`Jin et al.
`
`(10) Patent No.:
`(45) Date of Patent:
`
`US 7,916,781 B2
`Mar. 29, 2011
`
`US0079l678lB2
`
`€54)
`
`(75)
`
`SERIAL CONCATENATION OF
`INTERLEAVED CONVOLUTIONAL CODES
`FORMING TURBO-LIKE CODES
`
`_
`_
`Inventors: Hui Jin, Glen Gardner, NJ (US); Aamod
`Khandekar,PasadeI1a,CACUS);
`Robert J. McEliece, Pasadena, CA (US)
`
`(73) Assignee: California Institute of Technology,
`Pasadena’ CA(US)
`.
`.
`.
`.
`Subject. to any disclaimer, the term ofthis
`patent is extended or adjusted under 35
`U.S.C. 154(b) by 424 days.
`
`.
`( * ) Notice:
`
`(21) App1.N0.; 12/155,606
`~
`.
`Ffled‘
`
`(22)
`(65)
`
`Jun‘ 30’ 2008
`,
`,
`,
`Prior Publication Data
`US 2008/0294964 A1
`Nov. 27, 2008
`
`Related U.S. Application Data
`.
`.
`.
`.
`(63) Continuation of application No. 11/542,950, filed on
`Oct. 3, 2006, now Pat. No. 7,421,032, which is a
`continuation of application No. 09/861,102, filed on
`May 18, 2001, now Pat. No. 7,116,710, which is a
`continuation-in-part of application No. 09/922,852,
`filed on Aug. 18, 2000, now Pat. No. 7,089,477.
`(60) Provisional application No. 60/205,095, filed on May
`18, 2000.
`
`(51)
`
`Int. Cl.
`(2006.01)
`H043 1/66
`(52) U_s_ CL ...... N 375/240; 375/285; 375/296; 714/801;
`714/804
`(58) Field of Classification Search ................ .. 375/240,
`375/240.24, 254, 285, 295, 296, 260; 714/755,
`714/758, 800, 801, 804, 805
`See application file for complete search history.
`
`200N
`
`(56)
`
`References Cited
`
`U.S. PATENT DOCUMENTS
`
`5,181,207 A *
`5,392,299 A
`5,530,707 A
`5,751,739 A
`5,802,115 A
`2
`5
`5
`6,023,783 A
`6,031,874 A
`6,032,284 A
`5,044,115 A
`6,094,739 A
`6,195,396 B1*
`6,396,423 B1
`6,437,714 B1
`6,732,328 B1*
`6,859,906 B2
`7,089,477 B1
`
`.................... .. 714/755
`1/1993 Ch
`2/1995 Rhillaglirtlal.
`6/i996 Lin
`5/1998 Seshadri et al.
`9/1998 Meyer
`yang 61 31~
`1-
`ang
`2/2000 Divsalar etal.
`t
`2/2000 Ch
`ak h
`2/2000 Bliirsm es u e a
`3/3000 wang
`7/2000 Miller et al.
`.................. .. 375/261
`2/2001 Fang et al.
`5/2002 Laumen etal.
`8/2002 Kim et 211.
`5/2004 McEWen et al.
`2/2005 Hammons etal.
`8/2006 D"
`1
`t
`1.
`. W33 M e 3
`(Continued)
`
`............ .. 714/795
`
`1.
`
`OTHER PUBLICATIONS
`
`B
`
`, S.,
`
`f
`APP M d l
`S ft-O
`1., “A S ft-I
`d
`0 u e or
`utput
`0
`nput
`0
`et a
`ene em)
`£‘e:“iVe1Df?‘;°2'i‘;‘f‘°f C°‘1‘;*;‘7‘*“"“°'d C°d‘°'S’” 1555 C°”'”'“”i‘“’i°”S
`9 gm’
`(
`)’
`'
`’ an’
`’
`(Continued)
`
`Primary Examiner 7 Dac V Ha
`(74) Attorney] Agent, or Firm — Perkins Coje LLP
`
`ABSTRACT
`
`57
`
`(
`
`)
`serial concatenated coder includes an outer coder and an
`inner coder. The outer coder irregularly repeats bits in a data
`block according to a degree profile and scrambles the
`1ePea1ed b11S~ The 50131111131611 and 1ePea1ed 13115 3116 1111311110 an
`inner coder, which has a rate substantially close to one.
