通信路変動に対するロバスト性を有した符号化協調通信に関する研究

(IoT: Internet of

Things) IoT

Repeat-Accumulate(SC-RA: Spatially Coupled Repeat-Accumulate) SC-RA (SC-RA-CC: SC-RA Coded Cooperation)

RA

SC-RA-CC

CC(MD-CC: Multi-Dimensional

SC-RA-CC) MD-SC-RA-CC

MD-SC-RA-CC

MD-SC-RA-CC SNR(Signal-to-Noise Ratio)

(BEC: Block Erasure Channel) BEC MD-SC-RA-CC

1631090

1

Repeat-Accumulate (SC-RA: Spatially coupled SC-RA (SC-RA-CC: SC-RA Coded Cooperation)

SC-RA-CC SC-RA-CC(MD-SC-RA-CC: Multi-Dimensional SC-RA-CC)

MD-SC-RA-CC

2 5 3 Repeat-Accumulate 8 3.1 Repeat-Accumulate . . . 8 3.1.1 . . . 8 3.1.2 . . . 10 3.1.3 . . . 12 3.2 RA . . . 19 3.2.1 . . . 19 3.2.2 . . . 22 3.2.3 . . . 26 3.3 RA . . . 26 3.3.1 . . . 26 3.3.2 . . . 28 3.3.3 . . . 29 3.3.4 . . . 31 4 35 4.1 . . . 35 4.2 . . . 35 4.3 . . . 37 4.3.1 1-BEC . . . 37 4.3.2 T -BEC . . . 37 4.4 . . . 38 4.4.1 1-BEC . . . 38 4.4.2 T -BEC . . . 39 4.5 BEC . . . 39 4.5.1 1-BEC . . . 40 4.5.2 T -BEC . . . 41 4.6 MD-SC-RA-CC . . . 43 4.6.1 . . . 43 4.6.2 1-BEC . . . 44 4.6.3 T -BEC . . . 44 5 47

51

3.2 (Q, a)-regular RA . . . 11 3.3 3 . . . 12 3.4 . . . 14 3.5 (Q, a, L)- RA . . . 20 3.6 (Q, a)-regular RA . . . 23 3.7 (Q, a)-regular RA K . . . 24 3.8 RA . . . 25 3.9 j i ( C(j) = {c1, c2, . . . , c|C(j)|}) . . . 27 3.10 N T = 20 v = (1, 2, 3, 4) Q = 4 W = 4 SC-RA-CC . . . 28 3.11 N T = 20 v = (2, 6, 8, 9) Q = 4 W = 9 MD-SC-RA-CC . . . 29 3.12 N T = 20 v = (1, 2, 3, 4) Q = 4 W = 4 SC-RA-CC 1 . . . . 30 3.13 N T = 20 v = (2, 6, 8, 9) Q = 4 W = 9 MD-SC-RA-CC 1 . 31 3.14 N T = 20 v = (1, 2, 3, 4) Q = 4 W = 4 SC-RA-CC 2 . . . . 33 3.15 N T = 20 v = (2, 6, 8, 9) Q = 4 W = 9 MD-SC-RA-CC 2 . 34 4.1 N T = 20 v = (1, 2, 3, 4) Q = 4 W = 4 SC-RA-CC 1 4, 5 5, 6 4 12, 13, 14 14, 15, 16 6 . . . 38 4.2 1-BEC v = (2, 6, 8, 9) (1, 2, 3, 4) (1, 3, 4, 9) MD-SC-RA-CC DFP N = 5 T = 20 Q = 4 . . . 40 4.3 1-BEC v = (2, 6, 8, 9) (1, 2, 3, 4) (1, 3, 4, 9) MD-SC-RA-CC DFP . . . 44

. . . 48

5.2 T - v = (1, 3, 4, 9)

(2, 6, 8, 9) (1, 2, 3, 4) MD-SC-RA-CC SC-RA-CC PER . . . 49

1.1

(IoT: Internet

of Things)[1] IoT

IoT

(STBC:Space Time Block Code)[2] Alamouti

IoT [3] (Amplify-and-Forware) [4] (Decode-and-Forward) [5, 6] [7, 8] [9] [9] 2

(ANCC: Adaptive Network Coded Cooperation)[10] 2

ANCC

ANCC

1960 Gallager (LDPC: Low-Density Parity

Check) [11] (LDGM: Low-Density Generate Matrix)

ANCC 4 [12] ANCC [13] 4 ANCC Repeat-Accumulate(SC-RA: Spatially Coupled Repeat-Accumulate) [14] SC-RA

