MSIS 4523 Ch10.Error Detection Correction_Short

8/3/2019 MSIS 4523 Ch10.Error Detection Correction_Short

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Data Communications SystemsCh 10: Error Detection and Correction

JinKyu Lee, Ph.D.

[email protected]

Include the course code (MSIS4523) in every email subject!!

Layer2~4


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Topics

Error Detection vs. Correction vs. Retransmission

Block Codes Hamming Distance

Exclusive OR (XOR) operation

Linear Block Codes, Parity Check, Hamming Codes

Cyclic Codes

Cyclic Redundancy Check (CRC)

Checksums

Internet Checksum


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Error checking methods

Block codes

Cyclic codes

Checksums


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Single-bit error

Burst error of length 8

An error-detecting code can detectonly the types of errors for which it is

designed; other types of errors may

remain undetected.


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The structure of encoder and decoder


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Block Coding

In block coding, we divide our message into blocks,each of k bits, called datawords. We add r redundant

bits to each block to make the length n = k + r. The

resulting n-bit blocks are called codewords

Since n > k, the number of possible codewords isgreater than the number of datawords


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Datawords and codewords in block coding


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Process of error detection in block coding


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Table 10.1 A code for error detection (Example 10.2)


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Let us assume that k = 2 and n = 3. Table 10.1 shows thelist of datawords and codewords. Later, we will see how

to derive a codeword from a dataword.

Assume the sender encodes the dataword 01 as 011 and

sends it to the receiver. Consider the following cases:

1. The receiver receives 011. It is a valid codeword. Thereceiver extracts the dataword 01 from it.

Example 10.2


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2. The codeword is corrupted during transmission, and

111 is received. This is not a valid codeword and is

discarded.

3. The codeword is corrupted during transmission, and000 is received. This is a valid codeword. The receiver

incorrectly extracts the dataword 00. Two corrupted

bits have made the error undetectable.

Example 10.2 (continued)


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Error Correction

is NOT asking again when an error isdetected.

Receiver needs to be aware that an error

is present (error detection); then with the available data from the

sender, must identify the error and correct

it

Much more difficult than detection


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Process of error detection in block coding

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Structure of encoder and decoder in errorcorrection


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Let us add more redundant bits to Example 10.2 to see

if the receiver can correct an error without knowing

what was actually sent!

Add 3 redundant bits to the 2-bit dataword to make 5-bit

codewords (Table 10.2). If the dataword is 01, thesender creates the codeword 01011. The codeword is

corrupted during transmission, and 01001 is received.

Can we correct the error?What if we assume that there is only 1 bit corrupted?

Example 10.3


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Table 10.2 A code for error correction (Example 10.3)


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1. Comparing the received codeword with the first

codeword in the table (01001 versus 00000), the receiver

decides that the first codeword is not the one that was

sent because there are two different bits.

2. By the same reasoning, the original codeword cannot bethe third or fourth one in the table.

3. The original codeword must be the second one in the

table because this is the only one that differs from the

received codeword by 1 bit. The receiver replaces 01001with 01011 and consults the table to find the dataword

01.

Example 10.3 (solution)


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Hamming Distance

The Hamming Distance between two words (of

the same size) is the number of differences

between corresponding bits

The Minimum Hamming Distance is thesmallest Hamming Distance possible between

all pair in a set of words


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Modular Arithmetic

Need to embedded error detection in the message

Block coding adds redundant data to message; thenconducts a mathematical calculation

Modulo-N arithmetic uses only the integers in the range 0to N-1, inclusive

Often Modulo-2 is used with an XOR (exclusive OR)function If two bits are the same, result is a 0

If bits are different, result is 1


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XORing of two single bits or two words


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Let us find the Hamming distance between two pairs of

words.

1. The Hamming distance d(000, 011) is 2 because

Example 10.4

2. The Hamming distance d(10101, 11110) is 3 because


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Solution

We first find all Hamming distances.

Example 10.5 minimum Hamming distance of the coding scheme in Table 10.1?

The dmin in this case is 2.


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Parameters for Hamming Distances

To guarantee the detection of up to n errors inall cases, the minimum Hamming distance in ablock code must be

dmin = n + 1.

To guarantee correction of up to n errors in allcases, the minimum Hamming distance in ablock code must be

dmin >= 2n + 1.

dmin = minimum hamming distances

# of errors that can be detected = dmin -1

# of errors that can be corrected


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The minimum Hamming distance for our first code

scheme (Table 10.1) is 2. This code guarantees

detection of only a single error. For example, if the

third codeword (101) is sent and one error occurs, the

received codeword does not match any validcodeword. If two errors occur, however, the received

codeword may match a valid codeword and the errors

are not detected.

Example 10.7


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Our second block code scheme (Table 10.2) has dmin

= 3.

This code can detect up to two errors. Again, we see that

when any of the valid codewords is sent, two errors create a

codeword which is not in the table of valid codewords. The

receiver cannot be fooled.

However, some combinations of three errors change a

valid codeword to another valid codeword. The receiver

accepts the received codeword and the errors areundetected.

Example 10.8


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A code scheme has a Hamming distance dmin = 4. What is

the error detection and correction capability of this

scheme?

Solution

This code guarantees thedetection of up tothree errors

(s = 3), but it cancorrect up toone error.

If this code is used for error correction, part of its

capability is wasted. Error correction codes need to

have an odd minimum distance (3, 5, 7, . . . ).

