Digital Communication Networks

Transcription

1 Digital Communication Networks MIT PROFESSIONAL INSTITUTE, 6.20s July 24-28, 2006 Professor Muriel Medard, MIT Professor, MIT Slide 1

2 Digital Communication Networks Introduction Slide 2

3 Course syllabus Monday AM: Introduction to layering, the physical layer PM: The data link layer Tuesday AM: Delay models and queueing PM: Higher layer protocols: TCP/IP, ATM Wednesday AM: Routing PM: More routing, flow control Thursday AM: Multiple access PM: LANs, MANs, SANs, and switching Friday AM: Wireless Networks PM: Optical Networks Slide 3

4 Logistics Morning lectures: 9:00am - 12:00pm (break around 10:15) Lunch: on your own, 12:00-1:30pm Afternoon lecture: 1:30-4:30 (break around 2:45) Friday: start at 9:00 and end at 3:30pm (lunch: 11:45-12:30) Social: Dinner on Thursday evening Slide 4

5 Reference textbooks Computer Networks, by Peterson and Davie Most current on the internet Data Networks, by Bertsekas and Gallager Strong on probabilistic modeling and analysis Network Optimization: Continuous and Discrete Models, by Dimitri P. Bertsekas Network algorithms and routing, network flows, optimization theory Introduction to Probability, by Dimitri P. Bertsekas and John N. Tsitsiklis Slide 5

6 Data Networks Fundamental aspects of network Design and Analysis: Architecture Layering Topology design Protocols Pt.-to-Pt. Multiple access End-to-end Algorithms Error recovery Routing Flow Control Analysis tools Probabilistic modeling Queueing Theory Slide 6

7 Network Applications Resource sharing Computing Mainframe computer (old days) Today, computers cheaper than comm (except LANS) Printers, peripherals Information DB access and updates E.g., Financial, Airline reservations, etc. Services , FTP, Telnet, Web access Video conferencing DB access Client/server applications Slide 7

8 Network coverage areas Wide Area Networks (WANS) Span large areas (countries, continents, world) Use leased phone lines (expensive!) 1980 s: 10 Kbps, 2000 s: 2.5 Gbps User access rates: 56Kbps 155 Mbps typical Shared comm links: switches and routers E.g, IBM SNA, X.25 networks, Internet Local Area Networks (LANS) Span office or building Single hop (shared channel) (cheap!) User rates: 10 Mbps 1 Gbps E.g., Ethernet, Token rings, Apple-talk Metro Area networks (MANS) Storage area networks Slide 8

9 Network services Synchronous Session appears as a continuous stream of traffic (e.g, voice) Usually requires fixed and limited delays Asynchronous Session appears as a sequence of messages Typically bursty E.g., Interactive sessions, file transfers, Connection oriented services Long sustained session Orderly and timely delivery of packets E.g., Telnet, FTP Connectionless services One time transaction (e.g., ) QoS Slide 9

10 Switching Techniques Circuit Switching Dedicated resources Packet Switching Shared resources Virtual Circuits Datagrams Slide 10

11 Circuit Switching Each session is allocated a fixed fraction of the capacity on each link along its path Dedicated resources Fixed path If capacity is used, calls are blocked E.g., telephone network Advantages of circuit switching Fixed delays Guaranteed continuous delivery Disadvantages Circuits are not used when session is idle Inefficient for bursty traffic Circuit switching usually done using a fixed rate stream (e.g., 64 Kbps) Difficult to support variable data rates Slide 11

12 Problems with circuit switching Many data sessions are low duty factor (bursty), (message transmission time)/(message interarrival time) << 1 Same as: (message arrival rate) * (message transmission time) << 1 The rate allocated to the session must be large enough to meet the delay requirement. This allocated capacity is idle when the session has nothing to send If communication is expensive, then circuit switching is uneconomic to meet the delay requirements of bursty traffic Also, circuit switching requires a call set-up during which resources are not utilized. If messages are much shorter than the call set-up time then circuit switching is not economical (or even practical) More of a problem in high-speed networks Slide 12

