Handbook for Sound Engineers

Computer Aided Sound System Design 1357

to which sound sources are modeled plays a crucial role.
Accordingly, most simulation software packages have
continuously developed their capabilities of describing
loudspeakers by measurement data along with the com-
plexity of the loudspeaker systems themselves. At the
same pace, the availability and fast development of per-
sonal computers had a significant impact on acoustic
measurement systems and their accuracy on the one hand
and on the computing power available on the other hand.
In this sense, the measurement and simulation of
loudspeaker systems can be roughly divided into two
periods of time. The first period, until the late 1990s,
was characterized by the use of simplified far-field data
for almost any sort of loudspeaker and the assumption
of a point-source-like behavior. But with the advent of
modern line array technology, for both tour sound and
speech transmission applications, new concepts had to
be developed. These methods include the use of
multiple point sources as well as advanced mathemat-
ical models to image the complexity of today’s loud-
speaker systems. In addition to that, research was
further accelerated by the broad availability of DSP
platforms and the resulting need to simulate DSP
controlled loudspeakers as well as the virtual disappear-
ance of computer-based constraints, such as calculation
speed and memory.

35.2.1.1 Simulation of Point Sources

Theoretical Background. For many years the radia-
tion behavior of sound sources, and loudspeakers in par-
ticular, was basically described by a 3D matrix
containing magnitude data in a fixed spectral and spatial
resolution. Starting with the late 1980s, typical data files
contained directivity data for the audible octave bands,
such as from 63 Hz to 8 kHz, and for a spherical grid
with an angular spacing of 15°. Mostly, data was also
assumed to be symmetric in one or two planes. With the
need for higher data resolution and the limits of avail-
able PC memory and computing power changing at the
same time, more advanced data formats developed
eventually reaching a nowadays typical resolution of 5°
angular increments for -octave frequency bands.
Tables 35-1 and 35-2 show some of these typical loud-
speaker data formats and their resolutions.
Now let us look at the background for this develop-
ment. We express the complex sound pressure for the
time-independent propagation of a spherical wave:^29

(35-29)

where

is the receiver location,

f is the frequency,

is the complex radiation function of the

source depending on angles Mand (both being

functions of ) as well as on the frequency,

is the wave vector.

Loudspeaker measurements happen at discrete

angles , and frequencies fm , the simulation soft-

ware has to interpolate between such data points to

obtain a smooth response function:

(35-29A)

Here the interpolation function is represented by fInt.

The frequency resolution is basically given by the set of

available data points of fm, the angular resolution is

given by the density of data points

For a long time, most measurements were made to

acquire magnitude data only, and =

. In such a case, the simulation of interaction
between multiple sound sources yields a sound intensity,
ISum, that is derived either by power summation for inco-
herent sources n (located at ):

(35-30)

or by at least considering the run time phase

for coherent sources:

(35-30A)

This simulation model makes some significant

assumptions:

1. First, the use of a spherical waveform assumes that
   both measurement and simulation happen in the far
   field of the device, that is, at a distance where the
   sound source (normally a surface) can be consid-
   ered as a point source

(^1) » 3
p
pSphere rf A^ M-f^
r
=-------------------------exp – jkr
r
A M-f

• rre
  k

Aˆ

Mk -l

pSim rf m

fInt Aˆ Mk-lfm

r

= ---------------------------------------------exp – jkr

Mk-l.

Aˆ = Aˆ fInt Aˆ

fInt Aˆ

rn

ISum rf m pn

n

¦

2

=

>@fInt Aˆn Mk-lfm

2

rr– n

------------------------------------------------------ 2 -

n

=¦

)n –= kn rr– n

ISum rf m pn

n

¦

2

=

fInt Aˆn Mk-lfm

rr– n

------------------------------------------------

n

¦ exp>@–jkn^ rr– n

2

=

pReal rf |pSphere rf.