`
`22 Claims, 5 Drawing Sheets
`
`k 2
`
`02
`
`204
`
`206
`
`Apple 1101
`Apple 1101
`
`
`
`US 7,916,781 B2
`Page 2
`
`U.S. PATENT DOCUMENTS
`2001/0025358 A1
`9/2001 Eidson et al.
`
`OTHER PUBLICATIONS
`
`Benedetto, S., et al., “A Soft-Input Soft-Output Maximum A Poste-
`riori (MAP) Module to Decode Parallel and Serial Concatenated
`Codes,” The Telecommunications and Data Acquisition Progress
`Report (TDA PR 42-127), pp. 1-20, Nov. 1996.
`Benedetto, S., et al., “Bandwidth efficient parallel concatenated cod-
`ing schemes,” Electronics Letters, 31(24):2067-2069, Nov. 1995.
`Benedetto, S., et al., “Design of Serially Concatenated Interleaved
`Codes,” ICC 97, vol. 2, pp. 710-714, Jun. 1997.
`Benedetto, S., et al., “Parallel Concatenated Trellis Coded Modula-
`tion,” ICC 96, vol. 2, pp. 974-978, Jun. 1996.
`Benedetto, S., et al., “Serial Concatenated Trellis Coded Modulation
`with Iterative Decoding,” Proceedings 1997 IEEE International
`Symposium on Information Theory (ISIT), Ulm, Germany, p. 8, Jun.
`29-Jul. 4, 1997.
`Benedetto, S., et al., “Serial Concatenation of Interleaved Codes:
`Performace Analysis, Design, and Iterative Decoding,” The Telecom-
`munications andDataAcquisition Progress Report (TDA PR 42126),
`pp. 1-26, Aug. 1996.
`Benedetto, S., et al., “Serial concatenation of interleaved codes:
`performance analysis, design, and iterative decoding,” Proceedings
`1997 IEEE International Symposium on Information Theory (ISIT),
`Ulm, Germany, p. 106, Jun. 29-Jul. 4, 1997.
`Benedetto, S., et al., “Soft-Output Decoding Algorithms in Iterative
`Decoding of Turbo Codes,” The Telecommunications and Data
`Acquisition Progress Report (TDA PR 42-124), pp. 63-87, Feb. 1996.
`Berrou, C., et al., “Near Shannon Limit Error—Correcting Coding
`and Decoding: Turbo Codes,” ICC 93, vol. 2, pp. 1064-1070, May
`1993.
`Digital Video Broadcasting (DVB)—User guidelines for the second
`generation system for Broadcasting, Interactive Services, News
`Gathering and other broadband satellite applications (DVB-S2),
`ETSI TR 102 376 V1.1.1 Technical Report, pp. 1-104 (p. 64), Feb.
`2005.
`Divsalar, D., et al., “Coding Theorems for ‘Turbo-Like’ Codes,”
`Proceedings ofthe 3 6”’ AnnualAllerton Conference on Communica-
`tion, Control, and Computing, Monticello, Illinois, pp. 201-210, Sep.
`1998.
`
`Divsalar, D., et al., “Effective free distance ofturbo codes,” Electron-
`ics Letters, 32(5):445-446, Feb. 1996.
`Divsalar, D., et al ., “Hybrid Concatenated Codes and Iterative Decod-
`ing,” Proceedings 1997IEEE International Symposium on Informa-
`tion Theory (ISIT), Ulm, Germany, p. 10, Jun. 29-Jul. 4, 1997.
`Divsalar, D., et al., “Low-Rate Turbo Codes for Deep-Space Com-
`munications,” Proceedings 1995 IEEE International Symposium on
`Information Theory (ISIT), Whistler, BC, Canada, p. 35, Sep. 1995.
`Divsalar, D., et al., “Multiple Turbo Codes for Deep-Space Commu-
`nications,” The Telecommunications and Data Acquisition Progress
`Report (TDA PR 42-121), pp. 66-77, May 1995.
`Divsalar, D., et al., “Multiple Turbo Codes,”MILCOM ’95, vol. 1,pp.
`279-285, Nov. 1995.
`Divsalar, D., et al., “On the Design of Turbo Codes,” The Telecom-
`munications and Data Acquisition Progress Report
`(TDA PR
`42-123), pp. 99-121, Nov. 1995.
`Divsalar, D., et al., “Serial Turbo Trellis Coded Modulation with
`Rate-1 Inner Code,” Proceedings 2000 IEEE International Sympo-
`sium on Information Theory (ISIT), Sorrento, Italy, pp. 194, Jun.
`2000.
