dlc slides1

Chapter 3

Data Link Layer & Data Link Layer & Error Control CodingError Control Coding

ECE/CSC 570, NC State University

DATA LINK CONTROL (DLC)

DLC

FRAMEHEADERSEQUENCENUMBERS

TRAILERCRC

OBJECTIVE: PROVIDE ANERROR FREE BIT PIPEPACKET

PACKET


DATA LINK CONTROL (2)

DLC

MAC

PHYSICAL

DLC

MAC

PHYSICAL

UNRELIABLEBIT PIPE

RELIABLEBIT PIPE


Functions of the Data Link Layer (DLL)

n Provide service interface to the network layer

n Dealing with transmission errors error-control coding & retransmission

n Regulating data flowslow receivers not swamped by fast sendersdone by ARQ..


Data Link Layer Design Issues

n Service provided to the Network Layern Framingn Error Controln Flow Control


Packets vs. Frames

Relationship between packets and frames.


Services Provided to Network Layer

(a) Virtual communication.(b) Actual communication.


Service types

n Unacknowledged connectionless servicen Acknowledged connectionless servicen Acknowledged connection-oriented service


Services

n Unacknowledged connectionless service Source machine sends independent frames without having the

destination machine acknowledge them No logical connection is established If a frame is lost due to noise on the link, no detection or

recovery Appropriate for reliable links and leave the recovery to higher

layers. (LANs)

n Acknowledged connectionless service No logical connection is established Each frame sent is individually acknowledged Retransmit if time out Useful for unreliable channels (wireless systems!) Delay of a packet is long if many frames are retransmitted

Regular bulk postal mail

certified postal mail


Services (2)

n Acknowledged connection-oriented service Connection is established before any data are transferred Each frame is numbered and guaranteed to be received Each frame is received only once and all frames are

received in the right order. Provides the network layer processes with the equivalent of

a reliable bit stream

Reliable pipe !


Framing

n How to decide the successive frames start and stop ?

n Inserting time gaps between frames will not work. Why?

n Three types of framing used in practiceCharacter-based countFlag bytes with byte stuffingStarting and ending flags, with bit stuffing


Framing: Character-based count

A character stream. (a) Without errors. (b) With one error.

A header to specify the no. of characters in the frame

Even the destination knows that the frame is bad, it still has no way of telling where the next frame starts. Resending a frame back to the source

asking for retransmission does not help either. Rarely use any more


Framing: Flag bytes with byte stuffing

(a) A frame delimited by flag bytes.(b) Four examples of byte sequences before and after stuffing.

Use flag bytes to separate the frames

What if the data has the same byte pattern as flag bytes? Insert ESC as byte stuffing, but limited to 8-bit characters.


Framing: Starting/ending flags with bit stuffing

Bit stuffing

(a) The original data.

(b) The data as they appear on the line.

(c) The data as they are stored in receivers memory after de-stuffing.

a flag byte = 01111110

Delete 0 at the receiver

Many DLL protocols use combined method!

Error Control CodingError Control Coding


Error Detection and Correction

n The primary function of the DLC (Data Link Control) is to convert the unreliable bit pipe into a reliable bit pipe.

n This can be done by using FEC (Forward Error Correction) or ARQ (Automatic Repeat reQuest) for retransmission.

n The FEC relies upon some form of Error Control Coding and the ARQ relies on Error Control Coding but also on the introduction of Sequence Number(SN) and Request Numbers (RN).


Error Detection and Correction

n Error-Correcting CodesHamming Code

n Error-Detecting CodesParity CheckCyclic Redundancy Check (CRC)


Properties n Errors on a link tend to be dependent and occur in

burstsn For any reasonable code, the frequency of

undetectable errors is very small and thus difficult to measure and difficult to model analytically.

n The minimum distance of the coden The burst detecting capabilityn The probability that a completely random string will

be accepted as error free


Definitions n Codeword : n-bit unit (n = m + r ) containing m data and

r redundant, or check bits, is referred to as an n-bit codeword

n The minimum distance of the code : the smallest number of errors that can convert one codeword into another

n The length of a burst of errors : the number of bits from the first error to the last error (inclusive). The burst error detecting capability of a code is defined as the largest integer B such that a code can detect all bursts of length B or less.


