IN THE NAME OF GOD

COMPUTER NETWORKS

CHAPTER 3: THE DATA LINK LAYER (PART2)

Dr. Shahriar Bijani

Shahed University

March 2014

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References: A. S. Tanenbaum and D. J. Wetherall,

Computer Networks (5th Edition), Pearson Education, the book slides, 2011.

Chapter 6, Data Communications and Computer Networks: A Business User's Approach, 6th Edition

B. A. Forouzan, Data Communications and Networking, 5th Edition, Behrouz A. Forouzan, McGraw Hill, lecture slides, 2012.

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ERROR DETECTION AND CORRECTION

Noise is always present White Noise

(thermal or Gaussian noise)

Impulse Noise

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ERROR DETECTION AND CORRECTION

Two basic strategies to deal with errors:1. Include enough redundant information to enable the receiver

to deduce the original data: Error correcting codes.

2. Include only enough redundancy to allow the receiver to deduce that an error has occurred (but not which error): Error detecting codes.

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ERROR DETECTION & CORRECTION CODE

1.Hamming codes.2.Binary convolutional codes.3.Reed-Solomon codes.4.Low-Density Parity Check codes.

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All the codes presented in the previous slide add redundancy to the sent information.

A frame consists of m data bits (message) and r redundant bits (check).

Block code - the r check bits are computed solely as function of the m data bits with which they are associated. the m bits were looked up in a large table to find their corresponding r check bits.

Systemic code – the m data bits are send directly along with the check bits (rather than being encoded).

Linear code – the r check bits are computed as a linear function of the m data bits. XOR or modulo 2 addition is a popular choice.

ERROR DETECTION & CORRECTION CODE

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ERROR DETECTION & CORRECTION CODE

n – total length of a block (i.e., n = m + r) (n, m) code n –bit codeword containing n bits. m/n – code rate (range ½ for noisy channel and

close to 1 for high-quality channel).

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ERROR DETECTION & CORRECTION CODE

Example Transmitted: 10001001 Received: 10110001

XOR operation gives number of bits that are different. XOR: 00111000

Hamming Distance: the number of bit positions in which two codewords differ.

It shows that two codes are d distance apart = d errors to convert one into the other.

Minimum Hamming distance: the smallest Hamming distance between all possible pairs in a set of words.

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ERROR DETECTION & CORRECTION CODE

All 2m possible data messages are legal, but due to the way the check bits are computers not all 2n possible code words are used.

Only small fraction of 2m/2n=1/2r of possible messages will be legal codewords.

The error-detecting and error-correcting codes of the block code depend on this Hamming distance.

To reliably detect d error, we need a distance d+1 code. To correct d' error: we need a distance 2d' +1 code.

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ERROR DETECTION & CORRECTION CODE

Example: 4 valid codes:

0000000000 0000011111 1111100000 1111111111

The Minimal Distance is 5 => can correct 2 errors and detect 4 errors.

0000000111 => single or double – bit error. Hence the receiving end must assume the original transmission was 0000011111.

0000000000 had triple error => 0000000111 received The error can only be detected.

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ERROR DETECTION & CORRECTION CODE

Error correction requires evaluation of each candidate

codeword which may be time consuming search.

Through design this search time can be minimized.

In theory if n = m + r, a lower limit on the number of

check bits needed to correct single errors:

(m + r + 1) ≤ 2r

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1. THE HAMMING CODE Create the codeword:

1. Check bits (parity bits): All bit positions that are powers of 2: (p1, p2, p4, p8, p16, …).

2. The rest of the bit positions are filled with m data bits: (m3, m5, m6, m7, m9, m10, m11, m12, m13,…)

3. Each parity bit calculates the parity for some of the bits in the code word. The position of the parity bit determines the sequence of bits that it alternately checks and skips. Position 1: check 1 bit, skip 1 bit, check 1 bit, skip 1 bit, etc. (1,3,5,7,9,11,13,15,...)Position 2: check 2 bits, skip 2 bits, check 2 bits, skip 2 bits, etc. (2,3,6,7,10,11,14,15,...)Position 4: check 4 bits, skip 4 bits, check 4 bits, skip 4 bits, etc. (4,5,6,7, 12,13,14,15, 20,21,22,23,...)Position 8: check 8 bits, skip 8 bits, check 8 bits, skip 8 bits, etc. (8-15, 24-31, 40-47,...)etc.

