Building Acoustics

Oscillating systems 17

wheels or gears together with stochastic components from turbulent flow of air or water
in the cooling system. This will not cause any problem if the task is just to perform a
simple amplitude analysis, i.e. to ascertain the RMS-value of the signal.
For a spectrum analysis, however, one has to consider exactly what sort of
information is needed. Commercial frequency analysers may be divided roughly into two
groups: the analysers with a fixed set of filters and the FFT analysers. One may find
instruments where both types of analysis are implemented. The International
Electrotechnical Commission (IEC) specifies the requirements for such instruments.
Relevant examples are the standard for sound level meters, IEC 61672, Parts 1 and 2.
The octave-band and fractional octave-band filters are specified in IEC 61260.

1.4.5.1 Spectral analysis using fixed filters

Analysers using fixed filters may be divided into two groups depending on the analysis
being performed sequentially or in parallel. The latter type is called real time analyser
because the entire spectrum is scanned at the same time. These analysers have a set of
parallel filters, each filter covering a smaller or broader frequency band of the chosen
frequency range. On the other hand, performing the analysis sequentially, which implies
measuring one band at a time; one is dependent on the signal being stationary during the
whole time of measurement. This is of course a serious limitation and the real time
analysers now dominate the market.
In any case, we may for each frequency band determine a mean square value (or the
RMS-value) for the signal passing through, which gives us an estimate of the spectral
distribution. Measuring a stochastic signal this procedure will by definition give the
power spectrum if we let the bandwidth Δf of the filters go to zero. We may write

2

0

0

1

( ) lim ( , , ) d.

f

T

Gf tf f t

fT

ξ

∞

Δ⇒

⇒∞

=

Δ ∫

Δ (1.23)

The symbol ξ describes the result after passing the signal x(t) through the filter. In
practice the averaging time T must be chosen giving consideration to the accuracy
needed and the bandwidth must be adapted to the task. As for the latter, the question is
whether detailed frequency information or more approximate estimates, i.e. mean values
in octave or one-third-octave bands, are in demand.
The process of analysing a broadband signal using fixed filters with a given
frequency bandwidth Δf is illustrated in Figure 1.12, using a sound pressure signal p(t) as
an example. After passing one of the filters indicated by its centre frequency f 0 , we obtain
a new time signal p(t, f 0 , Δf), a function of the selected band. Normally, we want to
express the result in terms of the sound pressure level Lp in decibels (dB), as indicated by
the lowermost sketch, for which we calculate the RMS-value and thereafter write the
expression

2

rms 0

(^02)
0

(, )

p()10lg (dB).

pff

Lf

p

⎡⎤Δ

=⋅⎢⎥

⎣⎦

(1.24)

The quantity p 0 is the reference value for sound pressure equal to 2⋅ 10 -5 Pa. For
stochastic sound we observe from Equation (1.23) that p^2 rms(f 0 , Δf) in the expression is an
estimate of Gp(f 0 )⋅ Δf. As for the frequency bands normally used, an octave band will