Definitions (2)

n Hamming Distance d: the number of bit positions in which two codewords differGiven any two codewords, exclusive OR (XOR)

100010011011000100111000

n It requires d single-bit errors to convert one codeword to another.

0 for same values1 for different values


n The probability that all k bits in a message are received correctly is equal to ( )1- pe

k

0

1

0

11- pe

1- pe

pepe

Binary Symmetric Channel


Bit Error Raten A simple error correction coding method is to send every

packet three times.n Three values are then received for each bit and the receiver

uses a majority vote on each bit.n Evaluate the effectiveness of this method assuming that all

the bits errors are independent. The probability that a given bit is incorrectly received two or three

times is equal to e = 3 Pe2 (1-Pe) + Pe3

Therefore, the probability that at least one of the N bits of the packet is received incorrectly two or three times is equal to

PER (packet error rate) = 1 (1 e)N Ne 3N Pe2


Packet Error Rate (Contd)

n Compare the PERFor N = 105 and BER =Pe= 10-7, the PER of one time

transmission is 1- (1-BER)N N BER = 10-2.The PER of three times transmission is

PER Ne 3N Pe2 = 3 10-9

n Price? The useful transmission rate has been reduce by a factor

of 3.


Single Parity Coder

n Parity check append a single bit to the data. If the number of 1 bits in the codeword is even (or odd), a bit 0 (1) is appended. E.g., 1011010, a bit 0 will be added to the end and it makes

10110100 It can be used to detect single errors. Minimum distance of the code ?

1 2 ks s s 1 2 ks s s cCODER

where c = s1 s2 L sk , the modulo-2 sum


njjjjj

ikiiii

ssssrssssc

L

L

==

321

321

Horizontal and Vertical Parity Checksl A two-dimensional array with one parity check

for each row and one for each column.

s11 s12 s13 L s1k c1s21 s22 s23 L s2k c2s31 s31 s33 L s3k c3M M M M M M

sn1 sn2 sn3 L snk cnr1 r2 r3 L rk cn +1


njjjjj

ikiiii

ssssr

ssssc

L

L

=

=

321

321

Detects all single, double and triple errors and most quadruple but not all!

Horizontal and Vertical Parity Checks (2)

1 0 0 1 0 1 0 10 1 1 1 0 1 0 0 1 1 1 0 0 0 1 01 0 0 0 1 1 1 00 0 1 1 0 0 1 11 0 1 1 1 1 1 0

Horizontal checks

Vertical checks

1 0 0 1 0 1 0 10 1 1 1 0 1 0 0 1 1 1 0 0 0 1 01 0 0 0 1 1 1 00 0 1 1 0 0 1 11 0 1 1 1 1 1 0

If the bit in each circle is changed, all parity checks are still satisfied


Observationsn To detect d errors, we need a distance d+1 code so

that d single-bit errors cannot change a codeword to another.

n To correct d errors, we need a distance 2d+1 code so that even d-bit errors, the original codeword is still closer than any other codewords. E.g., there are 4 valid codewords: 0000000000, 0000011111,

1111100000, 1111111111. This code has distance of 5 (2d +1= 5). If a codeword 0000000111 arrives, the closest valid codeword is

0000011111 (2-bit error, d = 2). Then the receiver can determine the original codeword. If there are 3-bit errors, this code is unable to correct.


Hamming Code (Error Correcting Code)n The bits of the codeword are numbered consecutively,

starting from bit 1 at the left end. 1 2 3 4 5 6 7

Original code 1 0 0 1 0 0 0n The bits that are powers of 2 (1,2,4,8) are check bits, so

the codeword becomes1 2 3 4 5 6 7 8 9 10 11

Codeword x x 1 x 0 0 1 x 0 0 0 n The rest (3,5,6,7) are filled up with the m (=7) data bitsn Any k that is not a power of 2 can be written as a sum of

powers of 2. Data bit in k will contribute to some of the check bits. E.g., k = 11 = 1+2+8, which means it contributes to the check bits 1, 2, 8.