1999 [15] LDPC LDPC LDPC LDPC (BP: Belief Propagation) [16] [17] BP MAP(Maximum A Posteriori) [18] SC-RA Repeat-Accumulate(RA) [19] LDPC RA LDPC SC-RA LDPC RA

SC-RA (SC-RA-CC: SC-RA Coded Cooperation)[20] SC-RA-CC

BP [21] [21] BP SC-RA-CC SC-RA-CC(MD-SC-RA-CC: Multi-Dimensional SC-RA-CC) [22] [22]

MD-SC-RA-CC

SNR(Signal-to-Noise Ratio)

(BEC: Block Erasure Channel) BEC MD-SC-RA-CC

1.2

RA RA RA 6

2.1 N

(TDMA: Time-Division

Multiple Access) M [bits] T

. 2.2 T . N i, i = 1, 2, . . . , N i t i u(t)_i t = 1, 2, . . . , T u(t)_i i t j j = i + (t_{− 1)N} (2.1) t i uj ! u(t)_i i j i _{u1, . . . , uj−1} p_j p_j ! ⊙l∈C(j)ul (2.2) p_j M [bits] _⊙ ( ) _C(j) j ⊙ C(j) i 2M [bits]

BPSK(Binary Phase-Shift Keying)

t i x(t)_{i ∈ {+1, −1}}2M

2.1:

h(t)_i t i i

t h(t)_i 0 1

n(t)_i 2M

(AWGN: Additive White Gaussian Noise) 0, N0

T ∆T N T j C(j) (2.2) |C(j)| = 0 1-1 h(t)_i i t

2.2:

T

-T + ∆-T h(t)_i i

Repeat-Accumulate (SC-RA-CC) Repeat-Accumulate LDPC RA RA SC-RA-CC MD-SC-RA-CC

3.1

Repeat-Accumulate

RA 3.1.1 RA u(t)_i Q QM [bits] a[bits] Accumulator 1/(1− D) (1)

RA 3.1 D 1bit (Delay Device) ⊕

u = [u1, . . . , um, . . . , uM] Repetition Q QM u′ um u m QM [bits] u′ = [u1, . . . , u1, . . . , uM, . . . , uM] (3.1) u′ Interleaver H H 1 QM × QM w w = u′H (3.2)

3.1: RA (D 1bit (Delay Device) ⊕ ) w abits L = (Q/a)M s s s = [w1⊕ · · · ⊕ wa, . . . , w(QM−a+1)⊕ · · · ⊕ w(QM )] = [s1, . . . , sL] (3.3)

Combiner abits Combiner a

Com-biner Combiner s Accumulater p Accumulator 3.1 1/(1− D) Lbits p = [s1, s2⊕ p1, . . . , sL⊕ p(L−1)] = [p1, . . . , pL] (3.4) c RA u p c c =! u p " (3.5) (Q, a)-regular RA rRA . rRA= M M + L = M # 1 +Q a $ M = a a + Q (3.6) RA Repetition Q Combiner a

RA (Q, a)-regular RA HRA N × (M + L) L× M B L× N A HRA = ! B A " (3.7) B u′ w s Q a 0 1 B BT u s B Q a A L_{× L} A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 1 1 1 . .. . .. 1 1 1 1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.8) 2( 1 M ) Accumulator A HRA l_{∈ {2, . . . , L}} l_{− 1} A B (3.4) A Accumulator L× (M + L) HRA RA c HRAcT = 0 (3.9) 3.1.2 RA 3.2 HRA 3 u M M ( _◦) s N L ( (")) p L L ( •) m u(m) f (l) p(l) l∈ {1, . . . , L}

3.2: (Q, a)-regular RA u(m) m_{∈ {1, . . . , M}} f (l) l_{∈ {1, . . . , L}} p(l) l_{∈ {1, . . . , L}} HRA B l m u(m) f (l) ( ) Interleaver H Q a RA (Q, a)-regular RA RA Regular RA Irregular RA u(m) u um u(m) Q u um Q Interleaving H u(m) f (l′) ∈ D(u(m)) D(u(m)) u(m) Q f (l) u(m′)_{∈ E(f(l))\{p(l), p(l − 1)}} E(f (l)) f (l)

3.3: 3 a + 2 _\ l = 1 a + 1 f (l) um′ (3.2) p(l) f (l) f (l + 1) Accumulator (3.4) p RA 3.1.3 RA [23, 24] sum-product sum-product RA sum-product sum-product [25] f (x) D(v) (1) V ! {x1, . . . , xI} I xi i∈ {1, . . . , I} D(xi) V f (V ) = f (x1, . . . , xI) xi gi(xi)! + V\xi f (V ) (3.10) V\xi xi V ( ) f (V ) D(xi) ={0, 1}

