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CIS 6930 Powerline Communications Error Handling (c) 2013 Richard Newman

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Error Handling - Outline Analysis Forward Error Correction Copy codes Block codes Convolutional codes Scrambling Concatenated codes Turbo-codes Low Density Parity Check (LDPC) Backward Error Correction (BEC) Stop-and-Wait ARQ Conclusions

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Analysis Measures – Nature of errors detected/corrected – Efficiency Efficiency: U = Uf Up Ue * – Uf = framing efficiency = D/(D+H) ** where D = Data length and H = Header length – Up = protocol utilization Depends on protocol, propagation delay, transmission time – Ue = error caused utilization efficiency = 1-P where P = frame error rate

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Caveats Efficiency formula does not show – PHY layer efficiency – Variable data rate effects PHY Efficiency * – Must consider time spent to deliver frame – Include required gaps/spaces, preamble – Also includes PHY layer FEC Variable Data Rate Effects ** – Frame header/delimiter may be sent at lower modulation rate Esp. for wireless and PLC

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Error Correction Strategies Forward Error Correction – Include sufficient redundancy in transmitted units that errors can be corrected – Simplifies sender protocol – used in PHY Backward Error Correction – Include sufficient redundancy in transmitted units that errors can be detected – Retransmit damaged units – More efficient* – used in MAC and above Limitations – Hamming Distance of code limits capabilities – Always possible to “fool” receiver * Efficiency depends – if have to resend large frame for single error, maybe not so much...

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General ECC Considerations Systematic vs. non-systematic – Systematic = data bits appear in coded stream – Non-systematic = no data bits identifiable Hamming Distance – H(x,y) = number of bits where x and y differ – Code C = {x1, x2,..., xN} set of valid codewords – d = H(C) = min{H(x,y) | x and y are distinct codewords in C} – Maximum detection ability = d-1 – Maximum correction ability = (d-1)/2

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Error Correction/Detection Theory Code C is subset of n-bit binary words Received word R = T + E, for transmitted word T and error E H(C) determines maximum detection/correction capability – Max detection is d-1 – Max correction is floor (d-1)/2 – If correcting, then won't detect! – Can reduce correction to increase detection of uncorrectable errors, but correction + detection < d Efficiency – Detection can be done with fixed # parity bits Larger block has less overhead, but is more prone to errors – Correction requires parity bits scale with info bits Economy of scale still exists – may be non-linear

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Forward Error Correction Block vs. continuous Block = set number of information symbols encoded into set number of code symbols – Internal fragmentation – Need for delimitation Continuous = stream of information symbols encoded into stream of code symbols – Memory/constraint length – must “fill the pipeline” Linearity Sum of two code words is a code word Concatenation – Combine two codes (inner and outer) to increase correction capabilities

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Forward Error Correction Efficiency = code rate Rate = k/n for (n,k) code – k = “information bits” – n = total bits – t = n-k = redundant bits With continuous codes, need to account for “tail” - the number of bits in the memory

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Block Codes Copy codes LRC Hamming codes BCH Reed-Solomon LDPC

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Block Codes Copy Codes – Simplest code, still useful at times – Copy data bits r times to encode – Use received copies to “vote” for input value – Can survive a burst error if scrambled LRC – Longitudinal Redundancy Check – Information bits arranged in p-1 by q-1 matrix – Each row has parity bit at the end – Each column has parity bit at the bottom – n = pq, k = (p-1)(q-1), r = p+q-1 – Detects single bit errors

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LRC Example 1 0 1 1 0 1 1 0 1 0 1 1= information bits 1 0 1 1 _1 0 1 1 1 0 1 1 0 _ -> 0 1 1 0 0 1 0 1 1 _1 0 1 1 1 _ _ _ _ _0 1 1 0 0<- LRC ^ VRC 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 1 0 0 = code word

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LRC Example 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 1 0 0 = sent 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = error 1 0 1 1 1 0 1 0 0 0 1 0 1 1 1 0 1 1 0 0 = received 1 0 1 1 1 0 1 0 0 0 X 1 0 1 1 1 0 1 1 0 0 X errors in LRC and VRC locate bit error

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Hamming Codes – “perfect” 1-bit error correction – 2 n -1 bits per code word – n parity bits, remainder systematic information bits – Parity bit i is in position 2 i – Parity bit i checks even parity of bits in positions with ith bit of location non-zero

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Hamming Code Example n = 4, length N = 16-1 = 15 bits, k = 11, r = 4 information bits = 10110110101 f e d c b a 9 8 7 6 5 4 3 2 1bit positions 1 0 1 1 0 1 1 _ 0 1 0 _ 1 _ _info bits in posn ---------------->1parity bit 3 --------- ------->0parity bit 2 ----- ----- ----- -->0parity bit 1 -- -- -- -- -- -- -- >0parity bit 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0code word

