Handbook for Sound Engineers

1364 Chapter 35

(35-33)

for complex data

Hence, by using phase data the error is reduced by an
order of magnitude. In practice, it is most useful to
define a maximum acceptable phase error, such as

and use that to derive an upper (critical) frequency limit
based on the measurement setup. Fig. 35-29 shows this
critical frequency fCrit as a function of the distance
between POR and acoustic source.

We emphasize that the use of phase data does not
only reduce the error in the directivity data but it also
largely eliminates the need to define, find, and use the
so-called acoustic center, the imaginary origin of the
far-field spherical wave front.

Local Phase Interpolation. Once complex directivity
data is available for a loudspeaker, the next step is to
define an appropriate interpolation function for the
discrete set of data points to image the continuous
sound field of the source in the real-world
( ). An algorithm will have to work for both
magnitude and phase data in both domains, frequency
and space. While averaging, smoothing, and interpo-
lating magnitude data is usually a straightforward task,
the same is not true for phase data. Due to the mathe-
matical nature of phase, its data values are located on a
circle. Accordingly, when phase is mapped to a linear
scale the interpolation has to take wrapping into

account. In this regard, a variety of methods have been

proposed, such as use of group delay, unwrapped phase,

or the so-called continuous phase representation.

Although these methods have their advantages, it could

be shown that generally none of them is appropriate for

full-sphere radiation data of a real loudspeaker.^39

Alternatively, a method named local phase interpola-

tion can be applied successfully when some care is

taken about the resolution of the underlying data. This

method essentially interpolates phase on a local scale

rather than globally. For example, let the phase average

of two data points i and j be defined as

(35-34)

Then, it is assumed that the corresponding phase data

points are all located within a certain range:

(35-35)

In this respect, i and j may represent two angular

data points and or two frequencies f 1 and f 2.

Also, the averaging or interpolation function may

involve more than two points.

Note that in the above case we have assumed that for

calculating the absolute difference the maximum

possible difference is S. This can always be accom-

plished by shifting the involved phase values by multi-

ples of 2Srelative to each other.

From the condition above we can derive require-

ments directly for the measurement. Assuming that the

phase response will be usually dominated by a run-time

phase component due to one or several acoustic sources

being located away from the POR, conditions for the

spatial and spectral density of data points can be

computed.^39 With respect to frequency one obtains

(35-36)

where,

' f denotes the frequency resolution,

c the speed of sound.

Given these parameters, is the maximal

distance allowed between the POR and the acoustic

source at the given frequency resolution. With regard to

angle, one finds analogously

(35-37)

where,

Figure 35-29. Critical frequency for magnitude data and
complex data as a function of the distance x =z/2 between
POR and acoustic source, at a measuring distance of 3 m
and a maximal phase error of 45q.

G)^2 Sx

2

O 2 d

= ------------ 1sin–^2 -

argAˆz 0

G)Crit S

4

---=

fInt Aˆ oA

¢²)

1

2

---)i^1

2

+= ---)j

)i–)j S

2

---

• i -j

xcrit c

4 'f

|---------

xcrit

xcrit c

4 fsin '-

|--------------------------