130 Building acoustics
4.5.2.2 Reverberation time variance
Measurements of sound decay and reverberation time in rooms are performed either by
using a method based on an interrupted noise signal or by a method based on the
integrated impulse response, specifically by
- exciting the room using a stochastic noise signal, usually filtered in octave or
one-third-octave bands, and recording the sound pressure level after turning off
the source, i.e. the method outlined when deriving the reverberation time
formula in section 4.5.1.2 - measuring the impulse response, using either a maximum length sequence
signal (MLS signal) or a swept sine signal (SS signal), which again is filtered
in octave or one-third-octave bands, thereafter applying the method given in
section 4.3.1.
As for the first method concerned one will, due to the stochastic noise of the signal,
observe variations in the results when repeating the measurement. This will be the case
even if both source and microphone positions are exactly the same. The reason is that the
stochastic signal is stopped at an arbitrary time making the room excited by different
“members” of the ensemble of noise signals produced by the source. It makes no
difference if the stochastic signal in fact is pseudo stochastic, i.e. periodically repeats
itself, as the source normally is not stopped coincident with this period. The variance due
to the variation in the reverberation time measured at a given position we shall call an
ensemble variance. This quantity σe^2 (T) is therefore an analogue of the time variance
σt^2 (p^2 ) by a sound pressure measurement (see Equation (4.56)).
By measuring the reverberation time using M microphone positions, repeating each
measurement N times in each position, the relative variance in the average reverberation
will be given by
(^) ()
2
(^2) e
2 r
r
()
,
T
T N
M
σ
σ
σ+
= (4.58)
where the first term is the variance due to the spatial variation. It should be noted that the
last term will be zero when using an impulse response technique as the excitation signal
will be deterministic in this case. This does not, however, imply that systematic errors
cannot occur in this case if the system is not time invariant, e.g. due to temperature
changes etc. during the measurement. The SS technique is less prone to such errors than
the MLS technique.
Returning to the method of using interrupted noise, Davy et al. (1979) developed
theoretical expressions for the two contributions to the variance, applicable to frequencies above
the Schroeder frequency fS. In effect, they calculated the variance of the slope of the decay
curves but the results may easily be transformed to apply to the corresponding reverberation
time. As expected, these expressions are functions of the filter bandwidth and reverberation time
but also depends on the time constant (or “internal reverberation time”) of the measuring
apparatus together with the dynamic range available. It has to be remembered that at the time
when this work was performed the equipment available was of analogue type such as the level
recorder. We shall therefore just give an example applicable for one-third-octave measurements,
using a dynamic range of 30 dB and a RC detector (exponential averaging). The time constant
of this detector is assumed to be one-quarter of the equivalent time constant for the room. The
relative variance of the mean reverberation time may then be written (Vigran (1980)) as