Communication Theory II

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

2 Error Correction Techniques olinear Block Code Cyclic Code Reed-Solomon codes oconvolutional Codes oturbo Codes otrellis-coded Modulation Source of Information Source Encoder Channel Encoder Source Decoder Channel Decoder User of Information Modulator De-Modulator Channel 2

3 Linear Block Codes o Information is divided into blocks of length k o m parity bits or check bits are added to each block Total length: n = k + m o Code rate, code efficiency: R = k/n = k/(k + m) o Decoder looks for code word closest to received vector Received vector = code vector + error vector o Tradeoffs between Efficiency Reliability Encoding/decoding complexity o n digits of code (code vector c) is formed as a linear combination of k data digits (data vector d) 3

4 Systematic Linear Block Codes Systematic Identity matrix Generator matrix Knowing the data digits d, we can calculate the check digits 4

5 Example of a Linear Block Code Encoder c 1 = d 1 c 2 = d 2 c 3 = d 3 c 4 = d 1 + d 3 c 5 = d 2 + d 3 c 6 = d 1 + d 2 Generators: p 1 = p 2 = p 3 = d c p

6 Linear Block Code Decoder o Decoder looks for code word closest to received vector Received vector = code vector + error vector 6

7 Cyclic Codes o A subclass of linear block code o A procedure for selecting a generator matrix, most useful for higher-order correcting codes (t > 1, i.e., more than a single error) o Use shift registers to perform encoding and decoding o The code word with n bits is expressed as a polynomial c(x)=c 1 x n-1 +c 2 x n-2 + c n where each c i is either a 1 or 0. 7

8 Convolutional Codes o Among most commonly used channel codes E.g., in GSM and IS-95 o Encoding of information stream rather than information blocks Output (encoded bits) depends not only on current k input bits but also on past N 1 data bits (N > 1) Easy implementation using N-stage shift register and v-module 2-adder Input vector= a, output vector= v a inputs and v outputs. denote the number of bits in the vector Code rate (efficiency) a v o Decoding most often based on the Viterbi Algorithm a a = 1 bit v 1 v 2 v = 2 bit 8

9 Convolutional Codes Encoder o Output at a certain time depend on the current input + (N 1) previous inputs (N is the no. of shift registers) v = G d o Example: [output matrix] 2x1 = [generator Matrix] 2x3 [Input current state] 3x1 v 1 = g 1 v 2 g [s 1 s 2 s 3 ] 2 ogenerator Matrix [G] = g 1 g 2 ogenerators: g 1 =[1 0 1] g 2 =[1 1 1] input a s 1 a = 1 bit s 2 s 3 v 1 = s 1 + s 3 v = 2 bit v 2 = s 1 + s 2 + s 3 g 9 1 =[1 0 1] g 2 =[1 1 1]

10 How the encoder works? o Assume that we decode a message of k-bits input message=[11010], k = 5 o Initially, all the stages are clear d = [0 0 x] o Step -1: first input bit enter input a = 1 Current state [s 2 s 3 ]=00 d=[1 0 0] output v = [1 1] o Step -2: second input bit enter input a = 1 Current state [s 2 s 3 ]= 1 0 d=[1 1 0] output v = [1 0] input a a = 1 bit ostep -3: third input bit enter input a = 0 Current state [s 2 s 3 ]= 1 1 d=[0 1 1] output v = [1 0] s 1 s 2 s 3 v 1 = s 1 + s 3 v = 2 bit v 2 = s 1 + s 2 + s 3 g 10 1 =[1 0 1] g 2 =[1 1 1] 10

