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Today’s Schedule

Today’s Schedule. Reading: Lathi 10.2-10.4 Mini-Lecture 1: P E calculations Activity: Example Mini-Lecture 2: Joint distributions and moments, Marginal densities Mini-Lecture 3: Bivariate Gaussian and Central Limit Theorem. Normal Distribution. PDF CDF. Gaussian CDF Proof.

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Today’s Schedule

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  1. Today’s Schedule • Reading: Lathi 10.2-10.4 • Mini-Lecture 1: • PE calculations • Activity: Example • Mini-Lecture 2: • Joint distributions and moments, Marginal densities • Mini-Lecture 3: • Bivariate Gaussian and Central Limit Theorem Dickerson EE422

  2. Normal Distribution • PDF • CDF Dickerson EE422

  3. Gaussian CDF Proof • Start with definition of CDF: • Change variables, y=(m-x)/s: Dickerson EE422

  4. Detection: Unipolar Signaling s(t) • If “1”, send p(t) • If “0”, send nothing • Received signal: r(t) = s(t) + n(t) • Optimal threshold is Ap/2 • Noise is additive, Gaussian noise Peak Detect and Sample Detector + n(t) Dickerson EE422

  5. PE Calculations • P(1/0) – prob of detecting a ‘1’ when ‘0’ sent • P(n>Ap/2) • P(0/1) – prob of detecting a ‘0’ when ‘1’ sent • P(n<-Ap/2) Dickerson EE422

  6. Unipolar vs. Bipolar • Unipolar • Bipolar Dickerson EE422

  7. What if P(0) NE P(1)? 10.2-11 • Previously we assumed that the optimal threshold was zero, because both signals were equally likely. • What should the threshold, a, be if they are not? Goal is tominimize PE • Bipolar zero threshold Dickerson EE422

  8. Detection: Bipolar Signaling s(t) • If “1”, send p(t) • If “0”, send –p(t) • Received signal: r(t) = +/- p(t) + n(t) • Optimal threshold? • Noise is additive, Gaussian noise Peak Detect and Sample Detector + n(t) Dickerson EE422

  9. PE Calculations • P(1/0) – prob of detecting a ‘1’ when ‘0’ sent • P(n>Ap+a) • P(0/1) – prob of detecting a ‘0’ when ‘1’ sent • P(n<Ap-a) a Dickerson EE422

  10. Bipolar • Total Error Probability • Take derivative wrt a and set equal to zero Dickerson EE422

  11. Bipolar Uneven Probabilities • Take derivative wrt a and set equal to zero a -Ap Ap Dickerson EE422

  12. Bluetooth 802.11b (2.4 GHz) 802.11a (5.5 GHz) GSM CDMA (IS-95) Global Positioning System (GPS) HDTV Broadcasting SS-7 Satellite TV Satellite Modems or Satellite Digital Radio Fiber Optics Projects Dickerson EE422

  13. Phase 1: Gather Information Select group and topic Groups of 4 or less Web-based report on how the physical layer of the system works In-class presentation Phase 2: Analysis Analyze the effects of interference on your system Simulate a small part or a larger part if you can find a model Present results on web and to class Project Phases Dickerson EE422

  14. Joint or Multivariate CDFs • N-dimensional CDF is: • Joint CDF (two RVs) Dickerson EE422

  15. Joint or Multivariate PDFs • N-dimensional PDF is: • Joint CDF (two RVs) Dickerson EE422

  16. Correlation or Joint Mean • Correlation or Joint Mean is: • Two RV’s are uncorrelated if • NOTE: If RV’s are independent, then they are uncorrelated, not vice versa Dickerson EE422

  17. Covariance and Correlation Coefficient • Covariance • Correlation Coefficient Dickerson EE422

  18. Marginal PDFs • Can obtain individual or marginal PDFs by integrating across the entire range: • Independent RVs Dickerson EE422

  19. Bivariate Normal Distribution • If r=0, f(x,y) = f(x) f(y) Dickerson EE422

  20. Bivariate Normal Rho =0 rho = 0.5 Dickerson EE422

  21. Next Time • Reading: Lathi 10.4-10.5 • Mini-Lecture 1: • Central limit theorem • Mini-Lecture 2: • Correlation • Activity: • Mini-Lecture 3: • Solving problems Dickerson EE422

  22. Central Limit Theorem • Under certain conditions, sum of a large number of RVs tends to be a Gaussian RV • If z = x+y, then fz(z) = fx(x)* fy(y) • Why? Dickerson EE422

  23. Expected Value of a Function • The expected value (or ensemble average) of y=h(x) is: Dickerson EE422

  24. Example Rayleigh Distribution • Given two independent, identically distributed (IID) Gaussian RVs, x and y: • Find the PDFs of the amplitude and phase of these variables (polar coordinates): Dickerson EE422

  25. Example Rayleigh Distribution • Given two independent, identically distributed (IID) Gaussian RVs, x and y: • Find the PDFs of the amplitude and phase of these variables (polar coordinates): Dickerson EE422

  26. Error in PCM • Each sample encoded by a group of n binary pulses • Two sources of error • Quantization noise • Channel noise Dickerson EE422

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