Convergence Analysis and Design of Multiple Concatenated Codes

Convergence Analysis and Design
of Multiple Concatenated Codes
Fredrik Brännström
Department of Signals and Systems
Chalmers University of Technology
SE-412 58 Göteborg, Sweden
April 28, 2005
1. Background
2. System Model
3. Extrinsic Information Transfer (EXIT) Functions
4. EXIT Chart Projection
5. Optimal Puncturing Ratios
6. Optimal Energy Distribution
7. Conclusions

Background
Concatenated codes and iterative decoding
• Original turbo code by Berrou, Glavieux, and Thitimajshima (BGT) (1993).
• BGT: two punctured convolutional codes concatenated in parallel.
• Performance close to capacity.
EXIT charts for convergence analysis of iterative decoding
• Two serially/parallel concatenated codes by ten Brink in (1999, 2000).
• Multiple concatenated codes (MCCs) with N = 3 components in (2001, 2002).
This presentation:
• Multiple parallel concatenated codes (MPCCs).
• Convergence analysis using projected EXIT charts.
• Optimal puncturing ratios.
• Optimal energy distribution.
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 1

System Model
x ✲ π 0
✲ π 1
✲ π 2
✲ π 3
x 0 y 0 ✲
z
U 0 ✲ 0
x 1 y
✲ 1
C
✲ 1 U 1
x 2 y
✲ 2
C
✲ 2 U 2
x 3 y
✲ 3
C
✲ 3 U 3
❅
z 1 ✲
z 2 ✲
z 3
❅
❅
❅
M
✲
w
s✲
✓✏ ❄
r ✲
✒✑ M −1
Punctured Parallel Concatenated Code
❅
❅
❅
❅
E(z 0 ) U
−1
✲
0
E(z 1 )
✲ U
−1
1
E(z 2 )
✲ U
−1
2
E(z 3 )
✲ U
−1
3
A(y 0 )
A(y 1 )
✲
A(y 2 )
✲
A(y 3 )
✲
C −1
1
C −1
2
C −1
3
E(x 0 ) π
−1
✲
0
E 0 (x)
✲
E(x 1 )
✲ π
−1
E 1 (x)
✲
A(x
1
✛
1) A π1 ✛
1(x)
E(x 2 )
✲ π
−1
E 2 (x)
✲
A(x
2
✛
2) A π2 ✛
2(x)
E(x 3 )
✲ π
−1
E 3 (x)
✲
A(x
3
✛
3) A π3 ✛
3(x)
• Convolutional code (CC) C n , with code rate R n , maps the interleaved source
bits x n ∈ {−1, +1} L to y n = C(x n ) ∈ {−1, +1} L , for n = 1, 2, 3.
• U n denotes a random puncturer with puncturing ratio 0 ≤ δ n ≤ 1, n = 0, 1, 2, 3.
• δ n = 0.2 means that only 20% of y n is kept in z n and 80% is punctured.
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 2

System Model
x ✲ π 0
✲ π 1
✲ π 2
✲ π 3
x 0 y 0 ✲
z
U 0 ✲ 0
x 1 y
✲ 1
C
✲ 1 U 1
x 2 y
✲ 2
C
✲ 2 U 2
x 3 y
✲ 3
C
✲ 3 U 3
❅
z 1 ✲
z 2 ✲
z 3
❅
❅
❅
M
✲
w
s✲
✓✏ ❄
r ✲
✒✑ M −1
• Code rates: R = [R 0 , R 1 , . . . , R N ].
❅
❅
❅
❅
• Puncturing ratios: ∆ = [δ 0 , δ 1 , . . . , δ N ].
E(z 0 ) U
−1
✲
0
E(z 1 )
✲ U
−1
1
E(z 2 )
✲ U
−1
2
E(z 3 )
✲ U
−1
3
A(y 0 )
A(y 1 )
✲
A(y 2 )
✲
A(y 3 )
✲
C −1
1
C −1
2
C −1
3
E(x 0 ) π
−1
✲
0
E 0 (x)
✲
E(x 1 )
✲ π
−1
E 1 (x)
✲
A(x
1
✛
1) A π1 ✛
1(x)
E(x 2 )
✲ π
−1
E 2 (x)
✲
A(x
2
✛
2) A π2 ✛
2(x)
E(x 3 )
✲ π
−1
E 3 (x)
✲
A(x
3
✛
3) A π3 ✛
3(x)
• Energy distribution: Ψ = [ψ 0 , ψ 1 , . . . , ψ N ].
• Uniform energy distribution: Ψ 0 = [ 1
N+1 , 1
N+1 , . . . , 1
N+1
• SNR for component n: γ s,n =
ψ n
∑ Nj=0 δ j
R j
ψ j
γ b .
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 3
]
.

