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Error Control

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Why Error Control

Number of errors due to data transmission are orders ofmagnitude greater than those caused by hardware failures

Probability of bit error for internal circuits < 10-15

Probability of errors on an optical fiber link ~ 10-9

6 orders of magnitude more than hardware errors Probability of errors on a coaxial cable link ~ 10-6

Probability of errors on a switched telephone line ~ 10-4--10-5

What is the difference between error rates of 10-15 and 10-4

At transmission rate of 9600 bits/sec:

10-15 one error every 3303 years of continuous operation10-4 one error every second!

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Effects and Causes of Errors

Effects depend on line and network characteristics. Data may be:

Reordered

Distorted

Deleted

Inserted (due to noisy lines)

Two main causes of errors:

Linear distortion of the original data

E.g., caused by bandwidth limitations of the raw data channel

Non-linear distortion

Caused by echoes, cross-talk, white noise, and impulse noise

Effect of these errors can be remedied to some extent Cable insulations

Hardware compensation filters

Errors that remain have to be caught in software

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Error Characteristics of a Data Line

Characteristics related to average behavior:

Long-term average bit error rate

Characteristics related to overall line/network quality:

Percentage of time that the average bit error rate does not exceed

a given threshold value Percentage oferror-free seconds

Average bit error rate is commonly used to design an errorcontrol method

Provides an indication of performance

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Effective Error Control Mechanism

Should match the error characteristics of the channel

If channel introduces insertion errors, no need for a protocol thatprotects against deletion errors

If a channel produces independent, single bit errors, even a

simplest parity scheme will be effective If error rate of the channel is lower than peripheral equipment,

inclusion of any error control scheme results in performancedegradation

No error control method should be expected to catch all errors Goal is to increase the reliability of transmissions up to a required

level

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Error Model

Assume that probability of error is randomly distributed over asequence of bits Let a long-term average bit error rate of a channel =p

Probability ofnsubsequent bit errors in a message =pn

Probability of one or more bit errors in a message with nbits = 1-(1-p)n

Formal model of such a channel is called discrete memoryless channel This model ignores the affect of impulse noise

Discrete: because channel recognizes only a finite number of distinctsignal levels

Memoryless: because the probability of an error is assumed to beindependent of all occurrences of previous errors

Memoryless channel is also called a symmetric channel Other type is asymmetric channel:

Probability of an error may depend on the value of signal transmitted

Error distribution process may have memory i.e., if last nbits were in error, thenext few bits will likely be in error too

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Error Free Interval of Symmetric Channel

We can predict error free interval (EFI) using an error model

Expressed as probability of a series of at least ncontiguous error-free bittransmissions

Using binary symmetric channel model with long-term average bit errorrate ofb, the EFI is given as:

This means that the probability decreases linearly with length of interval

To express exponential decrease of probability with interval, we canuse Poisson distribution as:

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EFI (Contd)

Which model is more realistic: binary or Poisson

Need empirical studies

Such studies show that Poisson model gives a better errorprediction

A still better match can be found by adding a correction factor Benoit Mandelbrot approximation:

The parameter adetermines the impact of clustering effect

When a=0, model predicts same EFI as with Poisson distribution For non-zero a, probability of longer EFIs decreases more than the

probability of shorter EFIs

This affect is more pronounced with growing a

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Types of Transmission Errors Five general types:

Insertion

Deletion

Duplication

Distortion

Reordering

Insertions and deletions may be caused by the temporary loss ofsynchronization between sender and receiver Deletion errors may also be caused by inadequate flow control

Example: receiver may run out of buffer space to hold incoming messagesand may have to discard them

Duplication may be intentional

Example: a protocol that uses retransmissions Data reordering may occur if data are routed via many different routes

Sequencing problems (deletion, duplication, and reordering) are solvedby proper flow control Schemes

In case ofdata distortion or insertion, we need methods to verify theconsistency of the data

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Redundancy

Redundancy in messages is required for error detection

Two approaches: Forward error control

Large redundancy in message to help receiver reconstruct originalmessage

Corresponding transmission codes are called error-correcting codes

Feedback error control Based on retransmission request from receiver e.g., using NAK

Corresponding transmission codes are called error-detecting codes

Purpose of error control: reduce channel error rate

Not all errors can be detected There is always a residual error rate

If probability of a transmission error in a message =pandfraction of errors caught by the error control method = f, Then, residual error rate =p(1-f)

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How To Select An Error Control Scheme

General rule:

Ifp0, an error-correcting code is a bad choice

It merely slows down data transfer

Ifp1, a retransmission scheme is a bad choice

Almost every message will have to be retransmitted This includes retransmitted messages

Exceptions to this rule are possible

Ifpis small and cost of retransmission is high, a forward errorcontrol scheme may still be cost-effective

A combination of forward and feedback schemes may be used: Receiver corrects frequently occurring errors

Receiver asks for retransmissions of messages with less frequent errors

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Types of Error-Correcting and Error-Detecting Codes

