using the matlab communications toolbox

8/10/2019 Using the Matlab Communications Toolbox

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USING THE MATLABCOMMUNICATIONS TOOLBOX

TO LOOK AT CYCLIC CODING

Wm. Hugh Blanton

East Tennessee State Universityhttp://faculty.etsu.edu/blanton



The Theory

General Concepts



Channel Coding

Designed to improve communicationsperformance by enabling the

transmitted signals to better withstandthe effects of various channelimpairments, such as noise,

interference, and fading.



Hamming Distance

000 100

110

101

001

010

011

111

The number of bit positions in which two binary wordsdiffer.

This figure has a minimumdistance between code words of1.



Hamming Distance of 2

0000

r = 1

0011

0110

A code word that differs bya Hamming distance of 2

will detect an erroneouscode word.

0001 0111



1

1

1

If the minimum distance is 3, the code words are

separated by at least three edges.

Note that if a single error

occurs, the erroneous codeword will be closer to one of thecorrect code words.

The error is corrected by assigning thereceived code word to the nearest (inHamming distance) valid code word.



The important concepts associated

with Hamming distance are:1. When the Hamming distance is 1, it is

impossible to detect an error, much less

correct the error.2. When the Hamming distance is 2, the

error can be detected, but notcorrected.

3. When the Hamming distance is 3, asingle error can be detected andcorrected.



More Theory

Mathematical Concepts



LINEAR BLOCK CODES

Definition

– A systematic (n,k) linear block codeis a mapping from a k- dimensionalmessage vector to an n- dimensionalcodeword in such a way that part of

the sequence generated coincideswith the k message digits.



Information Code

0000 0000000

0001 1010001

0010 1110010

0011 0100011

0100 0110100

0101 1100101

0110 1000110

0111 0010111

1000 1101000

1001 0111001

1010 0011010

1011 1001011

1100 1011100

1101 0001101

1110 0101110

1111 1111111

The linear block

code that everyoneuses is the (7,4) code.

k -dimensional

information vector,where k = 4 .

n -dimensionalinformation vector,where n = 7 .



Generator Matrix

How do you generate a (7,4) code?

– Of course with a Generator Matrix!

1...00...

......

0...10...0...01...

21

22121

11211

nmnn

m

m

k

p p p

p p p p p p

I P G

Note that m = n - k ,

the difference



A (7,4) code is generated by thematrix

1000101

01001110010110

0001011

G

][Guv ][Guv



The (7,4) block code is generatedusing the matrix equation

– where

1. u is a [1 × k] vector representing theinformation word

2. v is a [1 × n] vector representing the codeword

3. [G] is a [k × n] generating matrix

][Guv ][Guv

][Guv



For the information word (0 0 0 1) ,the code word is computed as follows:

1000101

0100111

0010110

0001011

1000][Guv

1000101v

Transformationfrom 4-bit

information word to7-bit code word

Note that the additionsare modulo-2 additions



The Hamming distance for the (7,4)code is 3.

–Thus, this code can detect double biterrors and correct single bit errors



Parity Check Matrix

Another important matrix associatedwith block codes is the parity check

matrix , [H] .



The parity check matrix is formed bystarting with the identity matrix and

appending the transpose of thenonidentity portion of [G] :

nmmm

n

n

T

k

p p p

p p p

p p p

P I H

...1...00

......

...0...10

...0...01

21

22212

12111



For the (7,4) block code,

1110100

0111010

1101001

H



The Syndrome

The parity check matrix has the property

v[H] T = 0

That is, any errorless, received code wordmultiplied by the transpose of the parity

check matrix, [H], yields a zero vector, orsyndrome . – If the received code word contains an error, the

resulting vector will match the corresponding bitthat caused the error.

C d f h l d d d



Consider one of the valid code wordsof the (7,4) block code, v = 1100101 and the transpose of the (7,4) parity

check matrix.

101

111

110

011

100010

001

1010011T

H v



01000001

101

111

110

011

100

010

001

1010011

Modulo-2 Addition

00010010

101

111

110

011

100

010

001

1010011

011

110

101

000

001



01010000

101

111

110

011

100

010

001

1010011

000

101

111

110

011100

010

001

1010011

T

H v

Syndrome

The null vector

indicates no error.



Suppose the valid code words, v =1100101 gets corrupted to 1000101 .

101

111110

011

100010

001

1010001T H v

001



01000001

101

111

110

011

100

010

001

1010001

Modulo-2 Addition

10010000

101

111

110

011

100

010

001

1010001

011

110

101

000

001



01010000

101

111

110

011

100

010

001

1010001

010

101

111

110

011100

010

001

1010001

T

H v

Syndrome

Match syndrome withcorresponding code in

parity check matrix

The second rowcorresponds to thesecond bit from left.



Each row in the transposed paritycheck matrix corresponds to a bit in

the code word. – By matching the non-zero syndrome with

the contents contained in the rows of thetransposed parity matrix, thecorresponding corrupted bit can bedetected and and subsequently corrected.



The process will convert the erroneouscode 1000101 to the correct code

1100101 .