`Divsalar, D., et al., “Turbo Codes for PCS Applications,” IEEE ICC
`’95, Seattle, WA, USA, vol. 1, pp. 54-59, Jun. 1995.
`Jin, H., et al., “Irregular Repeat—Accumulate Codes,” 2nd Interna-
`tional Symposium on Turbo Codes, Brest, France, 25 pages, Sep.
`2000.
`
`Jin, H., et al., “Irregular Repeat—Accumulate Codes,” 2"“ Interna-
`tional Symposium on Turbo Codes &Related Topics, Brest, France, p.
`1-8, Sep. 2000.
`Richardson, T.J., et al., “Design of Capacity-Approaching Irregular
`Low-Density Parity-Check Codes,” IEEE Transactions on Informa-
`tion Theory, 47(2):619-637, Feb. 2001.
`Richardson, T.J., et al., “Efficient Encoding of Low-Density Parity-
`Check Codes,” IEEE Transactions
`on Information Theory,
`47(2):638-656, Feb. 2001.
`Wiberg, N., et al., “Codes and Iterative Decoding on General
`Graphs,” Proceedings 1 995 IEEE International Symposium on Infor-
`mation Theory (ISIT), Whistler, BC, Canada, p. 468, Sep. 1995.
`Aji, S.M., et al., “The Generalized Distributive Law,” IEEE Trans-
`actions on Information Theory, 46(2):325-343., Mar. 2000.
`Tanner, R.M., “A Recursive Approach to Low Complexity Codes,”
`IEEE Transactions on Information Theory, 27(5):533-547, Sep.
`1981.
`
`* cited by examiner
`
`
`
`U.S. Patent
`
`Mar. 29, 2011
`
`Sheet 1 of5
`
`US 7,916,781 B2
`
`FIG.1 (PriorArt)
`
`
`
`U.S. Patent
`
`Mar. 29, 2011
`
`Sheet 2 of5
`
`US 7,916,781 B2
`
`(.3
`L)
`<
`
`:
`
`Vl-
`
`Q5
`E
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`Q5
`F.
`
`E(
`
`52_
`
`|
`
`.2
`
`.2
`
`
`
`U.S. Patent
`
`Mar. 29, 2011
`
`Sheet 3 of5
`
`US 7,916,781 B2
`
`Variable Node
`
`Check Node
`
`Fraction of nodes
`degree i
`
`degree a
`
`2 9
`
`I-'"‘
`E
`3
`§
`
`E E
`
`8
`E
`<1:
`0:
`
`FIG. 3
`
`%
`
`2‘
`3
`
`f’
`1
`
`83. K»:
`
`III
`
`I-"""'*-.Zr—.- I
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`
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`
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`:
`:
`
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`I
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`
`U.S. Patent
`
`Mar. 29, 2011
`
`Sheet 4 of5
`
`US 7,916,781 B2
`
`
`
`FIG. 5A
`
`304
`
`304
`
`
`
`U.S. Patent
`
`Mar. 29, 2011
`
`Sheet 5 of5
`
`US 7,916,781 B2
`
`aGE
`
`
`
`US 7,916,781 B2
`
`1
`SERIAL CONCATENATION OF
`INTERLEAVED CONVOLUTIONAL CODES
`FORMING TURBO-LIKE CODES
`
`CROSS-REFERENCE TO RELATED
`APPLICATIONS
`
`This application is a continuation of U.S. application Ser.
`No. 11/542,950, filed Oct. 3, 2006 now U.S. Pat. No. 7,421,
`032, which is a continuation of U.S. application Ser. No.
`09/861,102, filedMay 18, 2001,nowU.S. Pat. No. 7,1 16,710,
`which claims the priority ofU.S. Provisional Application Ser.
`No. 60/205,095, filed May 18, 2000, and is a continuation-
`in-part ofU.S. application Ser. No. 09/922,852, filedAug. 18,
`2000, now U.S. Pat. No. 7,089,477. The disclosure of the
`prior applications are considered part of (and are incorporated
`by reference in) the disclosure of this application.
`
`GOVERNMENT LICENSE RIGHTS
`
`The U.S. Government has a paid-up license in this i11ven-
`tion and the right in limited circumstances to require the
`patent owner to license others on reasonable terms as pro-
`vided for by the terms of Grant No. CCR-9804793 awarded
`by the National Science Foundation.
`
`BACKGROUND
`
`10
`
`15
`
`20
`
`25
`
`Properties ofa channel affect the amount of data that can be
`handled by the channel. The so-called “Shannon limit”
`defines the theoretical limit of the amount of data that a
`
`30
`
`channel can carry.
