Convolutional Code(II)

7/30/2019 Convolutional Code(II)

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Chapt er10Convolut ional Codes

Dr. Chih-Peng Li ()


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Table of Cont ent s

10.1 Encoding of Convolutional Codes

10.2 Structural Properties of Convolutional Codes

10.3 Distance Properties of Convolutional Codes


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Convolut ional Codes

Convolutional codes differ from block codes in that the encoder

contains memory and the n encoder outputs at any time unit

depend not only on the kinputs but also on mprevious input

blocks.

An (n, k, m) convolutional code can be implemented with a k-

input, n-output linear sequential circuit with input memory m.

Typically, n and kare small integers with k


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Shortly thereafter, Wozencraft proposed sequential decoding as an

efficient decoding scheme for convolutional codes, and

experimental studies soon began to appear.

In 1963, Massey proposed a less efficient but simpler-to-

implement decoding method called threshold decoding.

Then in 1967, Viterbi proposed a maximum likelihood decodingscheme that was relatively easy to implement for cods with small

memory orders.

This scheme, called Viterbi decoding, together with improved

versions of sequential decoding, led to the application of

convolutional codes to deep-space and satellite communication in

early 1970s.

Convolut ional Codes


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Convolut ional Code

A convolutional code is generated by passing the information

sequence to be transmitted through a linear finite-state shift

register.

In general, the shift register consists ofK(k-bit) stages and n

linear algebraic function generators.


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Convolt uional Code

Convolutional codes

k= number of bits shifted into the encoder at one time

k=1 is usually used!!n = number of encoder output bits corresponding to the k

information bits

Rc = k/n = code rateK= constraint length, encoder memory.

Each encoded bit is a function of the present input bits and their

past ones.

Note that the definition of constraint length here is the same as

that of Shu Lins, while the shift registers representation is

different.


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Encoding of Convolut ional Code

Example 1:

Consider the binary convolutional encoder with constraint

length K=3, k=1, and n=3.

The generators are: g1=[100], g2=[101], and g3=[111].

The generators are more conveniently given in octal form

as (4,5,7).


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Encoding of Convolut ional Code

Example 2:

Consider a rate 2/3 convolutional encoder.

The generators are: g1=[1011], g2=[1101], and g3=[1010].In octal form, these generator are (13, 15, 12).


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Repr esent at ions of Convolut ional Code

There are three alternative methods that are often used

to describe a convolutional code:

Tree diagram

Trellis diagram

State disgram


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Tree diagram

Note that the tree diagram in theright repeats itself after the third

stage.This is consistent with the fact thatthe constraint length K=3.

The output sequence at each stage isdetermined by the input bit and thetwo previous input bits.

In other words, we may sat that the3-bit output sequence for each inputbit is determined by the input bit andthe four possible states of the shift

register, denoted as a=00, b=01,c=10, and d=11.

Tree diagram for rate 1/3,K=3 convolutional code.


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Trellis diagram

Tree diagram for rate 1/3, K=3 convolutional code.


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State diagram

0 1

0 1

0 1

0 1

a a a c

b a b c

c b c d

d b d d

State diagram for rate 1/3,

K=3 convolutional code.


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Example: K=2, k=2, n=3 convolutional code

Tree diagram


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Trellis diagram


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State diagram


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In general, we state that a rate k/n, constraint length K,

convolutional code is characterized by 2kbranches emanating

from each node of the tree diagram.

The trellis and the state diagrams each have 2k(K-1)possible states.

There are 2kbranches entering each state and 2kbranches leaving

each state.


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the encoder consists of an m= 3-stage shift register together withn=2 modulo-2 adders and a multiplexer for serializing theencoder outputs.

The mod-2 adders can be implemented as EXCLUSIVE-OR gates.Since mod-2 addition is a linear operation, the encoder is alinear feedforward shift register.

All convolutional encoders can be implemented using a linearfeedforward shift register of this type.

Example: A (2, 1, 3) binary convolutional codes:

10.1 Encoding of Convolut ional Codes


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The information sequence u =(u0, u1, u2, ) enters the encoder

one bit at a time.

Since the encoder is a linear system, the two encoderoutputsequence can

be obtained as the convolution of the input sequence u with the

two encoder impulse response.

The impulse responses are obtained by letting u =(1 0 0 ) and

observing the two output sequence.

