ECEN5533
Modern Communications Theory
Lecture #1
19 August 2014
Dr. George Scheets
www.okstate.edu/elec-engr/scheets/ecen5533
 Review
Chapter 1.1 - 1.4
Problems: 1.1a-c, 1.4, 1.5, 1.9
ECEN5533
Modern Communications Theory
Lecture #2
21 August 2014
Dr. George Scheets
Review Chapter 1.5 - 1.8
Problems: 1.13 - 1.16, 1.20
 Quiz #1
Local: Tuesday, 4 September, Lecture 6
Off Campus DL: < 11 September

ECEN5533
Modern Communications Theory
Lecture #3
26 August 2014
Dr. George Scheets
Review Appendix A
Problems: Quiz #1, 2011-2013
 Quiz #1
Local: Thursday, 4 September, Lecture 6
Off Campus DL: < 11 September

ECEN5533
Modern Communications Theory
Lecture #4
28 August 2014
Read: 5.1 - 5.3
 Problems: 5.1 - 5.3
 Quiz #1
Local: Thursday, 4 September, Lecture 6
Off Campus DL: < 11 September

www.okstate.edu/elec-engr/scheets/ecen5533/
ECEN5533
Modern Communications Theory
Lecture #5
2 September 2014
Dr. George Scheets
Read 5.4 & 5.5
 Problems 5.7 & 5.12
 Quiz #1
Local: Thursday, 4 September, Lecture 6
Off Campus DL: < 11 September
Strictly Review (Chapter 1)
Full Period, Open Book & Notes

Grading
In Class: 2 Quizzes, 2 Tests, 1 Final Exam
Open Book & Open Notes
WARNING!
Study for them like they’re closed book!
 Graded Homework: 2 Design Problems
 Ungraded Homework:
Assigned most every class
Not collected
Solutions Provided
Payoff: Tests & Quizzes

Why work the ungraded
Homework problems?
An Analogy: Commo Theory vs. Football
 Reading the text = Reading a playbook
 Working the problems =
playing in a scrimmage
 Looking at the problem solutions =
watching a scrimmage
 Quiz = Exhibition Game
 Test = Big Game

To succeed in this class...

Show some self-discipline!! Important!!
For every hour of class...
... put in 1-2 hours of your own effort.

PROFESSOR'S LAMENT
If you put in the time
You should do fine.
If you don't,
You likely won't.
Course Emphasis

Digital
Analog

Binary
M-ary

Wide Band
Narrow Band
French
Optical
Telegraph

Digital M-Ary System
M
= 8 x 8 x 4 = 256
Source:
January 1994
Scientific American
French System Map
Source:
January 1994
Scientific American
Trend is to Digital
Phonograph → Compact Disk
 Analog NTSC TV → Digital HDTV
 Video Cassette Recorder
→ Digital Video Disk
 AMPS Wireless Phone → 4G LTE
 Terrestrial Commercial AM & FM Radio
 Last mile Wired Phones

Review...
Fourier Transforms X(f)
Table 2-4 & 2-5
 Power Spectrum
Given X(f)
 Power Spectrum
Using Autocorrelation

 Use
Time Average Autocorrelation
Review of Autocorrelation

Autocorrelations deal with predictability
over time. I.E. given an arbitrary point
x(t1), how predictable is x(t1+tau)?
Volts
tau
time
t1
Review of Autocorrelation

Autocorrelations deal with predictability
over time. I.E. given an arbitrary
waveform x(t), how alike is a shifted
version x(t+τ)?
Volts
τ
255 point discrete time White
Noise waveform
(Adjacent points are independent)
Vdc = 0 v, Normalized Power = 1 watt
Volts
0
If true continuous time White Noise,
no predictability.
time
Rxx(0)
The sequence x(n)
x(1) x(2) x(3) ...
x(255)
 multiply it by the unshifted sequence x(n+0)
x(1) x(2) x(3) ...
x(255)
 to get the squared sequence
x(1)2 x(2)2 x(3)2 ...
x(255)2
 Then take the time average
[x(1)2 +x(2)2 +x(3)2 ... +x(255)2]/255

Rxx(1)
The sequence x(n)
x(1) x(2) x(3) ...
x(254) x(255)
 multiply it by the shifted sequence x(n+1)
x(2) x(3) x(4) ...
x(255)
 to get the sequence
x(1)x(2) x(2)x(3) x(3)x(4) ... x(254)x(255)
 Then take the time average
[x(1)x(2) +x(2)x(3) +... +x(254)x(255)]/254

Review of Autocorrelation

If the average is positive...
 Then
x(t) and x(t+tau) tend to be alike
Both positive or both negative

If the average is negative
 Then
x(t) and x(t+tau) tend to be opposites
If one is positive the other tends to be negative

