適用於最小和重組LDPC解碼演算法之補償技術

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(2). . . . . . LDPC .

(3). . . . . Compensation Technique for Min-Sum Shuffled LDPC decoding algorithm. .

(4). . . . . .

(5). . . . . . .

(6) LDPC .

(7). . . . . . . Compensation Technique for Min-Sum Shuffled LDPC decoding algorithm. . .

(8). . StudentMei-Yu Chen . AdvisorChih-Wei Liu . . . IC . . . . . . . . !. A Thesis Submitted to College of Electrical and Computer Engineering National Chiao Tung University in partial Fulfillment of the Requirements for the Degree of Master in Industrial Technology R & D Master Program on IC Design September 2008 Hsinchu, Taiwan, Republic of China. . . . . . . .

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(18) Compensation Technique for Min-Sum Shuffled LDPC decoding algorithm. Student : Mei-Yu Chen. AdvisorDr. Chih-Wei Liu. Industrial Technology R & D Master Program of Electrical and Computer Engineering College National Chiao Tung University ABSTRACT. Shuffled belief propagation (BP) algorithm for the decoding of low-density parity-check (LDPC) codes achieves a remarkable error performance and fast convergence. Nevertheless, it seems to be too complex for hardware implementation. The shuffled BP algorithm can be simplified by using the min-sum approximation, namely the min-sum shuffled BP algorithm; however, the min-sum shuffled BP algorithm suffers from remarkable performance degradation. In this thesis, to solve this problem, we explore some compensation techniques for the min-sum shuffled BP algorithm, including 1D-, 2D-normalization/-offset static schemes and the dynamic scaling approach. Simulations show that the compensated min-sum shuffled BP algorithm achieves the performance very close to that of the original shuffled BP algorithm in IEEE 802.11n system. ii.

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(21) . . . ................................................................................................. 1. . . . . . . . LDPC codes . 2.1. . LDPC }. .......................................................................... 3. .............................................................................................. 4 x. 2.1.1. . }. x. 2.1.2 }. i. ÷øù. x. ÷ø .................................................................................. 4 i. H. ú. .......................................................................... 5. 2.2 û. 2.3.

(22) ............................................................................................................... 10. ü. ý. 2.3.1. . BP decoding ........................................................................................ 12. 2.3.1-1. 2.3.4. BP decoding................................................................ 13. . 2.3.1-1-1 2.3.1-2. I(Quasi-cyclic structured)q

(23) .................................................. 7 þ. BP decoding . . tanh . . . .......................................... 17. BP decoding.................................................................... 23. 2.3.2. min-sum (MS) decoding ..................................................................... 25. 2.3.3. compensated min-sum (MS) decoding ............................................... 26. 2.3.3-1. \]. 2.3.3-2. 1D-normalized MS decoding .......................................................... 31. 2.3.3-3. 1D-offset MS decoding ................................................................... 32. 2.3.3-4. 2D-normalized MS decoding .......................................................... 32. 2.3.3-5. 2D-offset MS decoding ................................................................... 33. . . . ................................................................................. 27. shuffled BP decoding .................................................................................. 34 iv.

(24) 2.3.4-1. . . shuffled BP decoding v BP decoding . ............................... 38. . compensated min-sum (MS) shuffled decoding ....................... 42. 3.1. min-um (MS) shuffled decoding.................................................................... 42. 3.2. compensated MS shuffled decoding .............................................................. 42 3.2.1. 1D-normalized MS shuffled decoding................................................ 42. 3.2.2. 1D-offset MS shuffled decoding......................................................... 43. 3.2.3. 2D-offset MS shuffled decoding......................................................... 43. 3.2.4. 2D-normalized MS shuffled decoding................................................ 44. 3.2.5. static compensated MS shuffled decoding v dynamic compensated MS shufffled decoding........................................................................ 44. . . . . . . ..................................................................................... 45. . 4.1 g. h. \]. . . . ........................................................................... 45. 4.2 k. h. \]. . . . ........................................................................... 49. 4.2.1 k. 4.3 g. 4.4. \]. h h. \]. . \]. vk. . . f. \].

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(26). ........................................................ 53 . ............................................... 55. ............................................................................... 59. Summary...................................................................................... 62. . . . . . . !. " 802.11n # Parity Check Matrix .............................................. 66. . ..................................................................................................... 63. v.

(27) . . . 1. codeword=648 code rate=1/2d2/3d3/4d5/6 g. 2. codeword=1296 code rate=1/2d2/3d3/4d5/6 g. 3. codeword=648 code rate=1/2d2/3d3/4d5/6 k. 4. codeword=1296 code rate=1/2d2/3d3/4d5/6 k h. \]. . ...........61. 5. codeword=1944 code rate=1/2d2/3d3/4d5/6 k h. \]. . ...........61. vi. h. .............60. \] h. . ...........60. h. \] \]. . . .........61.

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(30) i. ...................................................................................................11. . O. O. . I ..........................................................16. : bit node .....................................................20 : check node ...................................................20. ø 14. y = ψ (x) . ø 15. shuffled BP decoding v BP decoding . ø 16. Tanner graph. ø 17. shuffled BP decoding v BP decding f". ø 18. shuffled BP decoding v BP decoding f. ø[12] .......................................................................................26. ú. ................................................37. shuffled BP decoding

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(33) . ø 20. compensated MS shuffled decoding ................................................................45. ø 21. codeword=648 f static compensated MS shuffled decoding . ø 22. codeword=1296 f static compensated MS shuffled decoding . ø 23. codeword=648 f dynamic compensated MS shuffled decoding . . . f. . ................41 . . ........47 . . ......48 . . . ..50. ø 24 codeword=1296 f dynamic compensated MS shuffled decoding . . ...51. ø 25 codeword=1944 f dynamic compensated MS shuffled decoding . . ...52. ø 26. codeword=648 fk. ø 27. codeword=1296 fk h. \]. . v SNR fë . ø.....................................54. ø 28. codeword=1944 fk h. \]. . v SNR fë . ø.....................................54. ø 29. codeword=648 coderate=1/2d2/3 f compensated MS shuffled decoding . ø 30. . v SNR fë . ø.......................................53 . codeword=648 coderate=3/4d5/6 f compensated MS shuffled decoding . .............................................................................................................57 . codeword=1296 coderate=1/2d2/3 f compensated MS shuffled decoding . ø 32. \]. .............................................................................................................56. . ø 31. h. . .............................................................................................................58. . codeword=1296 coderate=3/4d5/6 f compensated MS shuffled decoding . . . .............................................................................................................59. viii.

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(136) bit node v check node w 11. . . 5. O. 4 {. |. p.