Since the encoder has an m-time unit memory, the impulseresponses can last at most m+1 time units, and are written as :

),,,(and),,,( )()()()()()()()( 222

1

2

0

21

2

1

1

1

0

1 vv ==

),,,(

),,,()2()2(

1

)2(

0

)2(

)1()1(

1

)1(

0

)1(

m

m

ggg

ggg

=

=

g

g


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The encoder of thebinary (2, 1, 3) code is

The impulse response g(1) and g(2) are called the generator

sequences of the code.

The encoding equations can now be written as

where * denotes discrete convolution and all operations are mod-2.

The convolution operation implies that for all l 0,

where

)1111(

)1101()2(

)1(

=

=

g

g

)2()2(

)1()1(

guv

guv

==

( ) ( ) ( ) ( ) ( )

0 1 1

0

, 1,2,.m

j j j j j

l l i i l l l m m

i

u g u g u g u g j =

= = + + + =

0 for all .l iu l i = 1, there are n generator polynomials

for each of the kinputs.

Each set ofn generators represents the connections from one of

the shift registers to the n outputs.

The length Ki of the ith shift register is given by

where is the generator polynomial relating the ith input

to thejth output, and the encoder memory orderm is

( )( ) ,1,degmax1 kiDK jinji = g

( )( )Djig

( ) .degmaxmax

1

11

j

i

ki

njki

iKm g

==


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Since the encoder is a linear system, and u(i)(D) is the ith input

sequence and v(j)(D)is thejth output sequence, the generator

polynomial can be interpreted as the encoder transfer

function relating input i to outputj.

As with k-input, n-output linear system, there are a total ofkn

transfer functions.

These can be represented by the k n transfer function matrix

( )( )Djig

( )

( ) ( ) ( ) ( ) ( ) ( )( )

( )( )

( )( )

( )

( ) ( ) ( ) ( ) ( ) ( )

1 2

1 1 1

1 2

2 2 2

1 2

D

n

n

n

k k k

D D D

D D D

D D D

=

g g g

g g gG

g g g


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Using the transfer function matrix, the encoding equation for an

(n, k, m) code can be expressed as

where is the k-tuple of

input sequences and is

the n-tuple of output sequences.

After multiplexing, the code word becomes

( ) ( ) ( )DDD GUV =( ) (1) (2) ( )( ), ( ), , ( )kD D D D U u u u

( ) (1) (2) ( )( ), ( ), , ( )nD D D D

V v v v

( )( )

)( )

)( )

).121 nnnnn

DDDDDD vvvv

+++=

10 1 E di f C l i l C d


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Example 10.7

For the previous (3, 2, 1) convolutional code

For the input sequences u(1)(D) = 1+D2 and u(2)(D)=1+D, the

encoding equations are

and the code word is

( )

++

= 11

11

D

DDD

DG

( ) ( )( ) ( )( ) ( )( )[ ] [ ]

[ ]3233

2321

,D1,D1

11

111,1,,

DD

D

DDDDDDDDD

+++=

++++== vvvV

( ) ( ) ( ) ( )9 9 6 9 2

8 9 10 11

D 1 1

1

D D D D D D

D D D D D

= + + + + +

= + + + + +

v

1 0 1 1 1 1

0 1 1 1 0 0

10 1 E di f C l t i l C d


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Then, we can find a means of representing the code word v(D)

directly in terms of the input sequences.

A little algebraic manipulation yields

where

is a composite generator polynomial relating the ith inputsequence to v(D).

( ) ( ) ( ) ( )DD ink

i

i guv =

=1

D

( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 11 , 1 ,nn n n ni i i ig D D D D D D i k + + g g g

10 1 E di f C l t i l C d


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Example 10.8

For the previous (2, 1, 3) convolutional codes, the composite

generator polynomial is

and foru(D)=1+D2

+D3

+D4

, the code word is

again agreeing with previous calculations.

( ) ( ) ( ) 765432221 1 DDDDDDDDDD ++++++=+= ggg

( ) ( ) ( )

( ) ( )

2

4 6 8 3 4 5 6 7

3 7 9 11 14 15

1 1

1

D D D

D D D D D D D D D

D D D D D D D

=

= + + + + + + + + +

= + + + + + + +

v u g

10.2 St r uct ur al Pr oper t ies of Convolut ional


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Codes

Since a convolutional encoder is a sequential circuit, its

operation can be described by a state diagram.

The state of the encoder is defined as its shift register contents.

For an (n, k, m) code with k> 1, the ith shift register contains Kiprevious information bits.

Defined as the total encoder memory, the encoder

state at time unit l is the binary K-tuple of inputs

and there are a total 2Kdifferent possible states.