(3.10) I_{− 1} 2I−1 (3.10) I I f (V ) f (V ) = f1(A1) . . . fJ(AJ) (3.11) A1, . . . , AJ V fj(Aj) j∈ {1, . . . , J} f (V ) (1) fj(Aj) (2) xi ∈ V fj(Aj) Aj f (x1, x2, x3) = f1(x1)f2(x1, x2)f3(x2, x3) (3.12) 3.3 V = _{x1, x2, x3} A1 = {x1} A2 = {x1, x2} A3 ={x2, x3} (◦) (#) f2(x1, x2) x1 x2 (2) sum-product sum-product f (V ) fj(Aj) fj(Aj) ( ) f (x1, x2, x3, x4) f (x1, x2, x3, x4) = f1(x1)f2(x1, x2)f3(x1, x3)f4(x3, x4) (3.13) g1(x1) g1(x1) = + x2 + x3 + x4 f1(x1)f2(x1, x2)f3(x1, x3)f4(x3, x4) (3.14)

3.4: g1(x1) = f1(x1) , + x2 f2(x1, x2) - , + x3 f3(x1, x3) , + x4 f4(x3, x4) --(3.15) |D(xi)| = 10 i ∈ {1, . . . , 4} (3.14) 103− 1 = 999 (3.15) 108 (3.15) f (x1, x2, x3, x4) 3.4 x1 g1(x1) (3.15) sum-product

( ) f (V ) (3.11) G xi xi fj fj gr(xr) r ∈ {1, . . . , N} G xr G G v N (v) ( ) xk fi Mxk→fi(xk) = . a∈N(xk)\fi Ma→xk(xk) (3.16) xk Mxk→fi(xk) = 1 Mxk→fi(xk) xk → fi xk ( ) fi xk Mfi→xk(xk) = + N (fi)\xk fi(Ai) . a∈N(fi)\xk Ma→fi(a) (3.17) fi Mfi→xk(xk) = fi(xk) ( ) xr xr Mxr(xr) = . a∈N(xr) Ma→xr(xr) (3.18) xr gr(xr) sum-product 1. 2. 3.

5.

sum-product

(3) MAP

MAP(Maximum A posteriori Probability)

χ ={0, 1} Y PY|X(y|x) N C y∈ YN PX|Y(x|y) = PX(x)PY|X(y|x) PY(y) (3.19) X = (X1, . . . , XN) P_X|Y(x_{|y) = PX1...XN|Y}(x1, . . . , xN|y) (3.20) V V ! {x1, . . . , xN} Xn n∈ {1, . . . , N} P_Xn|Y(xn|y) = + V\xn P_X1...XN|Y(x1, . . . , xN|y) (3.21) PXn|Y(xn|y) xn= 0 P_Xn|Y(0|y) = 1 PY(y) + V\xn PX1...XN(x1, . . . , xn−1, 0, xn+1, . . . , xN) × PY|X1...XN(y|x1, . . . , xn−1, 0, xn+1, . . . , xN) (3.22) C 0 PX(x) = 0 (x /∈ C) P_Xn|Y(0_{|y) =} 1 PY(y) + x∈C,xn=1 PX(x)PY|X(y|x) (3.23) xn= 1 P_Xn|Y(1_{|y) =} 1 PY(y) + x∈C,xn=1 PX(x)PY|X(y|x) (3.24)

MAP n xˆn (3.23) (3.24) xn Ln= ln P_Xn|Y(0_|y) P_Xn|Y(1_|y) = ln / x∈C,xn=0PX(x)PY|X(y|x) / x∈C,xn=1PX(x)PY|X(y|x) (3.25) ˆ xn ˆ xn= ⎧ ⎨ ⎩ 0 Ln≥ 0 1 Ln< 0 (3.26) (3.23) (3.24) sum-product sum-product (4) sum-product sum-product RA (LDPC

: Low Density Parity Check)

sum-product

RA y∈ RM +L ₂

(2.3)

( ) f (l)

α_{f (l)→ψα}, ψα ∈ E(f(l)) 0

u(m) βu(m)→ψ ψ∈ D(u(m))

