2018/11/1 Thursday. YU Xiangyu

Transcription

1 2018/11/1 Thursday YU Xiangyu

2

3 Introduction ARQ FEC Parity Check Block Codes Cyclic Codes CRC (Cyclic Redundancy Check) Convolutional Codes Interleaving Turbo Codes LDPC

4 Information to be transmitted Source coding Channel coding Modulation Transmitter Antenna Air Antenna Information received Source decoding Channel decoding Demodulation Receiver

5 Channel coding: Purpose: to improve the reliability of signal transmission. Method: to add some redundant bits in order to discover or correct errors. Error control: all error correction measures including channel coding. Causations for producing error symbols: Intersymbol interference evoked by multiplicative interference. Signal to noise ratio reduction caused by additive interference. Fan P278

6 According to the statistical characteristics of the error symbols caused by additive interference: Random channel: Error symbols occur randomly, e.g., error symbols caused by additive noise. Burst channel: the occurrence is relatively concentrated, e.g., the error symbols caused by pulse interference. Mixed channel Fan P278

7 Terminal A Terminal B Transmission in only one direction (a) Terminal A Or Terminal B Transmission in either direction, but not simulaneously (b) Terminal A Terminal B Transmission in both directions simultaneously (c) Figure Terminal connectivity classifications. (a) Simplex. (b) Half-duplex. (c) Full-duplex. Sklar P

8 Error detection and retransmission: Utilizes redundant bits added to the data to detect that an error has been made. The error symbols can be discovered, but the locations of the errors can t be determined. The receiving terminal does not attempt to correct the error; it simply requests that the transmitter retransmit the data. The communication systems need to have the bidirectional channels. Sklar P315 Fan P

9 Feedback check: The received symbols will be returned to the transmitter for comparison of them with the original transmitted symbols. Disadvantages: need of bidirectional channel and rather low transmission efficiency. Error detection and deletion: When the error symbols are discovered in the receiver, they will be deleted immediately. It is suitable only in systems where a lot of redundancy exists in the transmitting symbols, and the deleted part of the received symbols doesn t influence the application. Fan P

10 Error detection codes Detects the presence of an error Automatic repeat request(arq) protocols Block of data with error is discarded Transmitter retransmits that block of data Error correction codes, or forward correction codes (FEC) Designed to detect and correct errors Stallings P193

11 FEC: utilizes the error control symbols attached not only to discover the error symbols, but also to correct the error symbols. requires a one-way link only not all error patterns can be corrected; errorcorrecting codes are classified according to their error-correcting capabilities!!!! Fan P

12 Source Transmitter Channel Receiver Destination Encoder Transmit controller Modulation Demodulation Decoder Transmit controller Acknowledge

13 Transmitting data Retransmission Transmitter Transmission Receiver 1 ACK Received data ACK NAK ACK ACK NAK Error Error (a) Output data ACK: Acknowledge Time NAK: Negative ACK Figure Automatic repeat request (ARQ). (a) Stop-and-wait ARQ (half-duplex). Fan P280 Sklar P

14 Tomasi Advanced Electronic Communications Systems, 6e FIGURE Example of stop-and-wait flow control

15 Transmitter Transmitting data Transmission Receiver 1 2 Received data ACK ACK ACK NAK ACK ACK ACK ACK ACK Error (b) Go-back 4 Go-back 7 ACK ACK ACK Error NAK ACK ACK ACK 7 8 ACK ACK Figure Continuous ARQ with pullback (full-duplex). Fan P280 Sklar P316

16 Transmitting data Retransmission Retransmission Transmitter Transmission Receiver ACK Received data ACK ACK NAK ACK ACK ACK ACK Error (c) ACK ACK ACK Error NAK ACK ACK ACK ACK ACK Figure Continuous ARQ With selective repeat (full-duplex). Sklar P316

17 Advantages Less parity symbols, higher code rate The calculation complexity of error detection is low. It can adapt the different characteristics of the channels Disadvantages It requires duplex channels cannot be used in the unidirection communication systems or broadcasting systems. The transmission efficiency is decreased due to retransmission. When the channel interference is serious, communication is virtually interrupted. Fan P

