Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 1 of 15
`Case 6:20-cv-01042—ADA Document 1-2 Filed 11/11/20 Page 1 of 15
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`6:20-cv-1042
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`6:20-cv- 1042
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`EXHIBIT B
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`EXHIBIT B
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`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 2 of 15
`Case 612°‘CV'01042'ADA D(“”1111llllfllllfllllllllll'lllll’llllfillllllllllllllllllllll'llll||||||||
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`US007421032B2
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`(12) United States Patent
`US 7,421,032 B2
`(10) Patent N0.:
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`Jin et al.
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`(45) Date of Patent: Sep. 2, 2008
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`(54) SERIAL CONCATENATION 0F
`INTERLEAVED CONVOLUTIONAL CODES
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`FORMING TURBO-LIKE CODES
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`(56)
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`References Cited
`U.S. PATENT DOCUMENTS
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`(73) Assignee:
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`Inventors: Hui Jin, Glen Gardner, NJ (US); Aamod
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`Khandekar, Pasadena, CA (US);
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`Robert J. McEliece, Pasadena, CA (US)
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`Callifornia Institute of Technology,
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`Pasadena, CA (US)
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`Subject to any disclaimer, the term of this
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`patent is extended or adjusted under 35
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`U.S.C. 154(b) by 0 days.
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`Notice:
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`Appl. No.:
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`11/542,950
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`Filed:
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`Oct. 3, 2006
`Prior Publication Data
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`US 2007/0025450 A1
`Feb. 1, 2007
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`Related US. Application Data
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`Continuation of application No. 09/861,102, filed on
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`May 18, 2001, now Pat. No. 7,116,710, and a continu-
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`ation-in—part of application No. 09/922,852, filed on
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`Aug. 18, 2000, now Pat. No. 7,089,477.
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`Provisional application No. 60/205,095, filed on May
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`18, 2000.
`Int. Cl.
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`(2006.01)
`H04L 5/12
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`....................... 375/262; 375/265; 375/348;
`US. Cl.
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`714/755; 714/786; 714/792; 341/52; 341/102
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`Field of Classification Search ................. 375/259,
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`375/262, 265, 285, 296, 341, 346, 348; 714/746,
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`714/752, 755, 756, 786, 792, 7947796; 341/51,
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`341/52,56,102,103
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`See application file for complete search history.
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`5,392,299 A
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`5,530,707 A *
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`5,751,739 A
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`5,802,115 A *
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`5,881,093 A
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`6,014,411 A
`6,023,783 A
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`6,031,874 A
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`6,032,284 A
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`6,044,116 A
`6,094,739 A *
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`2/1995 Rhines et a1.
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`6/1996 Lin ............................ 714/792
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`5/1998 Seshadri et a1.
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`9/1998 Meyer ........................ 375/341
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`3/1999 Wang et a1.
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`1/2000 Wang
`2/2000 Divsalar et a1.
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`2/2000 Chennakeshu et a1.
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`2/2000 Bliss
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`3/2000 Wang
`7/2000 Miller et a1.
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`................ 714/792
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`(Continued)
`OTHER PUBLICATIONS
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`Appendix A.l “Structure of Parity Check Matrices of Standardized
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`LDPC Codes,” Digital Video Broadcasting (DVB) User guidelines
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`for the second generation system for Broadcasting, Interactive Ser-
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`vices, News Gathering and other broadband satellite applications
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`(DVB-S2) ETSI TR 102 376 V1.1.1. (Feb. 2005) Technical Report,
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`pp. 64.
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`(Continued)
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`Primary ExamineriDac V. Ha
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`(74) Attorney, Agent, or FirmiFish & Richardson PC.
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`(57)
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`ABSTRACT
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`A serial concatenated coder includes an outer coder and an
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`inner coder. The outer coder irregularly repeats bits in a data
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`block according to a degree profile and scrambles the
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`repeated bits. The scrambled and repeated bits are input to an
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`inner coder, which has a rate substantially close to one.