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Hamming Code Example f e d c b a 9 8 7 6 5 4 3 2 1bit positions 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0code word 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0error 1 0 1 1 0 0 1 1 0 1 0 0 1 0 0received word ---------------->0 parity bit 3 X --------- ------->0parity bit 2 ----- ----- ----- -->1parity bit 1 X -- -- -- -- -- -- -- >0parity bit 0 Syndrome = 1010 = a = location of error – Bit error => invert received bit to correct it

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BCH Codes Bose, Chaudhuri, Hocquenghem ca. 1960 Easy to decode Syndrome decoding Easy to implement in hardware Cyclic code Polynomial code over a finite field Reed-Solomon codes a special form of BCH

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Reed-Solomon Codes Based on oversampled polynomial Redundant samples allow optimal polynomial to be recovered if most samples are good Symbols greater than 1 bit m-bit symbols => limit block size to 2 m -1 symbols 2 parity symbols needed to correct each error One to locate position (or none) One to specify correction to symbol 1 parity symbol needed to correct each erasure Handles small bursts Popular DVDs, CDs, Blu-Ray, DSL, WiMax, DVB, ATSC, Raid-6

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Low Density Parity Check Codes Linear code Capacity approaching code Can get near to Shannon limit in symmetric, memoryless channel Uses iterative belief propagation Defined by sparse parity check matrix No encumbering patents Used in DVB-S2 digital TV, ITU-T G.hn, 10GBase-T

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Low Density Parity Check Codes Forney Factor Graph representation (fragment) Inputs Constraint nodes Probability values associated with variable nodes used with constraints to update value and confidence in variable nodes iteratively Variable nodes

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Convolutional Codes May be systematic or not Shift register for information bits Each output bit has one or more taps into shift register Tapped values are XORed to produce output Outputs are sent round robin May “puncture” output to increase coding rate May “scramble” input to spread errors out May “puncture” (drop out bits) to increase rate

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Convolutional Codes + in out 1 0 0 1 1 0 1 -> = info bits 1 0 0 0 1 1 1 0 1 0 1 1 0 1 0 0 1 1 -> = output --------- tail Initialize shift register with 0’s, then shift in one bit at a time, then read one bit from each output

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Convolutional Codes Constraint length m (i.e., amount of memory or size of shift register) m = n+1, where n = degree of G(x)

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Decoding Convolutional Codes Maximum Likelihood Decoding Viterbi Algorithm – “Trellis” decoding – Dynamic programming – Number of states = 2 m, m=constraint length – State = contents of shift regisiter – Cost = HD for transition based on received bits http://www.cambridge.org/resources/0521882672/ 7934_kaeslin_dynpro_new.pdf

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Shift Register Trellis Graph

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Trellis edges depend on code Start state? Start state defined by code specification Usually all 0s

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Trellis Decoding of Convolutional Codes Illegal transitions? Only two legal transitions – 0 or 1 shift in G(x) will determine arc labels depending on state and input shifted in

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Trellis Decoding of Convolutional Codes Illegal transitions follow same edges, but cost according to errors e.g., if receive 01 or 10 in state 000 then 000 or 100 at cost 1

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Trellis Decoding – our code Illegal transitions follow same edges, but cost according to errors e.g., if receive 01 or 10 in state 000 then 000 or 100 at cost 1

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Trellis Decoding – our example On each pair of received bits, update minimum cost for each state, preserving path(s) that produce that cost – 4 bits later can correct

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Decoding Convolutional Codes Maximum Likelihood Decoding – At any point in received stream, output the stream of input bits most likely to have produced that output (i.e., fewest errors) Viterbi Algorithm – “Trellis” keeps track of state of memory – For each possible state, track least cost to get there – Accumulate costs – cost of state B is minimum over all previous states A of cost of A plus cost to transition to B from A given current input – If multiple ways to reach a state, take cheapest one – Only have to maintain costs for current state set, then update based on received bits

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Scrambling Convolutional codes correct well when errors are sparse Tend to have problems with burst errors Scramble bits after encoding, before decoding Concatenated codes – allow errors/resynch Scrambling Shuffle order of bits on the way out/in Interleaver depth = memory required to shuffle E.g., fill block in row order, read out column order

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Concatenated Codes Can increase correcting power by combining two codes – inner and outer Apply outer first when encoding, last decoding Apply inner last encoding, first decoding RSV – Reed-Solomon outer, Viterbi inner Very popular Convolutional inner code corrects sparse errors R-S outer code corrects bursts (n,k) and (n’,k’) code produce (nn’,kk’) code

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Concatenated Codes Outer Coder Inner Coder Scrambler Outer Coder Inner Coder Scrambler Channel Sender Receiver Noise

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Turbo Codes Essentially concatenating two convolutional codes (may be the same code) One code operates on straight input Other code operates on delayed and interleaved input Decoding involves iteration between the two codes Can approach Shannon Limit Patents held by French Telecom

