csc 535 communication networks i section 3.8 error detection and correction dr. cheer-sun yang

CSC 535 Communication Networks I

Section 3.8Error Detection and Correction

Dr. Cheer-Sun Yang

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Data Link Layer FunctionsFramingError detection and correction Flow controlError recovery (frame level)

3

Error DetectionAdditional bits added by transmitter for

error detection codeParity

Value of parity bit is such that character has even (even parity) or odd (odd parity) number of ones

Even number of bit errors goes undetected

4

Cyclic Redundancy CheckFor a block of k bits transmitter generates

n bit sequenceTransmit k+n bits which is exactly

divisible by some numberReceive divides frame by that number

If no remainder, assume no error For math, see Stallings chapter 7

5

Error ControlForward Error Correction (FEC): Detection

and correction of errors FEC is appropriate when a return channel is not

available, retransmission requests are not easily accommodated, or retransmission is not efficient.

Automatic repeat request (ARQ) Error detection Positive acknowledgment Retransmission after timeout Negative acknowledgement and retransmission

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Error Control TechniquesParity checking - a single bit error

detectionInternet checksumpolynomial codes (cyclic redundency

checking) - single bit error correction and multiple bit error detection

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Automatic Repeat Request (ARQ)Stop-and-wait ARQGo-Back-N ARQSelective Repeat ARQ ( also known as

selective reject or selective retransmission)

Copyright 2000 McGraw-Hill Leon-Garcia and Widjaja Communication Networks

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ChannelEncoderUserinformation

PatternChecking

All inputs to channel satisfy pattern/condition

Channeloutput Deliver user

informationor

set error alarm

Figure 3.49


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Calculate check bits

Channel

Recalculate check bits

Compare

Information bits Received information bits

Check bits

Information accepted if check bits match

Received check bits

Figure 350


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x = codewords o = non-codewords

x

x x

x

x

x

x

o

oo

oo

oo

o

oo

o

oxx x

x

xx

x

o oo

oo

ooooo

o

o

A code with poor distance properties A code with good distance properties(a) (b)

Figure 3.51


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1 0 0 1 0 0

0 1 0 0 0 1

1 0 0 1 0 0

1 1 0 1 1 0

1 0 0 1 1 1

Bottom row consists of check bit for each column

Last column consists of check bits for each row

Figure 3.52


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1 0 0 1 0 0

0 0 0 0 0 1

1 0 0 1 0 0

1 1 0 1 1 0

1 0 0 1 1 1

1 0 0 1 0 0

0 0 0 0 0 1

1 0 0 1 0 0

1 0 0 1 1 0

1 0 0 1 1 1

1 0 0 1 0 0

0 0 0 1 0 1

1 0 0 1 0 0

1 0 0 1 1 0

1 0 0 1 1 1

1 0 0 1 0 0

0 0 0 1 0 1

1 0 0 1 0 0

1 0 0 0 1 0

1 0 0 1 1 1

Two errors

One error

Three errors

Four errors

Arrows indicate failed check bits

Figure 3.53


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unsigned short cksum(unsigned short *addr, int count){

/*Compute Internet Checksum for “count” bytes * beginning at location “addr”.*/

register long sum = 0;while ( count > 1 ) {

/* This is the inner loop*/ sum += *addr++; count -=2;}

/* Add left-over byte, if any */if ( count > 0 )

sum += *addr;

/* Fold 32-bit sum to 16 bits */while (sum >>16)

sum = (sum & 0xffff) + (sum >> 16) ;

return ~sum;}

Figure 3.54


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(x7 x6 1) (x6 x5 ) x7 (1 1)x6 x 5 1

x7 x5 1

(x 1)(x2 x 1) x3 x 2 x x2 x 1 x3 1

Addition:

Multiplication:

Division: x3 + x + 1 ) x6 + x5

x3 + x2 + x

x6 + x4 + x3

x5 + x4 + x3

x5 + x3 + x2

x4 + x2

x4 + x2 + x

x

= q(x) quotient

= r(x) remainder

divisordividend

35 ) 1223

10517

Figure 3.55


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Steps:

1) Multiply i(x) by xn-k (puts zeros in (n-k) low order positions)

2) Divide xn-k i(x) by g(x)