0 p(l) βp(l)→ψβ ψβ ∈ {f(l), f(l + 1)} 0 ψ α_{f (l)→ψα} ! ln #_M f (l)→ψα(0) M_{f (l)→ψα}(1) $ (3.27) β_p(l)→ψβ ! ln , Mp(l)→ψβ(0) M_p(l)→ψβ(1) -(3.28) It = 0 Itmax

yj P (yj|xj) P (yj|xj) = 1 √ 2πσ2 exp # −(yj − xj) 2 2σ2 $ (3.29) yj λj λj ! ln P (yj|xj = 1) P (yj|xj =−1) = −(yj− 1) 2_{+ (y} j+ 1)2 2σ2 = 2yj σ2 (3.30) xj x j j∈ {1, . . . , M + L} AWGN λj yj σ2 j_{∈ {1, . . . , M + L}} (3.30) Λ_{∈ R}M +L y j yj λj 0 ( ) p(l) l_{∈ {1, . . . , L}} β βp(l)→f(l′)= ⎧ ⎨ ⎩ λ_p(l) l = L λ_p(l)+ α_{f (l}′′_)→p(l) (3.31) (f (l′), f (l′′)) = (f (l), f (l + 1)), (f (l + 1), f (l)) ( ) u(m) m∈ {1, . . . , M} β βu(m)→f(l)= λu(m)+ + f (l′_{)∈D(u(m))\f(l)} αf (l′)→u(m) (3.32) ( ) f (l) l _{∈ {1, . . . , L}} α αf (l)→p(l)= ⎛ ⎝ . ψ∈E(f(l))\p(l) sign(βψ→f(l)) ⎞ ⎠ f ⎛ ⎝ + ψ∈E(f(l))\p(l) f (|βψ→f(l)|) ⎞ ⎠ (3.33) α_{f (l)→u(m)} = ⎛ ⎝ . ψ∈E(f(l))\u(m) sign(β_ψ→f(l)) ⎞ ⎠ f ⎛ ⎝ + ψ∈E(f(l))\u(m) f (|βψ→f(l)|) ⎞ ⎠ (3.34)

sign(x)! ⎧ ⎨ ⎩ 1 x_{≥ 0} −1 x < 0 (3.35) f (x) f (x)! lnexp(x) + 1 exp(x)− 1 (3.36) ( ) u m um uˆm ˆ um = ⎧ ⎨ ⎩

0 sign7/_{f (l}′_)∈D(u(m))βu(m)→f(l′)

8 = 1 1 sign7/_{f (l}′_)∈D(u(m))βu(m)→f(l′)

8 =−1 (3.37) p l pl pˆl ˆ pl= ⎧ ⎨ ⎩ 0 sign7/_{f (l}′_{)∈{f(l),f(l+1)}}βp(l)→f(l′) 8 = 1 1 sign7/_{f (l}′)∈{f(l),f(l+1)}βp(l)→f(l′) 8 =−1 (3.38) l = L p(l) f (l′)_{∈ {f(l)}} ˆc ˆ c =! uˆ1 . . . ˆuM pˆ1 . . . ˆpL " (3.39) ( ) ˆc c ˆ c ˆ c HT_RA = 0 (3.40) ˆ c ( ) It 1 It < Itmax ˆ c

3.2

RA

RA [14] 3.2.1 RA

3.5: (Q, a, L)- RA (1) RA 3.5 M 2 F2 M uk∈ FM2 uk M uk′ k′ ∈ {1, . . . , k} Q k′ a a Q uk′ u′(a_l ′) l_{∈ {1, . . . , L}} L l_{∈ N} l 1 a_{− 1} a l L_{− a + 1} L a RA _M

Interleaving u′(a_l ′) a′ Interleaver Interleaving H_l(a′)

Interleaving H_l(a′) M_{× M}

(1) H

(3.2) w(a_l ′) Combiner a w(a_l ′) (3.3) sl= w(1)_l ⊕ · · · ⊕ w(a)_l ∈ FM2 (3.41) Accumulator (3.4) p_l Accumulator sl−1 sl l = 1 l∈ {1, . . . , L} RA C =! U P " (3.42) U = [u1, . . . , uK] P = [p1, . . . , pL] C U (2) RA RA HSC−RA HSC−RA= ! Π A " (3.43) A (2) (3.8) (1) sl s_l−1 sl RA A LM _{× LM} Accumulator A A′= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ A A . .. . .. A A ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.44) A M_{× M} L A

Π = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ H₁(1) H₂(1) H₂(2) .. . ... . .. H_Q(1) H_Q(2) . .. H_Q+1(1) . .. . .. H(a) L−Q−1 . .. H(a−1) L−Q H (a) L−Q . .. .._. .._. H_L(a−1)₋₁ H_L(a)₋₁ H_L(a) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.45) Interleaving H_l(a′) (1) Π Q H_l(a′) uk Q a′′ ∈ {1, . . . , a} H_l(a′) Combiner u’(a_l ′′) Interleaving H_l(a′) u′(a_l ′) H_l(a′) 3.2.2 RA RA RA RA (1) RA 3.6 (Q, a)-regular RA 3.1.2 3.2 (Q, a)-regular RA M N Q Q Q Q QM a a

