Download - Trellis Coded Modulation
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Trellis Coded Modulation
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Trellis Coded Modulation: Introduction Increases the constellation size compared to uncoded
communication Increases in throughput (b/s/Hz) Decline in BER performance due to decrease of dmin
Trellis Coded Modulation (TCM) is used to offset loss resulting from constellation size increase
TCM achieves this higher gain by jointly using the distance properties of the code and the distance properties of the constellation, by carefully mapping coded and uncoded bits to the constellation points.
TCM uses “set partitioning” to map the bits to the constellation points
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Ungerboek Set Partitioning Ungerboeck Set partitioning:
Partition a constellation such that in each partition the minimum distance increases.
For binary data, in each stage we partition the constellation (signal set) into two subsets.
The number of stages depends on the particular TCM scheme.
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Ungerboek Partitioning of 8-PSK Constellation
A0
B0 B1
C0 C1 C2C3
( )8πsinE2=d SA
0123
45 6 7
0
2
4
6
13
5 7
04
2
6
3
7
1
5
SB E2=d
SE
SC E2=d
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Convolutional Codes(Reminder)
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IntroductionBlock Codes: Code words are produced on a block by block basis.
In Block Codes, the encoder must buffer an entire block before generating the associated codeword.
Some applications have bits arrive serially rather than in large blocks
Convolutional codes operate on the incoming message sequence continuously in a serial manner
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Convolutional Codes SpecificationA convolutional code is specified by three parameters
(n, k, K), where k/n is the coding rate and determines the number of
data bits per coded bit K is called the constraint length of the encoder where
the encoder has K-1 memory elements
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Convolutional Encoder: Example
0 0
+
+
Input Output
Rate ½ Convolutional Encoder
1 0 1
1
1
c1
c2
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Convolutional Encoder: Example
1 0
+
+
Input Output
Rate ½ Convolutional Encoder
1 0 1
1
0
c1
c2
1 1
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Convolutional Encoder: Example
0 1
+
+
Input Output
Rate ½ Convolutional Encoder
1 0 1
0
0
c1
c2
1 0 1 1
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Convolutional Encoder: Example
1 0
+
+
Input Output
Rate ½ Convolutional Encoder
1 0 1c1
c2
0 0 1 0 1 1
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+
+
Input Output
b0 b1
States (b0b1)s0 00s1 10s2 01s3 11
S0
S3
S1S2
0/00
1/11
1/00
0/011/10
1/01
0/10
0/11
Input 0Input 1
State Diagram Representation
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s0 (0 0)
s1 (1 0)
s2 (0 1)
s3 (1 1)
00 00 00
11 11 11
01
10 10
10
01 01
1100
00 00 00
11 11 11
01 01 01
10 10 10
10 10 10
01 01 01
11 11 1100 00 00
S0
S3
S1S2
0/00
1/11
1/00
0/011/10
1/01
0/10
0/11Trellis Representation
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s0 (0 0)
s1 (1 0)
s2 (0 1)
s3 (1 1)
00 00 00 00 00 00
11 11 11 11 11 11
01 01 01 01
10 10 10 10 10
10 10 10 10
01 01 01 01 01
11 11 11 11
Input: 101 Output:
S0
S3
S1S2
0/00
1/11
1/00
0/011/10
1/01
0/10
0/11
00 00 00 00
001011
Trellis Representation
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Trellis Representation of QPSK
01
2 3SQPSK E2=d
SE
Trellis Representation01
23
01
23
01
23
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Trellis Representation of QPSKDmin for 3 Consecutive Symbols 0 0 0
01
23
01
23
01
23
01
23
01
23
01
23
SE
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Trellis Representation of QPSKDmin for 3 Consecutive Symbols 0 0 0
01
23
01
23
01
23
01
23
01
23
01
23
SE2
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Trellis Representation of QPSKDmin for 3 Consecutive Symbols 0 0 0
01
23
01
23
01
23
01
23
01
23
01
23
0 0 0
SE3
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Trellis Representation of QPSKDmin for 3 Consecutive Symbols 0 0 0
01
23
01
23
01
23
01
23
01
23
01
23
SE
0 0 0
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Trellis Representation of QPSKDmin for 3 Consecutive Symbols 0 0 0
01
23
01
23
01
23
01
23
01
23
01
