Error Detection and Correction

7/21/2019 Error Detection and Correction

http://slidepdf.com/reader/full/error-detection-and-correction-56d976d42845f 1/31

UCCN2043 – Lecture Notes

Page 1 of 31

5.0 Error Detection and Correction

In a digital communication system, totally error-free transmission is not possible due totransmission impairments. At the receiving end, there should be a mechanism for detection of

the errors and if possible for their correction. In this lesson, we will study the varioustechniques used for error detection and correction.

5.1 Need For Error Detection And Correction

Consider a communication system in which the transmitted bit stream is

1 0 1 1 0 1 1 1 0

The transmitted electrical signal corresponding to this bit stream and the received waveformare shown in Figure 5.1.

Figure 5.1: Errors introduced by transmission medium

Due to the noise introduced in the transmission medium, the electrical signal is distorted. Byusing a threshould, the receiver determines whether a 1 is transmitted or a 0 is transmitted. In

this case, the receiver decodes the bit stream as

1 0 1 0 0 1 0 1 0

At two places, the received bit is in error—1 has become 0 in both places.

In a digital communication system, some bits are likely to be received in error due to thenoise in the communication channel. As a result, 1 may become 0 or 0 may become 1. TheBit Error Rate (BER) is a parameter used to characterize communication systems.




Page 2 of 31

How many errors can be tolerated by a communication system? It depends on the application.For instance, if English text is transmitted, and a few letters are received in error, it istolerable. Studies indicate that even if 20% of the letters are missing, human beings canunderstand the text.

Suppose the communication system is used to transmit digitized voice from one place toanother. Studies indicate that even if the Bit Error Rate is 10-3, the listener will be able tounderstand the speech. In other words, a voice communication system can tolerate one errorfor every 1000 bits transmitted.

Note: The performance of a communication system can be characterized by the Bit ErrorRate (BER). If BER is 10-3, there is one error per 1000 bits.

Now consider the case of a banking application. Suppose you need to transfer $100 from youraccount to a friend's account through a data communication network. If the digit 1 becomes 3

due to one bit error during transmission, then instead of $100, $300 will be deducted fromyour account! So, for such applications, not even a single error can be tolerated. Hence, it isvery important to detect the errors for data applications.

Errors can be classified as random errors and burst errors.

Random Errors

• Appear at random intervals.

• Occur at random places in the bit stream

Burst Errors• Caused by sudden disturbances in the medium such as Lightning Sudden interference with the nearby devices

• Such disturbances cause many consecutive (sequence of) bits to be in error.

Detection and correction of errors is done through channel coding. In channel coding,additional bits are added at the transmitter end, and these additional bits are used at thereceiving end to check whether the transmitted data is received correctly or not and, if

possible, to correct the errors.




Page 3 of 31

5.2 Channel Coding

The following figure helps to identify the placement of channel coding and decoding modulein communication systems.

•

Channel coding is most often applied to communications links in order to improve thereliability of the information being transferred.

•

By adding additional bits to the transmitted data stream, it is possible to detect and evencorrect for errors in the receiver.

•

The added coding bits lower the raw data transmission rate through the channel (Codingexpands the occupied bandwidth for a particular message data rate).

• Channel codes that are used to detect errors are called error detection codes, while codesthat can detect and correct errors are called error correction codes. There are three generaltypes of channel codes:- Block codes, Convolutional codes and Concatenated Codes

• Because decoding is performed after the demodulation portion of the receiver, coding can be considered to be a post detection technique.

•

Channel coding is generally treated independently from the type of modulation used,although this has changed recently with the use of …

• Trellis coded modulation (TCM) schemes• OFDM, and•

New space-time processing that combines coding, antenna diversity, andmodulation to achieve large coding gains without any bandwidth expansion.

Information to betransmitted

Source

coding

Channel

codin

Modulation

Transmitter

Channel

Informationreceived

Source

decoding

Channel

decodin

Demodulation

Receiver

Channelcoding

Channel

decoding

Error Detection Codes

Error Correction Codes

(Forward Error Correction Codes)

Block Codes

Convolutional Codes

Concatenated Codes




Page 4 of 31

Error Detection

• In its most elementary form this involves recognizing which part of the receivedinformation is in error and, if appropriate or possible, requesting a repeat transmission –ARQ (Automatic Repeat Request Systems).

Error Detection and Correction

• With added complexity, it is possible not only to detect errors, but also to build in theability to correct errors without recourse to retransmission.