11 How the encoder works? o Step -4: fourth input bit enter input a = 1 Current state [s 2 s 3 ]= 0 1 d=[1 0 1] output v = [0 0] o Step -5: fifth input bit enter input a = 0 Current state [s 2 s 3 ]= 1 0 d=[0 1 0] output v = [0 1] ostep -6: first zero padding bit enter input a = 0 Current state [s 2 s 3 ]= 0 1 d=[0 0 1] output v = [1 1] ostep -7: second zero padding bit enter input a = 0 Current state [s 2 s 3 ]= 0 0 d=[0 0 0] output v = [0 0] o Input message=[11010], k = 5 o Output sequence: n = k+n-1=7 [11,11,10,00,11,11,00] o Code rate=1/2 (why?) Always k >> N othe input of length k is augmented (padded) with N-1 zeros (why?) To clear all stages before the next message enter The output for a message of length k is a sequence of n = k + N 1 digits each of v bits oeach input data digit influences N group of v digits in the output N is the number of shift registers, N is called also the constraint length input a s 1 a = 1 bit s 2 s 3 v 1 = s 1 + s 3 v = 2 bit v 2 = s 1 + s 2 + s 3 g =[1 0 1] g 2 =[1 1 1]

12 ogenerators: g 1 = [1 0 1] g 2 = [1 1 1] Convolutional Codes Encoder (Example) i/p Past stage o/p input a s 1 a = 1 bit s 2 s 3 v 1 = s 1 + s 3 v = 2 bit v 2 = s 1 + s 2 + s 3 g 9 1 =[1 0 1] g 2 =[1 1 1] New state 12

Convolutional Codes Encoder state diagram i/p Past stage o/p 0 0 0 0 0 1 0 0 1 1 0 0 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 1 1 0 1 New state odecoding using state diagram or truth table:

13 Convolutional Codes Encoder state diagram i/p Past stage o/p New state odecoding using state diagram or truth table: Input message=[11010], k = 5 Output sequence: n = k+n-1=7 [11,10,10,00,01,11,00] Input message padding g 1 = g 2 = [1 1 1] i/p Past stage o/p

14 Turbo Codes Brief history of turbo codes: First introduced by C. Berrou et al. only in 1993 Considered as the most efficient forward error correction (FEC) coding schemes Performance close to the (perfect) Shannon Limit at modest complexity! Turbo codes use known components Including: simple convolutional or block codes, interleaver, soft-decision decoder Turbo codes proposed for Low-power applications Such as deep-space and satellite communications, Interference-limited applications Such as 3G cellular, PCS (personal communication services), ad hoc networks and sensornets 14

15 Turbo Codes: Encoder Data Source X X Interleaving Convolutional Encoder 1 Convolutional Encoder 2 Y 1 Y Y 2 (Y 1, Y2) X: Information Y i : Redundancy Information 15

16 Interleaver o Interleaving is heavily used in wireless communication for protection against burst errors oburst errors that wipe out some or all a sequential set o digits due to: A stroke of lightening Human made electrical disturbance Magnetic tape defects on magnetic storage system 16

17 Concept of Interleaver 1) Sender writes row-byrow into buffer 2) Read col-by-col from buffer onto link 1) Write col-by-col from link into buffer 2) Receiver reads row-byrow from buffer 17

18 How Interleaver works? o Interleaver disperses burst error into 4 discrete errors It is much easier to correct 4 discrete 1-bit errors, than to correct 1 continuous 4-bit burst error 18

19 Trellis-Coded Modulation o Trellis codes for band-limited channels result from the treatment of modulation and coding as a combined entity rather than as two separate operations o The combination itself is referred to as trellis-coded modulation (TCM) o This form of signaling has three basic requirements: 1. Number of signal points in the constellation used is larger than what is required for the modulation format of interest with the same data rate; the additional signal points allow redundancy for forward error-control coding without sacrificing bandwidth 2. Convolutional coding is used to introduce a certain dependency between successive signal points, such that only certain patterns or sequences of signal points are permitted for transmission 3. Soft-decision decoding is performed in the receiver, in which the permissible sequence of signals is modeled as a trellis structure; hence the name trellis codes 19

20 Questions 20

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

Introduction to Wireless & Mobile Systems Chapter 4 Channel Coding and Error Control 1 Outline Introduction Block Codes Cyclic Codes CRC (Cyclic Redundancy Check) Convolutional Codes Interleaving Information