Oscillating systems 21
time T. In the same way as when applying the fixed filter analysis, outlined in section
1.4.5.1, we have to select a total measuring time giving the desired accuracy. Measuring
on a stationary signal we just have to repeat the process; performing an averaging over n
records and thus obtaining an estimate for a total measuring time of n⋅T.
We shall show a couple of examples of signal analysis using FFT both being
simulations based on data in a digital format. Comparing with the processing performed
in a commercial instruments this means that we are entering the process following the
AD-converter.
In the first example, given in Figure 1.14, we are using the same periodic signal as
shown in Figure 1.3 but now we have added a stochastic signal, a Gaussian distributed
noise signal with zero mean value and a standard deviation equal to 0.5. Performing the
Fourier transform we may see from Figure 1.15 that the periodic components show up
again but now together with a contribution due to the noise.
The second example shows an analysis of a Gaussian distributed noise signal where
the power spectrum is constant, i.e. G(f) has a constant value G 0 for “all” frequencies.
Such a signal is called white noise. This is of course an ideal concept, which is the reason
for the quotation marks on the word all. There must in practice certainly be an upper
limit in frequency. An example of the time signal of such noise is shown in Figure 1.16
whereas Figure 1.17 gives the corresponding power spectrum averaged using different
number of records n.
The spectra are shown using a relative scale to show clearly the improvements in
the accuracy by increasing the number n. Each curve is shifted by a factor of 10 from the
previous one. Without doing so the curves will lie on top of one another. (Are you able to
estimate the correct value of G 0 from Figure 1.17?)
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Time t (s)-5.0-4.0-3.0-2.0-1.00.01.02.03.04.05.0x(t)Figure 1.16 Noise signal in the time domain. The noise signal is “white” in the frequency range 0–500 Hz.