Gallaeeru Codes for CDMA Applications Vladislav Sorokine A t hesis submit ted in conformity with the requirement s for the Degree of Doctor of Philosophy, Graduate Department of Electrical asd Computer Engineering, University of Toronto @ Copyright by Vladislav Sorokine 1998 National Library Bibliothèque nationale du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 Weilingtan Street 395, nie WelIingtan OtrawaON KlAûN4 OttawaON K1AON4 Canada canada The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Libmy of Canada to Bibliothèque nationale du Canada de reproduce, loaq distribute or sell reproduire, prêter, distriîuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la fome de microfiche/nlm, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur consewe la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor subsîantial extracts fiom it Ni Ia thèse ni des extraits substantiels rnay be printed or othenvise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. To My Teachers Gallager Codes for CDMA Applications A thesis for the degree of Doctor of Philosophy, 1998 Depart ment of Elect rical and Computer Engineering University of Toront O Abstract In this thesis, we develop a new coding system for direct sequence code-division multiple access (DS-CDMA) that supports more multiple users than many previously reported DS-CDMA systems. The most important distinctive feature of this new approach is the use of powerful short fiame Gallager codes for data error protection. These codes combine strong protection against multiple access interference (MAI) and channel noise with small data delays. This combination makes these Gallager codes an excellent choice as an error-protecting scherne in speech-oriented CDMA applications. We investigate the theoretical performance of Gallager codes with maximum- likelihood decoder and derive new block error probability bounds for short frame Gallager codes. We show that these bounds are smaller than block error probabil- ity bounds for many orthogonal and superorthogonal convolutional codes that were previously suggested for similar applications. We propose a new construction for Gallager codes that results in a significant per- formance improvement for short frame Gdager codes and hence makes them strong candidates for CD MA applications. We discuss low complexity iterative decoding al- gorithrns (suboptimal a-pos teriori probability (APP ) algorithms) for Galtager codes. We show that the decoding complexity of Gallager codes obtained in the proposed construction is lower than the decoding complexity of turbo codes and other Gallager code constructions, a feature that leads to implementation of fast software-oriented decoders for Gdager codes. We also investigate decoder hardware implementation issues t hat exploit inherent parallelism of the decoding algorit hm. We present simulation results for Gallager codes in additive white Gaussian noise (AWGN) and fading channels. We show that the strong performance of Gdager codes in AWGN channels successfUy combats multiple access interference in CDMA systems. Gallager codes &O exhibit good performance in fading channels making them particularly suitable for mobile environment. We conclude that Gallager codes are strong candidates for wireless CDMA systerns that are emerging recently as a communication system of choice for the rapidly developing PCS market. ... Ill Acknowledgement s I wish to thank my supervisor Prof. F. Kschischang for his encouragement and in- tellectual challenge in my research. His Mendly advice and participation are among major factors that made my years in the University of Toronto fruitfùl and enjoyable. I wish to thank my co-supervisor Prof. S. Pasupathy for a great many thing that I have leanied from him. He taught me how to discover unexpected connections in the research work. My philosophical outiook would not be complete without a major influence fkom Prof. S. Pasupathy. My discussions with Dr. D. MacKay fiom Cambridge University proved to be extremely fixitfd for this work. 1 would also like to thank my good Glend Prof. S. Dumant of the Mernorial Uni- versity of Newfoundland whose constant encouragement of rny work 1 greatly value. 1am indebted to Communications Group students and faculty for creating a pos- itive atmosphere in the Group. 1 drew fiom multi-cultural experience of my friends and colleagues and 1 would like to thank them for being kind and understanding. Findy, I appreciate the financial support provided by the University of Toronto Open Fellowship, Idormation Technology and Research Center Scholatship, Edward (Ted) Rogers Scholarship and Ontario Graduate Student Scholarship. Contents Abstract Acknowledgements Contents v List of Figures ... List of Tables ~1l1 1 Introduction 1 1.1 Advant ages of error control coding in CDMA .............. 