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Bruce Hajek wrote:All rights reserved. Permission is hereby given to freely print and circulate copies of these notes so long as the notes are left intact and not reproduced for commercial purposes. Email to b-hajek at uiuc dot edu, pointing out errors or hard to understand passages or providing comments, is welcome.
Excerpts from the Preface:
From an applications viewpoint, the main reason to study the subject of these notes is to help deal with the complexity of describing random, time-varying functions
. A random variable can be interpreted as the result of a single measurement. The distribution of a single random variable is fairly simple to describe. It is completely specified by the cumulative distribution function F(x
), a function of one variable. It is relatively easy to approximately represent a cumulative distribution function on a computer. The joint distribution of several random variables is much more complex, for in general, it is described by a joint cumulative probability distribution function, F(x1, x2, . . . , xn
), which is much more complicated than n functions of one variable. A random process, for example a model of time-varying fading in a communication channel, involves many, possibly infinitely many (one for each time instant t within an observation interval) random variables. Woe the complexity!
These notes help prepare the reader to understand and use the following methods for dealing with the complexity of random processes:
- Work with moments, such as means and covariances.
- Use extensively processes with special properties. Most notably, Gaussian processes are characterized entirely be means and covariances, Markov processes are characterized by one-step transition probabilities or transition rates, and initial distributions. Independent increment processes are characterized by the distributions of single increments.
- Appeal to models or approximations based on limit theorems for reduced complexity descriptions, especially in connection with averages of independent, identically distributed random variables. The law of large numbers tells us that, in a certain context, a probability distribution can be characterized by its mean alone. The central limit theorem, similarly tells us that a probability distribution can be characterized by its mean and variance. These limit theorems are analogous to, and in fact examples of, perhaps the most powerful tool ever discovered for dealing with the complexity of functions: Taylor’s theorem, in which a function in a small interval can be approximated using its value and a small number of derivatives at a single point.
- Diagonalize. A change of coordinates reduces an arbitrary n-dimensional Gaussian vector into a Gaussian vector with n independent coordinates. In the new coordinates the joint probability distribution is the product of n one-dimensional distributions, representing a great reduction of complexity. Similarly, a random process on an interval of time, is diagonalized by the Karhunen-Lo`eve representation. A periodic random process is diagonalized by a Fourier series representation. Stationary random processes are diagonalized by Fourier transforms.
- Sample. A narrowband continuous time random process can be exactly represented by its samples taken with sampling rate twice the highest frequency of the random process. The samples offer a reduced complexity representation of the original process.
- Work with baseband equivalent. The range of frequencies in a typical radio transmission is much smaller than the center frequency, or carrier frequency, of the transmission. The signal could be represented directly by sampling at twice the largest frequency component. However, the sampling frequency, and hence the complexity, can be dramatically reduced by sampling a baseband equivalent random process.
These notes were written for the first semester graduate course on random processes, offered by the Department of Electrical and Computer Engineering
at the University of Illinois
at Urbana-Champaign. Students in the class are assumed to have had a previous course in probability, which is briefly reviewed in the first chapter of these notes. Students are also expected to have some familiarity with real analysis and elementary linear algebra, such as the notions of limits, definitions of derivatives, Riemann integration, and diagonalization of symmetric matrices. These topics are reviewed in the appendix. Finally, students are expected to have some familiarity with transform methods and complex analysis, though the concepts used are reviewed in the relevant chapters.
Hopefully some students reading these notes will find them useful for understanding the diverse technical literature on systems engineering, ranging from control systems, image processing, communication theory, and communication network performance analysis. Hopefully some students will go on to design systems, and define and analyze stochastic models. Hopefully others will be motivated to continue study in probability theory, going on to learn measure theory and its applications to probability and analysis in general.