(137) . i. . G Tanner ø. K H fB mÁ. u. i. K 1 P . [5)$. B. check node vP j. check node :. n Á. % $. . . ø7¥. # ,. * -. * ,. * +. F. $. +. *. Tanner Graph O. bit node ° 4. bit node :. H 8v. . * /. * .. #. ø 7. Tanner Graph. 2.3.1 BP decoding BP decoding K . .

(138) [2][4]$BP decoding Q . R . IH. v tanh . BP decoding(BP Based on the tanh rule)[9)$. G. min-sum decoding tanh . . . BP decoding(Belief Propagation Based on the Gallager’s Approach)[9]. 4S. . i. BP decoding 6. BP decoding 6G . shuffled BP . decoding$ BP decoding c node , ratio H. . -. . ( S E. F. w. . 0. . J LLR) ~ Sum Product F. propagation decoding)W eT .

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(140) $¥GJKTU. 1 (log-likelihood w. V.

(141) (belief.

(142) (message passing decoding)W $%&. ¥GÙ. algorithm SPA)[8]$ 12. Ji. m. . . . .

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(144)

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(153) Step 1 : 1.1 . -.(bit node)v. B. B. . i K 1. ò. I(bit node initialized). -.. &. -.(check node). . (iteration) I max K?. . iteration$ X. (0) Z mn = Fn. 1.2. (2-13). -.. (check node update) ò. ∏ sgn( Z. E ( i ) mn = Sign × Magnitude = {. n '∈N ( m ) \ n. n '∈ N (m) \ n Á. ( i −1). v check nodeTmWF . . ψ | Z mn (i −1) |) |}. )}.{ψ −1 | (. mn '. '. n '∈N ( m ) \ n. bit node ]. í ^. 5. | x| e| x| + 1 ψ (| x |) = ln{tanh ( )} = ln | x| 2 e −1. |x| e | x| − 1 ψ (| x |) = ln | tanh |= ln | x| 2 e +1. −1. −1. if Z mn > 0 , Sign = 1 else Sign = −1 1.3 . bit nodeTnW $. (2-14). -.(bit node update) B. (i ) Z mn = Fn +. E m( i'n) ;. m '∈M ( n ) \ m. Step 2 : /. V. . 2.1 /. . (hard decision). V p. . Z n( i ) = Fn +. `. v_. ¨. m∈M ( n ). (2-15). Ì. LLRTZnW . (i ) E mn. /. . V. . . c. ∧. B. . . $. ∧. if z n > 0 then wn = 0 ; if z n < 0 then wn = 1 2.2 _ ` G_. . `. . .

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(158) ∧ (i ). w i . . . . Fn(,. . 1 bit node *. . 3. . . . !. ". #. 8~ 11 . $%. &. '. bit node *. ().

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(160) LLR)0 check nodeStep1.2 check node /. 4.

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(168) check node ?. Zn(2)A 6 B. ; H OP c. #.

(169) Zmn(2)4 .

(170) 1 check node update 9 Emn(2)c #.

(171) bit node update J. ?. N :; H OP LQOP. bit node 7. b Step1.3bit node 1M. 9I. >. 1 bit node Zn(1)

(172) LLR. (). N ab Step1.2check node > 6. =. . . Step1.3 M. A.

(173) code wordWRQX V. H. K L Step2.1 M. 9I. L Step2.2 M. G. Fn $< . h. ?. 9I. . U. WRQX V. codeword ,i. 15. =. >. 2 ?. . ab Step2.11 bit bode Zn(2). fY. L Step3. *0 bit nod`a. Emn(2); Fn $< . check node. j. \ ?. :] k. l. ^ . _. C . m. . n. k. L ). :.

(174)

(175) %% $ $. # #. +. F2. ,. F2. F6 F4. F3. F1 *. *,. *+. F6. F5. *-. *.. */. # #. 8. BP decoding Step1.p ( bit node % &. '. q. q.

(176) !"#%$&#'(*)+,- %$ %$. # # +. ,. c1 E11. E11. E22. E32. E21 *. *,. *+. E36. E14. E33. E25. *-. E16 *.. Z14. b1. .0/2143. b4. Z16 .0/2165 b6. */. # #. 9. BP decoding Step1.2 ( check node update. 789:<; = > ?@,8"ABC9EDFAHG:ICIJ809 %$ %$. # # +. E11. *. E22. E21 Z1. F1. *+. F2. Z2. E32. E33. *,. ,. E36. E14. Z3. F3. *-. F4. E16. E25 Z4. *.. Z5. F5. #. 10. BP decoding Step1.3 ( Zn bit node update 16. */. F6. Z6.

(177) KIL,MONQPSR TVUXW0LYZ\[\M^]`_aY*b N [cdL(Mfe LhgMiYHjkMiY[*L0Z*ligMlimYZ\[\M %% $$. # # +. ,. c1. c2. Z11 Z11. Z22. Z21. Z25. Z33 Z14 *. *,. *+. E21. Z36. Z32. b1. Z16. *-. *.. F1. */. # #. 11. BP decoding Step1.3 ( Zmn bit node update. 2.3.1-1-1 BP decoding BP decoding r. Fs. t. s. p( w j = 1 r j , S j ). LLR posterior ( w j ) > 0S T y. l. 1

(178) x z. w

(179) LLR . . p( w j = 0 r j , , S j ). LLR posterior ( w j ) = ln. 0

(180) x. 2-16 F

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(187) M. . . . bit node +. module 2), ⊕ § . . p( s j w j = 1, r ). S j = {S 0 j , S 1 j ,..., S kj } ; Smj S. . (2-19). ( r j + (−1)1 ) 2. 2r j. =. .. . p( r w j = 1). = ln. . (2-18). . 1. (2-18)F ?. p (r w j = 0). ln. . >. p ( s j w j = 1, r ). .

(188) 5. . p ( s j w j = 0, r ). + ln. p(r w j = 1). (2-18)F

(189) . . p( w j = 1 | r , s j ). p ( r w j = 0). = ln. e. p( w j = 0 | r , s j ). LLR ( w j ) = ln. . (2-17)F

(190) APP d. ¨. \. J. ?. . . . J. (parity bit term) K. ¡ 2-12 F D. J. . . E. check node

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(196) (2-20)F¬. b=0 . parity bit term r. . D. «. 9 2-22 F. ∏ p( S. m∈M ( j ). mj. w j = 0, r j ). 1 0 (1 + ∏ (q mn − q 1mn )) m∈M ( j ) 2 n∈N ( m ) \ j. ∏. ∏ p(S. p ( S j w j = 1, r j ) = =. 9 2-21 F} b=1 «. F :. p ( S j w j = 0, r j ) = =. D. m∈M ( j ). mj. (2-21). w j = 1, r j ). 1 0 (1 − ∏ (q mn − q 1mn )) 2 m∈M ( j ) n∈N ( m ) \ j. ∏. 2-21 F; 2-22 Fr. J. J. K. 0

(197) x y. qi. Fs. t. L_. (2-22) . . J. K. xi 1

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(199) R. 2-24 F z. p( x1 ⊕ x 2 = 0) = (1 − 2 p1 ).(1 − 2 p 2 ). (2-23). p( x1 ⊕ x 2 = 1) = −(1 − 2 p1 ).(1 − 2 p 2 ). (2-24). . L+1 J. nodeF 2-25 T N. K. . . . J. . . ZL 0 ±

(200) ¶.