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`23 Claims, 5 Drawing Sheets
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`Check Node
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`degree a
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`Variable Node
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`RANDOMPEHMUTATION
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`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 3 of 15
`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 3 of 15
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`US 7,421,032 B2
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`Page 2
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`U.S. PATENT DOCUMENTS
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`5/2002 Laumen et a1.
`6,396,423 B1
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`8/2002 Kim et a1.
`6,437,714 B1
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`2/2005 Hammons et a1.
`6,859,906 B2*
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`9/2001 Eidson et a1.
`2001/0025358 A1
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`OTHER PUBLICATIONS
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`........... 714/786
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`Benedetto et al., “A Soft-Input Soft-Output Maximum A Posteriori
`
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`
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`(MAP) Module to Decode Parallel and Serial Concatenated Codes,”
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`The Telecommunications and Data Acquisition (TDA) Progress
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`
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`Report 42-127 for NASA and California Institute of Technology Jet
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`Propulsion Laboratory, Jospeh H. Yuen, Ed., pp. 1-20 (Nov. 15,
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`1996).
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`Benedetto et a1., “Bandwidth efficient parallel concatenated coding
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`schemes,” Electronics Letters 31(24): 2067-2069 (Nov. 23, 1995).
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`Benedetto et a1., “Design of Serially Concatenated Interleaved
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`Codes,” ICC 97, Montreal, Canada, pp. 710-714, ( Jun. 1997).
`Benedetto et a1., “Parallel Concatenated Trellis Coded Modulation,”
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`ICC ’96, IEEE, pp. 974-978, (Jun. 1996).
`Benedetto et a1., “Serial Concatenated Trellis Coded Modulation
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`With Iterative Decoding,” Proceedings from the IEEE 1997 Interna-
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`tional Symposium on Information Theory (ISIT), Ulm, Germany, p.
`8, Jun. 29-Ju1. 4, 1997.
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`Benedetto et a1., “Serial Concatenation of Interleaved Codes: Perfor-
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`mance Analysis, Design, and Iterative Decoding,” The Telecommu-
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`nications and Data Acquisition (TDA) Progress Report 42-126 for
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`NASA and California Institute of Technology Jet Propulsion Labo-
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`ratory, Jospeh H.Yuen, Ed., pp. 1-26 (Aug. 15, 1996).
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`Benedetto et a1., “Serial Concatenation of interleaved codes: perfor-
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`
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`mance analysis, design, and iterative decoding,” Proceedings from
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`the IEEE 1997 International Symposium on Information Theory
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`(ISIT), Ulm, Germany, p. 106, Jun. 29-Ju1. 4, 1997.
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`Benedetto et a1., “Soft-output decoding algorithms in iterative decod-
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`ing of turbo codes,” The Telecommunications and Data Acquisition
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`(TDA) Progress Report 42-124 for NASA and California Institute of
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`Technology Jet Propulsion Laboratory, Jospeh H. Yuen, Ed., pp.
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`63-87 (Feb. 15, 1996).
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`
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`Benedetto, S. et al., “A Soft-Input Soft-Output APP Module for
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`
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`Iterative Decoding of Concatenated Codes,” IEEE Communications
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`
`Letters 1(1): 22-24 (Jan. 1997).
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`
`
`Berrou et a1., “Near Shannon Limit Error-Correcting Coding and
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`Decoding: Turbo Codes,” ICC pp. 1064-1070 (1993).
`
`
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`
`
`
`
`
`
`Digital Video Broadcasting (DVB) User guidelines for the second
`
`
`
`
`
`
`
`generation system for Broadcasting, Interactive Services, News
`
`
`
`
`
`
`
`
`Gathering and other broadband satellite applications (DVB-SZ) ET SI
`
`
`
`
`
`
`
`
`
`
`TR 102 376V1.1.1. (Feb. 2005) Technical Report, pp. 1-104 (Feb. 15,
`
`2005).
