Handbook for Sound Engineers

Computer Aided Sound System Design 1363

35.2.1.3.1 Resolving the Far-Field Problem

One of the primary points to address in the simulation of
the acoustic sources is the correct application of the data
for the near field and far field. In the previous section we
have emphasized that many loudspeakers and loud-
speaker systems employed in the field today are actually
used mainly in their near field, that is, at distances where
the system cannot be approximated by a single point
source with a distance-independent directivity pattern.
Because of their size these systems can hardly be mea-
sured as a whole in their far field. However, measure-
ments at a near-field distance are only valid for use at
that distance and not beyond, see Eq. 35-23.
Principally, there are two solutions to that. On the
one hand, one can try to model the system as what it is,
namely a spatially extended source. It could be charac-
terized mathematically by an ideal straight or curved
line source with some correction factors derived from
measurements. On the other hand, already for the
purposes of practical assembly, transport, and mainte-
nance, almost all large-format loudspeakers are
composed of individual elements. For example, a
touring line array is built of multiple cabinets each of
which in turn house multiple transducers. Thus it seems
natural that the line array is described primarily by its
components and its overall radiation characteristics are
derived from that. In consequence, representing the
significantly smaller elements individually as point
sources, now the measurement and the simulation only
have to happen in the far field of the respective element.
Coherent superposition of the sound waves radiated by
these elements will then yield the correct behavior of
the entire system for both near and far field.

35.2.1.3.2 Acquisition and Interpolation of Complex
Data

Data. The issue of using complex data instead of magni-
tude-only data is closely related to finding an accurate
way to interpolate data points over angle and frequency
properly for both magnitude and phase data. In return,
using complex data on the level of individual elements
eliminates the need for higher precision when measur-
ing and interpolating data on the level of the loud-
speaker system as a whole.

Critical Frequency.
First, let us review the error that we make when measur-
ing a loudspeaker directivity balloon about a given point
of rotation (POR). Problems usually arise from the fact
that one or several sources are slightly off-set from the

POR and thus the measured data suffers a systematic

error. For a given setup, Fig. 35-28, we can estimate the

relative error for the magnitude data for large mea-

suring distances.^38

(35-31)

where,

x is the distance between the POR and the concerned

acoustic source,

d is the measurement distance between the POR and the

microphone (with d >> x),

is the measurement angle between the microphone

and the loudspeaker axis.

The error is maximal for measurements where the

connecting line between microphone and POR passes

through both POR and acoustic source, in this case, at

an angle of. Nevertheless, for all practical

cases the error is largely negligible. For example,

typical values of x= 0.1 m and d= 4 m yield an error of

only 0.2 dB.

When describing loudspeakers by magnitude data

only, the phase is neglected completely. To simulate the

interaction between coherent sources, often the run-time

phase calculated from the time of flight between POR

and receiver is used. As stated earlier, this assumes that

the inherent phase response of the system is negligible

and that there is an approximate so-called acoustic

center where the run-time phase vanishes and which

must be used as the POR. For this measurement situa-

tion the systematic error in the phase data,

,can be calculated as well.^38 For large

measuring distances d, see Fig. 35-28, it is given by

(35-32)

for magnitude-only data

where,

Odenotes the wavelength.

In contrast, by acquiring phase data in addition to the

magnitude data, this offset error can be minimized:

Figure 35-28. Typical setup for loudspeaker measurements.

'Aˆ

• Aˆ 1 x

2

2 d^2

-------- x

d

+–= ---sin-

θ

Microphone Loudspeaker

x

POR

d Source

• = 90 q

G) G= argAˆ

G)^2 S

O

= ------xsin-

argAˆ| 0