1

k

ii

K K=

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )kKl

k

l

k

lKlllKlll kuuuuuuuuu 21

22

2

2

1

11

2

1

1 21

10.2 St r uct ur al Pr oper t ies of Convolut ionalC d


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Codes

For a (n, 1, m) code, K= K1 = m and the encoder state at time

unit l is simply .

Each new block ofkinputs causes a transition to a new state.

There are 2kbranches leaving each state, one corresponding to

each different input block. Note that for an (n, 1, m) code, there

are only two branches leaving each state.

Each branch is labeled with the kinputs causing the transition

and n corresponding outputs .

The states are labeledS

0,S

1,,S

2K-1, where by conventionS

irepresents the state whose binary K-tuple representation

b0,b1,,bK-1 is equivalent to the integer

( )mlll uuu 21

)()2()1( k

lll uuu ))()2()1( nlll

0 1 1

0 1 1

2 2 2KK

i b b b

= + + +



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Assuming that the encoder is initially in state S0 (all-zero state),

the code word corresponding to any given information

sequence can be obtained by following the path through the

state diagram and noting the corresponding outputs on thebranch labels.

Following the last nonzero information block, the encoder is

return to state S0by a sequence ofm all-zero blocks appendedto the information sequence.

Codes



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Encoder state diagram of a (2, 1, 3) code

Ifu = (1 1 1 0 1), the code word

v = (1 1, 1 0, 0 1, 0 1, 1 1, 1 0, 1 1, 1 1)

Codes



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Encoder state diagram of a (3, 2, 1) code

Codes

10.2 St r uct ur al Pr oper t ies of Convolut ionalCodes


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The state diagram can be modified to provide a complete

description of theHamming weights of all nonzero code words

(i.e. a weight distribution function for the code).

State S0 is split into an initial state and afinal state, the self-

loop around state S0 is deleted, and each branch is labeled with

a branch gainXi,where i is the weight of the n encoded bits on

that branch.Eachpath connecting the initial state to the final state

represents a nonzero code word that diverge from and remerge

with state S0 exactly once.Thepath gain is the product of the branch gains along a path,

and the weight of the associated code word is the power ofXin

the path gain.

Codes



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Codes

Modified encoder state diagram of a (2, 1, 3) code.

The path representing the sate sequence S0S1S3S7S6S5S2S4S0has path gain X2X1X1X1X2X1X2X2=X12.



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Codes

Modified encoder state diagram of a (3, 2, 1) code.

The path representing the sate sequence S0S1S3S2S0has path gain X2X1X0X1 =X12.



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Codes

The weight distribution function of a code can be determined

by considering the modified state diagram as a signal flow

graph and applyingMasons gain formula to compute its

generating function

whereAi is the number of code words of weight i.

In a signal flow graph, a path connecting the initial state to the

final state which does not go through any state twice is called a

forward path.

A closed path starting at any state and returning to that state

without going through any other state twice is called a loop.

( ) ,ii

iXAXT =



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Codes

Let Cibe the gain of the ith loop.

A set of loops is nontouching if no state belongs to more than one

loop in the set.

Let {i} be the set of all loops, {i, j} be the set of all pairs of

nontouching loops, {i,j, l} be the set of all triples of

nontouching loops, and so on.

Then define

where is the sum of the loop gains, is the product of

the loop gains of two nontouching loops summed over all pairs of

nontouching loops, is the product of the loop gains of

three nontouching loops summed over all nontouching loops.

,1 ''''''''''

'''

''

'

,,,

++= ljlji ijji iii CCCCCC

i

iC '''

'

,

j

ji

iCC

''''

''''''

''

,,lj

ljii

CCC



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Codes

And i is defined exactly like , but only for that portion of the

graph not touching the ith forward path; that is, all states along

the ith forward path, together with all branches connected to

these states, are removed from the graph when computing i.

Masons formula for computing the generating function T(X) of

a graph can now be states as

where the sum in the numerator is over all forward paths and Fiis the gain of the ith forward path.

( ) ,i i

i

F

T X

=



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Codes

( )( )

( )( )( )

( )( )( )

( )( )

( )XCSSLoop

XCSSSSLoop

XCSSSSSLoop

XCSSSLoop

XCSSSSLoop

XCSSSSSSSLoop

XCSSSSSSSSLoop

XCSSSSSLoopXCSSSSSSSLoop

XCSSSSSSLoop

XCSSSSSSSSLoop

=

=

=

=

=

=

=

=

=

=

=

1177

4

103563

5

935673

8252

3

71421

861463521

9

514673521

2

414631

7

31425631

3

2146731

8

114256731

:11

:10

:9

:8

:7

:6

:5

:4:3

:2

:1Example (2,1,3) Code:

There are 11 loops in the

modified encoder state

diagram.