Hamming Code: Example

n Consider a (7,4) code: 4 bits message+3 parity bits 7 bits codeword 3 = 1+2 : 5 = 1+4 : 6 = 2+4 : 7 = 1+2+4

4(Even Parity)DDDP---

2(Even Parity)DD--DP-

1(Even Parity)D-D-D-P

Check bit contributed7-bit codewordDDDPDPP

7654321


Hamming Code: Example (contd)

4(Even Parity)1100---

2(Even Parity)11--11-

1(Even Parity)1-0-1-0

Check bit contributed7-bit codeword1100110

7654321

n Rule: Each circle must have EVEN parity.

n Message 1011 would be sent as 0110011, since


Hamming Code: Example (contd)n Single bit error (bit) occurs: 0110011 0110111

n The bad parity bits labeled 101 point directly to the bad bit (101 = 5 !)

(NOT!) 1(Even Parity)1110---

(OK!) 0(Even Parity)11--11-

(NOT!) 1(Even Parity)1-1-1-0

Parity computed7-bit codeword1110110

7654321


Hamming Code: Example (contd)

n Correction of single bit errorsn Detection of 2 bit errors (assuming no

correction is attempted)n Cost of 3 bits added to a 4-bit message.n In general, Hamming distance = 3The minimum distance between any two valid

codewords is 3 allows error correction of one bit and detection of 2 bits.


Hamming Code (contd)

Use of a Hamming code to correct burst errors.

When a codeword arrives, the receiver initializes a counter to zero. It then examines each check bit, k, to see if it has the correct parity. If k = 0 after the whole examination, the codeword is accepted. Otherwise, it contains the number of the incorrect bit, (1+2+8 = 11) Hamming codes can only correct single errors. To correct burst errors, transmit one column at a time.


Cyclic Redundancy Check (CRC)n The basic idea is to start with a bit string and to generate

parity checks on various subsets of the bits.n The most common code is the polynomial code Bit string is represented by a polynomial with coefficients of 0 and 1

only. A k-bit frame is regarded as a polynomial with k terms, ranging from

xk-1 to x0 . It is called a polynomial of degree (k-1) If the data bits are denoted as mK-1, mK-2, , m1,m0, the polynomial

M(x) representation of the string with coefficients is

M(x) = mK-1xK-1 + mK-2xk-2 + , +m1x + m0


CRC (Contd)n Both the sender and receiver agree upon a generator

polynomial, G(x) of degree r.

n G(x) = xr+gr-1xr-1++g1x+1 G(x) has at least two nonzero terms (i.e., xr and 1)

n Rule of the game: r(x) = Remainder[M(x)xr /G(x)] This is a ordinary long division of one polynomial by another, except

that the coefficients are restricted to be binary, and the arithmetic on coefficients is performed modulo 2.

n What is modulo 2 arithmetic?

n The CRC polynomial T(x) = M(x)xr r(x) = M(x)xr+r(x) The remainder is of degree at most r-1.

n If there is no error, T(x) must be divisible (modulo) by G(x)


Example of Polynomial

Example: r=3 and k=32 3

2

G(x) 1 x xM(x) 1 x

= + +

= +( m2m1m0=101)

x x3 2 1+ + x x5 3+x 2

x x x5 4 2+ +x x x4 3 2+ +

+ x

x x x4 3+ +x x2 + Remainder

Code-polynomial T(x) = x5 + x3 +x2 + x


Error-Detecting Codes

Calculation of the polynomial code checksum.

