EE 122 Error Detection and Reliable Transmission November

EE 122: Error Detection and Reliable Transmission November 19, 2003 Katz, Stoica F 04

EECS 122: Introduction to Computer Networks Error Detection and Reliable Transmission Computer Science Division Department of Electrical Engineering and Computer Sciences University of California, Berkeley, CA 94720 -1776 EECS 122 - UCB Katz, Stoica F 04

Today’s Lecture: 24 2 17, 18 6 19, 20 10, 11 7, 8, 9 Application 14, 15, 16 2 4 21, 22 23 Transport Network (IP) Link Physical Katz, Stoica F 04 3

High Level View § § Goal: transmit correct information Problem: bits can get corrupted - Electrical interference, thermal noise § Solution - Detect errors - Recover from errors • Correct errors • Retransmission already done this Katz, Stoica F 04 4

Overview Ø § Error detection & recovery Reliable Transmission Katz, Stoica F 04 5

Error Detection § § § Problem: detect bit errors in packets (frames) Solution: add extra bits to each packet Goals: - Reduce overhead, i. e. , reduce the number of redundancy bits - Increase the number and the type of bit error patterns that can be detected § Examples: - Two-dimensional parity Checksum Cyclic Redundancy Check (CRC) Hamming Codes Katz, Stoica F 04 6

Two-dimensional Parity § § § Add one extra bit to a 7 -bit code such that the number of 1’s in the resulting 8 bits is even (for even parity, and odd for odd parity) Add a parity byte for the packet Example: five 7 -bit character packet, even parity 0110100 1 1011010 0 0010110 1 1110101 1 1001011 0 1000110 1 Katz, Stoica F 04 7

How Many Errors Can you Detect? § § All 1 -bit errors Example: error bit 0110100 1 1011010 0 0000110 1 1110101 1 1001011 0 1000110 1 odd number of 1’s Katz, Stoica F 04 8

How Many Errors Can you Detect? § § All 2 -bit errors Example: error bits 0110100 1 1011010 0 0000111 1 1110101 1 1001011 0 1000110 1 odd number of 1’s on columns Katz, Stoica F 04 9

How Many Errors Can you Detect? § § All 3 -bit errors Example: error bits 0110100 1 1011010 0 0000111 1 1100101 1 1001011 0 1000110 1 odd number of 1’s on column Katz, Stoica F 04 10

How Many Errors Can you Detect? § § Most 4 -bit errors Example of 4 -bit error that is not detected: error bits 0110100 1 1011010 0 0000111 1 1100100 1 1001011 0 1000110 1 How many errors can you correct? Katz, Stoica F 04 11

Checksum § § Sender: add all words of a packet and append the result (checksum) to the packet Receiver: add all words of a packet and compare the result with the checksum Can detect all 1 -bit errors Example: Internet checksum - Use 1’s complement addition Katz, Stoica F 04 12

1’s Complement Revisited § § § Negative number –x is x with all bits inverted When two numbers are added, the carry-on is added to the result Example: -15 + 16; assume 8 -bit representation 15 = 00001111 -15 = 11110000 + 16 = 00010000 -15+16 = 1 1 0000 + 1 00000001 Katz, Stoica F 04 13

Cyclic Redundancy Check (CRC) § Represent a (n+1)-bit message as an n-degree polynomial M(x) - E. g. , 10101101 M(x) = x 7 + x 5 + x 3 + x 2 + x 0 § § Choose a divisor k-degree polynomial C(x) Compute reminder R(x) of M(x)*xk / C(x), i. e. , compute A(x) such that M(x)*xk = A(x)*C(x) + R(x), where degree(R(x)) < k § Let T(x) = M(x)*xk – R(x) = A(x)*C(x) § Then - T(x) is divisible by C(x) - First n coefficients of T(x) represent M(x) Katz, Stoica F 04 14

Cyclic Redundancy Check (CRC) § Sender: - Compute and send T(x), i. e. , the coefficients of T(x) § Receiver: - Let T’(x) be the (n+k)-degree polynomial generated from the received message - If C(x) divides T’(x) no errors; otherwise errors § Note: all computations are modulo 2 Katz, Stoica F 04 15

Arithmetic Modulo 2 § § Like binary arithmetic but without borrowing/carrying from/to adjacent bits Examples: 101 + 010 111 § 101 + 001 100 1011 + 0111 1100 101 010 111 101 001 100 1011 0111 1100 Addition and subtraction in binary arithmetic modulo 2 is equivalent to XOR a 0 0 1 1 b 0 1 a b 0 1 1 0 Katz, Stoica F 04 16

Some Polynomial Arithmetic Modulo 2 Properties § § If C(x) divides B(x), then degree(B(x)) >= degree(C(x)) Subtracting/adding C(x) from/to B(x) modulo 2 is equivalent to performing an XOR on each pair of matching coefficients of C(x) and B(x) - E. g. : B(x) = x 7 B(x) - C(x) = x 7 + x 5 + x 3 + x 2 + x 0 C(x) = x 3 + x 1 + x 0 (10101101) + x 5 (10100110) + x 2 + x 1 (00001011) Katz, Stoica F 04 17

Example (Sender Operation) § Send packet 110111; choose C(x) = 101 - k = 2, M(x)*x. K 11011100 § Compute the reminder R(x) of M(x)*xk / C(x) 101) 11011100 101 111 101 100 101 1 § § R(x) Compute T(x) = M(x)*xk - R(x) 11011100 xor 1 = 1101 Send T(x) Katz, Stoica F 04 18

Example (Receiver Operation) § Assume T’(x) = 1101 - C(x) divides T’(x) no errors § Assume T’(x) = 11001101 - Reminder R’(x) = 1 error! 101) 11001101 110 101 111 101 101 1 § R’(x) Note: an error is not detected iff C(x) divides T’(x) – T(x) Katz, Stoica F 04 19

