BSA is the best known computer searchalgorithm to find good mappings for BICM-ID. However, it becomes intractable to obtainMD mappings with a higher alphabet size, e.g., 6-D 64-QAM. In our simulations, weconsider a rate-1=2 convolutional code with the generator polynomial of (13P 15)8. Thelength of the used interleaver is about 10000 bits. All BER curves are presented with seveniterations, and all gains reported in this section are measured at a BER of 10â6.Table 4.12 indicates the values of Î¦Ë(P ) for different 2N -D (N = 2P 3) 2m-QAMmappings. Note that in the case of 6-D (N = 3) mappings using 2m-QAM (m S 4), theBSA result could not be obtained due to the computational time constraints. However,our proposed method yields efficient 2N -D mapping of 2m-QAM for any value of N andfor m (m â¤ 10). Table 4.12 shows that for all considered values of N and m, the resultingmappings with our proposed method offer greater values of Î¦Ë(P ) in comparison withthose of the BSA mappings. Moreover, for a given code and modulation, the mapping witha greater value of Î¦Ë(P ) achieves a lower error floor [34]. As a consequence, the proposedmappings are expected to offer lower error-floors than those of the BSA mappings. Thevalue of Î¦Ë(P ) for the proposed 4-D mapping using higher order 2m-QAM (m = 7P · · · P 10)is listed in Table 4.13.Table 4.12: Comparison of the harmonic mean of MSED, Î¦Ë(P ), for different mappings.Mapping N = 2 N = 3BSA MD 16-QAM 2.5814 2.8047Proposed MD 16-QAM 3.1622 3.3105BSA MD 32-QAM 2.8574 -Proposed MD 32-QAM 3.1677 3.3089BSA MD 64-QAM 2.8047 -Proposed MD 64-QAM 3.1683 3.2870Table 4.13: Î¦Ë(P ) for proposed 4-D mapping using higher order QAMs.Basic modulation Î¦Ë(P )128-QAM 3.2272256-QAM 3.2289512-QAM 3.28331024-QAM 3.2995564.3. AhmeeicTl eefhlgf Tad difchffibaEb/N0 (in dB)0 1 2 3 4 5 6 7 8 9 10BER10-610-510-410-310-210-1100BSA 4D 16-QAMProposed 4D 16-QAMBSA 6D 16-QAMProposed 6D 16-QAMCM capacityEb/N0 =3.98 dBFigure 4.3: BER performance of BICM-ID with 4-D and 6-D 16-QAM over Rayleigh fadingchannels.Fig. 4.3 plots the BER performance of BICM-ID employing different 2N -D mappingsof 16-QAM for N = 2P 3. From Fig. 4.3, it can be observed that our 4-D and 6-D 16-QAM mappings respectively achieve 0:85 dB and 1:5 dB gain over their BSA counterparts.The BER performance of BICM-ID with 4-D 32-QAM and 4-D 64-QAM is plotted in Fig.4.4. It can be seen from this figure that compared to the BSA mappings, the proposedmappings significantly improve the BER performance of BICM-ID using 4-D 32-QAM and4-D 64-QAM. In particular, our resulting mapping for 4-D 32-QAM offers a gain of 1:4dBwhile for 4-D 64-QAM it offers a gain of 2:6 dB compared with the BSA mappings.In Fig. 4.5, we compare the error bounds for BICM-ID presented in [15] using MD16-QAM mappings. This figure shows that our proposed mappings provide a gain of 0:9dB and 0:73 dB respectively over the BSA results for 4-D and 6-D 16-QAM. Fig. 4.6 plotsthe error bounds for BICM-ID with 4-D 32-QAM and 4-D 64-QAM. This figure shows thatin comparison with the BSA mappings, our proposed mappings improve the error boundrespectively by 0:5 dB and 0:53 dB.It is important to note that the BSA could not be applied to find a mapping for 6-D 32-QAM due to the computational time constraint. However, our proposed methodeasily finds suitable mappings for an unlimited dimension of modulations larger than 32-QAM. Moreover, for smaller MD modulations, our proposed mappings outperform the BSAmappings. This indicates the efficiency of our proposed method compared to the BSA.574.3. AhmeeicTl eefhlgf Tad difchffibaEb/N0 (in dB)0 2 4 6 8 10 12 14BER10-610-510-410-310-210-1100BSA 4D 32-QAMProposed 4D 32-QAMBSA 4D 64-QAMProposed 4D 64-QAMCM capacity (32-QAM)CM capacity (64-QAM)Eb/N0 =6.24 dBEb/N0 =5.02 dBFigure 4.4: BER performance of BICM-ID with 4-D 32- and 64-QAM over Rayleigh fadingchannels.Eb/N0 (in dB)2 3 4 5 6 7 8BER10-1010-910-810-710-610-510-4BSA 4D 16-QAMProposed 4D 16-QAMBSA 6D 16-QAMProposed 6D 16-QAMFigure 4.5: Error-floor bounds of BER for BICM-ID with 4-D and 6-D 16-QAM overRayleigh fading channels.584.3. AhmeeicTl eefhlgf Tad difchffibaEb/N0 (in dB)2 3 4 5 6 7 8BER10-1010-910-810-710-610-5BSA 4D 32-QAMProposed 4D 32-QAMBSA 4D 64-QAMProposed 4D 64-QAMFigure 4.6: Error-floor bounds of BER for BICM-ID with 4-D 32- and 64-QAM overRayleigh fading channels.59Uzsptwr GMulti-vimwnsionsl Msppiny oxM-cSM Uonstwllstions xorTIUM-IV ovwr SiGN UzsnnwlsThe AWGN channel is an effective model for many communication links such as satellitecommunication link. To improve the error performance and the data rate of BICM-IDover AWGN channels, efficient mappings of large constellations are required. However, asmentioned earlier, developing optimum mappings of large constellations for BICM-ID iscomplicated. This is because of the significantly large number of possible mappings for largeconstellations. In fact, for a constellation with order 2m, there are 2m! possible mappings,which quickly approaches infinity by increasing m. In this chapter, we first propose anoptimum MD mapping for 16-QAM. Then, employing an innovative transfer system, weconstruct highly efficient MD mappings for any order/dimension of QAM constellationsusing the proposed MD 16-QAM mapping. Throughout this chapter, we assume that ina 2m-ary QAM constellation: (i) the MSED between symbols is equal to 1 and (ii) thesymbol index in 2m-QAM starts from the top left corner of the constellation and increasesfrom top to bottom and from left to right (see for example Fig. 5.1 for 16-QAM, wherethe symbol with index i is indicated by hi).G.C aptimum MV CH-cSM msppinyIn a 2N -D 16-QAM mapping, a sequence of n binary bits is mapped to a vector of N16-QAM symbols, where n = 4N . The proposed method uses the precoding process givenin 5.1, followed by an intermediate mapping. Let l = [l1P · · · P ln] be an n-bit label andlË = [lË1P · · · P lËn] be the precoded version of l. Each element of lË is obtained according to thefollowing precoding process:lËi ={l (l) if i = chosen-indexPli âl (l) otherwiseP (5.1)605.1. Bcgimhm M7 16-D4M mTcciag 1S6S7S10S11S5S2S9S13S14S3S4S8S12S16S15SFigure 5.1: Symbol arrangement in a 16-QAM constellation.where l (x) denotes an indicator function that takes the value one if x has an odd Ham-ming weight, otherwise it is equal to zero. The chosen-index can also take a value from{1P 2P · · · P n}, andâ is the modulo-2 addition. The proposed MD 16-QAMmapping methodassigns the label l to the symbol-vector s = [s1P · · · P sa ] as followssj = (Ëlj)P (5.2)wherelËj= [lË42jâ1351P · · · P lË4j ]P (5.3)and j takes a value from {1P · · Â·N}, and  is the intermediate 16-QAM mapping function(which will be discussed later in this section).Let y2i indicate the ith unique squared Euclidean distance (SED) between the symbolsin a 16-QAM constellation, where y2i Q y2i51. As mentioned in 1.6, an optimum mappingoffers the maximum possible value of yË2min.broposition G.C. For v 2N BD 16BfVb mvppingA the mvximum yË2min is (N â 1)y25 + y24CeroofC If s is a central-symbol in a 16-QAM constellation, i. e., s â Υ = {h6P h7P h10P h11},there is only one 16-QAM symbol at the SED of y25 from s, and each of the remainingsymbols has a smaller SED from s. Assume that l = [l1P · · · P ln] is the assigned label to615.1. Bcgimhm M7 16-D4M mTcciagthe symbol-vector s = [s1P · · · P sa ], where âi; si â Υ. There are n distinct labels that aredifferent from l in only one bit position. Moreover, regarding the above discussion, thereis only one symbol-vector with SED of Ny25 from s. Consequently, the MSED between sand the symbol-vectors with a Hamming distance of one bit from s cannot be larger than(N â 1)y25 + y24. As a result, the maximum possible yË2min for a 2N -D 16-QAM mapping is(N â 1)y25 + y24.broposition G.D. aet us vssume thvt l = [l1P · · · P ln] is v pvrtixulvr nBwit sequenxe vnylk = [l1kP · · · P lnk ] is yifferent fiith l only in the kth wit