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Convolution Codes
ADVANCED COMMUNICATION SYSTEMS

PRATISHTHA SHIRA RAM
SEMESTER 8
DEPT. OF ECE

Pratishtha Shira Ram

Codes

Codes

Source Channel
Codes Codes

Linear Convolution
Block codes codes


Codes (contd.)

1 Source Codes
• The aim of source coding is to take the
source data and make it smaller.

2 Channel Codes
• The aim of channel coding theory is to
find codes which transmit
quickly, contain many valid code words
and can correct or at least detect many
errors.


Channel Codes
Channel codes are used to add redundancy to the codes.

Redundancy adds extra information about the data into the signal.

This extra information ensures that the data is received correctly at the
information sink.

Hence, channel codes increase the BER of the signal.

The aim of channel coding
The channel code is used to theory is to find codes which
protect data sent over it for transmit quickly, contain
storage or retrieval even in the many valid code words and
presence of noise (errors). can correct or at least detect
many errors.


Convolution Codes
Convolutional codes are characterized by
thee parameters:
(n, k, m)
Where,
n= Number of output bits
k= Number of input bits
m= Number of memory registers

Code Rate = k/n

=Number of input bits
/Number of output bits

Constraint length “L”= k(m-1)

L represents the no. of bits in the encoder
memory that affects the generation of n
output bits

An example: (2,1,4) Coder

n= Number of output bits 2
k= Number of input bits 1
m= Number of memory registers 4

Constraint length L=3

The no. of bits in the
shaded registers are called
states of code, and are
defined by
No. of states = 2L


(2,1,4) Coder

The constraint length = 3.

The shaded registers hold these bits, while
the un-shaded register holds the incoming
bit.

This means, 3 bits, or 8 different
combinations of the bits can be held in the
registers.

These 8 combinations determine what
output will be received as the the 2 output
bits v1 and v2.


Input sequence: 1,Output: 11 11 10 11

t=0; C
Input state
=000 t=2;
A
Input bit=1 Input state =010
Output bit=11 Input bit=0
Output bit=10

t=1;
B
Input state =100
Input bit=0
D
Output bit=11

t=3;
Input state =101
Input bit=0
Output bit=11

The convolution

Providing input of 1 to the coder provides what is called the ”impulse response”
of that coder.

Here, providing 1 as input (and then „flushing‟ it out from the registers using
zeros) gave the output sequence : 11 11 10 11.

Similarly, response for input bit 0 would be : 00 00 00 00.

The output sequence can be computed simply by convolving the input sequence
u with the impulse response g.

Or,
v=u*g

By the principle of linear superposition, a coded sequence can be generated
from the impulse response.


Example

To find the coded sequence for the input 1011, we just have to add the shifted
versions of the individual responses:

Input bit Its Impulse Response

1 11 11 10 11
0 00 00 00 00
1 11 11 10 11
1 11 11 10 11

1011 11 11 01 11 01 01 11

This result can be verified by the encoder model too.

Output bits and the encoder bits through
the(2,1,4) code:


Look–up table for the encoder


Understanding the encoder design

There are three graphical ways in which the encoder can be understood better:
1. State diagram
2. Tree diagram
3. Trellis diagram

State Diagram


Tree Diagram


Trellis Diagram


Trellis Diagram –Encoded Sequence


Deconvolution of codes

There are 2 basic methods of deconvolution

1. Sequential decoding: Fano algorithm
- It allows both forward and backward movement through the Trellis diagram
flow.
-If instead of 11 11 01 11 01 01 11, the code 01 11 01 11 01 01 11 is
received, the algorithm will take a start and tally with the outputs it finds on the
way. If an output does not tally, it retraces its position back to the previous
ambiguous decision.

2. Maximum –Likelihood decoding: Viterbi algorithm
-The viterbi decoder examines the entire received sequence of a given length.
-The decoder then computes a metric for the path and then makes a decision
based on it.
-The path with the higher metric is kept, and the one with the lower one is
discarded.
- Generally the Hamming distance is used as the metric here.

Sequential decoding: Fano algorithm


Maximum –Likelihood decoding:
Viterbi algorithm [Step 1]


Thank you!


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Convolution Codes

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