How About English? • 6.JQ4 ij a vondurfhl co8rse wibh sjart sthdenjs. • If every English letter had maximum uncertainty, log log(26) (26) average information / letter would be _________ 22 • Actually, English has only ______ bits of information per letter if last 8 characters are used as a predictor. 11 • English actually has _______ bit / character if even more info is used for prediction. 2 2
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Why Do These Work?
Redundancy Answer: They Lower _________________ Redundancy 6.004 Fall ‘98 L13 (10/27) Page 11
Data Compression Lot’s O’Redundant Bits Encoder Fewer Redundant Bits Decoder
Lot’s O’Redundant Bits 6.004 Fall ‘98 L13 (10/27) Page 12
An Interesting Consequence: • A Data Stream containing the most possible information possible (i.e. the least redundancy) has the statistics of Random ___________________ !!!!! RandomNoise Noise
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Digital Error Correction Original Message
Encoder Original Message + Redundant Bits
Corrector
Original Message 6.004 Fall ‘98 L13 (10/27) Page 14
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 ...
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The Effect of “Analog” Noise 01101110
01101110 6.004 Fall ‘98 L13 (10/27) Page 16
Max. Channel Capacity for Uniform, Bounded Amplitude Noise P Tx Noise Rx N P/N Max # Error-Free Symbols = ________________ P/N log Max # Bits / Symbol = _____________________ log22(P/N) (P/N)
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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 Wlog log2(P/N) 2(P/N)
____________________________ Note: This formula is slightly different for Gaussian noise. 6.004 Fall ‘98 L13 (10/27) Page 18
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.
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The mythical equipotential wire
V1
V2
V3
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But every wire has parasitics:
-
+
+ -
dI V =L dt
dV I =C dt 6.004 Fall ‘98 L13 (10/27) Page 21
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
Echoes _________________________ 6.004 Fall ‘98 L13 (10/27) Page 22
Fundamental Equations of Lossless Transmission Lines ∂V ∂x
V = V(x, t ) -
I = I(x, t)
x
+
...
...
dL l= dx
dC c= dx
∂I ∂x
∂V ∂ I = l ∂ x ∂t ∂ I ∂V = c ∂ x ∂t
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Transmission Line Math Lets try a sinusoidal solution for V and I:
j (ω t t + ω x x )
jω t t
jω x x
V = V0 e = V0 e e j (ω t t + ω x x ) jω t t jω x x I = I0 e = I0e e ∂V ∂ I = l ∂ x ∂t ∂ I ∂V = c ∂ x ∂t
jω xV0 = l jω t I 0 jω x I 0 = c jω tV0 6.004 Fall ‘98 L13 (10/27) Page 24
Transmission Line Algebra jω xV0 = l jω t I 0 jω x I 0 = c jω tV0 ωt = ωx
1 l c
ω x V0 = l ω t I 0 ω x I 0 = c ω t V0
V0 = I0
l c
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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:
Series or Parallel ?
– 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
