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6.004 Fall ‘98 L13 (10/27) Page 1
6.004 L13:Introduction to the Physics of
Communication
6.004 Fall ‘98 L13 (10/27) Page 2
The Digital Abstraction Part 1:The Static Discipline
Noise
Tx
Rx
Vol Voh
VihVil
6.004 Fall ‘98 L13 (10/27) Page 3
What is Information?
Information Resolves ______________UncertaintyUncertainty
6.004 Fall ‘98 L13 (10/27) Page 4
How do we measure information?
Error-Free data resolving 1 of 2 equally likely possibilities =
________________ of information.1 bit1 bit
6.004 Fall ‘98 L13 (10/27) Page 5
How much information now?
3 independent coins yield ___________ of information
# of possibilities = ___________
3 bits3 bits
88
6.004 Fall ‘98 L13 (10/27) Page 6
How about N coins ?
N independent coins yield
# bits = ___________________________
# of possibilities = ___________
........................
NN
2N2N
6.004 Fall ‘98 L13 (10/27) Page 7
What about Crooked Coins?
Phead = .75Ptail = .25
# Bits = - Σ pi log2 pi
(about .81 bits for this example)
6.004 Fall ‘98 L13 (10/27) Page 8
How Much Information ?
. . . 00000000000000000000000000000 . . .
None (on average)None (on average)
6.004 Fall ‘98 L13 (10/27) Page 9
How Much Information Now ?
. . . 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 . . .
. . . 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 . . .
Predictor
None (on average)None (on average)
6.004 Fall ‘98 L13 (10/27) Page 10
How About English?
• 6.JQ4 ij a vondurfhl co8rse wibh sjart sthdenjs.• If every English letter had maximum uncertainty,
average information / letter would be _________• Actually, English has only ______ bits of
information per letter if last 8 characters are usedas a predictor.
• English actually has _______ bit / character ifeven more info is used for prediction.
log2(26)log2(26)
22
11
6.004 Fall ‘98 L13 (10/27) Page 11
Why Do These Work?
Answer: They Lower _________________RedundancyRedundancy
6.004 Fall ‘98 L13 (10/27) Page 12
Data Compression
Lot’s O’ Redundant Bits
Encoder
Decoder
Fewer Redundant Bits
Lot’s O’ Redundant Bits
6.004 Fall ‘98 L13 (10/27) Page 13
An Interesting Consequence:
• A Data Stream containing the most possibleinformation possible (i.e. the leastredundancy) has the statistics of___________________ !!!!!Random NoiseRandom Noise
6.004 Fall ‘98 L13 (10/27) Page 14
Digital Error Correction
Encoder
Corrector
Original Message + Redundant Bits
Original Message
Original Message
6.004 Fall ‘98 L13 (10/27) Page 15
How do we encode digitalinformation in an analog world?
Once upon a time, there were these aliensinterested in bringing back to their planet the entirelibrary of congress ...
6.004 Fall ‘98 L13 (10/27) Page 16
The Effect of “Analog” Noise01101110
01101110
6.004 Fall ‘98 L13 (10/27) Page 17
Max. Channel Capacityfor Uniform, Bounded Amplitude Noise
P
N
Noise
Tx
Rx
Max # Error-Free Symbols = ________________
Max # Bits / Symbol = _____________________
P/NP/N
log2(P/N)log2(P/N)
6.004 Fall ‘98 L13 (10/27) Page 18
Max. Channel Capacity forUniform, Bounded Amplitude Noise (cont)
P = Range of Transmitter’s Signal SpaceN = Peak-Peak Width of NoiseW = Bandwidth in # Symbols / SecC = Channel Capacity = Max. # of Error-Free Bits/Sec
C = ____________________________
Note: This formula is slightly different for Gaussian noise.
W log2(P/N)W log2(P/N)
6.004 Fall ‘98 L13 (10/27) Page 19
Further Readingon 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.
6.004 Fall ‘98 L13 (10/27) Page 20
The mythical equipotential wire
V 1V 3V 2
6.004 Fall ‘98 L13 (10/27) Page 21
But every wire has parasitics:
- +V L
dIdt
=
I C dVdt
=+
-
6.004 Fall ‘98 L13 (10/27) Page 22
Why do wires act like transmissionlines?
Signals take time to propagate
Propagating Signals must have energy
Inductance and Capacitance Stores Energy
Without termination, energy reaching the end of a transmissionline has nowhere to go - so it
_________________________
......
EchoesEchoes
6.004 Fall ‘98 L13 (10/27) Page 23
Fundamental Equations of LosslessTransmission Lines
......
V V x t= ( , )
∂∂
∂∂
∂∂
∂∂
Vx
lIt
Ix
cVt
=
=
I I x t= ( , )
x
∂∂
Ix
∂∂
Vx
+-
cdCd x
=l
d Ld x
=
6.004 Fall ‘98 L13 (10/27) Page 24
Transmission Line Math
∂∂
∂∂
∂∂
∂∂
Vx
lIt
Ix
cVt
=
=
Lets try a sinusoidal solution for V and I:
I I e I e ej t x j t j xt x t x= =+0 0
( )ω ω ω ωV V e V e ej t x j t j xt x t x= =+
0 0( )ω ω ω ω
j V l j Ij I c j V
x t
x t
ω ωω ω
0 0
0 0
==
6.004 Fall ‘98 L13 (10/27) Page 25
Transmission Line Algebra
ωω
t
x l c= 1
j V l j Ij I c j V
x t
x t
ω ωω ω
0 0
0 0
==
ω ωω ω
x t
x t
V l II c V
0 0
0 0
==
VI
lc
0
0
=
6.004 Fall ‘98 L13 (10/27) Page 26
The Open Transmission Line
6.004 Fall ‘98 L13 (10/27) Page 27
The Shorted Transmission Line
6.004 Fall ‘98 L13 (10/27) Page 28
Parallel Termination
6.004 Fall ‘98 L13 (10/27) Page 29
Series Termination
6.004 Fall ‘98 L13 (10/27) Page 30
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
6.004 Fall ‘98 L13 (10/27) Page 31
When is a wire a transmission line?
t l vfl = /
Rule of Thumb:
t tr fl> 5t tr fl< 2 5.Transmission Line Equipotential Line
6.004 Fall ‘98 L13 (10/27) Page 32
Making Transmission LinesOn Circuit Boards
h
w
ε r
t
Voltage Plane
Insulating Dielectric
Copper Trace
cl
∝∝
Zv
0 ∝∝
ε r w/hε r w/h
h/wh/w
h / (w sqrt(ε r ) )h / (w sqrt(ε r ) )
1/sqrt(ε r )1/sqrt(ε r )
6.004 Fall ‘98 L13 (10/27) Page 33
Actual Formulas
6.004 Fall ‘98 L13 (10/27) Page 34
A Typical Circuit Board
w cmt cmh cm
c pF cml nH cm
===
==
0150 00380 038
192 75
...
. /
. /
G-10 Fiberglass-Epoxy
Z
v cmcm ns
0
10
38
1 4 1014
== ×
Ω. / sec
( / )
1 Ounce Copper
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