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Codes

Example (2,1,3) Code: (cont.)There are 10 pairs of nontouching loops :

( )

( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( )( ) ( )

( ) ( )4

118

4

117

4

107

7

82

8

97

9

116

3

114

3

84

8

113

4

82

11loop8loop:10pairLoop


8loop2loop:8pairLoop10loop7loop:7pairLoop







XCC,

XCC,

XCC,XCC,

XCC,

XCC,

XCC,

XCC,

XCC,

XCC,

=

=

=

=

=

=

=

=

=

=



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Codes

Example (2,1,3) Code : (cont.)

There are two triples of nontouching loops :

There are no other sets of nontouching loops. Therefore,

( )( ) ( )811107

4

1184

11loop10,loop7,loop:2tripleLoop11loop8,loop4,loop:1tripleLoop

XCCCXCCC

==

( )

( )( ) 3845247843384

453342738

21

1

XXXXXXXXXXXXXX

XXXXXXXXXXX

+=+++++++++++

++++++++++=



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Codes


There are seven forward paths in this state diagram :

( )( )

( )( )

( )( )( ).:7pathFoward

:6pathFoward:5pathFoward

:4pathFoward

:3pathFoward

:2pathFoward

:1pathFoward

7

704210

7

604635210

8

5046735210

6

4046310

11304256310

7

20467310

12

1042567310

XFSSSSS

XFSSSSSSSSXFSSSSSSSSS

XFSSSSSS

XFSSSSSSSS

XFSSSSSSS

XFSSSSSSSSS

=

==

==

=

=



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Codes


Forward paths 1 and 5 touch all states in the graph, and

hence the sub graph not touching these paths contains no

states. Therefore,

1 = 5 = 1.

The subgraph not touching forward paths 3 and 6:

3 = 6 = 1 -X



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Codes


The subgraph not touching

forward path 2:

2 = 1 -X


forward path 4:

4 = 1 (X + X) + (X2)

= 1 2X+X2



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forward path 7:

7 = 1 (X + X4 + X5) + (X5)

= 1 XX4



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The generating function for this graph is then given by

T(X) provides a complete description of the weight

distribution of all nonzero code words that diverge from andremerge with state S0 exactly once.

In this case, there is one such code word of weight 6, three of

weight 7, five of weight 8, and so on.

( )( ) ( )

( ) ( ) ( )

+++++=

+=

+++++++

=

1098763

876

3

477826

11712

25115321

21

11121111

XXXXXXXXXX

XX

XXXXXXXXX

XXXXX

XT



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There are eight loops, six pairs of nontouching loops, and one

triple of nontouching loops in the graph of previous modified

encoder state diagrams : (3, 2, 1) code, and

( )

( ) ( ).221

1

5432

6543426

3212342

XXXXX

XXXXXXX

XXXXXXXX

++=

++++++

+++++++=



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( ( (

( ) ( ) ( )( ) ( )

( ).2

111

111

11111

1111

876543

636

3743248

5626234

356245325

XXXXXXXXXX

XXXXXXXX

XXXXXXXXX

XXXXXXXXXXF

i

ii

+++=+++

++++

+++++

++++=

Example (3,2,1) Code : (cont.)There are 15 forward path in this graph, and

Hence, the generating function is

This code contains two nonzero code word of weight 3, five

of weight 4, fifteen of weight 5, and so on.

( ) +++=++

+++=543

5432

876543

1552221

2 XXXXXXXXXXXXXXXT



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Additional information about the structure of a code can beobtained using the same procedure.

If the modified state diagram is augmented by labeling each

branch corresponding to a nonzero information block with Yj

,wherej is the weight of the kinformation bits on the branch, and

labeling every branch with Z, the generating function is given

by

The coefficient Ai,j,l denotes the number of code words withweight i, whose associated information sequence has weightj,

and whose length is lbranches.

( ) .,,,,

,,

l

lji

ji

lji ZYXAZYXT =



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The augment state diagram for the (2, 1, 3) codes.



69/115

Example (2,1,3) Code:For the graph of the augment state diagram for the (2, 1, 3)

codes, we have:

( )( ) ( ) ( )( ) ( ) 749746384324

633334222

748744

435322424637748

744533633748744

324435233633

744422637533748

1

)

(

)

(1

ZYXZYZYXZZYX

ZYYZXZZYXZZXY

ZYXZYX

ZYXZYXZYXZYXZYX

ZYXZYXZYXZYXZYX

XYZZYXZYXXYZYZXZYX

ZYXZYXZYXZYXZYX

++=

+

++++++++++

++++++

++++=



70/115

Example (2,1,3) Code: (cont.)

