纜線數據機的更誤解碼與渦輪等化器研究

全文

(1)國立交通大學電子工程學系電子研究所碩士班碩. 士. 論. 文. 纜線數據機的更誤解碼與渦輪等化器研究. Research in Error-Correction Decoding and Turbo Equalization for Cable Modem Receivers. 研究生：劉明瑋指導教授：林大衛博士. 中華民國九三年六月.

(2) 纜線數據機的更誤解碼與渦輪等化器研究. Research in Error-Correction Decoding and Turbo Equalization for Cable Modem Receivers 研究生：劉明瑋. S t u d en t：Ming-Wei Liu. 指導教授：林大衛博士. Advisor：Dr. David W. Lin. 國立交通大學電子工程學系. 電子研究所碩士班碩士論文. A Thesis Submitted to Institute of Electronics College of Electrical Engineering and Computer Science National Chiao Tung University in Partial Fulfillment of Requirements for the Degree of Master of Science in Electronics Engineering June 2004 Hsinchu, Taiwan, Republic of China. 中華民國九三年六月.

(3) 纜線數據機的更誤解碼與渦輪等化器研究. 研究生: 劉明瑋. 指導教授: 林大衛博士. 國立交通大學電子工程學系電子工程研究所摘要由於高速傳輸需求的增加新的科技遂興起，一種基於正交調幅調變器 (quadrature amplitude modulation)纜線數據機(cable modem)的網路標準ITU-T J.83B 產生了。本論文研究基於 ITU-T J.83B 主要包含兩個方面：更誤解碼與渦輪等化器。 ITU-T J.83B 的前置更誤碼主要包含四個層次，分別為里德 - 所羅門碼 (Reed-Solomon code)，交錯器(interleaver)，隨機性發生器(randomizer)，和格狀編碼調變器(trellis-coded modulation)。我們首先分別對於每個層次分析它們的特性。對於里德-所羅門碼，我們得到它的錯誤檢查矩陣和理論錯誤率。對於格狀編碼調變器，我們解釋它具有的旋轉不變性，也藉由既有的編碼架構展延出 1024-正交調幅調變器。此外，我們透過其穿孔迴旋碼(punctured convolutional code)的重量分配頻譜得到她的理論錯誤率。最後，我們在模擬里德-所羅門碼和在白色高斯雜訊通道下的格狀編碼調變器，這些結果和分析值在高的訊號雜訊比下是相當接近的。關於渦輪等化器，我們結合格狀編碼調變器和使用最小平均平方演算法(least mean square)及多模量演算法(multi-modulus algorithm)的等化器得到兩種渦輪等化器的演算法。我們在纜線的通道下模擬而結果證明了使用軟性輸出的維特比解碼器(soft-output Viterbi decoding)可以獲得比傳統維特比解碼器多 0.2 dB 增益。. i.

(4) Research in Error-Correction Decoding and Turbo Equalization for Cable Modem Receivers. Student: Ming-Wei Liu. Advisor: Dr. David W. Lin. Department of Electronics & Institute of Electronics National Chiao Tung University. Abstract Due to the demand for high data transmission rate, a cable modem networks standard, ITU-T J.83B, based on QAM (quadrature amplitude modulation) technique is produced. This work mainly contains two parts established on ITU-T J.83B: the error correction decoding and turbo equalization for cable modem receivers. For forward error correction, we analyze the property of the coding schemes respectively containing Reed-Solomon code, interleaver, randomizer and trellis-coded modulation (TCM). As to the extended RS code, we derive its parity check matrix, symbol error rate bound. For the TCM we explain the rotational invariance characteristic of the QAM mapper, and also expand the exiting QAM modes scheme to 1024-QAM. Besides, we acquire its upper bound through the weight distribution spectrum of the punctured convolutional code and modulation mapping. Finally, we simulate the RS code and the TCM under AWGN, and the analytic results are close to the simulation. Concerning the turbo equalization, we combine the TCM and decision feedback equalizer using least mean square and multi-modulus algorithm to obtain two turbo equalization algorithms. We simulate these algorithms in the cable modem channel, and the results show that soft-output Viterbi decoding gains more than Viterbi algorithm in the decoding of TCM by 0.2 dB.. ii.

(5) 誌謝本論文承蒙恩師林大衛教授細心的指導與教誨，方得以順利完成。在研究所生涯的兩年中，林教授不僅在學術研究上給予學生指導，在研究態度亦給許相當多的建議，在此對林教授獻上最大的感謝之意。此外，感謝洪昆健學長平常能一起討論，並且在我遇到困難時熱心的給予幫助。也要感謝陳繼大學長在程式設計方面不吝指教；吳俊榮學長、林郁男學長與我分享研究與生活方面的經驗；還有筱晴、宗書、盈縈、明哲、建統、仰哲、岳賢等同學、學弟妹與我彼此砥礪、互相討論，讓兩年的研究生涯充滿歡樂與回憶。同時也要感謝朋友們能包容我不斷的多嘴，在生活中給予支持，使我能在辛苦的研究過程中保持心靈的平衡。最後，要感謝我的家人。如果我能有任何一絲絲成就，你們的栽培與鼓勵是這一切的基石。要感謝的人很多，僅以這篇論文獻給在研究所生涯中難忘的人，謝謝。. 劉明瑋民國九十三年六月於新竹. iii.

(6) Contents 1 Introduction. 1. 1.1. Scope of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2. Organization of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . .. 2. 2 Introduction to Cable Modem System. 3. 2.1. Cable Modem System . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2.2. Downstream Cable Modem Channel Characteristics . . . . . . . . . . . .. 3. 2.2.1. Transmission Loss . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2.2.2. Multipath Reflection . . . . . . . . . . . . . . . . . . . . . . . .. 4. Channel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 2.3. 3 Algorithm of FEC Encoder 3.1. 11. Reed-Solomon Encoder . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11. 3.1.1. Reed-Solomon Encoder in ITU-T J.83B . . . . . . . . . . . . . .. 12. 3.2. Interleaver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13. 3.3. Randomizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14. 3.4. Trellis-coded Modulation . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. 3.4.1. 64-QAM Modulation Mode . . . . . . . . . . . . . . . . . . . .. 15. 3.4.2. 256-QAM Modulation Mode . . . . . . . . . . . . . . . . . . . .. 15. 3.4.3. 1024-QAM Modulation Mode . . . . . . . . . . . . . . . . . . .. 17. 3.4.4. Binary Convolutional Coder . . . . . . . . . . . . . . . . . . . .. 17. 3.4.5. Rotationally Invariant Pre-coder . . . . . . . . . . . . . . . . . .. 18. 3.4.6. QAM Mapper . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. iv.

(7) 3.4.7. Rotational Invariance . . . . . . . . . . . . . . . . . . . . . . . .. 4 Algorithm of FEC Decoder 4.1. 19 27. Decoding of Trellis-Coded Modulation . . . . . . . . . . . . . . . . . . .. 27. 4.1.1. Decoding Architecture . . . . . . . . . . . . . . . . . . . . . . .. 27. 4.1.2. Decoding of Differential Precoder . . . . . . . . . . . . . . . . .. 27. 4.1.3. Decoding of PBCC . . . . . . . . . . . . . . . . . . . . . . . . .. 28. 4.1.4. Symbol Error Rate and Bit Error Rate . . . . . . . . . . . . . . .. 30. 4.2. De-randomizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36. 4.3. De-interleaver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36. 4.4. Reed-Solomon decoder . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37. 4.4.1. Extended RS codes . . . . . . . . . . . . . . . . . . . . . . . . .. 37. Concatenated Coding Simulation Results . . . . . . . . . . . . . . . . .. 41. 4.5. 5 Turbo Equalization. 44. 5.1. Principle of the Transmission Scheme . . . . . . . . . . . . . . . . . . .. 45. 5.2. Turbo-Equalizer with Viterbi Decoder . . . . . . . . . . . . . . . . . . .. 46. 5.2.1. Equalizer Structure . . . . . . . . . . . . . . . . . . . . . . . . .. 48. 5.2.2. Adaptatiive Algorithm . . . . . . . . . . . . . . . . . . . . . . .. 49. 5.2.3. Channel Decoder . . . . . . . . . . . . . . . . . . . . . . . . . .. 58. 5.2.4. Simulation Results of Turbo Equalizer with Viterbi Decoder . . .. 62. Turbo Equalizer with Soft-input Soft-output Decoder . . . . . . . . . . .. 65. 5.3.1. Symbol to Binary Converter (SBC) . . . . . . . . . . . . . . . .. 66. 5.3.2. Soft-input Soft-output (SISO) Channel Decoder . . . . . . . . . .. 66. 5.3.3. Introduction to SOVA Algorithm . . . . . . . . . . . . . . . . . .. 67. 5.3.4. Binary to Symbol Converter (BSC) . . . . . . . . . . . . . . . .. 69. 5.3.5. Simulation Results of Turbo Equalizer with SOVA Decoder . . .. 70. Turbo Equalizer Conclusion . . . . . . . . . . . . . . . . . . . . . . . .. 71. 5.3. 5.4. 6 Conclusion and Future Work. 73. v.