(201) ´ ·. ZL S µ. L+1 bit node T. }F 2-26 Y T. N. ZL 1 ±

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(203) check . ·. . z L = x1 ⊕ x 2 .... ⊕ x L = 0 ; z L −1 = x1 ⊕ x 2 ... ⊕ x L −1 (2-25). L. 2 p ( z L = 0) − 1 = ∏ ( q i − p i ) i =1. L. 2 p ( z L = 1) − 1 = −∏ (q i − p i ). (2-26). i =1. 19.

(204) 2-27 FT N. !. ¸. . check node C. .. p ( S mj w j = 0, r j ) =. 1 0 1 1 + ∏ (q mn ) − q mn 2 n∈N ( m ) \ j. p ( S mj w j = 1, r j ) =. 1 0 − q 1mn 1 − ∏ (q mn 2 n∈N ( m ) \ j. 12. 2-28 FT N. !. ¸. . ∏ p( S. mj. .. ^ L 12 M. N . (2-27). check nod C. m∈M ( j ). ?. .. ?. ¹ bit node. ¹ check node

(205) ¶. ^ L 13 M. N . w j = 0, r j ). 1 0 1 (1 + ∏ (q mn )) '− q mn ' m∈M ( j ) 2 n '∈N ( m ) \ j. ∏. ∏ p( S. p ( S j w j = 1, r j ) = =. . bit node C. p ( S j w j = 0, r j ) = =. ¸. ¹ bit node

(206) ¶ ?. m∈M ( j ). mj. w j = 1, rj ). 1 0 1 (1 − ∏ (q mn )) '− q mn ' ' m∈M ( j ) 2 n ∈N ( m ) \ j. ∏. 13. 1 2-28 FS º. ¸. . C. bit node . 2-18 F> ?. _. ». 20. .. ?. (2-28). ¹ check node.

(207) 2-29 F.

(208) LLR ( w ) = ln 0 j. ∏a. i. ¼. i. ½. p( w j = 0 | r , s j ) p( w j = 1 | r , s j ). = (∏ sgn( ai )) * exp( i. 2-30 F

(209) . LLR ( w ) =. +. ln |. 1−. ∏ δq. mn. (2-29). ln | ai |).. D. (2-30). 9 2-31 F '. ∏. sgn(δq mn ). exp(. ∏. sgn(δq mn ). exp(. n∈N ( m ) \ j. 1_. m∈M ( j ). ln m∈M ( j ). mn. i. 1+. σ2. +. σ2. ∏ δq. n∈N ( m ) \ j. n∈N ( m ) \ j. 2-29 F ¾. 2r j. 0 j. =. 1+. 2r j. n∈N ( m ) \ j. 0 1 0 δq mn = q mn − q mn ; LLR ( wmn ) = ln. ln | δq mn |). n∈N ( m ) \ j. ln | δq mn |). n∈N ( m ) \ j. |. (2-31). 0 0 q mn LLR ( wmn ) q = ; δ tanh( ). mn 1 2 q mn. (2-32) ¡ 2-32 F

(210) . 2r j. LLR ( w ) = 0 j. ∏sgn(δq. n∈N ( m ) \ j. s mj =. Amj =. σ2. mn. 2-31 F . +. ln |. 1 − s mj e. ∏sgn[LLR(w. 0 mn. n∈N ( m ) \ j. ∏ sgn[LLR(w. n∈N ( m ) \ j. ln | tanh n∈N ( m ) \ j. 9 2-33 F '. 1 + s mj .e. m∈M ( j ). )=. D. 0 mn. Amj. Amj. |. (2-33). )]. )]. 0 LLR ( wmn ) 0 |= ψ ( LLR ( wmn )) 2 n∈N ( m ) \ j. LLR ( w 0j ) = ChannelValue +. ln | m∈M ( j ). Smj=1 -1 If Smj=1 . . . s mj .e. Amj. +1. s mj .e. Amj. −1. . |. (parity bit term) . 21.

(211). 2-34 .

(212) ln | If. e. Amj. +1. e. Amj. −1. |= S mj ln | tanh −1 (. Amj 2. )|. Smj=-1. ln |. −e. Amj. +1. −e. Amj. −1. |= S mj . ln | tanh−1 (. Amj 2. )|. (2-34). LLR( w 0j ) = ChannelValue +. ln | m∈M ( j ). =. 2r j. σ. S mj {ψ −1 (. +. 2. m∈M ( j ). 2-35 .

(213). s mj .e. Amj. +1. s mj .e Amj − 1. |. (2-35). 0 Ψ ( LLR( wmn ))}. n∈N ( m ) \ j. 2-36 . LLR( w 0j ) =. 2r j. σ. 2. s mj =. S mj {ψ −1 (. +. m∈M ( j ). ∏ sgn[LLR(w. 0 mn. n∈N ( m ) \ j. Amj =. Amj )}. n∈N ( m ) \ j. (2-36). )]. 0 ψ ( LLR ( wmn )). (2-37). n∈N ( m ) \ j. e −1 | ex +1 x ex +1 ψ −1 ( x) = ln | tanh | −1 = ln | x | 2 e −1 x 2. ψ ( x) = ln | tanh |= ln |. x. 0 Z mn = LLR ( wmn ). 2-37 2-38 . smj =. ∏ sgn(Z. n '∈N ( m ) \ j. ). mn '. Amj =. ψ | Z mn | '. n '∈N ( m ) \ j. 22. (2-38).

