IV Convolutional Codes Puncturing of Convolutional Codes l

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IV. Convolutional Codes

IV. Convolutional Codes

Puncturing of Convolutional Codes l Increasing Code Rate by Puncturing of Some of the

Puncturing of Convolutional Codes l Increasing Code Rate by Puncturing of Some of the Outputs of Convolutional Encoder l Puncturing Rule selects the Outputs that are Eliminated l The Construction of a Punctured Convolutional Code is that its Trellis should maintain the Same State and Transition Structure of the Base Code © Tallal Elshabrawy 2

Puncturing Example c 2 c 1 Puncturing Rule c 1 c 2 Base Code

Puncturing Example c 2 c 1 Puncturing Rule c 1 c 2 Base Code Rate 1/2 Punctured Code Rate 2/3 s 0 (0 0) 00 00 11 11 11 s 1 (1 0) 01 s 2 (0 1) 10 © Tallal Elshabrawy 00 11 11 11 00 00 01 01 10 10 s 3 (1 1) 00 01 10 10 10 01 01 01 3

Puncturing Example c 2 c 1 Puncturing Rule c 1 c 2 Base Code

Puncturing Example c 2 c 1 Puncturing Rule c 1 c 2 Base Code Rate 1/2 Punctured Code Rate 2/3 s 0 (0 0) 00 00 11 11 11 s 1 (1 0) 0 s 2 (0 1) 1 © Tallal Elshabrawy 00 11 11 11 00 00 01 01 10 10 s 3 (1 1) 00 01 10 10 10 01 01 01 4

Puncturing Example c 2 c 1 Puncturing Rule c 1 c 2 Base Code

Puncturing Example c 2 c 1 Puncturing Rule c 1 c 2 Base Code Rate 1/2 Punctured Code Rate 2/3 s 0 (0 0) 00 0 11 1 11 s 1 (1 0) 0 s 2 (0 1) 1 © Tallal Elshabrawy 00 11 11 11 00 00 01 01 10 10 s 3 (1 1) 00 01 1 10 10 0 01 01 5

Puncturing Example c 2 c 1 Puncturing Rule c 1 c 2 Base Code

Puncturing Example c 2 c 1 Puncturing Rule c 1 c 2 Base Code Rate 1/2 Punctured Code Rate 2/3 s 0 (0 0) 00 0 11 1 11 s 1 (1 0) 0 s 2 (0 1) 1 © Tallal Elshabrawy 0 11 1 1 00 0 01 0 10 10 s 3 (1 1) 00 01 1 10 1 0 01 0 6

Puncturing Example Puncturing Rule c 2 Base Code Rate 1/2 Punctured Code Rate 2/3

Puncturing Example Puncturing Rule c 2 Base Code Rate 1/2 Punctured Code Rate 2/3 s 0 (0 0) Input : 1101 Output : 100011 c 1 00 0 11 1 11 s 1 (1 0) 0 s 2 (0 1) 1 © Tallal Elshabrawy 0 11 1 1 00 0 01 0 10 10 s 3 (1 1) 00 01 1 10 1 0 01 0 7

Codewords in Convolutional Codes l Convolutional codes constitute a special class of linear codes

Codewords in Convolutional Codes l Convolutional codes constitute a special class of linear codes l In linear block codes, all codewords have a fixed length l Convolutional codes depict serial encoding of bit streams l So, what is a codeword in convolutional codes? © Tallal Elshabrawy 8

Codewords in Convolutional Codes A code word in convolutional codes is any path on

Codewords in Convolutional Codes A code word in convolutional codes is any path on the trellis that starts at state s 0 and ends at state s 0 (0 0) 00 00 11 11 11 s 1 (1 0) 01 s 2 (0 1) 10 © Tallal Elshabrawy 00 11 11 11 00 00 01 01 10 10 s 3 (1 1) 00 01 10 10 10 01 01 01 9

Codewords in Convolutional Codes A code word in convolutional codes is any path on

Codewords in Convolutional Codes A code word in convolutional codes is any path on the trellis that starts at state s 0 and ends at state s 0 (0 0) 00 00 11 11 11 s 1 (1 0) 01 s 2 (0 1) 10 © Tallal Elshabrawy 00 11 11 11 00 00 01 01 10 10 s 3 (1 1) 00 01 10 10 10 01 01 01 10

Codewords in Convolutional Codes A code word in convolutional codes is any path on

Codewords in Convolutional Codes A code word in convolutional codes is any path on the trellis that starts at state s 0 and ends at state s 0 (0 0) 00 00 11 11 11 s 1 (1 0) 01 s 2 (0 1) 10 © Tallal Elshabrawy 00 11 11 11 00 00 01 01 10 10 s 3 (1 1) 00 01 10 10 10 01 01 01 11

Codewords in Convolutional Codes A code word in convolutional codes is any path on

Codewords in Convolutional Codes A code word in convolutional codes is any path on the trellis that starts at state s 0 and ends at state s 0 (0 0) 00 00 11 11 11 s 1 (1 0) 01 s 2 (0 1) 10 © Tallal Elshabrawy 00 11 11 11 00 00 01 01 10 10 s 3 (1 1) 00 01 10 10 10 01 01 01 12

Codewords in Convolutional Codes l Two Questions: l Why in convolutional codes is a

Codewords in Convolutional Codes l Two Questions: l Why in convolutional codes is a codeword defined to start as state s 0 and end at state s 0 l How could we guarantee that after a certain number of information bits, the trellis is at state s 0 © Tallal Elshabrawy 13

0/00 Signal Flow Graph S 0 1/11 0/11 D D s 3 S 2

0/00 Signal Flow Graph S 0 1/11 0/11 D D s 3 S 2 s 0 D 2 s 1 D S 1 0/01 1/10 0/10 D S 3 1 I 1/00 T(D) s 2 D 2 1/01 s 0 Solving © Tallal Elshabrawy 14

Free Distance of Convolutional Codes l The transfer function above says that the convolutional

Free Distance of Convolutional Codes l The transfer function above says that the convolutional code has l l l One codeword of weight 5 Two codewords of weight 6 Four codewords of weight 7 Eight codewords of weight 8 … l The free distance of convolutional codes is the minimum weight of any path on the trellis that starts and ends at state s 0 l The correction capability performance of convolutional codes is bounded by the free distance © Tallal Elshabrawy 15

TailBiting Convolutional Codes with Expurgation and RateCompatible PuncturingTailBiting Convolutional Codes with Expurgation and RateCompatible Puncturing
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