13 Circuit Switching Example L = message lengths λ = arrival rate of messages R = channel rate in bits per second X = message transmission delay = L/R R must be large enough to keep X small Bursty traffic => λx << 1 => low utilization Example L = 1000 bytes (8000 bits) λ = 1 message per second X < 0.1 seconds (delay requirement) => R > 8000/0.1 = 80,000 bps Utilization = 8000/80000 = 10% With packet switching channel can be shared among many sessions to achieve higher utilization Slide 13

14 Packet Switched Networks Messages broken into Packets that are routed To their destination PS PS PS PS Packet Network PS PS PS Buffer Packet Switch Slide 14

15 Packet Switching Datagram packet switching Route chosen on packet-by-packet basis Different packets may follow different routes Packets may arrive out of order at the destination E.g., IP (The Internet Protocol) Virtual Circuit packet switching All packets associated with a session follow the same path Route is chosen at start of session Packets are labeled with a VC# designating the route The VC number must be unique on a given link but can change from link to link Imagine having to set up connections between 1000 nodes in a mesh Unique VC numbers imply 1 Million VC numbers that must be represented and stored at each node E.g., ATM (Asynchronous transfer mode) Slide 15

16 Virtual Circuits Packet Switching For datagrams, addressing information must uniquely distinguish each network node and session Need unique source and destination addresses For virtual circuits, only the virtual circuits on a link need be distinguished by addressing Global address needed to set-up virtual circuit Once established, local virtual circuit numbers can then be used to represent the virtual circuits on a given link: VC number changes from link to link Merits of virtual circuits Save on route computation Need only be done once at start of session Save on header size Facilitate QoS provisioning More complex Less flexible 3 6 VC13 VC7 VC3 5 8 Node 5 table VC4 VC3 VC7 (3,5) VC13 -> (5,8) VC3 (3,5) VC7 -> (5,8) VC4 (6,5) VC3 -> (5,8) VC7 2 9 Slide 16

17 Circuit vs packet switching Advantages of packet switching Efficient for bursty data Easy to provide bandwidth on demand with variable rates Disadvantages of packet switching Variable delays Difficult to provide QoS assurances (Best-effort service) Packets can arrive out-of-order Switching Technique Network service Circuit switching => Synchronous (e.g., voice) Packet switching => Asynchronous (e.g., Data) Virtual circuits => Connection oriented Datagram => Connectionless Slide 17

18 Circuit vs Packet Switching Can circuit switched network be used to support data traffic? Can packet switched network be used for connection oriented traffic (e.g., voice)? Need for Quality of service (QoS) mechanisms in packet networks Guaranteed bandwidth Guaranteed delays Guaranteed delay variations Packet loss rate Etc... Slide 18

19 7 Layer OSI Reference Model Application Application Virtual network service Presentation Presentation Virtual session Session Session Virtual link for end to end messages Transport Transport Virtual link for end to end packets Network Network Network Network Data link Control Virtual link for reliable packets DLC DLC DLC DLC Data link Control physical interface Virtual bit pipe phys. int. phys. int. phys. int. phys. int. physical interface Physical link Slide 19 External Site subnet node subnet node External site

20 Layers Presentation layer Provides character code conversion, data encryption, data compression, etc. Session layer Obtains virtual end to end message service from transport layer Provides directory assistance, access rights, billing functions, etc. Standardization has not proceeded well here, since transport to application are all in the operating system and don't really need standard interfaces Focus: Transport layer and lower Slide 20

21 Transport Layer The network layer provides a virtual end to end packet pipe to the transport layer. The transport layer provides a virtual end to end message service to the higher layers. Slide 21 The functions of the transport layer are: 1) Break messages into packets and reassemble packets of size suitable to network layer 2) Multiplex sessions with same source/destination nodes 3) Resequence packets at destination 4) recover from residual errors and failures 5) Provide end-to-end flow control

22 Network layer The network layer module accepts incoming packets from the transport layer and transit packets from the DLC layer It routes each packet to the proper outgoing DLC or (at the destination) to the transport layer Typically, the network layer adds its own header to the packets received from the transport layer. This header provides the information needed for routing (e.g., destination address) Transport layer Each node contains one network Layer module plus one Link layer module per link Network layer DLC layer link 1 DLC layer link 2 DLC layer link 3 Slide 22

23 Link Layer Responsible for error-free transmission of packets across a single link Framing Determine the start and end of packets Error detection Determine which packets contain transmission errors Error correction Retransmission schemes (Automatic Repeat Request (ARQ)) Slide 23