(214)

(215). (check node update) 2-39 . . . check node update . . bit node update 2-40. (2-39) bit node updat . (2-40). Check node update : E ( i ) mn = Sign × Magnitude = {. ∏ sgn( Z. n '∈N ( m ) \ n. ( i −1) mn '. )}.{ψ −1 | (. ψ | Z mn (i −1) |) |} '. n '∈N ( m ) \ n. (2-39) Bit node update :. Z mn = Fn +. E. (2-40). m 'n m '∈M ( n ) \ m. 2.3.1-2 tanh BP decoding Check node update tanh . δq mn. . . BP decoding . . 2-41 . !" 2-43. 0 LLR ( wmn ) = tanh( ). 2. LLR ( w ) = 0 j. 2r j. σ2. (2-41). 1+ +. ln m∈M ( j ). ∏ δq. mn. ∏ δq. mn. n∈N ( m ) \ j. 1−. n∈N ( m ) \ j. LLR ( w ) = 0 j. 2r j. σ2. 1+ +. ∏. 0 LLR ( wmn ) ) 2. ∏. tanh(. 0 LLR ( wmn ) ) 2. n∈N ( m ) \ j. ln m∈M ( j ). 1−. (2-42). tanh(. n∈N ( m ) \ j. #. 2-42. 0 $ LLR ( w 0j ) = Z n % LLR ( wmn ) = Z mn Fn =. Tmn =. ∏ tanh(. n '∈N ( m ) \ n. Z mn' 2. ) 2-43 )*+. 23. 2r j. σ2. (2-43) &'. (. 2-44 . .

(216) Z n = Fn +. E mn. tanh (i ) mn. T. E mn. (i ). = ln. ln m∈M ( j ). 1 + Tmn. (i ). 1 − Tmn. (i ). 1 + Tmn 1 − Tmn. BP decoding ,-. =. ∏ tanh(. = ln. 1 + Tmn. (i ). 1 − Tmn. (i ). /. 0. 1. 2. (check node update). . ( i −1). Z mn'. n '∈N ( m ) \ n. (i ). .. (2-44). 2. ). 3 (i ). (i ). E mn = 2 tanh −1 (Tmn ). (2-45). 24.

(217) 2.3.2 min-sum (MS) decoding BP decoding . . | ln tanh −1 ( x) | :. 9. , check node update 4. ;!<=,>. M.P.C.FossorierM.Mihaljevic I ; bit node ,O. (ψ −1 | (. J. ?. @. A. B. $ 1999 LM K. Q( 'min {| Z mn' |} )R P. (6 2-14 )7 5. D. E. <=,F. G. H. . [9]N check node update. . check node update , magnitude Q S. n ∈N ( m ) \ n. . 8C. 8" ln|tanh(x)|. ψ | Z mn |) | )TN<=UV W ;C XY Z[Z\ ]. ln|tanh(x)|. '. n '∈N ( m ) \ n. | ln tanh −1 ( x) | :. 9. , ; ;< = >. min-sum decoding( * a. ^ 8 * ) _. `. a. b. c. d. e. 8. MS decoding) 3 uniformly most powerful( * a. UMP). BP-based decoding[10] 8C. f. min-sum decoding ' g. y = ψ (| x |) = ln(tanh. decoding ib l. |x|Qw. ,R. (. h. F. ij. ",ψ (| x |) : M. . |x| ) % 2. y ' = ψ −1 (| x |) = ln(tanh. S. ψ (| x |) :. mn. P xy. ,|y|Qw. t. u. pq. r. mv. t. . pq. r. 8ψ (| x |) ψ −1 (| x |) . 8z. ^z. ,o. i{. pq. r. . ψ (| x |) ,t. $:. . }. [10]. ψ −1 (ψ (| x |) =| x | #. [10][11s. | } 2-20 i~. 2-46 + :. | x | −1 ) kin-sum 2. (2-46) u. pq. r. ψ | Z mn | QO ^,ψ | Z mn | ,Q. . '. '. n '∈N ( m ) \ n. R. S. ^,ψ | Z mn ' | &. O. O. , | Z mn' | P. [10]' . (. . ψ | Z mn | ≈ max ψ | Z mn |= ψ { min | Z mn |} '. n '∈N ( m ) \ n. &7. 8:. '. n '∈N ( m ) \ n. ψ (| x |) ,t . pq. r. n '∈N ( m ) \ n. '. (. R. S. 8 2-57 . 25. . '. 2-56 . (2-47). ψ |{. ψ | Z mn |} |. n '∈N ( m ) \ n. . !. '. .

(218) ψ −1 | {. ψ | Z mn |} | '. n '∈N ( m ) \ n. ≈ψ. −1. | ψ { 'min | Z mn' |} |= 'min | Z mn' | n ∈N ( m ) \ n. n ∈N ( m ) \ n. Min Sum decoding . update 2-20 )*+. E mn = {. ∏ sgn(Z. n '∈N ( m ) \ n. Min Sum. (2-57). decoding. E mn = {. . . . . BP decoding , check node. 2-58 (2-58). )}.{ 'min | Z mn' |}. mn '. n ∈N ( m ) \ n. , check node update . ∏ sgn( Z. n '∈N ( m ) \ n. :. )}.{ 'min | Z mn' |}. mn '. n ∈N ( m ) \ n. y = ψ (x) :. } 14. }[12]. 2.3.3 compensated min-sum (MS) decoding min-sum decoding ;O =F. G. H . +. f. . P. ¢. [14]8C. U£. ¤. S. | b. , check node ;( ¡. R. ¥. ¦. a. BP decoding , ln|tanh(x)| | c. . . c. "D. E. <. D ;x min-sum decoding. ) normalization 3 b. . . . offset ),U. U8 compensated min-sum decoding(* 26.

(219) 8 compensated MS decoding) a. x$. , LDPC § ¨ ¥ . Compensation factor)ª x$¬ X² M. ¯ |. . . )3. check nod«. ¬. , LDPC § W . ³. N¬. ^,. ´. u. . . b. p¸. ¹. . W © R. u. [15][17]x$ . ¶. ®. m. , BP decoding ± °. J.Zhang M. Fossorier I µ. p(1D. BP decoding ,. pm¨. p(2D compensation factor)@. ;o. f ¼. ¨. m. J. K. $ 2005 L. x check node bit node ·. , LDPC º. ;o. . p». , check node bit node ,m± ,[16] ½ #. compensated min-sum (MS) decoding &@. $,¨. 8. 1D-normalized min-sum (MS) decoding ¾ 1D-offset min-sum (MS) decoding ¾ 2D-normalized min-sum (MS) decoding¾2D-offset min-sum (MS) decoding ¿ c. . 2.3.3-1 '. (. 1D-normalized MS decoding 8À. density evolution U[14] . 1D-normalized 8C. U£. . . )(normalization factor)'. . decoding , check node update 4 Â. 8 0 ÃÄ. " y n = s n + v n vn 8É ,.

(220). n , LLR Q(3a. 5. 9. Ê. 8-1Â. Ç. %Ë È. Ì. <. (. ª. ¡. BPSK BP. Á. U. . . 8 1 ÃÄ. 8 1Å. Æ AWGN Ç. Q8 0Í8 N 0 / 2 Fn 8f. Q). BP decoding , check node update : 27. È. Z. ! Î.

