Civil Engineering Reference
In-Depth Information
1.5.1
Probability density function. Autocorrelation
In
Figure 1.2,
the probability distribution was listed as one type of amplitude analysis. A
more precise term is the
probability density function p
(
x
), which gives us the probability
of finding the amplitude of a signal
x
(
t
) within a certain interval Δ
x
. It is given by a
limiting value as
Px
()
−
Px
(
+Δ
x
)
px
()
=
,
(1.25)
lim
x
Δ
x
Δ⇒
0
where
P
is the probability, a positive number between zero and one. In general,
p
(
x
) will
also be a function of time but for stationary and ergodic signals there will be no time
dependence. In the literature, one will find a huge number of mathematical density
functions with descriptions of their properties. The numbers of density functions
associated with physical phenomena are infinite. As for the stochastic noise signal we
have used up to now we have assumed it to be Gaussian distributed, which means that
the density function is given by
2
x
1
−
2
px
()
=
e
σ
σπ
2
,
(1.26)
2
where the standard deviation σ is a measure of the width of the distribution. The square
of this quantity is called
variance
and is given by
+∞
=
∫
2
2
σ
x px
()d.
x
(1.27)
−∞
In the case of oscillations, the mean value is zero and this equation gives us the
mean
square value
, i.e. the square of the
RMS-value
.
The density function, Equation
(1.26)
, gives us the well-known bell-shaped curve
as shown in
Figure 1.18
. It may, however, be interesting to see if this type of diagram
could give information on other types of signal. As an example we may calculate
p
(
x
) for
a sinusoidal signal superimposed on Gaussian noise signals. The sinusoidal signal is
given by
= + , where the phase angle θ is considered to be a random
variable. It may be shown that the probability density function (Bendat and Piersol
ˆ
ω θ
xt
()
x
sin(
)
2
⎛
⎞
xx
−⋅
ˆ cos
2
θ
−
⎜
⎟
π
⎜
⎟
σ
1
⎝
⎠
noise
∫
px
()
=
e
d.
θ
(1.28)
σππ
2
noise
0
ˆ
This function is shown in
Figure 1.18
when the amplitude is equal to one, i.e. the
RMS-value or σ
sinus
is approximately 0.71, using the RMS-value σ
noise
as a parameter.
When the latter is small one may see that the curve is approaching the density curve for a
sine or cosine function, a curve resembling a hyperbolic function with high values near
~
3
It is a misprint in Equation (2.35) in the reference.