positionA fihere k â {1P · · · P n}C VlsoAvssume thvt lË = [lË1P · · · P lËn] vny lËk = [lË1kP · · · P lËnk ] vre the prexoyey versions of l vny lkArespextivelyC aet us ye ne lËjvny lËjk vslËj= [lË42jâ1351P · · · P lË4j ]P (5.4)lËjk = [lË42jâ1351k P · · · P lË4jk ]Pfihere j â {1P · · · P N}C ihenA for vll vvlues of jA the Hvmming yistvnxe wetfieen lËj vny lËjkis either H or I witsCeroofC Suppose that the chosen-index in (5.1) is equal to p. Regarding Proposition 2.1, ifp is equal to k, then lË and lËk are different in all n bits; otherwise, they are the same in thekth bit position and are different in the remaining (nâ 1) bits. As a result, we can writeyH (ËljP lËjk) =8><>:4 if p = k , âjP4 if p Ì¸= k , j = qP3 if p Ì¸= k , j Ì¸= qP(5.5)where yH(aP w) is the Hamming distance between a and w and q is given byq = âk â 14â+ 1P (5.6)where â:â represents the floor function.broposition G.E. In (JCJ)A for vll vvlues of j vny kA the Hvmming yistvnxes of H vny Iwits wetfieen lËjvny lËjk oxxur n(N â 1) + 1 vny (nâ 1) timesA respextivelyCeroofC In Proposition 5.2, suppose that the chosen-index is equal to p. Then, regardingProposition 2.1, there are two possible cases as follows.625.1. Bcgimhm M7 16-D4M mTcciagCvse FO k = p. lË and lËk are different in all n bits. Then, for all values of j and k,there is a 4-bit difference between lËjand lËjk.Cvse GO k Ì¸= p. lËi and lËk are the same in kth bit and they are different in (n â 1)remaining bits. Then, using (5.5), the Hamming distance between lËjand lËjk is given byyH (ËljP lËjk) ={3 if j = qP4 otherwiseP(5.7)where q is defined in (5.6).For a given value of p and for all values of k, Cvse F occurs only one time. In this case,for j = 1P · · · P N , the Hamming distance of 4 bits between lËj and lËjk occurs N times overall.Similarly, since for a given p, there are (n â 1) possible values of k such that k Ì¸= p, thenCvse G occurs (nâ1) times. Each occurrence results in (Nâ1) Hamming distance of 4 bitsand one Hamming distance of 3 bits between lËjand lËjk, where j = 1P · · · P N . Therefore,for a particular l and for all values of j and k, between lËjand lËjk, a 4-bit distance occursn(N â 1) + 1 times and a 3-bit distance occurs (nâ 1) times.Suppose that in the proposed MD mapping, l is mapped to the symbol-vector s =[s1P · · · P sa ] and lk is mapped to sk = [s1kP · · · P sak ], where sj = (Ëlj) and sjk = (Ëljk) aretwo symbols in the 16-QAM intermediate mapping and they have a Hamming distance ofeither 3 or 4 bits from each other. As a result, to increase yË2min, i. e., the MSED betweens and sk, our approach is to increase the MSED between the symbols with a Hammingdistance of 3 or 4 bits in the intermediate 16-QAM mapping. Moreover, as Proposition 5.3declares, the 4-bit distance between lËjand lËjk is more often. Therefore, it is preferred thatthe MSED between the symbols with 4-bit distance in the intermediate mapping to be aslarge as possible.G.C.C CH-cSM intwrmwvistw msppinyEach symbol in a 16-QAM mapping is mapped by a 4-bit binary label. In the set of allpossible 4-bit labels, there are 5 labels with a Hamming distance of 3 or 4 bits from agiven label, l. We refer to these 5 labels as the forbidden-labels of label l. Let y2min bethe desired MSED between a particular symbol, h, and the symbols that have a Hammingdistance of 3 or 4 bits from h. We refer to symbols whose SED from h is less than y2minas the forbidden-symbols of symbol h and to the remaining symbols as the authorized-symbols of symbol h. As the main principle, if a label is mapped to the symbol h, noneof its forbidden-labels can be mapped to the forbidden-symbols of symbol h. We start themapping process by assigning labels to the central-symbols because central-symbols have635.1. Bcgimhm M7 16-D4M mTcciagthe smallest number of authorized-symbols; therefore, it is easier to map them at first.The maximum possible y2min for a 16-QAM constellation is y24. In fact, y24 is the maximumMSED between a central-symbol and any set of 5 symbols in a 16-QAM constellation.Let in Fig. 5.1, Ai indicate the set of authorized-symbols for symbol hi. Then, forthe central-symbols, i.e., h6, h7, h10, and h11, we have A6 = {h4P h12P h1=P h15P h16},A7 = {h1P h9P h1=P h14P h16}, A10 = {h1P h=P h4P h8P h16}, and A11 = {h1P h2P h4P h5P h1=}.The label of each central-symbol has 5 forbidden-labels. Moreover, each of the forbidden-labels is allowed to be mapped only to one of the corresponding authorized-symbols. Asa result, if hi and hj are two central-symbols with common members in their set ofauthorized-symbols, then they must have common members in their set of forbidden-labels as well. For a 16-QAM constellation, = 2. Therefore, the labels of two particularcentral-symbols must have two common members in their set of forbidden-labels. Conse-quently, regarding the definition of forbidden-labels, the Hamming distance between twocentral-symbols should be either one or two bits.The maximum Euclidean distance between a central-symbol and its authorized-symbolsis y5. Moreover, each central-symbol has only one authorized-symbol at the Euclideandistance of y5. In the proposed intermediate 16-QAM mapping, we constrain the Hammingdistance between a central-symbol and its only authorized-symbol at the Euclidean distanceof y5 to be 4 bits. For example, h16 is the only symbol from A6 at the Euclidean distancey5 from the central-symbol h6. Therefore, the labels mapped to h6 and h16 should bedifferent in all 4 bits. The symbol h16 is also a common member in both A7 and A10,which constrains both symbols h7 and h10 to have a 3-bit Hamming distance from h16.This causes the symbol h6 to have a one bit Hamming distance from both h7 and h10,which are at the Euclidean distance y1 from h6. There are the same conditions for othercentral-symbols whose Euclidean distance from each other is y1.In summary, the intermediate mapping is developed in the three following steps:1- The central-symbols are mapped such that there is a one bit Hamming distance betweentwo central-symbols with the Euclidean distance y1.2- Let h be a central-symbol that is mapped by the label l. The symbol at the Euclideandistance y5 from h should be mapped by the label l¯.3- Regarding the main principle, each of the eight remaining labels has two permittedsymbols to map. However, some of these labels are in the forbidden-labels of each other.As a result, mapping a label to one of the remaining symbols affects the rest of the mappingprocess. This is discussed further in the following.Assume that the chosen-index in the precoding function in (5.1) is equal to p and thetwo n-bit labels, i.e., l and lk, are different only in the kth bit position, where p Ì¸= k. Then,the precoded versions of l and lk, i. e., lË = [lË1P · · · P lËn] and lËk = [lË1kP · · · P lËnk ], are different645.1. Bcgimhm M7 16-D4M mTcciagin all bits except for in the kth bit position [23]. Suppose that l and lk are mapped tos = [s1P · · · P sa ] and sk = [s1kP · · · P sak ], respectively, where sj = (Ëlj), sjk = (Ëljk), and lËjand lËjk are defined in (5.4). Let lËqinclude both pth and kth bits of lË, i.e.,q = âpâ 14â+ 1 (5.8)= âk â 14â+ 1:Also assume that w = f(k) and z = f(p), where f(x) is defined asf(x) = xâ 4Ã (q â 1): (5.9)Then, lËq= [lËqN1P · · · P lËqN4] and lËqk = [lËqN1k P · · · P lËqN4k ] are the same in the wth bit position, i.e.,lËqNw = lËqNwk , and they are different in the remaining 3 bits including the zth bit. As a result,in the proposed method, for lËqthere is no lËqk that is different from lËqin all bits except for inthe zth bit. Therefore, in the third step of designing the intermediate mapping, the labelsthat are different in all bits except for in the zth bit are mapped to the symbol pairs withthe Euclidean distance y4 from each other. Then, these symbol pairs will never be usedas sq and sqk in the proposed MD mapping. Hence, some symbol pairs with the Hammingdistance of 3 bits and the Euclidean distance