Modulation and coding course- lecture 11

Modulation, Demodulation and
Coding Course
Period 3 - 2007
Catharina Logothetis
Lecture 11
Lecture 11 2
Last time, we talked about:

Another class of linear codes, known as
Convolutional codes.

We studied the structure of the encoder
and different ways for representing it.
Lecture 11 3
Today, we are going to talk about:

What are the state diagram and trellis
representation of the code?

How the decoding is performed for
Convolutional codes?

What is a Maximum likelihood decoder?

What are the soft decisions and hard
decisions?

How does the Viterbi algorithm work?
Lecture 11 4
Block diagram of the DCS
Information

source
Rate 1/n
Conv. encoder
Modulator
Information
sink
Rate 1/n
Conv. decoder
Demodulator
4434421
sequenceInput
21
,...),...,,(
i
mmm=m
4434421
444344421
bits) coded ( rdBranch wo
1
sequence Codeword
321

,...),...,,,(
n
nijiii
i
,...,u,...,uuU
UUUU
=
=

= G(m)U
,...)
ˆ
,...,
ˆ
,
ˆ
(
ˆ
21 i
mmm=m
{
4434421
444344421
dBranch worper outputs
1
dBranch worfor
outputsr Demodulato
sequence received
321

,...),...,,,(
n
nijii
i
i
i
,...,z,...,zzZ
ZZZZ
=

=Z
Channel
Lecture 11 5
State diagram

A finite-state machine only encounters a
finite number of states.

State of a machine: the smallest amount
of information that, together with a
current input to the machine, can predict
the output of the machine.

In a Convolutional encoder, the state is
represented by the content of the
memory.

Hence, there are states.
1
2
−K
Lecture 11 6
State diagram – cont’d

A state diagram is a way to represent
the encoder.

A state diagram contains all the states
and all possible transitions between
them.


Only two transitions initiating from a
state

Only two transitions ending up in a state
Lecture 11 7
State diagram – cont’d
10 01
00
11
outputNext
state
inputCurrent
state
101
010
11
011
100
10
001
110
01
111
000
00
0
S
1
S

2
S
3
S
0
S
2
S
0
S
2
S
1
S
3
S
3
S
1
S
0
S
1
S
2
S
3
S
1/11
1/00

1/01
1/10
0/11
0/00
0/01
0/10
Input
Output
(Branch word)
Lecture 11 8
Trellis – cont’d

Trellis diagram is an extension of the state
diagram that shows the passage of time.

Example of a section of trellis for the rate ½ code
Time
i
t
1+i
t
State
00
0
=S
01
1
=S
10
2

=S
11
3
=S
0/00
1/10
0/11
0/10
0/01
1/11
1/01
1/00
Lecture 11 9
Trellis –cont’d

A trellis diagram for the example code
0/11
0/10
0/01
1/11
1/01
1/00
0/00
0/11
0/10
0/01
1/11
1/01
1/00
0/00

0/11
0/10
0/01
1/11
1/01
1/00
0/00
0/11
0/10
0/01
1/11
1/01
1/00
0/00
0/11
0/10
0/01
1/11
1/01
1/00
0/00
6
t
1
t
2
t
3
t
4

t
5
t
1 0 1 0 0
11 10 00 10 11
Input bits
Output bits
Tail bits
Lecture 11 10
Trellis – cont’d
1/11
0/00
0/10
1/11
1/01
0/00
0/11
0/10
0/01
1/11
1/01
1/00
0/00
0/11
0/10
0/01
0/00
0/11
0/00
6

t
1
t
2
t
3
t
4
t
5
t
1 0 1 0 0
11 10 00 10 11
Input bits
Output bits
Tail bits
Lecture 11 11
Optimum decoding

If the input sequence messages are equally likely, the
optimum decoder which minimizes the probability of
error is the Maximum likelihood decoder.

ML decoder, selects a codeword among all the
possible codewords which maximizes the likelihood
function where is the received
sequence and is one of the possible codewords:
)(
)(m
p

′
U|Z
Z
)(m
′
U
)(max)( if Choose
)(
allover
)()( mmm
pp
(m)
U|ZU|ZU
U
=
′′
¾ML decoding rule:
codewords
to search!!!
L
2