paper-3 applications of rs code in real world implementation

7/30/2019 Paper-3 Applications of RS Code in Real World Implementation

1/12

International Journal of Computational Intelligence and Information Security, November 2012 Vol. 3, No. 9

ISSN: 1837-7823

20

Applications of RS code in Real World Implementation

Deepa TiwariMTech Scholar, Faculty of Engineering and Technology, Mewar University,

NH-79,Gangrar,Chittorgarh, Rajasthan,India

AbstractThe main aim of any communication system is to transmit maximum data without degrading the quality using

minimum bandwidth. A study of RS code, a branch of block code is given in this paper when used in a cell

phone in concatenated with a convolutional encoder, improves the sound quality. In this paper the no. of errors

and their magnitude is analysed through Matlab software in a digital communication system. The transmission

quality depends on the bit error probability Pb at the receiver. Therefore its BER curve for AWGN channel usingFSK modulation scheme is also analysed for different code rates.

Keywords- Galois field arithmetic, RS code, Interleaving, RS encoding and decoding.

1. Introduction

In digital communication there are two types of Forward error control codes.

1-Convolutional Codes

2-Block Codes

The convolutional codes are the codes which have a memory and used the property of convolution. It convolves

the message with the impulse response of the system. The common techniques to decode this coding are Viterbi

Algorithm.

In the block codes the stream of the bits is divided in fixed length blocks which are referred to a frame.

The code word is formed by adding proper redundant bits in the message bits. Since each code is different from

the previous one. Therefore the code is memory less. RS codes are linear block codes. In burst error channel the

code which suited best is RS code because of their capability to translate consecutive corrupted bits to few

erroneous symbols.

Reed Solomon code is the popular block codes which operate on m-bit symbols, not bits. As its name

implies it is discovered by Irving S. REED and GHUS SOLOMAN in 1960. The RS codes are a class of non-

binary cyclic codes[1]. These codes are detected and correct the errors in symbols. In binary codes the magnitude

of the error is always one, but in RS codes we have to find out the magnitude and location of the error both. This

difference can separate RS codes from Binary codes. The RS codes works in Galois field like other codes but themain difference between other codes and RS is here the base field and the extension field are taken to be the

same. RS codes are the most used codes in practical and real world implementations. Since RS codes are non

binary so the code word symbols are transmitted in the form of a string of bits across a binary channel Because

Reed-Solomon codes detect and correct symbol errors rather than bit errors. In fact the main difference in both

the codes is that the RS codes symbols are lies in GF (q), in which the q is a prime number or any power of a

prime number.

A cyclic RS code can be implemented for extending the length to q or even for q+ but the output code

will no longer be cyclic [2] for example the field GF (256) can be represented as an 8 bit sequence or byte oflength 255, so it is very popular error correction coding. Reed-Solomon codes can be used to perform a form of

forward error correction in computer networks. The specific type of correction is called erasure correction.

Mostly the channel error comes in bursts like in a CD. Any scratch can cause errors in several consecutive bursts

or in a hard disc or in a tape any blemish in magnetic medium may cause errors. The conventional block error

correction code is not sufficient for that purpose. A single error burst can lead to many errors in code words so it

needs a strong coding system which is suitable for bursting errors.

2. Finite field ArithmeticIt is the requirement of the RS codes principle to understand the operation in finite field called Galois field

named to its inventor ""EVARISTE GALOIS"(1811-1832 the fields which are having the finite number of

elements called finite field. The fields are expressed as GF (q) where q is the base field elements. The roots of

the polynomials are used to construct the field. RS code is a q -ary bch code having block length q-1,let q=2m so


2/12


ISSN: 1837-7823

21

symbols and the roots are lies in GF(2m), a t error correcting RS code is a non-binary cyclic code having

generator polynomial is the least degree polynomial with ,2,

3,.................

2tas its roots and the alpha belongs

to GF(2m) the minimal polynomial of GF(2

m) of an element alpha in GF(2

m) is the factor of r=X+ So the

generator polynomial of RS code is as G(X)=(X+)(X+2)................(X+2t) ,The degree of a RS(n,k) cycliccode is (n-k=2t).Since dmin=2t+1 (here 2t can be replaced by n-k) therefore dmin=n-k+1, so we can say that RS

codes takes the max distance[3]. Therefore we can say it belongs to a class of maximum distance code (MDS).