Two basic types: block and convolution codes

Block codes

All code words have same length

Encoding for each data message can statically be defined

Convolution codes Code word depends on data message and a given number of

previously encoded messages

Encoder changes its state with processing of each message

Length of the code words is usually constant

Other categorization of types of codes: linear, cyclic, andsystematic codes

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Linear, Cyclic, and Systematic Codes

Linear codes:Output is a linear combination of valid code words e.g., modulo-2 sum,produces another valid code word

Cyclic codes:A code in which every cyclic shift of a valid code word results in

another valid code word Systematic codes:

A code in which each code word includes data bits from originalmessage unaltered, either preceded or followed by a group of checkbits

In all cases, code words are longer than original data Let number of original data bits = dand number of additional bits = e

Code rate = d/(d+e)

Improving quality of a code high redundancy lower code rate

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Parity Check

Effective when probability of multiple bit errors is low

Useful for a binary symmetric channel

Add a single bit to every message that makes modulo-2 sum ofall bits equal to 1

Overhead of only one bit Error detection:

If any bit, including the check bit, is distorted, parity at the receiverwill come out to be wrong

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Residual Error Rate of a 1-bit Parity Check

Residual error rate (RER) of a binary symmetric channel:

Let probability of an error-free bit transmission = q = 1-p

Probability of error-free transmission of an n+1bits message = q(n+1)

Probability of a single bit error in n+1bits = (n+1).p. qn

Residual error rate of a 1-bit parity check = 1 - q

(n+1)

-(n+1).p.qn

This is equal to probability of non 1-bit errors

Example:

n = 15 and p = 10-4

Then RER is of the order of 10-6 per message or about 10-7 per bit

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RER of a 1-bit Parity Check: Example

Residual error rate of a binary symmetric channel per code word

increases as a function of bit error ratep When p0, parity check code significantly reduces error rate

Difference between corrected and uncorrected code ismaximum at p = 0.06

n=15

Error rate without parity bit: 1-q

n

Prob.of 1-bit error: (n+1)pqn

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Forward Error Correction

Forward error control scheme uses a small subset of availablebit combinations to encode messages

Codes are chosen such that it takes relatively large number of biterrors to convert one valid message into another

Receiver tries to correct a distorted message by mapping it ontothe closest valid message in that scheme

Closest = one that differs from received code word in fewest # of bits

Code rate is generally lower than that of an error-detectingcode

Useful when communication of control messages from receiverback to sender is difficult:

A very long transmission delay

Absence of a return channel

High bit-error rate

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Correcting 1-bit Errors Using LRC and VRC

Simple extensions of parity check per code word Longitudinal Redundancy Check (LRC):

Additional parity bit with a sequence of 7 bits new code word

Parity bit is called LRC

Vertical Redundancy Check (VRC) An extra sequence of 8 bits after a series of n code words

Each bit in this sequence works as parity for bits that occupy same position in ncode words

Example: ASCII coding for n=4:

Code rate = 28/(12+28) = 0.7

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Hamming distance

Minimum difference between two words in a code

The difference between two code words is the number of bits in whichthey differ

A code with a hamming distance ofn

any combination of up to n-1bit errors can be detected

any combination of up to (n-1)/2errors per code word can becorrected (if receiver interprets a non-valid code as the closest validcode)

Receiver will guess wrong for higher numbers of bit errors (probabilitylow overall error rate of channel may be reduced)

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Linear Block Code

Based on inserting a block ofccheck bits to protect mbits:

nmessages to be encoded

To be able to correct all single bit errors, we need Hammingdistance of at least 3 m+c+1= 2c

Example: Hamming code

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Bursts and Code Interleaving

In practice, transmission errors tend to come in bursts

Code interleaving is used to counter burst errors

Assume messages ofnbits each, protected against single bit errors

Assume traffic is non-interactive

Sender buffers each block ofksubsequent messages

Sender places these kmessages in a matrix of (k x n) bits

Sender transmits the bits in this matrix column by column

Receiver restores the matrix column by column

Receiver read row by row from matrix to get messages

Intercept burst errors up to a length ofkbits

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Cyclic Redundancy Checks (CRC)

Based on the addition of series of check bits to code words.

In absence of transmission errors, (code word + check bits) isdivisible by a given factor

Division method and factor used determine range of transmissionerrors that can be detected

A generator polynomial is used to form the checksum for a codeword

Example: Code word 10011 defines the polynomialx4+x+1.

Several standardized generator polynomials are available (e.g., CRC-12)

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Arithmetic Codes

A simpler version of CRC that uses only arithmetic operations(addition and modulo operations)

Much less overhead to compute the checksum

Good error detection capability

A version has been adopted for the ISO Class 4 transport protocolstandard (TP4)

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Summary Error Control

Motivation

Error Model

Types of Transmission Errors

Redundancy

Types of Error-Correcting and Error-Detecting Codes

Linear, Cyclic, and Systematic Codes

Parity Check and Residual Error Rate

Forward Error Correction

Linear Block Code Cyclic Redundancy Checks (CRC)

Arithmetic Codes

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