`Different techniques have been used to increase the data
`rate that can be handled by a channel. “Near Shannon Limit
`Error-Correcting Coding and Decoding: Turbo Codes,” by
`Berrou et al. ICC, pp 1064-1070, (1993), described a new
`“turbo code” technique that has revolutionized the field of
`error correcting codes. Turbo codes have sufiicient random-
`ness to allow reliable communication over the channel at a
`
`high data rate near capacity. However, they still retain sulfi-
`cient structure to allow practical encoding and decoding algo-
`rithms. Still, the technique for encoding and decoding turbo
`codes can be relatively complex.
`A standard turbo coder 100 is shown in FIG. 1. A block of
`
`k information bits is input directly to a first coder 102. A k bit
`interleaver 106 also receives the k bits and interleaves them
`prior to applying them to a second coder 104. The second
`coder produces an output that has more bits than its input, that
`is, it is a coder with rate that is less than 1. The coders 102,104
`are typically recursive convolutional coders.
`Three different items are sent over the charmel 150: the
`
`original kbits, first encoded bits 110, and second encoded bits
`112. At the decoding end, two decoders are used: a first
`constituent decoder 160 and a second constituent decoder
`
`162. Each receives both the original k bits, and one of the
`encoded portions 110, 112. Each decoder sends likelihood
`estimates of the decoded bits to the other decoders. The esti-
`mates are used to decode the uncoded information bits as
`
`corrupted by the noisy channel.
`
`SUMMARY
`
`A coding system according to an embodiment is config-
`ured to receive a portion of a signal to be encoded, for
`example, a data block including a fixed number of bits. The
`coding system includes an outer coder, which repeats and
`scrambles bits in the data block. The data block is apportioned
`
`2
`into two or more sub-blocks, and bits in different sub-blocks
`are repeated a different number of times according to a
`selected degree profile. The outer coder may include a
`repeater with a variable rate and an interleaver. Alternatively,
`the outer coder may be a low-density generator matrix
`(LDGM) coder.
`The repeated and scrambled bits are input to an inner coder
`that has a rate substantially close to one. The inner coder may
`include one or more accumulators that perform recursive
`modulo two addition operations on the input bit stream.
`The encoded data output from the inner coder may be
`transmitted on a charmel and decoded in linear time at a
`
`destination using iterative decoding techniques. The decod-
`ing techniques may be based on a Tanner graph representation
`of the code.
`
`BRIEF DESCRIPTION OF THE DRAWINGS
`
`FIG. 1 is a schematic diagram of a prior “turbo code”
`system.
`FIG. 2 is a schematic diagram of a coder according to an
`embodiment.
`
`FIG. 3 is a Tanner graph for an irregular repeat and accu-
`mulate (IRA) coder.
`FIG. 4 is a schematic diagram of an IRA coder according to
`an embodiment.
`
`FIG. 5A illustrates a message from a variable node to a
`check node on the Tarmer graph of FIG. 3.
`FIG. 5B illustrates a message from a check node to a
`variable node on the Tanner graph of FIG. 3.
`FIG. 6 is a schematic diagram of a coder according to an
`alternate embodiment.
`
`35
`
`FIG. 7 is a schematic diagram of a coder according to
`another alternate embodiment.
`
`DETAILED DESCRIPTION
`
`40
`
`45
`
`50
`
`55
`
`60
`
`65
`
`FIG. 2 illustrates a coder 200 according to an embodiment.
`The coder 200 may include an outer coder 202, an interleaver
`204, and inner coder 206. The coder may be used to format
`blocks of data for transmission, introducing redundancy into
`the stream of data to protect the data from loss due to trans-
`mission errors. The encoded data may then be decoded at a
`destination in linear time at rates that may approach the chan-
`nel capacity.
`The outer coder 202 receives the uncoded data. The data
`
`may be partitioned into blocks of fixed size, say k bits. The
`outer coder may be an (n,k) binary linear block coder, where
`n>k. The coder accepts as input a block u of k data bits and
`produces an output block v of 11 data bits. The mathematical
`relationship between u and v is v:TOu, where To is an n><k
`matrix, and the rate of the coder is k/n.