If the average is zero
 There
is no predictability
Autocorrelation Estimate
of Discrete Time White Noise
Rxx
0
tau (samples)
255 point Noise Waveform
(Low Pass Filtered White Noise)
23 points
Volts
0
Time
Autocorrelation Estimate of
Low Pass Filtered White Noise
Rxx
0
23
tau samples
Autocorrelation & Power
Spectrum of C.T. White Noise
Rx(τ)
A
0
Rx(τ) & Gx(f) form a
tau seconds
Fourier Transform pair.
They provide the same info G (f)
x
in 2 different formats.
A watts/Hz
0
Hertz
Autocorrelation & Power
Spectrum of White Noise
Rx(tau)
A
Average Power = ∞
D.C. Power = 0
A.C. Power = ∞
0
tau seconds
Gx(f)
0
A watts/Hz
Hertz
Autocorrelation & Power Spectrum
of Band Limited C.T. White Noise
Rx(tau)
A
2AWN
0
1/(2WN)
Average Power = 2AWN watts
D.C. Power = 0
A.C. Power = 2AWN watts
-WN Hz
tau seconds
Gx(f)
A watts/Hz
0
Hertz
Autocorrelations

Time Average Autocorrelation
 Easier
to use & understand than
Statistical Autocorrelation E[X(t)X(t+τ)]
 Fourier Transform yields GX(f)

Autocorrelation of a Random Binary
Square Wave
 Triangle
riding on a constant term
 Fourier Transform is sinc2 & delta function

Linear Time Invariant Systems
 If
LTI, H(f) exists & GY(f) = GX(f)|H(f)|2
Cosine times a Noisy Serial Bit Stream
Cos(2πΔf)
X
=
LTI
x(t)
y(t)
Filter
If input is x(t) = Acos(ωt)
output must be of form
y(t) = Bcos(ωt+θ)
RF Antenna Directivity
Maximum Power Intensity
Average Power Intensity
 WARNING!
Antenna Directivity is NOT =
Antenna Power Gain
10w in? Max of 10w radiated.
 Treat Antenna Power Gain = 1
 Antenna Gain = Power Gain * Directivity


High Gain = Narrow Beam
Directional Antennas
RF Antenna Gain
Antenna Gain is what goes in RF Link
Equations
 In this class, unless specified otherwise,
assume antennas are properly aimed.

 Problems

specify peak antenna gain
High Gain Antenna = Narrow Beam
source: en.wikipedia.org/wiki/Parabolic_antenna
Parabolic Directivity
Effective Isotrophic
Radiated Power

EIRP = PtGt

Path Loss Ls = (4*π*d/λ)2
Link Analysis

Final Form of Analog Free Space
RF Link Equation
Pr = EIRP*Gr/(Ls*M*Lo) (watts)

Derived Digital Link Equation
Eb/No = EIRP*Gr/(R*k*T*Ls*M*Lo)
(dimensionless)
Public Enemy #1: Thermal Noise
Models for Thermal Noise:
*White Noise & Bandlimited White Noise
*Gaussian Distributed
 Noise Bandwidth

 Actual
filter that lets A watts of noise thru?
 Ideal filter that lets A watts of noise thru?
 Peak value at |H(f = center freq.)|2 same?
 Noise

Bandwidth = width of ideal filter (+ frequencies).
Noise out of an Antenna = k*Tant*WN
Examples of Amplified Noise
Radio Static (Thermal Noise)
 Analog TV "snow"
2 seconds

of White Noise
Review of PDF's & Histograms

Probability Density Functions (PDF's), of
which a Histograms is an estimate of shape,
frequently (but not always!) deal with the
voltage likelihoods
Volts
Time
255 point discrete time White
Noise waveform
(Adjacent points are independent)
Vdc = 0 v, Normalized Power = 1 watt
Volts
0
If true continuous time White Noise,
No Predictability.
time
15 Bin Histogram
(255 points of Uniform Noise)
Bin
Count
Volts
Bin
Count
Volts
0
Time
Volts
15 Bin Histogram
(2500 points of Uniform Noise)
Bin
Count
200
When bin count range is from zero to max value, a
histogram of a uniform PDF source will tend to look
flatter as the number of sample points increases.
0
0
Volts
Discrete Time
White Noise Waveforms
(255 point Exponential Noise)
Volts
0
Time
15 bin Histogram
(255 points of Exponential Noise)
Bin
Count
Volts
Discrete Time
White Noise Waveforms
(255 point Gaussian Noise)
Thermal Noise is Gaussian Distributed.
Volts
0
Time
15 bin Histogram
(255 points of Gaussian Noise)
Bin
Count
Volts
15 bin Histogram
(2500 points of Gaussian Noise)
400
Bin
Count
0
Volts
Previous waveforms

Are all 0 mean, 1 watt, White Noise
0
0
Autocorrelation & Power
Spectrum of White Noise
Rx(tau)
A
0
The previous White
tau seconds
Noise waveforms all
have same Autocorrelation
Gx(f)
& Power Spectrum.
A watts/Hz
0
Hertz
Autocorrelation (& Power Spectrum)
versus
Probability Density Function





