Page 372 - Mechanical Engineers' Handbook (Volume 2)

8 Model Classifications 363

1 exp
( x ) 2
F(x) 2

2 2
where is the mean of the distribution and
is the standard deviation. The Gaussian
distribution has a number of important properties. First, if the input to a linear system is
Gaussian, the output also will be Gaussian. Second, if the input to a linear system is only
approximately Gaussian, the output will tend to approximate a Gaussian distribution even
more closely. Finally, a Gaussian distribution can be completely specified by two parameters,
and
, and therefore a zero-mean Gaussian variable is completely specified by its variance.
Random Processes
A random process is a set of random variables with time-dependent elements. If the statistical
parameters of the process (such as
for the zero-mean Gaussian process) do not vary with
time, the process is stationary. The autocorrelation function of a stationary random variable
x(t) is defined by
1 T
( ) lim x(t)x(t ) dt
xx
T→ 2T T
a function of the fixed time interval . The autocorrelation function is a quantitative measure
of the sequential dependence or time correlation of the random variable, that is, the relative
effect of prior values of the variable on the present or future values of the variable. The
autocorrelation function also gives information regarding how rapidly the variable is chang-
ing and about whether the signal is in part deterministic (specifically, periodic). The auto-
correlation function of a zero-mean variable has the properties
2

(0) ( ) ( ) ( )
xx xx xx xx
In other words, the autocorrelation function for 0 is identically the variance and the
variance is the maximum value of the autocorrelation function. From the definition of the
function, it is clear that (1) for a purely random variable with zero mean, ( ) 0 for
xx

0, and (2) for a deterministic variable, which is periodic with period T, (k2 T)
2
xx
for k integer. The concept of time correlation is readily extended to more than one random
variable. The cross-correlation function between the random variables x(t) and y(t)is
( ) lim x(t)y(t ) dt
xy
T→
For 0, the cross-correlation between two zero-mean variables is identically the covari-
ance. A final characterization of a random variable is its power spectrum, defined as
G( , x) lim T j t dt 2
1
T
T→ 2 T x(t)e
For a stationary random process, the power spectrum function is identically the Fourier
transform of the autocorrelation function
G( , x) j t
1
xx
( )e dt
with
(0)
xx
G( ,x) d