18 The major advantage of ARQ over forward error correction (FEC) is that error detection requires much simpler decoding equipment and much less redundancy than does error correction. Also, ARQ is adaptive in the sense that information is retransmitted only when errors occur. On the other hand, FEC may be desirable in place of, or in addition to, error detection, for any of the following reasons: 1. A reverse channel is not available or the delay with ARQ would be excessive. 2. The retransmission strategy is not conveniently implemented. 3. The expected number of errors, without corrections, would require excessive retransmissions. Sklar P

19 Error detection requires retransmission Detection inadequate for wireless applications Error rate on wireless link can be high, results in a large number of retransmissions Long propagation delay compared to transmission time

20 The key idea of FEC is to transmit enough redundant data to allow receiver to recover from errors all by itself. No sender retransmission required A simple redundancy is to attach a parity bit

21 No errors present Codeword produced by decoder matches original codeword Decoder detects and corrects bit errors Decoder detects but cannot correct bit errors; reports uncorrectable error Decoder detects no bit errors, though errors are present

22 Transmitter For a given frame, an error-detecting code (check bits) is calculated from data bits Check bits are appended to data bits Receiver Separates incoming frame into data bits and check bits Calculates check bits from received data bits Compares calculated check bits against received check bits Detected error occurs if mismatch Stallings P194

23 Error Detection Process Beard P276

24 Forward Error Correction Process Beard P283

25 Adopting error-correction coding for decrease of error symbol probability, the price paid is increase of bandwidth. Figure Performance of digital systems with and without coding. Fan P284 Couch P54

26 The information capacity (or channel capacity) C of a continuous channel with bandwidth B Hertz can be perturbed by additive Gaussian white noise of power spectral density N 0 /2, provided bandwidth B satisfies P C Blog 2 1 bits / sec ond N0B where P is the average transmitted power P = E b R b (for an ideal system, R b = C) E b is the transmitted energy per bit R b is transmission rate

27 log 1 1 log lim 1 1 log lim 1 1 log 1 lim 1 1 log lim 1 log lim 1 log lim 0 n S e n S X n S X n S S W n S n S W n S n S W n Wn S W N S W C X X X X S W n W W W W

28 R b /B Region for which R b >C Capacity boundary R b =C 1 Shannon Limit Region for which R b <C E b /N 0 db 0.1 E b is the transmitted energy per bit N 0 is the Gaussian white noise of power spectral density R b is the transmission rate B is the channel with bandwidth C is the channel capacity

29 ISBN Some cash

30 1 st generation Group identifier Publisher identifier Title identifier Check digit ISBN: ((7*10+8*9+0*8+2*7+2*6+5*5+3*4+2*3+1* 2)%11)=7 10->X 11->0

31 2 nd generation ((9*1+7*3+8*1+7*3+8*1+0*3+2*1+2*3+5*1+ 3*3+2*1+1*3)%10)=6 10->0

32 =7

33 =16 1+6= =34 3+4=7

34 The major categories of FEC codes are Block codes Cyclic codes Convolutional codes, Turbo codes, etc.

35 Parameters of the code sequence n - total number of symbols in the code sequence k - number of information symbols in the code sequence r - number of error control symbols in the code sequence k/n - code rate (n - k) / k = r / k - redundancy Fan P279

36 0-> >

37 echo on ep=0.3; for i=1:2:61 p(i)=0; for j=(i+1)/2:i p(i)=p(i)+prod(1:i)/(prod(1:j)*prod(1:(i-j)))*ep^j*(1-ep)^(i-j); echo off ; end end echo on ; pause % Press a key to see the plot. stem((1:2:61),p(1:2:61)) xlabel('n') ylabel('pe') title('error probability as a function of n in simple repetition code')

38 pe 0.35 Error probability as a function of n in simple repetition code n

39 Assume that a sequence of eight bits is encoded using a (5,1,5) repetition encoder. Suppose that we receive the following bit sequence at the output of a binary channel that introduces occasional bit errors. Assume bit index 0 corresponds to the start of the received codeword corresponding to the first bit. If at most two bit errors occur within each codeword, what was the transmitted sequence of eight bits? Edx

40 Parity bit appended to a block of data One dimensional parity-check code Classified into odd parity code and even parity code. Even parity Added bit ensures an even number of 1s Odd parity Added bit ensures an odd number of 1s Example, 7-bit character [ ] Even parity [ ] Odd parity [ ] Fan P Beard P