`
`
`
`
`
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`
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`
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`Divsalar et a1., “Coding Theorems for ‘Turbo-Like’ Codes,” Proceed-
`
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`
`
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`ings of the 36th Annual Allerton Conference on Communication,
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`
`
`Control, and Computing, Sep. 23-25, 1998, Allerton House,
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`
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`
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`Monticello, Illinois, pp. 201-210 (1998).
`Divsalar et a1., “Effective free distance of turbo codes,” Electronics
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`Letters 32(5): 445-446 (Feb. 29, 1996).
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`Divsalar, D. et a1., “Hybrid Concatenated Codes and Iterative Decod-
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`
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`ing,” Proceedings from the IEEE 1997 International Symposium on
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`
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`Information Theory (ISIT), Ulm, Germany, p. 10 (Jun. 29-Ju1. 4,
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`1997).
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`Divsalar, D. et a1., “Low-rate turbo codes for Deep Space Commu-
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`nications,” Proceedings from the 1995 IEEE International Sympo-
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`sium on Information Theory, Sep. 17-22, 1995, Whistler, British
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`Columbia, Canada, pp. 35.
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`Divsalar, D. et a1., “Multiple Turbo Codes for Deep-Space Commu-
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`nications,” The Telecommunications and Data Acquisition (TDA)
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`Progress Report 42-121 for NASA and California Institute of Tech-
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`nology Jet Propulsion Laboratory, Jospeh H. Yuen, Ed., pp. 60-77
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`(May 15, 1995).
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`Divsalar, D. et a1., “Multiple Turbo Codes,” MILCOM 95, San Diego,
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`CA pp. 279-285 (Nov. 5-6, 1995).
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`Divsalar, D. et a1., “On the Design of Turbo Codes,” The Telecom-
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`munications and DataAcquisition (TDA) Progress Report 42-123 for
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`ratory, Jospeh H.Yuen, Ed., pp. 99-131 (Nov. 15, 1995).
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`Symposium on Information Theory (ISIT), Italy, pp. 1-14 (Jun.
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`2000).
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`Divsalar, D. et a1., “Turbo Codes for PCS Applications,” ICC 95,
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`IEEE, Seattle, WA, pp. 54-59 (Jun. 1995).
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`Jin et a1., “Irregular RepeatiAccumulate Codes,” 2nd International
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`Symposium on Turbo Codes & Related Topics, Sep. 4-7, 2000, Brest,
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`France, 25 slides, (presented on Sep. 4, 2000).
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`Jin et al., “Irregular RepeatiAccumulate Codes,” 2nd International
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`France, pp. 1-8 (2000).
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`Richardson et a1., “Design of capacity approaching irregular low
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`* cited by examiner
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`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 4 of 15
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`Sheet 1 0f5
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`FIG.1 (PriorArt)
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`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 5 of 15
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`Sheet 2 0f5
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`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 6 of 15
`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 6 of 15
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`Sheet 3 0f5
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`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 7 of 15
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`Sheet 4 0f5
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`FIG. 5A
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`FIG. 53
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`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 8 of 15
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`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 9 of 15
`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 9 of 15
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`US 7,421,032 B2
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`1
`SERIAL CONCATENATION OF
`
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`INTERLEAVED CONVOLUTIONAL CODES
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`
`FORMING TURBO-LIKE CODES
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`CROSS-REFERENCE TO RELATED
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`APPLICATIONS
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`This application is a continuation of US. application Ser.
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`No. 09/861,102, filed May 18, 2001, now US. Pat.No. 7,116,
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`710, which claims the priority ofUS. provisional application
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`Ser. No. 60/205,095, filed May 18, 2000, and is a continua-
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`tion-in-part of US. application Ser. No. 09/922,852, filed
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`Aug. 18, 2000, now US. Pat. No. 7,089,477.
