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turbo2 - Turbo Coding and MAP Decoding Example 1 Intuitive...

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Turbo Coding and MAP Decoding - Example Copyright 2006 Charan Langton www.complextoreal.com 1 Intuitive Guide to Principles of Communications www.complextoreal.com Turbo decoding using the MAP algorithm Part 2 A Step by step example k u s k p k y y , 1 , 1 s k p k y y , 2 , 2 EC2 EC1 Figure 1 A rate 1/3 PCCC Turbo code In this example, we will use a rate 1/3 Turbo Code which has two identical convolutional codes. Note that since 2, s k y is a deterministically reshuffle version of 1, s k y , it is not transmitted. The second decoder is given 1, s k y , and it then de-interleaves it to get its copy of 2, s k y which it needs for decoding. The coding trellis for each is given by the figure below. The blue lines show transitions in response to a 0 and red lines in response to a 1. The notation 1/11, the first number is the input bit, the next are two code bits. Of these, the first is what we called the systematic bit , and as you can see, it is same as the input bit. The second bit is the parity bit. Each code uses this trellis for encoding.

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Turbo Coding and MAP Decoding - Example Copyright 2006 Charan Langton www.complextoreal.com 2 00 10 00 01 10 11 11 01 s1 s3 s2 s4 s1 s3 s2 s4 Figure 2 the trellis diagram of the rate 1/2 code RSC code For the example, we pick the information bit sequence: 1, 0, 1, 0, 1, 0, 0, which in bi-polar form is shown in Table 1. Code 1 encodes each of these bits as systematic and parity bits using the trellis in Fig. 7. The bits are coded in a bipolar format. Table 1 Coded bits through first Encoder Code 1 k=1 k=2 k=3 k=4 k=5 k=6 k=7 Systematic 1 -1 1 -1 1 -1 -1 Parity bit, ENC1 1 1 -1 1 1 -1 -1 The sequence looks like this on the trellis diagram. k=1 k=2 k=3 k=4 k=5 k=0 k=6 k=7 s1 s3 s2 s4 Figure 3 Trellis diagram for information (1, -1, 1, -1, 1, -1, -1) through first encoder The code 2 receives the same information bits but interleaved by a given pattern. Interleaving is an important issue in turbo coding. There are several different types of interleaving patterns other than the row column method you know. Here I have made up a “pseudo - random” interleaving patter n shown in Table 2. The deinterleaving pattern puts the bits back in the original order. Simulations shows that pseudorandom interleaving works best with turbo codes in AWGN channel. Table 2 Interleaving patter from Encoder 1 to Encoder 2 and then deinterleaving back to Encoder 1 from Encoder 2 Interleaving 1-2 1-3 2-4 3-1 4-5 5-2 6-6 7-7 Deinterleaving 2-1 1-3 2-5 3-1 4-2 5-4 6-6 7-7
Turbo Coding and MAP Decoding - Example Copyright 2006 Charan Langton www.complextoreal.com 3 After reordering the information sequence (1, -1, 1, -1, 1, -1, -1) into (1, 1, 1, -1, - 1, -1, -1) Encoder 2 codes the new sequence as in Table 3. Table 3 Interleaved bits coded by Encoder 2 Code 2 k=1 k=2 k=3 k=4 k=5 k=6 k=7 Systematic 1 1 1 -1 -1 -1 -1 Parity bit, ENC2 1 -1 1 -1 -1 -1 -1 On the trellis diagram this new sequence is coded as shown in Figure 8.

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turbo2 - Turbo Coding and MAP Decoding Example 1 Intuitive...

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