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12 - Random processes

from Part III - Random processes

Published online by Cambridge University Press:  05 June 2012

Hisashi Kobayashi
Affiliation:
Princeton University, New Jersey
Brian L. Mark
Affiliation:
George Mason University, Virginia
William Turin
Affiliation:
AT&T Bell Laboratories, New Jersey
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Summary

In this and following chapters, we will discuss random processes. After a brief introduction to this subject in Section 12.1, we will give an overview of various random processes in Section 12.2 and then discuss (strictly) stationary and wide-sense stationary random processes and introduce the notion of ergodicity. The last section focuses on complex-valued Gaussian processes, which will be useful in the study of communication systems and other applications.

Random process

There are many situations in which the time dependency of a set of probability functions is important. One example is a noise process that accompanies a signal process and should be suppressed or filtered out so that we can recover the signal reliably and accurately. Another example is the amount of outstanding packets yet to be processed at a network router or switch, which may lead to undesirable packet loss due to buffer overflows if not properly attended to in time.

Such a process can be conveniently characterized probabilistically by extending the notion of a random variable (RV) as follows: we assign to each sample point ω ∈ Ω a real-valued functionX(ω, t), where t is the time parameter or index parameter in some range T, which may be, for instance, T = (-∞, ∞) or T = {0, 1, 2, …} (see Figure 12.1). Imagine that we can observe this set of time functions {X(ω, t); ω ∈ Ω, tT} at some instant t = t1.

Type
Chapter
Information
Probability, Random Processes, and Statistical Analysis
Applications to Communications, Signal Processing, Queueing Theory and Mathematical Finance
, pp. 315 - 342
Publisher: Cambridge University Press
Print publication year: 2011

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  • Random processes
  • Hisashi Kobayashi, Princeton University, New Jersey, Brian L. Mark, George Mason University, Virginia, William Turin, AT&T Bell Laboratories, New Jersey
  • Book: Probability, Random Processes, and Statistical Analysis
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511977770.013
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  • Random processes
  • Hisashi Kobayashi, Princeton University, New Jersey, Brian L. Mark, George Mason University, Virginia, William Turin, AT&T Bell Laboratories, New Jersey
  • Book: Probability, Random Processes, and Statistical Analysis
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511977770.013
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Random processes
  • Hisashi Kobayashi, Princeton University, New Jersey, Brian L. Mark, George Mason University, Virginia, William Turin, AT&T Bell Laboratories, New Jersey
  • Book: Probability, Random Processes, and Statistical Analysis
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511977770.013
Available formats
×