CSCI 465 D ata Communications and Networks Lecture 9

1. CSCI 465Data Communications and NetworksLecture 9 Martin van Bommel CSCI 465Data Communications & Networks

2. Errors • An error occurs when a bit is altered between transmission and reception • binary 1 is transmitted and binary 0 is received or binary 0 is transmitted and binary 1 is received • Single bit error • isolated error that alters one bit but not nearby bits • caused by white noise • Burst error • contiguous sequence of B bits where first and last bits and any number of intermediate bits are received in error • caused by impulse noise or by fading in wireless • effects greater at higher data rates CSCI 465Data Communications & Networks

3. Error Detection • regardless of design you will have errors • can detect errors by using an error-detecting code added by the transmitter • code is also referred to as “check bits” • recalculated and checked by receiver • still chance of undetected error CSCI 465Data Communications & Networks

4. Parity Check • parity • parity bit set so character has even or odd # of ones • even parity – used in synchronous transmission • odd parity – used in asynchronous transmission • even number of bit errors goes undetected • problem • noise impulses often long enough to destroy more than one bit, especially at high data rates CSCI 465Data Communications & Networks

5. Cyclic Redundancy Check (CRC) • one of most common and powerful checks • for a block of k bits, transmitter generates an n-bit frame by adding an (n-k)-bit frame check sequence (FCS) • Transmits n bits which is exactly divisible by some predetermined number • receiver divides frame by that number • if no remainder, assume no error CSCI 465Data Communications & Networks

6. Side: Modulo-2 Arithmetic • Modulo-2 addition uses no carries • Addition and subtraction via exclusive-OR (XOR) 1100 0110 11011+ 1010 – 1100 X 101 –––––– –––––– –––––– 0110 1010 11011 11011 –––––––– 1110111 CSCI 465Data Communications & Networks

7. CRC Using Mod-2 Arithmetic • Define • T = n-bit frame to be transmitted • D = k-bit block of data (message), first k bits of T • F = (n – k)-bit FCS, last (n – k) bits of T • P = pattern of n – k + 1 bits (predetermined divisor) • Want T / P to have no remainder • T = 2n-kD + F (Note: 2n-kD shifts D (n-k) bits left) • F = remainder after dividing 2n-kD by P • Receiver will check that T / P has no remainder CSCI 465Data Communications & Networks

8. CRC Mod-2 Example • Given n = 15, k = 10, (n – k) = 5 • Message D = 1010001101 (10 bits)Pattern P = 110101 (6 bits)FCS F = to be calculated (5 bits)Transmission T = 2n-kD + F • Note: 2n-kD = 25D = 101000110100000 • 2n-kD / P = 1101010110 Remainder 01110 = F • Thus T = 1010001101 01110 CSCI 465Data Communications & Networks

9. CRC Mod-2 Example (2) • T / P Mod-2 should have no remainder • T / P = 1010001101 01110/ 110101110101 111011110101 111010110101 111110110101 101111110101 110101110101 00 CSCI 465Data Communications & Networks

10. CRC Polynomials • Express all values as polynomials in dummy variable X, with binary coefficients • E.g. for D = 110011, D(X) = X5 + X4 + X + 1 for P = 11001, P(X) = X4 + X3 + 1 • This gives R(X) = X3 + X2 + X and thus F = 1110 CSCI 465Data Communications & Networks

11. Error Detection Probability • An error E(X) will be undetectable only if it is divisible by P(X) • The following are detectable if suitable P(X) • All single-bit errors (if P has at least two terms) • All double-bit errors (if P “primitive”) • Any odd number of errors (if P has (X+1) as factor) • Burst error of length less than (n-k) – length of F • Many others CSCI 465Data Communications & Networks