41 In the even check code, the check bit is so designed that the summation of all the bits in the codeword yields an even result: an 1 an 2 a0 0 The check bit in odd check codes makes the number of 1s in the codeword be odd: an 1 an 2 a0 1 where a 0 is the check bit, other bits are information bits. In the parity-check code, there is only one check bit, so the code rate equals k/(k+1). Fan P

42 Consider a (n, k, 4) block code, where 6 parity bits [P1 P2 P3 P4 P5 P6] are generated for 6 data bits [D1 D2 D3 D4 D5 D6]. The values of the parity bits are computed by arranging the data and parity bits in a block as shown below. The first 5 parity bits are generated to ensure that all rows and columns containing data bits have even parity, and the last parity bit P6 ensures that the entire code word has even parity. The resulting codeword is[d1 D2 D3 D4 D5 D6 P1 P2 P3 P4 P5 P6]. Edx

43 What are the values of n and k for the code above? n=12 k=6 Assume we are interested in only detecting, but not correcting, bit errors. What is the maximum number of bit errors per codeword that can be detected using this coding scheme? 3 Edx

44 Suppose that the received codeword is [ ]. Assuming that at most one bit error occurred during transmission, what was the transmitted codeword? Edx

45 Suppose that the received codeword is [ ]. Assume that at most two bit errors have occurred. Which of the following statements is incorrect? The transmitted codeword could have been [ ]. The transmitted codeword could have been [ ]. The transmitted codeword could have been [ ]. The transmitted codeword could have been [ ]. Edx

46 Assume: there is a code composed of 3 binary symbols, so there are 2 3 = 8 different possible codewords: 000 fine 001 cloud 010 overcast 011 rain 100 snow 101 frost 110 fog 111 hail Now, if the error symbols occur, then error information will be received. Fan P281

47 If only 4 codewords among these 8 codewords are allowed to be used for transmission of the weather, e.g., let 000 fine 011 cloud 101 overcast 110 rain are permission codewords, other 4 kinds are forbidden codewords. Then, the receiver can detect one error symbol in a codeword. This code can only detect the error symbols, and cannot correct the error symbols. Fan P281

48 If only two permission codewords are defined: for example 000 fine 111 rain then the code can detect at most two error symbols, or correct one error symbol. Fan P281

49 Information is divided into blocks of length k, r parity bits or check bits are added to each block (total length n = k + r) Block codeword = information bits + check bits Expression of block code: (n, k) where n - total length of codeword k - number of information bits r = n k - number of check bits Code rate R = k/n Decoder looks for codeword closest to received vector (code vector + error vector) Fan P282

50 Consider a communication system using the block code illustrated below. Suppose we transmit bits using 25 samples per bit at a rate of Fs=1M samples per second. What is the net bit rate of our communication system in bits per second? Edx

51 a n-1 a n-2... a r a r-1 a n-2... a 0 t k information bits r check bits Code length n = k + r Fan P282

52 Tradeoffs between Efficiency Reliability Encoding/Decoding complexity Modulo 2 Addition

53 The codeword in the table is a (3, 2) code. Information bits Check bits Fine 00 0 Cloud 01 1 Overcast 10 1 Rain 11 0 Fan P282

54 Code weight: number of 1s in the codeword Code distance: the number of bits which have different values in the corresponding locations of two codewords, and also called Hamming distance Minimum code distance (d 0 ): minimum distance among the codewords Fan P282

55 Hamming distance for 2 n-bit binary sequences, the number of different bits E.g., v 1 =011011; v 2 =110001; d(v 1, v 2 )=3 Redundancy ratio of redundant bits to data bits Code rate ratio of data bits to total bits Coding gain the reduction in the required E b /N 0 to achieve a specified BER of an error-correcting coded system Beard P

56 Geometic meaning of code distance: by using a code with n = 3 as example a 1 (0,1,0) (1,1,0) (0,1,1) (1,1,1) (0,0,0) (1,0,0) a 2 a 0 (0,0,1) (1,0,1) General speaking, code distance is the Hamming distance between the vertexes of a unit regular polyhedron in an n dimensional space. 56 Fan P282

57 Error correction and detection abilities of a code are decided by the minimum code distance d 0. For detecting e error symbols, require d0 e A B e d 0 Hamming distance Two codewords with code distance 3 For correcting t error symbols, require d0 2t A B t d 0 t Hamming distance Fan P283

58 Figure (a) Hamming distance d(c i, c j ) 2t 1. (b) Hamming distance d(c i, c j ) 2t. The received vector is denoted by r. Haykin P638