(221) Tmn =. ∏. n '∈N ( m ) \ n. E mn = ln. <. 1 + exp(Z mn '). (2-59). 1 − Tmn 1 + Tmn. (2-60). : Á. if. 1 − exp(Z mn'). ∧. z n > 0 then w n = 1 ; if. ∧. z n < 0 then w n = 0. BP decoding check node update 8 E mn MS decoding check node update 8 '. z = Z mn '!" 2-61. E mn 2-59 BP decoding , check node updaet . ∏. n '∈N ( m ) \ n. 1 − exp(Z mn') 1 + exp(Z mn'). =. 1− ez 1+ ez. (2-61). '. Ï | E mn |=| z | ; | E mn |= 'min | Z mn ' | n ∈N ( m ) \ n. 1 − exp( ∈ min Z mn') 1− ez n ∈N ( m ) \ n < z 1 + exp( 'min Z mn' 1+ e n ∈N ( m ) \ n. (2-62). 1 − e x1 1 − e x2 < ⇔ x1 < x 2 1 + e x1 1 + e x2. Ð 2-63 !. (2-63). z < 'min Z mn' ' n ∈N ( m ) \ n. (. '. !" E mn > E mn Ñ«. m MS. , check node , magnitude Q^$ BP , check node , magnitudeBeta 8 ) BP decoding check node ^P Ë Ì Qx MS decoding , check node ^P Ë Ì Q,XQ. Beta =. E ( E mn ) '. E ( E mn ). 0 < Beta < 1. (2-64). 28.

(222) 1D-normalized factor ,Ò '. (. Ðt. u. Ó. ,ÔÕ. f. ¥. . . Ö. [10][13] !. )T7. 8½. x×. pÔÕ. f. ,Ø. Ù. O. ^ 8C. ß. n. Ú. 6'. (. Ï { X i : i = 1,2,.., W } = {Z mn ' : n '∈ N (m) \ n}W = λ − 1λ . p check node bit node ±Ü. Û Ê. £. à. H. )'. (pdf) SNR Q9 :. ,pXi m á. ±Ý. . (code rate)âã ä. ,É. Ê. @. Þ. TXi ,. âMS decoding ,å æ. Zmn=4/No (. 1 − exp( X i ) i =1 1 + exp( X i ) E ( E mn ) = E ln W 1 − exp( X i ) 1+ ∏ i =1 1 + exp( X i ) W. 1− ∏. (2-65). ( ) '. E E mn = E (min( X 1 , X 2 ,..., X W )) '. (. è. ê. Ö. é. '. E E mn , E E mn ç . ë. '. (. é. >. e. (2-66). 2-64 £. Ö. . Beta. '. E E mn Ï Yi = X i , i = 1,2,...,W TYi , pdf 8 2-67. . f Yi ( y ) = ( f X i ( y ) + f X i (− y )u ( y ) = 2 ⋅ f X i ( y )u ( y ) N 2-67 . f X i m Xi , pdfu(y)m y ,ì í. :. (2-67) . '. Pr( E mn > y ) = Pr(min(Y1 , Y2 ,..., YW ) > y ) = Pr{Y1 > y, Y2 > y,..., YW > y} = [Pr(Y1 > y )]W. (2-68). 29.

(223) '. x$ E mn > 0 !" 2-74 ∞. '. '. E ( E mn ) = Pr(|E mn |> y )dy (2-69). 0. ï 2-68 2-69 '. î. (. !". ∞. '. E (| E mn |) = [Pr(Y1 > y )]W dy 0. = =. ∞. ∞. 0. y. W. f Y1 ( y1 )dy1 u. 1 − Q(. 0 ∞. + [Q( u. u−y. σ y −u. σ. 2-70 ,t . ñ. @ u. '. ¬. P. E ( E mn ) ≈ [1 − Q( 0. 8C. W. α =∏ i =1. σ y +u. ) + Q(. σ. u− y. σ. ) + Q(. (. W. ). dy. )]W dy (2-70). (0). . ! E (| E mn |) ' Ö. u+ y. ) + Q(. 4 (Θ LLR ( Z mn ) = y n ) Q ( x) = N0. 8 4 σ 2 = u = N0 N0 ð. dy. . ò. u+ y. σ. ó. '. (. 1 2π. x2 2. ∞. e dx x. 2-70 )*!". )]W dy (2-71). ô. 1 − e Xi 1 + e Xi (2-72). n. õ. ln ð. ö. ÷. ]. . !. 1−α α3 α5 = −2(α + + + ...) 1+α 3 5 α , α 3 , α 5 ,... â± ,ø ù. (2-73) (sign) 2-65 2-73 ,ú. !" 2-74 . 30. ÷. ] .

(224) E ( E mn ) = E ln. 1− α 1+α. =2. Ï mk = E (| α | k ) ð. 1 − e Xi mk = E ∏ Xi i =1 1 + e W. { X i } mÉ Ê. (2-74). '. (. . !. k. W. Xi k. 1− e 1 + e Xi. = E. E (| α | 2 k −1 ) 2k − 1 k =1 ∞. [ ] = [E (tanh(Y / 2) )] = E (tanh(| X i | / 2) k ) k. W. W. (2-75). i. 2-75 . 2-74 !". E (| E mn |) = 2(m + m3 / 3 + m5 / 5 + ...) ^4. 5. 2-76 ,j ü. ý. «. E (| E mn |) O þ. '. ! E (| E mn |) 2-76 , E (| E mn |) . Ö ã. $ÔÕ. . ,û. (2-76). Å. ¬. )®. . . R. m». ¼. . ),ã ,. '. )T'. !",. . . '. 2-64 !" ( (. )xf. » Ñ. ¼. t . ,. ¨ ¡. 1D-normalized ± Ñ;Ö . ,U. Ó. _. u. Ó. ) Beta. ,ÔÕ. ,ÔÕ ¢. Ø. > ,Ó. Ù. 2-71 (. ¨. e. , >. ) e. . ^. density evolution Ë Ì. Q. 1D-offset2D-normalized2D-offset [14]. 2.3.3-2 1D-normalized MS decoding # . $ 1D-normalized MS decodingmx MS decoding , check node . ) [14] u. pª. decoding , check node update ª ¡. u. p. MS decoding 1D-normalized MS. 2-77 bit node update 4 31. 5. MS decoding.

(225) ± [14][17]. (i ). E mn. =[ '. ∏ sgn(Z. n ∈N ( m ) \ n (i ). Z mn. = Fn +. ( i −1) mn. '. )].[ min{ Z mn'. ( i −1). .β }]. '. n ∈N ( m ) \ n. | E m'n |(i ) ; Z n. (i ). m '∈M ( n ) \ m. = Fn +. Emn. (2-77). (i ). m∈M ( n ). 2.3.3-3 1D-offset MS decoding $ 1D-offset MS decodingmx MS decoding , check node. #.

