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Session 3. Digital Transmission Fundamentals

Explore the basics of digital transmission systems and networks, including the transmission infrastructure, communication channels, and the advantages of digital communication. Learn about line coding techniques such as NRZ, bipolar, Manchester, and more.

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Session 3. Digital Transmission Fundamentals

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  1. Stephen Kim dskim@iupui.edu Session 3.Digital Transmission Fundamentals

  2. Transmission Systems • Network – The Transmission Infrastructure • What to be transmitted? (object) • What to transmit through? (medium)

  3. Communication Channel transmitter Receiver Data Communication • Propagation of energy in the form of pulses. • Analog Communication • Transmit a waveform – varying continuously on time • Reproduce the waveform as exactly as possible at a remote site. • Digital Communication • Transmit a symbol from a finite set • Thru transmission medium or channel • Binary – {1, 0} • 1 – send a positive voltage for a certain period of time (t) • 0 – send a negative voltage for t • Receiver determine the input symbol with high probability

  4. Advantage of Digital Comm. • Can a digital signal go longer than an analog signal? • Can we make a digital signal to go longer than an analog signal? • Distortion of electrical signal along the distance • Attenuation • Noise • from other electrical systems • from other electrical cables • from nowhere (random noise) • Solutions • make the medium insensitive to noise • shielded wire • eliminate the noise source – how? • repeater – regenerate the signal

  5. S Repeater Repeater D Attenuated and distorted signal +noise Recovered signal +residual noise Amp Equalizer Repeater • Periodically regenerate the signal • Amplifier for attenuated signal • Equalizer to eliminate the distortion • Accumulation of noise

  6. S Repeater Repeater D Attenuated and distorted signal +noise Receptor Bit Generator Transmitter Recovered signal Repeater in Digital Transmission • The symbol is recovered every time by the repeaters. • No accumulation of noise • Quality of signal is independent of distance

  7. A(f) 1 0 Bandwidth • Range of frequencies passed by a channel • Attenuation is a function of frequency as well as distance • Amplitude-response function A(f) = Amplitude of output tone / Amplitude of input tone f

  8. 3 3 2 2 2 2 1 1 0 0 0 0 0 0 Nyquist’s Theorem • If a signal run thru a low-pass filter of bandwidth W, the filtered signal can be reconstructed by making only 2W samples per second. • The fastest pulse rate = 2W pulse/second. • Example • A noiseless 3-kHz channel can transmit binary signals at most at a rate of 6000 bit/sec. • Multilevel Transmission • Each pulse represents M levels, then the pulse can encode log2M binary. • A noiseless W-Hz channel can transmit at a bit rate of 2W·log2M.

  9. Low SNR t t t Signal-to-Noise Ratio (SNR) • Without noise, we can send information at any bit rate. • Large M • SNR = Signal power / Noise power • dB = 10 log10(S/N) High SNR t t t signal noise signal + noise

  10. Shannon Channel Capacity • Maximum bit rate, C=W·log2(1+S/N) • Example • Bandwidth 3400Hz, • SNR = 40dB • C= 3400 log2(1+10000)=45Kbps • No matter how often sample are taken or how many levels are encoded.

  11. Aoutcos (2ft + (f)) Aincos 2ft Channel t t Frequency Domain • Attenuation – Aout/Ain • Phase shift - (f) • Cascade of filters, or channels from various sources Output Signal of the Channel y(t)

  12. Fourier Analysis – 1 A periodic function g(t) is described by

  13. Fourier Analysis – 2 • Consider any 8-bit word, say 01100010 • T=8, f=1/8, • g(1)=0, g(2)=1, g(3)=1, g(4)=0, g(5)=0, g(6)=0, g(7)=1, g(8)=0

  14. 1 1 1 0 0 0 0 0 T Fourier Analysis – 3 N=1 N=2 N=4 N=8

  15. 1 0 1 0 1 1 1 0 0 Line Coding – 1 • How to convert a binary sequence into a digital signal • Cosinderation • Maximizing the bit rate • Easy to implementation • Cost of network systems • Embedding synchronization • Built-in error detection • Immunity to noise and interference • Unipolar NRZ encoding • simple • A binary 1 for +A voltage • A binary 0 for 0 voltage • Disadvantage – power for transmission

  16. 1 0 1 0 1 1 1 0 0 Line Coding – 2 • Polar NRZ Encoding • A binary 1 for +A/2 voltage • A binary 0 for –A/2 voltage • Efficient transmission power • Problem in NRZ Encodings • If there is a long sequence of 0’s, or a long sequence of 1’sthe encodings generate a low frequency • Some network components consider a low frequency as a noise, so eliminate it. • Example, telephone system filters out frequencies lower than 200Hz.