`GOVERNMENT LICENSE RIGHTS
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`The US. Government has a paid-up license in this inven-
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`tion and the right in limited circumstances to require the
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`patent owner to license others on reasonable terms as pro-
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`vided for by the terms of Grant No. CCR-9804793 awarded
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`by the National Science Foundation.
`BACKGROUND
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`Properties ofa channel affect the amount of data that can be
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`handled by the channel. The so-called “Shannon limit”
`defines the theoretical limit of the amount of data that a
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`channel can carry.
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`Different techniques have been used to increase the data
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`rate that can be handled by a channel. “Near Shannon Limit
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`Error-Correcting Coding and Decoding: Turbo Codes,” by
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`Berrou et al. ICC, pp 1064-1070, (1993), described a new
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`“turbo code” technique that has revolutionized the field of
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`error correcting codes. Turbo codes have sufficient random-
`ness to allow reliable communication over the channel at a
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`high data rate near capacity. However, they still retain suffi-
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`cient structure to allow practical encoding and decoding algo-
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`rithms. Still, the technique for encoding and decoding turbo
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`codes can be relatively complex.
`A standard turbo coder 100 is shown in FIG. 1. A block of
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`k information bits is input directly to a first coder 102. A k bit
`interleaver 106 also receives the k bits and interleaves them
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`prior to applying them to a second coder 104. The second
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`coder produces an output that has more bits than its input, that
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`is, it is a coder with rate that is less than 1. The coders 102,104
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`are typically recursive convolutional coders.
`Three different items are sent over the channel 150: the
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`original kbits, first encoded bits 110, and second encoded bits
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`112. At the decoding end, two decoders are used: a first
`constituent decoder 160 and a second constituent decoder
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`162. Each receives both the original k bits, and one of the
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`encoded portions 110, 112. Each decoder sends likelihood
`estimates of the decoded bits to the other decoders. The esti-
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`mates are used to decode the uncoded information bits as
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`corrupted by the noisy channel.
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`SUMMARY
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`A coding system according to an embodiment is config-
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`ured to receive a portion of a signal to be encoded, for
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`example, a data block including a fixed number of bits. The
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`coding system includes an outer coder, which repeats and
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`scrambles bits in the data block. The data block is apportioned
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`into two or more sub-blocks, and bits in different sub-blocks
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`are repeated a different number of times according to a
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`selected degree profile. The outer coder may include a
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`repeater with a variable rate and an interleaver. Alternatively,
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`the outer coder may be a low-density generator matrix
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`(LDGM) coder.
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`The repeated and scrambled bits are input to an inner coder
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`that has a rate substantially close to one. The inner coder may
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`include one or more accumulators that perform recursive
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`modulo two addition operations on the input bit stream.
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`The encoded data output from the inner coder may be
`transmitted on a channel and decoded in linear time at a
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`destination using iterative decoding techniques. The decod-
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`ing techniques may be based on a Tanner graph representation
`of the code.
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`BRIEF DESCRIPTION OF THE DRAWINGS
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`FIG. 1 is a schematic diagram of a prior “turbo code”
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`system.
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`FIG. 2 is a schematic diagram of a coder according to an
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`FIG. 3 is a Tanner graph for an irregular repeat and accu-
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`mulate (IRA) coder.
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`FIG. 4 is a schematic diagram of an IRA coder according to
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`FIG. 5A illustrates a message from a variable node to a
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`check node on the Tanner graph of FIG. 3.
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`FIG. 5B illustrates a message from a check node to a
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`variable node on the Tanner graph of FIG. 3.
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`FIG. 6 is a schematic diagram of a coder according to an
`alternate embodiment.
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`FIG. 7 is a schematic diagram of a coder according to
`another alternate embodiment.
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`DETAILED DESCRIPTION
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`FIG. 2 illustrates a coder 200 according to an embodiment.