4. Set a parity bit to 1 if the total number of ones in the positions it checks is odd. Set a parity bit to 0 if the total number of ones in the positions it checks is even.

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HAMMING CODE: EXAMPLE

m = 4 data bits (D) and r = 4 check bits, n = 7-bit codeword:

• This would be called a (7,4) code.

• The 3 bits to be added are 3 EVEN Parity bits (P), where the parity of each

is computed on different subsets of the message bits as shown below:

7 6 5 4 3 2 1

D

D

D P D

P P 7-BIT CODEWORD

D - D - D - P (EVEN PARITY)

D D - - D P - (EVEN PARITY)

D D D P - - - (EVEN PARITY)

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For example, the message 1101 would be sent as 1100110, since:

7 6 5 4 3 2 1

1 1 0 0 1 1 0 7-BIT CODEWORD

1 - 0 - 1 - 0 (EVEN PARITY)

1 1 - - 1 1 - (EVEN PARITY)

1 1 0 0 - - - (EVEN PARITY)

HAMMING CODE: EXAMPLE

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HAMMING CODE: PARITY CIRCLES

When these 7 bits are entered into the parity circles, it can be confirmed that the choice of these 3 parity bits ensures that the parity within each circle is EVEN:

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If an error occurs in any of the seven bits, it will affect different combinations of the three parity bits depending on the bit position.

E.g. a single bit error occurs:

transmitted message received message 1 1 0 0 1 1 0 1 1 1 0 1 1 0 BIT No: 7 6 5 4 3 2 1 BIT No.: 7 6 5 4 3 2 1

The above error (in bit 5) can be corrected by examining which of the three parity bits was affected by the bad bit:

HAMMING CODE: EXAMPLE

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7 6 5 4 3 2 1

1 1 1 0 1 1 0 7-BIT CODEWORD

1 - 1 - 1 - 0 (EVEN PARITY) NOT! 1

1 1 - - 1 1 - (EVEN PARITY) OK! 0

1 1 1 0 - - - (EVEN PARITY) NOT! 1

received message:

• The bad parity bits labeled 101 point directly to the bad bit since 101 binary equals 5.

• Examination of the 'parity circles' confirms that any single bit error could be corrected in this way.

HAMMING CODE: ERROR DETECTION

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Example of an (11, 7) Hamming code correcting a single-bit error.

HAMMING CODE: ERROR DETECTION

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HAMMING CODE: SUMMARY

The value of the Hamming code:

1. Detection of 2 bit errors (assuming no correction is

attempted);

2. Correction of single bit errors;

3. Cost of 3 bits added to a 4-bit message.

The ability to correct single bit errors comes at a cost which is

less than sending the entire message twice. (Recall that

simply sending a message twice accomplishes no error

correction.)

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2. ERROR DETECTION & CORRECTION:CONVOLUTIONAL CODES Not a block code There is no natural message size or encoding boundary as in

a block code. The output depends on the current and previous input bits.

Encoder has memory. Constraint length of the code: the number of previous bits on

which the output depends. They are deployed as part of the

GSM mobile phone system Satellite Communications, and 802.11 (see example in the previous slide).

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CONVOLUTIONAL ENCODERS

A convolutional encoder is a linear system.

A binary convolutional encoder can be represented as a shift register.

The outputs of the encoder: modulo 2 sums of the values in the certain register's

cells.

The input to the encoder is either the unencoded sequence (for non-recursive

codes) or the unencoded sequence added with the values of some register's

cells (for recursive codes).

Convolutional codes can be systematic or non-systematic.

Systematic codes: an unencoded sequence is a part of the output sequence.

Almost always recursive

Non-recursive codes are almost always non-systematic.

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CONVOLUTIONAL ENCODERS A combination of register's cells that forms one of the output

streams (or that is added with the input stream for recursive codes) is defined by a polynomial.

m: the maximum degree of the polynomials forming a code, then K =m+1 is a constraint length of the code.