Example 10.9


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10-3 LINEAR BLOCK CODES10-3 LINEAR BLOCK CODES

Almost all block codes used today belong to a subsetcalledlinear block codes. A linear block code is a code

in which the exclusive OR (XOR) of two valid

codewords creates another valid codeword.

Do the two codes in Table 10.1 and Table 10.2 belong to theclass of linear block codes? YES!


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Linear Block Codes

In a linear block code, the exclusive OR (XOR) of any

two valid codewords creates another valid codeword The Minimum Hamming Distance is the number of 1s

in the nonzero codeword with the smallest number of

1s. A simple parity-check code is a single-bit error-

detecting code

in which c = d+ 1 with dmin = 2.

* c = Length(codewords), d= Length(datawords)


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Parity Check

A simple parity-check code can detect an oddnumber of errors

Parity checks can be two dimensional

Note an even number of 1s is represented by 0

and an odd number of 1s is represented by 1 inparity bits The resulting codewords will always have even # of 1s!


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Simple parity-check code C(5, 4)


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Encoder and decoder for simple parity-check code


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Assume the sender sends the dataword 1011. The codeword

created from this dataword is 10111, which is sent to thereceiver.

1. No error occurs. 10111 is received.

The syndrome is 0. The dataword 1011 is created.

2. One single-bit error changes a1. - 10011 is received.

The syndrome is 1. No dataword is created.

3. One single-bit error changes r0. - 10110 is received.The syndrome is 1. No dataword is created.

Example 10.12


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4. An error changes r0 and a second error changes a3.

- 00110 is received.

The syndrome is 0. The dataword 0011 is created.

* Note that here the dataword is wrongly created

due to the syndrome value.

5. Three bitsa3, a2, and a1are changed by errors.

- 01011 is received.

The syndrome is 1. The dataword is not created.

The simple parity check, guaranteed to detect one single

error, can also find any odd number of errors.

Example 10.12 (continued)


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Two-dimensional parity-check code


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0 0 0


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0

0

0

0 0

00


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Hamming Codes

Originally designed with c = 7, d = 4, dmin=3

, where c = Length(codewords), d=

Length(datawords)

(See Table 10.4 )

Used for error correction Can detect two errors

Can correct 1 error


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Hamming code C(7, 4)


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The structure of the encoder and decoder for a Hamming code


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Generating & Checking Parity Bits

The parity check bits handle 3 of the 4 bits in the

dataword r0 = a2 + a1 + a0 modulo -2

r1 = a3 + a2 + a1 modulo -2

r2 = a1 + a0 + a3 modulo -2

The decoder creates a 3-bit syndrome in which each bit isthe parity check for the 4 (3 data+1 parity) of the 7 bitsreceived s0 = b2 + b1 + b0 + q0 modulo -2

s1 = b3 + b2 + b1 + b1 modulo -2 s2 = b1 + b0 + b3 + b2 modulo -2

Should always be 0s if no error occurred.


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Dataword: 1 1 0 1 0 1 1Dataword: 1 0 0 1 0 1 0

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Logical decision made by the correction logic analyzer

Hamming Code Error Correction Examples

Senders sidea3 a2 a1 a0 r2 r1 r0

Dataword: 0 0 0 1Dataword: 1 0 0 1

Receivers sideb3 b2 b1 b0 q2 q1 q0 s2 s1 s0

Dataword: 0 0 0 1 1 0 1Dataword: 1 0 0 1 0 1 1

1 0 1

1 1 0

0 0 0

0 0 0

1 1 0

1 0 0


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A CRC code with C(7, 4)


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CRC encoder and decoder


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Division in CRC encoder


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Division in the CRC decoder for two cases


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Checksum

An error detection method used in the

Internet: Internet Checksum Uses 1s Complement

1. Suppose our data is a list of five numbers (7, 11, 12, 0, 6) that

we want to send to a destination. The sum of these numbers is

36.2. When the sender transmits these numbers, it can include -36,

the negative of the sum (the complement), at the end of the five

numbers. In this case, we send (7, 11, 12, 0, 6, 36). The

attached 6th number is called the checksum.

3. The receiver can add all the received numbers including the

checksum. If the result is 0, it assumes no error, so accepts the

five numbers and discards the checksum. Otherwise, there is

an error somewhere, and the data are not accepted.


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Checksum concept


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Using a limited space: 4-bit word-wise Addition & Complements

0

1 111

1


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Sender site:1. The message is divided into 16-bit words.2. The value of the checksum word is set to 0.

3. All words including the checksum are

added using ones complement addition.4. The sum is complemented and becomes the

checksum.

5. The checksum is sent with the data.

Internet Checksum


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Receiver site:1. The message (including checksum) is

divided into 16-bit words.

2. All words are added using ones

complement addition.3. The sum is complemented and becomes the

new checksum.

4. If the new checksum value is 0, the

message is accepted; otherwise, it is

rejected.

Internet Checksum

I


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The header is divided

into 16-bit sections.

All the sections are

added and the sum iscomplemented. The

result is inserted in

the checksum field.

Internet

Checksum for

IP Header

(Sender side)

4

11

10001011 10110001

35,761

35761

Once the checksum iscalculated, the checksum

is plugged into the

checksum field, and then

the packet is sent off.

I t t


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10001011 10110001

35,761

11111111 1111111100000000 00000000

35761

On the receivers side

4Internet

Checksum for

IP Header

(Receiver side)

Date post:	07-Apr-2018
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