536
15 Random Vibrations
Some of the properties of R(τ ) is as follows:
(1) When τ = 0, R(0) = E[X
2
] = ensemble mean square of the process. Further,
for this process if E[X ] = 0, that is, the process has a zero mean, then the
mean square is identical to the variance, that is, R(0) = σ
2 .
(2) Autocorrelation function is always an even function of τ , that is, R(τ ) =
R(−τ ).
(3) If X (t) = X (t +τ ), then R(τ ) = E(X
2
). For this case, autocorrelation function
is independent of τ .
(4) If X(t) is independent of X (t + τ ), then R(τ ) = E [X (t) E [X (t + τ )]. In
this case, the autocorrelation function becomes the square of the mean. When
the mean is zero, R(τ ) = 0.
15.5.2 Temporal Statistics and Ergodic Hypothesis
So far, the discussion has been limited to the consideration of ensemble averages. For
obtaining necessary statistics, a large number of sample functions are required for
such cases. However, for certain stationary processes, it has been found that a single
but long sample function of time can provide the necessary statistical information,
when averaging is done with respect to time along the sample. Such an average is
called the temporal average.
Temporal mean and temporal mean square is defined as
X (t) = lim
T →∞
1
T
T /2
−T /2
x(t)dt
(15.21)
and
X (t)
2
= lim
T →∞
1
T
T /2
−T /2
x
2
(t)dt
(15.22)
Temporal autocorrelation is defined as
φ(τ ) = X (t)X (t + τ ) = lim
T →∞
1
T
T /2
−T /2
x(t)x(t + τ )dt
(15.23)
When τ = 0, φ(0) =
X
2
(t)
= temporal mean square.
If it is assumed that a single sample function is typical of all other sample functions,
the stationary random process is referred to as ergodic process. Therefore, ergodic
process is one for which ensemble averages are equal to temporal averages