59 For correcting t error symbols, and detecting e error symbols at the same time, require d e t 1 ( e ) 0 t A t e 1 B t Hamming distance (c) Two codewords with code distance (e + t + 1) Error correction and detection combination mode: When the number of error symbols is small, the system operates according to the FEC mode so as to save retransmission time and improve transmission efficiency. When the number of the error symbols is large, the system operates according to error detection with retransmission mode so as to reduce the total bit error probability. Fan P283

60 The figure illustrates, with a simple geometric analogy, the structure behind linear block codes. We can imagine the vector space V n comprising 2 n n-tuples. Within this vector space there exists a subset of 2 k n-tuples making up a subspace. These 2 k vectors or points, shown sprinkled among the more numerous 2 n points, represent the legitimate or allowable codeword assignments. Sklar P330

61 The basic goals in choosing a particular code, similar to the goals in selecting a set of modulation waveforms, can be stated as follows: 1. We strive for coding efficiency by packing the V n space with as many codewords as possible. This is tantamount to saying that we only want to expend a small amount of redundancy (excess bandwidth). 2. We want the codewords to be as far apart from one another as possible, so that even if the vectors experience some corruption during transmission, they may still be correctly decoded, with a high probability. Sklar P330

62 Examine the following coding assignment that describes a (6, 3) code. There are 2 k = 2 3 = 8 message vectors, and therefore eight codewords. There are 2 k = 2 6 = sixty-four 6-tuples in the V 6 vector space: Sklar P330

63 Suppose we receive the following bitstream If we assume that we can both detect and correct errors, what was the original bit stream?

64 For a (8, 4) block code with minimum code distance is 3, which one of the following statements is incorrect? Each codeword contains 4 message bits. The code rate is 0.5. We can detect 3 bit errors. We can detect and correct 1 bit errors. Edx

65 Consider a repetition code where codewords are formed by repeating each bit five times. What are the values of (n,k) and minimum code distance for this code? (5,1) D=5 Suppose we wish to detect, but not correct errors in each received codeword. What is the maximum number of bit errors that we can detect? 4

66 Suppose we wish to detect and correct errors in each received codeword. What is the maximum number of bit errors that we can detect and correct? 2

67 Which of the following statement(s) is(are) generally true about channel coding? Channel coding reduces the bit error rate of a communication system. Channel coding reduces bit errors by increasing the signal to noise ratio of the received signal. Channel coding relies on redundancy to detect or correct bit errors If there are two codewords in an error correcting code that differ by 3 bits, then this error correcting code can always detect errors in two bits per codeword. Edx

68 Designed to correct single bit errors Family of (n, k) block error-correcting codes with parameters: Block length: n = 2 m 1 Number of data bits: k = 2 m m 1 Number of check bits: n k = m Minimum distance: d min = 3 Single-error-correcting (SEC) code SEC double-error-detecting (SEC-DED) code Beard P

69 Concept of cyclic codes Cyclicity is designated in a way that any codeword obtained by an end-around shift of a codeword in a code is also a codeword in this code. Fan P

70 General condition If (a n-1 a n-2 a 0 ) is a codeword of a cyclic code, then the codewords after cyclic shift (a n-2 a n-3 a 0 a n-1 ) (a n-3 a n-4 a n-1 a n-2 ) (a 0 a n-1 a 2 a 1 ) are still the codewords of this code. Fan P292

71 Example: All codewords of a (7, 3) cyclic code are as follows: No. of codeword Information bit Paritycheck bit No. of codeword Information bit Paritycheck bit a 6 a 5 a 4 a 3 a 2 a 1 a 0 a 6 a 5 a 4 a 3 a 2 a 1 a If the second codeword in this table shifted one bit to the right, then it will become the fifth codeword; if the fifth codeword cyclic shifts one bit to the right, then it will become the seventh codeword. Fan P

72 Can be encoded and decoded using linear feedback shift registers (LFSRs) Takes fixed-length input (k) and produces fixedlength check code (n-k) In contrast, CRC error-detecting code accepts arbitrary length input for fixed-length check code Beard P291

73 Polynomial expression A codeword (a n-1 a n-2 a 0 ) with length n can be expressed as x in the above equation has not any meaning, and only its power is used to represent the location of the symbol. For example: the codeword can be expressed as ) ( a x a x a x a x T n n n n ) ( x x x x x x x x x x T Fan P

