Introduction to Random Processes (6): Stationarity

Luiz DaSilva Professor of Telecommunications dasilval@tcd.ie +353-1-8963660

Intuition A stationary process arises from a stable phenomenon that has evolved into a steady-state mode of behavior

… we make a distinction between strict stationarity and wide-sense stationarity A non-stationary process arises from an unstable phenomenon

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Strict-sense stationarity Defn: If for all n ≥ 1, the nth order distributions of a random process do not depend on the shift parameter t (for all t), then the process is said to be (strict-sense) stationary.

I.e., for all n ≥ 1, all time shifts t, all choices of sample times t1, t2, …, tn,

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),,(),,( 1)()(1)()( 11 ntXtXntXtX xxfxxfnn

…… …… ττ ++=

Notes � Intuitively, for stationary processes the “randomness” of the process does not change with time – a desirable characteristic

� In general, stationarity is a very restrictive

requirement. Often, we are happy with a weaker definition…

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Wide-sense stationarity Defn: A random process X(t) is wide-sense stationary if

� E[X(t)] = µx … mean function is a constant � E[X(t1)X(t2)] = E[X(t1+t)X(t2+t)]

… autocorrelation function is

shift-invariant Note: strict stationarity implies wide-sense stationarity

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Properties of WSS processes � Can write RX(t1,t2) = RX(τ), where τ = t1 – t2. � | RX(τ) | ≤ RX(0) for all τ.

� RX(τ) = RX(-τ) … an even function

Note: RX(0) represents the “mean square value” or “average power” of the process 6

Can a random process be WSS but not stationary? An example:

Let X[n] consist of two interleaved sequences of independent RVs:

For n even, X[n] assumes values +1 and -1 with probability ½.

For n odd, X[n] assumes values 1/3 and -3 with probability 9/10 and 1/10, respectively. Is this sequence stationary? Is it wide-sense stationary?

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Ergodicity Defn: A process is said to be ergodic if all orders of statistical averages and time averages are interchangeable. That is, (as said above, this also extends to higher-order statistics)

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∫∫−

∞→

∞

∞−

===T

TT

dttXT

dxxxftXE )(21lim)()]([η

∫∫ ∫−

∞→

∞

∞−

∞

∞−

+==+=T

TT

dttXtXT

dxdxxxfxxtXtXER )()(21lim);,()]()([)( 212121 ττττ

Ergodic processes q  Since we are averaging over absolute time, the statistical averages of all orders cannot depend on absolute time.

q  That is, ergodicity implies stationarity. q  However, the converse is not true.

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