Hence the generating function is

( )

( )( ) ( )

52847526

32447737

8483222526

731126378412

11

1)1(

111

ZYXYZXZYX

ZYXXYZYZXXYZZYX

ZYXZYXZZXYZYX

XYZZYXXYZZYXZYXFi

ii

+=

++++++

++=

( )

( )

( )+++++

+++=

+=

948474628

736347526

52847526

2

,,

ZYZYZYZYX

ZYZYYZXZYX

ZYXYZXZYXZYXT



71/115

Example (2,1,3) Code: (cont.)

This implies that the code word of weight 6 has length 5

branches and an information sequence of weight 2, one code

word of weight 7 has length 4 branches and information

sequence weight 1, another has length 6 branches and

information sequence weight 3, the third has length 7 branches

and information sequence weight 3, and so on.

The Tr ansf er Funct ion of a Convolut ional

Code


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The state diagram can be usedto obtain the distance property

of a convolutional code.

Without loss of generality, weassume that the all-zero code

sequence is the input to the

encoder.


Code


73/115

First, we label the branches of the state diagram as eitherD0=1,D1,D2, orD3, where the exponent ofD denotes the Hamming

distance between the sequence of output bits corresponding to

each branch and the sequence of output bits corresponding to theall-zero branch.

The self-loop at node a can be eliminated, since it contributes

nothing to the distance properties of a code sequence relative tothe all-zero code sequence.

Furthermore, node a is split into two nodes, one of which

represents the input and the other the output of the state diagram.


Code


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Use the modified state diagram, we can obtain four stateequations:

The transfer function for the code is defined as T(D)=Xe/Xa. Bysolving the state equations, we obtain:

3

2 2

2

c a b

b c d

d c d

e b

X D X DX

X DX DX

X D X D X

X D X

= +

= +

= +

=

( )6

6 8 10 12

2

6

2 4 8

1 2

d

d

d

DT D D D D D a D

D

=

= = + + + + =

( )( )

( )

=

dodd0

deven2 26d

da


Code


75/115

The transfer function can be used to provide more detailedinformation than just the distance of the various paths.

Suppose we introduce a factorNinto all branch transitions

caused by the input bit 1.Furthermore, we introduce a factor ofJinto each branch of the

state diagram so that the exponent ofJwill serve as a counting

variable to indicate the number of branches in any given pathfrom node a to node e.


Code


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The state equations for the state diagram are:

Upon solving these equations for the ratioXe/Xa, we obtain thetransfer function:

3

2 2

2

c a b

b c d

d c d

e b

X JND X JNDX

X JDX JDX

X JND X JND X

X JD X

= +

= +

= +

=

( )( )

3 6

2

3 6 4 2 8 5 2 8 5 3 10

6 3 10 7 3 10

, ,

1 1

2

J NDT D N J

JND J

J ND J N D J N D J N D

J N D J N D

=

+= + + +

+ + +


Code


77/115

The exponent of the factorJindicates the length of the path thatmerges with the all-zero path for the first time.

The exponent of the factorNindicates the number of 1s in the

information sequence for that path.

The exponent ofD indicates the distance of the sequence of

encoded bits for that path from the all-zero sequence.

Reference:

John G. Proakis, Digital Communications, Fourth Edition, pp.

477 482, McGraw-Hill, 2001.



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An important subclass of convolutional codes is the class of

systematic codes.

In a systematic code, the first koutput sequences are exact

replicas of the kinput sequences, i.e.,

v(i) = u(i), i = 1, 2, , k,

and the generator sequences satisfy( )

,...k,,, iij

ijj

i 21if0

if1=

==g



79/115

The generator matrix is given by

where I is the k kidentity matrix, 0 is the k kall-zero matrix,

and Pl is the k (nk) matrix

=

mmm

mm

m

P0P0P0PI

P0P0P0PI

P0P0P0PI

G 120

110

210

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

,

,

2

,

1

,

,2

2

,2

1

,2

,1

2

,1

1

,1

=

++

++

++

n

lk

k

lk

k

lk

n

l

k

l

k

l

n

l

k

l

k

l

l

ggg

gggggg

P



80/115

And the transfer matrixbecomes

Since the first koutput sequences equal the input sequences, they

are called information sequences and the last nkoutputsequences are calledparity sequences.

