# MODULE-12 - MODULE 12 DISCRETE-TIME RANDOM PROCESSES...

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MODULE 12 DISCRETE-TIME RANDOM PROCESSES Distribution and Density Autocorrelation and Power Spectrum Linear Filtering of Random Processes Optimal (Wiener) Filtering Page 12.1

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INDEX Discrete-Time Random Processes Random Variables Examples of Densities Statistical Independence Random Process Concepts Strict-Sense Stationarity Wide-Sense Stationarity Spectrum of Random Process Two Random Processes Linear Filtering of Random Processes Input-Output Cross-Correlation Output Autocorrelation Input-Output Power Spectra Example - Optimal (MMSE) Filtering Mean-Squared Error Minimizing the MSE Wiener Filter Example MAIN INDEX Page 12.2
12. DISCRETE-TIME RANDOM PROCESSES index READ : Section 2.10, Appendix A of Oppenheim & Schafer. This Module is intended as review . However, if the material is new to the student, this is a rather painless introduction to random processes. A sequence { x ( n )} is a discrete-time random process if each element x ( n ) is a random variable . If each x ( n ) can take on any value coming from a continuous range , then { x ( n )} is a discrete-time, continuous-amplitude random process. If each x ( n ) can take values only from a discrete range - a countable or (usually) finite set of real values, then { x ( n )} is a discrete-time, discrete-amplitude random process. In practice , any digital random process will be discrete- amplitude. However, in modeling, it is usually much simpler to assume continuous probability amplitude models. Page 12.3

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Random Variables index In the following, the notation used for a random variable will be x , y , etc. in defining functions such as F x , f x , etc. The same definitions apply for x ( n ) in defining F x ( n ) , f x ( n ) , etc. Definition - The cumulative distribution function ( cdf ) of the random variable x is the probability F x ( a ) = Pr{ x a } Properties - The cdf F x ( a ) satisfies: Monotone nondecreasing : if a b, then F x ( a ) F x ( b ) Right continuous : lim a a 0 + F x ( a ) = F x ( a 0 ) a lim F x ( a ) = 1 -∞ a lim F x ( a ) = 0 a a 0 1 F x ( a ) Page 12.4
index Definition - The probability density function (pdf) of a random variable x is: f x ( a ) = d da F x ( a ) and F x ( a ) = - a f x ( b ) db with appropriate generalized definitions for derivatives at jump discontinuities . Continuous-amplitude random variables have continuous cdf's. Properties - The pdf f x ( a ) satisfies: Non-negativity: f x ( a ) 0 for a R Unit integral: - f x ( b ) db = 1 Interval probabilities: Pr{ α < x β } = β α f x ( b ) db = F x ( β ) - F x ( α ) Note - The cdf (or pdf) is a complete description of the statistics of x . Given either F x (·) or f x (·), any probability of conceivable interest may be computed. Page 12.5

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Examples of Densities index Gaussian Density : f x ( a ) = σ π 2 1 e -( a- μ ) 2 /(2 σ 2 ) ; a R F x ( a ) = σ π 2 1 - a e -( b- μ ) 2 /(2 σ 2 ) db = η - σ μ a where η ( a ) = π 2 1 - a e - b 2 /2 db is the tabulated "normal probability integral." As shown: f x ( a ) a μ σ Page 12.6
Laplacian Density : index f x ( a ) = σ 2 1 e -| a- μ |/ σ ; a R

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