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l19_20_randsig

# l19_20_randsig - MIT OpenCourseWare http/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu HST.582J / 6.555J / 16.456J Biomedical Signal and Image Processing Spring 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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Random Signals (statistics and signal processing) Primary Concepts for Random Processes You should understand… • Random processes as a straightforward extension of random variables. • What is meant by a realization and an ensemble . • The importance of stationarity and ergodicity – Useful for estimating statistical properties of random processes. • Some idea of how the autocorrelation/autocovariance functions describe the statistical structure of a random signal. 4/24/2007 HST 582 © John W. Fisher III, 2002-2007 2 Harvard-MIT Division of Health Sciences and Technology HST.582J: Biomedical Signal and Image Processing, Spring 2007 Course Director: Dr. Julie Greenberg Cite as: John Fisher. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
4/24/2007 HST 582 © John W. Fisher III, 2002-2007 3 Random Processes • A random process, X , is an infinite-dimensional random variable. – If the size (dimension) is countably infinite then we have a discrete-time random process. – If the size (dimension) is uncountably infinite then we have a continuous-time random process. • As with any multi-dimensional random variable we can – compute marginal and conditional statistics/densities over subsets of the dimensions. • Random processes are interesting/tractable when there is some structure to the statistical relationships between various dimensions (i.e. the structure lends itself to analysis). • We’ll focus primarily on second-order statistical properties. 4/24/2007 HST 582 © John W. Fisher III, 2002-2007 4 Some Notation •Let denote a random process which can be thought of as the ensemble (or set) of all realizations • A single realization (a sample from the ensemble) can be denoted by: - is a random variable, which represents the 45 th sample over the ensemble •- i s a t w o - dimensional random variable, comprised of the 2 nd and 11 th samples over the ensemble. • When analyzing random processes, we are interested in the statistical properties of (potentially) all such combinations. 0 100 200 300 400 500 600 700 800 900 1000 -50 0 50 single realization 0 100 200 300 400 500 600 700 800 900 1000 -50 0 50 many realizations drawn from the ensemble Cite as: John Fisher. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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4/24/2007 HST 582 © John W. Fisher III, 2002-2007 5 Realization vs. Ensemble • Top plot: a single realization drawn from the ensemble.
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l19_20_randsig - MIT OpenCourseWare http/ocw.mit.edu...

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