CRC Properties § § Detect all single-bit errors if coefficients of xk and x 0 of C(x) are one Detect all double-bit errors, if C(x) has a factor with at least three terms Detect all number of odd errors, if C(x) contains factor (x+1) Detect all burst of errors smaller than k bits Katz, Stoica F 04 20

Code words § § § Combination of the n payload bits and the k check bits as being a n+k bit code word For any error correcting scheme, not all n+k bit strings will be valid code words Errors can be detected if and only if the received string is not a valid code word - Example: even parity check only detects an odd number of bit errors Katz, Stoica F 04 21

Hamming Distance § Given code words A and B, the Hamming distance between them is the number of bits in A that need to be flipped to turn it into B - E. g. , H(011101, 000000) = 4 § If all code words are at least d Hamming distance apart, then up to d-1 bit errors can be detected Katz, Stoica F 04 22

Error Correction § If all the code words are at least a hamming distance of 2 d+1 apart then up to d bit errors can be corrected - § § Just pick the codeword closest to the one received! How many bits are required to correct d errors when there are n bits in the payload? Example: d=1: Suppose n=3. Then any payload can be transformed into 3 other payload strings (e. g. , 000 into 001, 010 or 100). - Need at least two extra bits to differentiate between 4 possibilities In general need at least k ≥ log 2(n+1) bits A scheme that is optimal is called a perfect parity code Katz, Stoica F 04 23

Perfect Parity Codes § Consider a codeword of n+k bits - b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 b 9 b 10 b 11… § Parity bits are in positions 20, 21, 22 , 23 , 24… - b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 b 9 b 10 b 11… § A parity bit in position 2 h, checks all data bits bp such that if you write out p in binary, the hth place in p’s binary representation is a one Katz, Stoica F 04 24

Example: (7, 4)-Parity Code § n=4, k=3 - Corrects one error - log 2(1+n) = 2. 32 k = 3, perfect parity code b 1 b 2 Position 1 10 11 100 101 110 111 Check: b 1 x Check: b 2 § data payload = 1010 - For each error there is a unique combination of checks that fail - E. g. , 3 rd bit is in error, : 1000 both b 2 and b 4 fail (single case in which only b 2 and b 4 fail) b 3 b 4 x x b 5 x x Check: b 4 b 7 x x x x b 4 0 1 0 b 1 b 2 Position 1 10 11 100 101 110 111 Check: b 1 1 Check: b 2 Check: b 4 1 b 6 1 0 0 1 0 1 0 Katz, Stoica F 04 25

Overview § Ø Error detection & recovery Reliable transmission Katz, Stoica F 04 26

Reliable Transmission § § Problem: obtain correct information once errors are detected Solutions: - Use error correction codes • E. g. perfect parity codes, erasure codes (see next) - Use retransmission (we have studied this already) § Algorithmic challenges: - Achieve high link utilization, and low overhead Katz, Stoica F 04 27

Error correction or Retransmission? § Error Correction requires a lot of redundancy - Wasteful if errors are unlikely § Retransmission strategies are more popular - As links get reliable this is just done at the transport layer § Error correction is useful when retransmission is costly (satellite links, multicast) Katz, Stoica F 04 28

Erasure Codes n blocks Content Encoding n+k blocks Transmission >= n blocks Received Decoding Content (*Michael Luby slide) original n blocks Katz, Stoica F 04 29

Example: Digital Fountain Use erasure codes for reliable data distribution Intuition: - Goal is to fill the cup - Full cup = content recovered (*Michael Luby slide) Katz, Stoica F 04 30

Erasure Coding Approaches § Reed-Solomon codes -Complex encoding/decoding algorithm and analysis § Tornado codes -Simple encoding/decoding algorithm -Complexity and theory in design and analysis § LT codes -Simpler design and analysis (*Michael Luby slide) Katz, Stoica F 04 31

LT encoding Content Choose 2 random content symbols XOR content symbols 2 Choose degree Insert header, and send Degree Dist. Degree (*Michael Luby slide) Prob 1 0. 055 2 0. 3 3 0. 1 4 0. 08 100000 0. 0004 Katz, Stoica F 04 32

LT Encoding Content Choose 1 random content symbol Copy content symbol 1 Choose degree Insert header, and send Degree Dist. Degree (*Michael Luby slide) Prob 1 0. 055 2 0. 3 3 0. 1 4 0. 08 100000 0. 0004 Katz, Stoica F 04 33

LT Encoding Content Choose 4 random content symbols XOR content symbols 4 Choose degree Insert header, and send Degree Dist. Degree (*Michael Luby slide) Prob 1 0. 055 2 0. 3 3 0. 1 4 0. 08 100000 0. 0004 Katz, Stoica F 04 34

LT Decoding Content (unknown) 1. Collect enough encoding symbols and set up graph between encoding symbols and content symbols to be recovered 2. Identify encoding symbol of degree 1. STOP if none exists 3. Copy value of encoding symbol into unique neighbor, XOR value of newly recovered content symbol into encoding symbol neighbors and delete edges emanating from content symbol 4. Go to Step 2. (*Michael Luby slide) Katz, Stoica F 04 35

LT Encoding Properties § § § Encoding symbols generated independently of each other Any number of encoding symbols can be generated on the fly Reception overhead independent of loss patterns - The success of the decoding process depends only on the degree distribution of received encoding symbols. - The degree distribution on received encoding symbols is the same as the degree distribution on generated encoding symbols. (*Michael Luby slide) Katz, Stoica F 04 36

What Do You Need To Know? § Understand - 2 -dimensional parity - CRC - Hamming codes § Tradeoff between achieving reliability via retransmission vs. error correction Katz, Stoica F 04 37