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`The coder 200 may include an outer coder 202, an interleaver
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`204, and inner coder 206. The coder may be used to format
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`blocks of data for transmission, introducing redundancy into
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`the stream of data to protect the data from loss due to trans-
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`mission errors. The encoded data may then be decoded at a
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`destination in linear time at rates that may approach the chan-
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`nel capacity.
`The outer coder 202 receives the uncoded data. The data
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`may be partitioned into blocks of fixed size, say k bits. The
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`outer coder may be an (n,k) binary linear block coder, where
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`n>k. The coder accepts as input a block u of k data bits and
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`produces an output block v of 11 data bits. The mathematical
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`relationship between u and v is v:TOu, where T0 is an n><k
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`matrix, and the rate of the coder is k/n.
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`The rate of the coder may be irregular, that is, the value of
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`T0 is not constant, and may differ for sub-blocks of bits in the
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`data block. In an embodiment, the outer coder 202 is a
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`repeater that repeats the k bits in a block a number of times q
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`to produce a block with 11 bits, where n:qk. Since the repeater
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`has an irregular output, different bits in the block may be
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`repeated a different number of times. For example, a fraction
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`of the bits in the block may be repeated two times, a fraction
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`of bits may be repeated three times, and the remainder of bits
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`may be repeated four times. These fractions define a degree
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`sequence, or degree profile, of the code.
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`The inner coder 206 may be a linear rate-1 coder, which
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`means that the n-bit output block x can be written as x:T,w,
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`where T,is a nonsingular n><n matrix. The inner coder 210 can
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`have a rate that is close to 1, e.g., within 50%, more preferably
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`10% and perhaps even more preferably within 1% of 1.
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`In an embodiment, the inner coder 206 is an accumulator,
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`which produces outputs that are the modulo two (mod-2)
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`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 10 of 15
`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 10 of 15
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`US 7,421,032 B2
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`3
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`partial sums of its inputs. The accumulator may be a truncated
`rate-l recursive convolutional coder with the transfer func-
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`tion l/(l +D). Such an accumulator may be considered a block
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`coder whose input block [x1,
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`. ,yn] are related by the formula
`[y1, .
`Y1:X1
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`y2:x177x2
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`y3:x1"x2®x3
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`yn:xl@x2@x3@. .
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`. flax".
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`where “69” denotes mod-2, or exclusive-OR (XOR), addition.
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`An advantage of this system is that only mod-2 addition is
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`necessary for the accumulator. The accumulator may be
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`embodied using only XOR gates, which may simplify the
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`The bits output from the outer coder 202 are scrambled
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`before they are input to the inner coder 206. This scrambling
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`may be performed by the interleaver 204, which performs a
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`pseudo-random permutation of an input block v, yielding an
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`output block w having the same length as V.
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`The serial concatenation of the interleaved irregular repeat
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`code and the accumulate code produces an irregular repeat
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`and accumulate (IRA) code. An IRA code is a linear code, and
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`as such, may be represented as a set of parity checks. The set
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`of parity checks may be represented in a bipartite graph,
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`called the Tanner graph, of the code. FIG. 3 shows a Tanner
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`graph 300 of an IRA code with parameters (f1,
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`where 13:0, Eiffl and “a” is a positive integer. The Tanner
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`graph includes two kinds of nodes: variable nodes (open
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`circles) and check nodes (filled circles). There are k variable
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`nodes 302 on the left, called information nodes. There are r
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`variable nodes 306 on the right, called parity nodes. There are
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`r:(kZl.ifl.)/a check nodes 304 connected between the informa-
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`tion nodes and the parity nodes. Each information node 3 02 is
`connected to a number of check nodes 304. The fraction of
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`information nodes connected to exactly i check nodes is fi.
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`For example, in the Tanner graph 300, each of the f2 informa-
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`tion nodes are connected to two check nodes, corresponding
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`to a repeat of q:2, and each of the f3 information nodes are
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`cc a,
`connected to three check nodes, corresponding to q:3.