(8) List of Tables 2.1. Maximum Loss for Drop Cable (dB/100ft at 68F) with Different Cable Diameters and Frequency . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2.2. Multipath Echo Model According to DOCSIS 1.0 Standard . . . . . . . .. 6. 2.3. Cable Propagation Models . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 3.1. Differential Encoding Rule for QAM. . . . . . . . . . . . . . . . . . . .. 22. 4.1. Weight Distribution of the PBCC in TCM . . . . . . . . . . . . . . . . .. 34. 5.1. Conditions of Simulation 0.1 . . . . . . . . . . . . . . . . . . . . . . . .. 51. 5.2. Summary of Blind Algorithms . . . . . . . . . . . . . . . . . . . . . . .. 55. 5.3. Conditions of Simulation 1.1 . . . . . . . . . . . . . . . . . . . . . . . .. 57. 5.4. Conditions of Simulation 2.1 . . . . . . . . . . . . . . . . . . . . . . . .. 62. 5.5. Conditions of Simulation 3.1—Turbo Equalizer . . . . . . . . . . . . . .. 64. 5.6. Complexity Comparison of MAP and SOVA Algorithms Implemented in Turbo Equalization (from [22]) . . . . . . . . . . . . . . . . . . . . . . .. 67. 5.7. Conditions of Simulation 4.1.—Turbo Equalizer. . . . . . . . . . . . . .. 70. 5.8. Conditions of Simulation 4.2.—Turbo Equalizer. . . . . . . . . . . . . .. 72. vi.

(9) List of Figures 2.1. A cable plant model. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.2. A drop configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 2.3. Magnitude and impulse responses of filtered CH0. . . . . . . . . . . . . .. 10. 3.1. Layers of the FEC in J.83B. . . . . . . . . . . . . . . . . . . . . . . . . .. 12. 3.2. Interleaving functional block diagram [1, p. 22]. . . . . . . . . . . . . . .. 14. 3.3. Randomizer (7-bit symol scrambler [1, p. 25]. . . . . . . . . . . . . . . .. 15. 3.4. TCM encoder for 64-QAM [1, p. 26]. . . . . . . . . . . . . . . . . . . .. 16. 3.5. TCM encoder for 256-QAM [1, p. 28]. . . . . . . . . . . . . . . . . . . .. 16. 3.6. Proposed TCM encoder for 1024-QAM. . . . . . . . . . . . . . . . . . .. 17. 3.7. Punctured binary convolutional encoder. . . . . . . . . . . . . . . . . . .. 18. 3.8. Differential precoder. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. 3.9. 64-QAM constellation [1, p. 32]. . . . . . . . . . . . . . . . . . . . . . .. 20. 3.10 256-QAM constellation [1, p. 33]. . . . . . . . . . . . . . . . . . . . . .. 21. 3.11 Symbol map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. 3.12 256-state trellis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23. 3.13 256-state trellis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23. 3.14 Mapping of coded bit along in-phase axis of 64-QAM mapper. . . . . . .. 24. 3.15 Mapping of uncoded bits in the first quadrant of 64-QAM. . . . . . . . .. 25. 3.16 Mapping of uncoded bits in the first quadrant of 256-QAM.. 25. . . . . . . .. 3.17 Mapping of uncoded bits in other quadrants by rotating the mapping in. 4.1. the first quadrant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26. TCM decoding architecture. . . . . . . . . . . . . . . . . . . . . . . . .. 28. vii.

(10) 4.2. Performance of two 16-state decoders and one 256-state viterbi decoder in AWGN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29. 4.3. Effect on SER of Viterbi decoding depth. . . . . . . . . . . . . . . . . .. 30. 4.4. State transition diagram of example the PBCC. . . . . . . . . . . . . . .. 35. 4.5. SER performance of TCM in 64-QAM, 256-QAM, and 1024-QAM modes. 35. 4.6. BER performance of TCM in 64-QAM, 256-QAM, and 1024-QAM modes. 36. 4.7. (a) Performance of t=2 and t=3 for RS code. (b) Performance of t=3 with correct parity check symbol and ideal block code. . . . . . . . . . . . . .. 40. 4.8. 64-QAM trellis group [1, p. 27]. . . . . . . . . . . . . . . . . . . . . . .. 41. 4.9. (a) SER of 64-QAM FEC overall simulation results. (b) BER of 64-QAM FEC overall simulation results. . . . . . . . . . . . . . . . . . . . . . . .. 43. 5.1. Principle of the transmission scheme. . . . . . . . . . . . . . . . . . . .. 46. 5.2. Turbo equalizer principle. . . . . . . . . . . . . . . . . . . . . . . . . . .. 47. 5.3. Schematic diagram of module. . . . . . . . . . . . . . . . . . .. 47. 5.4. Proposed turbo equalizer structure. . . . . . . . . . . . . . . . . . . . . .. 47. 5.5. DFE structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48. 5.6. Adaptive DFE structure. . . . . . . . . . . . . . . . . . . . . . . . . . .. 50. 5.7. Simulation 0.1. LMS with variable step size. . . . . . . . . . . . . . . . .. 52. 5.8. Principle of blind algorithms. (a) RCA. (b) CMA. (c) MMA. . . . . . . .. 54. 5.9. MMA adaptive filter using DFE. . . . . . . . . . . . . . . . . . . . . . .. 56. 5.10 Simulation 1.1 Received symbols. . . . . . . . . . . . . . . . . . . . . .. 58. 5.11 Simulation 1.1 of MMA. (a) Before convergence. (b) After convergence. .. 59. 5.12 Simulation 1.1 of CMA. (a) Before convergence. (b) After convergence. .. 60. . . .. 5.13 Simulation 2.1 of MMA. (a) Case 1, input SNR = 25 dB. (b) Case 2, noise-free. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61. 5.14 SNR rise between MMA and LMS algorithms. . . . . . . . . . . . . . .. 63. 5.15 Simulation 3.1. Simultaion results of turbo equalizer with Viterbi decoder.. 65. 5.16 Schematic diagram of module. 65. . . . viii. . . . . . . . . . . . . . . . . . ..

(11) 5.17 Example of symbol to binary converter for 64-QAM. . . . . . . . . . . .. 66. 5.18 Transfor function from LLR to estimated symbol. . . . . . . . . . . . . .. 69. 5.19 Simulation 4.1. Simultaion results of turbo equalizer with SOVA decoder.. 71. 5.20 Simulation 4.1. Simultaion results of turbo equalizer with SOVA decoder.. 72. ix.

(12) Chapter 1 Introduction 1.1 Scope of the Work Due to the vigorous development of multi-media, the demand for high data transmission rate increases considerably. Since the general voice modem can not satisfies the data rate requirement, new technology like cable modem rises to offer high speed connections. Cable modem technology contains three main streams: DOCSIS, DAVIC, and IEEE 802.14, and our study is based on the DOCSIS standard (J.83B). This work studies two parts of J.83B: one is the forward error correction and the other is the turbo equalization. The forward error correction in J.83B consists of four layers as Reed-Solomon (RS) coding, interleaving, randomization, and trellis coding. We analysis the properties of each layer and then simulate by personal computer. The main challenges of the forward error correction consist in: The extend Reed-Solomon code is not a conventional Reed-Solomon code. Its single parity check byte extends the classical code by one byte. The trellis-coded modulation (TCM) is a combination of modulation and error control coding. Analysis of it requires jointly consider the contained punctured high rate convolutional code and the modulation mapper. Extend the coding scheme from the existing modulation modes to higher level mod1.

(13) ulation mode. The construction and simulation of the encoders and decoders in each layer. For turbo equalization, we first simulate the decision feedback equalizer with training based least-mean-square (LMS) algorithm and different blind equalization algorithms. Then we combine the TCM and equalizer to implement a turbo equalizer. Furthermore, we employ the soft-output Viterbi algorithm (SOVA) to substitute the conventional Viterbi algorithm and derive the improvement.. 1.2 Organization of This Thesis Organization of this thesis is: Chapter 1 is the scope of the work and the organization of this thesis. Chapter 2 introduces the cable modem system and its downstream channel characteristics. Chapter 3 briefly describes the encoding scheme of J.83B and analysis its property. Chapter 4 presents the decoding algorithms, their theoretical bounds, and simulation results. Chapter 5 contains two different turbo equalization algorithms, the simulation results and comparison. Chapter 6 sums up the conclusion and the future work.. 2.

