1. ECEN5533. Modern Communications Theory Lecture #6. 25 January 2016 Dr
ECEN Modern Communications Theory Lecture #6 25 January 2016 Dr. George Scheets
Read 5.4
Problems 5.1 – 5.3
Quiz #1
Next Time
Strictly Review (Chapter 1) Full Period, Open Book & Notes

2. ECEN5533. Modern Communications Theory Lecture #8. 27 January 2016 Dr
ECEN Modern Communications Theory Lecture #8 27 January 2016 Dr. George Scheets
Read 5.5
Problems 5.7 & 5.12
Exam #1
Wednesday, 10 February
Corrected Quiz #1 due < 5 February

3. Link Analysis
Final Form of Analog Free Space RF Link Equation Pr = EIRP*Gr / (Ls*M*Lo) (watts)
Digital Link Equation Eb/No = EIRP*Gr /(R*k*To*Ls*M*Lo) (dimensionless)
Very Accurate if LOS & No Reflections
Average if LOS and Reflections
Inaccurate if not LOS

4. Examples of Amplified Noise
Radio Static (Thermal Noise)
Analog TV "snow"
2 seconds
of White Noise

5. RF Public Enemy #1 Thermal Noise Models for Thermal Noise:
White Noise
Bandlimited White Noise
Gaussian Distributed Voltages
Ignored in most other classes
Can’t ignore on RF systems
Antenna is a Band Pass filter
Noise Bandwidth
Antenna Temperature

6. Review of Autocorrelation
Autocorrelations deal with predictability over time. I.E. given an arbitrary point x(t1), how predictable is x(t1+tau)?
time
Volts
tau t1

7. 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) x(255)2
Then take the time average [x(1)2 +x(2)2 +x(3) x(255)2]/255

8. 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

9. 255 point discrete time White Noise waveform (Adjacent points are independent)
Vdc = 0 v, Normalized Power = 1 watt
Volts
If true continuous time White Noise,
No Predictability.
time

10. Autocorrelation Estimate of Discrete Time White Noise
Rxx
tau (samples)

11. Autocorrelation & Power Spectrum of C.T. White Noise
Rx(tau) A
Rx(τ) & Gx(f) form a
Fourier Transform pair.
They provide the same info
in 2 different formats.
tau seconds
Gx(f)
A watts/Hz
Hertz

12. 255 point Noise Waveform (Low Pass Filtered White Noise)
23 points
Volts
Time

13. Autocorrelation Estimate of Low Pass Filtered White Noise
Rxx 23
tau samples

14. Autocorrelation & Power Spectrum of Band Limited C.T. White Noise
Rx(tau) A
2AWN
tau seconds
1/(2WN)
Average Power = 2AWN
D.C. Power = 0
A.C. Power = 2AWN
Gx(f)
A watts/Hz
-WN Hz
Hertz

15. Autocorrelation & Power Spectrum of White Noise
Rx(tau) A
tau seconds
Average Power = ∞
D.C. Power = 0
A.C. Power = ∞
Gx(f)
A watts/Hz
Hertz

16. 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
Time
Volts

17. 255 point discrete time White Noise waveform (Adjacent points are independent)
Vdc = 0 v, Normalized Power = 1 watt
Volts
If true continuous time White Noise,
No Predictability.
time

18. 15 Bin Histogram (255 points of Uniform Noise)
Count
Volts

19. Time
Volts
Volts
Count
Bin

20. 15 Bin Histogram (2500 points of Uniform Noise)
Count
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.
200
Volts

21. D.T. White Noise Waveforms (255 point Exponential Noise)
Time
Volts

22. 15 bin Histogram (255 points of Exponential Noise)
Count
Volts

23. D.T. White Noise Waveforms (255 point Gaussian Noise) Thermal Noise is Gaussian Distributed.
Time
Volts

24. 15 bin Histogram (255 points of Gaussian Noise)
Count
Volts

25. 15 bin Histogram (2500 points of Gaussian Noise)
Count
400
Volts

26. Previous waveforms
Are all 0 mean, 1 watt

27. Autocorrelation & Power Spectrum of White Noise
Rx(tau) A
The previous White
Noise waveforms all
have same Autocorrelation
& Power Spectrum.
tau seconds
Gx(f)
A watts/Hz
Hertz

28. Autocorrelation (& Power Spectrum) versus Probability Density Function
Autocorrelation: Time axis predictability
PDF: Voltage likelihood
Autocorrelation provides NO information about the PDF (& vice-versa)...
...EXCEPT the power will be the same... (i.e. PDF second moment E[X2] = A{x(t)2} = Rx(0))
...AND the D.C. value will be related. (i.e. PDF first moment squared E[X]2 = A{x(t)}2 constant term in autocorrelation )

29. Two serial bit streams….20 40 60 80
100 1
1.25 x i 50
100
150
200
250
300
350
400 1
1.25 x i

30. Random Bit Stream. Each bit S. I. of others
Random Bit Stream. Each bit S.I. of others. P(+1 volt) = P(-1 volt) = 0.5 20 40 60 80
100 1
1.25 x i
fX(x)
1/2 -1 +1
x volts

31. Voltage Distribution of domain behavior different.
Bit Stream. Average burst length of 20 bits. P(+1 volt) = P(-1 volt) = 0.5 50
100
150
200
250
300
350
400 1
1.25 x i
Voltage Distribution of
this signal & previous
are the same, but time
domain behavior different.
fX(x)
1/2 -1 +1
x volts

32. Autocorrelation of Random Bit Stream Each bit randomly Logic 1 or 020 40 60 80
100 1
1.25 x i 10 20 30 40 50 60 32 3 rx j 1

33. Bit Stream #2 Logic 1 & 0 bursts of 20 bits (on average)50
100
150
200
250
300
350
400 1
1.25 x i 10 20 30 40 50 60 32 6 rx j 1

34. Probability Density Function of Band Limited Gausssian White Noise
AC power =
4 watts
Volts
Time
fx(x)
.399/σx = .399/2 =
Volts

35. Autocorrelation & Power Spectrum of Bandlimited Gaussian White Noise
Rx(tau) 4
tau seconds
500(10-15)
Gx(f)
2(10-12) watts/Hz
-1000 GHz
Hertz

36. How does PDF, Rx(τ), & GX(f) change if +3 volts added
How does PDF, Rx(τ), & GX(f) change if +3 volts added? (255 point Gaussian Noise)
AC power = 4 watts
Volts 3
Time

37. Power Spectrum of Band Limited White Noise
Gx(f)
No DC
2(10-12) watts/Hz
-1000 GHz
Hertz
Gx(f)
3 vdc →
9 watts DC Power 9
2(10-12) watts/Hz
-1000 GHz
Hertz

38. Autocorrelation of Band Limited White Noise
Rx(tau)
No DC 4
tau seconds
500(10-15)
3 vdc →
9 watts DC Power
Rx(tau) 13 9
tau seconds
500(10-15)

39. How does PDF change if x(t) has 3 v DC?
σ2x = E[X2] -E[X]2 = 4
fx(x)
Volts
fx(x)
σ2x = E[X2] -E[X]2 = 4 3
Volts

40. Band Limited Continuous Time White Noise Waveforms (255 point Gaussian Noise)
AC power = 4 watts
DC power = 9 watts
Total Power = 13 watts
Volts 3
Time