Note that whereas in general kn generator sequences must be

specified to define an (n, k, m) convolutional code, only k(n k)sequences must be specified to define a systematic code.

Systematic codes represent a subclass of the set of all possiblecodes.

( )

( )( ) ( )( )( )( ) ( )( )

( )( ) ( )( )

=

+

+

+

DD

DD

DD

D

n

k

k

k

nk

nk

gg

gg

gg

G

1

2

1

2

1

1

1

100

010

001



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Example (2,1,3) Code:Consider the (2, 1, 3) systematic code. The generator

sequences are g(1)=(1 0 0 0) and g(2)=(1 1 0 1). The generator

matrix is

=

10001011

1000101110001011

G

The (2, 1, 3)

systematic code.



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Example (2,1,3) Code:

The transfer function matrix is

G(D) = [1 1 +D +D3].For an input sequence u(D) = 1 + D2 + D3, the information

sequence is

and the parity sequence is

( ) ( ) ( ) ( )( ) 323211 111 DDDDDDD ++=++== guv

( ) ( ) ( ) ( ) ( ) ( )( )65432

33222

1

11

DDDDDD

DDDDDDD

++++++=

++++== guv

g(1)=(1 0 0 0) and g(2)=(1 1 0 1).



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One advantage of systematic codes is that encoding is somewhatsimpler than for nonsystematic codes because less hardware is

required.

For an (n, k, m) systematic code with k > n k, there exits amodified encoding circuit which normally requires fewer than K

shift register states.

The (2, 1, 3) systematiccode requires only one

modulo-2 adder with

three inputs.



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Example (3,2,2) Systematic Code:Consider the (3, 2, 2) systematic code with transfer function

matrix

The straightforward realization of the encoder requires a total

of K= K1

+ K2

= 4 shift registers.

( )

+ ++= 2

2

110101

DDDDG



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Example (3,2,2) Systematic Code:Since the information sequences are given by v(1)(D) = u(1)(D)

and v(2)(D) = u(2)(D), and the parity sequence is given by

( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ,3223

1

13 DDDDD guguv +=

The (3, 2, 2) systematic encoder, and it requires

only two stages of encoder memory rather than 4.



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A complete discussion of the minimal encoder memory requiredto realize a convolutional code is given by Forney.

In most cases the straightforward realization requiring Kstates of

shift register memory is most efficient.In the case of an (n,k,m) systematic code with k>n-k, a simpler

realization usually exists.

Another advantage of systematic codes is that no inverting circuitis needed for recovering the information sequence from the code

word.

i d h h h d i i



87/115

Nonsystematic codes, on the other hand, require an inverter to

recover the information sequence; that is, an n kmatrix G-1(D)

must exit such that

G(D)G-1(D) = IDl

for some l 0, where I is the kkidentity matrix.

Since V(D) = U(D)G(D), we can obtain

V(D)G-1(D) = U(D)G(D)G-1(D) = U(D)Dl ,

and the information sequence can be recovered with an l-time-

unit delay from the code word by letting V(D) be the input to

the n-input, k-output linear sequential circuit whose transfer

function matrix is G-1(D).


F ( 1 ) d f f i i G(D) h


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For an (n, 1, m) code, a transfer function matrix G(D) has afeedforward inverse G-1(D) of delay l if and only if

GCD[g(1)(D), g(2)(D),, g(n)(D)] =Dl

for some l 0, where GCD denotes the greatest common divisor.For an (n, k, m) code with k> 1,

be the determinants of the distinct k ksubmatrices of the

transfer function matrix G(D).

A feedforward inverse of delay l exits if and only if

for some l 0.

let ( ), 1, 2, , ,in

D ik

=

kn

li DkniD = =,,2,1:)(GCD

E l (2 1 3) C d



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Example (2,1,3) Code:For the (2, 1, 3) code,

and the transfer function matrix

provides the required inverse of delay 0 [i.e., G(D)G1(D) = 1].

The implementation of the inverse is shown below

2 3 2 3 0GCD 1 , 1 1D D D D D D + + + + + = =

( )2

1

2

1 D DD

D D

+ += +

G

10.2 St r uct ur al Pr oper t ies of Convolut ional

Codes

E l (3 2 1) C d


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Example (3,2,1) Code:For the (3, 2, 1) code , the 2 2 submatrices ofG(D) yield

determinants 1 +D +D2, 1 +D2, and 1. Since

there exists a feedforward inverse of delay 0.

