# LLC: Frame Construction. Layer 2: Division into two Parts. Error-detecting and -correcting Codes. Error-correcting Codes

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2 Hamming Code Goal: Use of several parity bits, each of them considering several bits (overlapping). Errors can be identified and corrected by combining the parity bits. The Hamming code is the minimal code of this category. Idea: Representation of each natural number by sum of powers to two. In a code word w = z,, z n the parity bits are placed exactly at the k positions, for them the index is a power of two. At the remaining m = n - k positions the data bits are placed. Each of the k additional bits is a parity bit for all places, for which the representation in powers of two contains the position of the additional bit. Page 5 Hamming Code ASCII-Code H A M M I N G Parity bit : Data bit 3, 5, 7, 9, 3 = + 2 Parity bit 2: Data bit 3, 6, 7,, 5 = + 4 Parity bit 4: Data bit 5, 6, 7 6 = Parity bit 8: Data bit 9,, 7 = = + 8 Problem with Hamming code: errors involving several following bits are usually wrongly corrected = = Codeword Receiver: examine parity bits if necessary, sum up indices of the incorrect parity bits index of the incorrect bit -bit errors can definitely be identified and corrected Page 6 Hamming code Error Detection with Cyclic Codes Transmission error Receiver computes parity bits: Summing up the indizes, 2 and 4, bit 7 is detected as false Problem: how to recognize errors in several bits, especially sequences of bit errors? The use of simple parity bits is not suitable. However, in data communication (modem, telephone cables) such errors arise frequently. Weaknesses: 2-bit errors are not corrected (or wrongly corrected!) 3-bit errors are not recognized a) Bit 4 and bit inverted: parity bits, 2, 4, 8 are wrong bit 5 is to be corrected, but does not exist b) Bit 2 and bit 4 inverted parity bits 2, 4 wrong bit 6 is falsely recognized as incorrect c) Bits, 8, 9 inverted all parity bits are correct no error is recognized Page 7 Most often used: Polynomial Codes Idea: a k-bit PDU (a k-,, a ) is seen as a polynomial a k- x k- + + a with the coefficients and. Example: is seen as x 6... x x = x 6 + x 5 + x 2 + For computations, polynomial arithmetic modulo 2 is used, i.e. addition and subtraction without carriage. Both operations become Exclusive-OR operations. Page 8

3 Error Detection with Cyclic Codes CRC - Example Idea for error detection: Sender and receiver agree upon a generator polynomial G(x) = x r + + x. The first and the last coefficient have to be. The sender interprets a data block of length m as polynomial M(x). The sender extends M(x), i.e. adds redundant bits in a way that the extended polynomial M (x) is divisible by G(x). (Redundancy = remainder R(x) by division of the sequence with G(x)) The receiver divides the received extended M(x) by G(x). If the remainder is, there was no error, otherwise some error occurred. Name: Cyclic Redundancy Checksum (CRC) Note: also the parity bit can be seen as CRC, with generator polynomial x +! Data to be transmitted: Generator polynomial: x 4 + x + Sender: : = = x 3 + = R(x) CRC =, sending Receiver: Note: usually, the extra positions are preset with zeros but on some reason e.g. Ethernet uses the inverted bits as presets. : = Data received correctly Page 9 Page Computation of CRC: Shift Registers Shift Registers - Example Implementation by Shift Registers: XOR for substraction AND for applying substraction: first register = : no substraction first register = : substraction Data to be transmitted: Generator polynomial: x 4 + x + Generator polynomial x 4 + x +: R R R R R Simplified realization: R R R R R When no more input is given in the leftmost register, the other registers contain the CRC result. Page Page 2

4 Shift Registers - Example Shift Registers - Example Page 3 Page 4 Shift Registers - Example Shift Registers - Example x x x x x x x x x x x x x x x Page 5 Page 6

8 Step : Send without restrictions No transmission errors No flow control Page 29 Page 3 Step 2: Simple flow control No transmission errors Step 3: Error handling and flow control - sender Send-and-Wait as flow control Page 3 Page 32