•

This is particularly useful where there is no feedback path to the transmission source withwhich to request a resend. This process is known as FEC (Forward Error Correction).

• ARQ and FEC may be combined, such that minor errors are corrected without

retransmission, and major errors are corrected via a request for retransmission: this iscalled Hybrid Automatic Repeat-Request (HARQ).

5.2.1 Automatic Repeat Request Systems (ARQ)

The following figure shows the implementation of an ARQ system in communicationsystems.

There are basically three types of ARQ implementations

• Stop and Wait ARQ• Go Back N ARQ•

Selective ARQ

Source Transmitter Channel Receiver Destination

Encoder TransmitController

Modulation Demodulation Decoder TransmitController

Acknowledge




Page 5 of 31

Stop and Wait ARQ

• This is the simplest ARQ method where the transmitter waits after each message for anacknowledgement of correct reception (known as an ACK) from the receiver.

•

If the message is received in error, a negative acknowledgment (NAK) is returned. Whilethis process is taking place, any new messages must be stored in a buffer at thetransmitter site.

Transmitting

Data

1 32 3Time

Received Data 1 2 3Time

ACK

Output Data 1 2 3Time

Error

Retransmission

ACK: Acknowledge NAK: Negative ACK

ACK

NAK




Page 6 of 31

Go Back N ARQ

• As the name suggests, the transmitter in this case continues to transmit messages insequence until a NAK is received.

•

The NAK identifies which message was in error and the transmitter then 'back-tracks' tothis message, starting to retransmit all messages in the sequence from when the erroroccurred.

•

Clearly, this has less signalling overhead (no ACKs used) than the Stop and Wait protocol.

TransmittingData

1Time

Received Data

Output Data Time

Error

Go-back 3

2 3 4 5 3 44 5 6 7 5

1Time

2 3 44 5

Error

Go-back 5

1 2 3 44 5

N

A

K

N

A

K




Page 7 of 31

Selective Repeat ARQ

• By making the protocol slightly more complex, and by providing a buffer in the receiveras well as the transmitter, it is of course possible for the receiver to inform the transmitterof the specific message or packet that is in error.

•

The transmitter then needs to only send this specific message which the receiver canreinsert in the correct place in the receiver buffer.

•

Although the most complex, this is also the most efficient type of ARQ protocol and themost widely used. There are several variants of this protocol optimized for a given set ofchannel characteristics.

5.3 Error Detection Schemes

•

A basic requirement of the ARQ system is for the receiving equipment to be able to detectthe presence of errors in the received data.

• Error detection is most commonly realized using a suitable hash function (or checksumalgorithm).

• A hash function adds a fixed-length tag to a message, which enables receivers toverify the delivered message by re-computing the tag and comparing it with theone provided.

4

Transmitting

Data

1 Time

Received Data

Error

Retransmission

2 3 4 5 3 6 7 8 9 7

1Time

2 3 6 8 7

Error

Retransmission

5 9

Buffer 1 Time2 4 3 6 8 75 9

Output Data 1Time

2 43 6 875 9

N

A

K

N

A

K




Page 8 of 31

• There exists a variety of Error Detection Schemes which differ in terms of theircomplexity and their suitability for detecting certain kinds of errors

• Repetition Codes• Parity Bits•

Checksums

•

Cyclic Redundancy Checks (CRCs)

5.3.1 Parity

• One of the simplest yet most frequently used techniques for detecting errors is the paritycheck bit.

•

The parity check bit is usually a single bit (1 or 0) appended to the end of a data wordsuch that the number of 1s in the new data word is even for even parity, or odd for odd

parity.

Parity is used in serial communication protocols whereby we transmit one character at a time.For example, if the information bits are

1 0 1 1 0 1 0

then an additional bit is added, which is called a parity bit. The parity bit can be added insuch a way that the total number of ones becomes even. In such a case, it is called even parity.

In the above bit stream, already there are four ones, and hence a 0 is added as the parity bit.The bit stream transmitted is

1 0 1 1 0 1 0 0

In case of odd parity, the additional bit added will make the total number of ones odd. Forodd parity, the additional bit added in the above case is 1 and the transmitted bit stream is

1 0 1 1 0 1 0 1

At the receiving end, from the first 7 bits, the receiver will calculate the expected parity bit. If

the received parity and the calculated parity match, it is assumed that the character received isOK. Else, it can be concluded that at least one error has occurred during transmission and theARQ process can begin. Of course, if two bits are in error, the parity check will pass, and theerrors will go undetected.