(14) Chapter 2 Introduction to Cable Modem System 2.1 Cable Modem System With the increasing demands from users, the capacity of CATV changes from several channels to more than 100 channels. The transmission network also alters from star/tree coaxial network architecture to hybrid fiber coaxial (HFC) cell based network. The section closer to the end users and away from the head-end uses coaxial cables for signal transmission. The coaxial section architectures combination of three hierarchical system: trunk system, distribution (feeder) system and drop system. Our study concerns the coaxial section of the plant.. 2.2 Downstream Cable Modem Channel Characteristics 2.2.1 Transmission Loss The primary loss mechanisms in coaxial cables are frequency and temperature dependency of the inner conductor and the dielectric outer conductor losses. The frequency dependent transmission loss at RF frequency is influenced by the skin effect. While DC the current flows uniformly through the cross-section of the conductor, the current tends to crowd around the conductor surface as the RF current is increased. Companied with. 3.

(15) higher frequency, this effect increases impedance of a given conductor. A useful parameter is cable loss ratio (CLR) given by. . . (2.1). where and are two different frequencies. Generally speaking, cable plants are designed to operate over a wide range or temperature from -40 to 70 centigrade degree. With the temperature increases, the attenuation increases, resulting in the cable loss slowly and linearly increases [2, pp. 43–44]. Trunk and Feeder Cable To provide low RF loss, the cables with outer shield diameters ranging from 0.412 to 1.125 inches are used. Common diameters of this family of cables include 0.412, 0.500, 0.625, 0.750, 0.825, and 1.000 inch. They are generally labeled by the outer shield diameter in thousands of an inch; for example “500 cable” refers to a cable of 0.500 inch. Larger cables are used in longer distance connection, and smaller cables are used in shorter distance connection where the main losses are tap losses rather than cable losses [3, pp. 395–396] Taps are used to split the transmitted signals to drop cables. Drop Cable Drop cable has a smaller diameter than the feeder cable, and is used between the tap and subscriber home terminal. Table 2.1 shows the drop cable loss (dB/100feet) at 20 degree centigrade. The diameters of this family of drop cables include 0.240 (59 series foam), 0.272(6 series foam), 0.318 (7 series foam), and 0.395 (11 series foam) inch [2, pp. 43–44].. 2.2.2 Multipath Reflection The transmitted signal is partially reflected at where there is mismatch, resulting in prolonged channel impluse response. The reflection is due to impedance mismatch at various places along the transmission path, caused by cable imperfections and cable junctions, as 4.

(16) Table 2.1: Maximum Loss for Drop Cable (dB/100ft at 68F) with Different Cable Diameters and Frequency Frequency (MHz). 59Series Form 6Series Form 7Series Form 11Series Form. 5. 0.86. 0.58. 0.47. 0.38. 30. 1.51. 1.18. 0.92. 0.71. 40. 1.74. 1.37. 1.06. 0.82. 50. 1.95. 1.53. 1.19. 0.92. 110. 2.82. 2.24. 1.73. 1.36. 174. 3.47. 2.75. 2.14. 1.72. 220. 3.88. 3.11. 2.41. 1.96. 300. 4.45. 3.55. 2.82. 2.25. 350. 4.80. 3.85. 3.05. 2.42. 400. 5.10. 4.15. 3.27. 2.26. 450. 5.40. 4.40. 3.46. 2.75. 550. 5.95. 4.90. 3.85. 3.04. 600. 6.20. 5.10. 4.05. 3.18. 750. 6.97. 5.65. 4.57. 3.65. 865. 7.52. 6.10. 4.93. 3.98. 1000. 8.12. 6.55. 5.32. 4.35. 5.

(17) Table 2.2: Multipath Echo Model According to DOCSIS 1.0 Standard Echo Time Delay. Echo Magnitude Echo Magnitude (downstream). .

(18)

(19)

(20)

(21) . (upstream). . well as the inevitably used splitters and taps. This can be seen on an analog TV as ghosting, or it can result in a loss of the receiver synchronization in the digitally demodulated picture. Multipath reflections can be presented as:. !&% !

(22) "!$# ')(+* ,. or. ! &- "! # /' .10/23*546(7*98:. (2.2). (2.3). There are various discrete mutlipath reflection models such as IEEE 802.14 and DOCSIS 1.0 [2, p. 55] for both forward and return-path channels. While DOCSIS 1.0 model assumes a single dominant echo, the IEEE 802.14 assumes multiple echoes within the time-delay range. Table 2.2 shows the DOCSIS 1.0 model with maximum echo power and delay time. Trunk Amplifier, Bridger Amplifier, and Line Extender Trunk amplifiers, spaced 20–22 dB from one another, are moderate-gain amplifiers with a typical output of 30–36dBmV that are used to provide high CNR with low nonlinear distortion (-80 dBc) [2, p. 44]. Amplifiers are spaced in about 2000-ft depending on the bandwidth. The output power of bridger amplifier is in the range of 40–50 dBmV, but higher nonlinear distortions also exist [2, p. 44]. The nonlinearity in line extender amplifiers is also high. To reduce the effect of nonlinear distortions, a maximum of two to four 6.

(23) Fiber node. 4−port Tap. Trunk Amplifier. BridgerAmplifer Line Extender. Figure 2.1: A cable plant model. line-extender amplifiers are used, depending on the number of taps used. Typically, line extender amplifiers are spaced 120 to 350 m. A Cable Plant Model Since no two cable television companies are alike, there is no typical model. Therefore, we can suggest an architecture that fits the requirements mentioned above to be a model. The termination of fiber is a fiber node, out of which is coax distribution system extended. The distribution system has amplifiers and coax drops (typically 150 feet) that deliver the signal to home [2, p. 8] . Each fiber node can support 500–2000 households [4]. In our architecture, we pass the signal through only a few amplifiers between the headend and the home. The architecture is shown in Fig. 2.2 Each feeder distribution system consists of eight four-port taps and four feeder distribution systems are in a single leg extended from a fiber node. Besides, there are four legs in a fiber node. Therefore, this architecture can provide 512 households. We use 1’ cable to be the trunk cable which has 3.98-dB/100-m transmission loss at 750 MHz. The trunk amplifiers are spaced 550 m because there should be 20–22dB 7.

(24) Subscriber Outlet 1. Splitter at entry point Inside wiring Size 59 10−50’. Tap House drop cable Size 6 20−100’. Subscriber Outlet 2. Figure 2.2: A drop configuration. transmission loss between two trunk amplifiers. The space between two taps is 100 to 180 feet [5, pp. 305–306]. It is reasonable since in North America, many dwelling sections are 100 feet square blocks. The distance of line extenders is also reasonable for the requirement of 120 to 350 m mentioned above. 750 cables are used in feeder cables, 6-Series is used in drop cables, and 59-Series is used in home wiring.. 2.3 Channel Models Now we use the channel model shown in Fig. 2.1 to derive some channel models as shown in Table 2.3. We transmit an impulse from the transmitter and chose an outlet in home to be the receiver. The return loss of taps are assumed to be 16 dB, of outlet to be 4 dB, and the directivity is 10 dB. Since an outlet may be left unterminated, so it should be assumed that 100% reflection occurs at each subscriber outlet. Besides, the length of drop cables are different and the taps are assumed four-way 20-dB taps with through loss 1 dB [2, p. 46], [3, pp. 560-567]. Fig. 2.3 shows the frequency response and channel response of CH0 in Table 2.3. This figure is drawn by two times oversampling. And the frequency response is the equivalent low-pass signal.. 8.

(25) Table 2.3: Cable Propagation Models Magnitude Delay (ns) CH0. CH1. CH2. 0,9438. 0. 0.1263. Magnitude Delay (ns) CH3. 1. 0. 161.15. 0.1043. 210.7. 0.1699. 313.26. 0.0028. 421.4. 0.0622. -0.1696. 0.0037. 481.1. 0.0093. 511.7. 1. 0. 1. 0. 0.1411. 90.3. 0.0151. 331.1. 0.0063. 150.5. 0.0119. 391.3. 0.0255. 180.6. 0.0076. 421.4. 0.0021. 451.5. 1. 0. 0.1431. 90.3. 0.0050. 174.58. 0.0114. 180.6. 0.0052. 189.6. 0.0049. 195.65. CH4. 9.

(26) 2 Magnitude (dB). 0 −2 −4 −6 −8 −10 −12 6. 5. 4. 3 Freq (MHz). 2. 1. 0. 0.4. Magnitude. 0.3. 0.2. 0.1. 0. 0. 1. 2. 3. 4. 5 Time (ms). 6. 7. 8. Figure 2.3: Magnitude and impulse responses of filtered CH0.. 10. 9. 10.