System Model
x ✲ π 0
✲ π 1
✲ π 2
✲ π 3
x 0 y 0 ✲
z
U 0 ✲ 0
x 1 y
✲ 1
C
✲ 1 U 1
x 2 y
✲ 2
C
✲ 2 U 2
x 3 y
✲ 3
C
✲ 3 U 3
❅
z 1 ✲
z 2 ✲
z 3
❅
❅
❅
M
✲
w
s✲
✓✏ ❄
r ✲
✒✑ M −1
❅
❅
❅
❅
E(z 0 ) U
−1
✲
0
E(z 1 )
✲ U
−1
1
E(z 2 )
✲ U
−1
2
E(z 3 )
✲ U
−1
3
A(y 0 )
A(y 1 )
✲
A(y 2 )
✲
A(y 3 )
✲
C −1
1
C −1
2
C −1
3
E(x 0 ) π
−1
✲
0
E 0 (x)
✲
E(x 1 )
✲ π
−1
E 1 (x)
✲
A(x
1
✛
1) A π1 ✛
1(x)
E(x 2 )
✲ π
−1
E 2 (x)
✲
A(x
2
✛
2) A π2 ✛
2(x)
E(x 3 )
✲ π
−1
E 3 (x)
✲
A(x
3
✛
3) A π3 ✛
3(x)
Iterative Decoding
• The decoders are activated successively while passing reliability values of the
bits between each other.
• Decision statistics for the source bits: D(x) = E 0 (x) + E 1 (x) + E 2 (x) + E 3 (x),
D(x(i)) ˆx(i)=+1
≷ 0, for all i = 1, 2, . . . , L.
ˆx(i)=−1
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 4

Extrinsic Information Transfer (EXIT) Function
x(i)
✲
❥
✻
✲
D
✲
❄
❥✛
D
y(i)
✲
x(i)
✲
❥
✻
✲
D
✲
❥
✻
✲
D
✲
❥
✻
y(i)
✲
(I A(x) , I A(y)
)
I E(x) = Tx
1
0.8
0.6
0.4
0.2
1 0
0.8
0.6
I A(y)
0.4
0.2
0
0
0.2
0.4
0.8
0.6
I A(x)
(I A(x) , I A(y)
)
1
I E(x) = Tx
1
0.8
0.6
0.4
0.2
1 0
0.8
0.6
I A(y)
0.4
0.2
0
0
0.2
0.4
0.8
0.6
I A(x)
1
EXIT function for CC(4/7).
EXIT function for CC(7/5).
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 5

Extrinsic Information Transfer (EXIT) Function
)
(
I A(x 1) , I A(y1)
)
I E(x 1) = T x1
1
0.8
0.6
0.4
0.2
(
1 0
1 0
0.8
1 0.8
0.6
0.8
0.6
0.8
0.4
0.6
I 0.2
0.4
0.4
0.6
0.4
A(y1 ) 0.2
I 0.2
0 0 I A(y2 ) 0.2
A(x1 ) 0 0 I A(x2 )
⎛ ⎛
⎛ ⎛
√
⎞⎞
⎞
2
⎛
I E(xn ) = T xn ⎜
⎝ J ⎜
⎝√ J −1 ⎝ δ0 J⎝
ψ 0
N∑
8∑ N δ j
γ b
⎠⎠
+ J −1( I ) √
⎞
2 E(xj ) ⎟
j=0 R j
ψ j ⎠ , δ nJ⎝
ψ n
8∑ N δ j
γ b
⎠
j=0 R j
ψ j
J(σ) 1 − √ 1
+∞ ∫
2πσ
e −(ξ−σ2 /2) 2
2σ 2
log 2
(
1 + e
−ξ ) dξ
I A(x 2) , I A(y2)
I E(x 2) = T x2
1
0.8
0.6
0.4
0.2
j=1
j≠n
⎞
⎟
⎠ ,
1
−∞
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 6

Convergence Analysis and Design of Multiple Concatenated Codes
x ✲ π 0
✲ π 1
✲ π 2
✲ π 3
x 0 y 0 ✲
z
U 0 ✲ 0
x 1 y
✲ 1
C
✲ 1 U 1
x 2 y
✲ 2
C
✲ 2 U 2
x 3 y
✲ 3
C
✲ 3 U 3
❅
z 1 ✲
z 2 ✲
z 3
✲
❅
❅
❅
M
w
s✲
✓✏ ❄
r ✲
✒✑ M −1
❅
❅
❅
❅
E(z 0 ) U
−1
✲
0
E(z 1 )
✲ U
−1
1
E(z 2 )
✲ U
−1
2
E(z 3 )
✲ U
−1
3
A(y 0 )
A(y 1 )
✲
A(y 2 )
✲
A(y 3 )
✲
• Convergence analysis using projected EXIT charts, [1].
• Optimal puncturing ratios for all 1/4 ≤ R ≤ 1, [2].
C −1
1
C −1
2
C −1
3
E(x 0 ) π
−1
✲
0
E 0 (x)
✲
E(x 1 )
✲ π
−1
E 1 (x)
✲
A(x
1
✛
1) A π1 ✛
1(x)
E(x 2 )
✲ π
−1
E 2 (x)
✲
A(x
2
✛
2) A π2 ✛
2(x)
E(x 3 )
✲ π
−1
E 3 (x)
✲
A(x
3
✛
3) A π3 ✛
3(x)
• Optimal puncturing and energy distribution for all 1/4 ≤ R ≤ 1, [3].
[1] F. Brännström, L. K. Rasmussen, and A. J. Grant, “Convergence analysis and optimal scheduling for multiple concatenated codes,”
to appear in IEEE Trans. Inform. Theory, 2005.
[2] F. Brännström, L. K. Rasmussen, and A. Grant, “Optimal puncturing for multiple parallel concatenated codes,” in Proc. IEEE
Int. Symp. Inform. Theory (ISIT’04), Chicago, IL, June/July 2004, p. 154.
[3] F. Brännström, and L. K. Rasmussen, “Multiple parallel concatenated codes with optimal puncturing and energy distribution,” to
appear at IEEE Int. Conf. Commun. (ICC’05), Seoul, Korea, May 2005.
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 7