(226) [13]u. pª. node update ª. Emn. (i ). ∏ sgn( Z mn'. ( i −1). =[ '. n ∈N ( m ) \ n. Z mn. (i ). = Fn +. p. . Q. MS decoding 1D-offset MS decoding , check. 2-78 bit node update 4 ¡. u. 5. )].[min{ Z mn'. MS decoding ± [13] ( i −1). − α }]. '. n ∈N ( m ) \ n. | E m'n |( i ) ; Z n. (i ). m '∈M ( n ) \ m. = Fn +. E mn. (i ). m∈M ( n ). (2-78). 2.3.3-4 2D-normalized MS decoding u. p β1 ,. )o. p. 2D-normalized MS decoding m MS decoding , chcek node @. )x MS decoding , bit node ¶. ª. . check node bit node[16][17] 2D-normalized MS decoding , check. node update bit node update 2-79 . E mn Z mn. p β 2 , u. (i ). =[. ∏ sgn( Z mn'. ( i −1). n '∈N ( m ) \ n (i ). = Fn +. )].[min{ Z mn'. [16][17] ( i −1). .β1 }]. n '∈N ( m ) \ n. {| E m'n |(i ) .β 2 } ; Z n. m '∈M ( n ) \ m. 32. (i ). = Fn +. E mn. m∈M ( n ). (i ). (2-79).

(227) 2.3.3-5 2D-offset MS decoding 2D-offset MS decoding mx MS decoding , chcek node. x MS decoding , bit node. u. p α 2 , . o. p. p α 1 , u . @. ¶. . ª. . check node bit node[16] 2D-offset MS decoding , check node update bit node update 2-80 . E mn Z mn. (i ). =[. ∏ sgn(Z. n '∈N ( m ) \ n (i ). = Fn +. [16] ( i −1) mn '. )].[ min{ Z mn'. ( i −1). − α 1 }]. n '∈N ( m ) \ n. {| E m'n |( i ) −α 2 } ; Z n. m '∈M ( n ) \ m. 33. (i ). = Fn +. E mn. m∈M ( n ). (i ). (2-80).

(228) 2.3.4 shuffled BP decoding BP decoding ; node ¨ ;^ . ÔÕ. 8C. . ,. [19]ÔÕ. f. Ó. Ó. . =. ,U @. ZF. G. ,Ü. . f . . [20]b c. " . . . ©. . Y. ¯. ÔÕ. f. ;!<=F. J.Zhang M. ¾*)<=,>. . . . Ó G. H. node @ C. @. ,f. .. !. . . Ua. [21] . M. e. f. . 8 shuffled. BP decoding[18][19] ,Uâo @. partitioning) [21]o '. (. c. " è. §. c. m. U%;ÔÕ #. U @. é '. f. u. [5][21] u c. (. é. .. . Ó. -. c. m. .. !. $. f ). Ë '. ,&. * Ñ". #. U@ 1. ". shuffled BP decoding ,. U@. (vertical. (horizontal partitioning). . ,U+ @. #. i_. Ñ±S. [5][21]N(. shuffled BP decoding ,. H Tanner }f /. Æ0 Â. 1. 8 N N p bit node @ H. +. G Û u. â N G p bit nodes. N G = N / G (% N mod G=0) Step 1 : bit node update and check node update 1.1. bit node initialized. . i 8ÔÕ å. æ. Ó. (iteration) I max 8O. ^ iterationbit node Nt. 0Ó. ÔÕ. Ã. )8(2-81). Fn =. 2r j. σ2. ;. ( 0) Zmn = Fn. (2-81) 34.

(229) 1.2. Horizontal step : check node update For 1 ≤ g ≤ G m ∈ M (n) (i ) mn. T. E mn. ∏. =. tanh(. n '∈N ( m ) \ n n '≤( g −1). N G. (i ). = ln. 1 + Tmn. (i ). 1 − Tmn. (i ). (2-82) u. (i ) Z mn '. 2. ). ∏. ). (2-82). (2-84). check node update Ã2 5. (i ) 67 Z mn N4. (i), bit node 2. ( i −1) , bit node 867 Z mn >. 1.3. 2. n '∈N ( m ) \ n n '> ( g −1). N G. (i-1), bit node QN4 Ó. tanh(. ( i −1) Z mn '. b. e. 5 b. 3. ij u. ¶. ,¨. . . , n '≤ ( g − 1).N G ; u. , n ' > ( g − 1).N G ;u. i_. (i)3. ; Ó. Ó. Ó. ÔÕ. ÔÕ. (i-1). (i ) Tmn . Vertical step : bit node update m ∈ M (n) (i ) Z mn = Fn +. Step 2 : < 2.1. < <. Á. Z n( i ) = Fn +. E m( 'in). (i ) E mn. m∈M ( n ). m '∈M ( n ) \ m. :. 9. ;. (2-85). . (hard decision) Á. =. LLR>Zn?Å . Æ<. Á. @. ∧. A .

(230). = . . ∧. if z n > 0 then w n = 0 ; if z n < 0 then w n = 1. 2.2 9. :. B 9. :. ÔÕ. f. C. . ÔÕ. Step 3 : = . f. ∧ (i ). w .H T = 0 3 þ. . ". f. . ,Â. 35. ,O. ^ÔÕ. f. Ó. I max . .

(231) ∧ (i ). w D. 8'. iÓ. Nt. (iteration)f . ÔÕ. m shuffled BP decoding . « !. (. } 15 } 16 §. 8À. 1. G Ó. ". #. H. ÔÕ. I. f. J . ,2. H. Q M. L. . #. . . } 15 O j. , bit node 8Q(M. V. ,>. H. L. update ÃP"O. Y. O. O ¯. L. i-1 Ó. Ó. iÓ ¯. P"1. f. )· ÔÕ. f. . T. >. e. i-1 f. ¯. 2. ,Â. ,ÔÕ. j. Q. ±z. O. R. S. 8. check node update. u. @ V. O. _. V Q(K. 1 A. 2. O. V. ,. , bit node 2. Q. L . _. i-1 Ó ¯. ,> H. . f. . check node e. )· J. ÔÕ. >. e. . ( shuffled BP decoding . Ñ7. . . ,¨. Ó. V. ,U @. f. U. I. bit node update check node W . ,8 bit node 67(M. X shuffled BP decoding 1. ¯. , bit node 2 f. "â, check node % update NV ÔÕ. i. ,8Q. check node updateP"XY e. codeword +. bit node Nt J. ÔÕ. 67·. BP decoding â, bit node 8u. ;Ó. H. bit node check node ¯. update67,1. . ,ÔÕ. J . .. N check node update bit node update i_ . check node update Ã« e. . . QÉ. -. . , bit node update N,2. , bit node =. É. . F @. , check node update bit node update ,Uf u. $u. N>. ,(6,3)code ,. V. p bit node 8u. bit node Nt . Shuffled BP decoding u /. j. (. node update ,K . N. N'. Û @. ,N(8 6 } 15 H. . shuffled BP decoding Æ0 E. shuffled BP decoding H ·. ¼. ,Â. H f. 36. L . . )# T. Â. $671 . 2. W. . ©. .

