figure 6.1. a convolutional encoder. figure 6.2. structure of a systematic convolutional encoder of...
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)1(m
)2(m
)1(c
)2(c
)3(c
Figure 6.1. A convolutional encoder
m
)1(c
)2(c
Figure 6.2. Structure of a systematic convolutional encoder of rate 2/1Rc
m
)1(c
)2(c
1S 2S
Figure 6.3. Convolutional encoder of rate 2/1Rc
)1(m
)2(m
)1(c
)2(c
)3(c
)2(1S
)1(1S
convC )1,2,3( 3/2Rc Figure 6.4. Encoder of convolutional code of code rate
Sa = 0 0
0/00
1/0 1
Sb = 1 0 Sc = 0 1
Sd = 1 1
0/0 1
1/0 0
0/1 0
0/1 1
1/1 0
1/1 1
Figure 6.5. State diagram for the convolutional encoder of Figure 6.3
t1 t2 t3 t4 t5 t6 0/00 0/00 0/00 0/00 0/00
1/00 1/00 1/00
1/11 1/11 1/11 1/11 1/11
0/11 0/11 0/11
1/01 1/01 1/01
0/01 0/01 0/01 0/01
1/10 1/10 0/10
1/10 0/10 1/10
0/10
Sa = 00
Sb = 10
Sc = 01
Sd = 11
Figure 6.6. Trellis representation of the convolutional code of Figure 6.3
)1(m
)1(c
)2(c
Figure 6.7. A systematic convolutional encoder
2a 1a
0a
0S
1S 2S
na
m
nS
c
Figure 6.8. A FIR FSSM
2a 1a 0a
0S
1S 2S
1f
nf
na
m
nS
2f
c c
Figure 6.9. IIR FSSM
)k(s0 )k(s2 )k(s1
)k(m
)k(c )(1
)1k(s0 )2k(s0
Figure 6.10. FIR FSSM in the discrete time domain
)k(s2
)k(s1
)k(s0
)1k(s0
)2k(s0
)k(c
)k(m
Figure 6.11. An IIR FSSM
m
)1(c
)2(c
Figure 6.12. Equivalent systematic convolutional encoder of the encoder of Figure 6.3
t1 t2 t3 t4 t5 0/00 0/00 0/00 0/00
0/00 0/00 0/00
1/11 1/11 1/11 1/11 1/11
1/11 1/11 1/11
0/01 0/01 0/01
0/01 0/01 0/01 0/01
1/10 1/10 1/10 1/10 1/10
1/10 1/10
Sa = 00
Sb = 10
Sc = 01
Sd = 11
0/00
Figure 6.13. Trellis for the convolutional encoder of Figure 6.12
2a 1a
0a
0S
1S 2S
1f
nf
na
)1(c )2(c
m
nS
2f
Figure 6.14. General structure of systematic IIR convolutional encoders of rate 2/1Rc
aS bS cS
X
0X
X
2X
X dS
X
2X
aS
Figure 6.15. Modified state diagram
t1 t2 t3 t4 t5 t6
0 0 0
2 2 2 2
2 2 2
1 1 1
1 0 1 1
1 1 1 1 1 1 1
Sa = 00
Sb = 10
Sc = 01
Sd = 11
2
t7 0 0
0 0
0 0
0 0
0 0
Figure 6.16. Minimum free distance sequence evaluated on the trellis
t1 t2 t3 t4 t5 t6 2 1 1 0 2
1 0 2
0 1 2 0
1 2 0
0 1 1
0 0 1 1
2 2 2 1 1 1 1
Sa = 00
Sb = 10
Sc = 01
Sd = 11
1
t7 1 1
0 1
0 1
0 0
1 1
Figure 6.17. Hamming distance calculations for the Viterbi algorithm
t1 t2 t3 t4 2 3
1
0 3 1
2
0 3
2
Sa = 00
Sb = 10
Sc = 01
Sd = 11
4
1 1 0 1 0 1
4
4
t1 t2 t3 t4 t5 2 1
1
0 3 3
1
0 2
2 2
3
2 3
Sa = 00
Sb = 10
Sc = 01
Sd = 11
1 1
0 1 0 1 0 0
3
3
5
Figure 6.18. Survivor paths in the Viterbi algorithm
t1 t2 t3 2
1
0 3
0
2 2
3
Sa = 00
Sb = 10
Sc = 01
Sd = 11
1 1
0 1 0 1 t4 t5 1
3 1
2
2
0 0 t6 3
1
4
4 4
1 1
4
4
5 3dacc
1dacc
4dacc
4dacc
Figure 6.19. Viterbi decoding algorithm, time instant 6t
t1 t2 t3 Sa = 00
Sb = 10
Sc = 01
Sd = 11
1 1
0 1 0 1 t4 t5
0 0 t6
1 1
1dacc
4dacc
4dacc
5dacc
t7
0 1 t8
1 1
Figure 6.20. Viterbi decoding algorithm, decoded sequence at time instant 8t
aS bS cS
XY
YZ.1
XY
YX 2
XYZ dS
XYZ
YZX 2
aS
Figure 6.21. Extended and modified state diagram
0 1 1
1 0 1
0 0 0
1 1 0
1 1 1
0 1 0
1 0 0
0 0 1
x
y
z
Figure 6.22. Vector representation (polar format) of code vectors in a vector space of dimension
3n
1.6875
-1.0504
-0.1072
0 10 20 30 40 50 60 70 80 90-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Figure 6.23. Signal resulting from the transmission of the code vector )101(
in polar format over a Gaussian channel
0
1
0
1
2
3
Very reliable output for 0
Very reliable output for 1
