Principles of Communications pps

1
Principles of Communications
By: Dang Quang Vinh
Faculty of Electronics and Telecommunications
Ho Chi Minh University of Natural Sciences
Convolutional codes
09/2008
2
Introduction

In block coding, the encoder accepts k-bit message block
and generates n-bit codeword⇒Block-by-block basis

Encoder must buffer an entire message block before
generating the codeword

When the message bits come in serially rather than in large
blocks, using buffer is undesirable

Convolutional coding
3
Definitions

An convolutional encoder: a finite-state machine that
consists of an M-stage shift register, n modulo-2 adders

L-bit message sequence produces an output sequence
with n(L+M) bits

Code rate:


L>>M, so
ol)(bits/symb
)( MLn
L
r
+
=
ol)(bits/symb
1
n
r =
4
Definitions

Constraint length (K): the number of shifts over which a
single message bit influence the output

M-stage shift register: needs M+1 shifts for a message to
enter the shift register and come out

K=M+1
5
Example

Convolutional code (2,1,2)

n=2: 2 modulo-2 adders or 2 outputs

k=1: 1 input


M=2: 2 stages of shift register (K=M+1=2+1=3)
Path 1
Path 2
Output
Input
6
Example
Output
Input

Convolutional code (3,2,1)

n=3: 3 modulo-2 adders or 3 outputs

k=2: 2 input

M=1: 1 stages of each shift register (K=2 each)
7
Generations

Convolutional code is nonsystematic code

Each path connecting the output to the input can be
characterized by impulse response or generator
polynomial

denoting the impulse response of
the i
th
path


Generator polynomial of the i
th
path:

D denotes the unit-delay variable⇒different from X of
cyclic codes

A complete convilutional code described by a set of
polynomials { }
),,, ,(
)(
0
)(
1
)(
2
)( iiii
M
gggg
)(
0
)(
1
2)(
2
)()(
)(
iiiMi
M

i
gDgDgDgDg ++++=
)(), ,(),(
)()2()1(
DgDgDg
n
8
Example(1/8)

Consider the case of (2,1,2)

Impulse response of path 1 is (1,1,1)

The corresponding generator polynomial is

Impulse response of path 2 is (1,0,1)

The corresponding generator polynomial is

Message sequence (11001)

Polynomial representation:
1)(
2)1(
++= DDDg
1)(
2)2(
+= DDg
1)(
34

++= DDDm
9
Example(2/8)

Output polynomial of path 1:

Output sequence of path 1 (1001111)

Output polynomial of path 2:

Output sequence of path 2 (1111101)
1
1
)1)(1(
)()()(
236
2345456
234
)1()1(
++++=
++++++++=
++++=
=
DDDD
DDDDDDDD
DDDD
DgDmDc
1
)1)(1(
)()()(

23546
234
)2()2(
+++++=
+++=
=
DDDDD
DDD
DgDmDc
10
Example(3/8)

m= (11001)

c
(1)
=(1001111)

c
(2)
=(1111101)

Encoded sequence c=(11,01,01,11,11,10,11)

Message length L=5bits

Output length n(L+K-1)=14bits

A terminating sequence of K-1=2 zeros is
appended to the last input bit for the shift

register to be restored to its zero initial state
11
Example(4/8)

Another way to calculate the output:

Path 1:
11110000
10110100
10011001
11001001
00110010
01100100
11001001
output111m
c
(1)
=(1001111)
12
Example(5/8)

Path 2
m 101 output
001001 1 1
00100 11 1
0010 011 1
001 001 1 1
00 100 11 1
0 010 011 0
001 0011 1

c
(2)
=(1111101)
13
Example(6/8)

Consider the case of (3,2,1)

denoting the impulse
response of the j
th
path corresponding to i
th

input
Output
Input
),, ,,(
)(
0,
)(
1,
)(
1,
)(
,
)( j
i
j
i

j
Mi
j
Mi
j
i
ggggg
−
=
14
Example(7/8)
Output
Input
DDgg
DDgg
DDgg
Dgg
Dgg
DDgg
=⇒=
+=⇒=
=⇒=
=⇒=
=⇒=
+=⇒=
)()10(
1)()11(
)()10(
1)()01(
1)()01(

1)()11(
)1(
1
)3(
2
)1(
1
)3(
1
)2(
2
)2(
2
)2(
1
)2(
1
)1(
1
)1(
2
)1(
1
)1(
1
15
Example(8/8)

Assume that:


m
(1)
=(101)⇒m
(1)
(D)=D
2
+1

m
(2)
=(011)⇒m
(1)
(D)=D+1

Outputs are:

c
(1)
=m
(1)
*g
1
(1)
+m
(2)
*g
2
(1)
= (D
2

+1)(D+1)+(D+1)(1)
=D
3
+D
2
+D+1+D+1=D
3
+D
2
⇒c
(1)
=(1100)

c
(2)
=m
(1)
*g
1
(2)
+m
(2)
*g
2
(2)
= (D
2
+1)(1)+(D+1)(D)
=D
2

+1+D
2
+D=D+1 ⇒c
(2)
=(0011)

c
(3)
=m
(1)
*g
1
(3)
+m
(2)
*g
2
(3)
= (D
2
+1)(D+1)+(D+1)(D)
=D
3
+D
2
+D+1+D
2
+D=1=D
3
+1