y4 in the intermediate mapping will be unusedin the proposed MD mapping. As a consequence, in the resulting MD mapping, the MSEDof (N â 1)y5+ y4 between two labels with a Hamming distances of one bit will occur fewertimes. This diminishes NËmin (defined in 1.9) and makes the proposed mapping much closerto the optimum mapping.broposition G.F. huppose thvt in the thiry step of yesigning the intermeyivte mvppingAbi = [w1i P · · · P w4i ] vny bj = [w1j P · · · P w4j ] vre tfio winvry lvwels fiith the Hvmming yistvnxeof H witsC aet bi vny bj we mvppey to the symwols hi vny hjA fihixh vre vt the Euxliyevnyistvnxe y4 from evxh otherC VlsoA vssume thvt bx = [w1xP · · · P w4x] vny by = [w1yP · · · P w4y] vretfio winvry lvwelsA fihixh vre yifferent in only one wit positionC aet bx vny by we mvppeyto tfio xentrvlBsymwolsA iCeCA hx vny hyA fihixh vre vt the Euxliyevn yistvnxe y1 from evxhotherC boreoverA suppose thvt bx vny by hvve v 3Bwit Hvmming yistvnxe from bi vny bjArespextivelyC ihenA in oryer for bi vny bj to we the svme in the zth witA they neey bx vnyby to we yifferent in the zth wit positionCeroofC Suppose that hr and ht are two central-symbols, where hr is the neighbor of bothhx and ht, and ht is the neighbor of hy. Let hr and ht be mapped by the 4-bit binarylabels br and bt, respectively. Without loss of generality, assume that z = 1, bi = [0P 0P 0P 0]655.1. Bcgimhm M7 16-D4M mTcciagand bj = [0P 1P 1P 1]. Because bi and bj have not been mapped in the second step thenb¯i = [1P 1P 1P 1] and b¯j = [1P 0P 0P 0] cannot be used for the central-symbols in the first step.The labels bx and by, which have a 3-bits Hamming distance from bi and bj , respectively,will take a label from {[1P 0P 1P 1]P [1P 1P 0P 1]P [1P 1P 1P 0]} and {[1P 1P 0P 0]P [1P 0P 1P 0]P [1P 0P 0P 1]},respectively. Without loss of generality, let bx = [1P 0P 1P 1], then by can take a label from{[1P 0P 1P 0]P [1P 0P 0P 1]}. Let by = [1P 0P 1P 0], then bt, which should differ from by at one bitposition but not in the zth bit, will take a label from {[1P 1P 1P 0]P [1P 0P 0P 0]}. But, [1P 0P 0P 0]is different from bj in all bits, and thus, it is unusable for central-symbols. As a result, htis mapped by [1P 1P 1P 0]. Now, the only choices for br, which should be different from lxand lt at only one bit position but not in the zth bit, are [1P 0P 1P 0] and [1P 1P 1P 1]. However,[1P 0P 1P 0] cannot be used for br because it has already been mapped to by; moreover,[1P 1P 1P 1] is different from bi in all bits, and thus, it is not authorized to be mapped to acentral-symbol. Therefore, there is not an authorized label for hr. The same conditionsapply for the other scenarios in which hx or hy must select another label from their set ofauthorized-labels. Consequently, bx and by have to be different in the zth bit. 1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b 1 2 3 4b b b b 1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b( )a1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4b b b b 1 2 3 4b b b b 1 2 3 4bb b b1 2 3 4b b b b1 2 3 4b b b b1 2 3 4bb b b1 2 3 4b b b b1 2 3 4bb b b1 2 3 4b b b b( )bFigure 5.2: Proposed intermediate 16-QAM mapping for (a). z â {1P 2}, and (b). z â{3P 4}.Fig. 5.2. a and Fig. 5.2. b indicate our proposed intermediate mappings of 16-QAMwith z â {1P 2} and z â {1P 2}, respectively, where wi â {0P 1} for i = 1P · · · P 4. In whatfollows, we explain how the intermediate mapping of 16-QAM shown in Fig. 5.2. a isdeveloped.htep F : As aforementioned, in the first step, the central-symbols are mapped by the665.1. Bcgimhm M7 16-D4M mTcciagbinary labels with a Hamming distance of one bit. We first map the label [w1P w2P w=P w4] tothe symbol h6. Then, h7 and h10, whose Euclidean distances from h6 are y1, are mapped bythe labels with the Hamming distance one from h6, i. e, by [w1P w¯2P w=P w4] and [w¯1P w2P w=P w4],respectively. The symbol h11 is at the Euclidean distance y1 from both h7 and h10 thusits Hamming distance from each of h7 and h10 should be one bit. Therefore, it is mappedby [w¯1P w¯2P w=P w4].htep G : The symbols h1, h4, h1= and h16 are at the Euclidean distance y5 from thecentral-symbols h11, h10, h7 and h6, respectively. As a result, each of them takes a labelwith a Hamming distance of 4 bits from the label of the corresponding central-symbol atthe Euclidean distance y5.htep H : Each of the eight remaining labels for step 3 are forbidden by only one of thecentral-symbols, and each have two authorized-symbols to map. The first label is chosenrandomly and is arbitrarily mapped to one of its authorized-symbols. However, the restof the labels each one will have only one authorized-symbol to map. The only rule thatshould be considered is the main principle of the mapping method. For example, the label[w1P w2P w=P w¯4] has two authorized-symbols to map, i.e., h2 and h5. In this example, we mapit to h2. The label [w1P w¯2P w¯=P w4] is forbidden by h10 and it was authorized to be mappedto one of h= and h8. However, it cannot be mapped to h= anymore. This is because[w1P w¯2P w¯=P w4] has a Hamming distance of 3 bits from the label of h2, and the Euclideandistance of h= from h2 is smaller than y4. As a result, [w1P w¯2P w¯=P w4] is mapped to its otherauthorized-symbol, i.e., h8. In a similar way, the rest of the labels are mapped to theremaining symbols, as shown in Fig. 5.2. a.broposition G.G. In the proposey 2N BD 16BfVb mvppingA the bhED wetfieen symwolBvextors fiith v Hvmming yistvnxe of one wit is equvl to (N â 1)y25 + y24CeroofC Let l and lk be two 4N -bit labels that are different only in the kth bit position andthey are mapped to the symbol-vectors s = [s1P · · · P sa ] and sk = [s1kP · · · P sak ], respectively.The precoded versions of l and lk are lË = [Ël1P · · · P lËa ] and lËk = [Ël1kP · · · P lËak ], respectively,where lËjand lËjk are defined in (5.4). Using (5.2), for all values of j, sj = (Ëlj) and sjk =(Ëljk), where  is the proposed intermediate mapping function. Regarding the specificationsof the precoding function, lË and lËk are different either in 4N bits or (4N â 1) bits. If thisdistance is 4N bits, then for all values of j the Hamming distance between lËjand lËjk is 4bits. Otherwise, lËjand lËjk have a Hamming distance of 4 bits for (N â 1) values of j, andthey have a Hamming distance of 3 bits for the one remaining value of j. Moreover, theintermediate mappings indicated in Fig. (5.2) have two characteristics: (i) the Euclideandistance between two symbols with a Hamming distance of 4 bits is y5 and (ii) the Euclideandistance between two symbols with a Hamming distance of 3 is either y4 or y8. As a result,675.1. Bcgimhm M7 16-D4M mTcciagif lË and lËk have a Hamming distance of 4N and (4N â 1) bits, the MSED between s andsk is equal to Ny25 and (N â 1)y25+ y24, respectively. Consequently, overall in the proposed2N -D 16-QAM mapping, the MSED between s and sk is equal to (N â 1)y25 + y24.G.C.D broposwv F-V CH-cSM msppinyWxsmplw G.C. Let in the proposed MD 16-QAM mapping, N = 2, the chosen-index beequal to one (p = 1), l = [0P 1P 1P 1P 0P 0P 1P 1], and in Fig. 5.2, wi = 0 for i = 1P :::P 4. Using(5.9) we havez = f(p) = 1: (5.10)As a result, the mapping in Fig. 5.2(a) is used as the intermediate mapping. The precodedversion of l, i.e., lË = [1P 0P 0P 0P 1P 1P 0P 0], can be rewritten as lË = [Ël1P lË2], where lË1= [1P 0P 0P 0]and lË2= [1P 1P 0P 0]. Let in the proposed mapping method, l be mapped to s = [s1P s2];then, we can writes1 = (Ël1) = ([1P 0P 0P 0]) = h10Ps2 = (Ël2) = ([1P 1P 0P 0]) = h11: (5.11)As a consequence, in the proposed 4-D 16-QAM mapping, the label l = [0P 1P 1P 1P 0P 0P 1P 1]is mapped to the symbol-vector s = [h10P h11].Table 5.1 shows the proposed 4-D 16-QAM mapping. To achieve this mapping, it isTable 5.1: The proposed 4D mapping of 16-QAM for chosen-index equal to 1.h1 h2 h= h4 h5 h6 h7 h8 h9 h10 h11 h12 h1= h14 h15 h16h1 51 177 53 183 178 48 180 54 58 184 60 190 187 57 189 