`The rate of the coder may be irregular, that is, the value of
`To is not constant, and may differ for sub-blocks of bits in the
`data block. In an embodiment, the outer coder 202 is a
`repeater that repeats the k bits in a block a number of times q
`to produce a block with 11 bits, where n:qk. Since the repeater
`has an irregular output, different bits in the block may be
`repeated a different number of times. For example, a fraction
`of the bits in the block may be repeated two times, a fraction
`of bits may be repeated three times, and the remainder of bits
`may be repeated four times. These fractions define a degree
`sequence, or degree profile, of the code.
`The inner coder 206 may be a linear rate-1 coder, which
`means that the n-bit output block x can be written as x:T,w,
`where T,is a nonsingular n><n matrix. The inner coder 210 can
`
`
`
`US 7,916,781 B2
`
`3
`have a rate that is close to l, e.g., within 50%, more preferably
`10% and perhaps even more preferably within 1% of 1.
`In an embodiment, the inner coder 206 is an accumulator,
`which produces outputs that are the modulo two (mod-2)
`partial sums of its inputs. The accumulator may be a truncated
`rate-l recursive convolutional coder with the transfer func-
`
`tion 1/(l +D). Such an accumulator may be considered a block
`coder whose input block [x1,
`.
`.
`.
`, xn] and output block
`[y1, .
`.
`.
`, yn] are related by the formula
`Y1:X1
`
`Y2 :x1”x2
`
`Y3 :x1”x2@x3
`
`y,,:x1EBx2EBx3EB. .
`
`. 633:"
`
`where “G9” denotes mod-2, or exclusive-OR (XOR), addition.
`An advantage of this system is that only mod-2 addition is
`necessary for the accumulator. The accumulator may be
`embodied using only XOR gates, which may simplify the
`design.
`The bits output from the outer coder 202 are scrambled
`before they are input to the inner coder 206. This scrambling
`may be performed by the interleaver 204, which performs a
`pseudo-random permutation of an input block v, yielding an
`output block w having the same length as v.
`The serial concatenation of the interleaved irregular repeat
`code and the accumulate code produces an irregular repeat
`and accumulate (IRA) code. An IRA code is a linear code, and
`as such, may be represented as a set of parity checks. The set
`of parity checks may be represented in a bipartite graph,
`called the Tanner graph, of the code. FIG. 3 shows a Tanner
`graph 300 of an IRA code with parameters (fl, .
`.
`.
`, 1}; a),
`where fl.:0, Zl.fl.:l and “a” is a positive integer. The Tanner
`graph includes two kinds of nodes: variable nodes (open
`circles) and check nodes (filled circles). There are k variable
`nodes 302 on the left, called information nodes. There are r
`variable nodes 306 on the right, called parity nodes. There are
`r:(kZl.ifi)/a check nodes 304 connected between the informa-
`tion nodes and the parity nodes. Each information node 3 02 is
`connected to a number of check nodes 304. The fraction of
`
`information nodes connected to exactly i check nodes is fi.
`For example, in the Tanner graph 300, each of the f2 informa-
`tion nodes are connected to two check nodes, corresponding
`to a repeat of q—2, and each of the f3 information nodes are
`connected to three check nodes, corresponding to q:3.
`Each check node 304 is connected to exactly “a” informa-
`tion nodes 302. In FIG. 3, a:3. These connections can be
`made in many ways, as indicated by the arbitrary permutation
`of the ra edges joining information nodes 302 and check
`nodes 304 in permutation block 310. These connections cor-
`respond to the scrambling performed by the interleaver 204.
`In an alternate embodiment, the outer coder 202 may be a
`low-density generator matrix (LDGM) coder that performs
`an irregular repeat of the k bits in the block, as shown in FIG.
`4. As the name implies, an LDGM code has a sparse (low-
`density) generator matrix. The IRA code produced by the
`coder 400 is a serial concatenation ofthe LDGM code and the
`
`4
`
`accumulator code. The interleaver 204 in FIG. 2 may be
`excluded due to the randomness already present in the struc-
`ture of the LDGM code.
`
`If the permutation performed in permutation block 310 is
`fixed, the Tanner graph represents a binary linear block code
`with k information bits (ul,
`.
`.
`.
`, uk) and r parity bits
`(xl,
`.
`.
`.
`, x,), as follows. Each of the information bits is
`associated with one ofthe information nodes 302, and each of
`the parity bits is associated with one ofthe parity nodes 306.
`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
`connected to each of the check nodes 304 is zero. To see this,
`set xO:0. Then if the values ofthe bits on the ra edges coming
`out the permutation box are
`
`R
`
`[-1
`X1 = 961-1 + Z W1-1)R+z
`
`.
`(V1,
`l, 2, .
