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A stochastic process is a family of random variables indexed by a parameter t R for continuous-time stochastic processes or k Z + for discrete-time stochastic sequences For a deterministic signal such as x(t) = t2 the value of x is determined by the value of t Once t is known, the value of x(t) is known There is no element of chance Alternatively, signals such as v(t) = At2 where A N (0, 2 ) or v(t) = t2 + w(t) where w(t) has density pw (W, t), are stochastic processes In fact, in the above example, w(t) is also a stochastic process For a stochastic process v(t), the parameter t determines the distribution and statistics of the random variable v, but t does not determine the value of v; instead, given two times t1 and t2 we have two distinct random variables v(t1 ) and v(t2 ) The distributions for v(t1 ) and v(t2 ) may be the same or distinct, depending on whether or not the stochastic process is stationary Example 411 Let v(t) = t2 + w(t) where w(t) N (0, 2 ), the density for v is 2 V t2 1 exp pv (V, t) = 2 2 2 At each time, w(t) is a new Gaussian random variable with zero mean and variance 2 At each time, v(t) is a new Gaussian random variable with mean t2 and variance 2

The statistical quantities de ned previously for random variables become slightly more complicated when applied to stochastic processes, as they may depend on the parameter t or k For continuous-time processes, the de nitions are listed below Note that these de nitions are equally valid whether the parameter t represents a continuous or a discrete-time variable The cross correlation function between two random processes is de ned by (429) Rvw (t1 , t2 ) = cor (v(t1 ), w(t2 )) = E v(t1 )w(t2 ) The autocorrelation function of a random processes is de ned by Rv (t1 , t2 ) = cor (v(t1 ), v(t2 )) = E v(t1 )v(t2 ) (430)

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To get a handle on the new playlist so that you can work with it, assign it to a variable when you create it You can then add songs to the playlist by using the duplicate command, as in the following example:

The autocorrelation function quanti es the similarity of the random process to itself at two di erent times The cross covariance function between two random processes is de ned by cov (v(t1 ), w(t2 )) = E v(t1 )w(t2 ) E v(t1 ) E w(t2 ) (431)

set Listen to make new playlist with properties {name:"High Desert Rock", shuffle:true} duplicate (every track of playlist "Library" whose genre is "Stoner Rock") to Listen

The autocovariance function of a random processes is de ned by cov (v(t1 ), v(t2 )) = E v(t1 )v(t2 ) E v(t1 ) E v(t2 ) (432)

To delete a playlist, use a delete command and specify the playlist For example, the following statement deletes the playlist named High Desert Rock:

The above de nitions will be critically important in the analysis of subsequent chapters For example, the Kalman lter will propagate certain error covariance matrices through time as a means to quantify the relative accuracy of various pieces of information that are to be combined There are two forms of stationary random processes that will be used in this text De nition 41 The random process w(t) is stationary if its distribution is independent of time (ie, pw (W, t) = pw (W)) If a random process is stationary, then its expected value will be time invariant and both its autocorrelation function and its autocovariance function will only depend on the time di erence = t1 t2 (ie, Rv (t1 , t2 ) = Rv ( )) Often this strict sense of stationarity is too restrictive A relaxed sense of stationarity is as follows De nition 42 The random process w(t) is wide sense stationary (WSS) if the mean and variance of the process are independent of time Since a Gaussian random process is completely de ned by its rst two moments, a Gaussian process that is wide sense stationary is also stationary A wide sense stationary process must have a constant mean, and its correlation and covariance can only depend on the time di erence between the occurrence of the two random variables, ie cor (w(t1 ), w(t2 )) = Rww ( ) where = t1 t2 The Power Spectral Density (PSD) of a WSS random process w(t) is the Fourier transform of the autocorrelation function:

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Sw (j ) =

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