Building Acoustics

Oscillating systems 23

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 Δx. It is given by a
limiting value as

0

() ( )

() lim

x

Px Px x

px

Δ⇒ x

− +Δ

=

Δ

, (1.25)

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

1 2 2

() ,

2

x

px eσ

σπ

−

= (1.26)

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

σ^22 x px x()d.

+∞

−∞

=∫ (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 byxt()=+xˆsin(ωtθ), where the phase angle θ is considered to be a random
variable. It may be shown that the probability density function (Bendat and Piersol
(1980)),^3 in this case will be

2

noise

ˆcos

2

noise 0

1

() d.

2

xx

px e

θ

π σ

θ

σππ

⎛⎞−⋅

−⎜⎟⎜⎟

⎝⎠

= ∫ (1.28)

This function is shown in Figure 1.18 when the amplitude is equal to one, i.e. the
RMS-value or σsinus is approximately 0.71, using the RMS-value σnoise 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

xˆ

x~

(^3) It is a misprint in Equation (2.35) in the reference.