3.6: (Q, a)-regular RA aN Interleaver Accumulator (2) RA RA 3.7 3.6 (Q, a)-regular RA K k ∈ {1, . . . , K} Q ( ) a 3.7 3.8 M M 3.8 3.7 Kex RA rSC−RA

3.7: (Q, a)-regular RA K 3.8 L L = 9 Q aK : + (a_{− 1)} (3.46) ⌈x⌉ x_{∈ R} x Q ( ) a Q a Kex= L− K = 9 Q_{− a} a K : + a_{− 1} (3.47) l_{∈ {1, . . . , L}} l_{− 1} l (Q, a, K)- RA RA k uk

3.8: RA k Q uk Q Interleaving H_l(a′) u’(a_k′) l w(a_l ′) l l (3.3) sl l∈ {1, . . . L} Accumulator p_l C uk p_l (Q, a, K)- RA K M RA KM L M KM RA rSC−RA= _{KM + LM}KM = _{K + L}K = 9_{Q + a} K a K : + a− 1 (3.48) Q, a, K (Q, a, K)-RA

lim K→∞rSC−RA= a Q + a (3.49) 3.2.3 RA (4) sum-product (2) 3.8 RA 3.8 u(km) m ∈ {1, . . . , M} f (lm) p(lm) k l (2) 3.1.2 u(km) D(u(km)) f (lm) E(f (lm)) Y Y =! y₁ . . . y_K+L " ∈ R(K+L)M (3.50) y_i 2 (2.3) (4) sum-product ˆ C Cˆ ˆ C =! Uˆ Pˆ "=! uˆ1 . . . ˆuK pˆ1 . . . ˆpL " ∈ F(K+L)M2 (3.51)

3.3

RA

MD-SC-RA-CC SC-RA-CC MD-SC-RA-CC 3.3.1 3.9 j i W M Q ; D Accumulator j _M j W _{uj′ | j−W ≤ j′ ≤ j − 1, j′ _{∈ N } N 1 2} p_j C(j) MD-SC-RA-CC _C(j) C(j) = {{j − v1, . . . , j− vq, . . . , j− vQ} ∩ N} (3.52)

3.9: j i ( C(j) = {c1, c2, . . . , c_|C(j)|}) q = 1, 2, . . . , Q _{C(j) N} 0≤ |C(j)| ≤ Q |C(j)| = 0 vq j vq v v = (v1, . . . , vq, . . . , vQ) (3.53) vq {1, . . . , W } W vQ C(j) j v MD-SC-RA-CC j V(j) V(j) = {{j + v1, . . . , j + vq, . . . , j + vQ}} (3.54) |V(j)| Q M C(j) ; S- [26] |C(j)| M [bits] Accumulator p_j vj pj SC-RA-CC v = (1, . . . , Q) MD-SC-RA-CC SC-RA-CC j j−Q SC-RA-CC MD-SC-RA-CC

3.10: N T = 20 v = (1, 2, 3, 4) Q = 4 W = 4 SC-RA-CC 3.3.2 MD-SC-RA-CC 3.10 N T = 20 v = (1, 2, 3, 4) Q = 4 W = 4 v SC-RA-CC 1 (N T + W )× NT I2 (N T + W )_{× (NT + W )} 1 M_{× M} 1 1 1 I2 M× M 2 2 2 2 Accumulator 1 |C(j)| Q I2 1 p₁ p₁ 9 p₉ v = (1, 2, 3, 4) (3.52) C(9) = {9 − 1, 9 − 2, 9 − 3, 9 − 4} = {5, 6, 7, 8} 3.10 9 5, 6, 7, 8 9

3.11: N T = 20 v = (2, 6, 8, 9) Q = 4 W = 9 MD-SC-RA-CC MD-SC-RA-CC v 3.11 v = (2, 6, 8, 9) W = 9 MD-SC-RA-CC v SC-RA-CC 1 5 25, . . . , 29 5 12 p₁₂ C(12) = {12 − 2, 12 − 6, 12 − 8, 12 − 9} = {3, 4, 6, 10} (3.52) 1 2 |C(1)| = 0, |C(2)| = 0 3.3.3 MD-SC-RA-CC 3.12 N T = 20 v = (1, 2, 3, 4) Q = 4 1 3.10 v MD-SC-RA-CC SC-RA-CC