It is very easy to verify that parity can detect errors only if there is an odd number of error(s);if the number of errors is 1, 3, or 5, the error can be detected. If the number of errors is even,

parity bit cannot detect the error.

• Parity check is best suited to low noise, low distortion links where the error rate is knownto be very low.




Page 9 of 31

• For links with a high probability of error, more sophisticated error checking methodsmust be used – such as the block or convolutional codes which require the addition oflarger numbers of redundant bits.

5.3.2 Checksum

Checksum of information bits is calculated using simple binary arithmetic. Checksum is usedextensively because its computation is very easy. However, checksum cannot detect all errors.

Suppose we want to send two characters, C and U.

The 7-bit ASCII values for these characters areC 1 0 0 0 0 1 1U 1 0 1 0 1 0 1

In addition to transmitting these bit streams, the binary representation of the sum of these twocharacters is also sent.

The value of C is 67 and the value of U is 85.The sum is 152.The binary representation of 152 is

1 0 0 1 1 0 0 0.

This bit stream is also attached to the original binary stream, corresponding to C and U, whiletransmitting the data.

So, the transmitted bit stream is1 0 0 0 0 1 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0

At the receiving end, the checksum is again calculated. If the received checksum matches thiscalculated checksum, then the receiver assumes that the received data is OK. The checksumcannot detect all the errors.

Also, if the characters are sent in a different order, i.e., if the sequence is changed, thechecksum will be the same and hence the receiver assumes that the data is correct.

However, checksum is used mainly because its computation is very easy, and it provides areasonably good error detection capability.

Note: Checksum is used for error detection in TCP/IP protocols to check whether packetsare received correctly. Different algorithms are used for calculation of checksum.




Page 10 of 31

5.3.3 Cyclic Redundancy Checks (CRCs)

• One of the most common, and one of the most powerful, error-detecting codes is theCyclic Redundancy Check (CRC). CRC is well suited for detecting burst errors and is

particularly easy to implement in hardware, and is therefore commonly used in digitalnetworks and storage devices such as hard disk drives.

• Implementation of CRC can be described as follow:

• Given a k bit block of bits, or message, the transmitter generates an (n-k)-bitsequence, known as a Frame Check Sequence (FCS), such that the resulting frame,consisting of n bits, is exactly divisible by some predetermined number.

•

The receiver then divides the incoming frame by that number and, if there is noremainder, assumes there was no error.

Additional bits added to the information bits are called the CRC bits. These bits can be 16 or32. If the additional bits are 16, the CRC is represented as CRC-16. CRC-32 uses 32additional bits. There are international standards for calculation of CRC-16 and CRC-32.

Since CRC calculation is very important, the C programs to calculate the CRC are given inAppendix 1 and 2 at the end of the lecture note. When these programs are executed, theinformation bits and the CRC in hexadecimal notation will be displayed.

CRC – Code Creation & Detection

• The CRC creation process is defined as follows:•

Get the block of raw message (k bits).• Left shift the raw message by (n-k) bits and then divide it by G (r bits).• Get the remainder R as FCS (number of bits, r=n-k bits)•

Append the R to the raw message. The result (n bits) is the frame to be transmitted.

•

CRC is checked using the following process:• Receive the frame (n bits)• Divide it by G (r bits)

•

Check the remainder. If the remainder is not zero, then there is an error in theframe.

• These procedures can be further clarified in three ways: modulo-2 arithmetic, polynomials, and digital logic.

Note: In CRC-polynomial calculation, a standard polynomial is used. This polynomial isdifferent for CRC-16 and CRC-32. The bit stream is divided by this polynomial tocalculate the CRC bits.




Page 11 of 31

CRC – Code Creation & Detection – Modulo-2 Arithmetic

• Modulo-2 arithmetic is in fact just the exclusive-OR (XOR) operation.

• Now define

T = n-bit frame to be transmittedD = k-bit block of data, or message, the first k bits of TF = (n - k)-bit FCS, the last (n - k) bits of TP = pattern of (n - k + 1) bits; this is the predetermined divisor

F DT F

DT k n

bitsk n

bitsk

bitsn +=⇒= −

−

)(

)(

)(

)( 2

By multiplying D by 2(n-k), we have in effect shifted it to the left by (n – k) bits and paddedout the result with zeroes. Adding F yields the concatenation of D and F, which is T.