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Recall Error Correction Strategies Forward Error Correction – Include sufficient redundancy in transmitted units that errors can be corrected – Simplifies sender protocol – used in PHY Backward Error Correction – Include sufficient redundancy in transmitted units that errors can be detected – Retransmit damaged units – “More efficient” – used in MAC and above Limitations – Hamming Distance of code limits capabilities – Always possible to “fool” receiver – Efficiency of BEC depends on size of frame & FER

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Backward Error Correction (BEC) Received data cannot be corrected Include checksum/redundancy check to detect errors (detection is cheap) Retransmit frames that have errors How does sender know which to resend? ACK – OK, don’t resend NAK – Received damaged frame No response – time out and resend ACKs Cumulative vs. individual vs. SACK

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BEC Requirements Sender – Must add error detection code to frames – Must store frames sent until ACKed – Must know which frames receiver got correctly – Must know when to resend frame – Must have timer for ARQ Receiver – Must check error detection code on frames – Must be able to distinguish duplicate frames – Must be able to signal sender on state – May buffer out-of-order frames

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Error Detection Codes Generally fixed length field (longer = stronger) – Fixed overhead => low %age for longer frames – But longer frame more likely to have error! Some error detection code types – Parity – single bit, used for characters – Arithmetic Redundancy Check (ARC) – treat transmission as n-bit numbers, add them modulo 2 n, may shift in carry, may shift words – Cyclic Redundancy Check (CRC) – easy to implement in H/W using shift register, can design code to detect errors of specific types – Cryptographic Checks – one-way hash

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Acknowledgements ACK types Individual ACK – just the SN indicated Cumulative ACK – indicates next expected unit Beneficial when ACKs are lost - redundancy SACK – indicates multiple good/back units Sequence Number (SN) per unit Units may be frames, bytes, cells, etc. SNs eventually wrap around Need to avoid confusion – send/receive window Larger SN = more framing overhead

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Error Detection Parity ARC – Arithmetic Redundancy Check Checksum adds values (with carry) Drop overflow May rotate addends CRC – Cyclic Redundancy Check Linear code, generated by polynomial Easy to implement in feedback shift register Cryptographic Hashes – MD5, SHA, HMAC Useful for secure integrity checks

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Stop-and-Wait ARQ One unit sent at a time Wait for response (ACK/NAK) Units have sequencing information (1 bit) Accept received units in strict sequence – Gaps are not allowed – Duplicates are discarded One-bit acknowledgement (ACK) Resend unit on time-out or NAK

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Stop-and-Wait ARQ L/R 2T L = Unit length (bits) R = Transmission rate (bps) T = Propagation delay (sec) Let a = T/(L/R) = propagation delay in frames Cycle time = Tc = 1+2a Data Tx time = 1 Utilization due to protocol = Up = 1/(1+2a) 1 2a

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Stop-and-Wait ARQ Receiver needs way to distinguish duplicate frame from new frame, since ACK may be lost.... Frame 1 Frame 2 Frame 1 Resent Frame 1 X

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Go-Back-N ARQ Transmit units according to send window, k Units have sequencing information Accept received units in strict sequence – Gaps are not allowed – Duplicates, out of order units are discarded Cumulative acknowledgement (ACK) Resend un-acked units – Includes those received after a lost unit

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GBN ARQ k 2a Cycle time = Tc = 1+2a Data Tx time = k Up = k/(1+2a) if no errors, and k <= 1+2a If errors at probability P, Expect to resend k units per error, so if k <= 1+2a, Up = k/(2a+1)(1-P-kP) U = k(1-P)/(2a+1)(1-P-kP) 1 2a

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Selective Repeat ARQ Transmit units according to transmit window Units have sequencing information Store received units according to sequencing information – Gaps are allowed – Duplicates are discarded Selective acknowledgement (SACK) Resend un-acked units

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Selective Repeat ARQ Efficiency is best possible – Only redundancy is sequence info and error detection – Never resend units unnecessarily If P = prob(unit is lost), k = Tx window, then Utilization U = (1-P)k/(2a+1) for k <= 1+2a (U = (1-P) if k > 1+2a)

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Observations w.r.t. PLC Need a variety of techniques to deal with PL channel impairments – Modulation and coding used to get near channel capacity means there will be errors – Channel changes means there will be too many errors sometimes – If raw error rate too low, then get closer!!! Longer frames mean greater error probability – Want to correction unit size to ensure success

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Observations w.r.t. PLC Different coding levels needed for different PPDU/MPDU parts – Frame delimiter needs to be heard by all, so must be very robust – Payload can be adapted to highest rate path can handle Coding method needs to deal with impulse noise – Want to be selective about what must be resent Cross-layer design issues

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Coding and Error Control

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