2.1 Additive and multiplicative inverseFor learning finite field arithmetic we need to learn the additive and multiplicative inverse of a finite field.

The additive inverse of a is -a and the multiplicative inverse of a is a- 1 (a~=0).Therefore subtraction of two field

elements can be done as adding the additive inverse of one to other for example a-b=a+(-b), the division of two

elements is as multiplying one by the multiplicative inverse of other. for example a/b=a*b-1.In GF(q) If q is a

prime number or any powers of prime number then this field is called the primitive field.

Table 1: Addition and multiplication in Galois Field(GF)

2.2 Irreducible PolynomialA polynomial is called irreducible over GF (2) if it is not divisible by any polynomial over GF (2) of a

positive & non zero degree having lower than m. For example if a polynomial is having even no. of terms then it

is divisible by X+1.

The polynomial of degree 2 X2,X

2+1,X

2+X are divisible by X or X+1 so the polynomials are not

irreducible. But X2+X+1 is an irreducible polynomial because they are not divisible by any polynomial of degree

one .so the polynomial X2+X+1 are an irreducible of degree 2. Similarly the polynomial X3+X+1 and X4+X+1 is

an irreducible over GF (2) of degree 3 and 4 continuously. So we can say that if m>=1 the irreducible

polynomial does exists.

2.3 Irreducible and Primitive PolynomialsThe polynomials for GF (23) GF (24) and GF (25) is X3+X+1, X4+X+1 and X5+X2+1 continuously. The each

polynomial can be factorized by only 1 and the polynomial itself. So these polynomials are irreducible and

primitive polynomials. The table is shown below for these irreducible primitive polynomials of degree m.


3/12


ISSN: 1837-7823

22

Table 2: List of some Primitive Polynomial.

For any positive integer m there must be at least one irreducible polynomial of degree m. The irreduciblepolynomial of degree m divides Xr +1 where r=2m-1.This test can be used to identify the irreducible polynomial

(but this condition is not for some special cases). For example X3+X+1 will divide X7+1 and gives X4+X2+X+1

quotient and remainder zero so X3+X+1 is irreducible [7]. Similarly X

4+X+1 will divide X

15+1 and gives the

quotient X11+X8+X7+X5+X3+X2+X+1 and remainder zero. So X4+X+1 is an irreducible polynomial.

2.4 Test for primitiveIn continuation of test for irreducible we can now able to show that X3+X+1 should not divide other than

X7+1 we divide X

6+1 by X

3+X+1 and find that remainder is X

2it means full division is not possible so we can

say that X3+X+1 is primitive.

3. A case study

Figure1: Block diagram of speech coding system for a mobile

The microphone transforms the sound waves of our voices (and other generated noise) into an electrical signal

Since GSM is a digital system, so this analog signal cant be used in GSM network. Therefore it has to be

converted into digital form [4]. This is accomplished by A/D converter.

We have to compress the input data of sound wave, so the analog signal is inputted into a band pass filter. Band

pass filter only allows a desirable frequency range to pass through it, and all undesired signals are cut off,or attenuated. This compressed analog signal is processed to A/D converter. In the cell phone, the analog signal

which is generated by a microphone is passed to the A/D converter. The A/D converter measures the analog

signal, or samples it 8000 times per second. ADC takes a sample of the output of the Band pass signal every .125

sec or 125 s. Each sample is quantized with a 14-bit data block. 14 bits per sample at 8000 samples per second,


4/12


ISSN: 1837-7823

23

we calculate a data rate of 112,000 bits per second, or 112 kb/s. A data rate of 112 kbps is very large. and is

impractical to handle by a radio transmitter. For reducing the bitrate, the signal is passed through a speech

encoder.

The speech encoder compresses the data of a speech signal. Many speech encoding schemes are available.