Example:




Page 12 of 31

CRC – Code Creation & Detection - Polynomials

• In polynomials approach, the CRC process can be viewed by expressing all values as polynomials in a dummy variable X, with binary coefficients. The coefficients correspondto the bits in the binary number.

The code word with n bits is expressed as

c(X) = c1Xn-1 + c2X

n-2 + … + cnX0

• A generator polynomial with constant coefficients, is used as the divisor in a polynomiallong division over a finite field.

•

Taking the input data (blocks of input bits called message polynomial) as the dividendwhere the remainder (the FCS) becomes the result.

Note: Even parity is a special case of a single-bit CRC, where the single-bit FCS isgenerated by the generator polynomial (divisor) x+1.

•

Message polynomial is divided by Generator polynomial giving quotient and remainder,the coefficients of the remainder form the bits of final CRC.




Page 13 of 31

• Define:

M - The original frame (k bits) to be transmitted before adding the Frame CheckSequence (FCS).

F - The resulting FCS of r bits to be added to M (usually r=8, 16, 32).

T - The cascading of M and F.

G - The predefined CRC generating polynomial with pattern of r+1 bits.

• The main idea in CRC algorithm is that the FCS is generated so that the remainder of T/Gis zero.

Example:

Find the codewords c(x) ifm(x) = 1 + x + x2 and g(x) = 1 + x + x3 for (7,4) CRC.

Answer:

We have…n = Total number of bits = 7,k = Number of information bits = 4,r = Number of parity bits = n - k = 3.

Remember

c(X) = c1Xn-1 + c2X

n-2 + … + cnX0

m(x) = (0)x3 + (1)x2 + (1)x + (1)1 and g(x) = (1)x3 + (0)x2 + (1)x + (1)1

0 1 1 1 1 0 1 1




Page 14 of 31

CRC – Code Capabilities

• An error E(X) will only be undetectable if it is divisible by G(X). It can be shown that allof the following errors are not divisible by a suitably chosen G(X) and hence aredetectable:

•

All single-bit errors, if G(X) has more than one non-zero term.• All double-bit errors, as long as G(X) has a factor with at least three terms.• Any odd number of errors, as long as P(X) contains a factor (X + 1)• Any burst error for which the length of the burst is less than or equal to (n – k);

that is, less than or equal to the length of the FCS• A fraction of error bursts of length n - k + 1; the fraction equals (1 - 2-(n-k-1))• A fraction of error bursts of length greater than n - k + 1; the fraction equals (1 -

2-(n-k))

•

Common CRC Codes

Code Generator polynomial g(x) Parity check bits

CRC-12 1+x+x2+x3+x11+x12 12

CRC-16 1+x2+x15+x16 16

CRC-CCITT 1+x5+x15+x16 16

•

The CRC-12 system is used for transmission of streams of 6-bit characters and generatesa 12-bit FCS.

•

Both CRC-16 and CRC-CCITI are popular for 8-bit characters, in the United States andEurope, respectively, and both result in a 16-bit FCS.

5.4 Error Detection & Correction Schemes (Forward Error Correction Schemes)

• Correction of errors using an error detection code, requires that block of data be

retransmitted, using the ARQ disciplines explained in Section 5.2.1. For wirelessapplications this approach is inadequate for two reasons.

• The bit error rate on a wireless link can be quite high, which would result in alarge number of retransmissions.

• In some cases, especially satellite links, the propagation delay is very longcompared to the transmission time of a single frame. The result is a veryinefficient system.

•

Thus, Error Detection & Correction Schemes or more commonly called Forward ErrorCorrection (FEC) Schemes are inevitable.




Page 15 of 31

• The key idea of FEC is to transmit enough redundant data to allow receiver to recoverfrom errors all by itself. No sender retransmission required.

• There are two main types of forward error correction coding scheme:

•

Block coding – where a group (block) of bits is processed as a whole in order togenerate a new (longer) coded block for transmission. A complementary blockdecoder is used in the receiver.

•

Convolutional coding – operates on the incoming serial bit stream generating areal-time encoded serial output stream. A complementary serial (convolutional)decoder is used in the receiver.

• In Forward Error Correction Schemes, there are 4 possible outcomes:

• If there are no bit errors, the input to the FEC decoder is identical to the originalcodeword, and the decoder produces the original data block as output.

• For certain error patterns, it is possible for the decoder to detect and correct those

errors. Thus, even though the incoming data block differs from the transmittedcodeword, the FEC decoder is able to map this block into the original data block.

• For certain error patterns, the decoder can detect but not correct the errors. In thiscase, the decoder simply reports an uncorrectable error.

