# Lecture 10

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```					    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

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How do we measure information?

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

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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 = ___________

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What about Crooked Coins?
Ptail = .25                  Phead = .75

# Bits = -    S pi log2 pi

(about .81 bits for this example)

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How Much Information ?

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

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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

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