`
`.
`
`.
`
`.
`
`.
`
`, vm), then we have the recursive formula for j’
`, r. This is in effect the encoding algorithm.
`
`Two types of IRA codes are represented in FIG. 3, a non-
`systematic version and a systematic version. The nonsystem-
`atic version is an (r,k) code, in which the codeword corre-
`sponding to the information bits (ul, .
`.
`. ,uk) is (xl, .
`.
`.
`, x, .
`The systematic version is a (k+r, k) code, in which the code-
`wordis(u1, .
`.
`. ,uk;x1, .
`.
`. ,x4.
`
`The rate of the nonsystematic code is
`
`Rn sy:
`
`The rate of the systematic code is
`
`61
`
`Rm =
`
`a+Zifi
`i
`
`For example, regular repeat and accumulate (RA) codes
`can be considered nonsystematic IRA codes with a:l and
`exactly one fl. equal to 1, say fq:l, and the rest zero, in which
`nsys
`case R
`simplifies to R:l/q.
`
`The IRA code may be represented using an alternate nota-
`tion. Let Kl. be the fraction of edges between the information
`nodes 302 and the check nodes 304 that are adjacent to an
`information node of degree i, and let pl be the fraction of such
`edges that are adjacent to a check node of degree i+2 (i.e., one
`that is adjacent to i information nodes). These edge fractions
`may be used to represent the IRA code rather than the corre-
`sponding node fractions. Define }»(x):Zl.}»l.xi‘1 and p(x):
`Zl.pl.xi‘1 to be
`
`the generating functions ofthese sequences. The pair (K, p) is
`called a degree distribution. For L(x):Zl.flxl.,
`
`10
`
`15
`
`20
`
`25
`
`30
`
`35
`
`40
`
`45
`
`50
`
`55
`
`60
`
`65
`
`
`
`US 7,916,781 B2
`
`5
`The rate of the systematic IRA code given by the
`
`L(x)=fl(t)dt/fll(t)dt
`o
`/
`o
`
`ZPj/f 71
`Rate: 1+ J
`_
`J
`Z/lj/J
`
`degree distribution is given by
`“Belief propagation” on the Tanner Graph realization may
`be used to decode IRA codes. Roughly speaking, the belief
`propagation decoding technique allows the messages passed
`on an edge to represent posterior densities on the bit associ-
`ated with the variable node. A probability density on a bit is a
`pair of non-negative real numbers p(0), p(l) satisfying p(0)+
`p(1):1, where p(0) denotes the probability of the bit being 0,
`p(l) the probability of it being 1. Such a pair can be repre-
`sented by its log likelihood ratio, m:log(p(0)/p(l)). The out-
`going message from a variable node u to a check node V
`represents information about u, and a message from a check
`node u to a variable node V represents information about u, as
`shown in FIGS. 5A and SB, respectively.
`The outgoing message from a node u to a node v depends
`on the incoming messages from all neighbors w ofu except v.
`Ifu is a variable message node, this outgoing message is
`
`watv
`m(I4 —> v) = Zm(w —> 14) + m0(14)
`
`where mO(u) is the log-likelihood message associated with u.
`If u is a check node, the corresponding formula is
`
`m(I4 —> v)
`2
`
`tanh
`
`Watv
`_ H tanh
`
`m(w 4 14)
`2
`
`10
`
`15
`
`20
`
`25
`
`30
`
`35
`
`40
`
`Before decoding, the messages m(w—>u) and m(uev) are
`initialized to be zero, and mO(u) is initialized to be the log-
`likelihood ratio based on the charmel received information. If
`
`45
`
`the channel is memoryless, i.e., each channel output only
`relies on its input, and y is the output of the channel code bit
`u, then mO(u):log(p(u:0|y)/p(u:1|y)). After this initializa-
`tion, the decoding process may run in a fully parallel and local
`manner. In each iteration, every variable/check node receives
`messages from its neighbors, and sends back updated mes-
`sages. Decoding is terminated after a fixed number of itera-
`tions or detecting that all the constraints are satisfied. Upon
`termination, the decoder outputs a decoded sequence based
`on the messages
`
`m(14) = Z wm(w ~> 14).
`
`Thus, on various charmels, iterative decoding only differs
`in the initial messages m0(u). For example, consider three
`memoryless charmel models: a binary erasure channel
`(BEC); a binary symmetric channel (BSC); and an additive
`white Gaussian noise (AGWN) channel.