The scheme used in GSM is referred as the Linear Predictive Coding (LPC). The ADC quantifies each audio

sample by a 14-bit "word". In LPC, 280 of the 14-bit samples from the converter are saved up and stored intoshort-term memory. The sample is taken at every 125 s, this audio block having 2240 bits. LPC analyses each

block of data and finds 8 coefficients which are used for filtering. LPC identifies some specific bits that can be

correlate with specific characteristics of human voice, like a vocal system of human (teeth, tongue, etc.) and

relate coefficients to them. The excitation signal can be relate to the things such as pitch and loudness. LPC can

recognize a number of parameters of human voice and redundancies in human speech and removes them. TheLPC sequence is sent to a Long-Term Prediction (LTP) Analysis tool. The LTP function compares the sequence

it receives with the reference stored in its memory and chooses the sequence which most resembles the sequence

of the current analysis. The LTP function now calculates the difference between these two sequences. The LTP

function now translates the difference value with a pointer indicating earlier sequence it used for doing

comparison. this action prevents encoding of the redundant data using LPC and LTP, the speech encoder reduces

(compresses) the 20ms block from 2,240 bits to 280 bits.as we see that this is a reduction by eight times. 280 bits

every 20ms gives us a net data rate of 14 kilobits per second (kbps).

To prevent the sound quality degradation from handset to base station transmitter using radio link, the data

stream is put through a series of error detection and error correction procedures called error correction coding.

The first phase of channel coding is called linear block coding. In this module RS coding is used now RS coding

adds 3 parity bits with 4 tail bits, so, a 280-bit code becomes a 287-bit code. This block of 287-bit is then fed into

a convolutional code. When the last data bit enters the register, it uses the remaining four bits to calculate the

redundancy bit for the last data bit. After convolutional coding, now the audio block is 456 bits, nothing can be

done if the entire 456-bit block of data is lost or some burst of errors is occur. For protecting data against burst

errors, the bits have to arrange in different order and partitioned to eight individual sub-blocks. If one sub-block

will be lost or corrupted then only one-eighth of the data of each audio block will be lost and it can be recovered

through the convolutional code on the decoder. This process of changing the order and making partition is called

interleaving. Interleaving is done in a block called Interleaver and in the decoder the de-interleaving process is

done.

4. InterleavingPractically the best method to correct burst errors is interleaving but in theory it is the worst method to deal

with burst errors. The best error structure can be spoiled by the interleave[5] and it make the error random. An

interleave takes a sequence of symbols and permutes them. At the receiver the sequence is permuted back into

original order by a de-interleaver. Interleaver has a good ability in dealing with burst of errors. because it

shuffles the symbols at the receiver. Block interleaver is very popular for this purpose. block interleaving is also

a permutation formatting that is opposite of the permutation done for interleaving, this ensures that the original

data is correctly restored from the interleavers output data. Interleaving is of two kinds one is block interleaving

and other is cross interleaving or convolutional interleaving. Here we explained only block interleaving.

4.1 Block InterleavingBlock interleaving can be done in two types in row and in column. A block is an array of n Rows and m

columns.

Figure 2: Block Interleaving


5/12


ISSN: 1837-7823

24

4.2 Row Interleaving

The row 2 is replaced by row 1 and row 3 is replaced by row 2 and row 1 is replaced by row 3 in the encoding

end.

4.3 Column Interleaving

The column (4,2,1,3) means column 4 is replaced by column 1, column 2 is not changed. column 1 is replaced by

column 3 and column 3 is replaced by column 4.


6/12


ISSN: 1837-7823

25

4.4 De-Interleaving in the decoding end

In de-interleaving first we revert back the changes in column matrix then in row and see how we get the original

data back. The biggest side effect of the interleaving techniques is the latency of the system it increases latency.

Also structure of the error is not defined. Other advanced decoding algorithms can be used in place of inerleaver

and achieve more reliable communication.

5. RS EncoderThe RS Encoding has the 3 most important parameters that is (n,k,t). where n is the number of symbols per

code word, k is the number of message symbols and t is the capability of the encoder to correcting the symbol

error. Since n-k=2t so t=n-k/2. The generator polynomial can therefore be expressed as below

G(X) = g0 + g1X+ g2X2

+ + g2t- 1X2t- 1

+X2t

for example for an (15,11) error correcting code t=n-k/2, t=15-11/2=2, so it will correct two errors. Since 2t=4

so it will have 4 roots and the generator polynomial will look like this G(X)=(X-) (X-2) (X-

3) (X-4)

by solving this we get the generator polynomial as follows

G(X)=3 - X + X

2 - 3X3 + X4..................................(a)

but we know = -, putting it in equation (a) we get

G(X)=3+ X + X

2+ 3X3 + X4...................................(b)

5.1 Systematic EncoderThe systematic encoding of RS codes is done in Linear Feedback Shift Register (LFSR) the message

polynomial m(x) is shifted into the right most k stages of a code word [6]. After that the code word is appended

by a parity polynomial p(x) in the leftmost (n-k) stages.