24 Physical Layer Responsible for transmission of bits over a link Propagation delays Time it takes the signal to travel from the source to the destination Signal travel approximately at the speed of light, C = 3x10 8 meters/second E.g., LEO satellite: d = 1000 km => 3.3 ms prop. delay GEO satellite: d = 40,000 km => 1/8 sec prop. delay Ethernet cable: d = 1 km => 3 µs prop. delay Transmission errors Signals experience power loss due to attenuation Transmission is impaired by noise Simple channel model: Binary Symmetric Channel P = bit error probability Independent from bit to bit In reality channel errors are often bursty 0 1 P 1-P 1-P P 0 1 Slide 24

25 Internet Sub-layer A sublayer between the transport and network layers is required when various incompatible networks are joined together This sublayer is used at gateways between the different networks It looks like a transport layer to the networks being joined It is responsible for routing and flow control between networks, so looks like a network layer to the end-to-end transport layer In the internet this function is accomplished using the Internet Protocol (IP) Often IP is also used as the network layer protocol, hence only one protocol is needed Slide 25

26 Internetworking with TCP/IP FTP client FTP Protocol FTP server TCP TCP Protocol TCP ROUTER IP IP Protocol IP Protocol IP IP Ethernet driver Ethernet Protocol Ethernet driver token ring driver token ring Protocol token ring driver Ethernet token ring Slide 26

27 Encapsulation user data Appl header user data Application IP header TCP header TCP header IP datagram application data TCP segment application data Ethernet IP TCP Ethernet header header header application data trailer Ethernet frame 46 to 1500 bytes TCP IP Ethernet driver Ethernet Slide 27

28 Physical Layer and Coding Muriel Médard EECS LIDS MIT

29 Overview A variety of physical media: copper, free space, optical fiber Unified way of addressing signals at the input and the output of these media: basic models Nyquist sampling theorem How we use the channels: modulation Modeling the net effect of channels: transition probabilities and errors Making up for channel errors: principles of coding theoretical limits MIT

30 Signals Channel has a physical signal traversing a physical medium Electromagnetic signal Polarization of the wave TM TE Propagation of the wave MIT

31 Signals What do we mean by frequency? We can express a signal by a superposition of sinusoids, each of them with its own frequency A continuous signal is bandlimited to a bandwidth W at carrier frequency f 0 if it can be expressed in terms of sinusoids in a range of frequencies of width W around a frequency f 0 If we multiply the signal by a frequency shift, then we operate around 0, the baseband representation passband baseband f 0 -W/2 f 0 f 0 +W/2 -W/2 0 W/2 MIT

32 Nyquist sampling theorem We can wholly reconstruct a signal from its periodic samples as long as those samples are within 1/W of each other Continuous signal x(t), discrete time signal is x[n] = x(n T) where T = 1/W x(t) 0 T 2T 3T 4T 5T 6T t MIT

33 Modulation: creating signals How we generate signals, by varying amplitude, phase, frequency On-off keyed (OOK): send 1, 0, a special case of amplitude modulation Frequency-shift keyed (FSK), change frequencies: Phase-shift key (PSK): 2 PSK 4 PSK MIT

34 Modulation Pulse position modulation (PPM) Combining phase and amplitude, for instance: PAM 5x5 symbol code for 100BASE-T2 PHY MIT

35 Channels Canonical channel model: signal goes into channel, there is a non-additive effect and addition of noise N[n] Tx X[n] G G(X[n]) Y[n] Rx Very commonly, G is a multiplicative effect: Y[n] = g X[n] + N[n] Y[n] = g -1 X[n-1] + g 0 X[n] + N[n] INTERSYMBOL-INTERFERENCE (ISI) MIT

36 Effect of the channel If we put in a certain string of input signals X[1], X[2],, then we have a certain probability of having a certain output string Y[1], Y[2], The net effect is what is the probability of having a certain output when a given input is given MIT

37 Channel model The channel is described by an input alphabet, an output alphabet, a set of probabilities on the input alphabet and a set of channel transitions, for a memoryless channel (on every bit the channel behaves independently of other times): Tx Rx k m Channel with input alphabet of size k+1 and output alphabet of size m+1 p(y x) is transition probability, probability of getting output y for input x MIT

