The Art of Error Correcting Coding phần 1 ppsx

[...]... following: s ¯ 00 01 10 11 00 11 10 01 0000 10 00 010 0 00 01 011 0 11 10 0 010 011 1 10 11 0 011 11 11 1 010 11 01 010 1 10 01 110 0 Decoding with the standard array proceeds as follows Let r = v + e be the received ¯ ¯ ¯ word Find the word in the array and output as decoded message u the header of the column ¯ in which r lies Conceptually, this process requires storing the entire array and matching ¯ the received word... of correctable error patterns, t i=0 i Example 1. 3.5 The binary repetition (3, 1, 3) code has generator matrix G = 1 1 1 and parity-check matrix 1 1 0 H = 1 0 1 Accordingly, its standard array is the following: s ¯ 0 1 00 11 10 01 000 10 0 010 0 01 111 011 10 1 11 0 The four vectors in the second column of the array (i.e., the coset leaders) are the elements of the Hamming sphere S1 (000) in Figure 1. 5,... Sent (1. 22) Received 1- p 0 0 p p 1 1-p 1 Figure 1. 6 A binary symmetric channel model Table 1. 1 The standard array of a binary linear block code s ¯ ¯ u0 = 0 ¯ u2 ¯ ··· uk 1 ¯ ¯ 0 s1 ¯ s2 ¯ ¯ v0 = 0 ¯ e1 ¯ e2 ¯ v1 ¯ e 1 + v1 ¯ ¯ e 2 + v1 ¯ ¯ ··· ··· ··· v2k 1 ¯ e1 + v2k 1 ¯ ¯ e2 + v2k 1 ¯ ¯ s2n−k 1 ¯ e2n−k 1 ¯ e2n−k 1 + v1 ¯ ¯ ··· e2n−k 1 + v2k 1 ¯ ¯ INTRODUCTION 11 where H is the. .. recommend The Art of Error Correcting Coding as an excellent introductory and reference book on the principles and applications of error correcting codes Professor Hideki Imai The University of Tokyo Tokyo, Japan The ECC web site A companion web site for the book The Art of Error Correcting Coding has been set up and is located permanently at the following URL address: http:/ /the- art- of- ecc.com The ECC... v0 ¯ v1 ¯   v0,0 v1,0       G= =    vk 1 ¯ vk 1, 0 v0 ,1 v1 ,1 vk 1, 1 (1. 11) ··· ··· v0,n 1 v1,n 1 · · · vk 1, n 1      (1. 12) 8 INTRODUCTION Due to the fact that C is a k-dimensional vector space in V2 , there is an (n − k)-dimensional dual space C , generated by the rows of a matrix H , called the parity-check matrix, such that GH = 0, where H denotes the transpose of H... 2.2.3 10 INTRODUCTION Example 1. 3.2 Consider the binary linear (4, 2, 2) code from Example 1. 3 .1 Let messages ¯ and code words be denoted by u = (u0 , u1 ) and v = (v0 , v1 , v2 , v3 ), respectively From ¯ (1. 20) it follows that v2 = u 0 + u 1 v3 = u0 The correspondence between the 22 = 4 two-bit messages and code words is as follows: (00) → (0000) ( 01) → ( 011 0) (10 ) → (10 11) (11 ) → (11 01) (1. 21) 1. 3.2... 1e-08 1e-09 1e -10 0 .1 0. 01 0.0 01 0.00 01 1e-05 p Figure 1. 7 Exact value and upper bound on the probability of undetected error for a binary linear (4,2,2) code over a BSC Example 1. 4 .1 For the binary linear (4, 2, 2) code of Example 1. 3.2, W(C) = (1, 0, 1, 2, 0) Therefore, Equation (1. 28) gives Pu (C) = p2 (1 − p)2 + 2 p3 (1 − p) Figure 1. 7 shows a plot of Pu (C) compared with the upper bound in the right-hand... their algorithmic aspects (the “how” they work), rather than to their theoretical aspects (the ‘why’ they work) Finally, combinations of codes and interleaving for iterative decoding and of coding and modulation for bandwidth-efficient transmission are the topic of the last part of the book 1. 1 Error correcting coding: Basic concepts All error correcting codes are based on the same basic principle: redundancy... apart as possible ¯ Consider two vectors x1 = (x1,0 , x1 ,1 , , x1,n 1 ) and x2 = (x2,0 , x2 ,1 , , x2,n 1 ) in ¯ ¯ ¯ ¯ ¯ V2 Then the Hamming distance between x1 and x2 , denoted dH (x1 , x2 ), is defined as the number of elements in which the vectors differ, n 1 ¯ ¯ dH (x1 , x2 ) = i : x1,i = x2,i , 0≤i . convolutional codes . . . 11 5 5.6.2 RCPCcodes 11 6 Problems 11 6 6 Modifying and combining codes 11 9 6 .1 Modifyingcodes 11 9 6 .1. 1 Shortening 11 9 6 .1. 2 Extending 12 1 6 .1. 3 Puncturing 12 2 6 .1. 4 Augmenting,. h1" alt="" The Art of Error Correcting Coding The Art of Error Correcting Coding, Second Edition Robert H. Morelos-Zaragoza  2006 John Wiley & Sons, Ltd. ISBN: 0-470- 015 58-6 The Art of Error. 8 1. 3 .1 Encoding with G and H 8 1. 3.2 Standardarraydecoding 10 1. 3.3 Hamming spheres, decoding regions and the standard array . . . . . 12 1. 4 Weightdistributionanderrorperformance 13 1. 4 .1 Weight