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1.1.1 Decoding of Convolutional Codes

One of the advantages of Convolutional codes in comparison with Block codes is that

they are simpler to decode. The process of decoding Convolutional codes is finding the

path that the encoder has traversed. The path is the way that the encoder starts from state

[0 0] to the end state [0 0] in the Trellis diagram which has been introduced in the last

section. There are many algorithms using to decode Convolutional codes, such as:

sequential decoding, majority-logic decoding, or Viterbi decoding. In this project, I will

focus on Viterbi algorithm, which has become the most widely used convolutional

decoding algorithm, because it reduces the computational complexity with satisfied

performance. Viterbi algorithm was first introduced by Viterbi in 1967, and now it becomes

the most popular one in application.

Now we look again at Figure 2.1.3, note that at a particular time t, the encoder will

have one state S[x x], so now we will denote a node by (S, t).For example, at time t = Si+1,

the encoder will have the state [0 1], so we will call it ([0 1], Si+1). As previously stated, the

decoder will have to find the path that the encoder had done before, so as to decode to

take right data symbols. In this project, I just introduce the idea of the Viterbi algorithm.

Detail information as well as the way to design a Viterbi decoder will be discussed more

clearly in the thesis.

The idea of Viterbi algorithm is to find the local survivor path, which is the path that

has the smallest distance between the received sequence and the code sequence. This

algorithm has the same idea with maximum likelihood decoding algorithm, which is used

to find the closest code sequence to the received sequence. The process is based on Trellis

diagram to tracks the states of the encoder.To achieve maximum likelihood decoding each branch in the Trellis diagram is

assigned a metric measured by the Hamming distance between the received word (with

noise) and ideal word (00, 01, 10, 11). Therefore, the metric can be 0, 1 or 2 (the Hamming

distance between two words is the difference between the bits of those code words, as

described in the first section). At each time t, the decoder will receive a pair of channel

symbols which are already changed by noise. The algorithm will base on Trellis diagram to

find the shortest way (minimum metric) so as to get the word which is closest to the code

word. For example, the input sequence of the encoder is [0 1 1 0 1 0], and the channel

noise is [00 01 00 00 01 00], so we will have the code word [00 11 10 10 00 01] and thereceived word [00 1010 10 0101]. It means there are 2 errors happened during the

process of transmission (at bit 2 and 5). Now let us look at the Trellis diagram to find out

how the decoder works (Figure 2.1.4):

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Figure 2.1.4: Viterbi algorithm decoding process

According to Figure 2.1.4, the part in the left hand side describe how a state transit

to another, for example, state [0 0] will stay at [0 0] if input bit is 0, and become state [1 0]

if input is 1. Now I will describe how the right hand side Figure goes. Firstly, let us look at

the symbol above each path vector, which has the form (XX/X). The first 2 symbols will be

the output corresponding to the input, and the last symbol will denote the Hamming

distance between the received word and output. Note that the received words are written

below each transition. As we discussed above, the initial state will always be state [0 0], so

there are 2 paths can happen: [0 0] and [1 0]. The corresponding output will be 00 and 11

respectively, while the received word in the decoder is 00, so the Hamming distance will be

0 and 2 as described in the Figure. According to Viterbi algorithm, the decoder will choose

the path which has smaller Hamming distance to continue the process. However, in the

second path, the Hamming distances are both 1 and 1, so the decoder must do both paths

to find which one is shorter. The process continues, and the red line in the Figure is the

shortest way that the decoder can realize. Therefore, the decoded word will be [00 11 10

10 00 01] which is exactly similar to the transmitted code word.

In my thesis, this process will be discussed again with more details. Naturally,

convolutional codes will be a part of my final thesis, because they represent one important

technique within the general class of channel codes that students should know.

Convolutional codes have found many applications, such as deep-space communications,

speech transmission, or in digital modulation communication systems. They work well with

00

10

01

11

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00/0 00/1

11/1

00/1

11/1

01/2

10/010/0

01/2

11/1

00/1

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00 10 10 10 01 01Received word

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the channels which have noise and other external factors which can change the bit

sequence.