Trellis Diagram&

The Viterbi Decoding Algorithm

景文科技大學 電子工程系

張明化 資料來源: 1.Error control coding: fundamentals and applications, ShuLin / D. J. Costello, Jr. Prentice Hall, 2004. 2.Digital Communications, John. Proakis, McGRAW-Hill, 4th-edition, 2000. 3. http://www.dyu.edu.tw/~thhu(老胡小鋪)

(2,1,2) convolutional code

Fig. 1

State diagram of a (2,1,2) convolutional code

Fig. 2

The trellis for a (2,1,2) convolutional code

Fig. 3

The trellis for a (2,1,2) convolutional code

Fig. 4

Example 1:

• We consider the (2, 1, 2) convolutional code given in Fig. 1. The trellis diagram corresponding to a message of 5 bits long plus 2 zeros:

• U= (1 0 1 1 1 0 0 )

• The trellis has a depth of 7 as shown in the following page.

A terminated trellis diagram with a depth of 7

Fig. 5

The Viterbi Decoding Algorithm

Basic Concepts •Generate the code trellis at the decoder. •The decoder penetrated through the code trellis level

by level in search for the transmitted code sequence. •At each level of the trellis, there are 3 steps involved in

the decoding procedure: computation, addition and selection (CAS).

•The decoder stores the partial path with the largest metric and eliminates all the other partial paths. The stored partial path is called the survivor.

Two kinds ofViterbialgorithm

decoding: HDD & SDD • Hard Decision Decoding (HDD): the metric is

to compute the Hamming distance d(V, Z),

• Soft Decision Decoding (SDD) over AWGN channel: the metric is to compute the Euclidean distance.

d(s(V), r)

Example 2:

• V = (1, 0, 0, 1, 1, 0, 0), and its signal waveform is s(v) = (-1, 1, 1, -1, -1, 1, 1).

• r = (-0.3, 1.5, 2.3, -0.8, 0.4, 0.5, 0.7), and its hard decision output is z = (1, 0, 0, 1, 0, 0, 0).

• The Hamming distance is d(v, z) = 1.

• The Euclidean distance is d(v, r) -5.7.

Example 3:

Consider the (2, 1, 2)convolutional code given in Example 2. The received sequence r is

r = (1.3, -0.2, -0.8, -0.9, -1.5, 0.03, -0.4, 0.5, 0.6, 0.7, -0.6, -0.9, -1.7, 1.2).

Its hard decision limiter output is

Z = (01, 11, 10, 10, 00, 11, 10).

The decoding procedures of HDD and SDD are illustrated at

Fig. 6 ~ 15.

The trellis diagram of a (2, 1, 2) convolutional code with L = 7 ( message length)

Fig. 6

Fig. 6

HDD process at level 2 (comparison and elimination)

Fig. 7

HDD process at level 3 (comparison and elimination)

Fig. 8

HDD process at level 4 (comparison and elimination)

Fig. 9

HDD process at level 5 (comparison and elimination)

Fig. 10

HDD process at level 6 (comparison and elimination)

Fig. 11

HDD process at level 7

(comparison and elimination)

Fig. 12

HDD termination

Fig. 13

SDD process at level 2

Fig. 14

SDD termination

Fig. 15

(2, 1, 2) convolutional code over AWGN with HDD & SDD

Fig. 16