EXIT chart projection of PCC(1 + 5/7 + 7/6 + 7/4)
PCC(1 + 5/7 + 7/6 + 7/4)
Target BER: P b ≤ 10 −5
Uniform energy distribution
R = 1/4
γ b = 0.25 dB
Ψ 0 = [0.25, 0.25, 0.25, 0.25]
∆ ⋆ = [1.0, 1.0, 1.0, 1.0]
R = 1/3
γ b = −0.25 dB
Ψ 0 = [0.25, 0.25, 0.25, 0.25]
∆ ⋆ = [0.15, 0.85, 1.0, 1.0]
I E(x 1)
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
I E(x2 )
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 8

Achievable SNR-Rate Region
1
0.9
0.8
Code rate, R
0.7
0.6
0.5
0.4
0.3
C BPSK
PCC(1 + 5/7 + 7/6 + 7/4) Ψ 0
0.2
−1 0 1 2 3 4 5 6 7 8
SNR: γ b = E b /N 0 [dB]
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 9

Achievable SNR-Rate Region
1
0.9
0.8
Code rate, R
0.7
0.6
0.5
0.4
C BPSK
0.3
PCC(1 + 21/37 + 21/37) Ψ 0
PCC(1 + 5/7 + 7/6 + 7/4) Ψ 0
0.2
−1 0 1 2 3 4 5 6 7 8
SNR: γ b = E b /N 0 [dB]
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 10

EXIT chart projection of PCC(1 + 5/7 + 7/6 + 7/4)
PCC(1 + 5/7 + 7/6 + 7/4)
Target BER: P b ≤ 10 −5
R = 1/4
γ b = −0.36 dB
Ψ 0 = [0.25, 0.25, 0.25, 0.25]
∆ ⋆ = [1.0, 1.0, 1.0, 1.0]
R = 1/4
γ b = −0.36 dB
Ψ ⋆ = [0.08, 0.34, 0.24, 0.34]
∆ ⋆ = [1.0, 1.0, 1.0, 1.0]
I E(x 1)
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
I E(x2 )
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 11

Achievable SNR-Rate Region
1
0.9
0.8
Code rate, R
0.7
0.6
0.5
0.4
C BPSK
PCC(1 + 21/37 + 21/37) Ψ 0
0.3
PCC(1 + 5/7 + 7/6 + 7/4) Ψ 0
PCC(1 + 5/7 + 7/6 + 7/4) Ψ ⋆
0.2
−1 0 1 2 3 4 5 6 7 8
SNR: γ b = E b /N 0 [dB]
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 12

Performance of PCC(1 + 5/7 + 7/6 + 7/4) after 20 and 80 activations at R = 1/4
10 0 SNR: γ b = E b /N 0 [dB]
10 −1
C BPSK with Ψ 0
R = 1/4 and Ψ 0
C BPSK with Ψ ⋆
R = 1/4 and Ψ ⋆
10 −2
BER
10 −3
10 −4
10 −5
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 13

Conclusions
• A method for designing efficient multiple concatenated coding schemes.
– Efficient: good performance (BER, convergence threshold) and low decoding
complexity.
• EXIT functions are powerful tools to be used in all steps of designing multiple
concatenated codes.
– Code searches
– Convergence analysis with projected EXIT chart
– Optimal activation schedule
– Optimal puncturing ratios
– Optimal energy distribution
– Achievable SNR-rate region
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 14

Conclusions
• Other systems/components:
– Multiple serially concatenated codes
– Rate-compatible codes
– Multi-user decoding
– Bit-interleaved coded modulation (BICM)
– Multiple-input multiple-output (MIMO) systems
– Channel equalization
– Cross layer design
– ...
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 15

Optimal puncturing ratios for PCC(1 + 4/7 + 7/5)
1
0.8
δ0, δ1, δ2
0.6
0.4
0.2
δ 0
δ 1
δ 2
0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Code rate, R
Convergence Analysis and Design of Multiple Concatenated Codes, Fredrik Brännström 2005 16