63h2 147 17 149 23 18 144 20 150 154 24 156 30 27 153 29 159h= 83 209 85 215 210 80 212 86 90 216 92 222 219 89 221 95h4 243 113 245 119 114 240 116 246 250 120 252 126 123 249 125 255h5 163 33 165 39 34 160 36 166 170 40 172 46 43 169 45 175h6 3 129 5 135 130 0 132 6 10 136 12 142 139 9 141 15h7 195 65 197 71 66 192 68 198 202 72 204 78 75 201 77 207h8 99 225 101 231 226 96 228 102 106 232 108 238 235 105 237 111h9 220 94 218 88 93 223 91 217 213 87 211 81 84 214 82 208h10 124 254 122 248 253 127 251 121 117 247 115 241 244 118 242 112h11 188 62 186 56 61 191 59 185 181 55 179 49 52 182 50 176h12 28 158 26 152 157 31 155 25 21 151 19 145 148 22 146 16h1= 76 206 74 200 205 79 203 73 69 199 67 193 196 70 194 64h14 236 110 234 104 109 239 107 233 229 103 227 97 100 230 98 224h15 44 174 42 168 173 47 171 41 37 167 35 161 164 38 162 32h16 140 14 138 8 13 143 11 137 133 7 131 1 4 134 2 128685.2. M7 mTcciag bf 2m-D4Massumed that the chosen-index in (5.1) is equal to one, and wi in Fig. 5.2 is equal to 0for i = 1P :::P 4. In this table, the label in the (j + 1; k + 1)th entry is mapped to thesymbol-vector x = (hj P hk). For example, the label 115, corresponding to the binary labell = [0P 1P 1P 1P 0P 0P 1P 1], is the (11P 12)th entry of Table 5.1. This means that 115 is mappedto the symbol-vector x = [h10P h11].G.D MV msppiny ox 2m-cSMWe propose a transfer system that takes the proposed MD 16-QAM as the input andconstructs a desired MD mapping of rectangular 2m-QAM through m â 3 steps (m S 4).In this approach, 16 symbols are first chosen from the 2m-QAM constellation. Then, anMD mapping is developed employing these selected symbols and using the proposed MD16-QAMmapping. It is important to note that the proposed method is specifically designedfor the rectangular 2m-QAM constellations. Such constellations are composed of 2ât2â rowsand 2mâât2â columns of signal points (symbols). In what follows, we describe the proposedmethod in two main sections.G.D.C MV msppiny usiny CH symtols xrom 2m-cSMThe 16 chosen symbols from 2m-QAM in the first step of the proposed method is shownin Table 5.2. In this Table, Îj represents the 2m-QAM symbol with index j. These 16symbols are selected such that their structure in the 2m-QAM constellation is equivalentto a 16-QAM constellation (see Fig. 5.3 as an example for m = 6).Let in the proposed method, l = [l213P · · · P l2ma3] be anmN -bit sequence that is mappedto a vector of N symbols from 2m-QAM. In step i, li = [l213i P · · · P l22i5=3a3i ] denotes the(i+ 3)N least significant bits of l, where lji is given bylji = l2maâ2i5=3a5j3: (5.12)Table 5.2: The 16 chosen symbols from 2m-QAM in the first step of the proposed method.1 Î1 9 Î152tâ)2 Î152ât2 ââ2 10 Î152tâ)52ât2 ââ23 Î152ât2 ââ) 11 Î152tâ)52ât2 ââ)4 Î15=Ã2ât2 ââ2 12 Î152tâ)5=Ã2ât2 ââ25 Î152tâ2 13 Î15=Ã2tâ26 Î152tâ252ât2 ââ2 14 Î15=Ã2tâ252ât2 ââ27 Î152tâ252ât2 ââ) 15 Î15=Ã2tâ252ât2 ââ)8 Î152tâ25=Ã2ât2 ââ2 16 Î15=Ã2tâ25=Ã2ât2 ââ2695.2. M7 mTcciag bf 2m-D4M 37Î46Î47Î54Î55Î45Î38Î53Î61Î62Î39Î40Î48Î56Î64Î63Î5Î14Î15Î22Î23Î13Î6Î21Î29Î30Î7Î8Î16Î24Î32Î31Î33Î42Î43Î50Î51Î41Î34Î49Î57Î58Î35Î36Î44Î52Î60Î59Î1Î10Î11Î18Î19Î9Î2Î17Î25Î26Î3Î4Î12Î20Î28Î27ÎFigure 5.3: 64-QAM constellation. The black symbols represent the 16 selected symbolsin the first step of the proposed mapping method.In step i = 1, the proposed MD 16-QAM mapping method is applied to map l1 to anN -tuple vector of 16-QAM symbols. Next, each 16-QAM symbol in the resulting mappingis replaced by one of the 16 chosen symbols from 2m-QAM constellation, according to Table5.3. In this table, for example, hi â Îj means that in a given symbol-vector, the 16-QAMsymbol hi should be replaced by the 2m-QAM symbol Îj .Table 5.3: The chart of substitution of 2m-QAM symbols for the 16-QAM symbols.h1 â Î1 h9 â Î152tâ)h2 â Î152ât2 ââ2 h10 â Î152tâ)52ât2 ââ2h= â Î152ât2 ââ) h11 â Î152tâ)52ât2 ââ)h4 â Î15=Ã2ât2 ââ2 h12 â Î152tâ)5=Ã2ât2 ââ2h5 â Î152tâ2 h1= â Î15=Ã2tâ2h6 â Î152tâ252ât2 ââ2 h14 â Î15=Ã2tâ252ât2 ââ2h7 â Î152tâ252ât2 ââ) h15 â Î15=Ã2tâ252ât2 ââ)h8 â Î152tâ25=Ã2ât2 ââ2 h16 â Î15=Ã2tâ25=Ã2ât2 ââ2705.2. M7 mTcciag bf 2m-D4MAccording to proposition 5.5, in the proposed 2N -D 16-QAM mapping, yË2min = (N â1)y25 + y24, where y4 and y5 are equal to the Euclidean distance of h1 from h7 and h11,respectively. Furthermore, in Table 5.3, the 16-QAM symbols h1, h7 and h11 are re-placed by 2m-QAM symbols Î1, Î152tâ)52ât2 ââ) and Î152tâ252ât2 ââ) , respectively. As aconsequence, yË2min for the obtained mapping using 16 symbols from 2m-QAM is equal to(Nâ1)yË25+yË24, where yË4 and yË5 represent the Euclidean distance of Î1 from Î152tâ)52ât2 ââ)and Î152tâ252ât2 ââ) , respectively. In other words, we can writeyË24 = 22mâ2ât2ââ4 + 22ât2ââ2P (5.13)yË25 = 22mâ2ât2ââ2 + 22ât2ââ2:G.D.D frsnsxwrriny xrom 16-cSM to 2m-cSMIn step iâ1 (i S 1), let liâ1 be mapped to siâ1 = [s1iâ1P · · · P saiâ1], where sjiâ1 = xjiâ1+Iyjiâ1and I2 = â1. In step i, li is mapped to si = [s1i P · · · P sai ], where sji = xji + Iyji and xjiand yji are calculated using Table 5.4. In this table, ei = [z1i P · · · P zai ] represents the Nmost significant bits of li, i.e., zji = lji , wi denotes the Hamming weight of ei, E and Orepresent the set of even and odd integers, respectively, and sgn(:) is the sign function,which is defined assgn(m) =8><>:1 if m S 0P0 if m = 0Pâ1 if m Q 0:(5.14)For example, in step i, if wi â E, zji = 1, and i â E, in order to calculate the real part ofsji , i.e., xji , one needs to use xji = xjiâ1 + 2mâât2âââ j2ââ2 from Table 5.4.Table 5.4: Transfer system to generate symbol coordinates for MD 2m-QAM mappings.wi â E wi â Ozji = 0xji = xjiâ1 xji = xjiâ1 â sgn(xjiâ1)2mâât2ââ1yji = yjiâ1 yji = yjiâ1 â sgn(yjiâ1)2ât2ââ1zji = 1i â E xji = xjiâ1 + 2mâât2âââ i2ââ2 xji = xjiâ1 + (2ââ i2ââ1 â sgn(xjiâ1))2mâât2ââ1yji = yjiâ1 yji = yjiâ1 â sgn(yjiâ1)2ât2ââ1i â O xji = xjiâ1 xji = xjiâ1 â sgn(xjiâ1)2mâât2ââ1yji = yjiâ1 â 2ât2âââ iâ)2ââ2 yji = yjiâ1 â (2ââiâ)2ââ1 + sgn(yjiâ1))2ât2ââ1715.2. M7 mTcciag bf 2m-D4Mbroposition G.H. aet li = [l1i P · · · P l2i5=3ai ] vny liNk = [l1iNkP · · · P l2i5=3aiNk ] we tfio lvwels instep i thvt vre yifferent only in the kth wit position vny vre mvppey to symwolBvextorssi = [s1i P · · · P sai ] vny siNk = [s1iNkP · · · P saiNk]A respextivelyC For vll iA the bhED wetfieen sivny siNkA iCeCA yË2minNiA is grevter thvn or equvl to (N â 1)yË25 + yË24CeroofC See Appendix B.Wxsmplw G.D. In the proposed MD mapping method, let us set m = 6 (64-QAM), N = 2,and l = [1P 1P 1P 0P 0P 1P 1P 1P 0P 0P 1P 1]. In the following three steps, l is mapped to a vectorof 64-QAM symbols, i.e., s = [s1P s2].htep F : Let l1 = [0P 1P 1P 1P 0P 0P 1P 1] be the eight least significant bits of l and thechosen-index be equal to one (p = 1). The sequence l1 is mapped to the symbol-vectors1 = [s11P s21], where s11 = x11 + Iy11 and s21 = x21 + Iy21 belong to the 64-QAM signal set. Inthe proposed 4-D 16-QAM mapping when p = 1, the sequence [0P 1P 1P 1P 0P 0P 1P 1], i.e., l1,is mapped to the symbol-vector [h10P h11] (c.f. Example 5.1, Section 5.1.2). From Table5.3, the equivalent 64-QAM symbols for h10 and h11 are Î=5 and Î=7, respectively. As aresult, s11 = Î=5 and s21 = Î=7.htep G : Let l2 = [1P 0P 0P 1P 1P 1P 0P 0P 1P 1] be the ten least significant bits of l and bemapped to s2 = [s12P s22]. Considering l1 from step 1, l2 can be rewritten as l2 = [e2P l1],where e2 = [1P 0]. Table 5.4 is used to obtain the symbols s12 and s22 using the symbols s11and s21 from step 1, respectively. Since i â E (i = 2), the Hamming weight of e2 is odd(wi â O), and z12 = 1, then the following equations from Table 5.4 are used to calculatethe real and imaginary parts of s12.xji = xjiâ1 + (2ââ i2ââ1 â sgn(xjiâ1))2mâât2ââ1 (5.15)yji = yjiâ1 â sgn(yjiâ1)2ât2ââ1:Let us set x11 = 0:5 and y11 = 1:5 (s11 = Î=5 = 0:5 + I1:5) in the above equations, whichresults in x12 = â2:5 and y12 = â2:5. Similarly, since z22 = 0, then the following equationsfrom Table 5.4 are used to