1 1.2 Delays associated with FEC ....................... 3 1.3 Turbo codes, Gdager codes and iterative decoding algorithms .... 4 1.4 Organization of the thesis ........................ 5 2 Gdager codes: theory, generalizations and constructions 7 v 2.1 Background ................................ I 2.2 Notation and definitions ......................... II 2.3 Constructions for Gallager codes ..................... 14 2.4 Generator matrices for Gallager codes .................. 21 2.5 {N, -,3) Gallager codes: minimum Hamming distance and usehl bounds ................................... 22 2.6 Average probability of maximum-likelihood decoding error ...... 33 3 Iterative decoding algorithms: formulations. enhancements. imple- mentations and complexity 43 3.1 Background ................................ 43 3.2 The sum-product algorithm ....................... 47 3.3 The min-sum algorithm ......................... 49 3.4 Algorithm modifications ......................... 51 3.4.1 Computations in the log domain ................. 52 3.4.2 Re-iteration techniques ...................... 54 3.5 Software impIementation issues ..................... 59 3.6 Hardwareimplementationissues ..................... 62 3.7 Encoding and decoding comput ationd complexity ........... 71 3.7.1 Encoding ............................. 71 3.7.2 Decoding ............................. 73 4 CDMA system capacity and simulation results 4.1 Background ................................ 4.2 Combined error-control coding and DS-spreading ............................... 4.3 Capacity of a CDMA cell ......................... 4.4 Successive user cancellation ....................... 4.5 Simulation results for AWGN and fading channels ........... 4.5.1 Performance of Gdager codes in AWGN channels ....... 4.5.2 Performance of Gdager codes in fully interleaved flat Rayleigh- fading channels .......................... 4.5.3 Decoder complexity ........................ 5 Summary of the thesis and future work 92 5.1 Summary of results ............................ 92 5.2 Suggestions for future research ...................... 94 A The min-sum algorithm for Gallager codes 98 References vii List of Figures 1.1 CDMA Systern model: dK(t)is the data stream of K-th user. CK is the K-th user's spreading code, encoder and decoder provide error protection against channel noise and multiple access interference. 2.1 An example of {20,3,4) Gallager code. 2.2 Construction of a general Gallager code. 2.3 Parity-check matrix of Construction 1. - . 2.4 Himination of short cycles in Construction 1. 2.5 Parity-check matrix of Construction 2. 2.6 Exponent &(A) in the upper bound on the average number of codewords of Hamming weight 1 in an {N,-, 3) code. - . - . 2.7 Exponent B2(4in the upper bound on the average nurnber of codewords of Hamrning weight 1 in an {N,-,3) code. - . 2.8 Dependence of function &(A) on the parameter a in (2.30): (a) a = 50. (b) a! = 100. (c) a = 150. - . - . 2.9 Dependence of function Bz(X)on the parameter a in (2.30): (a) a = 50. (b) cr = 100. (c) a = 150. - . - 2.10 Function B(X):(a) obtained by direct enurneration of sequences (Propo- sitions 4-5). (b) obtained by bounding techniques of Propositions 1-2. 2.11 Number of sequencesof weight I = 1536 x X versus A. Left graph: to see the initial growth in the number of sequences of the Hamrning weight I, only Hamming weights such that the corresponding number of codewords is fewer than 1000, have been shown. The typical minimum Hamming distance is 0.15 x 1536 = 230 for {N,-, 3) codes. Right graph: as the Hamming weight of codewords increases. the number of codewords with this Hamming weight grows exponentially. The majority of codewords in the code have a Hamming weight greater than 0.35 x 1536 = 538. ... 2.12 Upper bounds on the maximum-likelihood block error rate versus signal-te noise ratio: (-) (1536, -, 3) Gallager codes; superorthogonal codes [l]: (x) d=3, (+) d=4, (O) d=5.. ....................... 2.13 Bounds on the maximum-likelihood error probability: (-) the union bound; (O) the Gallager bounds with parameter values given in Table 1. ..... Tanner graph of Construction 2 (Fig. 2.5); 0 denotes a site s{1-9) and @ denotes a check cil-?}. ........................ Likelihoods of two possible values for each site given observed values for each site. .................................. Downstream flow of the information in the min-sum