• For certain, typically rare, error patterns, the decoder does not detect that anyerrors have occurred and maps the incoming n-bit data block into a k-bit blockthat differs from the original k-bit block.




Page 16 of 31

5.4.1 FECC – Block Codes

• Information is divided into blocks of length k. r parity bits or check bits are added to each block (total length n = k+r ).

• Each k -bit sequence are mapped into a unique n-bit codeword

• An (n, k) block code encodes k data bits into n-bit codewords.

• There are 2k valid codewords out of a total of 2n possible codewords.

• The ratio of redundant bits to data bits, (n - k) / k , is called the Redundancy of thecode.

• The ratio of data bits (k ) to total bits (n), is called the Code Rate, R = k/n.

•

Decoder looks for codeword closest to received vector (code vector + error vector)

•

Tradeoffs between Efficiency, Reliability and Encoding/Decoding complexity.

Block Codes - Principles

•

The Hamming distance d(v1, v2) between two n-bit binary sequences v1 and v2 is thenumber of bits in which v1 and v2 disagree.

For example, if v1 = 011011, v2 = 110001. Then d(v1, v2) = 3

•

For a code consisting of the codewords w1, w2,…, ws where s = 2k , the MinimumDistance dmin of the code is defined as:

[ ]),(min

min ji wwd ji

d ≠

=

•

The number of errors, t , that can be detected satisfies:

t = d min - 1

• The maximum number of guaranteed correctable errors per codeword satisfies:

−=

2

1mind t




Page 17 of 31

Linear Block Codes and Examples

• A Block Code is linear if and only if the modulo-2 sum of any two codewords is also avalid codeword.

Linear Block Codes – Hamming Codes

•

Hamming codes are a family of (n, k) block error-correcting codes that have thefollowing parameters:

• Block length: n = 2 x - 1

• Number of data bits: k = 2 x - x - 1

•

Number of check bits: r = n – k = x

• Minimum distance: dmin = 3

•

Where x ≥ 3.

Linear Block Codes – Cyclic Codes

•

Most of the error-correcting block codes that are in use are in a category called cycliccodes.

• For such codes, if the n-bit sequencec = (c0, c1, ... , cn-1)

is a valid codeword, then(cn-1, c0, c1, … , cn-2),which is formed by ,cyclically shifting c one place to the right, is also a valid codeword.

• This class of codes can be easily encoded and decoded using linear feedback shiftregisters (LFSRs).

•

Examples of cyclic codes include the Bose-Chaudhuri-Hocquenhem (BCH) and Reed-Solomon codes.




Page 18 of 31

Block Codes – Codewords Generation and Error Detection

• The uncoded k data bits be represented by the m vector:m = (m1, m2, …, mk )

•

The corresponding codeword be represented by the n-bit c vector:c = (c1, c2, …ck , ck+1, …, cn-1, cn)

• Each parity bit consists of weighted modulo-2 sum of the data bits represented bysymbol.

⊕⊕⊕=

⊕⊕⊕=

=

=

=

++++

knk nnn

k k k k k k

k k

pm pm pmc

pm pm pmc

mc

mc

mc

...

......

...

2211

)1()1(22)1(111

22

11

• The block length C of the Linear Block Code isC = m G

where m is the information codeword block length, G is the generator matrix.

•

Generator matrixG = [Ik | P]k×n

where pi = Remainder of

−+−

)(

1

x g

x ik n

for i=1, 2, .., k , and I is unit matrix.

•

The parity check matrix

H = [PT | In-k ]

where PT

is the transpose of the matrix P.




Page 19 of 31

• Operations of the generator matrix and the parity check matrix

The parity check matrix H is used to detect errors in the received code by using the fact thatc * HT = 0 ( null vector)

Let x = c ⊕ e be the received message where c is the correct code and e is the error.

Compute

( )

T

T T

T

T

H e

H e H c

H ec

H x s

=

⊕=∗⊕=

∗=

If s is 0 then message is correct else there are errors in it, from common known error patternsthe correct message can be decoded.

Block Codes – Generator Matrix (Example)

Find linear block code encoder G if code generator polynomial g(x)=1+x+x3 for a (7, 4)code.

We have n = Total number of bits = 7,k = Number of information bits = 4,r = Number of parity bits = n - k = 3.

[ ] ,

100

010001

|2

1

==

k p

p p

P I G

L

LLLL

L

L

where

pi = Remainder of

−+

)(

13

x g

x i

for i=1, 2, .., k , and I is unit matrix.