  17. 1 0 1 0 1 1 1 0 0 Line Encoding – 3 • Bipolar Encoding • A binary 0 is mapped to 0 voltage. • A consecutive 1s are alternatively mapped to +A/2 and –A/2. • No low frequency signal • Synchronization • Determine the boundary between bits. • Using NRZ encodings or Bipolar encoding, need of separate synchronization signal

  18. 1 0 1 0 1 1 1 0 0 Manchester encoding Differential Manchester encoding Line Encoding – 4 • Manchester Encoding • A binary 1 is denoted by a transition from +A/2 to –A/2 in the middle of the bit time interval. • A binary 0 is denoted by a transition from –A/2 to +A/2. • Every bit contributes a transition that is used for synchronization. • Doubled pulse rate • Applied to Ethernet LAN • Differential Manchester Encoding • A transition in the middle of every bit time • Transition in the beginning of the bit interval represents a binary 0. • No transition in the beginning of the bit interval represents a binary 1. • Applied to Token-ring

  19. Line Encoding – 5 • The Manchester code is an 1B2B code scheme. • 1 bit information is encoded 2-bit signal. • 0 is mapped to 01, 1 is mapped to 10. • FDDI (Fiber Distributed Data Interface) • use 4B5B • 4 bit information is encoded 5-bit signal

  20. Property of Communication MediaThe Big Picture • Propagation speed  c/f • c – speed of light (3x108m/sec), f – frequency • Guided Media – Wired media • Point-to-point, so discrete network topologies • Unlimited resource • simply install more wires, expensive though. • Attenuation  10^<distance>, exponential • Twisted pair, Coaxial cable, Fiber optic • Unguided Media – wireless media • Broadcast with limited directionality • Broad and continuous network topologies • Limited resource, so regulated by some organizations • Inexpensive to be installed • Attenuation  <distance>^2, quadratical • Radio, Infrared light

  21. Reflected path Direct path Wired Media • Twisted Pair • Two parallel insulated wires • one for signal, another for reference (ground) • Crosstalk • Many cables are usually bundled • A cable talks to other adjacent cable. • Twist the pair of wires • Let them (signal&reference) catch same noise • Coaxial cable • Better immunity to noise and interference • Optical fiber • On-Off pulse on the guided medium • No interference with electromagnetic • Multimode mode and Single mode

  22. Wireless Media • Radio • Frequency • Low frequency – can pass obstacles, but quick power dissipation • High frequency – can travel in straight; bounce off obstacles; absorbed by rain; Multipath fading • Multipath fading – bounced signal are cancelled direct signal. • Regulated by government agency • Infrared Light • Cannot penetrate walls • Directional • No regulation

  23. Summary of Media

  24. Overview of Error Detection and Correction • Highly reliable modern communication • Non-zero error probability • Error-sensitive digital communication • Compressed information, no redundancy, distinct roles of data • How to handle errors • ARQ • Automatic Retransmission reQuest • detect and send a request for retransmission • bandwidth redundancy • FEC • Forward Error Correction • detect and correct • embedded redundancy

  25. Error Detection – Single Parity Check • Codeword = information bits & check bit • Two Choices • Even parity - # of 1s in the codeword is 2n. • Odd parity - # of 1s in the codeword is 2n+1. • mostly use the even parity method. • Modulo k arithmetic - concern with remainder alone. • k  2k  3k  …, k+1  2k+1  3k+1  … • ex) (modulo 3) 3693n, 47103n+1, 58113n+2 • The parity is (modulo 2) operation for individual bits in the codeword. • ex) 1001010  (1+0+0+1+0+1+0) modulo 2 = 1 • A sender attaches the result as an additional bit. • A recipient computes the even # of 1s in every codeword. • Ability of the single parity check

  26. Internet CheckSum • A Checksum is calculated for the header • The header is modified by an Internet router, sothe checksum must be re-calculated at every router • Seek for an efficient software • Assume a header contain L words(b0 to bL-1), and then add a word (checksum word, bL). • x = b0 + b1 + … + bL-1 modulo 2L – 1 • bL = -x

  27. data data/alarm data+redn (data . redn)error S R + error Principle of Error Detection • Hamming Distance • Hypercube • Valid codeword or Invalid codeword

  28. A Hypercube 1110 0110 0111 1111 1100 0100 0101 1101 1011 0011 1001 0010 0001 1010 0000 1000

  29. d Distance between 2 Valid Codeword • The minimum distance is d , so as to detect d-1 errors • Can we make codeword to detect 3 bit errors? • 00000011, 00001100, 00110000, 11000000 • Eg) 8-bit word => 256 possible codes • Select codewords as many as possible so that the minimum distance is 4.

