Lecture Error

7/31/2019 Lecture Error

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1

Error Detection and Correction


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2

Handling Errors (1): Ignore Them!

data in

data out

Tx -DATA

frame

Rx

Some errors are not significant

audio

video


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3

Handling Errors (2): Do something

data in

data out

Lets meet at1pm tomorrow Tx

-DATAframe

RxLets meet at

3pm tomorrow

Most errors are significant must be detected

If only detect errors Reverse Error Correction (REC)

must ask for retransmission ARQ protocol

unsuited to real-time applications

unsuited to simplex communication

If detect and correct errors Forward Error Correction (FEC)


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4

A Simple REC System

data in


-

-

Lets meet at

3pm tomorrow

Lets meet at1pm tomorrow

NAK

Rx

Send data twice

Rx delivers data if two copies identical and generates ACK-frame

Rx sends NAK-frame if two copies differ


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5

A Simple FEC System

data in data out


-

-

-

Lets meet at

3pm tomorrow



RxLets meet at

1pm tomorrow

Send data thrice

Rx delivers the majority


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6

Defeating Error Correction

data in data out


-

-

-

Lets meet at

3pm tomorrow



RxLets meet at

3pm tomorrow

More severe errors may not be detected


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7

The Basis for Error Handling

add redundancy to data to form codewords

handle two types of errors:

random errors bit error rate (BER)

error bursts up to B bits in error

Codewords should have enough redundancy to detect probable errors


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9

Coordination Failure with Communication Failure

H1

m1 -H2

...

H1mn1 H2

H1mn - H2


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10

Hamming Distance

- b00 1

- b0

6

b1

00 01

10 11

-b0

6b1

>b2

000 001

010 011

100 101

110 111

HD is number of bits between codewords

e.g. 0002 is HD of two from 0112, 1012, and 1102

Error detection/correction based on HD between valid codewords


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11

Hamming Distance of Two

000 001

010 011

100 101

110 111

000 001

010 011

100 101

110 111

One bit error will change valid codeword to invalid codeword

Two bit error will change valid codeword into different valid codeword


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Hamming Distance of Three

000 001

010 011

100 101

110 111

One bit error will change valid codeword to invalid codeword

invalid code HD of one from only one valid codeword

Two bit error will change valid codeword to invalid codeword


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Using Hamming Distance

To reliably detect up to e bits in error

bits in error must be less than HD

e = HD 1

HD = e + 1

To reliably detect and correct up to e bits in error

bits in error must be less than halfway of HD

e = HD 1

2

HD = 2e + 1

5


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15

Parity Bits

codeword

r m6 m5 m4 m3 m2 m1 m0

r = m0 +2 . . . +2 m6

ASCII X (10110002)

1 1 0 1 1 0 0 0

modulo arithmetic

no carry

0 +2 0 = 0

1 +2 0 = 1

0 +2 1 = 1

1 +2 1 = 0

this defines even parity

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16

Parity Bits

codeword

r m6 m5 m4 m3 m2 m1 m0

r = m0 +2 . . . +2 m6

ASCII X (10110002)

1 1 0 1 1 0 0 0

1 1 0 0 1 0 0 0

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Parity Bits

codeword

r m6 m5 m4 m3 m2 m1 m0

r = m0 +2 . . . +2 m6

ASCII X (10110002)

1 1 0 1 1 0 0 0

1 1 0 1 1 0 1 0

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18

Parity Bits

codeword

r m6 m5 m4 m3 m2 m1 m0

r = m0 +2 . . . +2 m6

ASCII X (10110002)

1 1 0 1 1 0 0 0

0 1 0 1 1 0 0 0

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Odd and Even Parity

000 001

010 011

100 101

110 111 even parity

r = m0 +2 . . . +2 mn

hardware: XOR returns 0

by eye: even number of one bits

000 001

010 011

100 101

110 111 odd parityr = m0 +2 . . . +2 mn + 1

hardware: XOR returns 1

by eye: odd number of one bits

Dont need to know which bit is redundant bit

Implements HD=2 reliably detects one bit errors

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Worksheet: Hamming Distance and Parity

1. Draw arcs between the codewords HD=1

1011 1100 1001

1111 1010 1000

0111 0000

2. If1000 is to be a valid codeword, shade codewords which maintain HD=2.

3. What type of parity do codewords have?

4. If1010 was received, with single bit, what codewords might have been sent?

5. How bit errors can parity check detect, and how many can it correct?


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Hamming Code

b12 b11 b10 b9 b8 b7 b6 b5 b4 b3 b2 b1

20

21

22

23

m8 m7 m6 m5 r4 m4 m3 m2 r3 m1 r2 r1

X 0 1 0 1 ? 1 0 0 ? 0 ? ?

