# A Game of Numbers (Understanding Directivity

```A Game of Numbers
(Understanding Directivity Specifications)
José (Joe) Brusi, Brusi Acoustical Consulting
Loudspeaker directivity is expressed in many different ways on specification sheets and
marketing materials provided by loudspeaker manufacturers. Until there is a
benchmark for expressing this data, it will continue to be difficult to compare products
produced by companies who utilize different methods. This article will define the most
commonly used terms and explain the different ways loudspeaker manufacturers’
directivity information is expressed.
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The end result is a set of frequency response
curves for each point of the measurement sphere,
with resolution anywhere from 1/24th octave to 1
octave, and angular intervals anywhere from 1º to
10º. One these sets can be plotted as a "waterfall"
display (Figure 2), where the high resolution data
shows the transition from a flat on-axis frequency
response (0º) to a response dominated by bass
frequencies behind the cabinet (180º).
Level in dB
How directivity is measured
of the sound pressure level (SPL) at all points on a sphere
surrounding the speaker system, in other words the speaker is
at the center of that sphere and the distance between the
speaker and the surface of this sphere would be equal at all
points and should be large when compared to the dimensions of Figure 1. Setup for
the speaker. The set-up for this type of measurement is shown
directivity measurement
in Figure 1. These measurements are made at all frequencies
so we can ascertain the frequency response at any angle. Typically this measurement
is accomplished by placing a microphone at a practical distance (usually around 4
meters or 12 feet) and then rotating the loudspeaker to achieve the different angles.
Most often this rotation happens around one axis, so two sets of measurements must
be made, one for the vertical and one for the horizontal. The most sophisticated
systems can rotate around two axes, so that all points of the sphere can be measured
in the same run, providing not only vertical and horizontal polar plots (drawing an
analogy with the Globe, that would be the equator and two opposite meridians), but all
the polars at oblique angles as well.
On ax
is
angle
off ax
is
Figure 2. Waterfall display showing a
loudspeaker's frequency response
Page 1. Joe Brusi, A game of numbers. Understanding Directivity Specifications
When the SPL at all points of the sphere is plotted in 3D for a
given frequency; we get what is commonly referred to as the
"directivity balloon". The reasons for that name are clear when
we see the accompanying figure. A "squashed balloon", the plot
looks lopsided because it is displaying the effects of a horn that
provides narrower dispersion vertically than horizontally. This
horn was part of a two-way 15” cabinet, the most common
configuration for sound reinforcement, which will be used as an
Figure 3. Directivity
balloon for the 4 kHz
octave bands
In 1994, when working for DAS Audio, I developed the AutoPol
platform for measurement and post-processing of high
resolution directivity information. It was a very advanced system even by today’s
standards, providing 2-degree angular and 24 point-per-octave frequency resolution.
AutoPol extracted comprehensive directivity information, displayed it in graphical as
well as tabular form, converted to a range of file formats for modeling and boasted an
integrated module for in-house high-resolution polar prediction.
Polar Plots
A polar plot is the result of 360º rotation around one axis. The horizontal polar would
be similar to walking around a speaker placed on a tripod or, following the Globe
analogy, taking measurements along the equator. Turning the speaker 90 degrees
would then allow us to measure the vertical polars (which would correspond to the two
meridians directly in front and behind the speaker). Typically, only vertical and
horizontal polar plots are included in specification sheets. A graphical example of these
can be seen in Figure 4a.
Polars are not always as well behaved though. At the crossover region, interference
between bands results in an irregular polar response, which can be clearly seen on the
vertical polar in Figure 4b. Also, if the polar does not point straight, that is an indication
that there is no alignment between adjacent bands. Moreover, constant directivity (CD)
horns often show SPL maxima off-axis (Figure 4c).
Figure 4a. Vertical and
horizontal polars for the 6.3 kHz
1/3-octave band
Page 2. Joe Brusi, A game of numbers. Understanding Directivity Specifications
Figure 4b. Polars at 2 kHz
Figure 4c. Polars at 10 kHz
Isobar Plots
If we join all the points where the level is the same on
the measurement sphere, we end up with an isobar plot.
Figure shows a plot with -3dB, -6dB, and -9dB isobars.
The Coverage Angle
The coverage angle is defined as the angle enclosed by
the -6dB points on a loudspeaker polar plot. The reason
the -6dB point is used is to avoid overlap when a
number of sources are splayed, as, ideally, having the
splay angle between adjacent boxes within an array be Figure 5. Isobar plot for the 4 kHz
octave band
the same as the coverage angle of a single box would
result in seamless coverage between speaker systems (assuming perfect summation).
