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Analog Signaling
x(t + )
a x(t )
inputs
are
functions
of time
xˆ (t + )
a xˆ (t )
xˆ (t ) + yˆ (t )
linear
x(t ) + y (t )
A linear time-invariant system is characterized by the eigenfunctions
eiwt
w = 2f
f = frequency
t = time
i
fix w
Ae = e
ln A
e
i
iwt
phase
gain
iwt
i
iwt
→ e = Ae e
e = e
ln A+ i
=e
i ( − i ln A+ )
=e
A(w) = amplitude response – limits bandwidth
Θ(w) = phase shift – distorts pulse shape
iF ( w ) transfer
function
limits
signaling
capacity
Sampling Theorem: A band-limited signal of duration T and of
bandwidth W can be reconstructed perfectly by 2WT samples, at
evenly spaced intervals. The sample vector (x1, …, x2WT) can be
viewed as a point in 2WT-dimensional space.
A.9

These samples can tell us the signal’s energy and their radius
2W T
(distance from origin)
1 2W T 2
2
E=
x
r
=
x
2WE
n
n =
2W n =1
n =1
And the signal power (energy per unit time) is S = E r = 2WST
T
The noise added to the channel has power N, and a corresponding
radius 2WNT . The total power (signal + noise) has radius 2WT (S + N ) .
How many spheres of noise can fit in?
[2WT ( S + N )]
1
2W T
2
S+N
=
N
WT
WT
S
# of
M
= 1 + .
1
messages
2W T
N
2
(2WTN )
signal- to- noise ratio
ratio of volumes
S
I = log M = WT log 1 + ,
The amount of information sent is
N
I
S
And the rate of info is
= W log 1 +
E.g. (telephone)
T
N
Appendix
(end.)

Information & Coding Theory
Channel
(errors)
Information
Symbols
Source
s1,…,sq
signal
Encoding
Source/Channel
Information
Symbols Destinsignal
Decoding
+ noise Channel/Source
ation
s1,…,sq
Noise
Example: Morse Code
transmitter
A, …, Z
Encoding
keyer
telegraph wire
dots, dashes
spaces
receiver
∙ ─ _ Decoding
shortwave radio
A, …, Z
recognizer
Example: ASCII Code
Character
keyboard
seven-bit
Telephone
seven-bit terminal
modem
modem
blocks
wire
blocks screen
character

Information Source – the symbols are undefined, and the “meaning” of the
information being sent is not dealt with – only an abstract measure of the
“amount” or “quantity.
Examples
text of various forms – reports, papers, memos, books, scientific data
(numbers)
pictures of various forms – diagrams, art, photographic images, scientific
data (e.g. from satellites)
sound of various forms – music, speech, noises, recorded sound, radio
animation of various forms – moving pictures, film, video tape, video
camera, television
equations representing mathematical ideas or algorithms – two textual
representation systems with graphical output: Tex & Mathematica
continuous
analog
waveforms
and shapes
discrete
digital
sampled and
quantized

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