The required transfer function matrix is

given by:

2 21 , 1 , 1 1GCD D D D + + + =

( )10 01 1

1

D D

D

= +

G


T d t d h t h h f df d i d t


91/115

To understand what happens when a feedforward inverse does notexist, it is best to consider an example.

For the (2, 1, 2) code with g(1)(D) = 1 +D and g(2)(D) = 1 +D2 ,

and a feedforward inverse does not exist.

If the information sequence is u(D) = 1/(1 +D) = 1 +D +D2

+ , the output sequences are v(1)

(D) = 1 and v(2)

(D) = 1 +D;that is, the code word contains only three nonzero bits eventhough the information sequence has infinite weight.

If this code word is transmitted over a BSC, and the three

nonzero bits are changed to zeros by the channel noise, thereceived sequence will be all zeros.

2

GCD 1 , 1 1 ,D D D + + = +


A MLD will then produce the all zero code word as its


92/115

A MLD will then produce the all-zero code word as itsestimated, since this is a valid code word and it agrees

exactly with the received sequence.

The estimated information sequence will beimplying an infinite number of decoding errors caused by a

finite number (only three in this case) of channel errors.

Clearly, this is a very undesirable circumstance, and thecode is said to be subject to catastrophic error propagation,

and is called a catastrophic code.

and

for a code to be noncatastrophic.

( ) 0,D =u

( )

( )( )

( )( )

( )1 2

GCD , , ,

n l

D D D D

= g g g ( )GCD i D( ): 1,2,...., are necessary and sufficient conditionslni Dk = =


Any code for which a feedforward inverse exists is


93/115

Any code for which a feedforward inverse exists isnoncatastrophic.

Another advantage of systematic codes is that they are always

noncatastrophic.A code is catastrophic if and only

if the state diagram contains a loop

of zero weight other than theself-loop around the state S0.

Note that the self-loop around

the state S3 has zero weight.

State diagram of a (2, 1, 2)

catastrophic code.


In choosing nonsystematic codes for use in a


94/115

In choosing nonsystematic codes for use in acommunication system, it is important to avoid the

selection of catastrophic codes.

Only a fraction 1/(2n 1) of (n, 1, m) nonsystematic

codes are catastrophic.

A similar result for (n, k, m) codes with k> 1 is stilllacking.

The performance of a convolutional code depends on the

10.3 Dist ance Pr oper t ies of Convolut ional

Codes


95/115

The performance of a convolutional code depends on thedecoding algorithm employed and the distance properties of the

code.

The most important distance measure for convolutional codes isthe minimum free distance dfree, defined as

where v and v are the code words corresponding to the

information sequences u and u, respectively.

In equation above, it is assumed that ifu and u are ofdifferent lengths, zeros are added to the shorter sequence so

that their corresponding code words have equal lengths.

( )freemin , : ,d d v v u u

dfree is the minimum distance between any two code words in the


Codes


96/115

dfree is the minimum distance between any two code words in thecode. Since a convolutional code is a linear code,

where v is the code word corresponding to the information

sequence u.

dfree is the minimum-weight code word of any length produced by

a nonzero information sequence.

( ){ }( ){ }

( ){ }

free

min :

min : 0

min : 0 ,

d w

w

w

= +

=

=

v v u u

v u

uG u

Also it is the minimum weight of all paths in the state diagram


Codes


97/115

Also, it is the minimum weight of all paths in the state diagramthat diverge from and remerge with the all-zero state S0, and it

is the lowest power ofXin the code-generating function T(X).

For example, dfree = 6 for the (2, 1, 3) code of example 10.10(a),and dfree = 3 for the (3, 2, 1) code of Example 10.10(b).

Another important distance measure for convolutional codes is

the column distance function (CDF).Letting

denote the ith truncation of the code word v, and

denote the ith truncation of the code word u.

[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )1 2 1 2 1 20 0 0 1 1 1, , ,n n ni i ii v v v v v v v v v=v

[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )1 2 1 2 1 20 0 0 1 1 1, , ,k k ki i ii u u u u u u u u u=u

The column distance function of order i di is defined as


Codes


98/115

The column distance function of orderi, di, is defined as

where v is the code word corresponding to the information

sequence u.

di is the minimum-weight code word over the first (i + 1) time

units whose initial information block is nonzero.

