Ch. 6 Error Detection and Correction

1
Ch. 6 Error Detection and Correction

• By William Stallings (Tenth Edition)

2
6.1 Types of Errors

• An error occurs when a bit is altered.
• A single bit error is an isolated error.
• Error Bursta group of bits in which two
  successive erroneous bits are always separated by
  less than a given number x of correct bits.
• See Figure 6.1.

3
6.2 Error Detection

• Probabilities of Error Types
• Pb Probability of a single bit error (BER).
• P1 A frame arrives with no bit errors
• P2 A frame arrives with some undetected bit
  errors.
• P3 A frame arrives with some detected bit
  errors.
• Assume that no parity is sent (P30)
• P1 (1-Pb) F, where F is the frame size.
• P2 1 - P1
• Example If Pb or F is "large" then P2 could
  become a problem.
• Fig. 6.3--The error detection process.

4
6.3 Parity Checks

• Parity CheckSee Figure 6.3
• The simplest scheme.
• Even number of errors is undetectable.
• Cyclic Redundancy Check (CRC)
• Very powerful and often used.
• Given k message bits, n "check" bits are
  generated.
• kn bits are transmitted together.

5
6.5 Cyclic Redundancy Check-CRC

• Modulo 2 Arithmetic
• Binary addition without carries (exclusive or
  operation.)
• The simple view is to consider variable, binary
  strings
• D-- data or message (k-bits)
• F--frame check sequence (n-k bits)
• P--pattern of n-k1 bits
• T--transmitted frame (n bits)

6
6.5 CRC (cont.)

• Consider the following three step process
• STEP 1 Multiply D by 2n-k.
• n-k 0's will be shifted in, from the right.
• STEP 2 Divide by P
• This will produce a quotient and a remainder.
• The quotient (Q) will not be used, but the
  remainder (R) is used as F.
• STEP 3 Add R to the shifted version of D to
  produce T.
• R replaces the n 0's that were added to D.
• T will be transmitted.

7
6.5 CRC

• Why does this work?
• We are looking for something that is evenly
  divisible by P.
• T/P (D2n-k R)/P (D2n-k/P) R/P
  QR/P R/P.
• But R/P R/P 0, because of modulo 2 arithmetic.
• Hence T/PQ which is evenly divisible by P
  (remainder is 0).
• Example 6.6--page 195.

8
6.5 CRC(cont.)

• Suppose that bit errors are possible during
  transmission.
• Model the error pattern (E) as a binary string
  with a 1 in the position of a bit that has been
  received incorrectly.
• Received frame Tr T E.
• Divide by P Tr/P T/P E/P Q E/P
• If the remainder is 0, then assume E0.
• If P is chosen properly, very few error patterns
  will be evenly divisible by P (i.e. have a
  remainder of 0) and very few undetected errors
  will occur.

9
6.5 CRC

• Polynomials (Example 6.7 and Fig. 6.5)
• The CRC process can be viewed as binary
  polynomials of X.
• The coefficients of the polynomials correspond to
  the previously defined bit-string variables.
• Arithmetic operations use modulo-2 (this makes
  it an "algebra.")
• Generating Polynomials
• Carefully chosen polynomials can allow the
  detection of many types of errors(p.193).
• Ex CRC-12, CRC-16, CRC-CCITT, CRC-32.

10
6.5 CRC

• CRCs can be generated using hardware (See Fig.
  6.6 and 6.7 and Example 6.8).
• Basic components
• D Type flip-flops (or J-K Equivalents)-- one for
  each CRC bit.
• Exclusive OR gates (or equivalents)
• Operation
• Flip-Flops are initialized to 0.
• Message bits are clocked into the circuit.

11
6.4 Error Correction

• Fig. 6.7 Error Correction Process
• k bits are mapped into n bit block.
• Result is called a codeword.
• Example 6.7--Forward Error Correction
• (n-k)/k is called the redundancy.
• k/n is called the code rate.
• Fig. 6.8 How Coding Improves System Performance

12
6.5 Line Configurations

• Topology refers to the physical arrangement of
  stations on a link.
• Point-to-point
• Multipoint--saves you money!
• Duplexity refers to the direction and timing of
  signal flow.
• Full-duplex digital lines generally require
  4-wires.
• Half-duplex digital lines require 2 wires
• Fig. 6-9 Traditional Computer/Terminal
  Configurations.

13
6.5 Line Discipline

• Point-to-Point Three Phases (not in 8th
  Edition)
• Establishment
• Data Transfer
• Termination
• Multipoint Links
• Poll--the primary requests data from secondary
• Select--the primary has data to send and informs
  secondary that data are coming.
• Contention--no primary a station can transmit
  when the line is free used in LANs and satellite
  systems.

14
Appendix G Interfacing (Fig.G.1)

• DTE--Data Terminal Equipment (not in 8th Edition)
• Equipment consisting of digital end instruments
  that convert the user information into data
  signals for transmission, or reconvert the
  received data signals into user information.
• DCE--Data Circuit-terminating Equipment
• In a data station, the equipment that provides
  the signal conversion and coding between the data
  terminal equipment (DTE) and the line.
• DCE may be separate equipment or an integral part
  of the DTE or intermediate equipment.

15
G.1 Interfacing (cont.)

• Interchange Circuits
• The connection between the DTE and DCE.
• Standards--Physical Layer of the OSI Model
• V.24/EIA-232-F (RS-232--1962)
• X.21--15 wire interface for public switched
  network interfacing.
• ISDN Physical Interface (8 wire interface).

16
G.1 Four Characteristics

• Mechanical
• Pertain to the actual physical connection of the
  DTE and DCE (the terminator plugs and sockets).
• Electrical
• The voltage levels and timing of voltage changes.
• Functional
• The functions performed by various interchange
  circuits data, control, timing and ground.
• Procedural
• The sequence of events for transmitting data.

17
G.1 EIA-232-F

• Mechanical (ISO 2110)
• DB-25 connector (a 25 pin connector)
• Fig. G.2.
• Electrical(V.28)
• Digital signaling up to 20 kbps up to 15m.
• Logic 1 and OFF less than -3 volts
• Logic 0 and ON greater than 3 volts
• And more (C, R, short circuit current, max
  voltages, slew rate, etc.)

18
G.1 EIA-232-F (p.2)

• Functional (V.24)
• Table G-1--Interchange Circuits
• Procedural (V.24)
• Fig. G.4

19
G.1 Loopback Testing

• EIA-232-F control circuits assist in loopback
  testing and fault isolation.
• Local loopback tests are used to check the
  functioning of the local interface and the local
  DCE.
• Remote loopback tests are used to check the
  transmission channel and the remote DCE.
• Figure G.3 Local and remote loopback.

20
G.1 The Null Modem

• Used to connect two DTEs directly (no DCEs used).
• It is not a real modem, but simply a cable that
  rewires the circuits to trick the DTEs into
  thinking that they are talking with DCEs.
• Fig. G.5 illustrates the null modem wiring.

21
G.2 ISDN Physical Interface

• X.21--15 pin connection for digital interface to
  public switched networks.
• ISDN--ISO 8877 specifies an 8 pin connector.
• The reduction of interface circuits forced
  greater complexity in the logic circuits at each
  end of the cable, but integrated circuits have
  become cheap whereas wire remains relatively
  expensive.
• Fig. G.6 shows the ISDN Interface.