  30. Principle of Error Correction • If recipient receives an invalid code, make it go to the nearest valid code. • To correct t errors, the distance is at least 2t+1. • Correction of single bit errors • m message bits, r check bits • Each valid codeword is surrounded by m+r invalid codeword. • So, there are at least (m+r+1)2mcodeword • In addition, m+r bits can make 2 (m+r) combinations • (m+r+1)  2 r • m=7, then 8+r  2 r, the smallest r is 4.

  31. Error Models • To measure the effectiveness of error detection/correction,need to know the behavioral property of the errors. • Error is considered as a binary vector of length n (size of codeword) • Model • Random error vector model • All 2m possible error vectors are equally likely to occur. • P(0000)=P(0001)=P(1111) • Random bit error model • The bit errors occur independently of each other. • p = a single bit error, P(j errors) = nCj pj (1-p)n-jP(no error) = (1-p)n, P(any error) = 1–(1-p)n  np • Burst error model • low error period + high error period • random bit error model & random error vector model • Close to most communication channel

  32. Error Detection Failure for Single Parity • In Random error vector model • In Random bit error model • In Burst error model • Hard to say

  33. Cyclic Redundancy Check (CRC) • S and R agree upon a generator function g(x) of degree n. • Use binary and modulo-2 arithmetic • no carry for addition, no borrow for subtraction • addition = subtraction = exclusive OR. • n is the degree of g(x). f(x) Is xnf(x)-r(x)+e(x) divisible by g(x) ? S g(x) R g(x) xnf(x)-r(x) xnf(x)-r(x)+e(x) + g(x)*s(x) g(x)*s(x)+e(x) xnf(x)=g(x)*s(x)+r(x) e(x)

  34. 111 1011 1100000 1011 111000 1011 10100 1011 10 Example of CRC • g(x) = x3+x+1 = 1011 • f(x) = x3+x2 = 1100 • xnf(x) = x6+x5 = 1100000 xnf(x)+r(x)=x6+x5+x=1100010 Transmit No error 1100010 is divisible by 1011

  35. Capability of CRC • Detect all single errors if g(x) cannot divide xi. • Detect all double errors if g(x) cannot divide 1+xm , where m is the bit distance b/w two error bits. • For a primitive polynomial of degree N, the smallest m is 2N-1, for which 1+xm is divisible by the primitive. • ex) g(x) has a factor of a primitive polynomial of degree 15, it will not divide 1+xm for any m below 32768. x15+x+1 is such a polynomial. • Detect all triple error if g(x) contains x+1 as a factor. • Detect all burst error of length  n, where n is the degree of the generator function. • e(x) is not divisible by g(x) if deg[e(x)] < deg[g(x)] • All generator function is (x+1)p(x). • (CRC-8) x8+x2+x+1 • (CRC-16) x16+x15+x2+1

  36. Advanced Topic: Graph Coloring • The error detection (or correction) problem can be mapped to a graph coloring problem. • General Graph Coloring • For an undirected graph G=(V,E) • Assign a color to each vertex such that the color is different from its neighbors with the minimum number of color used. • The color index will be the redundancy. • It is known to unsolvable. • For a planar graph, the number of colors seems to be 4. • Nobody proved it yet, many believed so. • For a hypercube, the number of colors is 2 (0 or 1). • n-D hypercube (a binary number of length n) a=[an an-1…a1] • The color is a*[1111…1]T

  37. Expanded Graph Coloring • Xk = the minimum number of colors in a vertex-coloring such that any two vertices with distance exactly k. • Yk = the minimum number of colors in a vertex-coloring such that any two vertices within distance k. • For a binary a of length n, consider nlog(n) m If the distance of a and b is 2, then am  bm Proof? (do it yourself if you are interesting at)

  38. Expanded Graph Coloring • The matrix T will assign different colors within distance 3. • So, we solved Xk and Yk for k3. • Why? • How about Xk and Yk for k>3 ? • There is a solution, but it is complicated.

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