<p> Error Detection/Correction</p><p>· Messages are a sequence of bits, 0’s and 1’s · Bit corruption – channel changes the value of some bits 01 10 · Corruption may be random for each bit (thermal noise for example ). · Corruption also may occur in bursts · Corruption burst: smallest subsequence containing all corrupted bits</p><p> msg sent: 01101110011011 msg recv: 01111110111001 corrupted bits X X X corruption burst: |------|</p><p>1 Definitions · error detection: detect if a message is corrupted (NOT detect which individual bits were corrupted). The ability to perform error detection is usually measured in two ways:</p><p>1. We say that a protocol performs x-bit detection, iff, for ANY message, and for ANY y number of corrupted bits in the message, where y  x, the protocol will detect the corruption.</p><p>2. We say that a protocol performs x-burst detection, iff, for ANY message, and for ANY corruption burst of size y in the message, where y  x, the protocol will detect the corruption.</p><p>· error correction: correct a corrupted message (restore its original value).</p><p>We say a protocol performs x-bit correction iff, in ANY message, if ANY y number of bits are corrupted, where y  x, the protocol can restore the original contents of the message.</p><p>Note that error bursts are not considered.</p><p>2 Accomplished by adding redundant check bits</p><p>3 Code Words</p><p>· code word: sequence M, |M| = m, data bits followed by a sequence R, |R|</p><p>= r, check bits (usually m >> r)</p><p>M R m data r check bits</p><p> bits </p><p>· the R check bits are a function of the M data bits</p><p>· I.e., there is a standard function f, known by both the sender and receiver.</p><p>· A code word is valid iff R = f(M). I often use “legal” and “valid”</p><p> interchangeably</p><p>4 Detecting Corruption</p><p>· Sender computes R = f(M), and sends M;R (; = concatenation)</p><p>· Channel transforms (corrupts) M;R into M';R', and receiver receives M';R'</p><p>· Receiver computes X = f(M')</p><p>· If X=R', the receiver accepts the msg., if X  R', the receiver rejects the</p><p> msg..</p><p>· Note, X  R'  corruption, but (X = R'  no corruption), since the channel</p><p> may turn a valid code word into another valid code word.</p><p>5 Abundance of Illegal Code Words</p><p>· code = a collection of legal code words · Assume for simplicity that you always send m data bits.</p><p>· m data bits 2m legal code words.</p><p>· i.e. 2m+r - 2m = number of illegal code words</p><p>· E.g., if m = 8, and r = 2, 256 valid code words, 1024 - 256 invalid code words.</p><p>· The hope is that if corruption occurs, the channel turns a valid code word into an invalid code word (not another valid code word), since there are more invalid code words than valid ones.</p><p>· The larger the number of illegal code words (i.e. larger r) the better.</p><p>6 Hamming distance</p><p>· Hamming distance of two equal-length code words: #different bits in two</p><p> code words.</p><p>· c1 = 1000100</p><p> c2 = 1011001</p><p> xor = 0011101</p><p>Hamm(c1, c2) = 4</p><p>· Hamming distance of a code = minimum hamming distance of any pair of</p><p> equal-length code words in the code.</p><p>7 Detecting Corruption · To perform x-bit detection, Hamm(code)  x+1</p><p>Why? Assume Hamm(code) ≥ x+1, then all valid code words are separated by at least x+1 bit changes ≥ x+1 ≥ x+1 |<------>|<------>| c2---|------c1------|----c3 e e’ If sender sends c1, and channel corrupts x bits or less (resulting in words e or e’), you end up with an invalid word, because other valid words (e.g. c2 and c3) are at least x+1 bits away from c1. · E.g. byte parity bit (“even”parity bit of a sequence of bits is the XOR of these bits) m = 8 data bits and r = 1 parity bit byte parity (even) code word 10111010 1 single bit error 11111010 1 = illegal code word double bit error 11011010 1 = another valid code word</p><p>Hamm dist of byte parity bit code = 2, and it performs 1-bit detection.</p><p>8 Error Correction To perform x-bit correction, you replace the received invalid code word by the closest legal codeword (in terms of Hamming distance).</p><p>E.g. valid code words 00000 00000 00000 11111 11111 00000 11111 11111 code Hamm dist = 5</p><p>Sender sends 00000 00000 Receiver receives 00000 00011 Receiver corrects to 00000 00000 (closest legal code word to received code word)</p><p>9 Hamming Distance for Correction To perform x-bit correction, Hamm dist of the code  2x + 1</p><p>Why? If x bits corrupted, there is only one valid code word within a distance of x bits of the invalid codeword. </p><p>Assume Hamm(code)  2x + 1, and sender sends code word c1. </p><p>|<------2x+1---->|<------2x+1------>| c2------c1------|------c3 ≤ x ≥ x+1 |<------>|<------>| e e = c1 corrupted by x bits, c2 and c3 are at least x+1 away from e c1 is only x or less away from e (thus correct to c1). </p><p>Assuming only x bits or less can get corrupted, the closest word to e is c1, and the receiver correctly corrects e to c1. </p><p>10 Example</p><p>In the earlier code example, we can always correct  2 bit errors, since Hamm(code) = 5.</p><p>E.g. sender sends 00000 11111 Receiver receives 00000 00111 Receiver corrects to 00000 11111</p><p>This is assuming  2 bits are corrupted.</p><p>If sender sends 00000 11111 And the receiver receives 00000 00011 (3 bit errors) The receiver incorrectly corrects the code word to 00000 00000</p><p>11 Parity Detection</p><p>· Assume each message is of size m*n (m is usually a constant) · Below, b = bit. · Visualize the message (which is just a bit string) as a 2-dimentional array. b[0,0], b[0,1], . . . b[0,m-1] b[1,0], b[1,1], . . . b[1,m-1] . . . b[n-1,0], b[n-1,1], . . . b[n-1,m-1] p[0], p[1] . . . p[m-1] p[i] = XOR of b[0,i], b[1, i], … b[n-1,i] i.e., p[i] = XOR of column i (this yields even parity)</p><p>· Receiver receives data bits and parity bits · checks if p[i] = XOR of b[0,i], b[1, i], … b[n-1,i] · if not, the message is thrown away.</p><p>12 Transmission Order</p><p>· How to send the bits ? ( in what order?) Row by row or column by column?</p><p>· We transmit row by row</p><p> b[0,0], … b[0,m-1], b[1,0], … b[1,m-1], … , b[n-1,0], … b[n-1,m-1], p[0], p[1], … p[m-1]</p><p>Why? Any burst size  m will be detected (the line in the figure represents a corruption burst)</p><p> burst size  m only 1 bit/column may be corrupted</p><p> burst size  m 2 bits/column may be corrupted, i.e., error may not be detected.</p><p>13 Row and column parity bits b[0,0], b[0,1], . . . b[0,m-1] q[0] b[1,0], b[1,1], . . . b[1,m-1] q[1] . . b[n-1,0], b[n-1,1], . . . b[n-1,m-1] q[n-1] p[0], p[1] . . . p[m-1]</p><p>· q[i] = XOR of b[i,0], … b[i,m-1] · p[i] as before · property: any two-bit error is detected (burst size  m+1 errors are detected). · What if we add a bit r on right bottom corner, where r = parity on q’s ? · property with r: any three-bit error is detected</p><p>14</p>