(27) Chapter 3 Algorithm of FEC Encoder The forward error correction (FEC) definition is composed of four layers in J.83B, as shown in Fig. 3.1. The FEC section uses various types of error correction algorithms and interleaving techniques by the cable channel situation. Reed-Solomon (RS) Coding — Provides block coding and decoding. Interleaving — Disperses the symbols to protect bursty errors. Randomization — Randomizes the data from interleaver to admit effective QAM demodulator synchronization. Trellis Coding — Combines the punctured convolutional encoding and modulation.. 3.1 Reed-Solomon Encoder Due to the enormous advances in the digital techniques for computers, new digital communication systems are now rapidly replacing the former analogue systems. The ReedSolomon code is one of the most powerful techniques to ensure the completeness of transmitted or stored digital information. The Reed-Solomon codes are nonbinary codes with code symbols from a Galois field. They were discovered in 1960 by I. Reed and G. Solomon. the work was done when they were at MIT Laboratory. In the decades since their discovery, RS codes have had 11.

(28) FEC Encoding. Reed− Solomon encoder. Inter− leaver. Ran− domizer. Trellis encoder channel. Reed− Solomon decoder. De− ran− domizer. De− inter− leaver. Trellis decoder. FEC Decoding. Figure 3.1: Layers of the FEC in J.83B. countless applications from compact discs and digital TV in living room to space craft and satellite in outer space.. 3.1.1 Reed-Solomon Encoder in ITU-T J.83B In ITU-T J.83B recommendation, the MPEG-2 transport stream is Reed-Solomon (RS). code with capability of correcting up to symbol errors per block over . Although the FEC frame format differs between 64-QAM and encoded using. 256-QAM, each modulation type utilizes the same RS code.. , The primitive polynomial forms the field over A systematic encoder implements a. where.

(29)

(30). extended RS code over . .. as:. . . The generator polynomial is:. . .

(31)

(32)

(33)

(34)

(35)

(36)

(37) .

(38) ! . 12.

(39)

(40) . . (3.1).

(41) The message polynomial consists of 122, 7-bit symbols, and is described as below:. . .

(42). The message polynomial is first multiplied by mial . . ))

(43)

(44). .

(45) . . , then divided by the generator polyno-. to form a remainder, described by the following: . . .

(46) .

(47) . . .

(48) .

(49) . . . The remainder polynomail provides 5 parity symbols, and then is added to the message polynomial to form a 127-symbols codeword. The following polynomial describes the generated code word.. .

(50)

(51) !

(52) )

(53)

(54)

(55)

(56)

(57) An extended code, , is evaluated through the code word at the sixth power of .. + . (3.2). This extended code is then added to the last symbol of a transmitted Reed-Solomon block. The extended code word appears as follows: . . .

(58).

(59) )

(60). .

(61) .

(62) .

(63) .

(64) . .

(65) . .

(66) . .

(67). .

(68). . (3.3). The order of transmitted RS code block from the encoder output is constructed as (order sent is left to right):. ! ) . . . . 3.2 Interleaver Interleaving is a form of time diversity that mitigates the effects of bursty errors. Several techniques aim at reducing channel effets either by supplying block interleaving or convolutional interleaving. A channel is considered fully interleaved when consecutive symbols 13.

(69) Interleaver. De−interleaver. 1 2. 7 bits. I. 1 2. J J J. J J J J J J 1. 2. 1. 2. I−3 I−2 I−1. J J J J J J. J J J J J J. 7 bits. J J J. J J J J J J. I. I−3 I−2 I−1. Channel. Figure 3.2: Interleaving functional block diagram [1, p. 22]. are rearranged to be affected independent by channel erros and partially interleaved when consecutive symbols of the received sequence are affected by the same bursty errors. Both 64-QAM and 256-QAM employ a convolutional interleaver between the RS block coding and the randomizer. As Fig. 3.2 depicts, the interleaving commutator initially points to the top-most branch, and the RS code symbols are sequentially shift into the bank of registers. The first interleaver has zero delay, the second one has a symbol. . period of delay, the third has . symbol periods of delay, and so on.. To resist different channel conditions, users can choose only one depth of interleaving in 64-QAM. , while five alternative modes in 256-QAM.. 3.3 Randomizer Both 64-QAM and 256-QAM introduce a randomizer shown in Fig. 3.3 to uniformize the distribution in the constellation. The randomizer adds a Pseudorandom Noise (PN) sequence of 7-bit symbols over . . (i.e., bit-wise exclusive-OR) to the symbol from. the convolutional interleaver. The randomizer is initialized as pre-loading to the “all-ones” state during the FEC frame trailer, and is enabled at the first data symbol. A linar feedback shift register is applied to specify a follows:. 14. . polynomial defined as.

(70) −1. −1. Z. −1. Z. 7. 7 Data out. Z 7. α. Data In. 3. Figure 3.3: Randomizer (7-bit symol scrambler [1, p. 25].. where.

(71)

(72) .

(73)

(74). . 3.4 Trellis-coded Modulation 3.4.1 64-QAM Modulation Mode For 64-QAM, the input of the trellis coded modulator is a 28-bits sequence labeled in pairs of A symbols and B symbols as shown in Fig. 3.4. As depicted, all 28 bits are assigned to a trellis group of five 6-bits QAM symbols and therefore the overall rate of the TCM code. . is . The TCM encoding can be approximately divided into three parts which will be discussed in further detail later. The first part is the differential precoder used to produce . . . rotational invariance, the second part is a rate- binary convolutional encoder with . puncturing (PBCC), and the third part is the QAM mapper.. 3.4.2 256-QAM Modulation Mode 256-QAM modulation mode employs a similar trellis coding as 64-QAM with the same PBCC. As description in Fig. 3.5, all 38 bits are assigned to a trellis group of five 8-bits. . QAM symbols and the overall rate of the TCM code is therefore .. 15.

(75) Time. Uncoded MSBs of ’A’ A13,A11,A8,A5,A2 C5 MSBs of ’B’. A12,A10,A7,A4,A1 C4 B13,B11,B8,B5,B2 C2. 28 bits. B12,B10,B7,B4,B1. C1. Coded LSB of ’A’. A9,A6,A3,A0. QAM mapper 64−QAM. U5,U4,U3,U2,U1. W. X. PBCC. C3. output. Differential Precoder LSB of ’B’. B9,B6,B3,B0. Z. Y. PBCC. V5,V4,V3,V2,V1 C0. Figure 3.4: TCM encoder for 64-QAM [1, p. 26].. Time. Uncoded MSBs of ’A’. A18,A15,A11,A7,A3 C7 A17,A14,A10,A6,A2 C6 A16,A13,A9,A5,A1 C5. MSBs. B18.B15,B11,B7,B3. of ’B’. B17,B14,B10,B6,B2. 38 bits. C3 C2. B16,B13,B9,B5,B1. C1. Coded LSB of ’A’. A12,A8,A4,A0. QAM mapper 256−QAM. U5,U4,U3,U2,U1. W. X. C4. PBCC. Differential Precoder LSB of ’B’. B12,B8,B4,B0. Z. PBCC. Y. V5,V4,V3,V2,V1. Figure 3.5: TCM encoder for 256-QAM [1, p. 28].. 16. C0. output.

(76) Time. Uncoded A23,A19,A14,A9,A4. C9. A22,A18,A13,A8,A3 C8 MSBs of ’A’. A21,A17,A12,A7,A2 C7 A20,A16,A11,A6,A1 C6 B23,B19,B14,B9,B4. MSBs. B22,B18,B13,B8,B3. of ’B’. B21,B17,B12,B7,B2. 48 bits. C4 C3 C2. B20,B16,B11,B6,B1. C1. Coded LSB of ’A’. QAM mapper. A15,A10,A5,A0. W. PBCC. X. 1024−QAM. U5,U4,U3,U2,U1. C5. output. Differential Precoder LSB of ’B’. B15,B10,B5,B0. Z. PBCC. Y. V5,V4,V3,V2,V1. C0. Figure 3.6: Proposed TCM encoder for 1024-QAM.. 3.4.3 1024-QAM Modulation Mode Although ITU-T J.83B recommendation only support 64-QAM and 256-QAM modulation modes, we can develop a 1024-QAM modulation mode from its regular coding rule of the two existing modes for further study. The 1024-QAM mode is shown in Fig. 3.6. We label and order the input sequence in pairs of A symbols and B symbols like those in 64-QAM and 256-QAM modes, and 48 bits are group into five 10-bits QAM symbols.. . The overall rate of the 1024-QAM modulation mode TCM codes are . Furthermore, we ultilize the same differential precoder and the same PBCC.. 3.4.4 Binary Convolutional Coder. . . The TCM includes a rate- binary convolutional code based on a rate- convolutional. . encoder to introduce redundancy into the LSBs of the trellis group. The rate- convolutional encoder is a 16-state encoder with generator =[25,37] (octal) and puncturing. matrix. =[0001;1111] (where “0” denotes punctured bit positions). The structure of the 17.