Autocorrelation: Time axis predictability
PDF: Voltage liklihood
Autocorrelation provides NO information about
the PDF (& vice-versa)...
...EXCEPT the power will be the same...
PDF second moment E[X2] = Rx(0) = area
under Power Spectrum = A{x(t)2}
...AND the D.C. value will be related.
PDF first moment squared E[X]2 = constant
term in autocorrelation = E[X]2δ(f) = A{x(t)}2
Satellite vs Sun, Daytime, Northern
Hemisphere
x
Spring Sun
→ same
plane as
Satellite.
Winter
Sun is
below
satellite
orbital
plane.
Summer
Sun is
above
satellite
orbital
plane.
x
x
Fall Sun
→ same
plane as
satellite.
x
2013 Fall Sun Outage, Microspace's AMC-1
x
Source: www.ses.com/4551568/sun-outage-data
Band Limited Continuous Time
White Noise Waveforms
(255 point Gaussian Noise)
If AC power = 4 watts &
BW = 1,000 GHz...
Volts
0
Time
Probability Density Function of
Band Limited Gausssian White Noise
Volts
0
fx(x)
.399/σx = .399/2 = 0.1995
0
Time
Volts
Autocorrelation & Power Spectrum of
Bandlimited Gaussian White Noise
Rx(tau)
4
0
500(10-15)
tau seconds
Gx(f)
2(10-12) watts/Hz
-1000 GHz
0
Hertz
How does PDF, Rx(τ), & GX(f)
change if +3 volts added?
(255 point Gaussian Noise)
AC power = 4 watts
Volts
3
0
Time
Power Spectrum of Band Limited
White Noise
Gx(f)
No DC
2(10-12) watts/Hz
-1000 GHz
3 vdc →
9 watts DC Power
-1000 GHz
0
Hertz
Gx(f)
2(10-12) watts/Hz
9
0
Hertz
Autocorrelation of Band Limited
White Noise
Rx(tau)
No DC
4
0
500(10-15)
3 vdc →
9 watts DC Power 13
tau seconds
Rx(tau)
9
0
500(10-15)
tau seconds
How does PDF change if x(t) has 3 v DC?
0
σ2x = E[X2] -E[X]2 = 4
fx(x)
0
σ2x = E[X2] -E[X]2 = 4
Volts
fx(x)
3
Volts
Band Limited Continuous Time
White Noise Waveforms
(255 point Gaussian Noise)
Volts
AC power = 4 watts
DC power = 9 watts
Total Power = 13 watts
3
0
Time
Model for an Active Device
+
Sin
&
Nin
+
G>1
Namp = kTampWn
GSin
&
G(Nin + Nai)
Noise Figure

F = SNRin/SNRout
 WARNING!
Use with caution.
If input noise changes, F will change.

F = 1 + Tamp/Tin

Tin = 290o K (default)
Model for a Passive Device
+
Sin
&
Nin
+
G<1
Namp = kTpassiveWn
Tpassive = (L-1)Tphysical
GSin
&
G(Nin + Nai)
Temperatures...
 Active
Device (Tamp)
From
 Passive
Spec Sheet (may have F)
Device (Tcable or T passive)
 (L-1)*Tphysical
System Noise (Actual)
Noise Antenna "Sees"
Noise Striking Antenna
= Noise exiting antenna
= NoWThermal
= NoWAntenna
9
= kTsurroundings1000*10
≈ kTant1000*109 9
= k*290*1000*10
= 2.07 n watts
= 4.00 n watts
(Tantenna = 150 Kelvin)
System
Much of this noise
doesn't
exit system.
Cable
+ Amp
Blocked by system filters. kTantWN = ???
Noise exiting Antenna that will exit the System =
kTant6*106 = 12.42*10-15 watts
System Noise (Simplified Model)
Noise Actually Exiting Antenna
= Noise Antenna "Sees"
≠ Noise Exiting Antenna
that will exit the System
= kTantWN
= 12.42*10-15 watts
Antenna
Power
Gain = 1
Signal Power in =
Signal Power out
System
Cable + Amp
This is the
model we use.
We don't worry about
noise that won't make the output.
SNR Considering all the noise
Noise Seen by Antenna
= NoWAntenna
= kTant1000*109 = 2.07 n watts
Signal Power Picked Up by Antenna
= 10-11 watts
System
Cable + Amp
SNR at "input" of antenna = 10-11/(4*10-9) = 0.0025
SNR at output of antenna = 10-11/(2.07*10-9) = 0.004831
SNR at System Output = 43.63
SNR Considering Noise Hitting Antenna That
Can Reach the Output
Noise seen by Antenna TCRO
= NoWN
= kTant6*106 = 12.42 femto watts
Signal Power Picked Up by Antenna
= 10-11 watts
System
Cable + Amp
SNR at output of antenna = 805.2
SNR at System Output = 43.63
This is the
noise we're
worried about.
SNR of Actual System Improves
Filtering...
Removes noise power outside signal BW
Lets the signal power through
System
Cable + Amp
SNR at Antenna Input = 0.0025
SNR at Antenna Output = 0.004831
SNR at System Output = 43.67
SNR of Model Worsens
Only considers input noise that is in
the signal BW & can reach the output.
Cable & electronics dump in more
noise.
System
Cable + Amp
SNR at antenna output = 805.2
SNR at System Output = 43.67