calculate the real and imaginary parts of s22.xji = xjiâ1 â sgn(xjiâ1)2mâât2ââ1 (5.16)yji = yjiâ1 â sgn(yjiâ1)2ât2ââ1:From step 1, we set x21 = 0:5 and y21 = â0:5 (s21 = Î=7 = 0:5âI0:5) in the above equations,which results in x22 = â3:5 and y22 = 3:5.725.3. AhmeeicTl eefhlgfhtep H : In this step, l= = l and it can be rewritten as l= = [e=P l2], where e= = [1P 1].Let l= be mapped to s= = [s1=P s2=]. The symbols s1= and s2= are obtained as follows. Sincei â O (i = 3), the Hamming weight of e= is even (wi â E), and z1= = z2= = 1, then thefollowing equations from Table 5.4 are used to calculate the real and imaginary parts of s1=and s2=.xji = xjiâ1 (5.17)yji = yjiâ1 â 2ât2âââ iâ)2ââ2:Substituting x12, x22, y12, and y22 in the above equations gives s1= = â2:5â I3:5 (= Î16) ands2= = â3:5 + I2:5 (= Î2). Consequently, l is mapped to s = (Î16PÎ2).G.E Numwriusl rwsultsThis section provides a selection of numerical examples to illustrate the performance andadvantage of our proposed MD mapping for BICM-ID systems over the AWGN channel.We compare our resulting mappings with the mappings that are optimized for the AWGNchannel employing the well-known BSA [9]. From many aspects, the BSA is the bestknown computer search technique to find desired mappings for BICM-ID. However, theBSA becomes intractable to achieve suitable mappings for larger MD constellations, e.g.,6-D 64-QAM, due to computational time constraints. We consider a rate-1=2 convolutionalcode with the generator polynomial of (13P 15)8. An interleaver length of about 10000 bitsis used. All gains reported in this section are measured at a BER of 10â6.Table 5.5 compares the values of yË2min and NËmin for our proposed mappings and theBSA mappings of MD 2m-QAM (m = 4P 5P 6). In the case of MD mappings of higher orderTable 5.5: yË2min and NËmin for different mappings.Modulations yË2min NËminBSA 4-D 16-QAM 1.2 2Proposed 4-D 16-QAM 2.60 6:40Ã 102BSA 6-D 16-QAM 1.33 1:9Ã 102Proposed 6-D 16-QAM 2.8 1:6Ã 104BSA 4-D 32-QAM 0.8 1Proposed 4-D 32-QAM 2.15 1:02Ã 10=BSA 4-D 64-QAM 1.2 2Proposed 4-D 64-QAM 2.48 1:02Ã 104735.3. AhmeeicTl eefhlgfTable 5.6: yË2min and NËmin for MD mapping of higher order modulations.Modulations yË2min NËminProposed 4-D 128-QAM 2.11 1:6Ã 104Proposed 4-D 256-QAM 2.45 1:6Ã 105Proposed 4-D 512-QAM 2.10 2:6Ã 105Proposed 4-D 1024-QAM 2.44 2:6Ã 106QAMs, the BSA results could not be obtained due to the computational time constraints.Therefore, in Table 5.6, we report the values of yË2min and NËmin only for the proposedMD mapping of larger constellations. Table 5.5 shows that in comparison with the BSAmappings, the proposed mappings offer greater values of yË2min. As a result, it is expectedthat the proposed mappings improve the error performance of BICM-ID over the AWGNchannel. This is confirmed by the simulation results plotted in Fig. 5.4 and Fig. 5.5. Itcan be observed from Fig. 5.4 that the proposed 4-D 16-QAM and 6-D 16-QAM mappingsoutperform their BSA counterparts by 1:4 dB and 2:5 dB, respectively. Fig. 5.5 indicatesthat the proposed 4-D 32-QAM and 4-D 64-QAM mappings offer a gain of 2:55 dB and3:6 dB, respectively, over the corresponding BSA mappings. The error bounds for differentmappings are plotted in Fig. 5.6 and Fig. 5.7. These figures show that the proposedmappings offer lower error floors in comparison with the BSA mappings.Please note that the BSA becomes intractable when finding a MD mapping of a largemodulation. But, our proposed method is a heuristic method and generates good MDmappings of large modulations instantaneously. Furthermore, for smaller MD modulations,our proposed mappings improve the system performance compared to the BSA mappings.This shows the efficiency of our proposed method compared to the BSA.745.3. AhmeeicTl eefhlgfEb/N0 (in dB)0 1 2 3 4 5 6 7 8BER10-610-510-410-310-210-1100BSA 4D 16-QAMProposed 4D 16-QAMBSA 6D 16-QAMProposed 6D 16-QAMCM capacityEb/N0 =2.09 dBFigure 5.4: BER performance of BICM-ID over the AWGN channel.Eb/N0 (in dB)0 2 4 6 8 10 12 14BER10-610-510-410-310-210-1100BSA 4D 32-QAMProposed 4D 32-QAMBSA 4D 64-QAMProposed 4D 64-QAMCM capacity (32-QAM)CM capacity (64-QAM)Eb/N0 =3.13 dBEb/N0 =4.28 dBFigure 5.5: BER performance of BICM-ID over the AWGN channel.755.3. AhmeeicTl eefhlgfEb/N0 (in dB)0 1 2 3 4 5 6 7 8 9 10BER10-10010-8010-6010-4010-20100BSA 4D 16-QAMProposed 4D 16-QAMBSA 6D 16-QAMProposed 6D 16-QAMFigure 5.6: Error-floor bounds over the AWGN channel.Eb/N0 (in dB)0 5 10 15BER10-10010-8010-6010-4010-20100BSA 4D 32-QAMProposed 4D 32-QAMBSA 4D 64-QAMProposed 4D 64-QAMFigure 5.7: Error-floor bounds over the AWGN channel.76Uzsptwr HUonulusionsH.C iork suuompliszwv in tzis tzwsisIn this thesis, we have studied the mapping problem of BICM-ID for a wide range ofmodulations. In particular, we have focused on the 2-D and MD mapping of 2m-ary (un-limited m) modulations. We have proposed heuristic methods as well as computer searchtechniques to achieve efficient mappings for BICM-ID. Presented analytic and simulationresults show that in comparison with the-state-of-the-art mappings such as the BSA map-pings, our proposed mappings significantly improve the systemâs error performance overthe AWGN and Rayleigh fading channels.In Chapter 2, we have proposed a novel mapping method to find efficient 2-D mappingsof higher order QAM and PSK modulations for BICM-ID. Our method generates mappingsthrough a systematic approach. Two main qualities of our proposed method are as follows:(i) it is a very simple method, and (ii) it generates efficient mappings for 2m-ary QAM andPSK with an unlimited value of m. Simulation results show that compared to the bestpreviously known mappings at a target BER rate of 10â6, our proposed mappings cansave up to 5:7 dB and 4:4 dB of the transmit signal over the AWGN and Rayleigh fadingchannels, respectively.In Chapter 3, we have proposed a heuristic method to design MD mappings for BICM-ID systems using 16- and 64-QAM. The innovation of the proposed method in this chapteris that it can efficiently generate MD mappings using 16- and 64-QAM. Presented nu-merical results show that in comparison with the well-known BSA mappings and randommappings, our mappings outperform significantly over AWGN and Rayleigh fading chan-nels. Compared to the BSA mappings for a target BER of 10â6, our mappings can saveup to 3:5 dB and 3 dB transmit signal energy over AWGN and Rayleigh fading channels,respectively. The corresponding performance gains are larger compared to random map-pings. The proposed mappings also have improved the error-floor performance comparedto random mappings and the mappings obtained by the BSA.In Chapter 4, we have introduced a novel mapping method to construct efficient MDmappings to improve the error performance of BICM-ID over Rayleigh fading channels. Wehave broken the MD mapping design problem into four distinct 2-D mapping functions.776.2. Fhghee jbekThen, we have developed cost functions, which are optimized to minimize the error-floorof MD mappings. Due to the lower complexity of 2-D space, the optimization approachis very simple and results in excellent MD mappings of higher order constellations suchas 2m-QAM for m = 7P 8P 9P 10. However, the well-known BSA becomes intractable infinding suitable MD mappings of these large constellations. In the case of MD mappingof 2m-QAM for m = 4P 5P 6, our proposed mappings outperform the BSA mappings at theBER of practical interest, i.e., 10â6, by up to 2:6 dB. In addition, the proposed mappingsoffer lower error-floors compared to their BSA counterparts. Consequently, our proposedmappings outperform the best previously known mappings, i.e., the BSA mappings, inboth low and high SNR regions.In chapter 5, we have proposed an optimum MD mapping of 16-QAM for BICM-ID performance over the AWGN channel. Then, a transferring system is developed toconstruct MD mappings of higher order QAMs using the proposed MD mapping of 16-QAM. It is proven that