Error Control Coding - Digital Communication: Principles and System Modelling - page 196

The nodes on the left denote four possible current states and those on the right
represent the next state. The solid transition line represents input m = 0, and hash-

broken line represents input m 1. Along with each transition line the output is

represented during that particular transition. As for an example, if the encoder be in
state 'a' currently, for input m

=

= 0, the state will be transited to state 'a' (00) with

output v 1 v 2 = 00 and for input m = 1, the state will be transited to state 'b' (01)

with output v 1 v 2 = 11. Thus the trellis diagram is the compact and convenient

representation.

9.4.5 Decoding of Convolutional Code by Viterbi

Let's assume that, a coded signal is received as Y. Convolutional encoding operates
continuously on input data. Hence there are non-code vectors and blocks as such.
Here equiprobable (transmission error probability for symbol '1' and '0' are equal)
transmission is considered. Let's define a matrix as under.

9.4.5.1 Metric

It is the discrepancy between the received signal Y and the decoded signal at a
particular node. This metric may be added over few nodes for a particular path.

9.4.5.2 Surviving Path

This is the path of the decoded signal with minimum metric.
In viterbi decoding, a metric is assigned to each surviving path (i.e., metric of a
particular path is obtained by adding individual metric on the nodes along the path)
Y is decoded as the surviving path with smallest metric.

9.4.5.3 Principle of Decoding

Let us consider the following example of viterbi decoding. Let the signal being
received is encoded by the encoder shown in the Fig. 9.3 . For this encoder, we have
obtained trellis of Fig. 9.9 . Let the 1st six received bits be

Y = 11 01 11

It is to be noted that, for single bit input, the encoder transmits 2 bits (2, 1, 2) as
output. These output bits are received at the decoder and represented by Y. Hence,
Y given above represents the output bits for three successive message bits. Assume
that the decoder is at state a 0 .
Let us look at the trellis diagram of Fig. 9.9 for this encoder. It signifies, if the
current state is 'a', then the next state will be 'a' or 'b'. This has been shown in