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`Each check node 304 is connected to exactly a informa- 50
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`tion nodes 302. In FIG. 3, a:3. These connections can be
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`made in many ways, as indicated by the arbitrary permutation
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`of the ra edges joining information nodes 302 and check
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`nodes 304 in permutation block 310. These connections cor-
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`respond to the scrambling performed by the interleaver 204.
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`In an alternate embodiment, the outer coder 202 may be a
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`low-density generator matrix (LDGM) coder that performs
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`an irregular repeat of the k bits in the block, as shown in FIG.
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`4. As the name implies, an LDGM code has a sparse (low-
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`density) generator matrix. The IRA code produced by the 60
`coder 400 is a serial concatenation ofthe LDGM code and the
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`accumulator code. The interleaver 204 in FIG. 2 may be
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`excluded due to the randomness already present in the struc-
`ture of the LDGM code.
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`If the permutation performed in permutation block 310 is 65
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`fixed, the Tanner graph represents a binary linear block code
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`with k information bits (ul,
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`(x1,
`,x,), as follows. Each of the information bits is
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`associated with one ofthe information nodes 302, and each of
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`the parity bits is associated with one of the parity nodes 306.
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`The value of a parity bit is determined uniquely by the con-
`dition that the mod-2 sum of the values of the variable nodes
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`connected to each of the check nodes 304 is zero. To see this,
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`set x0:0. Then if the values ofthe bits on the ra edges coming
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`out the permutation box are (v1, .
`, vm), then we have the
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`recursive formula
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`i
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`z:1
`xj =Xj71+2 mg)“;
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`.
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`, r. This is in effect the encoding algorithm.
`forj:l, 2, .
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`Two types of IRA codes are represented in FIG. 3, a non-
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`systematic version and a systematic version. The nonsystem-
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`atic version is an (r,k) code, in which the codeword corre-
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`sponding to the information bits (ul, .
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`. ,uk) is (x1, .
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`. ,x,).
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`The systematic version is a (k+r, k) code, in which the code-
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`word is (ul, .
`. ,uk; x1, .
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`, x,).
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`The rate of the nonsystematic code is
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`Rnsys —
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`The rate of the systematic code is
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`a
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`Rsys =
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`a+Zifi
`i
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`For example, regular repeat and accumulate (RA) codes
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`can be considered nonsystematic IRA codes with a:l and
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`exactly one fl. equal to 1, say fq:l, and the rest zero, in which
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`case Rwy: simplifies to R:l/q.
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`The IRA code may be represented using an alternate nota-
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`tion. Let Al. be the fraction of edges between the information
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`nodes 302 and the check nodes 304 that are adjacent to an
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`information node of degree i, and let pl. be the fraction of such
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`edges that are adjacent to a check node of degree i+2 (i.e., one
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`that is adjacent to i information nodes). These edge fractions
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`may be used to represent the IRA code rather than the corre-
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`sponding node fractions. Define }t(x):2i}tixi'l and p(x):2ipi
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`xi"l to be the generating functions of these sequences. The
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`pair (2», p) is called a degree distribution. For L(x):2ifl.xl.,
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`x
`1
`o
`o
`L(x) = f Mndt/f MIMI
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`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 11 of 15
`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 11 of 15
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`5
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`The rate of the systematic IRA code given by the degree
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`distribution is given by
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`US 7,421,032 B2
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`6
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`relies on its input, and y is the output of the channel code bit
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`u, then m0(u):log(p(u:0|y)/p(u:l |y)). After this initializa-
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`tion, the decoding process may run in a fully parallel and local
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`manner. In each iteration, every variable/check node receives
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`messages from its neighbors, and sends back updated mes-
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`sages. Decoding is terminated after a fixed number of itera-
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`tions or detecting that all the constraints are satisfied. Upon
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`termination, the decoder outputs a decoded sequence based
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`on the messages m(u):2wm(w—>u).
