ECEN5533. Modern Communications Theory Lecture #6. 25 January 2016 Dr ECEN5533 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

ECEN5533. Modern Communications Theory Lecture #8. 27 January 2016 Dr ECEN5533 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Time Volts Volts Count Bin

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

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

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

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

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

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

Previous waveforms Are all 0 mean, 1 watt

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

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 )

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

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

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

Autocorrelation of Random Bit Stream Each bit randomly Logic 1 or 0 20 40 60 80 100 1 1.25 x i 10 20 30 40 50 60 32 3 rx j 1

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

Probability Density Function of Band Limited Gausssian White Noise AC power = 4 watts Volts Time fx(x) .399/σx = .399/2 = 0.1995 Volts

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

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

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

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)

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

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