Introduction to Wireless & Mobile Systems. Chapter 4. Channel Coding and Error Control Cengage Learning Engineering. All Rights Reserved.

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1 Introduction to Wireless & Mobile Systems Chapter 4 Channel Coding and Error Control 1

2 Outline Introduction Block Codes Cyclic Codes CRC (Cyclic Redundancy Check) Convolutional Codes Interleaving Information Capacity Theorem Turbo Codes ARQ (Automatic Repeat Request) Stop-and-wait ARQ Go-back-N ARQ Selective-repeat ARQ 2

3 Introduction Information to be transmitted Source coding Channel coding Modulation Transmitter Antenna Air Antenna Information received Source decoding Channel decoding Demodulation Receiver 3

4 Forward Error Correction (FEC) Introduction to Wireless & Mobile Systems, 4 th Edition 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 The major categories of FEC codes are Block codes Cyclic codes Reed-Solomon codes (Not covered here) Convolutional codes, and Turbo codes, etc. 4

5 Linear Block Codes Information is divided into blocks of length k r parity bits or check bits are added to each block (total length n = k + r) Code rate R = k/n Decoder looks for codeword closest to received vector (code vector + error vector) Tradeoffs between Efficiency Reliability Encoding/Decoding compleity Modulo 2 Addition

6 p Linear Block Codes: Eample Eample: Find linear block code encoder G if code generator polynomial g()=1++ 3 for a (7, 4) code; n = Total number of bits = 7, k = Number of information bits = 4, r = Number of parity bits = n - k = 3 Coefficients of = Re = p p 4 = Re = [ 110] = Re = [ 011] = 2 p4 Re = [ 101] [ 111] G = = [ I I is the identity matri P is the parity matri P] 6

7 Linear Block Codes: Eample The Generator Polynomial can be used to determine the Generator Matri G that allows determination of parity bits for a given data bits of m by multiplying as follows: m. G = [1011] = [ ] Data Data Parity Can be done for other combination of data bits, giving the code word c Other combinations of m can be used to determine all other possible code words 7

8 c c ck... cn = m = m c k = m k + 1 = Linear Block Codes k- data and r = n-k redundant bits 1 2 = m p k + m p k +... mk pk k + m p 1 1 1n 1( 1) m 2 p 2n 2 2( 1)... m k p kn ( 1) 8

9 Linear Block Codes The uncoded k data bits be represented by the m vector: m = (m 1, m 2,, m k ) The corresponding codeword be represented by the n-bit c vector: c = (c 1, c 2, c k, c k+1,, c n-1, c n ) Each parity bit consists of weighted modulo 2 sum of the data bits represented by symbol for Eclusive OR or modulo 2 addition 9

10 Linear Block Codes Linear Block Code The block length C of the Linear Block Code is C = mg where m is the information codeword block length, G is the generator matri. G = [I k P] k n where P i = Remainder of [ n-k+i-1 /g()] for i=1, 2,.., k, and I is unit or identity matri. At the receiving end, parity check matri can be given as: H = [P T I n-k ], where P T is the transpose of the matri p. 10

11 Linear Block Codes Eample: Find linear block code encoder G if code generator polynomial g() with k data bits, and r parity bits = n - k 10 0P 1 [ ] 01 0P 2 G = I P = k 0 0 1P where n k+ i 1 i P = Re mainder of, for i = 1, 2,, k g ( ) 11

12 Linear Block Codes Eample: The parity matri P (k by n-k matri) is given by: p11 p12 p P 1 1( n k ) = p21 p22 p 2( n k ) P 2 P = pk1 pk1 pk ( n k ) k P where n k+ i 1 i P = Re mainder of, for i = 1, 2,, k g ( ) 12

