# ECE 60000 - Random Variables and Signals

Credits: 3## Areas of Specialization(s):

BioengineeringCommunications, Networking, Signal & Image Processing

Counts as:

Normally Offered: Each Fall, Spring

Catalog Description:

Engineering applications of probability theory. Problems on events, independence, random variables, distribution and density functions, expectations, and characteristic functions. Dependence, correlation, and regression; multi-variate Gaussian distribution. Stochastic processes, stationarity, ergodicity, correlation functions, spectral densities, random inputs to linear systems; Gaussian processes.

- Probability, Random Variables, and Stochastic Processes, 4th Edition, A. Papoulis and S. U. Pillai, McGraw-Hill, 2001, ISBN No. 9780073660110.

Recommended Text(s): None.

Lecture Outline:

Week | Topic |

1 | 1. The Meaning of Probability A. Preliminary Remarks B. The Various Definitions of Probability C. Determinism versus Probability |

2. The Axioms of Probability A. Set Theory B. Probability Space C. Conditional Probabilities and Independent Events D. Summary | |

2 | 3. The Concept of a Random Variable A. Random Variables; Distributions, Densities B. Examples of Distribution and Density Functions C. Conditional Distributions and Densities D. Bayes' Theorem in Statistics (re-examined) |

3 | 4. Functions of One Random Variable A. Concept of a Function of One Random Variable B. Determination of the Distribution and Density of y=g(x) C. Applications D. Expected Value; Dispersion; Moments E. Characteristic Functions |

5. Two Random Variables A. Joint Distribution and Density Functions B. Conditional Distributions and Densities C. Independent Random Variables D. Jointly Normal Random Variables | |

4 | 6. Functions of Two Random Variables A. One Function of Two Random Variables B. Two Functions of Two Random Variables C. Expected Value: Moments; Characteristic Functions D. Mean-square Estimation; the Orthogonality Principle E. More on Normal Random Variables |

5 | 7. Sequences of Random Variables A. General Concepts B. Mean; Mean-square Estimation; Moments; Characteristic Functions |

6 | 8. Sequences of Random Variables A. Applications B. Normal Random Variables C. Convergence Concepts and the Law of Large Numbers D. The Central-limit Theorem |

RANDOM PROCESSES | |

7 | 9. General Concepts A. Introductory Remarks B. Special Processes C. Definitions D. Stationary Processes E. Transformation of Stochastic Processes (Systems) F. Stochastic Continuity and Differentiation G. Stochastic Differential Equations H. Stochastic Integrals; Time Averages; Ergodicity |

8,9 | 10. Correlation and Power Spectrum of Stationary Processes A. Correlation B. Power Spectrum C. Linear Systems D. Hilbert Transforms; Shot Noise; Thermal Noise E. Mean-square Periodicity and Fourier Series F. Band-limited Processes G. An Estimate of the Variation of a Band-Limited Process |

10 | 11. Linear Mean-square Estimation A. Introductory Remarks B. The Orthogonality Principle in Linear Mean-square Estimation C. The Wiener-Kilmogoroff Theory |

11,12 | 12. Linear Mean Square Estimation A. The Filtering Problem B. The Prediction Problem C. Wide-sense Markoff Sequences and Recursive Filtering |

13 | 13. Nonstationary Processes; Transients in Linear Systems w/Stochastic Inputs A. Transients in Linear Systems with Stochastic Inputs B. Two-dimensional Fourier Transforms C. Time Averages |

14 | 14. Harmonic Analysis of Stochastic Processes A. Series Expansions B. Approximate Fourier Expansion with Uncorrelated Coefficients C. Fourier Transforms of Stochastic Processes D. Generalized Harmonic Analysis |

Three one-hour Exams plus Final Exam. |