ABSTRACT
A random process X(t) is the ensemble of all possible realizations (sample functions) X(t, σ) as shown in Fig. 4.1. Here t denotes the independent variable (usually identified with time) and σ denotes chance (randomness). For any given value of t, X(t) is a random variable. By taking ensemble averages at
a fixed value of t, we can define expected values of the random process X(t). These expectations are first of all the mean value function
X¯(t) = E[X(t)] (4.1) and the auto-covariance function
RXX(t, s) = E[(X(t)− X¯(t))(X(s)− X¯(s))] (4.2) From Eq. (4.2) we obtain as a special case for t= s
RXX(t, t) = E[(X(t)− X¯(t))2] = σ2X(t) (4.3)
An enhanced description of a random process involves the probability distribution functions. This includes the one-time distribution function
FX(x, t) = Prob[X(t) < x] (4.4)
and, moreover, all multi-time distribution functions
FX(x1, t1;x2, t2; . . . ;xn, tn)
= Prob[(X(t1) < x1) ∧ (X(t2) < x2) ∧ . . . ∧ (X(tn) < xn)] (4.5)
for arbitrary n∈N. If all these distribution functions are (multidimensional) Gaussian distributions, then the process X(t) is called Gaussian process. This class of random processes has received particular attention in stochastic dynamics since its properties are easily described in terms of the mean value function and the auto-covariance function only. A random process is called weakly stationary if its mean value function X¯(t) and
auto-covariance function RXX(t, s) satisfy the relations
X¯(t) = X¯ = const. RXX(t, s) = RXX(s− t) = RXX(τ) (4.6)
For weakly stationary processes we have
RXX(τ) = RXX(−τ) max τ∈R
|RXX(τ)| = RXX(0) = σ2X (4.7)
Intuitively, one may expect that for large time separation (i.e. for τ →±∞) the autocovariance function should approach zero. If this is actually the case, then the Fourier transform of the auto-covariance functions exists, and we define the auto-power spectral density SXX(ω) of the weakly stationary random process X(t) in terms of
SXX(ω) = 12π ∞∫
RXX(τ)eiωτdτ (4.8)
By inverting this transformation we can recover the auto-covariance function in terms of
RXX(τ) = ∞∫
−∞ SXX(ω)e−iωτdω (4.9)
These equations are frequently called Wiener-Khintchine-relations. Specifically, for ω=0 we obtain from the previous equation
σ2X = RXX(0) = ∞∫
−∞ SXX(ω)dω (4.10)
This leads to the interpretation of the power spectral density (PSD) as the distribution of the variance of a process over the frequency axis. It forms the basis of the so-called power spectral method of random vibration analysis. According to the range of frequencies covered by the PSD, the extreme cases of
wide-band and narrow band random processes may be distinguished. The qualitative relation between the PSD and the respective auto-covariance functions is shown in Fig. 4.3.