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`Thus, on various channels, iterative decoding only differs
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`in the initial messages m0(u). For example, consider three
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`memoryless channel models: a binary erasure channel
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`(BEC); a binary symmetric channel (BSC); and an additive
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`white Gaussian noise (AGWN) channel.
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`In the BEC, there are two inputs and three outputs. When 0
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`is transmitted, the receiver can receive either 0 or an erasure E.
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`An erasure E output means that the receiver does not know
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`how to demodulate the output. Similarly, when l is transmit-
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`ted, the receiver can receive either 1 or E. Thus, for the BEC,
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`y e {0, E, l}, and
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`"10(14):
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`+00 ify=0
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`ifyzE
`0
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`ify=l
`—oo
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`In the BSC, there are two possible inputs (0,1) and two
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`possible outputs (0, l). The BSC is characterized by a set of
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`conditional probabilities relating all possible outputs to pos-
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`sible inputs. Thus, for the BSC y e {0, l},
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`log
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`"1004) =
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`—log
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`ify=O
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`if y = l
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`and
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`In the AWGN, the discrete-time input symbols X take their
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`values in a finite alphabet while channel output symbole can
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`take any values along the real line. There is assumed to be no
`distortion or other effects other than the addition of white
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`Gaussian noise. In an AWGN with a Binary Phase Shift
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`Keying (BPSK) signaling which maps 0 to the symbol with
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`amplitude E and l to the symbol with amplitude fifg,
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`output y e R, then
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`m0(u)fily\/E/NO
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`where NO/2 is the noise power spectral density.
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`“Belief propagation” on the Tanner Graph realization may
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`be used to decode IRA codes. Roughly speaking, the belief
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`propagation decoding technique allows the messages passed
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`on an edge to represent posterior densities on the bit associ-
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`ated with the variable node. A probability density on a bit is a
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`pair of non-negative real numbers p(0), p(l) satisfying p(0)+
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`p(l):l, where p(O) denotes the probability of the bit being 0,
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`p(1) the probability of it being 1. Such a pair can be repre-
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`sented by its log likelihood ratio, m:log(p(0)/p(l)). The out-
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`going message from a variable node u to a check node v
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`represents information about u, and a message from a check
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`node u to a variable node v represents information about u, as
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`shown in FIGS. 5A and 5B, respectively.
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`The outgoing message from a node u to a node v depends
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`on the incoming messages from all neighbors w ofu except v.
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`If u is a variable message node, this outgoing message is
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`waev
`m(u —> v) = Z m(w —> 14) +m0(u)
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`where m0(u) is the log-likelihood message associated with
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`u. If u is a check node, the corresponding formula is
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`tanh
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`m(u —> v)
`2
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`tanhm(w —> 14)
`2
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`=
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`waev
`I I
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`25
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`Before decoding, the messages m(w—>u) and m(uev) are
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`initialized to be zero, and m0(u) is initialized to be the log-
`likelihood ratio based on the channel received information. If
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`the channel is memoryless, i.e., each channel output only
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`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 12 of 15
`Case 6:20-cv-01042-ADA Document 1-2 Filed 11/11/20 Page 12 of 15
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`US 7,421,032 B2
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`7
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`The selection of a degree profile for use in a particular
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`transmission channel is a design parameter, which may be
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`affected by various attributes of the channel. The criteria for
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`selecting a particular degree profile may include, for example,
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`the type of channel and the data rate on the channel. For
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`example, Table 1 shows degree profiles that have been found
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`to produce good results for an AWGN channel model.
`
`a
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`A2
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`A3
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`A5
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`A6
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`MO
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`211
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`A12
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`213
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`M4
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`M6
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`A27
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`A28
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`Rate
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`OGA
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`0*
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`(1311010) * (dB)
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`S.L. (dB)
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`TABLE 1
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`2
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`0.139025
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`0.2221555
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`0.638820
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`3
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`0.078194
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`0.128085
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`0.160813
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`0.0