`In the BEC, there are two inputs and three outputs. When 0
`is transmitted, the receiver can receive either 0 or an erasure E.
`
`50
`
`55
`
`60
`
`65
`
`6
`An erasure E output means that the receiver does not know
`how to demodulate the output. Similarly, when 1 is transmit-
`ted, the receiver can receive either 1 or E. Thus, for the BEC,
`ye{0, E, 1}, and
`
`111004) =
`
`+oo if y=0
`if y=E
`0
`if y=1
`—oo
`
`In the BSC, there are two possible inputs (0,1) and two
`possible outputs (0, 1). The BSC is characterized by a set of
`conditional probabilities relating all possible outputs to pos-
`sible inputs. Thus, for the BSC ye{0, 1},
`
`In the AWGN, the discrete-time input symbols X take their
`values in a finite alphabet while charmel output symbolsY can
`take any values along the real line. There is assumed to be no
`distortion or other effects other than the addition of white
`
`Gaussian noise. In an AWGN with a Binary Phase Shift
`Keying (BPSK) signaling which maps 0 to the symbol with
`amplitude \/3 and 1 to the symbol with amplitude —\/K,
`output yeR, then
`
`mo(14)0
`
`where N0/2 is the noise power spectral density.
`The selection of a degree profile for use in a particular
`transmission charmel is a design parameter, which may be
`affected by various attributes of the charmel. The criteria for
`selecting a particular degree profile may include, for example,
`the type of charmel and the data rate on the charmel. For
`example, Table 1 shows degree profiles that have been found
`to produce good results for an AWGN charmel model.
`
`a
`
`A2
`)3
`L5
`A6
`M0
`M1
`M2
`M3
`M4
`M6
`A27
`A28
`Rate
`OGA
`0*
`(Eb/N0) * (dB)
`S.L. (dB)
`
`TABLE 1
`
`2
`
`0.139025
`0.2221555
`
`0.638820
`
`3
`
`0.078194
`0.128085
`0.160813
`0.036178
`
`0.108828
`0.487902
`
`0.333364
`1.1840
`1.1981
`0.190
`—0.4953
`
`0.333223
`1.2415
`1.2607
`-0.250
`—0.4958
`
`4
`
`0.054485
`0.104315
`
`0.126755
`0.229816
`0.016484
`
`0.450302
`0.017842
`0.333218
`1.2615
`1.2780
`-0.371
`—0.4958
`
`Table 1 shows degree profiles yielding codes of rate
`approximately 1/3 for the AWGN channel and with a:2, 3, 4.
`For each sequence, the Gaussian approximation noise thresh-
`old, the actual sum-product decoding threshold and the cor-
`responding energy per bit (Eb)-noise power (NO) ratio in dB
`are given. Also listed is the Sharmon limit (S.L.).
`
`
`
`US 7,916,781 B2
`
`:4
`
`99
`
`the performance
`increased,
`is
`a
`As the parameter
`improves. For example, for a:4, the best code found has an
`iterative decoding threshold of Eb/NO:—0.371 dB, which is
`only 0.12 dB above the Shannon limit.
`The accumulator component of the coder may be replaced
`by a “double accumulator” 600 as shown in FIG. 6. The
`double accumulator can be viewed as a truncated rate 1 con-
`
`volutional coder witl1 transfer function 1/(1 +D+D2).
`Alternatively, a pair of accumulators may be the added, as
`shown in FIG. 7. There are three component codes:
`the
`“outer” code 700, the “middle” code 702, and the “iimer”
`code 704. The outer code is an irregular repetition code, and
`the middle and inner codes are both accumulators.
`
`IRA codes may be implemented in a variety of channels,
`including memoryless channels, such as the BEC, BSC, and
`AWGN, as well as charmels having non-binary input, non-
`symmetric and fading channels, and/or charmels with
`memory.
`A number of embodiments have been described. Neverthe-
`
`less, it will be understood that various modifications may be
`made without departing from the spirit and scope of the
`invention. Accordingly, other embodiments are within the
`scope of the following claims.
`What is claimed is:
`
`1. A method of encoding a signal, comprising:
`receiving a block of data in the signal to be encoded, the
`block of data including information bits;
`performing a first encoding operation on at least some of
`the information bits, the first encoding operation being a
`linear transform operation that generates L transformed
`bits; and
`performing a second encoding operation using the L trans-
`formed bits as an input, the second encoding operation
`including an accumulation operation in which the L
`transformed bits generated by the first encoding opera-
`tion are accumulated, said second encoding operation
`producing at least a portion of a codeword, wherein L is
`two or more.