(77) G1 : 25 (octal). + D. Puncture Matrix. D. D. 0001. D. 1111. + G2 : 37 (octal). Figure 3.7: Punctured binary convolutional encoder. punctured binary convolutional coder is shown in Fig. 3.7. We notice from Figs. 3.4, 3.5, and 3.6 that there are two separate PBCC coding branches. Therefore, we can treat the QAM as two separate PAMs. According to this characteristic, the complexity of analysis can be reduced to two of a. . . . -state trellises instead. -state trellis. The simulation proof will be discussed in next chapter.. 3.4.5 Rotationally Invariant Pre-coder Both 64- and 256-QAM modulation use the differential pre-coder to cause the trellis coded modulation rotationally invariant. With the differential precoder, the information is carried in the phase change rather than the absolute phase. The LSB of the in-phase and the quadrature components are differentially encoded as follows:.

(78) . . .

(79). .

(80) .

(81) . . . .

(82) .

(83) . . . .

(84)

(85) . . (3.4). . (3.5). where the subscripts are time indexes, as illustrated in Fig. 3.8. In 64-QAM constellation, as shown in Fig. 3.9, constellation, in Fig. 3.10,. . and are differentially encoded, while in 256-QAM and are encoded instead.. 18.

(86) Wj. Xj Differential Precoder. Zj. Yj. Figure 3.8: Differential precoder.. 3.4.6 QAM Mapper To implement rotationally invariant TCM, we use a certain mapping pattern [9], [10]. The way to map has two steps. From Figs. 3.4 and 3.5, we see that, for each QAM symbol, only two bits are encoded while the rest are not. Hence, we map the coded and the uncoded bits separately. The coded bits are mapped according to set partitioning with two levels, which results in the enlargement of the minimum free distance [9], [10]. Secondly, the uncoded bits are mapped with rotational invariance. We can extend those regularities above to extend to larger constellations, like 1024QAM.. 3.4.7 Rotational Invariance Due to the phase ambiguity in the signal space, it is desirable to design the code to be transparent to signal element rotations. The trellis-coded modulation in J.87B ingeniously involves this characteristic called rotational invariance. This characteristic is useful in blind equalization and will be discussed in next chapter. We analysis the composition elements of rotational invariance in the following discussion. Differential Precoder The physical meaning of (3.4) and (3.5) can be understood with the help of Fig. 3.11 and Table 3.1. Table 3.1 and Fig. 3.11 explain the relationship between the input bit pair and the corresponding phase change from the previous symbol. For instance, let , and let the current input bit pair the previous output bit pair be be . Then form Table 3.1, the phase rotate clockwise and obtain the current output bit pair as . By this, an error caused by a carrier phase. . . . . . . . . . rotation will not propagate. 19. .

(87) Q. 110 111. 111 011. 010 111. 011 011. 100 101. 101 111. 110 101. 111 111. 110 100. 111 000. 010 100. 011 000. 100 000. 101 010. 110 000. 111 010. 101 011. 000 111. 001 011. 000 101. 001 111. 010 101. 011 111. 100 100. 101 000. 000 100. 001 000. 000 000. 001 010. 010 000. 011 010. 010 011. 011 001. 000 011. 001 001. 000 001. 001 101. 100 001. 101 101. 010 110. 011 100. 000 110. 001 100. 000 010. 001 110. 100 010. 101 110. 100 011. 111 001. 100 011. 101 001. 010 001. 011 101. 110 001. 111 101. 110 110. 111 100. 100 110. 101 100. 010 010. 011 110. 110 010. 111 110. 100 111. C5C4C3 C2C1C0. I. Figure 3.9: 64-QAM constellation [1, p. 32].. 20.

(88) Q 1110. 1111. 1110. 1111. 1110. 1111. 1110. 1111. 1111. 1101. 1011. 1001. 0111. 0101. 0011. 0001. 1111. 1111. 1100. 1101. 1100. 1101. 1100. 1101. 1100. 1101. 0000. 0011. 1110. 1100. 1010. 1000. 0110. 0100. 0010. 0000. 1100. 1100. 1010. 1011. 1010. 1011. 1010. 1011. 1010. 1011. 0000. 0011. 1111. 1101. 1011. 1001. 0111. 0101. 0011. 0001. 1011. 1000. 1001. 1000. 1001. 1000. 1001. 1000. 1001. 0000. 1110. 1100. 1010. 1000. 0110. 0100. 0010. 0000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 0110. 0111. 0110. 0111. 0110. 0111. 0110. 0111. 0000. 0011. 0100. 0111. 1000. 1011. 1100. 1111. 1111. 1101. 1011. 1001. 0111. 0101. 0011. 0001. 0111. 0111. 0111. 0111. 0111. 0111. 0111. 0111. 0100. 0101. 0100. 0101. 0100. 0101. 0100. 0101. 0000. 0011. 0100. 0111. 1000. 1011. 1100. 1111. 1110. 1100. 1010. 1000. 0110. 0100. 0010. 0000. 0100. 0100. 0100. 0100. 0100. 0100. 0100. 0100. 0010. 0011. 0010. 0011. 0010. 0011. 0010. 0011. 0000. 0011. 0100. 0111. 1000. 1011. 1100. 1111. 1111. 1101. 1011. 1001. 0111. 0101. 0011. 0001. 0011. 0011. 0011. 0011. 0011. 0011. 0011. 0011. 0000. 0001. 0000. 0001. 0000. 0001. 0000. 0001. 0000. 0011. 0100. 0111. 1000. 1011. 1100. 1111. 0000. 0011. 0100. 1011. 1100. 1000. 1111. 1111. 1111. 1111. 1111. 1111. 0100. 0111. 1000. 1011. 1100. 1111. 1100. 1100. 1100. 1100. 1100. 1100. 0100. 0111. 1000. 1011. 1100. 1111. 1011. 1011. 1011. 1011. 1011. 1011. 1011. 0011. 0100. 0111. 1000. 1011. 1100. 1111. 1110. 1100. 1010. 1000. 0110. 0100. 0010. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 1110. 1101. 1010. 1001. 0110. 0101. 0010. 0001. 0000. 0001. 0000. 0001. 0000. 0001. 0000. 0001. 0001. 0001. 0001. 0001. 0001. 0001. 0001. 0001. 0001. 0011. 0101. 0111. 1001. 1011. 1101. 1111. 1110. 1101. 1010. 1001. 0110. 0101. 0010. 0001. 0010. 0011. 0010. 0011. 0010. 0011. 0010. 0011. 0010. 0010. 0010. 0010. 0010. 0010. 0010. 0010. 0000. 0010. 0100. 0110. 1000. 1010. 1100. 1110. 1110. 1101. 1010. 1001. 0110. 0101. 0010. 0001. 0100. 0101. 0100. 0101. 0100. 0101. 0100. 0101. 0101. 0101. 0101. 0101. 0101. 0101. 0101. 0101. 0001. 0011. 0101. 0111. 1001. 1011. 1101. 1111. 1110. 1101. 1010. 1001. 0110. 0101. 0010. 0001. 0110. 0111. 0110. 0111. 0110. 0111. 0110. 0111. 0110. 0110. 0110. 0110. 0110. 0110. 0110. 0110. 0000. 0010. 0100. 0110. 1000. 1010. 1100. 1110. 1110. 1101. 1010. 1001. 0110. 0101. 0010. 0001. 1000. 1001. 1000. 1001. 1000. 1001. 1000. 1001. 1001. 1001. 1001. 1001. 1001. 1001. 1001. 1001. 0001. 0011. 0101. 0111. 1001. 1011. 1101. 1111. 1110. 1101. 1010. 1001. 0110. 0101. 0010. 0001. 1010. 1011. 1010. 1011. 1010. 1011. 1010. 1011. 1010. 1010. 1010. 1010. 1010. 1010. 1010. 1010. 0000. 0010. 0100. 0110. 1000. 1010. 1100. 1110. 1110. 1101. 1010. 1001. 0110. 0101. 0010. 0001. 1100. 1101. 1100. 1101. 1100. 1101. 1100. 1101. 1101. 1101. 1101. 1101. 1101. 1101. 1101. 1101. 0001. 0011. 0101. 0111. 1001. 1011. 1101. 1111. 1110. 1101. 1010. 1001. 0110. 0101. 0010. 0001. 1110. 1111. 1110. 1111. 1110. 1111. 1110. 1111. 1110. 1110. 1110. 1110. 1110. 1110. 1110. 1110. 0000. 0010. 0100. 0110. 1000. 1010. 1100. 1110. Figure 3.10: 256-QAM constellation [1, p. 33].. 21. C7C6C5C4 C3C2C1C0. 1111. 0111. I.