3.12: N T = 20 v = (1, 2, 3, 4) Q = 4 W = 4 SC-RA-CC 1 (2.1) 1 1 jc jr jc jr jc= 1, 2, . . . , N T jr = 1, 2, . . . , N T + W Q |V(j)| = Q j |C(j)| 2 I2 Accumulator 5 1, 2, 3, 4 MD-SC-RA-CC j C(j) C(j) v N T Q W MD-SC-RA-CC v 3.11 v = (2, 6, 8, 9) W = 9 MD-SC-RA-CC 1 v

3.13: N T = 20 v = (2, 6, 8, 9) Q = 4 W = 9 MD-SC-RA-CC 1 5 2 3.14 3.15 j i t 2 i t 3.12 3.13 N = 5 3.14 3.15 Q 2 MD-SC-RA-CC 3.14 3.15 ” ” 3.3.4 3.12 2 1 10 Q = 4 MD-SC-RA-CC [16]

MD-SC-RA-CC r = N T M N T M + (N T + vQ− v1)M = N T 2N T + vQ− v1 (3.55) N T M (N T +vQ−v1)M N T (vQ−v1) r 1/2 W (vQ− v1) MD-SC-RA-CC r 0 MD-SC-RA-CC MD-SC-RA-CC v = (1, 2, . . . , Q) SC-RA-CC

3.14: N T = 20 v = (1, 2, 3, 4) Q = 4 W = 4 SC-RA-CC 2

3.15: N T = 20 v = (2, 6, 8, 9) Q = 4 W = 9 MD-SC-RA-CC 2

MD-SC-RA-CC 1- T -MD-SC-RA-CC

v LDPC

(DE: Density Evolution) [27]

v

4.1

(SNR: Signal-to-Noise Ratio)

(BEC: Block Erasure Channels)[28] BEC

ϵB 1− ϵB

2

1-BEC T -BEC 2 1-BEC t i

ϵB j

T -BEC t i

ϵB

1-BEC T -BEC MD-SC-RA-CC

v

4.2

l l x(2)(j)(q)_l j q k = (j_{− 1)Q + q} j q x(1)(k)_l ! x(1)(j)(q)_l x(2)(k)_l ! x(2)(j)(q)_l y(1)(j)_l j y(2)(j)_l x y j _V′(j) V′(j) =_{{(j − 1)Q + 1, . . . , (j − 1)Q + Q}.} (4.1) j _C′(j) C′(j) ={{(j − v1)Q + Q, . . . , (j− vQ)Q + 1} ∩ N} . (4.2) j ϵ(j) BEC j _{{0, 1}} 2 1 ϵB l + 1 x(2)(k)_l+1 x(2)(k)_l+1 = ⎧ ⎨ ⎩1− 7 1− y(1)(j)l 82 . n∈C′_(j)\k 7 1− x(1)(n)l 8⎫⎬ ⎭ (4.3) j k 2 (1− y(1)(j)l ) 2 y(2)(j)_l+1 y(2)(j)_l+1 = ⎧ ⎨ ⎩1− 7 1− y(1)(j)l 8 . n∈C′_(j) 7 1− x(1)(n)l 8⎫⎬ ⎭ (4.4) l + 1 x(1)(k)_l+1 x(1)(k)_l+1 = ϵ(j) _· . m∈V′(j)\k x(2)(m)_l+1 (4.5) j k

y(1)(j)_l+1 y(1)(j)_l+1= ϵ(j) _{· y(2)}(j)_l+1 (4.6) 2 x(1)(k)_l+1 v BEC 0 MD-SC-RA-CC (DFP:

Decoding Failure Probability)Pd v

4.3

v 1-BEC T -BEC MD-SC-RA-CC 4.3.1 1-BEC 1-BEC j ϵ(Q+1)_B 1-BEC Pd≥ 1 − (1 − ϵQ+1B )N T (4.7) 4.3.2 T -BEC ϵ 1− ϵ T -BEC N = 5 T -BEC 5 ϵB 5 2 2/5 = 0.4 T -BEC 1_{− 0.4 = 0.6} (3.55) MD-SC-RA-CC 0.5 5 3 3/5 = 0.6 T -BEC 1− 0.6 = 0.4 MD-SC-RA-CC

4.1: N T = 20 v = (1, 2, 3, 4) Q = 4 W = 4 SC-RA-CC 1 4, 5 5, 6 4 12, 13, 14 14, 15, 16 6 T -BEC MD-SC-RA-CC Pd≥ N + l=⌈(1−r)N⌉ NCl ϵl (1− ϵ)N−l (4.8) ⌈(1 − r)N⌉ rN 2 ⌊(1 − r)N⌋ ” ”