(232) $( (. .. 1. 1. /. + 1/ 1+ ,. n odp. 1,. 0. 1. 2. 3. &. 4. 1-. } 15. ê u. ë. shuffled BP decoding BP decoding ,XY. } 16 Tanner graph ê ë. f. ·. Æ0± Ñmt. u. shuffled BP decoding ,f. N check node update bit node update/. , check node update bit node update# ÔÕ. f. node 2. M. H. L . Q ;H. l X. i-1 Ó , node . Æ0} 15 ; H. ÔÕ. " 6 %·. bit node 8QK. N update , node. 37. H. N8: L . V. e. i Ó. N+ ÔÕ. · u. Ó. bit.

(233) } 16. Tanner graph ê. shuffled BP decoding f ë. Æ0. 2.3.4-1 shuffled BP decoding BP decoding Nb Íè o. é. u. c. ' d. 19 . e. / '. (. (. f. ·. . Y ©. Z. \ XY. shuffled BP decoding BP decoding o [. 0}] ^ (. ,f. o. c. d. &. e '. , 1. . ¶. 4. } 17 . 9. ÔÕ. f. Ó. d T. e. , . '. (. } 18¾. . } 17 8 shuffled BP decoding BP decoding ,f. . c. ¶. ;j. (i ). N$ check node update Tmn ,¨ u. Ó. ÔÕ. Æ0½. (. ,f. Æ0±. BP decoding , check node update _. ( i −1) > , bit node Q Z mn. e 38. Shuffled BP decoding . 2. 3. .

(234) ,¨. . . ;j. u. ( i −1) bit node Q Z mn 3b. Ó. u. (i ) , bit node Q Z mn >. Ó. check e. node update %$. 5. 1. ; ( (i −1). 0. Z ' BP : T = ∏ tanh( mn 2 n'∈N ( m)\ n (i ) mn. & 0. 1. & 1. ) (i). (i−1) Zmn' Zmn ' ShiffledBP : T = ∏ tanh( ) ∏ tanh( ) 2 n'∈N(m)\n 2 n'∈N(m)\n (i) mn. n'<n. 1. n'>n. & 1 0 8%$ # 1 8 # 1 89 #. 4. 99:. 2 ( 5( 3 & 6)7. 5((. } 17. } 18 1. Ð} 18 ß . P. . . ' @. Í. 8 64`code rate 8 1/a¾ab c ¾c b d ¾5/eO. , shuffled BP decoding BP decoding ,f Ó. '. shuffled BP decoding BP decding i. (. ]. 1. '. &. ,. ^ÔÕ. XY. Ó. 8 15. }. shufled BP decoding %X BP decoding ,f. , shuffled BP decding T. (error performance)!f. . 39. I. X¨. @. , BP decodingf. & &. '.

(235) } 18. shuffled BP decoding BP decoding i. } 19 1. 8 64`code rate 8 1/a¾ab c ¾c b d ¾5/6 O. 8 15shuffled BP decoding BP decoding N¨ XY. . ^f. SNR ,f. (iteration)Ó. (Imax) Ó. (iteration)Ó . ÔÕ. (. ]. "N± , SNR Qi. %X BP ,Ò ¥. . . î. B. shuffled BP decoding ,f gïÅ. Æh. >. 64`code rate 8 1/a¾ab c ¾c b d ¾5/6 , shuffled BP decoding Ë BP decoding Ë. f Ó. ÔÕ. } Ð} 19'. Õ. B î. Ì. ÔÕ. f. Æ} 19'. Ó. (i. j. Ó. X BP decoding . n. o . R. (. ' Ì. ( ÔÕ. ÔÕ. !"1 f. Ó. 8. 8. , 57%~89% Ñkl" shuffled BP decoding N code rate Y D. Y. ¯ É. . p. 40. code rate ,m. [o. c. f. P. ,ÔÕ. ÃÔ f. .

(236) } 19. shuffled BP decoding BP decoding f 41. ÔÕ. Ó. i. î. B.

(237)

(238) 3.1. min-um (MS) shuffled decoding MS decoding ± >. q bit node ,O. MS decoding ¨. P. QR. check node update ÃMS Shuffled decoding Ñm. shuffled BP decoding ,^P. UmMS shuffled decoding N>. ,Q. (i ) ( i −1) Q8Q( Z mn )i@ ', Z mn '. , bit node Qâ2. update ,%mu. S. e. Ó. ÔÕ. (magnitude)4 e. ð. s 4. (i ) E mn =[. 5. ®. m. check node update Ã. MS decoding> e. check node. ( i −1) ,8 bit node Q Z mn ' . 3-1 8 MS shuffled decoding , check node updater ii. 5. C. check node update. % shuffled BP decoding ± . ∏ sgn(Z. n '∈N ( m ) \ n. ( i ) or ( i −1) mn '. ( i ) or ( i −1) ].[ 'min {| Z mn |}] ' n ∈N ( m ) \ n. (3-1). 3.2 compensated MS shuffled decoding compensated MS decodin ± compensated MS shuffled decoding @ +. 1D-normalized MS shuffled decoding ¾ 1D-offset MS shuffled decoding ¾ 2D-normalized MS shuffled decoding¾2D-offset MS shuffled decoding. 3.2.1 1D-normalized MS shuffled decoding 1D-normalized MS shuffled decoding m MS shuffled decoding , chcek node. 42.