E.g. the polynomials of Figure 1:

g1(z)=1+z+z2+z3+z6

g2(z)=1+z2+z3+z5+z6

Figure 1: A standard NASA convolutional encoder with polynomials (171,133).

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CONVOLUTIONAL ENCODERS: EXAMPLE 1

Example:

g1(z)=1+z+z2+z3+z6

g2(z)=1+z2+z3+z5+z6

A code rate is an inverse number of output polynomials. For the sake of clarity, here we restrict ourselves to the codes

with rate R=1/2. Decoding procedure for other codes is similar.

Encoder polynomials are usually denoted in the octal notation. For the above example: “1111001” = 171 and “1011011” = 133.

The constraint length of this code is 7.

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CONVOLUTIONAL ENCODER: EXAMPLE 2

Figure 2. A recursive convolutional encoder.

An example of a recursive convolutional encoder is on the Figure 2.

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TRELLIS DIAGRAM A convolutional encoder is often seen as a finite state machine.

Each state corresponds to some value of the encoder's register. Given the input bit value, from a certain state the encoder can move to two other states.

A solid line= input 0, a dotted line = input 1 (the rightmost bit is the newest one).

Any valid sequence from the encoder's output can be represented as a path on the trellis diagram. One of the possible paths is denoted as red (as an example).

Figure 3. A trellis diagram corresponding to the encoder on the Figure 2.

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TRELLIS DIAGRAM Each state transition on the diagram corresponds to a pair of

output bits. There are only 2 allowed transitions for every state (2 allowed

pairs of output bits, and the 2 other pairs are forbidden) If an error occurs, it is very likely that the receiver will get a set

of forbidden pairs, which don't create a path on the trellis diagram.

So, the task of the decoder is to find a path on the trellis diagram which is the closest match to the received sequence.

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VITERBI ALGORITHM

A convolutional code is decoded by finding the sequence of

input bits that is most likely to have produced the observed

sequence of output bits (which includes any errors).

Viterbi algorithm reconstructs the maximum-likelihood path for

a given input sequence.

The input sequence requiring the fewest errors at the end is

the most likely message.

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3. ERROR DETECTION & CORRECTION: REED-SOLOMON

Like Hamming codes, Reed-Solomon codes are linear block codes, and they are often systematic too.

Unlike Hamming codes, which operate on individual bits, Reed-Solomon codes operate on m bit symbols.

based on the fact that every n degree polynomial is uniquely determined by n + 1 points.

Example ax + b is determined by two points. Extra points on the same

line are redundant, which is helpful for error correction. 2 data points represent a line. we send those two data points

plus two check points on the same line. If one of the points is received in error, we can still recover the

data points by fitting a line to the received points. 3 points will lie on the line, and 1 error point will not.

By finding the line we have corrected the error

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ERROR-DETECTING CODES

Linear, systematic block codes

1.Parity.

2.Checksums.

3.Cyclic Redundancy Checks (CRCs).

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1. PARITY BITS

Detects 1-bit errors and some 2-bit errors Not reliable against bursty errors

Idea: add extra bits to keep the number of 1s even Example: 7-bit ASCII characters + 1 parity bit

0101001

1 0 1 1 11011110

0110100

1101001

0001110110

TWO DIMENSIONAL PARITY

Can detect all 1-, 2-, and 3-bit errors, some 4-bit errors 14% overhead

010100111010011011110000111001101001011111

101110

1111011 0

Parity bit for each

row

Parity bit for each column

Parity bit for the

parity byte

2. CHECKSUMS Idea:

Add up the bytes in the data Include the sum in the frame

Use ones-complement arithmetic Lower overhead than parity: 16 bits per frame But, not resilient to errors

Why? Used in UDP, TCP, and IP

DataSTART ENDChecksum

0101001

1101001= 10010010+ 01

3. CYCLIC REDUNDANCY CHECK (CRC)

Uses field theory to compute a semi-unique value for a given message

In a cyclic code, rotating a codeword always results in another codeword

Example:

Much better performance than previous approaches Fixed size overhead per frame (usually 32-bits) Quick to implement in hardware Only 1 in 232 chance of missing an error with 32-bit CRC

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CRC ENCODER/DECODER

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CYCLIC REDUNDANCY CHECK (CRC)

Example calculation of the CRC