Civil Engineering Reference
In-Depth Information
important function used to characterize stochastic functions in the frequency domain, the
power spectral density function
,
or in short, the
power spectrum
. Since
x
T
(
t
) only exist in
the time interval
T
we may calculate the mean square value as follows
T
+∞
∞
1
1
2
2
∫
2
()
∫
2
()
∫
2
x
()
t
=
x
t
d
t
=
x
t
d
t
=
|
X
( )| d
f
f
,
(1.18)
T
T
T
T
T
T
T
0
−∞
0
where the last expression is calculated using Equation
(1.16)
. By assuming that the
function is stationary (and ergodic) we may show that the mean square value of
x
(
t
) is
∞
∞
2
⎡
⎤
2
∫
∫
2
2
x
( )
t
=
lim
x
( )
t
=
lim
X
(
f
)
d
f
=
G f
(
) d ,
f
(1.19)
⎢
⎥
T
T
T
T
⇒∞
⎣
⎦
T
⇒∞
0
0
where
G
(
f
) is the
power spectral density function
mentioned above and defined by the
Fourier transform
2
⎡
⎤
2
Gf
() im
=
X
() .
f
⎦
(1.20)
⎢
⎥
T
T
⎣
T
⇒∞
Equation
(1.20)
gives the formal definition of the function but the reason for using such a
term should be more obvious looking at Equation
(1.19)
. Here one may see that
G
(
f
) is
the contribution per Hz to the mean square value, a quantity generally proportional to the
power, hence the term
power spectrum
or more correctly
power spectral density.
1.4.4
Discrete Fourier transform (DFT)
When processing data digitally one has, as pointed out above, to use a discrete set of
frequency values. Calculating a finite Fourier transform using Equation
(1.17)
will give a
Fourier series with period
T.
Here we shall just give a summary of this discrete type of
transformation and point to some relationships important in measurement applications.
When presenting the examples on spectral analysis (see section
1.4.5
below), we shall
use this technique to simulate the results from a digital frequency analysis.
The first two steps in a digital analysis are 1)
sampling
and 2)
quantization
. The
first step means that the signal
x
(
t
) is substituted by a number
N
samples separated by a
time step of Δ
t
as illustrated in
Figure 1.10
. The length of the signal being analysed is
then
T
=
N
⋅ Δ
t
, and the calculations are performed treating the data as periodic with a
time period of
T
. According to the
sampling theorem
this implies that the upper
frequency limit in the analysis is
f
co
= 1/(2Δ
t
). This frequency is called the
Nyquist
frequency
or
cut-off frequency
.
The quantization, performed by an analogue to digital converter (AD-converter),
means an allocation of a finite set of numbers to the amplitudes of the sampled signal.
This is illustrated in
Figure 1.11
, which shows the numbers or estimates allocated to the
samples. The number of bits handled by the AD-converter gives the accuracy of these
estimates. A 12-bit converter resolves the signal into 2
12
= 4096 steps, a 16-bit into
65536 steps etc.