⇒c
(3)
=(1001)

Output c=(101,100,010,011)
16
State diagram

4 possible states

Each node has 2 incoming
branches, 2 outgoing branches

A transition from on state to
another in case of input 0 is
represented by a solid line and
of input 1 is represented by
dashed line

Output is labeled over the
transition line
state Binary description
a 00
b 10
c 01
d 11
00
10
01

11
a
b
d
c
0/00
1/11
0/10
1/01
1/10
0/01
1/00
0/11
Consider convolutional code (2,1,2)
17
Example

Message 11001

Start at state a

Walk through the
state diagram in
accordance with
message sequence
00
10
01
11
a

b
d
c
0/00
1/11
0/10
1/01
1/10
0/01
1/00
0/11
00
a
10 01 00
b
c
a
State
Input 1 0 0
Output
10
b
1
11
01
c
0
01
00
a

0
11 11 10 11
11
d
01
1
18
Trellis(1/2)
a=00
b=10
c=01
d=11
0/00
1
/
1
1
1
/
1
1
0/00
0
/
1
0
1
/
0
1

1
/
1
1
0/00
0
/
1
0
1
/
0
1
0
/
1
1
1
/
0
0
1/10
0
/
0
1
1
/
1
1

0/00
0
/
1
0
1
/
0
1
0
/
1
1
1
/
0
0
1/10
0
/
0
1
1
/
1
1
0/00
0
/
1

0
1
/
0
1
0
/
1
1
1
/
0
0
1/10
0
/
0
1
1
/
1
1
0/00
0
/
1
0
1
/
0

1
0
/
1
1
1
/
0
0
1/10
0
/
0
1
0/00
0
/
1
0
0
/
1
1
0
/
0
1
0/00
0
/

1
1
Level j=0 1 5432 L-1 L L+1 L+2
19
Trellis(1/2)

The trellis contains (L+K) levels

Labeled as j=0,1,…,L,…,L+K-1

The first (K-1) levels correspond to the
encoder’s departure from the initial state a

The last (K-1) levels correspond to the
encoder’s return to state a

For the level j lies in the range K-1≤j≤L, all
the states are reachable
20
Example

Message 11001
a=00
b=10
c=01
d=11
0/00
1
/
1

1
0/00
0
/
1
0
1
/
1
1
0/00
0
/
1
0
1
/
0
1
0
/
1
1
1
/
0
0
1/10
1
/

1
1
0/00
0
/
1
0
1
/
0
1
1
/
0
0
1/10
0
/
0
1
0/00
0
/
1
0
1
/
0
1
0

/
1
1
1
/
0
0
1/10
0
/
0
1
0/00
0
/
1
1
0
/
0
1
0/00
Level j=0 1 5432
Input
Output
1 1 0 0 1 0 0
6 7
1
/
1

1
1
/
0
1
0
/
0
1
0
/
1
1
1
/
1
1
0
/
1
0
0
/
1
1
11 01 01 11 11 10 11
21
Maximum Likelihood Decoding
of Convolutional codes


m denotes a message vector

c denotes the corresponding code vector

r denotes the received vector

With a given r , decoder is required to make estimate
of message vector, equivalently produce an estimate
of the code vector

otherwise, a decoding error
happens

Decoding rule is said to be optimum when the
propability of decoding error is minimized

The maximum likelihood decoder or decision rule is
described as follows:

Choose the estimate for which the log-likelihood
function log
p(r/c)
is maximum
m
ˆ
c
ˆ
ccmm ==
ˆ
ifonly

ˆ
c
ˆ
22
Maximum Likelihood Decoding
of Convolutional codes

Binary symmetric channel: both c and r are binary
sequences of length N

r differs from c in d positions, or d is the Hamming
distance between r and c
∏
=
=
N
i
ii
crpcrp
1
)|()|(
∑
=
=⇒
N
i
ii
crpcrp
1
)|(log)|(log




=
≠
−
=
ii
ii
ii
cr
cr
p
p
crp
if
if

1
)|(with
)1log(
1
log
)1log()(log)|(log
pN
p
p
d
pdNpdcrp
−+









−
=
−−+=⇒
23
Maximum Likelihood Decoding
of Convolutional codes

Decoding rule is restated as follows:

Choose the estimate that minimizes the
Hamming distance between the received vector
r

and the transmitted vector
c

The received vector r is compared with each
possible code vector c, and the one closest
to r is chosen as the correct transmitted
code vector
c
ˆ

24
The Viterbi algorithm

Choose a path in the trellis whose coded sequence differs
from the received sequence in the fewest number of
positions
25
The Viterbi algorithm

The algorithm operates by computing a
metric for every possible path in the trellis

Metric is Hamming distance between coded
sequence represented by that path and
received sequence

For each node, two paths enter the node, the
lower metric is survived. The other is
discarded

Computation is repeated every level j in the
range K-1≤ j≤L

Number of survivors at each level ≤2
K-1
=4