6.004 l13: introduction to the physics of...
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6.004 L13:Introduction to the Physics of
Communication
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The Digital Abstraction Part 1:The Static Discipline
Noise
Tx
Rx
Vol Voh
VihVil
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What is Information?
Information Resolves ______________UncertaintyUncertainty
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How do we measure information?
Error-Free data resolving 1 of 2 equally likely possibilities =
________________ of information.1 bit1 bit
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How much information now?
3 independent coins yield ___________ of information
# of possibilities = ___________
3 bits3 bits
88
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How about N coins ?
N independent coins yield
# bits = ___________________________
# of possibilities = ___________
........................
NN
2N2N
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What about Crooked Coins?
Phead = .75Ptail = .25
# Bits = - Σ pi log2 pi
(about .81 bits for this example)
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How Much Information ?
. . . 00000000000000000000000000000 . . .
None (on average)None (on average)
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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)
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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
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Why Do These Work?
Answer: They Lower _________________RedundancyRedundancy
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Data Compression
Lot’s O’ Redundant Bits
Encoder
Decoder
Fewer Redundant Bits
Lot’s O’ Redundant Bits
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An Interesting Consequence:
• A Data Stream containing the most possibleinformation possible (i.e. the leastredundancy) has the statistics of___________________ !!!!!Random NoiseRandom Noise
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Digital Error Correction
Encoder
Corrector
Original Message + Redundant Bits
Original Message
Original Message
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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 ...
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The Effect of “Analog” Noise01101110
01101110
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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)
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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)
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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.
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The mythical equipotential wire
V 1V 3V 2
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But every wire has parasitics:
- +V L
dIdt
=
I C dVdt
=+
-
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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
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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
=
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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
==
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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
=
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The Open Transmission Line
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The Shorted Transmission Line
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Parallel Termination
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Series Termination
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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
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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
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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 )
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Actual Formulas
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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