4.4

v 4.4.1 1-BEC 4.1 v = (1, 2, 3, 4) Q = 4 SC-RA-CC 4 4 4 4

MD-SC-RA-CC 4 5 4.3.1 9 ϵ3 ϵQ+1= ϵ5 6 6 12, 13, 14 18 ϵ4 ₄ 1-BEC MD-SC-RA-CC 4 6 4 6 4.4.2 T -BEC T -BEC 1-BEC 3.15 MD-SC-RA-CC 2 1 1 2, 3, 4, 5 1 3

(v_m(k)mod N )̸=(vn(k)mod N ), (vm(k)mod N ) > 0, ∀m, n∈[1, q], m̸=n, N >q (4.9)

4.5

BEC

2 BEC MD-SC-RA-CC MD-SC-RA-CC N = 5 T = 20 Q = 4 rmin = 0.48 r > 0.48 1 [29] v = (1, 3, 4, 9) 2 v = (2, 6, 8, 9) v = (1, 3, 4, 9) 1-BEC v = (2, 6, 8, 9) T -BEC 1-BEC

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4.2: 1-BEC v = (2, 6, 8, 9) (1, 2, 3, 4) (1, 3, 4, 9) MD-SC-RA-CC DFP N = 5 T = 20 Q = 4 4.5.1 1-BEC 4.2 (1, 2, 3, 4), (1, 3, 4, 9), (2, 6, 8, 9) MD-SC-RA-CC 1-BEC DFP ϵB 1-BEC DFP 4.2 v = (1, 3, 4, 9) DFP 1-BEC 1 v = (2, 6, 8, 9) DFP ϵB = 0.1 v = (1, 3, 4, 9) DFP 0.001 DFP SC-RA-CC v = (1, 2, 3, 4) DFP 0.1 DFP SC-RA-CC BEC

4.1: T -BEC v = (2, 6, 8, 9) (1, 2, 3, 4) (1, 3, 4, 9)MD-SC-RA-CC DFP N = 5 T = 20 Q = 4 v (1, 2, 3, 4) (1, 3, 4, 9) (2, 6, 8, 9) Coding rate 0.493 0.480 0.483 DFP 4.50× 10−2 _{8.56 × 10}−3 _{8.56× 10}−3 _{8.56 × 10}−3 E0 0 0 0 0 E1 0 0 0 0 E2 5 0 0 0 E3 10 10 10 10 E4 5 5 5 5 E5 1 1 1 1 4.5.2 T -BEC T -BEC N ” ” ” ” 2 T -BEC 2N N l N − l l = 0, 1, . . . , N l N − l NCl l MD-SC-RA-CC El Eall Eall = N + l=0 El (4.10) ϵB N l DFP ϵl_B(1− ϵB)l T -BEC DFP PD(v, ϵB) = N + l=0 El ϵlB(1− ϵB)l (4.11) El v ϵB v DFP El ϵB v ϵB∈ [0, 1] DFP T -BEC El v 4.1 T -BEC (1, 2, 3, 4), (1, 3, 4, 9), (2, 6, 8, 9) MD-SC-RA-CC DFP ϵB= 0.10 DFP

4.2: T -BEC v = (1, 3, 6, 10, 11), (1, 4, 8, 9, 11)MD-SC-RA-CC DFP N = 6 T = 15 Q = 5 v (1, 3, 6, 10, 11) (1, 4, 8, 9, 11) Coding rate 0.480 0.480 DFP 5.64× 10−3 _{1.27 × 10}−3 _{1.27× 10}−3 E0 0 0 0 E1 0 0 0 E2 0 0 0 E3 6 0 0 E4 15 15 15 E5 6 6 6 E6 1 1 1 4.1 v = (1, 2, 3, 4) E2 = 5 DFP v = (1, 3, 4, 9) (2, 6, 8, 9) DFP DFP 4.3.2 MD-SC-RA-CC N = 5 r = 0.480 0.60 5 2 0.60_{− 0.48 = 0.12} 2 MD-SC-RA-CC MD-SC-RA-CC MD-SC-RA-CC N = 6 T = 15 Q = 5 rmin= 0.48 0.50 6 3 1 v = (1, 3, 6, 10, 11) 2 v = (1, 4, 8, 9, 11) 4.2 T -BEC (1, 3, 6, 10, 11), (1, 4, 8, 9, 11) MD-SC-RA-CC DFP ϵB= 0.10 DFP 4.2 2 v = (1, 4, 8, 9, 11) E3 = 0 DFP 2 T -BEC DFP