(239) . u. ) β ·. p. ª. u. MS shuffled decoding. pª. 1D-normalized MS shuffled decoding , check node update ª update . E mn Z mn. t. u. (i ). =[. ¨. +. 3-a bit node. . ∏ sgn( Z mn'. ( i ) or ( i −1). n '∈N ( m ) \ n (i ). ¡. = Fn +. )].[min{ Z mn'. ( i ) or ( i −1). .β }]. n '∈N ( m ) \ n. {| E m'n | (i ) } ; Z n. (i ). m '∈M ( n ) \ m. = Fn +. E mn. (i ). m∈M ( n ). (3-2). 3.2.2 1D-offset MS shuffled decoding 1D-offset MS shuffled decodingmx MS shuffled decoding , check node. u. p. . Q

(240) u. update ª. MS shuffled decoding½. pª. 3-3 bit node update 4 ¡. Emn. (i ). =[ '. ∏ sgn( Z. n ∈N ( m ) \ n. Z mn. (i ). = Fn +. ( i ) or ( i −1) mn. '. t 5. u ¨. , check node. . )].[min{ Z mn'. ( i ) or ( i −1). − α }]. '. n ∈N ( m ) \ n. | Em'n |( i ) ; Z n. (i ). = Fn +. m'∈M ( n ) \ m. Emn. (i ). m∈M ( n ). (3-3). 3.2.3 2D-offset MS shuffled decoding 2D-offset MS shuffled decodingmx MS shuffled decoding , chcek node. u p. p α1 . x MS shuffled decoding , bit node. . @. E mn Z mn. (i ). =[. ¶. ª. check node bit nodev. ∏ sgn(Z. n '∈N ( m ) \ n (i ). = Fn +. ( i ) or ( i −1) mn '. ]. )].[ min{ Z mn'. u. pα 2 . o. 3-4 . ( i ) or ( i −1). − α 1 }]. '. n ∈N ( m ) \ n. {| E m'n |( i ) −α 2 } ; Z n. m '∈M ( n ) \ m. 43. (i ). = Fn +. E mn. m∈M ( n ). (i ). (3-4).

(241) 3.2.4 2D-normalized MS shuffled decoding. . 2D-normalized MS shuffled decoding m MS shuffled decoding , chcek node u. ) β1x MS shuffled decoding , bit node . p. o. E mn Z mn. p@. (i ). ¶. ∏ sgn( Z. =[. n '∈N ( m ) \ n (i ). = Fn +. check node bit nodev ª. ( i ) or ( i −1) mn. '. )].[ min{ Z mn '. ( i ) or ( i −1). . p β 2 u. ). 3-5 ]. .β1 }]. n '∈N ( m ) \ n. {| E m'n |(i ) .β 2 } ; Z n. (i ). m '∈M ( n ) \ m. = Fn +. E mn. (i ). m∈M ( n ). (3-5). 3.2.5 static compensated MS shuffled decoding dynamic compensated MS shufffled decoding w. &@. +. ;x. . z. ,y. ;¨. ,{. z. . {. z. âo. node update O ¤. ¥. ¦. . P t. c. ¨. . Ó c. Ut P. U·. 3.2.1~3.2.4 /. M. u. [23]t · {. z. ",¿. mx MS shuffled decoding , check c. . c. 8x¨. , SNR | }. ¨. . c. compensated MS shuffled decoding;{. d z. MS shuffled decodin~p. 44. e. ;y. a. z. 8 static. 8 dynamic compensated.

(242) . . . . . ; Standard 802.11n ,. '. (. -. ( !. .. 1)@ . x codeword 1 ¶. H. 8 648¾1296¾1944code rate 8 1/2¾2/3¾3/4¾5/6 , compensated MS shuffled. decoding¾MS shuffled decoding shuffled BP decoding 8 compensated MS shuffled decoding ,Ö Å . #. ^4. . B î. ». BP decoding , BER-SNR ~ . . SB(Shuffled BP) Emn , Zmn. e. . ·. . . m. M. ,T. ». . f ¬. ,. . ?. @. B. . Z mn. =[. ∏ sgn( Z. n '∈N ( m ) \ n (i ). = Fn +. ( i ) or ( i −1) mn '. )].[min{ Z mn '. . 1D-offset Min Sum Shuffled decoding :. ( i ) or ( i −1). (i ). Emn = [. .B1}]. {| E m'n | ( i ) } ; Z n. (i ). m '∈M ( n ) \ m. = Fn +. E mn. (i ). m∈M ( n ). (i ). Zmn = Fn +. 2D-normalized Min Sum Shuffled decoding :. E mn Z mn. (i ). =[. ∏ sgn( Z. n '∈N ( m ) \ n (i ). = Fn +. ( i ) or ( i −1) mn '. )].[ min{ Z mn ' (i ). = Fn +. .B1 }]. E mn. 20. E mn. m∈M ( n ). (i ) or(i −1) mn'. )].[min{ Zmn'. (i )or(i −1). − A1}]. n'∈N ( m) \ n. (i ). | Em'n |(i ) ; Zn = Fn +. m'∈M ( n) \ m. Emn. (i ). m∈M ( n). 2D-offset Min Sum Shuffled decoding :. n '∈N ( m ) \ n. {| E m'n | (i ) .B2 } ; Z n. m '∈M ( n ) \ m. ( i ) or ( i −1). ∏sgn(Z. n'∈N ( m)\ n. n '∈N ( m ) \ n. (i ). ". shuffled R. MF(Min-Sum Shuffled) Emn , Zmn. 1D-normalized Min Sum Shuffled decoding : (i ). . ¼ x$ Standard 802.11n LDPC cod § . A1 = SB_Emn - MF_Emn A2 = SB_Zmn - MF_Zmn B1 = SB_Emn / MF_Emn B2 = SB_Zmn / MF_Zmn. E mn. } a. . , compensated MS shuffled decoding , BER-SNR ~ 5. d. kl compensated MS shuffled decoding ,f . shuffled BP decoding R. c. Z mn. (i ). =[. ∏ sgn( Z. n '∈N ( m ) \ n. (i ). = Fn +. ( i ) or ( i −1) mn '. )].[ min{ Z mn'. ( i ) or ( i −1). {| E m'n | (i ) − A2 } ; Z n. (i). m '∈M ( n ) \ m. compensated MS shuffled decoding. 4.1 . 45. − A1}]. n '∈N ( m ) \ n. = Fn +. E mn. m∈M ( n ). (i ).

(243) . . MS shuffled decoding

(244). . . . . . . . 648 . . . . 2. . 1296code rate 1/22/33/45/!

(245). ". #. $ static 1D- normalized MS shuffled decodingstatic 1D-offset MS shuffled . decodingstatic 2D-normalized MS shuffled decodingstatic 2D-offset MS shuffled decodingMS Shufled decoding %& ./0. 1. . <=$>. . '. . ?. @ABC. 2. 3. 45. D E. . . 6. 78. 9:

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(248) Static. 2D-normalized > Static 1D-offset> Static 2D-offset > Static 1D-normalized. Static 2D-normalized > Static 2D-offset >Static 1D-offset> Static 1D-normalized. Static 2D-normalized > Static 1D-offset > Static 2D-offset = Static 1D-normalized. Static 2D-normalized = Static 2D-offset = Static 1D-offset = Static 1D-normalized. 21. codeword=648 static compensated MS shuffled decoding ABC 47. D.