Lecture 10

							    How Computer Work
        Lecture 10

Introduction to the Physics of
       Communication


              How Computer Work Lecture 10 Page 1
     The Digital Abstraction Part 1:
         The Static Discipline
         Vol                            Voh
Tx



                                              Noise
Rx



               Vil                Vih

                     How Computer Work Lecture 10 Page 2
   What is Information?




Information Resolves ______________
                       Uncertainty




                How Computer Work Lecture 10 Page 3
How do we measure information?




Error-Free data resolving 1 of 2 equally likely possibilities =
   1 bit
________________ of information.

                              How Computer Work Lecture 10 Page 4
How much information now?




                             3 bits
 3 independent coins yield ___________ of information
                       8
 # of possibilities = ___________



                        How Computer Work Lecture 10 Page 5
     How about N coins ?


                 ........................
N independent coins yield

# bits = ___________________________
                        N


                         2 N
# of possibilities = ___________



                        How Computer Work Lecture 10 Page 6
What about Crooked Coins?
   Ptail = .25                  Phead = .75




   # Bits = -    S pi log2 pi

          (about .81 bits for this example)

                        How Computer Work Lecture 10 Page 7
How Much Information ?


. . . 00000000000000000000000000000 . . .
         None (on average)




                        How Computer Work Lecture 10 Page 8
How Much Information Now ?
  ...0101010 1010101010101...




  ...0101010 1010101010101...


                      Predictor
  None (on average)


                      How Computer Work Lecture 10 Page 9
        How About English?
• 6.JQ4 ij a vondurfhl co8rse wibh sjart
  sthdenjs.
• If every English letter had maximum
  uncertainty, average information / letter would
         log (26)
  be _________
          2



• Actually, English has only ______ bits of
                                  2

  information per letter if last 8 characters are
  used as a predictor.
• English actually has _______ bit / character if
                            1

  even more info is used for prediction.
                       How Computer Work Lecture 10 Page 10
Data Compression
 Lot’s O’ Redundant Bits


      Encoder

                Fewer Redundant Bits


      Decoder



 Lot’s O’ Redundant Bits

                How Computer Work Lecture 10 Page 11
 An Interesting Consequence:
• A Data Stream containing the most
  possible information possible (i.e. the
  least redundancy) has the statistics of
  ___________________ !!!!!
       Random Noise




                     How Computer Work Lecture 10 Page 12
Digital Error Correction
      Original Message


       Encoder
              Original Message + Redundant Bits




         Corrector


     Original Message
                 How Computer Work Lecture 10 Page 13
   How do we encode digital
information in an analog world?

Once upon a time, there were these aliens interested in
bringing back to their planet the entire library of congress ...




                                How Computer Work Lecture 10 Page 14
The Effect of “Analog” Noise
                      01101110




                       01101110

             How Computer Work Lecture 10 Page 15
              Max. Channel Capacity
      for Uniform, Bounded Amplitude Noise
                          P

Tx

                                          Noise

Rx

                                  N
                                         P/N
     Max # Error-Free Symbols = ________________

     Max # Bits / Symbol = _____________________
                              log2(P/N)

                          How Computer Work Lecture 10 Page 16
       Max. Channel Capacity for
Uniform, Bounded Amplitude Noise (cont)

     P = Range of Transmitter’s Signal Space
     N = Peak-Peak Width of Noise
     W = Bandwidth in # Symbols / Sec
     C = Channel Capacity = Max. # of Error-Free Bits/Sec
    C=              W log2(P/N)
          ____________________________

Note: This formula is slightly different for Gaussian noise.

                             How Computer Work Lecture 10 Page 17
            Further Reading
         on Information Theory


The Mathematical Theory of Communication,
Claude E. Shannon and Warren Weaver, 1972, 1949.

Coding and Information Theory, Richard Hamming,
Second Edition, 1986, 1980.




                         How Computer Work Lecture 10 Page 18
The mythical equipotential
          wire




  V1       V2                    V3




            How Computer Work Lecture 10 Page 19
But every wire has parasitics:

                          dI
                     V L
   -       +              dt

                         dV
                     IC
       +

       -
                         dt

               How Computer Work Lecture 10 Page 20
 Why do wires act like transmission
              lines?

...                                                                 ...


      Signals take time to propagate

      Propagating Signals must have energy

              Inductance and Capacitance Stores Energy

      Without termination, energy reaching the end of a transmission
      line has nowhere to go - so it
                              Echoes
                        _________________________
                                   How Computer Work Lecture 10 Page 21
      Fundamental Equations of Lossless
            Transmission Lines
       V  V ( x, t )       V           I  I ( x, t )
                            x
                        -        +                              x
...                                                                       ...




                                                           V      I
                                    dC             I       x
                                                               l
                                                                   t
                                 c
             l
                dL                  dx             x       I    V
                                                               c
                dx                                          x     t

                                         How Computer Work Lecture 10 Page 22
Transmission Line Math
    Lets try a sinusoidal solution for V and I:

              j ( t t  x x )         j t t   j x x
 V  V0 e                    V0e e
          j ( t t  x x )      j t t j x x
 I  I0 e                    I0e e

V      I
 x
    l
        t
                                  jxV0  l jt I0
 I
    c
       V                         jx I0  c jtV0
 x     t
                          How Computer Work Lecture 10 Page 23
Transmission Line Algebra
jxV0  l jt I0                     x V0  l t I0
jx I0  c jtV0                     x I0  c t V0


    t             1                     V0             l
                                           
    x             lc                    I0             c
      Propagation Velocity             Characteristic Impedence


                             How Computer Work Lecture 10 Page 24
Parallel Termination




         How Computer Work Lecture 10 Page 25
Series Termination




        How Computer Work Lecture 10 Page 26
              Series or Parallel ?
• Series:
   – No Static Power Dissipation
   – Only One Output Point
   – Slower Slew Rate if Output is Capacitively Loaded
• Parallel:
   – Static Power Dissipation
   – Many Output Points
   – Faster Slew Rate if Output is Capacitively Loaded
• Fancier Parallel Methods:
   – AC Coupled - Parallel w/o static dissipation
   – Diode Termination - “Automatic” impedance matching



                            How Computer Work Lecture 10 Page 27
When is a wire a transmission
              line?
          t fl  l / v

                 Rule of Thumb:

 tr  2.5 t fl                      tr  5 t fl
Transmission Line              Equipotential Line



                      How Computer Work Lecture 10 Page 28
    Making Transmission Lines
        On Circuit Boards
                                                 Insulating Dielectric
Copper Trace
                                                               r
                w


       t

                          h
                                                        Voltage Plane



                                               Z0  h / (w sqrt(            ))
               c    r w/h
                                                                         r


                                               v  1/sqrt( )
               l   h/w
                                                                 r



                              How Computer Work Lecture 10 Page 29
Actual Formulas




       How Computer Work Lecture 10 Page 30
   A Typical Circuit Board
1 Ounce Copper
                      G-10 Fiberglass-Epoxy
w  015cm
      .
t  0.0038cm
h  0.038cm

c  19 pF / cm
     .                Z0  38 
l  2.75 nH / cm      v  1. 4  1010 cm / sec
                           (14 cm / ns )


                   How Computer Work Lecture 10 Page 31