Generatormatrix

G

CodeVector

C

Messagevector

m

Paritycheckmatrix

HT

CodeVector

C

Nullvector

0

n

k




Page 20 of 31

[ ]11011

Re3

3

1 →+=

++= x

x x

x p

[ ]011

1Re 2

3

4

2 →+=

++= x x

x x

x p

[ ]11111

Re 2

3

5

3 →++=

++= x x

x x

x p

[ ]10111

Re 2

3

6

4 →+=

++= x

x x

x p

Therefore

=

=

1011000

1110100

0110010

1100001

101

111

011

110

Gand P

Block Codes – Codewords Generation (Example)

Consider a (7,4) linear block code, where

=

1011000

1110100

01100101100001

G

Show that the codeword c = [1 0 1 1 0 0 1] for input m = [1 0 1 1].

[ ]

[ ]1001101

)1(1)1(1)0(0)1(1

),0(1)1(1)1(0)1(1),1(1)1(1)1(0)0(1

),1(1)0(1)0(0)0(1),0(1)1(1)0(0)0(1

),0(1)0(1)1(0)0(1),0(1)0(1)0(0)1(1

1011000

1110100

0110010

1100001

1101,

=

⊕⊕⊕

⊕⊕⊕⊕⊕⊕

⊕⊕⊕⊕⊕⊕

⊕⊕⊕⊕⊕⊕

=

== mGcCodeword




Page 21 of 31

Block Codes – Error Detection with No Error (Example)

Assuming the information is m = [1 0 1 1].Hence the data with FECC to be transmitted is c = [1 0 1 1 0 0 1]

If there is no error in transmission, error vector, e = [0 0 0 0 0 0 0]

The received vector,

x = c ⊕ e

= [ 1 0 1 1 0 0 1 ] ⊕ [ 0 0 0 0 0 0 0 ]= [ 1 0 1 1 0 0 1 ]

=> x = c

Calculate s.

T xH s =

where

I P H T |=

As calculated earlier

=⇒

=

1101

0111

1110

101111

011

110

T P P

Therefore

[ ]

=⇒

==

100

010

001

101

111

011

110

1001101

0100111

0011110

| T T H I P H

[ ]

=

100

010

001

101

111

011

110

1001101 s




Page 22 of 31

s = [ 1(0) ⊕ 0(1) ⊕ 1(1) ⊕ 1(1) ⊕ 0(1) ⊕ 0(0) ⊕ 1(0),

1(1) ⊕ 0(1) ⊕ 1(1) ⊕ 1(0) ⊕ 0(0) ⊕ 0(1) ⊕ 1(0),

1(1) ⊕ 0(0) ⊕ 1(1) ⊕ 1(1) ⊕ 0(0) ⊕ 0(0) ⊕ 1(1) ]

s = [ 0 0 0 ]

Block Codes – Error Detection with 1-bit Error (Example)

Now, assuming the 4th bit is corrupted at the receiving end.In other words, the error vector e = [0 0 0 1 0 0 0]

This, the received vector,

x = c ⊕ e

= [ 1 0 1 1 0 0 1 ] ⊕ [ 0 0 0 1 0 0 0 ]= [ 1 0 1 0 0 0 1 ]

=> x ≠ c

The position of the erroneous bit can be identified by matrix calculation of s and withreference to H

T .

[ ]

==

100

010

001101

111

011

110

1000101

T

xH s

s = [ 1(0) ⊕ 0(1) ⊕ 1(1) ⊕ 0(1) ⊕ 0(1) ⊕ 0(0) ⊕ 1(0),

1(1) ⊕ 0(1) ⊕ 1(1) ⊕ 0(0) ⊕ 0(0) ⊕ 0(1) ⊕ 1(0),

1(1) ⊕ 0(0) ⊕ 1(1) ⊕ 0(1) ⊕ 0(0) ⊕ 0(0) ⊕ 1(1) ]

s = [ 1 0 1 ]

Referring to H T where

=

100

010

001

101

111

011

110

T H

It is thus identified that the erroneous bit is bit 4.




Page 23 of 31

Block Codes – Error Detection with > 1-bit Error (Example)

Now, assuming the 2nd bit and 4th bit are corrupted at the receiving end.In other words, the error vector e = [0 1 0 1 0 0 0]

This, the received vector,x = c ⊕ e

= [ 1 0 1 1 0 0 1 ] ⊕ [ 0 1 0 1 0 0 0 ]= [ 1 1 1 0 0 0 1 ]

=> x ≠ c

Try to identify the erroneous bits!