(89) Table 3.1: Differential Encoding Rule for QAM . Phase Change from Previous Symbol. . . . . Xj Yj 00. 10. 01. 11 Figure 3.11: Symbol map. PBCC To see why the two PBCCs can satisfy rotational invariance, consider a part of the. !. . -state. trellis of the encoder in Figs. 3.12, 3.13 In Fig. 3.13,. . . ) 9 . . , represent the state of the convolutional encoder. Let. the current transmitted signal be A and the next signal be C as shown in the figure. If the. , then they will become B and D instead as # 9 shown in Fig. 3.12. The code is rotationally invariant because for each path 9 of -rotated symbols through the trellis [7, p. there exists a valid path. received signals are rotated clockwise by. . . . . 463]. QAM Mapper To analyze the mapping, we first disregard the uncoded bits and examine the in-phase axis of the QAM mapper given in (3.9) and (3.10), from [1, pp. 32, 33]. The coded bits alternates in the fashion shown in Fig. 3.14. By this mapping methodology, the minimum. 22.

(90) A D. C. B. Figure 3.12: 256-state trellis.. A S0 ....S3 S4....S7. C theta=−pi/2. theta=−pi/2. D. _ _ S4....S7 S0... .S3. B. Figure 3.13: 256-state trellis.. 23.

(91) 0. 1. 0. 1. 0. 1. 0. 1. Figure 3.14: Mapping of coded bit along in-phase axis of 64-QAM mapper. intra-set (within set of 0s or set of 1s) distance is enlarged, which is important considering the spirit of set partitioning. The quadrature axis is the same. Consequently, by the binary convolutional encoding, we can make the coded signals have a larger minimum free distance than the original so as to reduce the symbol error rate at the same SNR. Now we disregard the coded bits and examine the mapping of the ncoded bits. In the first quadrant of the 64-QAM mapper shown in (3.9), we can discern a pattern of bit mapping consisting of four symbols as a unit and four units organized a quadrant, depicted in Fig. 3.15. As presented in the figure, those uncoded bits are arranged in a regular form which could be used to understand the regularity of the mapping of 256-QAM in Fig. 3.10. In 256-QAM constellation, a similiar mapping rule is used as shown in Fig. 3.16. Different from 64-QAM mapper, the bits arranged in the same unit is not both As or Bs in Fig. 3.5 while in 64-QAM both are. But how to make the mapping rotationally invariant? For this, if the constellation is rotated by. . ,. an integer, the mapping must appear like it is not rotated. For the coded. bits, the differential precoder is used to achieve rotational invariance. For the uncoded . bits, we just rotate the bits mapping in the first quadrant by. counterclockwise to get. mapping in the th quadrant, as shown in Fig. 3.17. (For convenience, we have used decimal numbers in place of binary numbers in this figure, as the MSB. . to LSB.) By this, if the constellation is rotated by. . , where. . . is any integer, the mapping. of the uncoded bits are still the same as if it is not rotated. Note that the mapping in the four-symbol unit follows Gray coding. In addition, each partitioned subset in the second level, or third level in 256-QAM, has Gray-coded bit mapping. Therefore, the intra-set or inter-set symbols errors correspond to a single bit error, which is the minimum amount of bit erros possible.. 24.

(92) C2C1 C5C4. 10. 11. 00. 01. 10. 11. 00. 01. Figure 3.15: Mapping of uncoded bits in the first quadrant of 64-QAM.. C5C1 C7C3. 01 11 00 10. 01. 01. 11. 11 00. 10 C6C2. 00. 10. Figure 3.16: Mapping of uncoded bits in the first quadrant of 256-QAM.. 25.

(93) 15. 13. 7. 5. 10. 11. 14. 15. 14. 12. 6. 4. 8. 9. 12. 13. 11. 9. 3. 1. 2. 3. 6. 7. 10. 8. 2. 0. 0. 1. 4. 5. 5. 4. 1. 0. 0. 2. 8. 10. 7. 6. 3. 2. 1. 3. 9. 11. 13. 12. 9. 8. 4. 6. 12. 14. 15. 14. 11. 10. 5. 7. 13. 15. Figure 3.17: Mapping of uncoded bits in other quadrants by rotating the mapping in the first quadrant.. 26.

(94) Chapter 4 Algorithm of FEC Decoder 4.1 Decoding of Trellis-Coded Modulation 4.1.1 Decoding Architecture The TCM decoder in Fig. 4.1 is used in the discussion below. The demodulator produces soft in-phase and quadrature outputs from the received siganls through the cable network. Before the signals are fed into the rate-1/2 Viterbi decoder, each branch will be depuntured with depunturing matrix: [0001:1111] (“0” means no transmission, and “1” denotes transmission). Those decoded bits, the LSBs of the two branches, are encoded and punctured again to map the estimated received symbol through a mapping table. The mapping table uses the punctured signals and (suboptimal) four symbls nearest to the received symbol to decide those uncoded bits. The differential decoder is used to decode those LSBs.. 4.1.2 Decoding of Differential Precoder. . . . . The differential precoder operats on the pairs and to obtain an estimate on the transmitted pair . Therefore, the pairs and can be used to derive as (4.1) and (4.2).. . . . . . .

(95)

(96)

(97) . 27. . .

(98)

(99) . . . . . . (4.1).

(100) Viterbi. Encoder. Punc. Viterbi. Encoder. Punc. Depunc. Demod. Differential Decoder. Table. Figure 4.1: TCM decoding architecture. . .

(101)

(102) . .

(103). .

(104) . .

(105) . +. (4.2). 4.1.3 Decoding of PBCC From Figs. 3.4, 3.5, and 3.6, we notice that there are two separate PBCC coding branches. Therefore, we can treat the QAM encoding as two separate PAMs. According to this characteristic, the complexity of analysis can be reduced to two of a. . . . -state trellises instead. -state trellis.. Viterbi Decoder The Viterbi algorithm was proposed in 1967 for decoding convolutional codes. Our application of the Viterbi algorithm to decode TCM is based on the trellis diagram representing tht TCM encoder. In the presence of parallel transitions, that is, when more than one signal is associated with the transition from one trellis state to the next, a minor modification should be made in the algorithm. In this situation, the branch metric computation is based on the preliminary detection of the symbol in the branch subconstellation that lies closest to the received signal. Its metric can be used thereafter for that branch, and the Viterbi algorithm can proceed conventionally. In this situation we have a one-to-one correspondence between the sequence of source symbols and the path through the trellis. Thanks to the set partitioning, the nearest Euclidean distance of the same class is twice the minimum distance of those without set partitioning and hence the degradation of the performance is little compared with the influence of the complexity of convolutional code decoder. 28.

(106) 0. 10. two 16−state viterbi decoders one 256−state viterbi decoder. −1. 10. −2. SER. 10. −3. 10. −4. 10. −5. 10. 8. 8.5. 9. 9.5. 10. 10.5 Eb/N0. 11. 11.5. 12. 12.5. 13. Figure 4.2: Performance of two 16-state decoders and one 256-state viterbi decoder in AWGN. According to the TCM encoder, the LSBs of the two branches are encoded separately with convolutional encoding, hence we may decode TCM by one Viterbi decoder which decode two branches together. However, decode the two brances takes 256-states Viterbi decoder introducing vast complexity, so we try to decode the two branches with two soft decision Viterbi decoders with 16-states each separately as shown in Fig. 4.1. The simulation results in Fig. 4.2 depict similar performance, which implies the possibility to use two one-dimensional Viterbi decoder with 16 states each instead of one two-dimensional Viterbi decoder with 256 states. Viterbi Decoder Length Usually the Viterbi decoder length of 5.8 times the constraint length is enough. However, the punctured convolutional codes may require longer length for its puncture. Therefore, we simulation with different lengths and find a suitable one. Fig. 4.3 shows the error performance of the TCM with 64-QAM using various decoding depths by two one29.

(107) 0. 10. length 144 length 72 length 36 length 16 length 4. −1. 10. −2. SER. 10. −3. 10. −4. 10. −5. 10. −6. 10. 9. 8. 10. 12. 11. 13. 14. 15. Eb/N0. Figure 4.3: Effect on SER of Viterbi decoding depth. dimensional Viterbi decoders. In the figure, the depth means the numbers of de-punctured symbols pairs with 2 bits in a symbol. For example, since the convolutional encoder is a rate-1/2 coder, there should be 144 symbols in a depth-72 decoder. However, due to the puncturation, there are only 90 received symbols in depth-72 decoder and the rest 54 symbols are inserted with don’t cares. Based on the simulation results, a decoder of depth-72 is considered sufficient to achieve good performance.. 4.1.4 Symbol Error Rate and Bit Error Rate Since the two LSBs partition the QAM constellation into four subsets, we examine the coding gain of each separately. For memoryless channels an union bound on the bit error probability of a convolutional code can be obtained. The bound could be derived from the transfer function. . . . of the code which describes the weight distribution, or wight. spectrum, of the incorrect code words and the number of bit errors on the path [11]. Here,. represents the number of “1” source bits and. . means the number of “1” coded bits.. Punctured codes are considered high-rate codes and hence their performance can be 30.