7/12


ISSN: 1837-7823

26

Figure 3: Systematic Encoder

The operation of the LFSR is explained below [7]

When switch (1) is closed, the first k clock cycles is allowed to shift the message symbols in the shift register.

When switch 2 is in downward Position, the first k close cycles allow the direct transfer of the message symbols

to the output of the LFSR. When switch 1 open & switch 2 in upward position (closed). Now the parity symbolsin the shift register by shifting them to the output of the LFSR with the help of the (n-k) clock cycles. When the n

clock cycles ends .Then the o/P contents of the LFSR is p(X)+Xn-k m(X) where p(X) is the parity symbols and

m(X) is the message polynomials.

For Example if we take the symbols 010,110,111 it means 010 , and 110 means 3,and 111 means

5 the

rightmost symbol is the first symbol.

Table 3: LFSR Operation

Input Clock cycles Register Contents feedback 0 0 0 0 0

5

1 1

5

1

0

2 3 0

2

2

4

- 3 -

When the third clock cycle is over, then the register contains the four parity symbols, 0,

2,

4, and

6, which is

shown in the table above. After that switch 1 of the LFSR circuit is opened, switch 2 is connected to the upward

position, and the register contents which are four parity symbols shifted to the output of the LFSR. So the output

of the LFSR having the code word, C(X), which can be written in the polynomial form.

So the output word is

C(X)= 0

+ 2X +

4X

2+

6X

3+

1X

4+

3X

5+

5X

6

= (100) + (001) X + (011) X2

+ (101) X3

+ (010) X4

+ (110) X5

+ (111) X6

So we can say that the roots are same for the codeword as well as for generator polynomial. So a valid code word

is in the following form C(X)=m(X)*G(X) and the other evaluated code word at any root of G(X) must yields

zero.


8/12


ISSN: 1837-7823

27

Table 4: Some standard RS code with their parameters

6. Decoding of RS CodesA logical approach for decoding the RS code is shown in the form of a flow chart in the figure below.

Figure 4: Steps in the decoding of the RS code


9/12


ISSN: 1837-7823

28

The decoder receives the encoded code word and detects the error with the help of check symbols [8]. As soon

as the decoder receives the encoded code word it detect the errors with the help of checksymbols and correctsthem. Due to the noise present in the channel the decoded code word may differ from the original code word,

now using the different polynomials to detects the location and magnitude of the error.

The Received code word is in the form , R(x)=Transmitted code word R(x)+E(x)

The following steps are used in the decoding of the RS encoded code word.1. In the first step a syndrome polynomial is to be generated. Then an error polynomial is generated. Sothat we can locate the error.

The error can be located using any of the 4 methods given.

Figure 5: Taxonomy of Error Locating Polynomial

2. Computation of the roots of the error locator polynomial. The method used in this process is ChienSearch. When the decoder determines the error locator polynomial then the decoder calculates the

polynomial so that the location of the error can be find out. The method generally used is a trial and

error method referred as chien search. In which the decoder replaces the every possible root -e

the e is

the location in the code word into the error locator polynomial. If the result is zero it means that there is

an error in that particular location.

3. When the error location is known the decoder uses the forneys equation and find the value of the error.Since RS codes are non-binary therefore the error values must also be computed by the followingformula.

Then the equation of the error locator polynomial can be expressed as

The error evaluator polynomial can be expressed as

Where t is the no. of check symbols divide by 2

7.Analysis and findings

7.1 Analysis of errors using Matlab software.The simulation is run for 2 times. One time sample message is taken for GF( 2

5) and the sample of

the transmitted message taken is 0, the length of the code is 25-1=31.the other time the transmitted

message sample is taken 1, the length of the code is 24-1=15,the error is introduced. After simulating


10/12


ISSN: 1837-7823

29

the error magnitude ,error location and the corrected message is find out. We transmitted the message

in first case zero(0) since the length of the message is 31,so the message is 31 times 0.the error is

introduced in the 3,5,6 locations ,and the magnitude of the error is 2,3,2 continuously, so after error

correction the original message is recovered similarly in the other case the message.