11 Explicit Reservation Uses two frame types: reservation frame (very small) in the first phase data frame (constant length) in the second phase Variant : without contention Only suitable for small number of users Each user i is assigned the i-th slot in the reservation frame. If it wants to send data, it sets the i-th bit in the reservation frame to. After the reservation phase, all stations having set their reservation bit can send their data in the order of their bits in the reservation frame. reservation frame data frames of stations having reserved Explicit Reservation Uses two frame types: reservation frame (very small) in the first phase data frame (constant length) in the second phase Variant 2: with contention For higher number of users The reservation frame consists of a limited number of contention slots (smaller than the number of participating stations) Users try to get a contention slot (and by that make a reservation for a data slot) by random choice, writing their station number into a slot If there is no collision in the reservation phase, a station may send. reservation frame with contention slots data frames of stations having reserved This procedure is called Bitmap Protocol Page 4 Page 42 Implicit Reservation Implicit Reservation No reservation slots, only data slots of certain length. A window consists of N data slots, windows a cyclically repeated The duration of the window must be longer than the round-trip time Procedure: A station which wants to send observes N slots without doing anything and marks the slots as follows:, if the slot if empty or collided, if the slot is used by somebody else In the following window the station randomly chooses one of the slots marked with (Simplification: choose the first slot marked with ) Two cases: conflict: try again successful transmission: slot reserved for the station as long as it sends data. If the station is not using its slot in one window, the reservation is dismissed. Example: 8 data slots, Stations A - F Reservation Slots: Window Window 2 A C D A B A B/D F ACDABA-F Window 3 A C A B A AC-ABA-- Window 4 A B/F B A F A---BAF- Window 5 A B A F D A---BAFD Window 6 A C E E B A F D t Stations observing 8 slots Collision within slot 7; the other slots are reserved Page 43 Page 44

12 MAC Decentralized Protocols Best known protocol has the name ALOHA Developed on the Hawaiian islands: stations are connected by satellite Very simple principle, no coordination: Stations are sending completely uncoordinated, all using the same frequencies When two (or more) stations are sending at the same time, a collision occurs: both messages are destroyed. Problem: collisions occur even with very small overlaps! Vulnerability period: 2 times the length of a frame When a collision occurs, frames are repeated after a random time Problem: since traffic runs over a satellite, a sender only hears after very long time, whether the transmission was successful or not. Collision MAC Decentralized Protocols Problem with ALOHA: even small laps already lead to transmission conflicts. Therefore often collisions arise, causing many repetitions: No guaranteed response times Low throughput Improvement: Slotted ALOHA The whole time axis is divided into time slots (similar to TDMA, but time slots are not firmly assigned to stations) The transmission of a block starts at the beginning of a time slot fewer collisions, vulnerability period of one frame length But: the stations must be synchronized! Collision Sender A Sender A Sender B Sender B Sender C t Sender C t Page 45 Page 46 How to estimate the Efficiency of ALOHA? Randomness Which possible states do we have in ALOHA? thinking users backlogged users common channel total traffic G no conflict conflict G: average number of frames per time unit S: throughput, i.e. rate of successful transmissions Relation between G and S? Depends significantly on the traffic structure. Model A: only one sender. No collisions, so S = G is possible Model B: many users, each of them inactive most time new arrivals are totally random Suppose that the total traffic is absolutely random. What is the biggest randomness possible? arrival Random arrivals: t consider time interval of length h. Observe arrivals in that interval. h Arrivals have a certain intensity G (average rate per time unit) Randomness : probability of an arrival in a (small) time interval h is proportional o(h): disturbance function with to the intensity G o( h) for h to h h... to G h For very small h, only or arrivals are possible: Prob(exactly one arrival within interval length h) = G h + o(h) Prob(no arrivals within interval length h) = G h + o(h) Page 47 Page 48

13 Use for ALOHA Throughput vs. Offered Traffic These conditions/requirements lead to the Poisson distribution. Consider the total number i of events in an interval of length T i ( G T) G T Prob(exactly i arrivals in [;T]) = e Back to ALOHA: relation between S and G Suppose that G follows the poisson distribution, i.e. G is totally random. i! S = G Prob(no collision occurs) = G Prob(nobody else started in my vulnerability period) = G Prob(no arrival in interval of length T = 2) for pure ALOHA G Prob(no arrival in interval of length T = ) for slotted ALOHA Prob( arrivals in [;T]) = e -G T = G e -2G for pure ALOHA G e -G for slotted ALOHA Note: randomness (i.e. Poisson) is not valid for high traffic and if waiting time after retransmission is short Page 49 Analytical computation: Both, Pure ALOHA and Slotted ALOHA cannot achieve a high throughput But: simple principle, no coordination necessary between the stations Page 5 MAC Decentralized Protocols MAC Coordination by using a Token Variant of ALOHA for networks with small range exists Similar to ALOHA: no coordination of the stations But: each station which wants to send first examines whether already another station is sending If no sending takes place, the station begins to send. (Carrier Sense Multiple Access, CSMA) Notice: this principle only works within networks having a short transmission delay using the principle within satellite systems is not possible because there would be no chance to know whether a conflict occurred before end of the transmission Advantages: simple, because no master station and no tokens are needed; nevertheless good utilization of the network capacity Disadvantage: no guaranteed medium access, a large delay up to beginning a transmission is possible Introduction of a token (determined bit sequence) Only the holder of the token is allowed to send Token is cyclically passed on between all stations particularly suitable for ring topologies Token Ring (4/6/ Mbit/s) Characteristics: Guaranteed accesses, no collisions Very good utilization of the network capacity, high efficiency Fair, guaranteed response times Possible: multiple tokens But: complex and expensive Passing on of the token Page 5 Page 52

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