1 Block Coding Messages are made up of k bits. Transmitted packets have n bits, n > k: k-data bits...

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Block Coding

Messages are made up of k bits.

Transmitted packets have n bits, n > k:k-data bits and r-redundant bits.

n = k + r

Modulo-2 Arithmetic

Addition and subtraction are described by the logical exclusive-or operation.

Modulo-2 Arithmetic (logical xor)

Chapter 10

Error Detection and

Correction

Interference

Example causes of interference:HeatNoise due to interference (EM fields)AttenuationDistortion

Interference causes bit errors.

Bit Error Classifications

Single bit error

Burst error

Single Bit Error

Single bit errorA one is interpreted as a zero (or vice-versa)Refers to only one bit modified in a specified

transmission unit of dataAn uncommon type of error for serial data due to

the duration of a bit being much less than the duration of interference.

Figure 10.1 Single-bit error

Burst Error

More than one bit is damaged by interference.

Figure 10.2 Burst error of length 8

Error Detection

Detection of errors is necessary to determine if data should be rejected.

Possible responses to error detection include:Retransmission request Forward error correction (corrected on the receiving

end)Forward correction saves bandwidth & time

Redundancy

Extra bits can be included with the data transmission to assist in the detection and correction of errors

– For digital transmissions that implies using block-coding

Redundant Bits

Detection vs Correction

Detection is easier than correction

Data Coding Schemes

Block coding – described in this course

– Convolution coding – this is covered in advanced Math or EE signal processing courses.

Block Coding

Data-words are made up of k bits.

Codewords are made up of k data bits (a data-word) and r redundant bits.

n = k + r

Where n is the number of bits in a codeword

Figure 10.5 Datawords and codewords in block coding

Examples

No block coding for 10base-T Ethernet, Manchester coding.

4B/5B block coding (Fast Ethernet), 8B6T or MLT-3 line coding.

8B/10B block coding (Gigabit Ethernet), PAM-5 line coding.

Error Detection With Block Coding

• The receiver can detect an error if

– The receiver has a list of valid code words

– A received codeword is not a valid code word

Table 10.2 A code for error correction (Example 10.3)

Possible Transmission Outcomes

• A codeword is sent and received without incident

• A codeword is sent but is modified in transmission. The error is detected if the codeword is not in the valid codeword list.

• A codeword is sent but is modified in transmission. The error is not detected if the resulting new codeword is in the list of valid codewords.

Error-Detecting Code

• Error detecting code can only detect the types of errors it was designed to detect. Other types of errors may go undetected.

The Hamming Distance

• The Hamming Distance, d(x,y), between two words of the same size is the number of differences between corresponding bits.

• Examples

d(000,011) = 2

d(10101,11110) = 3

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

The minimum Hamming distance is the smallest Hamming distance between all possible pairs in a set of words.

Table 10.1 A code for error detection (Example 10.2)

Find the minimum Hamming distance of the coding scheme in Table 10.1.

SolutionWe first find all Hamming distances.

Example 10.5

The dmin in this case is 2.

Table 10.2 A code for error correction (Example 10.3)

Find the minimum Hamming distance of the coding scheme in Table 10.2.

SolutionWe first find all the Hamming distances.

The dmin in this case is 3.

Example 10.6

To guarantee the detection of up to s errors in all cases, the minimum

Hamming distance in a block code must be dmin = s + 1.

Hamming Code Notation

What is the maximum number of detectable errors for each of the two previous coding schemes?

– d-min = 2, s =

– d-min = 3, s =

Figure 10.8 Geometric concept for finding dmin in error detection

To guarantee correction of up to t errors in all cases, the minimum Hamming distance in

a block code must be dmin = 2t + 1.

Figure 10.9 Geometric concept for finding dmin in error correction

How many errors can be corrected for the two example coding schemes:

d-min = 2, t =

d-min = 3, t =

To guarantee correction of up to t errors in all cases, the minimum Hamming distance in

a block code must be dmin = 2t + 1.

Example

• A code scheme has dmin = 5.

– What is the maximum number of detectable errors?

– What is the maximum number of correctable errors?

Linear Block Codes

• A linear block code is a code where the logical exclusive-or of any two valid codewords creates another valid codeword.

Linear Block Codes

• The min Hamming distance for a LBC is the minimum number of ones in a non-zero valid codeword.

Exercise:

For each of the next two examples,

Compute the • Hamming distance,

• the number of detectable errors,

• number of correctable errors

• Show that the code is linear

Table 10.1 A code for error detection C(3,2)

Table 10.2 A code for error correction C(5,2)

Parity Check

• Count the number of ones in a data word.

• If the count is odd, the redundant bit is one

• If the count is even, the redundant bit is zero

A simple parity-check code is a single-bit error-detecting

code in which n = k + 1 with dmin = 2.

A simple parity-check code can detect an odd number of errors.

Table 10.3 Simple parity-check code C(5, 4)

How Useful is a Parity Check?

• Detecting any odd number of errors is pretty good, can we do better?

2-Dimensional Parity Check

• It is possible to create a 2-D parity check code that detects and corrects errors.

Figure 10.11 Two-dimensional parity-check code

Interleaving

● By interleaving the columns into slots, it becomes possible to

● detect up to n-row errors.● The example is 70%efficient. The efficiency can

be improved by adding more rows.

Cyclic Codes

• If a codeword is shifted cyclically, the result is another codeword.

– (highest order bit becomes the lowest order bit)

– Cyclic codes are linear codes

C(7,4)

Assume d-min = 3.

Answer the following about table 10.6:

● What is the codeword size?● What is the data-word size?

Table 10.6 is this C(7, 4) cyclic?

C(7,4)

Is the table 10.6 code cyclic?

Is it linear?

What is d-min?

How many detectible errors?

How many correctible errors?

Hamming Codes

d-min >= 3

Minimum number of detectable errors: 2

Minimum number of correctable errors: 1

For C(n,k), • n = 2^r – 1

• r = n – k (number of redundant bits)

• r >= 3 (author uses m = r)

Which are Hamming Codes?

C(7,4)

C(7,3)

Error Correction

1. Using CRC codes computing bit syndromes

2. Using interleaving with multiplexing.

– Use a parity bit in each frame

– Check for invalid code words

(see example exercises)

Checksum

• Adding the codewords together at the source and destination.

• If the sum at the source and destination match, there is a good chance that no errors occurred.

• Checksums are not as reliable as the CRC.

Checksum Examples

• 1's complement

• 16 bit checksum used by the Internet

Sender: Checksum Calculation

1. Divide into 16 bit unsigned words

2. Add the 16 bit words using 1’s complement addition.

3. Complement the total

4. Send all the above words.

Receiver: Checksum Calculation

1. Divide into 16 bit words

1.Add the 16 bit words and the checksum value using 1’s complement arithmetic.

2.The complement of the total should be zero.

Example: Sender

0x466f

0x726f

0x757a

+_______

0x12e58 partial sum

0x2e59 sum

0xd1a6 complement

Example: Receiver

0x466f

0x726f

0x757a

0xd1a6 (sender’s checksum)

+_______

0x1fffe (partial sum)

0xffff (sum)

0x0000 (complement)

1 Block Coding Messages are made up of k bits. Transmitted packets have n bits, n > k: k-data bits...

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