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ECEN5533 Modern Communications Theory Lecture #1 20 August 2013 Dr. George Scheets www.okstate.edu/elec-engr/scheet

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## ECEN5533 Modern Communications Theory Lecture #1 20 August 2013 Dr. George Scheets www.okstate.edu/elec-engr/scheet

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**ECEN5533 Modern Communications TheoryLecture #1 20**August 2013Dr. George Scheetswww.okstate.edu/elec-engr/scheets/ecen5533 • Review Chapter 1.1 - 1.4Problems: 1.1a-c, 1.4, 1.5, 1.9**ECEN5533 Modern Communications TheoryLecture #2 22**August 2013Dr. George Scheets • Review Chapter 1.5 - 1.8Problems: 1.13 - 1.16, 1.20 • Quiz #1 • Local: Tuesday, 10 September, Lecture 7 • Off Campus DL: No later than 17 Sept.**ECEN5533 Modern Communications TheoryLecture #3 27**August 2013Dr. George Scheets • Review Appendix AProblems: Quiz #1, 2010-2012 • Quiz #1 • Local: Tuesday, 10 September, Lecture 7 • Off Campus DL: No later than 17 Sept.**ECEN5533 Modern Communications TheoryLecture #4 5**September 2013 • Read: 5.1 - 5.3 • Problems: 5.1 - 5.3 • Quiz #1 • Local: Tuesday, 10 September, Lecture 7 • Off Campus DL: No later than 17 Sept. www.okstate.edu/elec-engr/scheets/ecen5533/**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 LAMENTIf you put in the timeYou should do fine.If you don't,You likely won't.**Course Emphasis**• DigitalAnalog • Binary M-ary • Wide BandNarrow 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**Review...**• Fourier Transforms X(f)Table 2-4 & 2-5 • Power SpectrumGiven X(f) • Power SpectrumUsing Autocorrelation • Use Time Average Autocorrelation**Volts**time Review of Autocorrelation • Autocorrelations deal with predictability over time. I.E. given an arbitrary point x(t1), how predictable is x(t1+tau)? tau 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)? τ Volts**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 sequencex(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 sequencex(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 alikeBoth positive or both negative • If the average is negative • Then x(t) and x(t+tau) tend to be oppositesIf 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)**Autocorrelation & Power Spectrum of C.T. White Noise**Rx(τ) A 0 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 0 Hertz**Autocorrelation & Power Spectrum of White Noise**Rx(tau) A 0 tau seconds Average Power = ∞ D.C. Power = 0 A.C. Power = ∞ Gx(f) A watts/Hz 0 Hertz**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 Band Limited C.T. White**Noise Rx(tau) A 2AWN 0 tau seconds 1/(2WN) Average Power = 2AWN watts D.C. Power = 0 A.C. Power = 2AWN watts Gx(f) A watts/Hz -WN 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**Filter x(t) y(t) If input is x(t) = Acos(ωt)output must be of formy(t) = Bcos(ωt+θ)**RF Antenna Directivity**• Maximum Power Intensity Average Power Intensity • WARNING!Antenna Directivity is NOT = Antenna Power Gain10w in? Max of 10w radiated. • Treat Antenna Power Gain = 1 • Antenna Gain = Power Gain * Directivity • High Gain = Narrow Beam**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**Effective Isotrophic Radiated Power**• EIRP = PtGt • Path Loss Ls = (4*π*d/λ)2 • Received Signal Power = EIRP/Ls**Link Analysis**• Final Form of Analog Free Space RF Link EquationPr = EIRP*Gr/(Ls*M*Lo) (watts) • Derived Digital Link EquationEb/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 Time Volts 0**15 Bin Histogram(2500 points of Uniform Noise)**Bin 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 0 0 Volts**Discrete TimeWhite Noise Waveforms(255 point Exponential**Noise) Volts 0 Time**15 bin Histogram(255 points of Exponential Noise)**Bin Count Volts**Discrete TimeWhite 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)**Bin Count 400 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 Noise waveforms all have same Autocorrelation & Power Spectrum. tau seconds Gx(f) 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)... ...EXCEPTthe 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**Band Limited Continuous TimeWhite 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 Time fx(x) .399/σx = .399/2 = 0.1995 0 Volts**Autocorrelation & Power Spectrum of Bandlimited Gaussian**White Noise Rx(tau) 4 0 tau seconds 500(10-15) 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 0 Hertz Gx(f) 3 vdc → 9 watts DC Power 9 2(10-12) watts/Hz -1000 GHz 0 Hertz