the proposed transferring system guarantees efficient MD mappingsof 2m-QAM (unlimited value of m). Simulation results show that our proposed mappingssave up to 3:6 dB of the transmit power for a target BER of 10â6 compared to the well-known BSA mappings. Moreover, our proposed mappings improve the error-floor of BICM-ID. Consequently, for MDmapping of medium constellations such as 64-QAM, our proposedmappings outperforms the BSA mappings, in both low and high SNR region over theAWGN channel. Moreover, our proposed method constructs efficient unlimited dimensionmappings of unlimited order QAMs for AWGN channels.H.D Futurw workFor all the results and methods presented in this thesis, it is assumed that the CSI isperfectly known at the receiver side. However, for many practical applications, the CSI ispartially known at the receiver. In this case, all mapping design guidelines might be subjectto change. As a result, new mapping methods will be required for different modulationsbased on the new mapping guidelines.This thesis addresses the mapping problem for the constellations with fixed signalpoints. In particular, the proposed mapping methods in this thesis mostly consider theQAM and PSK constellations. However, changing the signal pointsâ positions in the con-stellations can improve the system performance significantly. Therefore, optimizing signalconstellations for BICM-ID as well as finding suitable mappings for the optimized constel-lations are interesting problems to address in future work.78Titlioyrspzy[1] E. Zehavi, â8-PSK trellis codes for a Rayleigh channel,â IEEE irvnsC CommunC, vol.40, pp. 873-884, May 1992.[2] A. Alvarado âTowards fully optimized BICM transmissions,â Ph.D. dissertation, Dept.Sign. and syst., Chalmers Univ. of Tech., Goteborg, Sweden 2010.[3] X. Li and J. A. 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Available:https://www.dropbox.com/s/8xl1al1nmaxmleo/Report.pdf.815iUlibgeTchl Bibliography[33] G. Ungerboeck, âChannel coding with multilevel/phase signals,â IEEE irvnsC InformCiheory, vol. 28, pp. 56-67, Jan. 1982.[34] A. Chindapol and J. A. Ritcey, âDesign, analysis and performance evaluation forBICMID with square QAM constellations in Rayleigh fading channels,â IEEE JC helextCVrevs CommunC, vol. 19, pp. 944-957, May 2001.82Sppwnviuws83Sppwnvix Sbroox xor broposition D.EeroofC We prove Proposition 2.3 for two groups of modulations as follows: (i) PSK andsquare QAM and (ii) cross QAM.beK snv squsrw cSML Perfect Gray mappings can be obtained from the natural binarylabeling for PSK and square QAM constellations using the simple procedure described in[29] and [30]. It is well-known that a given symbol in a Gray mapping has the Hammingdistance of one bit from each of its adjacent symbols.If lk = [l1kP l2kP · · · P lmk ] and ln = [l1nP l2nP · · · P lmn ] are two m-bit labels, the proposedmapping method maps lk and ln to the symbols whose labels in the corresponding Graymapping are lËk = [lË1kP lË2kP · · · P lËmk ] and lËn = [lËnl P lË2nP · · · P lËmn ], respectively. Thus, if lk andln are adjacent symbols in the resulting mapping, lËk and lËn are adjacent symbols in theGray mapping and have the Hamming distance of one bit from each other. According toProposition 2.2 (c.f., eq. (2.12)), the original label lt can be obtained uniquely from itsprecoded version lËt using the reverse process, i.e., lt = Î¨â1(Ëlt). Hence, in order to provethat a given symbol in the resulting mapping, e.g., lk, has the Hamming distance of eithertwo, (m â 1) or m bits from its neighbour, e.g., ln, we need to prove the following. If lËkand lËn have the Hamming distance of one bit from each other, lk and ln should have theHamming distance of either two, (mâ 1) or m bits from each other. Using eq. (2.12), wecan rewrite lËk and lËn as follows:lik ={lËik â lËqk if i Ì¸= qPl (Ëlk)âA(m)Ã lËqk otherwiseP(A.1)lin ={lËin â lËqn if i Ì¸= qPl (Ëln)âA(m)Ã lËqn otherwise:(A.2)Since both of lËk and lËn have the same length, A(m) takes the same value in eq. (A.1) andeq. (A.2). Assume that lËk and lËn differ only at the jth bit position and chosen-index isequal to q. There are two cases as follows:844cceadik 4. Cebbf fbe Cebcbfigiba 2.3Cvse FO j = q. Since lk and ln differ in the jth bit position and when j = q, we can write{lËik =¯Ëlin if i = qPlËik = lËin otherwiseP(A.3)then if lËqk = W, using eq. (A.3) we have lËqn =¯Ëlqk = W¯. Since lËk and lËn differ in one bitposition, one of them has an odd Hamming wight and the other has an even Hammingwight. As a result, if l (Ëlk) = X, l (Ëln) = X¯. Hence, when i Ì¸= q, by replacing lËik by lËin, lËqkby W, l (Ëlk) by X, lËqn by W¯, and l (Ëln) by X¯ we can rewrite eq. (A.1) and eq. (A.2) asfollows:lik ={lËin âW if i Ì¸= qPX âA(m)ÃW otherwiseP (A.4)lin ={lËin â W¯ if i Ì¸= qPX¯ âA(m)Ã W¯ otherwise: (A.5)Let us assume that lx = [l1xP l2xP · · · P lmx ] and lx = lkâ ln, i.e., lix = likâ lin for all i. Usingeq. (A.4) and eq. (A.5), we can writelix = lik â lin ={lËin âW â lËin â W¯ if i Ì¸= qPX â (A(m)ÃW)â X¯ â (A(m)Ã W¯) otherwise: (A.6)Using the fact lËin âW â lËin â W¯ = (lËin â lËin)â (W â W¯) = 0â 1 = 1 and X â (A(m)ÃW)âXÂ¯ â (A(m)Ã W¯) = (X â X¯)â A(m)Ã (W â W¯) = (1â A(m)) = A¯(m), we can rewrite eq.(A.6) aslix ={1 if i Ì¸= qPA¯(m) otherwise:(A.7)If m is odd, A¯(m) = 1 (c.f., Proposition 2.2) and lx = lk â ln has a Hamming weightof m, which implies that the labels of two adjacent symbols, lk and ln, in the resultingmapping are different in all m bit positions. On the other hand, if m is even, A¯(m) = 0(c.f., Proposition 2.2) and lx has the Hamming weight of (mâ 1). This implies that lk andln are different in (mâ 1) bits.Cvse GO j Ì¸= q . In this case using eq. (2.1), we can rewrite eq. (A.1) and eq. (A.2) as854cceadik 4. Cebbf fbe Cebcbfigiba 2.3follows:lik =8><>:lËjk â lËqk if i Ì¸= qP i = jPlËik â lËqk if i Ì¸= qP i Ì¸= jPl (lËk)âA(m)Ã lËqk if i = qP(A.8)lin =8><>:lËjn â lËqn if i Ì¸= qP i = jPlËin â lËqn if i Ì¸= qP i Ì¸= jl (lËn)âA(m)Ã lËqn if i = q:(A.9)Since lËk and lËn differ only in the jth bit position, if lËjk = W, we can write lËjl =¯Ëljk = W¯.When i Ì¸= j, we have lËik = lËin, which yields lËqk = lËqn as well. Moreover, Since lËk and lËn differin one bit position, one of them has an odd Hamming weight and the other one has an evenHamming weight. As a result, if l (Ëlk) = X, we can write l (Ëln) = X¯. Thus, by replacinglËjk by W, lËjn by W¯, lËik by lËin, lËqk by lËqn, l (Ëlk) by X and l (Ëln) by X¯, we can rewrite eq. (A.8)and eq. (A.9) as follows:lik =8><>:W â lËqn if i Ì¸= qP i = jPlËin â lËqn if i Ì¸= qP i Ì¸= jPX âA(m)Ã lËqn if i = qP(A.10)lin =8><>:W¯ â lËqn if i Ì¸= qP i = jPlËin â lËqn if i Ì¸= qP i Ì¸= jPX¯ âA(m)Ã lËqn if i = q:(A.11)Let us assume that lx = [l1xP l2xP · · · P lmx ] and lx = lk â ln, i.e., lix = lik â lin for all i. As aresult, using eq. (A.10) and eq. (A.11) we can writelix = lik â lin =8><>:W â lËqn â W¯ â lËqn if i Ì¸= qP i = jPlËin â lËqn â lËin â lËqn if i Ì¸= qP i Ì¸= jPX â (A(m)Ã lËqn)â X¯ â (A(m)Ã lËqn) if i = q:(A.12)Using the facts W â lËqn â W¯ â lËqn = (W â W¯) â (lËqn â lËqn) = 1 â 0 = 1, lËin â lËqn â lËin â lËqn =(lËin â lËin) â (lËqn â lËqn) = 0 â 0 = 0 and X â (A(m) Ã lËqn) â X¯ â (A(m) Ã lËqn) = (X â X¯) âA(m)Ã (lËqn â lËqn) = 1â (A(m)Ã 0) = 1, we rewrite eq. (A.12) aslix = lik â lin =8><>:1 if i Ì¸= qP i = jP0 if i Ì¸= qP i Ì¸= jP1 if i = q:(A.13)864cceadik 4. Cebbf fbe Cebcbfigiba 2.3Thus, the Hamming weight of lx = lk â ln is two, which implies that the labels of twoadjacent symbols in the resulting mapping, i.e., lk and ln, are different in two bit positions.In summary, if lk and ln are the labels of two adjacent symbols in the resulting squareQAM or PSK constellations with our proposed mapping and lËk and lËn are different in onebit position, say the jth bit position, the Hamming distance between lk and ln denoted byyH(lkP ln) is given by (from the above discussions):yH(lkP ln) =8><>:m if j = qP and m is oddPmâ 1 if