(108) derived from the trellis codes of the high-rate code, whose nodes depths are multiple of b for a rate-b/v code [11]. The transfer function of a convolutional code may be evaluated by solving equations describing the transitions between those states. In ITU-T J.83B, the punctured convolutional code is of the rate 4/5, memory-4 punctured code with generator. . 2 . and puncture matrix [P1,P2] = [0 0 0 1, 1 1 1 1] (“0” denotes no trans-. mission, “1” denotes transmission). The self-loop at state zero is eliminated by splitting that state into two states and . The rest states are naming . The exponent of B indicates the number of information bits “1” causing the transition and the exponent of D indicates the Hamming weight associated with the transition. We mention this example for the reason that it has a similiar encoder and most important simliar puncture matrices which can help us understand the trellis expanded by puncturation [11]. The transition behavior could be described by the matrix notation [11]. # . . . or.

(109). . .

(110) . . . . . . . . . . . . . Combine the two equation above, we get a equation. . # . The inverse of matrix as. . . . . . . # . may be expanded as an infinite series of power of matrices. # . #.

(111).

(112). . . #

(113). # . ) .

(114). Substituting into the equation above, we have. . . . .

(115) . #. . .

(116). 31. . . #

(117). . . # . . .

(118). ) .

(119) . As a result, the terms. . . . . . and. . . . . . . . . . . .

(120)

(121)

(122). . . . . of the weight spectrum from. . . . . . .

(123)

(124). . . . . . . . .

(125)

(126). . .

(127).

(128) . .

(129).

(130). . .

(131).

(132) . . In the above expressions,. . are given by.

(133). )).

(134). ) . . According to the argorithm above, we obtain the coefficients of Table 4.1.. .

(135). . ,. is the free distance of the code and. . . )) . , . as shown in. is the number of. , that split from the correct. incorrect paths or adversaries of Hamming weight , path and remerge with it sometime later. As for , it is simply the total number of bit is the number of bit errors of errors in the adversaries with Hamming distance and. . the source on the branch. The derivation of the union bound depends on the pairwise error probability. . as-. sociated to the channel condition. Since our simulation channel is an AWGN channel therefore we can write as. . . $%. . ! " #. . &. . '. +*. !)(. $%. . (4.3). where. In eq.. (4.3),. -,. .. /. . 1032465. 8 7 3 9 7 . !:(. is the Euclidean distance between the pairwise symbols and '. is. the minimum Euclidean distance between two adjacent symbols in the constellation. Using the weight spectrum in Table 4.1, a union bound of symbol error rate is obtained as. <;. . .

(136)

(137). 32. . . . (4.4).

(138) We can derive upper bound of the bit error rate of the punctured convolutional code as . . . .

(139)

(140). . . (4.5). where is of the rate- convolutional code. Similary to (4.5), the bit error rate of the uncoded part is . where *. (. . . . . . *. . is the number uncoded bits in a symbol.. Combining (4.4) and (4.5), we obtain the bit error rate of one axis as. . In (4.6),. . . ( .

(141) . . . . 5 . .

(142). . 9. . (4.6). represents the average bit different number of the uncoded bits of two. adjacent symbols in the constellation. Fig. 4.4 illustrates an example of bit errors of an original all-zeros path with an error path diverging from the original path and remerging to it. In this example, there are one 1 and two 0 input bits and therefore cause one bit error in the punctured convolutional code. The number of error symbols, with uncoded bits, is related to the number of 1s in the output of the punctured convolutional code and there are three. If there are average two uncoded-bit errors between two adjacent symbols, the total number of bit errors should be. .

(143). . . .. The equations above are the union bound for binary convolutional codes. However, there are still parallel transitions in TCM. Therefore, both the parallel transitions of correct path and that of the incorrect path are considered. However, the later is not the dominative. !. term in symbol error rate, so we only analyze approximation of the former as: )(. <;. . . . . . . '. $%. ! "#. . (4.7). !. Sum up the above equation and (4.4) we obtain the total symbol error rate as: )(. ;. . .

(144)

(145). .

(146). 33. . . . ' ! " #. $%. . (4.8).

(147) Table 4.1: Weight Distribution of the PBCC in TCM. . . . 3. 2. 4. 6. 4. 10. 48. 40. 5. 78. 528. 390. 6. 528. 5123. 3168. 7. 3148. 40144. 22036. 8. 19722. 3.1102e+5. 1.5778e+5. 9. 1.2227e+5. 2.2811e+6. 1.1004e+6. 10 7.6337e+5. 1.6495e+7. 7.6339e+6. 11 4.7431e+6. 1.1634e+8. 5.1274e+7. 12 2.9417e+7. 8.0622e+8. 3.53e+8. 13 1.8033e+9. 5.4327e+9. 2.3443e+9. 14 1.0833e+9 3.5319e+10 1.5166e+10 15 6.3041e+9. 1.4e+11. 9.4561e+10. And with (4.6), similar result can be also derived. Figs. 4.5, and 4.6 depict the simulation results and our analytic results of symbol error . rate and bit error rate in AWGN separately. In each plot,. is the bit energy in the. . information source. We have to notice that since there are two independent PBCC in. . each the TCM scheme. Therefore, the total coding rate is . . 256-QAM mode, and . in 64-QAM mode, . . . in. in 1024-QAM mode. In Fig. 4.5, there is a 5 dB gain between. classical QAM modulation and TCM. And our simulation shows that in 64-QAM mode, the analytic results are 0.5 dB more than simulation results. In 256-QAM mode and 1024QAM mode, the simulation results are 0.5 dB more than the analytic results. Similar in Fig. 4.6, we have 4 dB gains between classical QAM modulations and TCM schemes. The analytic results in 64-QAM mode are 0.5 dB more than the simulation results. And in 256-QAM and 1024-QAM modes, the simulation results are 0.25 dB more than the analytic results. In these observation, we also find that the analytic values are very close to the simulation results in high SNR situations.. 34.

(148) X0/0. X0/0. 00/0. 11/0 X1/1 X0/0. Figure 4.4: State transition diagram of example the PBCC.. 2. 10. 0. 10. −2. 10. −4. SER. 10. −6. 10. −8. classic 64−QAM TCM 64−QAM u.b. TCM 64−QAM sim classic 256−QAM TCM 256−QAM u.b. TCM 256−QAM sim classic 1024−QAM TCM 1024−QAM u.b. TCM 1024−QAM sim. 10. −10. 10. −12. 10. 5. 10. 20. 15. 25. 30. Eb/N0. Figure 4.5: SER performance of TCM in 64-QAM, 256-QAM, and 1024-QAM modes.. 35.

(149) 0. 10. −2. 10. −4. SER. 10. −6. 10. −8. 10. TCM 1024−QAM u.b. TCM 1024−QAM sim. TCM 256−QAM u.b. TCM 256−QAM sim. TCM 64−QAM u.b. TCM 64−QAM sim. classic 64−QAM classic 256−QAM classic 1024−QAM. −10. 10. −12. 10. 5. 10. 15. Eb/N0. 20. 25. 30. Figure 4.6: BER performance of TCM in 64-QAM, 256-QAM, and 1024-QAM modes.. 4.2 De-randomizer Because the function of the randomizer is to add PN sequence to the FEC frame, the de-randomizer adds an identical PN sequence to recover the orignal FEC frame. To generate an identical PN sequence, we use Fig. 3.3 as the de-randomizer. Differentially, “Data In” represents the received symbol and “Data Out” indicates the output symbol to the de-interleaver. The operation is similar to the randomizer. Firstly, the initialization is defined as pre-loading to the “all-ones” state. Secondly, the de-randomizer begins when the first information symbol arrives.. 4.3 De-interleaver As shown in Fig. 3.2, the de-interleaver is the reverse of the interleaver. Besides, the de-interleaver commutator first points to the top-most branch with the longest delay. This convolutional interleaver and de-interleaver pair are better than conventional block interleaver in two ways: 1) half of the memory units creat the same interleaving 36.

(150) distance, and 2) the received information can be outputed continuously after the first valid symbol instead of waiting for the complete arrival of the whole block information.. 4.4 Reed-Solomon decoder Both 64-QAM and 256-QAM mode utilize a (128,122) code over GF(128). The decoding capability is up to t = 3 symbol errors per RS block. This code is extended from a (127,122) code by adding a parity check symbol as shown in (3.3).. 4.4.1 Extended RS codes. . Any narrow-sense. noncyclic . -ary Reed-Solomon code. can be extended to form a. -ary maximum-distance separable (MDS) code by adding a parity. check [12, p. 191]. From (3.1) and (3.2), we can further derive a parity-check matrices without and with the single parity check symbol as in (4.9) and (4.10) respectively:. . . . . 8. H . . . . . . H. 8. . .. . .. .. . . . . . . . .. . .. .. .. . .. .. ... .. . .. .. . . . . ... ... . . . . . . . . . . . (4.9). . . . .. . .. .. ... . . . . . . . .. . .. .. .. . . . .. . .. .. . . (4.10) . . . . . !. . From (4.9) to (4.10), the number of syndromes is extended from five to six. In (4.10), , of the the first five rows can be used to compute the syndromes for received RS code as follows [7, p. 258]:. 37. . .

