Civil Engineering Reference
In-Depth Information
to the maximum values. This is easily seen looking at a sinusoidal function. The function
“spends more time” around the maximum values than around zero. For increasing σ
noise
the function will, however, approach the common Gaussian curve again.
1
0.9
0.8
0.7
0.1
0.6
0.5
0.3
0.4
0.5
0.3
0.2
0.1
0
-4
-3
-2
-1
0
1
2
3
4
Amplitude x
Figure 1.18
Probability density functions of a sum of a sinusoidal signal and a Gaussian noise signal. Thick
curve - Gaussian noise only with σ = 1. Thin curves - sum of sinusoidal, having amplitude equal 1 and noise
with σ equal 0.1, 0.3 and 0.5, respectively.
A probability density function thus gives information on how the amplitude is
distributed. It does, in other words, tell us the amount of time it is expected to be within a
certain range and furthermore, in which part of the time it is larger or smaller than some
given value. It cannot, however, tell us anything about the time coherence in the signal.
As an example it may tell us that the sound pressure is above 1 Pa during 1% of the time
but it will not tell us whether this occurs due to pulses having a duration of 10 ms or 100
ms. We may obtain information of the latter type by looking at the signal at certain
intervals τ in time.
We define the
autocorrelation function R
(τ
) as
T
1
∫
R
()
τ
=
x t
() (
⋅
x t
+
ττ
)d
.
(1.29)
lim
T
T
⇒∞
0
As is apparent from Equation
(1.29)
, we take the value of the function
x
(
t
) at a given
time
t
and multiply it with the value at a later time
t
+ τ , thereafter taking the mean value
of all such products. Before showing some examples we shall point to some important
properties. First, this function gives a description of the signal in the time domain as
equivalent to the
spectral density function
in the frequency domain. These functions
comprise a Fourier transformation pair.
Assuming, as before, a stationary signal and performing the averaging process over
a sufficiently long time, the function
R
will be independent of time and a function of τ
only. Furthermore, the function is symmetric,
R
(τ)
= R
(
-
τ) and it has a maximum for τ =
0. In the latter case, we see from Equation
(1.29)
that this maximum is nothing more
