Channel-coding-BAN-Khalid. Communication

Channel coding.
Presentation by master student:
Ban Khalid Ammar.
Najaf technical college.

Channel coding.
Seminar topics:
▪ Block code
Parity check , Binary code space.
▪ Linear block code
Systematic linear , Hamming single error correction.
▪ Cyclic code
Systematic cyclic cod , error correction .

Channel coding:
communication theory
▪ Probability of repeat a symbol in a message is P(x).
▪ The length of code to represent this symbol is:
I=log2(1/p(x))
▪ Source Entropy is the average of information in
message H(s):

Binary channel: symmetric and
non- symmetric.
BSC:
If condition probabilities
Are symmetric:
P(0|1)=P(1|0)=P
P(1|1)=P(0|0)=1-P
BNC:
If condition probabilities
Are non- symmetric:
P(0|1)≠P(1|0)
P(1|1)≠P(0|0)

Source
coding
Channel
coding
[D/C]
error
RX
source
Stream of pulses.
(probability)
message Redundant
bits for error
detection.
Dependent
on source
coding
channel

Channel coding.
1. Block code:
Block Code meaning that segmentation the data into
segments called block, and each block consist of
(k bits) encoded into a new block (n bits) called a
Code word, if the redundant bits added to beginning
Of code word called systematic code, n>k

1.Block code:
Parity check code.
The Block code with check code of one Bit , if the
number of one bits in the message is even it is:
Even parity check code. And if the number of one bits
in the Message is odd it is: odd parity check code.
Note that if single error occurs in received message can be
detect but, can’t be corrected.
Ex: 111 it is odd parity(r=1)
11011 it is even parity (r=0)

1.Block code:
Binary code space.
Hamming weight: defined as the number of non-zero
bits in a code word.
Ex:
00000 (weight=0)
00111 (weight=3)
11011 (weight=4)

Binary space code:
Hamming distance : error control capability, the hamming distance
between two code words is equal to the number of difference
between each bit of them…
Ex:10011011
11010010 the hamming distance is (3)
The minimum distance of a Block code is the smallest distance
between any pair of code words in the code.
In general if (n) minimum distance of block code then:
1. Error can be corrected if (n) is odd.
2. error can be corrected and, can be detected if(n) is even.

Binary space code:
The relation between the number of bit error correction (t) and
the minimum distance should be:
Relation between and error detection capability (q) ,(where q
number of bit error detection possible) is:
to find the number of maximum possible code word of the length
(n) and minimum distance this is given by B(n,d) and
hamming give the following values:
B(n,1) =
B(n,2)=
B(n,3)=

Binary code space:
B(n,4)=
B(n,2k)= B
B(n,2k+1)=
The equality in B(n,3) is valid when is an integer
such codes are referred to as:
Close packed codes,These are obtained by selecting
K is positive integer ,for example :k=2,3,4 and n=3,7,15 resulting in
block packed codes are: B(3,3)=

Binary block code:
B(7,3)= ,B(15,3)=
When the number of maximum possible code words is
found out from :
For n=5:
gives:
B(6,3) (m=3)
B(5,3) (m=2)
100110
000000
00000
110011
010101
01101
011110
111000
10110
001011
101101
11011

2. Linear block code:
If we have a stream of bits we segment it to blocks(k)
and added to them a redundant bits (r), we get the
encoded bits(n).
The conversion operation from message(k-bits) to
encoder(n-bits) by using Generators matrix.
n>k, if n=k mean that no encoding
r=n-k
Generators
matrix
Message
K-bits
Encoded
bits(n)

2.Linear block code:
Generators Matrix.
G= ,P: parity check, I: identity matrix.
G=
Ex:(7,4) linear block code and you have m=(1101)
solution: n=7 , k=4 , then r=7-4=3 and G(k*n)
G=
Where dependent on
original message ( k-bits)

systematic code
V: represent the code word for message (m),
At receiver the received message (r) may be equal to
the transmitted message(v) or not,

at Rx
To detect the error in received message and correcting it:
If vector S=0 meaning that no error occurred during the
transmission in the channel.
If S≠0 meaning that the error occurred.
If we received (R=1001001) to find the error must be find the
matrix H?
R (n bits)
S(r bits)
(syndrome)

at Rx error correction.
S=R* = [1001001]
S= [1001001]* =[ 111] if we compare this data with
we found it in fifth row then the
error occurred in the fifth bit into
the received message.
The correction is be : [1001101]

2.Linear code:
Hamming single error correction code:
Hamming code is a Linear block code (n,k) bits.
The code word
Ex: data(1011)
solution: d3=1 , d5=0 ,d6=1 , d7=1
P1= 1 0 1 =0 , p2=1 1 1=1 , p4= 0 1 1=0
Where d3,d5,d6,d7 is data
message bits.

2. Linear code:
Hamming code:
At receiver use to detect the error and correction it.
S= * R if results of parity bits in syndrome equal to parity bits in
the received message that meaning no error occurred , if not equal
the error occurred.
Ex: R=1001110 , to detect the error calculate parity bits:
P1=0 1 0=1, p2= 0 1 0=1 , p4= 1 1 0=0 p=[1 1 0] not equal to
parity bits in R then the error occurred in the sixth bit in R,
The correct cod word is [1001100]

Cyclic code.
The code word form of cyclic code is:
where D is data
is polynomial of the degree (n-k) and its factor (n-k).
Ex: D= 1101 ,
Solution:
v=1010001

Systematic Cyclic code:
is a remainder of division
Ex: D= 1101 ,
1.
2.Long division:
= 001
= 0011101

Error correction in systematic cyclic
code:
To find Syndrome from received message R(x) from the remainder
of division , if S(x) =0 meaning no error occurred ,and if
S(x)≠0 will correct the error by shifting the received message with [n]
repeating shift and correct [n-1-i] from shift.
Ex:
S(x)= rem
The remainder not equal to zero ,
To correct the error make a cyclic shift:

Cyclic code error correction:
R(X)= 1 0 1 0 0 1 0 after shift (n-1-i) correct the error.
1st shift= 0 1 0 1 0 0 1
2nd shift= 1 0 1 0 1 0 0
3rd shift= 0 1 0 1 0 1 0
4th shift= 0 0 1 0 1 0 1 (n-1-i)=7-1-2=4
5th shift= 0 0 1 0 1 0 7-1-1 =5
6th shift = 0 0 0 1 0 1
7th shift= 1 1 0 0 0 1 0, the correction is [1100010]

Reference :
1.elements of digital communication (sarkar )
2.youtube – channel coding.
Thank you..

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