1. Add (r=4) zero bits at the end of frame polynomial

2. Divide by G(x)3. Subtract (or add) the remainder

from the bit string corresponding to xrM(x)

11010110110000- 1110

11010110111110


Error Detection with CRCn T(x) is divisible by G(x), so the remainder should be zero for correct

transmission.

1101011011111010011

1001110011

00001000000001000000

00101000000101100000

10111100110100100000

10011

100111100001010

Assume there is an error occurs, so the received bit string is T(x) + E(x) (in module 2!!)

Each 1 bit in E(x) corresponds to a bit that has been inverted. If there are k 1 bits in E(x), k single-bit errors have occurred.


Error Detection with CRC (2)

n If no errors occur, then E(x) = 0 remainder =0n E(x) 0 is undetectable if and only if

E(x) = G(x) z(x), where z(x) is nonzero polynomial.

])()([Re

])()(

)()([Re])(

)([Re

xGxEmainder

xGxE

xGxTmainderxG

xYmaider

=

+=

T(x) Y(x)=T(x)+E(x)Channel


n If there is a single error, then E(x)=xj for some j . Since G(x) has at least two non- zero terms xr and 1,it follows that G(x) z(x) must also have two non-zero terms for any non-zero z(x) . Consequently G(x)z(x)cannot equal E(x) .

This means that all single errors can be detected.

Single Error


Double Errors

n We know that xj is not divisible by G(x). So only if xi-j +1 is divisible by G(x) will we fail to detect double errors. G(x) is carefully chosen such that it does not divide xk+1 for any kup to the maximum value of (i-j) (the max. frame length) a primitive polynomial

Double errors can be detected.

n E.g., x15+x14+1 will not divide xk+1 for any value of k below 32,768!

jixxxxxE jijji >+=+= - where)1()(


CRC Propertiesn In most CRC implementations the G(x) used is of the form

G(x)=(x+1) { primitive polynomial of degree r-1 }.n A polynomial E(x) is divisible by x+1 if and only if E(x)

contains an even number of non-zero coefficients. If E(x) has an odd number of terms, substituting 1 for x everywhere will always yield 1 as the result. Thus, no polynomial with an odd number of terms is divisible by x+1.

n This nice property ensures that all odd number of errors can be detected.


CRC Properties (Contd)

n The degrees of the highest and lowest order terms of G(x) differ by r.

n For G(x)z(x), it must be that the difference between the highest and lowest degree is at least r. Thus if E(x) is a error polynomial, then the burst length of errors must be at least r+1. This means that all bursts of length r can be detected because the burst stream has 1 at the first and last bits.


Standards

n Standard CRC polynomials: G(x) = x16+x15+x2+1 CRC-16G(x) = x16+x12+x5+1 CRC-ITU-TG(x)

=x32+x26+x23+x22+x16+x12+x11+x10+x8+x7+x5+x4+x2+x1+1 IEEE 802


Block Codesn The previous discussion is about how to transform

a message of k bits into a codeword of n (n=k+r) bits with n > k. Accordingly, there are 2k codewordsin a set of 2n binary words.

n A good code is one where the codewords are as far from each other as possible.

n Easy to use: coding and decoding. These codes are called block codes because they group the source bits into blocks of k bits that they encode and decode independently of one anotherone word at a time.

n Probability of failing to detect errors in random string = 2-r


Practical Implementations of CRC

Feedback Shift Register

rM(x) xR(x) Rem{ }G(x)

=

g g g1 2 m -1In p u t

++ + + . . .

How the long division (by modulo 2) is done?


Example: r = 3 and k = 32 3

2

G(x) 1 x xM(x) 1 x

= + +

= +

Input 101000

First bit into the circuit

Initial values

Remainder 110011111101010001000

+ +Input

r(x) = x2+x


Other Error-Detecting Codes

n Convolutional Codesn Decoding:Maximum likelihood detectionTrellis decoding

n Usually involves number theory!

dlc slides1

Documents

nc state universityfunctions

nc state universitypackets

nc state universityframingn

data link layer dlln

use flag bytes

logical connection

independent frames

successive frames