Download EET 465 LAB #2 - Pui Chor Wong

EET 465 LAB #2
Linear Block Code
Introduction
An (n,k) block code is completely defined by M = 2k binary sequences each of fixed
length n.
Each of the sequences of fixed length n is referred to as a code word. k is the number of
information bits.
The Code C thus consists of M code words
C = {c1, c2, c3, c4, …cM}
Practical codes are normally block codes. A block code converts a fixed length of k data
k
bits to a fixed length n codeword, where n > k. The code rate Rc is: Rc  , and the
n
k
redundancy of the code is 1  .
n
Generator and Parity check Matrices
The output of a linear binary block encoder (i.e a codeword ) for an (n,k) linear block
code, is a linear combination of a set of k basis vectors, each of length n, denoted by g1,
g2, …..gk.
The vectors g1, g2, …..gk are not unique.
From linear algebra, the basis vectors, g1, g2, …..gk, can be represented as a matrix G
defined as:
g 
 1
 g 2
G 
 
 
 
 g k 
The k-by-n G matrix is called the generator matrix.
Since the vectors g1, g2, …..gk are not unique, the matrix G is also not unique. The G
matrix has the property that any linear combination of the rows is a code word
In an (n,k) linear block code let the code word corresponding to an input information
sequence Xm be denoted by Cm.
Example
1000110
0100011
  0000000
0000 
0010111


0001101
1000110
0100011
  0001101
0001 
0010111


0001101
1000110
0100011
  0010111
0010 
0010111


0001101
1000110
0100011
  0010111  0001101  0011010
0011 
0010111


0001101
Parity Check Matrix
Take the Parity matrix in the Generator matrix and transpose it, add an addition
Identity matrix:
1011 100 


H  1110 010
0111 001
 110 
 011 


 111 


101 
T

H 
 100 


 010 
 001 




The transpose of the Parity Check Matrix can now be used as it is a 7×3 matrix:
110 
 011


111
0001101 101  101  100  001  000 ,
100 


010
 001


a zero vector indicate the codeword with a valid parity check.
Syndrome
In general we know with successful parity check, cHT should be zero. Since error
is added due to noise to the codeword, the received codeword r=c+e is the
codeword to go through parity check:
rHT=(c+e)HT=cHT+eHT using matrix operations
The results is called the syndrome
Due to the fact that cHT is zero, the syndrome becomes eHT.
Error in each bit position generates a different syndrome:
110 
 011


111
0000001 101  001
100 


010
 001


110 
 011


111
0000010 101  010
100 


010
 001


110 
 011


111
0000100 101  100
100 


010
 001


Purpose of the lab:
1. Given the following generator matrix:
1000110
0100011

G
0010111


0001101
write a matlab function:
function codeword=lbc_gen(message)
% This is my 7,4 linear block code generator function
%
...
based on the generator matrix G given.
test your function with a few message as done in class.
2. Create another function using the parity check matrix H to determine the
syndrome of the received codeword:
function syndrome=syndrome_gen(codeword)
% This is my 7,4 linear block code syndrome generator function
%
...
3. Determine the function which will correct the error in the received codeword
based on the syndrome.
4. Report with screenshots to describe any finding and suggest another
generator matrix G which is different from the one used in this lab.