Convolutional Codes

Basic Definitions

- k 1, n 2 , (2,1) Rate-1/2 convolutional code
- Two-stage register ( M2 )
- Each input bit influences the output for 3

intervals (K3) - K constraint length of the code M 1

Generator Polynomial

- A convolutional code may be defined by a set of n

generating polynomials for each input bit. - For the circuit under consideration
- g1(D) 1 D D2
- g2(D) 1 D2
- The set gi(D) defines the code completely. The

length of the shift register is equal to the

highest-degree generator polynomial.

State Diagram Representation

- The output depends on the current input and the

state of the encoder ( i. e. the contents of the

shift register).

Trellis Diagram Representation

- Expansion of state diagram in time.

Decoding

- A message m is encoded into the code sequence c.
- Each code sequence represents a path in the

trellis diagram. - Minimum Distance Decoding
- Upon receiving the received sequence r, search

for the path that is closest ( in Hamming

distance) to r .

The Viterbi Algorithm

- Walk through the trellis and compute the Hamming

distance between that branch of r and those in

the trellis. - At each level, consider the two paths entering

the same node and are identical from this node

onwards. From these two paths, the one that is

closer to r at this stage will still be so at any

time in the future. This path is retained, and

the other path is discarded. - Proceeding this way, at each stage one path will

be saved for each node. These paths are called

the survivors. The decoded sequence (based on

MDD) is guaranteed to be one of these survivors.

The Viterbi Algorithm (contd)

- Each survivor is associated with a metric of the

accumulated Hamming distance (the Hamming

distance up to this stage). - Carry out this process until the received

sequence is considered completely. Choose the

survivor with the smallest metric.

- 6.3 The Viterbi Algorithm
- The viterbi algorithm is used to decode

convolutional codes and any structure or system

that can be described by a trellis. - It is a maximum likelihood decoding algorithm

that selects the most probable path that

maximizes the likelihood function. - The algorithm is based on add-compare-select the

best path each time at each state.

- Example For the convolutional code example in

the previous lecture, starting from state zero,

Decode the following received sequence.

At the end of the trellis, select the path

with the minimum cumulative Hamming weight

This is the survival path in this example

Decoded sequence is m10 1110

Compute the two possible paths at each state and

select the one with less cumulative Hamming

weight

Add the weight of the path at each state

This is called the survival path

Distance Properties of Conv. Codes

- Def The free distance, dfree, is the minimum

Hamming distance between any two code sequences. - Criteria for good convolutional codes
- Large free distance, dfree.
- Small Hamming distance (i.e. as few differences

as possible ) between the input information

sequences that produce the minimally separated

code sequences. dinf - There is no known constructive way of designing a

conv. code of given distance properties. However,

a given code can be analyzed to find its distance

properties.

Distance Prop. of Conv. Codes (contd)

- Convolutional codes are linear. Therefore, the

Hamming distance between any pair of code

sequences corresponds to the Hamming distance

between the all-zero code sequence and some

nonzero code sequence. Thus for a study of the

distance properties it is possible to focus on

the Hamming distance between the all-zero code

sequence and all nonzero code sequences. - The nonzero sequence of minimum Hamming weight

diverges from the all-zero path at some point and

remerges with the all-zero path at some later

point.

Distance Properties Illustration

- sequence 2 Hamming weight 5, dinf 1
- sequence 3 Hamming weight 7, dinf 3.

Modified State Diagram

- The span of interest to us of a nonzero path

starts from the 00 state and ends when the path

first returns to the 00 state. Split the 00 state

(state a) to two states a0 and a1. - The branches are labeled with the dummy variables

D, L and N, where - The power of D is the Hamming weight ( of 1s)

of the - output corresponding to that branch.
- The power of N is the Hamming weight ( of 1s)

of the - information bit(s) corresponding to that branch.
- The power of L is the length of the branch

(always 1).

Modified State Diagram (contd)

Properties of the Path

- Sequence 2
- code sequence .. 00 11 10 11 00 ..
- state sequence a0 b c a1
- Labeled (D2LN)(DL)(D2L) D5L3N
- Prop. w 5, dinf 1, diverges from the allzero

path by 3 branches. - Sequence 3
- code sequence .. 00 11 01 01 00 10 11 00 ..
- state sequence a0 b d c b c

a1 - Labeled (D2LN)(DLN)(DL)(DL)(LN)(D2L)

D7L6N3 - Prop. w 7, dinf 3, diverges from the allzero

path by 6 branches.

Transfer Function

- Input-Output relations
- a0 1
- b D2LN a0 LNc
- c DLb DLNd
- d DLNb DLNd
- a1 D2Lc
- The transfer function T(D,L,N) a1 /a0

Transfer Function (contd)

- Performing long division
- T D5L3N D6L4N2 D6L5N2 D7L5N3 .
- If interested in the Hamming distance property of

the code only, set N 1 and L 1 to get the

distance transfer function - T (D) D5 2D6 4D7
- There is one code sequence of weight 5. Therefore

dfree5. - There are two code sequences of weight 6,
- four code sequences of weight 7, .