[RP]Lecture Note XI - Kyung Hee University Department of...

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) XI-1 C1002900 RP Lecture Handout XI: Stochastic Processes Reading: Chapter 9.1 In preceding lectures, we have focused on random variables and random vectors, and their manipulation in problems of detection and estimation. We now want to broaden our development to accommodate sequences and waveforms that are random as well. Sequences (discrete-time) and waveforms (continuous-time) of this type are re- ferred to as random or stochastic processes. 11.1. Definitions A random process   X t is an indexed collection , : , X t w t   of random va- riables, all on the same probability space , , P . Example 11.1: AC voltage   cos X t A t    . A random process is a family or an ensemble of time functions depending on the outcome , or equivalently, a function of t and . If   is fixed, it is a single time function, called the sample path correspond- ing to . If t is fixed,   X t is a random variable equal to the state of the given process at time t . If t and are fixed,   X t is a number. 11.2. Statistics of Stochastic Processes In general, probabilistic characterizations of a stochastic process involve specifying the joint probabilistic description of the process at different points in time. In particular,
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) XI-2 stochastic processes are completely characterized by the collection of the n th-order densities , , , , n n X t X t f x x 1 1 (XI.1) or distributions , , , , , , n n n n X t X t F x x X t x X t x 1 1 1 1 (XI.2) for every possible choice of n and the time instants , , , n t t t 1 2 . Very difficult. In practice, the development of stochastic processes follows one of two approaches: (i) Only partial statistical descriptions of the processes are pursued; or (ii) The focus is restricted to processes with special structures or properties that substantially simplify their descriptions. One of the simplest stochastic processes is a discrete-time white noise. Example 11.1: A particular simple discrete-time white noise corresponds to the case in which the samples X n are zero-mean and have identical variances: . X n X n X m n m    2 0 (XI.3) If , X n 2 0 , i.e., X n is a discrete-time Gaussian white noise, we have , , , , ; , N N N i X n X n i f x x x 1 1 1 0 (XI.4) which completely characterizes the process.
Image of page 2
Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB)
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern