1. Subject Name: Information Theory Coding Subject Code: 10EC55
Prepared By: Shima Ramesh,Pavana
Department: ECE
Date: 10/11/2014
11/20/2018
MVJCE

2. UNIT 8 Convolution codes
11/20/2018
MVJCE

3. Convolution codes Time domain approach Transfer domain approach

4. Convolution codes
Convolution codes can be designed to detect or correct errors.
The encoder is labeled as (n,k,m) =(2,1,2) encoder
At each input bit time one bit is shifted into the left most stage nad the bits that were
Present in the register shifted to right by one position.

5. Encoding procedure.
consider the input sequence u=101
(i) Initially the registers are in reset mode (0,0).
(ii) at first time unit the input bit is 1.the bit enters the first register and pushes out the previous content ‘0’ which will now enter the 2nd register and pushes out the previous content.
(iii) all these bits are passed to xor-gate and out pair (1,1) is obtained.
(iv)the same steps are repeated until time unit 4 and zeros are introduced to clear reg content producing two more output pairs.
(v) hence after (L+m ) time units i.e (3+2)=5 time unit the output sequence will be v=(11,10,00,10,11)
m=memory order
L= length of input sequence.

6. Problem illustration where the input is u=101

7. Convolution encoding time domain approach
(2,1,3) encoder
Here the encoder consists of m=3 stage shift register , n=2 modulo 2 adders
And multiplexer to serial the encoder output.

8. For the above given encoder (2,1,3) two impulse response
The information sequence u=(u1,u2,u3……) enters encoder one bit at a time starting from u1. specifically the encoder output sequence in this is case
Are obtained by discrete convolution of information sequence with encoder
Impulse response. The impulse response are also called as generator sequence.
Impulse response can last upto m+1 time units.
For the above given encoder (2,1,3) two impulse response

9. By inspection the impulse response can be written as
The encoder output are defined as
Hence for the above given encoder the output is given as
And the out sequence is given by

10. Suppose for the above example the input u=(10111) the output sequence is
The output sequence is given by

11. Generation matrix can be written as
If the information sequence u is of finite length L the generation matrix G has L rows and n*(m+L) columns.and output v has length of n*(m+L).
Generation matrix can be written as
The encoding equation in matrix form is given by

12. Consider the example u=(10111)
Performing multiplication v=uG we get v=(11, 01, 00, 01, 01, 00,11)

13. Consider example (3,2,1)
Here as k=2, the encoder consists of 2 m=1 shift register, n=3 three modulo-2-adders, two multiplexers.

14. The input sequences is given by
The generator sequences for the encoder are
The encoding equations can be written as

15. The convolution operation implies
Code word can be written as

16. Consider the example u=(110110) hence
V=u.G

17. u=(110110) Hence the output is given by Code rate is given by R= k/n
Fractional rate loss given by=

18. Encoding of convolutional codes; Transform domain approach
Example (2,1,m): the encoding equations become
Where u(X) is the informational polynomial
V(X) is the encoded polynomial

19. Generator polynomials are written as
After multiplexing the code word become
The indeterminate X can be regarded as a unit-delay operator

20. Consider the example as (2,1,3) The impulse response is
The generator polynomials
The information sequence can be written as
u=10111 the information polynomial is
The two code polynomials are then

21. From the polynomials so obtained can be written as
The code word can be written as

22. OR We may use the multiplexing technique and write
The code polynomial is
The code word can be written as

23. The transfer function matrix (k*n) is given by
Using the transfer function matrix, the encoding equations for (n,k,m )
Code can be expressed
Is the k vector , representing information polynomials
n-vector represents encoded sequences
The code word becomes

24. The encoder we have
For the information sequence

25. Hence the code word is

26. Encoder state at time unit T’
State Diagram
The state of an encoder is defined as shift register contents.
For an (n,k,m) code with K>1 i-th shift register contains Ki previous information bits
Is the encoder memory
Encoder state at time unit T’
The encoder inputs
Are the binary k-tuple of inputs
There are 2^k different possible states
Each new block of k-inputs causes a transition to a new state.
Hence there are 2^k branches leaving each state. One each corresponding
To the input block

27. Consider the example (2,1,3)
State table for the (2,1,3) encoder can be written as

28. The two output sequence

30. Catastrophic code
For catastrophic code “the state diagram consists of a loop of zero weight other than the self loop around the state So.
The characteristic of a catastrophic code that an information
sequence of infinite weight produce a finite code word.
Non-Catastrophic code
It contains no zero weight loops other than the self loop around the all
Zero state So.
All infinite weight information sequence must generate infinite
weight code words.

31. State diagram of (2,1,2) catastrophic code

32. Procedure for writing tree-graph:
If the input is 0 the upper path is followed and if the input is 1 then the lower path is followed.
The vertical line is called node and horizontal line is called branch.
The output code words for each input bit are shown on the branches.
The encoder output for any information sequence can be traced through tree paths.

33. Tree graph for the (2,1,2) convolution encoder

34. To obtain complete code sequence corresponding to information sequence of length kL.
The tree graph is to extended by n(m-l) time units. The extended part unit is called “tail of the tree”.2kL right most nodes are called “terminal nodes”.
Illustration of tail of the tree
The completed sequence is