3) Add remainder r(x) to xn-k i(x) (puts check bits in the n-k low order positions):

quotient remainder

transmitted codewordb(x) = xn-ki(x) + r(x)

xn-ki(x) = g(x) q(x) + r(x)

Figure 3.56


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Generator polynomial: g(x)= x3 + x + 1Information: (1,1,0,0) i(x) = x3 + x2

Encoding: x3i(x) = x6 + x5

1011 ) 1100000

1110

1011

1110

1011

10101011

010

x3 + x + 1 ) x6 + x5

x3 + x2 + x

x6 + x4 + x3

x5 + x4 + x3

x5 + x3 + x2

x4 + x2

x4 + x2 + x

x

Transmitted codeword:b(x) = x6 + x5 + xb = (1,1,0,0,0,1,0)

Figure 3.57


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g (x ) x 3 x 1

reg 0 reg 1 reg 2++

g3 1

i (x )

g0 1 g1 1 i (x ) x 3 x 2

Encoder for

clock input reg 0 reg 1 reg 20 - 0 0 01 1=i3 1 0 0

2 1=i2 1 1 0

3 0=i1 0 1 1

4 0=i0 1 1 1

5 0 1 0 16 0 1 0 07 0 0 1 0check bits: r0 = 0 r1 = 1 r2 = 0

r(x) = x

Figure 3.58


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b(x)

e(x)

R(x)+ (Receiver)(Transmitter)

Error pattern

Figure 3.59


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1. Single errors: e(x) = xi 0 i n-1If g(x) has more than one term, it cannot divide e(x)

2. Double errors: e(x) = xi + xj 0 i < j n-1

= xi (1 + xj-i )If g(x) is primitive, it will not divide (1 + xj-i ) for j-i 2n-k1

3. Odd number of errors: e(1) =1 If number of errors is odd.If g(x) has (x+1) as a factor, then g(1) = 0 and all

codewords have an even number of 1s.

Figure 3.60


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4. Error bursts of length b: 0000110 • • • • 0001101100 • • • 0

e(x) = xi d(x) where deg(d(x)) = L-1g(x) has degree n-k; g(x) cannot divide d(x) if deg(g(x))>

deg(d(x))

L = (n-k) or less: all will be detected L = (n-k+1): deg(d(x)) = deg(g(x)) i.e. d(x) = g(x) is the only undetectable error pattern, fraction of bursts which are undetectable = 1/2L-2

L > (n-k+1): fraction of bursts which are undetectable

= 1/2n-k

L

ithposition

error pattern d(x)

Figure 3.61


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b

e

r+ (receiver)(transmitter)

error pattern

b

e

r+ (receiver)(transmitter)

error pattern

(a) Single bit input

(b) Vector input

Figure 3.62


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0010000

s = H e = =101

single error detected

0100100

s = H e = = + =011

double error detected100

1 0 1 1 1 0 01 1 0 1 0 1 00 1 1 1 0 0 1

1110000

s = H e = = + + = 0 110

triple error notdetected

011

101

1 0 1 1 1 0 01 1 0 1 0 1 00 1 1 1 0 0 1

1 0 1 1 1 0 01 1 0 1 0 1 00 1 1 1 0 0 1

111

Figure 3.63


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s = H r = He

s = 0 s = 0

no errors intransmission

undetectableerrors

correctableerrors

uncorrectableerrors

(1-p)7 7p3

1-3p 3p

7p

7p(1-3p) 21p2

Figure 3.64


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If dmin= 2t+1, non-overlapping spheres of radius t can be drawn around each codeword; t=2 in the figure

b1 b2o o o o

set of all n-tupleswithin distance t

set of all n-tupleswithin distance t

Figure 3.66


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b1 b2 b3 b4 bL-3 bL-2 bL-1 bL. . .

L codewordswritten verticallyin array; thentransmitted rowby row

b1 b2 b3 b4 bL-3 bL-2 bL-1 bL

. . .

A long error burst produceserrors in two adjacent rows

Figure 3.66

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Required ReadingSection 3.8

csc 535 communication networks i section 3.8 error detection and correction dr. cheer-sun yang

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mcgrawhill leongarcia

xffff sum

long sum

addr count

bits transmitter

failed check bitsfigure

int count

count bytes