4.3: BP MD-SC-RA-CC SC-RA-CC (1) (2) (3) 0.4926 BP 0.4768 0.4748 0.4694 0.4867

4.6

MD-SC-RA-CC

(3.55) MD-SC-RA-CC SC-RA-CC MD-SC-RA-CC

SC-RA-CC MD-SC-RA-CC MD-SC-RA-CC 4.6.1 MD-SC-RA-CC 3 1. _{v1, . . . , N + vQ}; 2. {W − v1, . . . , N − (W + vQ)}; 3. _{v1, . . . , W − v1, N − (W + vQ), . . . , N + vQ}; ϵ 3 4.2 N = 5 T = 20 Q = 4 v = (2, 6, 8, 9) BP ϵth ϵth! sup ? ϵ > 0| lim l→∞x(2) (∀_k) l = 0 @ (4.12) 4.3 BP SC-RA-CC BP SC-RA-CC (1) BP LDPC [27]

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パンクチュア MD-SC-RA-CC 4.3: 1-BEC v = (2, 6, 8, 9) (1, 2, 3, 4) (1, 3, 4, 9) MD-SC-RA-CC DFP 4.6.2 1-BEC 4.3 MD-SC-RA-CC 1-BEC DFP (1, 2, 3, 4), (1, 3, 4, 9), (2, 6, 8, 9) N = 5 T = 20 Q = 4 r = 0.493 4.3 MD-SC-RA-CC DFP 0.07 MD-SC-RA-CC DFP SC-RA-CC DFP 0.2 MD-SC-RA-CC DFP DFP 4.6.3 T -BEC 4.4 4.5 (1, 2, 3, 4), (1, 3, 4, 9), (2, 6, 8, 9), (1, 2, 3, 4, 5), (1, 3, 6, 10, 11), (1, 4, 8, 9, 11) MD-SC-RA-CC DFP N = 5

4.4: T -BEC v = (2, 6, 8, 9) (1, 2, 3, 4) (1, 3, 4, 9) MD-SC-RA-CC DFP N = 5 T = 20 Q = 4 SC-RA-CC MD-SC-RA-CC v (1, 2, 3, 4) (1, 3, 4, 9) (2, 6, 8, 9) Coding rate 0.493 0.493 0.493 DFP 4.50 _{× 10}−2 8.56 _{× 10}−3 8.56 _{× 10}−3 8.56_{× 10}−3 E0 0 0 0 0 E1 0 0 0 0 E2 5 0 0 0 E3 10 10 10 10 E4 5 5 5 5 E5 1 1 1 1 r = 0.493 N = 6 r = 0.489 ϵB= 0.10 DFP 4.4 4.1 MD-SC-RA-CC MD-SC-RA-CC DFP MD-SC-RA-CC SC-RA-CC 4.5 MD-SC-RA-CC 4.2 (1, 3, 6, 10, 11), (1, 4, 8, 9, 11) DFP E3 = 20 6 3 (1, 3, 6, 10, 11), (1, 4, 8, 9, 11) DFP MD-SC-RA-CC SC-RA-CC MD-SC-RA-CC E2= 0 DFP

4.5: T -BEC v = (1, 2, 3, 4, 5), (1, 3, 6, 10, 11), (1, 4, 8, 9, 11) MD-SC-RA-CC DFP N = 6 T = 15 Q = 5 SC-RA-CC MD-SC-RA-CC v (1, 2, 3, 4, 5) (1, 3, 6, 10, 11) (1, 4, 8, 9, 11) Coding rate 0.489 0.489 0.489 DFP 5.38 _{× 10}−2 1.59_{× 10}−2 1.59 _{× 10}−2 1.27_{× 10}−3 E0 0 0 0 0 E1 0 0 0 0 E2 4 0 0 0 E3 20 20 20 0 E4 15 15 15 15 E5 6 6 6 6 E6 1 1 1 1

2 1- T

-1 (PER:

Packet Error Rate) 5.1

S-S 7 Sum-Puroduct 120 v = (1, 3, 4, 9) v = (2, 6, 8, 9) SC-RA-CC v = (1, 2, 3, 4) 2 MD-SC-RA-CC 3 r = 0.493 5.1 1- MD-SC-RA-CC SC-RA-CC PER Eb/N0 Eb 1

MD-SC-RA-CC SC-RA-CC PER = 10−4

1.5dB 4.3 MD-SC-RA-CC 1-5.2 T - MD-SC-RA-CC SC-RA-CC PER Eb/N0 = [10, 16] 3 PER SNR Eb/N0 = [17, 20] v = (2, 6, 8, 9) SC-RA-CC v = (1, 2, 3, 4) 0.5dB ?? MD-SC-RA-CC T

-5.1: MD-SC-RA-CC SC-RA-CC v (1, 3, 4, 9) (2, 6, 8, 9) (1, 2, 3, 4) r 0.493 N 5 T 20 Q 4 [bits] M 100

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• , , ”

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