Building Acoustics

Oscillating systems 25

than the mean square value of the signal. Lastly, the value of the function will, for
stochastic signals, approach zero with increasing τ.
Two examples of autocorrelation are shown in Figure 1.19, a pure sinusoidal signal
and a Gaussian noise signal having variance equal to one. Mathematically speaking, the
first one should turn out as a sinusoidal function but in practice the signal must be of
finite length, therefore also in the numerical calculation used here. Both functions will
therefore approach zero when increasing τ. (You may show that R(0) is correct by
applying the information above.)

-1000 -500 0 500 1000

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

tau (sample)

R(tau)

Figure 1.19 Autocorrelation function of stochastic noise and sinusoidal signal of length 5⋅π.

1.5.2 Test signals

As pointed out in the introduction to this section, the reason for the relatively broad
coverage of Gaussian noise signal is their use as test signals. With their broadband
character it may cover any frequency interval and the spectrum may be shaped if
necessary. One will find an example of the latter in so-called pink noise, shaping the
signal such that the energy in the test signal is constant when using filters having a
constant relative bandwidth; octave or fractional-octave filters. Real test signals must of
course have a finite length; they are pseudo stochastic repeating themselves after a given
time. This property is normally not exploited when using this common type of noise
signal generator. Using pseudo stochastic signals that exactly repeat themselves were
first implemented applying MLS signals, a type that became very popular after being
introduced in measuring equipment well over a decade ago.
We have called MLS signals as pseudo stochastic due to their noise-like properties
but it must be stressed that they are wholly deterministic. There are, however, a multitude
of deterministic signals that have been or are still being used for testing. These may be
periodic (purely sinusoidal, amplitude modulated as well as frequency modulated signals
etc.) or transient. As an example here we shall use the swept sine (SS) signal, a type also