(151) ! . In the equation,. . . . !. . !. . . is the received . . . . . . . . for . . . (4.11). symbol in a RS block where. . is the received. extended parity symbol. Furthermore, the sixth syndrome is obtained from the sixth row of (4.10) as: . . . . .

(152). . . . . . . (4.12). Performance Comparison The software program for the following simulation is adapted from the public-domain software at the website [8]. In Fig. 4.7(a), the error performance of a two-error correcting RS decoder is compared with that of a three-error decoder for the outer RS(128,122) code. This decoder receives 128 symbols and decodes two random errors in a RS block using first four syndromes in (4.11). In the simulation, symbol error rate after the three-error decoder is is almost 1 dB better with respect to the double-error decoder when added symbol errorrate is. . In Berlekamp’s RS code decoding algorithm, the complexity is. is the. , where . . . .. number of error-correcting capability. Therefore, a double-error decoder is ! (0.49%) less complex than a triple-error decoder. As a result, when complexity becomes an issue, the proposed decoder gives a trade-off between complexity and coding gain. Ideally, a (128,122) block code can correct up to three errors in a block, and a symbol. !. error bound can be shown as in Fig. 4.7(a) and below:. <; In this equation,. ;. ! . . . 5. 9 . ;. . ;. !. . (4.13). . is the symbol error probability after decoding and. . ;. is the added. random error probability of the received RS symbol before decoding. However, the simulation result of the triple-error RS code decoder is worse than that of the ideal (128,122) block code. Because once an error occurs in the parity check symbol, the error location or evaluation computation will fail and undermine the performance. The performance bound of the three-error decoder is depicted in Fig. 4.7(a) and can be derived 38.

(153) as:. ;. . . ; !. . . . 5. . 9. ! . ;. . ;. . . !

(154) . ; . (4.14). . This bound is obtained if all RS block with an erred parity check symbol is detected and then skip the decoding procedure because the decoding procedure must be wrong. If we force the parity check symbol to be correct, the performance will approach the ideal block code as illustrated in Fig. 4.7(b). One may wonder whether we can use t=3 mode at first and change to t=2 mode if the decoder detect the error number is lager than 3. By this idea, the decoder may ignore the parity check symbol error and derives t=2 correction ability when errors in other symbols are less or equal to 2. We simulate this idea and find although the parity check symbol error may be avoided, the error detection ability also is reduced. The reduction undermines the performance and therefore do not improve. We conjecture that the decoder might misjudge the error number and causes more error after decoding. Thesis [24] also elaborates a similar idea about decoding the extended RS code by two possible paths. In one path, the extended parity check symbol is correct and all 6 syndromes are used, while in the other path, the extended parity check symbol in incorrect and only five syndromes are used. This approach first detects the order of the error location polynomial. When the fifth syndrome is used (sixth syndrome is not used), the decoder detects the number. If the order is less then 3 and the error quantity. is zero,. 5 syndromes are used in decoding. If not, they check at the sixth step (sixth syndrome is used). If the order is 3, 6 syndromes are used in decoding. In that thesis, the author observes that sometimes while 4 symbols are incorrect, one of the error is in extended parity check symbols, the decoder may misjudge that there are only 3 errors. This misjudgement causes more errors after decoding. Unfortunately, the simulation in that thesis is taken under AWGN while our simulation is under binary symmetric channel (BSC) and hence no comparison could be made directly. To sum up, the extended RS code can extend the parity check matrix and our decoder provides error correction ability close to t=3. However, the complexity is 2.25 times than t=2. 39.

(155) 0. 10. RS (127,122) t=2 RS (128,122) t=3 block code (128,122) t=3 bound of RS (128,122) t=3. −1. 10. −2. SER of RS Code. 10. −3. 10. −4. 10. −5. 10. −6. 10. −7. 10. −3. 10. 0. −1. −2. 10. 10 10 Symbol Error Prob of Received Symbol. (a) 0. 10. RS (128,122) t=3 block code (128,122) t=3 −1. 10. −2. SER of RS Code. 10. −3. 10. −4. 10. −5. 10. −6. 10. −7. 10. −3. 10. −1. −2. 10 10 Symbol Error Prob of Received Symbol. 0. 10. (b) Figure 4.7: (a) Performance of t=2 and t=3 for RS code. (b) Performance of t=3 with correct parity check symbol and ideal block code.. 40.

(156) QAM symbols. Bits input to BCC. T0. T1. T2. T3. T4. B2. B5. B8. B11. B13. B1. B4. B7. B10. B12. A2. A5. A8. A11. A13. A1. A4. A7. A10. A12. B0. B3. B6. B9. A0. A3. A6. A9. Time 28 bits MSB. LSB MSB. LSB MSB. LSB MSB. LSB. A10 A8 A7 A5 A4 A2 A1 A9 A6 A3 A0 A13 A12 A11 B10 B8 B7 B5 B4 B2 B1 B9 B6 B3 B0 B13 B12 B11. RS0. RS1. RS2. RS3. Figure 4.8: 64-QAM trellis group [1, p. 27].. 4.5 Concatenated Coding Simulation Results In this section, we integrate and then simulate the four layers of forward error correction under 64-QAM mode. The trellis group is formed from RS symbols as Fig. 3.4 and Fig. 4.8. Similar to encoding procedure, the decoded bits after TCM is grouped to RS symbols in the same approach. As for the interleaver, parameters I=128 and J=1 are chosen. Since the 64-QAM mode runs at 5.056941 Mbaud [14] and 4 RS symbols are group into 5 TCM. symbols. Therefore, the symbol time of each RS symbol is 0.2472 s, the burst protection time is 0.2472 s 128=31.6416 s, and the latency time is I (I-1) J 0.2472 s=4.018 ms.. Fig. 4.9 shows the symbol error rate (SER) of the simulation results. The simulation results of 64-QAM TCM from Fig. 4.9 (a) is shown in the ‘*’ line, the simulation results of the overall FEC is in the ‘o’ line, and the theoretical performance of the overall FEC is in the ‘+’ line. The theoretical performance is derived by the combination of (4.14) and the simulation results of 64-QAM TCM. The combination is constructed under the assumption that the errors of the RS decoder input are white. This assumption is reasonable because the Euclidean distance of the first 41.

(157) several nearest pairwise error symbol of TCM is not deeper than the depth of the (I=128, J=1) interleaver. The transformation from TCM results to SER of RS code decoder is related to Fig. 4.8. As in the figure, a TCM symbol error can cause the uncoded-bits part error. And on average a single TCM symbol error results 1.2 RS symbols errors. For example, if a single TCM symbol error causes decoding errors of A12 and A13, then RS1 is incorrect. But if the single TCM symbol error causes the incorrection of A10 and A11, both R0 and R1 are erroneous. From Fig. 3.9, however, we observe that the average number of bit difference between two adjacent symbols is 1. Hence, in the last example, only R0 or R1 is erroneous. As a result, on the average, only on symbol will be effected by the uncoded part if a single TCM symbol error happens. Although the error of uncodedpart is related to coded-part, we can assume the interleaver causes the independence of them. To sum up, the SER relationship is as follows: . . ;. . ;. .

(158) . is the SER of the input of the RS decoder,. . 2.

(159) . . 2 is the BER of the coded part.. . . is the SER of TCM and. The BER of the overall simulation is shown in Fig. 4.9 (b). The analysis of RS decoder is similar to above. However, the direct transformation from SER to BER is complex. Therefore we derive the theoretical performance of BER by dividing the SER theoretical performance by 3.6. Here 3.6 is an experience parameter. From our simulation results in Fig. 4.9 (b), the coding gain of overall FEC at BER=. . is 4.7 dB. In thesis [24], the author derived simulation coding gain by 4.3 dB. The difference may be caused by the different trancation length of the Viterbi decoder, RS decoder, and interleaver. The author used trancation length of 32 and his proposed RS decoder, and interleaver of I=16 and J=8.. 42. .

(160) 0. 10. classic 64−QAM TCM 64−QAM TCM 64−QAM + IL (128,1) + RS theo TCM 64−QAM + RS. −2. 10. −4. SER. 10. −6. 10. −8. 10. −10. 10. −12. 10. 8. 10. 12. 14 Eb/N0. 16. 18. 20. (a) 0. 10. classic 64−QAM TCM 64−QAM TCM 64−QAM + IL (128,1) + RS theo TCM 64−QAM + RS. −2. 10. −4. 10. −6. BER. 10. −8. 10. −10. 10. −12. 10. −14. 10. 8. 10. 12. 14 Eb/N0. 16. 18. 20. (b) Figure 4.9: (a) SER of 64-QAM FEC overall simulation results. (b) BER of 64-QAM FEC overall simulation results.. 43.