Each bit position bn where n = 2x used for redundancy bit

Parity generated on all bits bm where m has x bit set

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Numbering bits means each has unique identifier

b12 1 1 0 0

b11 1 0 1 1

b10 1 0 1 0

b9 1 0 0 1

b8 1 0 0 0

b7 0 1 1 1

b6 0 1 1 0

b5 0 1 0 1

b4 0 1 0 0

b3 0 0 1 1

b2 0 0 1 0

b1 0 0 0 1

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26

Hamming Code: Compute 20 Value

b12 b11 b10 b9 b8 b7 b6 b5 b4 b3 b2 b1

20 1 1 1 0 0 ?

21

22

23

m8 m7 m6 m5 r4 m4 m3 m2 r3 m1 r2 r1

X 0 1 0 1 ? 1 0 0 ? 0 ? ?

The value ofr1

is the even parity for all bn

where n has b1 = 1

Thus r1 = 1

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b12 b11 b10 b9 b8 b7 b6 b5 b4 b3 b2 b1

20

21 1 0 1 0 0 ?

22

23

m8 m7 m6 m5 r4 m4 m3 m2 r3 m1 r2 r1

X 0 1 0 1 ? 1 0 0 ? 0 ? 1

The value ofr2

is the even parity for allbn

wheren

hasb2 = 1

Thus r2 = 0

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b12 b11 b10 b9 b8 b7 b6 b5 b4 b3 b2 b1

20

21

22 0 1 0 0 ?

23

m8 m7 m6 m5 r4 m4 m3 m2 r3 m1 r2 r1

X 0 1 0 1 ? 1 0 0 ? 0 0 1

The value ofr3

is the even parity for allbn

wheren

hasb3 = 1

Thus r3 = 1


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Worksheet: Hamming Code

m8 m7 m6 m5 r4 m4 m3 m2 r3 m1 r2 r1

1. Write message 11101110 into the message bit boxes above.

2. Compute using even parity the bits r4r3r2r1

3. Invert message bit m6, and write down in the order r4r3r2r1 a 1 ifrn parity is

now incorrect, or a 0 if correct.

4. Invert the bit position in the codeword which the number represents.

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Cyclic Redundancy Check

If we take a number n and divide by divisor d, result is

integer quotient q

integer remainder r

obeys the equation n = qd + r

If we add an error e to n, then n+

e=

q

d+

r

e < d r = r

r = r e must be an integer multiple ofd

CRC uses this idea (but using modulo 2 division)

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Examples of CRC Checking

n d q r e n + e r OK?

245 161 1 84 0 245 84 Yes

1 246 85 No

2 247 86 No

160 405 83 No

161 406 84 Yes

162 407 85 No

321 566 83 No

322 567 84 Yes

323 568 85 No

First error pattern not detected = 16110 = 101000012

Second error pattern not detected = 32210 = 1010000102

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Properties of 16 bit CRC

name polynomial binary divisor application

CRC-16 X16 + X15 + X2 + 1 11000000000000101 USB

CRC-CCITT X16 + X12 + X5 + 1 10001000000100001 Bluetooth

Choosing values for polynomials need care to achieve the best performance. Those

above detect:

error bursts of 16 bits

all error bursts with an odd number of bits

99.997% of 17 bit errors

99.998% of 18 or longer bit errors

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Properties of 32 bit CRC

IEEE 802.3 CRC-32 is

x32 + x26 + x23 + x22 + x16 + x12 + x11 + x10 + x8 + x7 + x5 + x4 + x2 + x + 1

As a shorthand write as a number representing polynomials (missing off the

last +1). e.g. IEEE 802.3 0x82608EDB, iSCSI 0x8F6E37A0

HD max no message bits

0x82608EDB 0x8F6E37A0 0xBA0DC66B 0xD419CC15 0x80108400

8 91 177 152 58 0

7 171 (177) (152) 81 0

6 268 5,243 16,360 1,060 0

5 2,974 (5,243) (16,360) 65,505 65,5054 91,607 128K 114,663 (65,505) (65,505)

3 128K unknown (114,663) (65,505) (65,505)

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Error Bursts

message1+redundant1 = n15 n14

n13

n12

n11


n23

n22

n21

message3+redundant3 = n35 n34 n

33 n

32 n

31


n43

n42

n41

- . . . n13 n42 n

32 n

22 n

12 n

41 n

31 n

21 n

11

transmitted data interleaves codewords

If basic error checking can detect B bit bursts, interleaving N codewords

allows N B bursts to be detected

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