23
SE2
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Trellis Representation of QPSKDmin for 3 Consecutive Symbols 0 0 0
01
23
01
23
01
23
01
23
01
23
01
23
SE2
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Trellis Representation of QPSKDmin for 3 Consecutive Symbols 0 0 0
01
23
01
23
01
23
01
23
01
23
01
23
0 0 00 0 1SE2
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Trellis Representation of QPSK Summary)dmin for 3 Consecutive Symbols 0 0 0
01
23
01
23
01
23
01
23
01
23
01
23
01
2 3
=QPSK Sd 2E
SE
= + + =2 2min QPSK S Sd 0 0 2E 2E
S0
S0
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Four State Trellis with Parallel Paths0 0 04 4 4
26
26
26
2 6 2 6 2 604
04
04
15
15
15
15
15
15
3 73 7
3 7
3 7 3 7 3 7
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Four State Trellis with Parallel Paths
0 0 04 4 4
26
26
26
2 6 2 6 2 604
04
04
15
15
15
15
15
15
3 73 7
3 7
3 7 3 7 3 7
Dmin for 3 Consecutive Symbols 0 0 0
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Four State Trellis with Parallel Paths
0 0 04 4 4
26
26
26
2 6 2 6 2 604
04
04
15
15
15
15
15
15
3 73 7
3 7
3 7 3 7 3 7
Dmin for 3 Consecutive Symbols 0 0 0
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Four State Trellis with Parallel Paths
0 0 04 4 4
26
26
26
2 6 2 6 2 604
04
04
15
15
15
15
15
15
3 73 7
3 7
3 7 3 7 3 7
Dmin for 3 Consecutive Symbols 0 0 0
Distance between 0 0 0 and 2 1 2
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Four State Trellis with Parallel PathsDmin for 3 Consecutive Symbols 0 0 0
Distance between 0 0 0 and 2 1 2
Is this Dmin?[ ] [ ] ( ) ( )( ) ( )= + + =
2 2 2
S S S S0,0,0 , 2,1,2d 2E 2 E sin π 8 2E 2.14 E
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Four State Trellis with Parallel Paths
0 0 04 4 4
26
26
26
2 6 2 6 2 604
04
04
15
15
15
15
15
15
3 73 7
3 7
3 7 3 7 3 7
Dmin for 3 Consecutive Symbols 0 0 0
Distance between 0 0 0 and 0 0 4
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Four State Trellis with Parallel PathsDmin for 3 Consecutive Symbols 0 0 0
Distance between 0 0 0 and 0 0 4
[ ] [ ] ( ) S2
S4,0,0,0,0,0 E2=E2+0+0=d
Is this Dmin? YES & it is better than that of uncoded QPSK
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Coding Gain Four State Trellis TCM
31
Union Bound
mine
0
d1P M 1 erfc2 2 N
Coding Gain Four State Trellis TCM
2
min 8PSK TCM min 8PSK TCM8PSK TCM 10 10 2
min QPSK min QPSK
d dG dB 20 log 10 log
d d
S8PSK TCM 10
S
4EG dB 10 log 3 dB
2E
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Eight State Trellis without Parallel Paths
32
S0
S1
S2
S3
S4
S5
S6
S7
0 0 0
6
7
6
0 4 2 6
1 5 3 7
4 0 6 2
5 1 7 3
2 6 0 4
3 7 1 5
6 2 4 0
7 3 5 1
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Eight State Trellis without Parallel Pathsdmin for 3 Consecutive Symbols 0 0 0
[ ] [ ] ( ) ( )( ) ( )-= = + + =2 2 2
min 8PSK TCM S S S S0,0,0 , 6,7,6d d 2E 2 E sin π 8 2E 2.14 E
Distance between 0 0 0 and 6 7 6
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Coding Gain Eight State Trellis TCM
34
Union Bound
mine
0
d1P M 1 erfc2 2 N
Coding Gain Four State Trellis TCM
2
min 8PSK TCM min 8PSK TCM8PSK TCM 10 10 2
min QPSK min QPSK
d dG dB 20 log 10 log
d d
S8PSK TCM 10
S
4.585EG dB 10 log 3.6 dB
2E
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Encoder for Four State Trellis TCM
35
+
+
m1
m2
u1
u2
u3
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Encoder for Four State Trellis TCM
36
0
+
+
m1
m2
u1u2
u3
0
S0 (00)
S1 (10)
S2 (01)
S3 (11)
0 0 0
1 0 0 0 1 1 1 1 1
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Encoder for Four State Trellis TCM
37
0
+
+
m1
m2
u1u2
u3
1
S0 (00)
S1 (10)
S2 (01)
S3 (11)
0 0 0
1 0 0
0 1 0 1 1 0
0 1 1 1 1 1
0 0 1 1 0 1
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Encoder for Four State Trellis TCM
38
1
+
+
m1
m2
u1u2
u3
0
S0 (00)
S1 (10)
S2 (01)
S3 (11)
0 0 0
1 0 0
0 1 1
1 1 1
0 1 0 1 1 0
0 1 1 1 1 1
0 0 0
1 0 0
0 0 1 1 0 1
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Encoder for Four State Trellis TCM
39
1
+
+
m1
m2
u1u2
u3
S0 (00)
S1 (10)
S2 (01)
S3 (11) 1
0 0 0
1 0 0
0 1 1
1 1 1
0 1 0 1 1 0
0 0 1
1 0 1
0 1 1 1 1 1
0 0 0
1 0 0
0 0 1 1 0 1 0 1 0
11 0
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Encoder for Four State Trellis TCM
40
1
+
+
m1
m2
u1u2
u3
S0 (00)
S1 (10)
S2 (01)
S3 (11) 1
0 0 0
1 0 0
0 1 1
1 1 1
0 1 0 1 1 0
0 0 1
1 0 1
0 1 1 1 1 1
0 0 0
1 0 0
0 0 1 1 0 1 0 1 0
11 0
0 0 0 1 0 0
0 1 1
1 1 1
0 1 0
1 1 0
0 0 1
1 0 1