38 How do we determine the probabilities? We want to get to p(x y) Use Bayes s rule: p(x y) = p( x, y) / p(y) Also p(x, y) = p(y x) p(x) p(y) = p(y x i ) p(x i )! i For the case where all the inputs have the same probability, then p(x y) = 1/n p(y x) / p (y x i )! i How can we try to make up for the effect of the channel? MIT

39 When do we use codes Two different types of codes: source codes: compression channel codes: error-correction Source-channel separation theorem says the two can be done independently for a large family of channels stream Source encoder s Channel encoder x y Channel decoder Source decoder MIT Modulator, channel, receiver, etc...

40 What is a reasonable way to construct codes? Suppose that errors occur independently from symbol to symbol: a reasonable way to code is to make codewords as dissimilar as possible Number of bits in red: Hamming distance x 1 = x 2 = If we received y = , we would map to the first codeword - we ll make this more precise later MIT

41 What do we mean by constructing a code? Block code: map a block of bits to another, generally larger, block of bits (n, k) linear block code encodes k-bit message into n-bit code vector, rate of code is R = n/k In general, code maps a message from set M= 2 k onto a codeword of length n Every codeword is one-to-one correspondence with a message An error occurs in decoding if we map a received codeword to the wrong message or to more than one message MIT

42 What do we mean by decoding? We choose most likely input to have yielded observed output - Maximum Likelihood Decoding Given that we observe the output vector y, the probability that x 1 was transmitted is p(x 1 y) and the probability that x 2 was transmitted is p(x 2 y) (let s look at 2 codewords) Then we pick x 1 when p(x 1 y) / p(x 2 y) > 1 or equivalently when ln(p(x 1 y) ) - ln(p(x 1 y) ) > 0 in terms of log likelihoods MIT

43 Decoding example Consider the binary symmetric channel (BSC) 0 Tx 1-p Rx p 0 1 p 1 1-p The probabilities are: p(x 1 y) = (1-p) 13 p 13 no errors and 1 error p(x 1 y) = (1-p) 11 p 3 11 no errors and 3 errors The BER is p MIT

44 Hamming distance and code capability Minimum Hamming Distance d min of a code is the minimum Hamming Distance between any two codewords A code can detect up to t errors iff d min t + 1 A code can correct up to t errors iff d min 2t + 1 A code can correct up to t errors and detect up to t > t errors iff d min 2t + 1 and d min t + t + 1 How does d min relate to r = n-k, the number of redundant bits? Singleton bound: r + 1 d min Example take strings of k bits and add a parity check code: r is 1 and d min is 2 These are bounds, most codes don t achieve these bounds MIT!!!!!

45 Linear codes: how do we build them? Code word x = s G where for (n,k) code G is a matrix with k rows and n columns Linear code because sums of codewords are codewords How can we find d min? The Hamming distance between two binary codewords is the same as the Hamming weight of their sum Thus, the minimum Hamming distance is the minimum weight of non-zero code vectors MIT

46 Syndromes and parity-check For any code, it is always possible to find a systematic code, i.e. a code for which G = [I k x k P k x r ]by manipulating the rows using linear operations We define the parity-check matrix H to be: & P r x k # $ - I! % k x k " We have x H = 0 Syndrome s = y H Define error sequence e = y + x Then s = e H Decoding is done by finding the minimum Hamming weight sequence that satisfies s = e H and decoding to the codeword y + z MIT

47 Let s think in 3-D Hamming distance between codes is related to t, the number of errors to correct in a (n, k) linear block code Example: rate (3,1) repetition code: can correct at most 1 error and detect 2 errors MIT

48 Further bounds Hamming bound: r Gilbert bound: there exists a code such that: MIT

49 Cyclic Codes Cyclic codes are codes with cyclic shift property in code words: if (a, b, c, d, e) is a codeword, so is (b, c, d, e, a) A generator matrix is then Can we make use of the regular structure of this matrix to simplify things? MIT