13 Block Codes: Linear Block Codes Message vector m Generator matri G Code Vector C Air Code Vector C Parity check matri H T Null vector 0 Transmitter Receiver Operations of the generator matri and the parity check matri Consider a (7, 4) linear block code, given by G as G = H T = P I n k For convenience, the code vector is epressed as c = mc p Where c mp p = = is an (n-k)-bit parity check vector T 13

14 Block Codes: Linear Block Codes Define matri H T T P as H = I n k Received code vector = c e, here e is an error vector, the matri H T has the property P T ch = m c p I n k = mp c = c c = 0 The transpose of matri H T is H T p p p T = PI n k Where I n-k is a n-k by n-k unit matri and P T is the transpose of parity matri P. H is called parity check matri. Compute syndrome as ( ) s = H = c e H = ch eh = eh T T T T T 14

15 Linear Block Codes If S is 0 then message is correct else there are errors in it, from common known error patterns the correct message can be decoded. For the (7, 4) linear block code, given by G as G = H For m = [ ] and c = mg = [ ] If there is no error, the received vector = c, and s=ch T =[0, 0, 0] = 15

16 Linear Block Codes Let c suffer an error such that the received vector =c e =[ ] [ ] =[ ] Then, Syndrome s = H T = [ ] T = [101] = (eh ) This indicates error position, giving the corrected vector as [ ] 16

17 Cyclic Codes It is a block code which uses a shift register to perform encoding and decoding the code word with n bits is epressed as: c()=c 1 n-1 +c 2 n-2 + c n Input D1 where each coefficient c i (i=1,2,..,n) is either a 1 or 0 The codeword can be epressed by the data polynomial m() and the check polynomial c p () as c() = m() n-k + c p () where c p () = remainder from dividing m() n-k by generator g() if the received signal is c() + e() where e() is the error To check if received signal is error free, the remainder from dividing c() + e() by g() is obtained(syndrome). If this is 0 then the received signal is considered error free else error pattern is detected from known error syndromes Output C D 1, D 2 - Registers D2 17

18 Introduction to Wireless & Mobile Systems, 4 th Edition Cyclic Code: Eample Eample: Find the code words c() if m() = and g() = for (7, 4) cyclic code We have n = total number of bits = 7, k = number of information bits = 4, r = number of parity bits = n - k = 3 rem g m rem c k n p = = = 1 ) ( ) ( ) ( Then, ) ( ) ( ) ( c m c p k n = + = 18

19 Cyclic Redundancy Check (CRC) Introduction to Wireless & Mobile Systems, 4 th Edition Cyclic Redundancy Code (CRC) is an error-checking code The transmitter appends an etra 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 19

20 Cyclic Redundancy Check (CRC) 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 eactly divisible by P 20

21 Cyclic Redundancy Check (CRC) 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. n-k +F J should be eactly divisible by P Dividing Q. n-k by P gives Q. n-k /P=Q+R/P Where R is the reminder J=Q. n-k +R. This value of J should yield a zero reminder for J/P 21

22 Common CRC Codes Code-parity check bits CRC-12 Generator polynomial g() CRC CRC-CCITT CRC

23 Common CRC Codes Code Generator polynomial g() Parity check bits CRC CRC CRC-CCITT

24 Convolutional Codes Most widely used channel code Encoding of information stream rather than information blocks Decoding is mostly performed by the Viterbi Algorithm (not covered here) The constraint length K for a convolution code is defined as K=M+1 where M is the maimum 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 A convolution code encoder with n=2 and k=1 or code rate r=1/2 is shown net 24

25 Convolutional Codes: (n=2, k=1, M=2) Encoder 1 0 Input 1 0 D1D 01 1 D2D y 1 y1 2 Output c D 1, D 2 - Registers Input : Output y 1 y 2 : Input : Output y 1 y 2 :

26 State Diagram 10/1 01/ /0 10/ /1 11/1 11/ /0 26

27 0 First input Tree Diagram First output

28 Trellis

29 Interleaving Input data a1, a2, a3, a4, a5, a6, a7, a8, a9, Interleaving Read 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 Write 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, 29

