The structure of encoder and decoder To detect

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The structure of encoder and decoder To detect or correct errors, we need to

The structure of encoder and decoder To detect or correct errors, we need to send extra (redundant) bits with data. 10. 1

Figure 10 -6 Binary symmetric channel. Principles of Communications, 5/E by Rodger Ziemer and

Figure 10 -6 Binary symmetric channel. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10 -12 Rb = Cc relationship for additive white Gaussian noise channel. Principles

Figure 10 -12 Rb = Cc relationship for additive white Gaussian noise channel. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10. 10 Encoder and decoder for simple parity-check code 10. 4

Figure 10. 10 Encoder and decoder for simple parity-check code 10. 4

We can count the number of 1 s in the Xoring of two words

We can count the number of 1 s in the Xoring of two words 1. The Hamming distance d(000, 011) is 2 because 2. The Hamming distance d(10101, 11110) is 3 because 10. 5

Example 10. 6 Find the minimum Hamming distance of the coding scheme in Table

Example 10. 6 Find the minimum Hamming distance of the coding scheme in Table 10. 2. Solution We first find all the Hamming distances. The dmin in this case is 3. 10. 6

Figure 10 -14 Example of rate 1/3 repetition mode. Principles of Communications, 5/E by

Figure 10 -14 Example of rate 1/3 repetition mode. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10 -15 Coder for (7, 4) cyclic code. Principles of Communications, 5/E by

Figure 10 -15 Coder for (7, 4) cyclic code. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10 -16 Decoder for a (7, 4) cyclic code. Principles of Communications, 5/E

Figure 10 -16 Decoder for a (7, 4) cyclic code. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10 -18 Performance of repetition codes on AWGN and Rayleigh fading channels. Principles

Figure 10 -18 Performance of repetition codes on AWGN and Rayleigh fading channels. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10 -20 Performance of noncoherent FSK in a Rayleigh fading channel. Principles of

Figure 10 -20 Performance of noncoherent FSK in a Rayleigh fading channel. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10 -21 General convolutional encoder. Principles of Communications, 5/E by Rodger Ziemer and

Figure 10 -21 General convolutional encoder. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10 -22 A rate 1/3 convolutional encoder. Principles of Communications, 5/E by Rodger

Figure 10 -22 A rate 1/3 convolutional encoder. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10 -23 Code tree. Principles of Communications, 5/E by Rodger Ziemer and William

Figure 10 -23 Code tree. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10 -24 State diagram for the example convolutional encoder. Principles of Communications, 5/E

Figure 10 -24 State diagram for the example convolutional encoder. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10 -25 Trellis diagram. Principles of Communications, 5/E by Rodger Ziemer and William

Figure 10 -25 Trellis diagram. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10 -26 Termination of the trellis diagram. Principles of Communications, 5/E by Rodger

Figure 10 -26 Termination of the trellis diagram. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Example: For the convolutional code example in the previous lecture, starting from state zero,

Example: For the convolutional code example in the previous lecture, starting from state zero, Decode the following received sequence. At the end of the trellis, select the path with the minimum cumulative Hamming weight This is the survival path in this example Add the weight of the path at each state Compute the two possible paths at each state and select the one with less cumulative Hamming weight Decoded sequence is m=[10 1110] This is called the survival path

The largest metric, verify that you get the same result! Note also the Hamming

The largest metric, verify that you get the same result! Note also the Hamming distances!

Problem of optimum decoding is to find the minimum distance path from the initial

Problem of optimum decoding is to find the minimum distance path from the initial state back to the initial state (below from S 0 to S 0). The minimum distance is one of the sums of all path metrics from S 0 to S 0 Exhaustive maximum likelihood method must search all the paths in phase trellis (2 k paths emerging/ entering from 2 L+1 states for an (n, k, L) code) The Viterbi algorithm gets improvement in computational efficiency via concentrating into survivor paths of the trellis The Viterbi algorithm

Assume for simplicity a convolutional code with k=1, and thus up to 2 k

Assume for simplicity a convolutional code with k=1, and thus up to 2 k = 2 branches can enter each state in trellis diagram Assume optimal path passes S. Metric comparison is done by adding the metric of S 1 and S 2 to S. At the survivor path the accumulated metric is naturally smaller (otherwise it could not be the optimum path) For this reason the non-survived path can be discarded -> all path alternatives need not to be further considered Note that in principle the whole transmitted sequence must be received before decision. However, in practice storing of states for input length of 5 L is quite adequate

After register length L+1=3 branch pattern begins to repeat Smaller accumulated metric selected 1

After register length L+1=3 branch pattern begins to repeat Smaller accumulated metric selected 1 1 (Branch Hamming distances in parenthesis) First depth with two entries to the node The decoded ML code sequence is 11 10 10 11 00 00 00 whose Hamming distance to the received sequence is 4 and the respective decoded sequence is 1 1 0 0 0 (why? ). Note that this is the minimum distance path. (Black circles denote the deleted branches, dashed lines: '1' was applied) 22

Figure 10 -27 Communication system with interleaving. Principles of Communications, 5/E by Rodger Ziemer

Figure 10 -27 Communication system with interleaving. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10 -28 Turbo coder. Principles of Communications, 5/E by Rodger Ziemer and William

Figure 10 -28 Turbo coder. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10 -29 Recursive, systematic, convolutional coder. Principles of Communications, 5/E by Rodger Ziemer

Figure 10 -29 Recursive, systematic, convolutional coder. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.

Figure 10 -30 Performance curves for turbo code. Principles of Communications, 5/E by Rodger

Figure 10 -30 Performance curves for turbo code. Principles of Communications, 5/E by Rodger Ziemer and William Tranter Copyright © 2002 John Wiley & Sons. Inc. All rights reserved.