j = qP and m is evenP2 if j Ì¸= qP(A.14)where q is the chosen-index, and j = 1P 2P · · · Pm, is the bit position in which the labels oftwo adjacent symbols in the Gray mapping are different from each other. Let us assume thatthe chosen-index is equal to q. Then, if j = q, the Hamming distance between two adjacentsymbols in our resulting PSK and square QAM mappings is yH(lkP ln) = m or (mâ1) (c.f.,eq. (A.14)). However, for a given value of q, the fraction of the time that we get j = q fortwo adjacent symbols is less than 1m . Therefore, the fraction of adjacent symbols that willhave the Hamming distance of m or (mâ 1) bits is less than 1m , which tends to be smallerfor higher order constellations. Also, the fraction of adjacent symbols that will have theHamming distance of two bits from each other is larger than (mâ1m ), which tends to belarger for higher order constellations. Therefore, our proposed mapping method yields asmaller average Hamming distance between adjacent symbols in the resulting PSK andsquare QAM mappings especially for larger constellations as shown in Table A.1.Table A.1: Hamming distance between adjacent symbols for proposed mappings.Modulation mTglad fme-bej gf ad-bacefl sye-bgdsHejceflage gf adbaceflsyebgds oalh Haeeafgdaslafce gf 2 balsHejceflage gf adbaceflsyebgds oalh Haeeafgdaslafce gf (tâ )) balsHejceflage gf adbaceflsyebgds oalh Haeeafgdaslafce gf t balsAnejage Hae-eafg daslafcebeloeef adbaceflsyebgds16-QAM 4 24 83.33% 16.67% 0% 2.1732-QAM 5 52 88.46% 0% 11.54% 2.3564-QAM 6 112 92.86% 7.14% 0% 2.21128-QAM 7 232 94.83% 0% 5.17% 2.26256-QAM 8 480 96.67% 3.33% 0% 2.17512-QAM 9 976 97.54% 0% 2.46% 2.171024-QAM 10 1984 98.39% 1.61% 0% 2.1116-PSK 4 16 87.50% 12.5% 0% 2.1332-PSK 5 32 93.75% 0% 6.25% 2.1964-PSK 6 64 96.87% 3.13% 0% 2.09128-PSK 7 128 98.44% 0% 1.56% 2.08256-PSK 8 256 99.22% 0.78% 0% 2.04512-PSK 9 512 99.61% 0% 0.39% 2.031024-PSK 10 1024 99.80% 0.20% 0% 2.01874cceadik 4. Cebbf fbe Cebcbfigiba 2.3Cross cSML Perfect Gray mappings cannot be defined for cross QAM constellations.However, pseudo-Gray mappings can be obtained for such constellations using the proce-dure described in [29] and [31]. It is worth noting that the Hamming distance betweentwo adjacent symbols in a pseudo-Gray mapping of a cross QAM constellation is at most2 bits.Let lk and ln be the labels of two adjacent symbols in our proposed mapping for across QAM. Thus, lËk and lËn are the labels of the adjacent symbols in the correspondingpseudo-Gray mapping of the constellation and have the Hamming distance of either oneor two bits from each other. In order to prove that the label of a symbol in the proposedmapping e.g., lk, has the Hamming distance of either two, (mâ1) or m bits from the labelof its neighbouring symbols, e.g., ln, it suffices to prove the following. If lËk and lËn have theHamming distance of one or two bits from each other, then the Hamming distance betweenlk and ln is either two, (m â 1), or m bits. It is already proven that if lËk and lËn have aHamming distance of one bit, lk and ln have a Hamming distance of either two, (mâ 1),or m from each other. As a result, we need to prove that, if lËk and lËn have the Hammingdistance of two bits from each other, then lk and ln have the Hamming distance of eithertwo, (mâ 1), or m bits from each other. This proof is given below.Assume that lËk and lËn are different in two bit positions, i.e., the hth and the jth bitpositions, and the chosen-index is equal to q. Two cases can be discussed as follows.Cvse FO h Ì¸= q vny j Ì¸= q. In this case, we can rewrite eq. (A.1) and eq. (A.2) as follows:lik =8>>>><>>>>:lËhk â lËqk if i Ì¸= qP i = hPlËjk â lËqk if i Ì¸= qP i = jPlËik â lËqk if i Ì¸= qP i Ì¸= hP i Ì¸= jPl (lËk)âA(m)Ã lËqk if i = qP(A.15)lil =8>>>><>>>>:lËhn â lËqn if i Ì¸= qP i = hPlËjn â lËqn if i Ì¸= qP i = jPlËin â lËqn if i Ì¸= qP i Ì¸= hP i Ì¸= jPl (lËn)âA(m)Ã lËqn if i = q:(A.16)Since lËk and lËn are different in the hth and the jth bit positions, if lËhk = W, then lËhn =¯Ëlhk = W¯,and if lËjk = X, then lËjn =¯Ëljk = X¯. Also if i Ì¸= h and i Ì¸= j, then lËik = lËin, which resultsin lËqk = lËqn. Moreover, since lËk and lËn differ in two bit positions, both of them has eitheran odd Hamming weight or an even Hamming weight. As a result, if l (Ëlk) = D, thenl (Ëln) = D as well. Thus, in eq. (A.15) and eq. (A.16), by replacing lËhk by W, lËhn by W¯, lËjkby X, lËjn by X¯, lËik by lËin, lËqk by lËqn, l (Ëlk) by D, and l (Ëln) by D, we can rewrite eq. (A.15)884cceadik 4. Cebbf fbe Cebcbfigiba 2.3and eq. (A.16) as follows:lik =8>>>><>>>>:W â lËqn if i Ì¸= qP i = hPX â lËqn if i Ì¸= qP i = jPlËin â lËqn if i Ì¸= qP i Ì¸= hP i Ì¸= jPD âA(m)Ã lËqn if i = qP(A.17)lin =8>>>><>>>>:W¯ â lËqn if i Ì¸= qP i = hPX¯ â lËqn if i Ì¸= qP i = jPlËin â lËqn if i Ì¸= qP i Ì¸= hP i Ì¸= jPD âA(m)Ã lËqn if i = q:(A.18)Let us assume that lx = [l1xP l2xP · · · P lmx ] and lx = lk â ln so that âi; lix = lik â lin. As aresult, using eq. (A.17) and eq. (A.18) we havelix = lik â lin =8>>>><>>>>:W â lËqn â W¯ â lËqn if i Ì¸= qP i = hPX â lËqn â X¯ â lËqn if i Ì¸= qP i = jPlËin â lËqn â lËin â lËqn if i Ì¸= qP i Ì¸= hP i Ì¸= jPD â (A(m)Ã lËqn)âD â (A(m)Ã lËql ) if i = q:(A.19)In eq. (A.19), W â lËqn â W¯ â lËqn results in (W â W¯) â (lËqn â lËqn), which equals to 1 â 0 = 1;Xâ lËqnâXÂ¯â lËqn results in (XâXÂ¯)â(lËqnâ lËqn), which equals to 1â0 = 1; lËinâ lËqnâ lËinâ lËqn resultsin (lËinâ lËin)â(lËqnâ lËqn), which equals to 0â0 = 0. Moreover, Dâ(A(m)Ã lËqn)âDâ(A(m)Ã lËqn)results in (D âD) â A(m) Ã (lËqn â lËqn), which equals to 0 â A(m) Ã 0 = 0. Consequently,we can rewrite eq. (A.19) aslix = lik â lin =8>>>><>>>>:1 if i Ì¸= qP i = hP1 if i Ì¸= qP i = jP0 if i Ì¸= qP i Ì¸= hP i Ì¸= jP0 if i = q:(A.20)Thus, the Hamming weight of lx = lkâ ln is two, which implies that in this case, the labelsof two adjacent symbols in the proposed mapping, i.e., lk and ln, are different only in twobit positions.Cvse GO dne of h or j is equvl to q. Without losing generality, we assume that h = q894cceadik 4. Cebbf fbe Cebcbfigiba 2.3and j Ì¸= q4. In this case, we can rewrite eq. (A.1) and eq. (A.2) as follows:lik =8><>:lËjk â lËqk if i Ì¸= qP i = jPlËik â lËqk if i Ì¸= qP i Ì¸= jl (lËk)âA(m)Ã lËqk if i = qP(A.21)lin =8><>:lËjn â lËqn if i Ì¸= qP i = jPlËin â lËqn if i Ì¸= qP i Ì¸= jl (lËn)âA(m)Ã lËqn if i = q:(A.22)Since lËk and lËn differ in the qth and the jth bit positions, if lËqk = W, then lËqn =¯Ëlqk = W¯, andif lËjk = X, then lËjn =¯Ëljk = X¯. Also, if i Ì¸= q and i Ì¸= j, then lËik = lËin. Moreover, since lËk andlËn differ in two bit positions, then both of them has either an odd Hamming weight or aneven Hamming weight. As a result, if l (Ëlk) = D, then l (Ëln) = D as well. Hence, byreplacing lËqk by W, lËqn by W¯, lËjk by X, lËjn by X¯, lËik by lËin, l (Ëlk) by D, and l (Ëln) by D, wecan rewrite eq. (A.21) and eq. (A.22) as follows:lik =8><>:X âW if i Ì¸= qP i = jPlËin âW if i Ì¸= qP i Ì¸= jD âA(m)ÃW if i = qP(A.23)lin =8><>:X¯ â W¯ if i Ì¸= qP i = jPlËin â W¯ if i Ì¸= qP i Ì¸= jD âA(m)Ã W¯ if i = q:(A.24)Suppose that lx = [l1xP l2xP · · · P lmx ] and lx = lk â ln such that âi; lix = lik â lin. As a result,using eq. (A.23) and eq. (A.24) we can writelix = lik â lin =8><>:X âW â X¯ â W¯ if i Ì¸= qP i = jPlËin âW â lËin â W¯ if i Ì¸= qP i Ì¸= jPD â (A(m)ÃW)âD â (A(m)Ã W¯) if i = q:(A.25)In eq. (A.25), X â W â X¯ â W¯ results in (W â W¯) â (X â X¯), which equals 1 â 1 = 0;lËinâWâ lËinâ W¯ results in (lËinâ kËin)â (Wâ W¯), which equals 0â1 = 1; Dâ (A(m)ÃW)âDâ(A(m)Ã W¯) results in (DâD)âA(m)Ã (Wâ W¯), which equals to 0â (A(m)Ã 1) = A(m).It is worth noting that since m is odd for non-square QAM, then A(m) is equal to 0.4It is importvnt to note thvt it hvs no importvnxe thvt whixh of h or j is equvl to q. Woth of them giveexvxtly the svme result.904cceadik 4. Cebbf fbe Cebcbfigiba 2.3Consequently, we can rewrite eq. (A.25) aslix = lik â lin =8><>:0 if i Ì¸= qP i = jP1 if i Ì¸= qP i Ì¸= jP0 if i = q:(A.26)As