50 Polynomials on fields Galois field: a field with a finite number of elements We can define a polynomial over a finite field (Galois Field) GF(q) as f(d) = f n D n + f n-1 D n f 0 D is not a variable in the field, it is an indeterminate - think of the D-transform or the Z-transform For polynomials, addition and multiplication can be defined, they are commutative, associative, closure holds, an addition inverse is defined BUT NO MULTIPLICATIVE INVERSE Instead, we have the following theorem: let f(d) and g(d) be polynomials over GF(q) and let g(d) have degree at least 1. Then there exist unique polynomials h(d) and r(d) over GF(q) for which the degree of r(d) is less than that of g(d) and f(d) = g(d) h(d) + r(d) (Euclidean division algorithm) MIT

51 Relation between polynomials and cyclic codes Represent a vector x = (x n-1 x 0 ) as x(d) = x n-1 D n x 0 We say that x(d) is a codeword when the coefficients for the letters of a codeword If we have a cyclic code, then it is the case that the remainder of dividing D x(d) by D n - 1 ( also called the remainder of D x(d) modulo D n - 1 ) is also a code word To see this: Dx(D) = x n-1 D n + + x 0 D = x n-1 (D n - 1) + + x 0 D + x n-1 Define g(d) to be the lowest degree monic polynomial which is a code word in a cyclic code define m to be its degree Because of linearity, any D j g(d) is also a codeword, so in general a(d) g(d) is a codeword Moreover all code words have this form MIT

52 Implementing systematic cyclic codes In polynomial form, a systematic binary code is of the form: x(d) = D r t(d) - d(d) = D r t(d) + d(d) The check bits are d(d) and d(d) has degree less than r Let s take d(d) to be the remainder of D r t(d) / g(d) What about the syndrome? s(d) = x(d) / g(d) = D r t(d) / g(d) + d(d) / g(d) = 2 d(d) = 0 For some q(d), we have g(d) q(d) + d(d) = D r t(d) So g(d) q(d) = D r t(d) - d(d) = D r t(d) + d(d) which is our codeword x(d) MIT

53 Implementing cyclic codes Divide by g(d) circuit D r s(d) g g g r-1... s 1 + s s r-1 + D r s(d)... Divide by g(d) MIT + Encoder Circuits are easily implemented using shift registers to represent polynomials in D

54 Decoding Brute force: divide y(d) by g(d), from there get syndrome, check syndrome against syndrome table, generate n-bit error pattern, then add to y(d) to obtain corrected word However, we know that we can only correct a number t of errors Let s define the Meggitt set to be the set of error patterns such that e n-1 = 1 Use Meggitt s theorem: suppose that g(d) h(d) = D n - 1 and e(d) / g(d) = s(d) then [D b e(d) mod (D n - 1)]/g(D) = [D b s(d)]/g(d) So only need to store syndromes for error patterns in the Meggitt set MIT

55 MIT Different types of codes: Hamming codes Hamming codes are some of the few perfect codes Sphere of radius v: set of all sequences at distances v or less from a sequence A v-error-correcting sphere-packed code has the property that the spheres of size v around the code are non-overlapping and that every sequence is at most v+1 in distance from some code word A v-error-correcting sphere-packed code for which every sequence is at most v+1 in distance from some code word is perfect Hamming codes: rows of H are all the different nonzero sequences of length n-k Thus n = 2 n-k -1 For all Hamming codes, the minimum Hamming distance is 3, so they can correct 1 error or detect 2 The rate of a Hamming code is R = (2 r - r -1 )/(2 r -1 )

56 Cyclic Redundancy Check Codes (CRCs) Normally look only to see whether the syndrome is 0 or not, which is why they are error-detecting This is not a function of the CRC code, it s more of the way it s decoded Usually used in a concatenated fashion with an error correcting code g(d) = (x+1) p(d) where p(d) is a primitive polynomial, which means that it divides D 2r but no lower order polynomial of the form D v -1 Length n = 2 r-1-1 Usually used in shortened mode: for a systematic code, stuff with 0s and those 0s need not be transmitted but must be taken into account at the decoding MIT

57 Convolutional codes Rather than map a block to a block, we code continuously Tend to be used in lower BER channels We can define the code as a set of states: for M memory elements in the encoder, there are 2 M states The trellis structure is as follows, say M=2 Decoding 11 We use a time between known states 01 Possible ways of getting from one state 00 to another are called the adversary paths 10 t t + 1 Trellis MIT Viterbi algorithm is a means of applying dynamic programming to path selection