Cyclic Codes



CYCLIC CODES

The implementation of linear blockcodes requires circuits capable of

performing matrix multiplication and ofcomparing the result of various binarynumbers.

– Although integrated circuits have beendeveloped to implement the mostcommon codes, the circuitry can becomequite complex for very long blocks of

code.



A special case of the block code, thecyclic code can be electronically

implemented relatively easily.



A (7,4) cyclic code generating

matrix

1011000

0101100

0010110

0001011

G



A (7,4) cyclic code generating

matrix

1011000

0101100

0010110

0001011

G

Non-systematic —No Identity

matrix.



A nonsystematic (7,4) cyclic codegenerating matrix is made systematic by

performing row, column operations untilyou obtain the Ik partition.

1000101

0100111

00101100001011

sysG



MATLAB COMMUNICATIONS

TOOLBOX DERIVATION



MATLAB COMMUNICATIONSTOOLBOX DERIVATION

Although row,column matrixmanipulation is a straightforward

process for students that have taken alinear algebra course, manyengineering technology students are

not familiar with the process.



Although, the discussion of linearblock codes is not mathematically

complicated, it is mathematicallytortuous with many mathematicalturns.



Finally, the discussion for the (7,4)cyclic code is relatively benign with a

manageable number of calculations.



For larger codes, the calculationsbecome overwhelming, especially for

the engineering technology student.



In order to

– better manage larger codes,

–relieve the mathematical stress onengineering technology students, and

– focus on the application of block codes

rather than the mathematics; the MATLAB Communications Toolbox

provides a set of cyclic code functions.

Consider the problem with a (15 7)



Consider the problem with a (15,7) cyclic code with the following generatorpolynomial.

875431 x x x x x x

110111011

thatcorresponds tothe code word

1 x1 x2 x3 x4 x5 x6 x8x7



The nonsystematic cyclic codegenerating matrix, G , is:

110111011000000

011011101100000

001101110110000

000110111011000

000011011101100

000001101110110

000000110111011

G



And the systematic cyclic code

generating matrix is

100000011011101

010000010110011

001000010000100

000100001000010

000010000100001

000001011001101

000000110111011

sysG



By using the function cyclpoly , MATLAByields a set of all generating polynomials

>> p=cyclpoly(15,7,'all')

p =

1 0 0 0 1 0 1 1 11 1 1 0 1 0 0 0 1

1 1 0 1 1 1 0 1 1

875431 x x x x x x



By using the function cyclgen , the nonsystematicgenerating matrix, genmat , is derived as follows:

>> genpoly=[1 1 0 1 1 1 0 1 1];

>>[parmat,genmat]=cyclgen(15,genpoly,'nonsys')

genmat =



Columns 1 through 131 1 0 1 1 1 0 1 1 0 0 0 00 1 1 0 1 1 1 0 1 1 0 0 00 0 1 1 0 1 1 1 0 1 1 0 0

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

Columns 14 through 150 00 00 00 0

0 01 0



110111011000000

011011101100000

001101110110000

000110111011000

000011011101100

000001101110110

000000110111011

genmat

which is

in matrix form.

000000110111011



110111011000000

011011101100000

001101110110000

000110111011000

000011011101100

000001101110110

000000110111011

genmat

110111011000000

011011101100000

001101110110000

000110111011000

000011011101100

000001101110110

000000110111011

G



Likewise, the systematic generating matrix,genmatsys , can be computed using cyclgen

without the conditional parameter (nonsys):

>> genpoly=[1 1 0 1 1 1 0 1 1];>>[parmat,genmat]=cyclgen(15,genpoly)

genmatsys



genmatsys =

Columns 1 through 12

1 1 0 1 1 1 0 1 1 0 0 01 0 1 1 0 0 1 1 0 1 0 01 0 0 0 0 1 0 0 0 0 1 00 1 0 0 0 0 1 0 0 0 0 10 0 1 0 0 0 0 1 0 0 0 0

1 1 0 0 1 1 0 1 0 0 0 01 0 1 1 1 0 1 1 0 0 0 0

Columns 13 through 150 0 0

0 0 00 0 00 0 01 0 00 1 00 0 1



which is

in matrix form.

100000011011101

010000010110011

001000010000100

000100001000010

000010000100001

000001011001101

000000110111011

genmatsys



The (15,7) code produces 32,768(215) possible code words of which

128 (27) are valid code words.

– Needless to say, picking the right 128

valid codes out of 32,768 possibilities is afrightening task.



The Communications Toolbox provides anencode function that alleviates much of this

mathematical anxiety.

>> data=(0:127);

>> c=de2bi(data,'left-msb');>> valid_code=encode(c,15,7,'cyclic',genpoly)



valid_code =0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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

information code



Conclusion



The MATLAB Communications Toolboxderives the same generating matrices

that are derived by mathematicalmanipulation. – The underlying result is that the

electronic engineering technology student

can apply more effort in understandingthe theory rather than the mathematics,especially when considering morecomplex problems.



By reducing the mathematical anxiety,students can concentrate more

attentively on the actual nuances ofblock coding, and the instructor canmove from theoretical foundation of

cyclic codes to the circuitimplementation of cyclic codes.

using the matlab communications toolbox

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