a result, the Hamming weight of lx = lk â ln is m â 2; in this case, the labels of twoadjacent symbols in the proposed mapping, i.e., lk and ln, are different only in (m â 2)bit positions. However, in a pseudo-Gray mapping of a cross QAM constellation, if S isthe set of all possible values of j and h, then the cardinality of the set S is equal to 3.Therefore, by choosing q (chosen-index) such that q Ì¸= j and q Ì¸= h, there will not be anyHamming distance of (mâ2) bits between two adjacent symbols in the proposed mappingsof a cross-QAM constellation. As a result, all two bit Hamming distances between lËk andlËn in the pseudo-Gray mapping result in a two bit Hamming distances between lk and ln inthe proposed mappings. However, using eq (A.14), when the Hamming distance betweenlËk and lËn in the pseudo-Gray mapping is one bit, there will be a Hamming distance of mbits between two adjacent symbols in the proposed mappings. This happens only whenlËk and lËn are different at the qth bit position. As a result, for the proposed cross QAMmappings, the fraction of adjacent symbols with Hamming distance m can be at most 1m ,and consequently, the fraction of adjacent symbols with Hamming distance two is largerthat (mâ1m ).91Sppwnvix Tbroox xor broposition G.HLet us define li = [eiP liâ1], where ei = [z1i P · · · P zai ], in which zji = lji , and liâ1 =[l1iâ1P · · · P l2i523aiâ1 ], in which ljiâ1 = la5ji . Similarly, we define liNk = [eiNkP liâ1Nk], whereeiNk = [z1iNkP · · · P zaiNk], in which zjiNk = ljiNk, and liâ1Nk = [l1iâ1NkP · · · P l2i523aiâ1Nk ], in which ljiâ1Nk =la5jiNk . Suppose that in step iâ 1, liâ1 and liâ1Nk are mapped to siâ1 = [s1iâ1P · · · P saiâ1] andsiâ1Nk = [s1iâ1NkP · · · P saiâ1Nk], respectively. We group all possible cases for li and liNk in 10cases (B.1) and investigate them in the following. When k â¤ N (Cvse F to Cvse L ), eiand eiNk are different only in the kth bit position. Without loss of generality, it is assumedthat in Cvse FBL, the Hamming weight of ei and eiNk is even and odd, respectively.8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:k 6 N8>>>>>>>>>>><>>>>>>>>>>>:j Ì¸= k8><>:zji = zjiNk = 0; Xasz 1zji = zjiNk = 1{i â E; Xasz 2i â O; Xasz 3j = k8>>>><>>>>:zji = z¯jiNk = 0{i â E; Xasz 4i â O; Xasz 5zji = z¯jiNk = 1{i â E; Xasz 6i â O; Xasz 7k S N8><>:wi â E; Xasz 8wi â O{sgn(xjiâ1) = sgn(xjiâ1Nk); Xasz 9sgn(xjiâ1) = âsgn(xjiâ1Nk); Xasz 10(B.1)Cvse F : k 6 N , j Ì¸= k, zji = zjiNk = 0. Using Table 5.4 we can writexji â xjiNk = sgn(xjiâ1Nk)2mâât2ââ1 (B.2)yji â yjiNk = sgn(yjiâ1Nk)2ât2ââ1Pand as a result,924cceadik 5. Cebbf fbe Cebcbfigiba 5.6|sji â sjiNk|2 = |xji â xjiNk|2 + |yji â yjiNk|2 (B.3)= 22mâ2ât2ââ2 + 22ât2ââ2= yË25:Cvse G : k 6 N , j Ì¸= k, zji = zjiNk = 1, i â E. Using Table 5.4 we can writexji â xjiNk = sgn(xjiâ1Nk)2mâât2ââ1 (B.4)yji â yjiNk = sgn(yjiâ1Nk)2ât2ââ1Pand as a result,|sji â sjiNk|2 = |xji â xjiNk|2 + |yji â yjiNk|2 (B.5)= 22mâ2ât2ââ2 + 22ât2ââ2= yË25:Cvse H : k 6 N , j Ì¸= k, zji = zjiNk = 1, i â O. Using Table 5.4, we can writexji â xjiNk = sgn(xjiâ1Nk)2mâât2ââ1 (B.6)yji â yjiNk = sgn(yjiâ1Nk)2ât2ââ1Pand as a result,|sji â sjiNk|2 = |xji â xjiNk|2 + |yji â yjiNk|2 (B.7)= 22mâ2ât2ââ2 + 22ât2ââ2= yË25:Cvse I : k 6 N , j = k, zji = z¯jiNk = 0, i â E. Using Table 5.4, we can writexji â xjiNk = (sgn(xjiâ1Nk)â 2ââj2ââ1)2mâât2ââ1 (B.8)yji â yjiNk = sgn(yjiâ1)2ât2ââ1:Since |sgn(xjiâ1Nk)â 2ââj2ââ1| â¥ 2â1, then934cceadik 5. Cebbf fbe Cebcbfigiba 5.6|sji â sjiNk|2 = |xji â xjiNk|2 + |yji â yjiNk|2 (B.9)> 22mâ2ât2 ââ4 + 22ât2 ââ2= yË24:Cvse J : k 6 N , j = k, zji = z¯jiNk = 0, i â O. Using Table 5.4, we can writexji â xjiNk = sgn(xjiâ1Nk)2mâât2ââ1 (B.10)yji â yjiNk = 2ât2ââ1(2ââjâ)2ââ1 + sgn(yjiâ1Nk))PSince |2ââ jâ)2 ââ1 + sgn(yjiâ1Nk)| â¥ 2â1, then|sji â sjiNk|2 = |xji â xjiNk|2 + |yji â yjiNk|2 (B.11)> 22mâ2ât2 ââ2 + 22ât2 ââ4= yË24:Cvse 6 : k 6 N , j = k, zji = z¯jiNk = 1, i â E. Using Table 5.4, we can writexji â xjiNk = 2mâât2ââ1(2ââj2ââ1 + sgn(xkiâ1)) (B.12)yji â yjiNk = sgn(yjiâ1Nk)2ât2ââ1:Since |2ââ j2 ââ1 + sgn(xjiâ1)| â¥ 2â1, then|sji â sjiNk|2 = |xji â xjiNk|2 + |yji â yjiNk|2 (B.13)> 22mâ2ât2 ââ4 + 22ât2 ââ2= yË24:Cvse L : k 6 N , j = k, zji = z¯jiNk = 1, i â O. Using Table 5.4 we can writexji â xjiNk = sgn(xjiâ1Nk)2mâât2ââ1 (B.14)yji â yjiNk = 2ât2ââ1(â2ââ jâ)2 ââ1 + sgn(yjiâ1Nk)):944cceadik 5. Cebbf fbe Cebcbfigiba 5.6Since | â 2ââ jâ)2 ââ1 + sgn(yjiâ1Nk)| â¥ 2â1, then|sji â sjiNk|2 = |xji â xjiNk|2 + |yji â yjiNk|2 (B.15)> 22mâ2ât2 ââ2 + 22ât2 ââ4= yË24:From Cvse F to Cvse L, we conclude that{|sji â sjiNk|2 = yË25 if j Ì¸= kP k 6 N|sji â sjiNk|2 > yË24 if j = kP k 6 NP(B.16)which results inâsi â siNkâ2 =aâj=1âsji â sjiNkâ2 (B.17)= (N â 1)yË25 + |ski â skiNk|2> (N â 1)yË25 + yË24:It is worth noting that in Cvse MBFE, k is greater than N ; therefore, ei = eiNk.Cvse M : k S N , wi â E. Let us define sji = sjiâ1 + ji and sjiNk = sjiâ1Nk + jiNk, whereji = (xji âxjiâ1)+ I(yji â yjiâ1) and jiNk = (xjiNkâxjiâ1Nk)+ I(yjiNkâ yjiâ1Nk). Using Table 5.4,for all values of i we can writejiNk = ji =8><>:0 if zji = 02mâât2âââ j2ââ2 if zji = 1P i â EâI2ât2 âââ jâ)2 ââ2 if zji = 1P i â OP(B.18)and as a result,|sji â sjiNk| = |sjiâ1 + ji â sziâ1Nk â jiNk| (B.19)= |sjiâ1 â sjiâ1Nk|:When k S N and wi â O, using Table 5.4 we can write954cceadik 5. Cebbf fbe Cebcbfigiba 5.6xji â xjiNk = (xjiâ1 â xjiâ1Nk) + (sgn(xjiâ1Nk) (B.20)âsgn(xjiâ1))2mâât2ââ1Pyji â yjiNk = (yjiâ1 â yjiâ1Nk) + (sgn(yjiâ1Nk)âsgn(yjiâ1))2ât2ââ1:Let us define âx = m â âm2 â and âxNi = (xji â xjiNk). Without loss of generality, assumethat xjiâ1 S xjiâ1Nk and therefore âxNiâ1 S 0. In Cvse N and Cvse FE, we prove that{2âxâ2 â¤ |âxNi| â¤ 3Ã 2âxâ2 if 2âxâ2 â¤ |âxNiâ1| â¤ 3Ã 2âxâ2|âxNi| = 2âxâ1 if |âxNiâ1| = 2âxâ1:(B.21)Cvse N : k S N , wi â O, sgn(xjiâ1) = sgn(xjiâ1Nk). Using (B.20) we have (xji â xjiNk) =(xjiâ1 â xjiâ1Nk) and therefore |âxNi| = |âxNiâ1|. As a result, |âxNi| possesses all the featuresof |âxNiâ1|. This indeed proves (B.21) for this case.Cvse FE : k S N , wi â O, sgn(xjiâ1) = âsgn(xjiâ1Nk). Without loss of generality, assumethat sgn(xjiâ1) = 1. From (B.20) we can writexji â xjiNk = xjiâ1 â xjiâ1Nk â 2âx P (B.22)which results in|âxNi| = ||âxNiâ1| â 2âx |: (B.23)If 2âxâ2 6 |âxNiâ1| 6 3Ã 2âxâ2, we can use (B.23) to write2âxâ2 6 |âxNi| 6 3Ã 2âxâ2: (B.24)In particular, if |âxNiâ1| = 2âxâ1, from (B.23) we have|âxNi| = 2âxâ1: (B.25)Suppose ây = âm2 â, âyNi = (yji â yjiNk), and yjiâ1 S yjiâ1Nk. Following the same approachin the above two cases, we can prove{2âyâ2 â¤ |âyNi| â¤ 3Ã 2âyâ2 if 2âyâ2 â¤ |âyNiâ1| â¤ 3Ã 2âyâ2|âyNi| = 2âyâ1 if |âyNiâ1| = 2âyâ1:(B.26)964cceadik 5. Cebbf fbe Cebcbfigiba 5.6Assume that in step (i â 1) and for j Ì¸= jâ² (jâ² â {1P · · · P N}), (i) |sjiâ1 â sjiâ1Nk| = yË5,i.e., |âxNiâ1| = 2âxâ1 and |âyNiâ1| = 2âyâ1, and (ii) for j = jâ² we have either{2âxâ2 â¤ |âxNiâ1| â¤ 3Ã 2âxâ2|âyNiâ1| = 2âyâ1(B.27)or{|âxNiâ1| = 2âxâ12âyâ2 â¤ |âyNiâ1| â¤ 3Ã 2âyâ2:(B.28)It is important to note that the above assumption is already true for step 1 (see proposition5.5) and for any step when k 6 N (see Cvse FBL ). Using (B.27) and (B.28), in step i, wehave |sji â sjiNk| = yË5 for j Ì¸= jâ²; and when j = jâ², one of the followings will be satisfied:{2âxâ2 â¤ |âxNi| â¤ 3Ã 2âxâ2|âyNi| = 2âyâ1(B.29)or{|âxNi| = 2âxâ12âyâ2 â¤ |âyNi| â¤ 3Ã 2âyâ2:(B.30)From (B.29) and (B.30), we conclude that |sjâ²i â sjâ²iNk| > yË4. As a consequence, we can write{|sji â sjiNk| = yË5 if j Ì¸= jâ²|sji â sjiNk| > yË4 if j = jâ²:(B.31)From (B.31), we haveâsi â siNkâ2 =aâj=1âsji â sjiNkâ2 (B.32)= (N â 1)yË25 + |sjâ²i â sjâ²iNk|2> (N â 1)yË25 + yË24:974cceadik 5. Cebbf fbe Cebcbfigiba 5.6Finally, using (B.17) and (B.32) it is concluded that in step i, we haveâsi â siNkâ2 > (N â 1)yË25 + yË24P (B.33)which implies that yË2minNi â¥ (N â 1)yË25 + yË24.98
Thesis/Dissertation
eng
Electrical and Computer Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Signal mapping for bit-interleaved coded modulation with iterative decoding
Text