[ ]

==

100

010

001

101111

011

110

1000111T xH s

s = [ 1(0) ⊕ 1(1) ⊕ 1(1) ⊕ 0(1) ⊕ 0(1) ⊕ 0(0) ⊕ 1(0),

1(1) ⊕ 1 (1) ⊕ 1(1) ⊕ 0(0) ⊕ 0(0) ⊕ 0(1) ⊕ 1(0),

1(1) ⊕ 1 (0) ⊕ 1(1) ⊕ 0(1) ⊕ 0(0) ⊕ 0(0) ⊕ 1(1) ]

s = [ 0 1 1 ]

Referring to H T where

=

100010

001

101

111

011

110

T H

Oops! Something is wrong! => We have hit the limitation of (7, 4) block code!

(Refer to section Block Codes – Principles)




Page 24 of 31

5.4.2 FECC – Convolutional Codes

• Encoding of continuous information stream rather than information blocks.

• Easy implementation using shift register

=> Assuming k inputs and n outputs

• The encoder has memory and stores the past input data in the shift registers and performsmodulo-2 sum of “some” of the past inputs with the current input to form the output bits.

• The encoder is written as (n, k, K) encoder, where K is the constraint length defined as the“number of shift registers+1”.

•

Generator vectors are commonly used to represent the actual behaviour of the encoder.

• Several algorithms exist for decoding convolutional codes. For relatively small values of K , Viterbi Algorithm is widely used. Longer constraint length codes are more practicallydecoded with any of several sequential decoding algorithms.

Convolutional Codes – Encoder

• The following example shows a (2,1,3) convolutional encoder• (n, k, K) = (2, 1, 3) => 2 outputs, 1 input, 3 constraint length

• with generator vectors g1 = [1 0 1] and g2 = [1 1 0]

•

From the above figure•

There are two generator vectors (each is K bits long) corresponding to the (n=2)output streams.

• The bits from left to right represent the connections in the encoder circuit.• A ‘1’ represent a link (which performs the modulo-2 sum) and• A ‘0’ represent no link.

• Depending on its generator vectors, A (2,1,3) encoder may take various forms

• Generators are often expressed in octal format by grouping 3 bits together.• Therefore, the generator for the above example is (101

2, 110

2) = (5

8,6

8).




Page 25 of 31

Convolutional Codes – State Table

• To describe the bahaviour of a convolutional encoder when a particular bit stream isapplied to the input, a time-indexed state table may be handy.

•

The following example shows the state table when the bit stream [1 0 0 1 1 0] is appliedto the input of the convolutional encoder where

• (n, k, K) = (2, 1, 3)• Generator = (58,68)

(In other word generator vectors g1 = [1 0 1] and g2 = [1 1 0])

Time index, i 1 2 3 4 5 6

Encoder State, m1, m2 00 10 01 00 10 11

Input bit, x 1 0 0 1 1 0

Output, c = y1 y2 11 01 10 11 10 11

• To generally describe the behaviour of a convolutional encoder, three alternative methodsare often used:

•

State Diagram• Trellis Diagram

•

Tree Diagram

Convolutional Codes – State Diagram

• Operation of a convolutional encoder can be described by a state diagram. The state of theencoder is defined as the content of its shift register.

•

The following example shows the state diagram of the convolutional encoder where• (n, k, K) = (2, 1, 3)• Generator = (58,68)

(In another word generator vectors g1 = [1 0 1] and g2 = [1 1 0])

0 1 1 0 0 1




Page 26 of 31

Convolutional Codes – Trellis Diagram

•

The trellis diagram is another way to present the general behaviour of a convolutionalencoder. In addition, it can be used to show the path of an actual input sequence.

•

Referring to the convolutional encoder described above, if a bit stream of [1 0 0 1 1] isapplied to the input. The Trellis diagram would looks like…

1 1 0 0 1




Page 27 of 31

Convolutional Codes – Tree Diagram

• Just like the Trellis diagram, a Tree diagram can present the general behaviour of aconvolutional encoder as well as showing the path of an actual input sequence.