74 Cyclic Redundancy Code (CRC) is an error-checking code The transmitter appends an extra n-bit sequence to every frame called Frame Check Sequence (FCS). The FCS holds redundant information about the frame that helps the receivers detect errors in the frame CRC is based on polynomial manipulation using modulo arithmetic. Blocks of input bit as coefficient-sets for polynomials is called message polynomial. Polynomial with constant coefficients is called the generator polynomial

75

76 Generator polynomial is divided into the message polynomial, giving quotient and remainder, the coefficients of the remainder form the bits of final CRC Define: Q The original frame (k bits) to be transmitted F The resulting frame check sequence (FCS) of n-k bits to be added to Q (usually n = 8, 16, 32) J The cascading of Q and F P The predefined CRC generating polynomial The main idea in CRC algorithm is that the FCS is generated so that J should be exactly divisible by P Beard P

77 The CRC creation process is defined as follows: Get the block of raw message Left shift the raw message by n bits and then divide it by p Get the remainder R as FCS Append the R to the raw message. The result J is the frame to be transmitted J=Q.x n-k +R J should be exactly divisible by P Dividing Q.x n-k by P gives Q.x n-k /P=Q+R/P Where R is the reminder J=Q.x n-k +R. This value of J should yield a zero reminder for J/P Beard P

78 Widely used versions of P(X) CRC 12 X 12 + X 11 + X 3 + X 2 + X + 1 CRC 16 X 16 + X 15 + X CRC CCITT X 16 + X 12 + X 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 + X + 1 Beard P280

79 Dividing circuit consisting of: XOR gates Up to n k XOR gates Presence of a gate corresponds to the presence of a term in the divisor polynomial P(X) A shift register String of 1-bit storage devices Register contains n k bits, equal to the length of the FCS Beard P280

80 Tomasi Advanced Electronic Communications Systems, 6e FIGURE CRC-16 generating circuit

81 Input msg: ( ) Using CRC-16 S(D)=D 7 +D 5 +D 4 +D 2 +D 1 +1 (K=8) g(d)=d 16 +D 15 +D 2 +1 (L=16)

82 1 1) ( ) ( Remainder 1 Remainder ) ( ) ( Remainder ) ( D D D D D D D D D D D D D D D D D D D D D D D D D D D g D D S D C L =D 9 +D 8 +D 7 +D 5 +D 4 +D = 0 D D D D D D D 9 +1 D 8 +1 D 7 +0 D 6 +1 D 5 +1 D 4 +0 D 3 +0 D 2 +1 D 1 +0

83 ( ) ( ) 0 ) ( ) ( mainder Re D g D R ) ( r r D D r D r D R L K L K L K L K

84 Most widely used channel code Encoding of information stream rather than information blocks Decoding is mostly performed by the Viterbi The constraint length K for a convolution code is defined as K=M+1 where M is the maximum number of stages in any shift register The code rate r is defined as r = k/n where k is the number of parallel information bits and n is the the number of parallel output encoded bits at one time interval

85 Generates redundant bits continuously Error checking and correcting carried out continuously (n, k, K) code Input processes k bits at a time Output produces n bits for every k input bits K = constraint factor k and n generally very small n-bit output of (n, k, K) code depends on: Current block of k input bits Previous K-1 blocks of k input bits Beard P299

86 Convolution codes Features of convolution code: The parity-check bits are not only related to the present information segment of k bits, and also related to the previous m = (N 1) information segments. N is called constraint length of the codeword. Convolution codes are denoted as (n, k, m), where the code rate is k/n. Fan P

87 Figure Convolutional encoding (k = 3, n = 4, K = 5, and R = 3/4). Couch P50

88 1 0 x Input 1 0 D1 D0 1 1 D2 D y 1 y1 2 Output c D 1, D 2 - Registers Input x: Output y 1 y 2 : Input x: Output y 1 y 2 :

89 10/1 01/ / /0 00/1 11/1 11/ /0

90 0 00 First input First output

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92 Trellis diagram expanded encoder diagram Viterbi code error correction algorithm Compares received sequence with all possible transmitted sequences Algorithm chooses path through trellis whose coded sequence differs from received sequence in the fewest number of places Once a valid path is selected as the correct path, the decoder can recover the input data bits from the output code bits Beard P

