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2 Fundamentals of Information, Signal and System Theory

moments, i.e. the expected value and the variance of the signal are time invariant.

E(Xt) = μX = const.

Var(Xt) = σ²

X ^{= const.}

(2.44)

and the covariance between Xt and Xt+τ depends only on τ, and not on time t:

Cov(Xt, Xt+τ) = Cov(X0, Xτ) .

(2.45)

From the property E(Xt) = const. it can be concluded that a stationary process has

no trend. A trend in this context is a long-term movement around which the process

fluctuates. As shown in Figure 2.18, this can be both linear (Vt) and non-linear (Wt)

in nature. The constant variance condition V(Xt) = const., on the other hand, implies

that the signal amplitude of a stationary process does not increase or decrease. In the

case where only E(Xt) = const. is required, it is called mean-stationarity.

time t/s

time t/s

time t/s

time t/s

Fig. 2.18: Examples of a stationary signal Ut (top left) and non-stationary signals with linear trend Vt

(top right), non-linear trend Wt (bottom left) and increasing variance Xt (bottom right).

The class of strongly stationary signals satisfies a more fundamental requirement: the

distribution functions themselves must not depend on the shift. An identical, i.e. sta-

tionary, distribution of the random variables Xt thus means that in a stationary pro-

cess all realisations of Xt have the same distribution:

P(Xt1 ≤x1, . . . , Xtn ≤xn) = P(Xt1+τ ≤x1, . . . , Xtn+τ ≤xn) .

(2.46)

With this definition, the joint distribution of the random variables Xt1, . . . , Xtn and

Xt1+τ, . . . , Xtn+τ is equal. However, strong stationarity is analytically more difficult to

handle than weak stationarity.