In terms of the generator matrix of the code,

[ ] [ ]( ) [ ] [ ]{ }

[ ] [ ]{ }

0 0

0

min , : ,

min : 0

i i i

i

d d

w

=

v v u u

v u

[ ] [ ] [ ]i i i

=v u G

[G]i is a k(i + 1) n(i + 1) submatrix of G with the form


Codes


99/115

[G]i is a k(i + 1) n(i + 1) submatrix ofG with the form

or

[ ]0 1

0 1

0

, ,i

i

i

i m =

G G GG G

G

G

[ ]

0 1 1

0 2 1

1

1

10 1

0 1 10

1

0 1

0

, .

m m

m m m

m

m

mmi

m

i m

= >

G G G G

G G G GGG G

G GG GG

G G GG

GG G

G

Then[ ] [ ]( ) [ ]{ }


Codes


100/115

Then

is seen to depend only on the first n(i + 1) columns ofG and this

accounts for the name column distance function.The definition implies that di cannot decrease with increasing i

(i.e., it is a monotonically nondecreasing function ofi).

The complete CDF of the (2, 1, 16) code with

is shown in the figure of the following page.

[ ] [ ]( ) [ ]{ }0min : 0i i id w= u G u

( ) 2 5 6 8 13 16

3 4 7 9 10 11 12 14 15 16

1 ,

1

D D D D D D D D

D D D D D D D D D D

= + + + + + + +

+ + + + + + + + + +

G

Column distance function of a (2,1,16) code


Codes


101/115

Co u d sta ce u ct o o a ( , , 6) code

Two cases are of specific interest: i = m and i.


Codes


102/115

pFori = m, dm is called the minimum distance of a convolutional

code and will also be denoted dmin.

From the definition di = min{w([u]i[G]i):[u]00}, we see thatdmin represents the minimum-weight code word over the first

constraint length whose initial information block is nonzero.

For the (2,1,16) convolutional code, dmin = d16 = 8.Fori, limidi is the minimum-weight code word of any

length whose first information block is nonzero.

Comparing the definitions of limidi and dfree, it can be shownthat for noncatastrophic codes

freelim ii

d d

=

di eventually reaches dfree and then increases no more. Thisll h ithi th t f t i t l th (i


Codes


103/115

yusually happens within three to four constraint lengths (i.e.,

when i reaches 3m or 4m).

Above equation is not necessarily true for catastrophic codes.Take as an example the (2,1,2) catastrophic code whose

state diagram is shown in Figure10.11.

For this code, d0 = 2 and d1 = d2 = = limidi =3, sincethe truncated information sequence [u]i =(1, 1, 1, , 1)

always produces the truncated code word [v]i =(1 1, 0 1, 0 0,

0 0,, 0 0), even in the limit as i.

Note that all paths in the state diagram that diverge from and

remerge with the all-zero state S0 have weight at least 4, and

dfree = 4.

Hence, we have a situation in which limidi = 3 dfree =


Codes


104/115

, i i free4.

It is characteristic of catastrophic codes that an infinite-weight

information sequence produces a finite-weight code word.

In some cases, as in the example above, this code word can

have weight less than the free distance of the code. This is due

to the zero weight loop in the state diagram.

In other words, an information sequence that cycles around this

zero weight loop forever will itself pick up infinite weight

without adding to the weight of the code word.

In a noncatastrophic code, which contains no zero weight loopother than the self loop around the all zero state S all infinite


Codes


105/115

p g pother than the self-loop around the all-zero state S0, all infinite-

weight information sequences must generate infinite-weight

code words, and the minimum-weight code word always has

finite length.

Unfortunately, the information sequence that produces the

minimum-weight code word may be quite long in some cases,

thereby causing the calculation ofdfree to be a rather formidable

task.

The best achievable dfree for a convolutional code with a given

rate and encoder memory has not been determined exactly.

Upper and lower bound on dfree for the best code have been

obtained using a random coding approach.

A comparison of the bounds for nonsystematic codes with theb d f t ti d i li th t f di t i


Codes


106/115

bounds for systematic codes implies that more free distance is

available with nonsystematic codes of a given rate and encoder

memory than with systematic codes.This observation is verified by the code construction results

presented in succeeding chapters, and has important

consequences when a code with large dfree must be selected foruse with either Viterbi or sequential decoding.

Heller (1968) derived the upper bound on the minimum free

distance of a rate 1/n convolutional code:

( )1

free1

2min 1

2 1

l

lld K l n

+

Maximum Free Dist ance Codes

Rate


107/115

Rate


Rate 1/3


108/115

Rate 1/3


R t 1/4


109/115

Rate 1/4



110/115

Rate 1/5


Rate 1/6


111/115


Rate 1/7


112/115


Rate 1/8


113/115


Rate 2/3


114/115

Rate k/5


Rate 3/4 and 3/8


115/115

Rate k/7

Documents

Convolutional Code(II)