30 Interleaving (Eample) 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 Write 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 30

31 Information Capacity Theorem (Shannon Limit) 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 B log2 1 bits ond N B = + / sec 0 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 31

32 Shannon Limit 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 32

33 Turbo Codes 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 compleity! 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 33

34 Turbo Codes: Encoder Data Source Convolutional Encoder 1 y 1 Interleaving Convolutional Encoder 2 y 2 y (y 1, y2) : Information y i : Redundancy Information 34

35 Turbo Codes: Decoder De-interleaving y 1 Convolutional Decoder 1 Interleaver Interleaving Convolutional Decoder 2 De-interleaving y 2 : Decoded Information 35

36 Automatic Repeat Request (ARQ) Source Transmitter Channel Receiver Destination Encoder Transmit controller Modulation Demodulation Decoder Transmit controller Acknowledge 36

37 Stop-And-Wait ARQ (SAW ARQ) Introduction to Wireless & Mobile Systems, 4 th Edition Retransmission Transmitting data Time Received data Error Time Output data Time ACK: Acknowledge NAK: Negative ACK 37

38 Stop-And-Wait ARQ (SAW ARQ) Given n = number of bits in a block, k = number of information bits in a block, D = round trip delay, R b = bit rate, P b = BER of the channel, and P ACK (1- P b ) n Throughput: S SAW = (1/T SAW ). (k/n) = [(1- P b ) n / (1 + D * R b / n) ] * (k/n) where T SAW is the average transmission time in terms of a block duration T SAW = (1 + D.R b / n). P ACK + 2 (1 + D R b / n) P ACK (1- P ACK ) + 3 (1 + D R b / n) P ACK (1- P ACK ) = (1+ D R b /n) P ACK i (1-P ACK ) i-1 = (1+ D R b /n) P ACK /[1-(1-P ACK )] 2 = (1 + D R b / n) / P ACK i=1 38

39 Go-Back-N ARQ (GBN ARQ) Go-back 3 Go-back 5 Transmitting data Time Received data Error Error Time Output data Time 39

40 Go-Back-N ARQ (GBN ARQ) Throughput S GBN = (1/T GBN ) (k/n) where = [(1- P b ) n / ((1- P b ) n + N (1-(1- P b ) n ) )] (k/n) T GBN = 1. P ACK + (N+1). P ACK. (1- P ACK ) +2. (N+1). P ACK. (1- P ACK ) 2 +. = P ACK +P ACK [(1-P ACK )+(1-P ACK ) 2 +(1-P ACK ) 3 + ]+ P ACK [N (1-P ACK )+2 N (1-P ACK ) 2 +3 N (1-P ACK ) 3 + ] = P ACK +P ACK [(1-P ACK )/{1-(1-P ACK )}+ N (1-P ACK )/{1-(1-P ACK )} 2 ] = 1+ N(1-P ACK )/P ACK 1 + (N [1 - (1- P b ) n ])/ (1- P b ) n 40

41 Selective-Repeat ARQ (SR ARQ) Retransmission Retransmission Transmitting data Time Received data Error Error Time Buffer Time Output data Time 41

42 Selective-Repeat ARQ (SR ARQ) Throughput S SR = (1/T SR ) * (k/n) = (1- P b ) n * (k/n) where T SR = 1.P ACK + 2 P ACK (1- P ACK ) + 3 P ACK (1- P ACK ) 2 +. = P ACK i (1-P ACK ) i-1 i= 1 = P ACK /[1-(1-P ACK )] 2 = 1/(1- P b ) n where P ACK (1-P b ) n 42

Communication Theory II

Communication Theory II Lecture 24: Error Correction Techniques Ahmed Elnakib, PhD Assistant Professor, Mansoura University, Egypt May 14 th, 2015 1 Error Correction Techniques olinear Block Code Cyclic