Less reliable ouptut for 0
Less reliable ouptut for 1
Figure 6.24. A soft decision channel
-2 0 2 4 6 8 10 12 1410
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Eb/No [dB]
Pb
Repetition code, soft decision Repetition code, hard decision Uncoded transmission
Figure 6.25. A comparison between hard and soft decision decoding of the triple repetition code
(n=3), and uncoded transmission
t1 t2 t3 t4 1 2
1
1 2 3
4
1 3
3
Sa = 00
Sb = 10
Sc = 01
Sd = 11
2
1 0 0 1 0 0
4
3
4
Erroneous decision
Figure 6.26. Hard decision decoding of example of Section 16.3
t1 t2 t3 t4 t5 -1-1 -1-1 -1-1 -1-1
-1-1 -1-1 -1-1
+1+1 +1+1 +1+1 +1+1 +1+1
+1+1 +1+1 +1+1
-1+1 -1+1 -1+1
-1+1 -1+1 -1+1 -1+1
+1-1 +1-1 +1-1 +1-1 +1-1
+1-1 +1-1
Sa = 00
Sb = 10
Sc = 01
Sd = 11
-1-1
Figure 6.27. Trellis of the convolutional encoder of Figure 6.3 with output values in polar format
t1 t2 t3 t4 6.245 12.0675
1.445 11.467 6.299
13.499
1.667 12.699
12.267
Sa = 00
Sb = 10
Sc = 01
Sd = 11
12.900
+1.35 -0.15 -1.25 +1.40 -0.85 –0.10
16.7
15.699
16.499
2.499
Figure 6.28. Soft decision decoding to determine the survivor at time instant
4t on the corresponding trellis
t1 t2 6.245 12.0675
1.445 11.467
1.667
12.267
Sa = 00
Sb = 10
Sc = 01
Sd = 11
+1.35 -0.15 -1.25 +1.40
t3 t4
6.299
13.499
12.699
-0.85 –0.10
2.499
t5
24.064
21.064
10.064
6.864
-0.95 –1.75
17.664
6.864
17.864
13.264
Figure 6.29. Soft decision decoding to determine the survivor at time instant
5t on the corresponding trellis
t1 t2 6.245
1.445 11.467
1.667
12.267
Sa = 00
Sb = 10
Sc = 01
Sd = 11
+1.35 -0.15
-1.25 +1.40 t3
6.299
13.499
12.699
-0.85 –0.10
2.499
t4 t5
10.064
6.864
-0.95 –1.75
6.864
13.264
t6
9.204
15.604
14.404
+0.50 +1.30
7.204
18.804
12.404
17.604
10.404
Figure 6.30. Soft decision decoding to determine the survivor at time instant
6t on the corresponding trellis
t1 t2
1.445
1.667
Sa = 00
Sb = 10
Sc = 01
Sd = 11
+1.35 -0.15
-1.25 +1.40 t3
6.299
-0.85 –0.10
2.499
t4 t5
10.064
6.864
-0.95 –1.75
6.864
t6
9.204
+0.50 +1.30
7.204
12.404
10.404
Figure 6.31. Soft decision decoding to determine the final survivor on the corresponding trellis
)1(m
)1(bc
)2(bc
1S
2S
Puncturing of
outputs )1(
bc
and )2(
bc
of the base convolucional code
)1(c
)2(c
3/2Rc
2/1Rc
Figure 6.32. Punctured convolutional encoder of rate
based on a convolutional code of rate
t1 t2 t3 t4 t5 t6 0/00 0/0 0/00 0/0 0/00
1/00 1/0 1/00
1/11 1/1 1/11 1/1 1/11
0/11 0/1 0/11
1/01 1/0 1/01
0/0 0/01 0/0 0/01
1/1 1/10 0/10
1/1 0/1
1/10 0/10
Sa = 00
Sb = 10
Sc = 01
Sd = 11
3/2Rc
2/1n/k
Figure 6.33. Trellis for a punctured convolutional code of rate
based on a convolutional code of rate
Figure 6.34. Trellis for a convolutional code of rate 3/2Rc
constructed in the traditional way
11/110
00/000
01/001
10/111
11/101
01/010
10/100
11/0110
01/100
10/010
11/000
01/111
10/001
00/01100/101
00/110
11/110
00/000
01/001
10/111
Sa=00
Sb=10
Sc=01
Sd=11
m
)1(c
)2(c
Figure P.6.1. Convolutional encoder, Problem 6.3
m
)1(c
)2(c
)3(c
Figure P.6.2. Convolutional encoder, problem 6.4
m
)1(c
)2(c
Figure P.6.3. Convolutional encoder, Problem 6.5
m
)1(c
)2(c
Figure P.6.4. Convolutional encoder, Problem 6.6
0 0
1 1
00/0
10/1
01/0 11/1
Figure P.6.5. Trellis diagram, problem 6.7
2a 1a
0a
0S
1S 2S
1f
)1(c )2(c
m
2f
Figure P.6.6. Convolutional encoder, Problem 6.8
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