58 Beyond codes based on distance What is the best performance we can get with a coded system? Let us consider a channel with bandwidth W, with noise energy N per Hz, and with a maximum E on the mean transmitted Shannon s limit is that if we have no delay constraint and no complexity constrain on the coder and decoder, the best achievable error-free rate (capacity) is given by Bits per second A modulation to achieve this limit is a one that itself looks like noise and the codes are random MIT

59 Recall the BSC 0 X p 1-p Y 0 1 p 1 1-p Capacity is: I(X; Y) = H(Y) - H(Y X) H is entropy H(Z) = - Σ p Z (z) log(p Z (z) ) Let s work it out... MIT

60 Random codes The codes are random in that the codewords are selected at random from a large number of codewords The coding theorem gives results based on the fact that an error is unlikely because the for long enough codewords, two codewords are unlikely to be the close enough to be confused Codes have emerged which work on that principle Examples: Turbo Codes and Low Density Parity Check Codes MIT

61 Data Link Layer (DLC) Responsible for reliable transmission of packets over a link Framing: Determine the start and end of packets Error Detection: Determine when a packet contains errors Error recovery: Retransmission of packets containing errors DLC layer recovery May be done at higher layer Slide 1

62 Framing Where is the DATA?? Three approaches to find frame and idle fill boundaries: 1) Character oriented framing 2) Length counts - fixed length 3) Bit oriented protocols (flags) Slide 2

63 Character Based Framing Frame SYN SYN STX Header Packet ETX CRC SYN SYN SYN is synchronous idle STX is start text ETX is end text Standard character codes such as ASCII and EBCDIC contain special communication characters that cannot appear in data Entire transmission is based on a character code Slide 3

64 Issues With Character Based Framing Character code dependent How do you send binary data? Frames must be integer number of characters Errors in control characters are messy NOTE: Primary Framing method from 1960 to ~1975 Slide 4

65 Length field approach (DECNET) Use a header field to give the length of the frame (in bits or bytes) Receiver can count until the end of the frame to find the start of the next frame Receiver looks at the respective length field in the next packet header to find that packet s length Length field must be log 2 (Max_Size_Packet) + 1 bits long This restricts the packet size to be used Issues with length counts Difficult to recover from errors Resynchronization is needed after an error in the length count Slide 5

66 Fixed Length Packets (e.g., ATM) All packets are of the same size In ATM networks all packets are 53 Bytes Requires synchronization upon initialization Issues: Message lengths are not multiples of packet size Last packet of a message must contain idle fill (efficiency) Synchronization issues Fragmentation and re-assembly is complicated at high rates Slide 6

67 Bit Oriented Framing (Flags) A flag is some fixed string of bits to indicate the start and end of a packet A single flag can be used to indicate both the start and the end of a packet In principle, any string could be used, but appearance of flag must be prevented somehow in data Standard protocols use the 8-bit string as a flag Use (<16 bits) as abort under error conditions Constant flags or 1's is considered an idle state Thus is the actual bit string that must not appear in data INVENTED ~ 1970 by IBM for SDLC (synchronous data link protocol) Slide 7

68 BIT STUFFING (Transmitter) Used to remove flag from original data A 0 is stuffed after each consecutive five 1's in the original frame Stuffed bits Original frame Why is it necessary to stuff a 0 in ? Slide 8

69 DESTUFFING (Receiver) If 0 is preceded by in bit stream, remove it If 0 is preceded by , it is the final bit of the flag. Example: Bits to be removed are underlined below flag Slide 9

70 Overhead In general with a flag 01 K 0 the bit stuffing is require whenever 01 k-1 appears in the original data stream For a packet of length L this will happen about L/2 k times E{OH} = L/ 2 k + (k+ 2) bits For 8 bit flag OH ~ 8 + L/64 For large packets efficiency ~ 1-1/64 = 98.5 (or 1.5% overhead) Optimal flag length If packets are long want longer flag (less stuffing) If packets are short want short flag (reduce overhead due to flag) K opt ~ log 2 (L) Slide 10