93 The Viterbi algorithm removes from consideration those trellis paths that could not possibly be candidates for the maximum likelihood choice. When two paths enter the same state, the one having the best metric is chosen; this path is called the surviving path. This selection of surviving paths is performed for all the states. The decoder continues in this way to advance deeper into the trellis, making decisions by eliminating the least likely paths. Sklar P401

94 The early rejection of the unlikely paths reduces the decoding complexity. Note that the goal of selecting the optimum path can be expressed, equivalently, as choosing the codeword with the maximum likelihood metric, or as choosing the codeword with the minimum distance metric. Sklar P401

95 Sklar P

96 Sklar P404

97 Sklar P403

98 Some channels suffer from switching noise and other burst noise (e.g., telephone channels or channels disturbed by pulse jamming All of these time-correlated impairments result in statistical dependence among successive symbol transmissions. That is, the disturbances tend to cause errors that occur in bursts, instead of as isolated events. Sklar P461

99 Data written to and read from memory in different orders Data bits and corresponding check bits are interspersed with bits from other blocks At receiver, data are deinterleaved to recover original order A burst error that may occur is spread out over a number of blocks, making error correction possible Beard P298

100 Block Interleaving Beard P298

101 Write Read Input data a1, a2, a3, a4, a5, a6, a7, a8, a9, Interleaving a1, a2, a3, a4 a5, a6, a7, a8 a9, a10, a11, a12 a13, a14, a15, a16 Write Transmitting data Received data a1, a5, a9, a13, a2, a6, a10, a14, a3, Through Air a1, a5, a9, a13, a2, a6, a10, a14, a3, De-Interleaving a1, a2, a3, a4 a5, a6, a7, a8 a9, a10, a11, a12 a13, a14, a15, a16 Read Output data a1, a2, a3, a4, a5, a6, a7, a8, a9,

102 Write Burst error Transmitting data on air 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, Received data 0, 1, 0, 0 De-Interleaving 0, 1, 0, 0 0, 1, 0, 0 1, 0, 0, 0 Read Output data 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, Discrete errors

103 消息分组 交织 交织后的消息分组 第 1 帧 第 2 帧 第 3 帧 传输中第 2 帧丢失 解交织后的消息分组

104 A brief historic of turbo codes: The turbo code concept was first introduced by C. Berrou in Today, Turbo Codes are considered as the most efficient coding schemes for FEC Scheme with known components (simple convolutional or block codes, interleaver, soft-decision decoder, etc.) Performance close to the Shannon Limit (E b /N 0 = -1.6 db if R b 0) at modest complexity! Turbo codes have been proposed for low-power applications such as deep-space and satellite communications, as well as for interference limited applications such as third generation cellular, personal communication services, ad hoc and sensor networks Agrawal

105 Alain Glavieux Claude Berrou

106 Figure Block diagram of turbo encoder. Haykin P674

107 y 1 Convolutional Decoder 1 De-interleaving Interleaver x Interleaving Convolutional Decoder 2 De-interleaving y 2 x x : Decoded Information Agrawal

108

109 Figure Noise performances of 1/2 rate, turbo code and uncoded transmission for AWGN channel; the figure also includes Shannon s theoretical limit on channel capacity for code rate r 1/2. Haykin P677

110 Figure Results of the computer experiment on turbo decoding, for increasing number of iterations. Haykin P683

111 Developed in 1960s and re-discovered in 1990s Approach Shannon s limit Use very long codes(normally longer than 1000 bits) Check for errors by using many equations that each add at least three bits together Variable nodes correspond to bits Constraint nodes implement equations Uses iterative decoding Variable nodes estimate the bits And estimate the probabilities of being those bits Constraint nodes combine the estimates to see if they satisfy the equations If not, they determine which variable nodes are likely different than their estimates. Beard P297

112 Tanner Graph for LDPC Iterative Decoding Beard P297

113 Iterative procedure continued Estimates from several constraint nodes are sent back to each variable node to create new estimates These are sent again to constraint nodes to check against equations Procedure is called Message passing Belief propagation Beard P297

114 Introduction ARQ FEC Parity Check Block Codes Cyclic Codes CRC (Cyclic Redundancy Check) Convolutional Codes Interleaving Turbo Codes

115 Beard s book P Review Questions 10.2 What is the CRC? Explain how go- back- N ARQ works Problems 10.5 For P = and M = , find the CRC Calculate the Hamming pairwise distances among the following codewords: a , 10101, b , , ,

116