`
`2. The method of claim 1, further comprising:
`outputting the codeword, wherein the codeword comprises
`parity bits.
`3. The method of claim 2, wherein outputting the codeword
`comprises:
`outputting the parity bits; and
`outputting at least some of the information bits.
`4. The method of claim 3, wherein outputting the codeword
`comprises:
`outputting the parity bits following the information bits.
`5. The method of claim 2, wherein performing the first
`encoding operation comprises transforming the at least some
`of the information bits via a low density generator matrix
`transformation.
`
`6. The method of claim 5, wherein generating each ofthe L
`transformed bits comprises mod-2 or exclusive-OR summing
`of bits in a subset of the information bits.
`7. The method of claim 6, wherein each ofthe subsets ofthe
`information bits includes a same number of the information
`bits.
`8. The method of claim 6, wherein at least two of the
`information bits appear in three subsets of the infonnation
`bits.
`9. The method of claim 6, wherein the information bits
`appear in a variable number of subsets.
`10. The method of claim 2, wherein performing the second
`encoding operation comprises using a first ofthe parity bits in
`the accumulation operation to produce a second of the parity
`bits.
`
`5
`
`10
`
`15
`
`20
`
`25
`
`30
`
`35
`
`40
`
`45
`
`50
`
`55
`
`60
`
`65
`
`8
`11. The method of claim 10, wherein outputting the code-
`word comprises outputting the second ofthe parity bits imme-
`diately following the first of the parity bits.
`12. The method of claim 2, wherein performing the second
`encoding operation comprises performing one of a mod-2
`addition and an exclusive-OR operation.
`13. A method of encoding a signal, comprising:
`receiving a block of data in the signal to be encoded, the
`block of data including information bits; and
`performing an encoding operation using the information
`bits as an input, the encoding operation including an
`accumulation of mod-2 or exclusive-OR sums of bits in
`
`subsets of the information bits, the encoding operation
`generating at least a portion of a codeword,
`wherein the information bits appear in a variable number of
`subsets.
`
`14. The method of claim 13, further comprising:
`outputting the codeword, wherein the codeword comprises
`parity bits.
`15. The method of claim 14, wherein outputting the code-
`word comprises:
`outputting the parity bits; and
`outputting at least some of the information bits.
`16. The method of claim 15, wherein the parity bits follow
`the information bits in the codeword.
`
`17. The method of claim 13, wherein each of the subsets of
`the information bits includes a constant number of the infor-
`mation bits.
`
`18. The method of claim 13, wherein performing the
`encoding operation further comprises:
`performing one of the mod-2 addition and the exclusive-
`OR summing of the bits in the subsets.
`19. A method of encoding a signal, comprising:
`receiving a block of data in the signal to be encoded, the
`block of data including information bits; and
`performing an encoding operation using the information
`bits as an input, the encoding operation including an
`accumulation of mod-2 or exclusive-OR sums of bits in
`
`subsets of the information bits, the encoding operation
`generating at least a portion of a codeword, wherein at
`least two of the information bits appear in three subsets
`of the information bits.
`
`20. A method of encoding a signal, comprising:
`receiving a block of data in the signal to be encoded, the
`block of data including information bits; and
`performing an encoding operation using the information
`bits as an input, the encoding operation including an
`accumulation of mod-2 or exclusive-OR sums of bits in
`
`subsets of the information bits, the encoding operation
`generating at least a portion of a codeword, wherein
`performing the encoding operation comprises:
`mod-2 or exclusive-OR adding a first subset of information
`bits in the collection to yield a first sum;
`mod-2 or exclusive-OR adding a second subset of infor-
`mation bits in the collection and the first sum to yield a
`second sum.
`
`21. A method comprising:
`receiving a collection of infonnation bits;
`mod-2 or exclusive-OR adding a first subset of information
`bits in the collection to yield a first parity bit;
`mod-2 or exclusive-OR adding a second subset of infor-
`mation bits in the collection and the first parity bit to
`yield a second parity bit; and
`outputting a codeword that includes the first parity bit and
`the second parity bit.
`
`
`
`US 7,916,781 B2
`
`9
`22. The method of claim 21, wherein:
`the method further comprises mod-2 or exclusive-OR add-
`ing additional subsets of information bits in the collec-
`tion and parity bits to yield additional parity bits; and
`
`10
`the information bits iii the collection appear in a Variable
`number of subsets.
`
`*
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`*
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`*
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`*
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`*