71 Framing Errors All framing techniques are sensitive to errors An error in a length count field causes the frame to be terminated at the wrong point (and makes it tricky to find the beginning of the next frame) An error in DLE, STX, or ETX causes the same problems An error in a flag, or a flag created by an error causes a frame to disappear or an extra frame to appear Flag approach is least sensitive to errors because a flag will eventually appear again to indicate the end of a next packet Only thing that happens is that an erroneous packet was created This erroneous packet can be removed through an error detection technique Slide 11

72 Error detection techniques Used by the receiver to determine if a packet contains errors If a packet is found to contain errors the receiver requests the transmitter to re-send the packet Error detection techniques Parity check single bit Horizontal and vertical redundancy check Cyclic redundancy check (CRC) Slide 12

73 Effectiveness of error detection technique Effectiveness of a code for error detection is usually measured by three parameters: 1) minimum distance of code (d) (min # bit errors undetected) The minimum distance of a code is the smallest number of errors that can map one codeword onto another. If fewer than d errors occur they will always detected. Even more than d errors will often be detected (but not always!) 2) burst detecting ability (B) (max burst length always detected) 3) probability of random bit pattern mistaken as error free (good estimate if # errors in a frame >> d or B) Useful when framing is lost K info bits => 2 k valid codewords With r check bits the probability that a random string of length k+r maps onto one of the 2 k valid codewords is 2 k /2 k+r = 2 -r Slide 13

74 Parity check codes k Data bits r Check bits Each parity check is a modulo 2 sum of some of the data bits Example: c 1 = x 1 + x 2 + x 3 c 2 = x 2 + x 3 + x 4 c 3 = x 1 + x 2 + x 4 Slide 14

75 Single Parity Check Code The check bit is 1 if frame contains odd number of 1's; otherwise it is > > Thus, encoded frame contains even number of 1's Receiver counts number of ones in frame An even number of 1 s is interpreted as no errors An odd number of 1 s means that an error must have occured A single error (or an odd number of errors) can be detected An even number of errors cannot be detected Nothing can be corrected Probability of undetected error (independent errors) Slide 15 P(undet ected ) =! N$ ' # & p i (1 ( p) N (i N = packet size " i % p = error prob. i even

76 Horizontal and Vertical Parity Horizontal checks Vertical checks The data is viewed as a rectangular array (i.e., a sequence of words) Minimum distance=4, any 4 errors in a rectangular configuration is undetectable Can you find an example of a pattern of 6 errors that cannot be detected? Slide 16

77 Cyclic Redundancy Checks (CRC) M k Data bits T R r Check bits M = info bits R = check bits T = codeword T = M 2 r + R A CRC is implemented using a feedback shift register Bits in Bits out Slide 17

78 Cyclic redundancy checks T = M 2 r + R How do we compute R (the check bits)? Choose a generator string G of length r+1 bits Choose R such that T is a multiple of G (T = A*G, for some A) Now when T is divided by G there will be no remainder => no errors All done using mod 2 arithmetic T = M 2 r + R = A*G => M 2 r = A*G + R (mod 2 arithmetic) Let R = remainder of M 2 r /G and T will be a multiple of G Choice of G is a critical parameter for the performance of a CRC Slide 18

79 Example r = 3, G = 1001 M = => M2 r = Slide = R (3 bits) Modulo 2 Division

80 Checking for errors Let T be the received sequence Divide T by G If remainder = 0 assume no errors If remainder is non zero errors must have occurred Example: Send T = Receive T = (no errors) No way of knowing how many errors occurred or which bits are In error Slide => No errors

81 Performance of CRC For r check bits per frame and a frame length less than 2 r-1, the following can be detected 1) All patterns of 1,2, or 3 errors (d > 3) 2) All bursts of errors of r or fewer bits 3) Random large numbers of errors with prob r Standard DLC's use a CRC with r=16 with option of r=32 CRC-16, G = X 16 + X 15 + X 2 +1 = Slide 21

82 Physical Layer Error Characteristics Most Physical Layers ( communications channels) are not well described by a simple BER parameter Most physical error processes tend to create a mix of random & bursts of errors A channel with a BER of 10-7 and a average burst size of 1000 bits is very different from one with independent random errors Example: For an average frame length of 10 4 bits random channel: E[Frame error rate] ~ 10-3 burst channel: E[Frame error rate] ~ 10-6 Best to characterize a channel by its Frame Error Rate This is a difficult problem for real systems Slide 22