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DIGITAL COMMUNICATION Packet error detection (CRC) November 2011 A.J. Han Vinck

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Some books

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Cyclic Redundancy Check Add n-k bits of extra data (the CRC field) to an k-bit message to provide error detection function –For efficiency, want n-k << n –e.g., n-k = 32 for Ethernet and n = 12,000 (1500 bytes) HeaderBody CRC Beginning sequence Ending sequence

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Automatic Repeat reQuest IDEA: ask for retransmission if error detected ERROR ? If Yes: ask retransmission channelPass data if no error

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Some properties of ARQ Automatic Repeat-reQuest (ARQ) send ACK: packet OK send NACK: packet in error Some modes: –Resend the packet if no ACK received before timeout –Use a predetermined maximum # of retransmissions until it is correctly received Some ARQ protocols are –Stop-and-wait ARQ –Go-Back-N ARQ –Selective Repeat ARQ. –Hybrid ARQ is a combination of ARQ and forward error correction.

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A pictorial view of coding 2 n vectors 2 k code words (code) Replace k information bits by a unique n bit code word

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Error detection 2 n vectors transmitted word received received vector not equal to one of the 2 k code words.

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Misdetection 2 n vectors transmitted word received

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Consider the following example transmitted received one error Property: one error can be corrected two errors can be detected CHECK!

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Error detection capability code words differ in at least d min positions 2 k vectors up to d min –1 errors are detectable 0xx1x0x0 4 differences 1xx0x1x1 < = 3 errors can be detected

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Misdetection 2 n vectors 2 k code words for a random vector: the probability of false acceptance is (2 k -1)/2 n = probability of hitting the wrong code word

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A polynomial description of the procedure Let a = (a 0 a 1,..., a k-1 ) be k information bits the information is represented byA(X) = X n-k (a 0 +a 1 X +...+a k-1 X k-1 ) the check polynomial isF(X) = f 0 +f 1 X +...+f n-k X n-k. The transmitted code polynomial or encoding procedure is C(X) = ( A(X) mod F(X) + A(X) ) Note that C(X) mod F(X) 0, and thus is C(X) a multiple of F(X).

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A non-systematic procedure Transmit C(X) = F(X) (a 0 +a 1 X +...+a k-1 X k-1 ) then, C(X) modulo F(X) = 0!

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Some examples All coefficient operations are modulo-2 –( 1+ X + X 3 ) (X + X 2 ) = X+ X 2 + X 4 + X 2 + X 3 + X 5 = X + X 3 + X 4 + X 5 errors change 0 1 and 1 0 Ex: ( X + X 4 + X 5 ) modulo-(1 + X 2 ) = 1 –Subtract as often as possible ( 1 + X 2 ), but keep calculating mod-2

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Ex: ( X + X 4 + X 5 ) Modulo- (1 + X 2 ) X + X 4 + X 5 X 3 + X 5 X 3 X + X 3 + X 4 X 2 + X 4 X 2 X + X 2 + X 3 X + X 3 X X X 2 1 1

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detection procedure Assume that we receive the polynomial R(X)= C(X) + E(X) where E(X) is a binary error polynomial i.e. an error vector ( 0,0,0,1,0) C(X) = X 3 the decoder calculates R(X) mod F(X) - if the result = 0 no error detected - if the result ≠ 0, then an error is detected. The polynomial R(X) mod F(X) = 0 if and only if E(X) is a multiple of F(X).

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performance ASSUME: the polynomial F(X) has the form F(X) = X n-k - degree n-k; - and a nonzero constant term, THEN: any error burst of length n-k has a polynomial representation that looks likeE(X) = X i ( X n-k-1 ) and can thus never be a multiple of F(X). F(X) is capable of detecting any burst of length n-k !

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Performance cont‘d A nonzero burst of length (n-k) can never give a zero result. This can also be seen as follows n - k E(X) = X i F(X) = n-k+1 WHY: the binary representation of F(X) has length (n - k + 1) cancellation of the highest degree nonzero component in E(X) always gives a non-zero result.

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CRC standard polynomials CRC-8: x 8 +x 2 +x 1 +1 CRC-10: x 10 +x 9 +x 5 +x 4 +x 1 +1 CRC-12: x 12 +x 11 +x 3 +x 2 +1 CRC-16: x 16 +x 15 +x 2 +1 CRC-CCITT: x 16 +x 12 +x 5 +1 CRC-32: x 32 +x 26 +x 23 +x 22 + x 16 +x 12 +x 11 +x 10 +x 8 x 7 +x 5 +x 4 +x 2 +1 Ethernet uses CRC-32 HDLC: CRC-CCITT ATM: CRC-8, CRC-10, and CRC-32

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Why these polynomials (CRC-16 and CRC-CCITT) ? 1st property: Multiples of these polynomials have even weight Hence: error vectors of odd weight cannot be a multiple! 2nd property: error vectors of the form E(X) = (1 + X i ) are not divisible by F(X) for i < L, L larger than word length Try to check this property! Conclusion: 1, 2 and 3 errors are detectable. CHECK!