Figure 7: Screen Shot of transmitted and received sequence using Matlab

7.2 Analysis for BER Vs Eb/No using Matlab

BER Vs Eb/No curve for different RS Codes using Matlab [14] over AWGN channel, for FSK

modulation is shown in figure. 8


11/12


ISSN: 1837-7823

30

Figure 8: BER Vs Eb/No for RS Code over AWGN channel, for FSK modulation

As we can see from the BER Vs Eb/No curve that the BER is an exponentially decreasing function of

block length.so with the increasing code length the BER improves. It is observed from the above curve

that large code size reduces the noise means increases the signal. observation is taken for 5 different

types of codes shown in Table 5.

Table 5: The Analysed data

Code length RS code Set Code Rate Error correcting

capability(t)

7 RS(7,3) .428 2

15 RS(15,11) .738 2

31 RS(31,21) .677 5

63 RS(63,39) .619 12

255 RS(255,183) .717 36

8. ConclusionIn mobile communication the signal travels from handset to the base station transmitter because of the data

loss due to fading and other atmospheric conditions sound quality degrades. Many softwares and hardwaresare available in the market in different forms for both the encoder and decoder. These soft encoding and

decoding process can be implemented in different supported platforms like c/c++ , matlab, code composer

studio, and in VHDL. Which may use many digital signal processing architectures like TMS320C5X series,

TMS320C6X series.

For protecting the sound quality in cell phone the data stream is passed through the RS error correction

modules which is used as a concatenated code with an interleaver to lower the error rate and enhances the quality

of data transmission.RS codes are also used in deep space communication, Consumer Electronics(like in CDs,

DVDs, Blue ray disks. A pair of cross-interleaved Reed- Solomon codes is used in a compact disk system),

Spread spectrum systems, data transmission technologies-DSL and wimax, in broadcast system -DVB, ATSC.

The encoding and decoding techniques of RS code is very efficient. It has a good burst error correcting

capability. For making the mobile and optical communication system reliable communication medium the RS

codes are the best error correcting codes that can give the reliability performance in the field of mobilecommunication. Since RS codes are very nearer to obtaining the Shanons limit, so in future it can also be used


12/12


ISSN: 1837-7823

31

in every digital communication field including in the next generation of mobile phones ,other space missions and

the next generation of RAID(after RAID 6).

References

[1] I.S. Reed and G. Solomon,(1960) Polynomial Codes Over Certain Finite Fields, SIAM Journal of Applied

Mathematics, vol. 8, pp. 300304.[2] S. B. Wicker,(1995), Error control systems for digital communication and storage, Prentice hall.

[3] Salvatore Gravano Introduction to Error Control Codes,Oxford university of Press.[4] www.gsmfordummies.com

[5] Todd K. Moon,(1995), Error correction coding Mathematical Methods and Algorithms, JOHN WILEY & SONS,INC., PUBLICATION.

[6] .Shu Lin and D.J.Costello ,(1983), Error Control Coding fundamental and Applications, Prentice hall.

[7] Bernard Sklar,(2001), Digital Communications (Fundamental and Applications), 2nd Edition. Prentice

Hall

[8] Lionel Biard, Dominique Noguet (2008) Reed-Solomon Codes for Low Power Communications,Journalof communications, Vol. 3, No. 2.

[9] Anish S. Sanghvi, Nishant B. Mishra, Rajkumar Waghmode and Kiran T. Talele (2011) Performance of

Reed-Solomon Codes in AWGN Channel, International Journal of Electronics and Communication

Engineerin, Volume 4, pp. 259-266.

[10] Reed-Solomon II (May 2011),Mega-Core Function User Guide Altera

[11] Saurabh Mahagan and Gurupadam Singh (2011), Reed-Solomon Code Performance for M-ary Modulation

over AWGN Channel, International Journal of Engineering Science and Technology (IJEST),Vol. 3 No.1.

[12] NPTEL lesson 34 block codes (IIT Kharagpur).

[13] www.wikipedia.com

[14]www.mathworks.com
http://www.mathworks.com/http://www.mathworks.com/http://www.mathworks.com/http://www.mathworks.com/

paper-3 applications of rs code in real world implementation

Documents