•

Referring to the convolutional encoder described above, if a bit stream of [1 0 0 1 1] isapplied to the input. The Tree diagram would looks like…

y1 y2




Page 28 of 31

5.4.3 FECC – Interleaving

a1, a2, a3, a4, a5, a6, a7, a8, a9, …Input Data

a1, a2, a3, a4a5, a6, a7, a8a9, a10, a11, a12a13, a14, a15, a16

WriteR ead

Interleaving

a1, a5, a9, a13, a2, a6, a10, a14, a3, …TransmittingData

a1, a2, a3, a4a5, a6, a7, a8a9, a10, a11, a12a13, a14, a15, a16

Read

r it

e

De-Interleaving

a1, a2, a3, a4, a5, a6, a7, a8, a9, …Output Data




Page 29 of 31

Note: It needs to be noted that in source coding techniques, removing the redundancy in thesignal reduces the data rate.For instance, in voice coding, low-bit rate coding techniques reduce the redundancy.In contrast, in error correcting codes, redundancy is introduced to facilitate errorcorrection at the receiver.

0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0,…Transmitting

Data

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

Read

r ite

De-Interleaving

0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, …Output Data

Burst error

Discrete error




Page 30 of 31

Appendix 1

Program for calculation of CRC-16.

#include <stdio.h>

#include <stdlib.h>

#include <string.h>long CRC = 0x0000;

long GenPolynomial = 0x8005; //Divisor for CRC-16 Polynomial

void bitBybit(int bit);

int main()

{

unsigned int MsgLength;

int i=0,j=0;

char SampleMsg[] = "Hello World";

char tempBuffer[100];

MsgLength = sizeof(SampleMsg)-1;

printf("\nActual Message: %s\n",SampleMsg);

strcpy(tempBuffer, SampleMsg);

tempBuffer[MsgLength] = 0x00;

tempBuffer[MsgLength+1] = 0x00;

tempBuffer[MsgLength+2] = '\0';

printf("\nAfter padding 16 0-bits to the Message:");

for(i=0;i<MsgLength+2;++i){

unsigned char ch = tempBuffer[i];

unsigned char mask = 0x80;

for(j=0;j<8;++j)

{

bitBybit(ch&mask);

mask>>=1;

}

printf(" ");

}

printf("\n\nCalculated CRC:0x%x\n\n",CRC);

return 0;

}

void bitBybit(int bit)

{

long firstBit = (CRC & 0x8000);

CRC = (CRC << 1);

if(bit)

{

CRC = CRC ^ 1;

printf("1");

}

else

{

CRC = CRC ^ 0;

printf("0");

}

if(firstBit)

{

CRC = (CRC^GenPolynomial);

}

}

In this listing, the actual message to be transmitted is "Hello World". The message is paddedwith sixteen 0 bits, and the message bit stream is

01001000 01100101 01101100 01101100 01101111 00100000 0101011101101111 01110010 01101100 01100100 00000000 00000000

The calculated CRC value in hexadecimal notation is 0x303f70c3.




Appendix 2

Program for calculation of CRC-32.

#include <stdio.h>#include <stdlib.h>

#include <string.h>

long CRC = 0x00000000L;

long GenPolynomial = 0x04c11db7L; //Divisor for CRC-32 Polynomial

void bitBybit(int bit);

int main()

{

unsigned int MsgLength;

int i=0,j=0;

char SampleMsg[] = "Hello World";

char tempBuffer[100];

MsgLength = sizeof(SampleMsg)-1;

printf("\nActual Message: %s\n",SampleMsg);

strcpy(tempBuffer, SampleMsg);

tempBuffer[MsgLength] = 0x00;




tempBuffer[MsgLength+4] = '\0';

printf("\nAfter padding 32 0-bits to the Message:");

for(i=0;i<MsgLength+4;++i)

{

unsigned char ch = tempBuffer[i];

unsigned char mask = 0x80;

for(j=0;j<8;++j)

{

bitBybit(ch&mask);

mask>>=1;

}

printf(" ");

}

printf("\n\nCalculated CRC:0x%x\n\n",CRC);

return 0;}

void bitBybit(int bit)

{

long firstBit = (CRC & 0x80000000L);

CRC = (CRC << 1);

if(bit)

{

CRC = CRC ^ 1;

printf("1");

} else

{

CRC = CRC ^ 0;

printf("0");

}

if(firstBit)

{ CRC = (CRC^GenPolynomial);

}

}

This listing gives the C program to calculate CRC-32. In this program the message for whichCRC has to be calculated is "Hello World". The message bit stream is

01001000 01100101 01101100 01101100 01101111 00100000 0101011101101111 01110010 01101100 01100100 00000000 00000000 0000000000000000

The calculated CRC is 0x31d1680c.

Documents

Error Detection and Correction