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SUMMARY for ERROR DETECTION Remark: We do all operations modulo 2 (XOR). F(X) = X n-k A packet has the form C(X) = A(X) mod F(X) A(X) = X n-k (a 0 + a 1 X a k-1 X k-1 checksum information packet of length k. C =checksum a 0 a 1 a 2... a k-1 n-k bitsk bits data To calculate A(X) mod F(X), cancel highest nonzero term with F(X) until result has degree less than n-k

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Cont‘d invert C at positions where the binary error vector E has ones: R = C E EXAMPLE C = checksum a 0 a 1 a 2... a k C = c 0 c 1 c c n E = e 0 e 1 e e n R = r 0 r 1 r r n check sum If no errors C(X) mod F(X) = A(X) mod F(X) A(X) mod F(X) = 0! If errors{C(X) E(X) } mod F(X) = E(X) mod F(X).

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Shift-register The following shift register can be used to calculate A(X) modulo ( 1 + X + X 3 ) a 0 a 1 a 2... a k-1 Homework: give a description of the shift control to obtain the result

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Cont‘d Example: calculate ( 1 +X 2 +X 4 +X 5 ) modulo ( 1 +X + X 3 ) –In binary: ( ) modulo ( ) The operations in binary are as follows X 2 +X 4 +X X 2 ( 1 +X + X 3 ) X ( 1 +X + X 3 ) ( 1 +X + X 3 ) RESULT: X 2 Homework: draw the corresponding shift register

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CRC cont‘d The following „clocked“ shift register can be used to implement CRC16: F(X) = 1 +X 2 + X 15 + X 16 Data in 13 flip-flops Example: X 16 modulo F(X) = 1 + X 2 + X 15 Question: When is the result A(X) mod F(X) equal to 0? What does this mean?

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A simple method row parity Fill row wise Transmit column wise RESULT: RESULT: any burst of length L can be detected L

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A simple code: the single parity check code Make the number of ones in a code word even: –Method 1 Take F(X) = 1 + X, i.e. C(X) = F(X) A(X). C(X= 1) adds up the number of ones and since X = 1 is a root of F(X), the result is 0. –Method 2. Use the n-th bit to make the parity of a word even CONCLUSION: detection of any number of odd errors –Efficiency: R = (n-1)/n

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ISBN checksum: checksum for 10-digit ISBN - Given 9 digit product code. Starting at leftmost digit: - multiply corresponding digit by 10, 9, 8,... down to 2 inclusive - add the resulting numbers: add digit 10 such that the result is divisible by 11 the number 10 is written as X e.g., X is valid. The last digit has to be 10 (= X). 10*0 + 9*2 + 8*0 + 7*1 + 6*6 + 5*1 + 4*5 + 3*8 + 6*2 + 1*10 = = 132 = 12*11 detects transpositions and single digit errors - an error e at position I gives as result the value eI modulo 11 0 - detection of an transposition error of A and B: A I + B (I-1) BI + A(I-1) = AI + B(I-1) – A + B the value ( – A + B ) 0 modulo 11 for A B

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A.J. Han Vinck29 Combining error detection-correction G1(X) G2(X) generates a code with minimum distance D2 G1(X) generates a code with minimum distance D1 < D2 C(X) = I(X) G1(X) G2(X) = I’(X)G1(X) decoding: step 1: correct a maximum of ( D1 – 1 )/2 errors with G1(X) step 2: detect the remaining errors with G1(X)G2(X) properties: 1. t ( D1 – 1 )/2 errors correctable 2. t D ( D1 – 1 )/2 errors detectable Homework: construct some examples

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A.J. Han Vinck30 Combining error detection-correction Example: nk d min generator polynomial (octal) 63573G1 = 103 (octal) 515G2 x G1 = G1 x (127) 457G3 x G2 x G1= G2 x G1 x (147)

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A.J. Han Vinck31 Example: bluetooth Three error correction schemes are defined for Bluetooth: –1/3 rate FEC, a simple repetition –2/3 rate FEC, a shortened Hamming Code (10,15) –ARQ scheme for data (automatic retransmission request) FEC schemes reduce the number of retransmissions A CRC code decides whether a packet/header contains errors, i.e. transmit C(X) = I(X) g(X) receive R(X) = C(X) + E(X) check R(X) modulo g(X) = 0 ?

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Bookland EAN Symbol

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Proper codes Linear block codes do not necessarily obey the 2 -(n-k) bound. In other words, a code is proper if P U is monotonically increasing in p, for 0.0 p 0.5. Hamming codes and primitive double error-correcting BCH codes are proper. To illustrate the above behavior, we compare the standard CRC-16 for a length 80 packet with a double error correcting BCH code of length 81 and 17 redundant digits. Detection error probability as a function of the channel error probability.

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Unidirectional errors detection: Berger codes application: memory systems n binary data bits log 2 (n+1) check bits indicate the # of zeros in the first part Example: n = 7; log 2 (n+1) = Suppose errors from 1 to 0 in the first part only: number of zeros does not match check in the second part only: idem dito in the first and second part: the first part gets more zeros the second part # gets smaller CONCLUSION: any number of errors detectable! Errors: unidirectional, i.e. In a word only from 0 to 1 (or from 1 to 0)

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Some practical info INTEL website: Xilinx website: Altera website:

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