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beamwidth in degrees
Figure 4a (above) shows graphically how the coverage angle is calculated. In this case
we have about an 80º x 35º (this a common way of expressing the horizontal and
vertical coverage angles for a speaker, by the way) pattern for the 6.3 kHz 1/3rd octave
band polar shown. Vertical and horizontal coverage angles with frequency can be seen
in Figure 6. This 2-way system has a 360º coverage at low frequencies, which
gradually decreases until the crossover frequency, where the horn and driver take over
to achieve fairly constant coverage up to 12 kHz. Both graphs reflect the same set of
data, but the top one uses a linear vertical
400
scale while the bottom one uses a
300
200
logarithmic scale. The latter tends to be
100
the most common, and reveals the
0
performance of the horn a bit better
400
200
visually. Since the coverage angle varies
100
50
with frequency, specification sheets
20
normally provide a value that has been
averaged over a specified frequency
range, typically that of the high frequency
Figure 6. Horizontal and vertical coverage angles
horn. This value often conforms to
with frequency. Linear (top) and logarithmic (bot.)
"nominal values" such as 90º, 60 or 40º.
Our example speaker would most likely
Page 3. Joe Brusi, A game of numbers. Understanding Directivity Specifications
be rated as a 90º X 60º (horizontal x vertical) system.
There is some disagreement among specification writers as to what the 0-dB reference in the
calculation of the coverage angle should be. Some manufacturers take the on-axis level, while
some others use the maximum output level exhibited by the polar. If the polar is nice and smooth
both criteria will yield the same result, but if they are irregular, such as those in Figures 4b and 4c,
resulting coverage angle values may differ significantly.
The Q Factor
This is a mathematical expression of how directive a source is. Larger Q factor values
denote more directional sources. The Q factor is calculated by comparing the on-axis
level with the average level for all the points in the measurement sphere. In practice, Q
is often derived from the horizontal and vertical polar plots. A spherical source (a
source that has the same output level at all angles, close to a subwoofer at low
frequencies) has a Q factor of 1. A hemispherical source, akin to a spherical one that is
placed against a wall (this condition is referred to as 2 pi or half space), has a Q of 2.
As with coverage angle, different evaluation methods are used which can result in
slightly different Q results.
The directivity index (DI)
DI is the same as Q factor, but expressed in a logarithmic fashion as follows:
DI (in dB) = 10*log (Q)
Q
DI in dB
Thus, a spherical source has a Q factor of 1 and a directivity factor of 0 dB, and a
hemispherical (half-sphere) source has a Q factor of 2 and a directivity factor of 3 dB.
Typical directivity indexes for a horn would spread from 10 to 20 dB, corresponding to
Q factors of 10 to 100. The DI for the polars represented in Figure 4 is 12 dB (Q factor
of 16). On-axis sensitivity and DI have a direct correlation. For a compression
driver/horn from our 2-way speaker example, an X dB more directional horn will result
in X dB more sensitivity. Using a subwoofer as an example, when it is placed against a
wall it will increase in sensitivity by 3 dB (DI will change from 0 to 3 dB, Q factor from 1
to 2) as compared to its free-field sensitivity. Figure 7 show the DI and Q with
frequency for our example 2-way system. We can see a rising response from the 15”
woofer, which corresponds to the narrowing of the cone's directivity, as frequency
increases (also referred to as “beaming”). In contrast, the directivity that the horn
exhibits is fairly constant with frequency, commonly referred to as a CD (constant
20
100
directivity) horn. DI (Q) and
15
coverage angles are
10
10
typically plotted for 1/3rd5
octave bands.
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Figure 7. Q and DI with frequency
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Relative off-axis frequency response
Manufacturers may provide off-axis frequency response measurements on their data
sheets. These measurements are generally made at intervals of 10 or 15 degrees offaxis (from the 0º angle, which is the reference measurement).
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off axis dB
Vertical (up and down) and
horizontal relative frequency
responses in 15-degree intervals
(0-15-30-45 degrees) can be seen
on Figure 8 for our example
loudspeaker. When examining
these off-axis frequency
responses, it is important to
realize that they are relative. This
means that the 0 degree
response would be a straight line
going through 0 dB, since it has
become the reference for the
other responses. If we were able
to equalize our speaker to be flat
at 0 degrees, these would be our
frequency responses at the specified
angles.
Figure 8. Relative frequency responses.
Vertical (top) and horizontal (bottom)
0-15-30-45 degrees
If responses cross each other, that reveals polar response irregularities. Our example
system shows this at the crossover region (around 2 kHz, due to interference between
the highs and the lows; this system is not too bad as there was a reasonable amount of
passive filtering, but it can be really nasty for shallow filters) and around 10 KHz (due to
the CD horn). Generally, specification sheets show plots of only one half of the vertical
and one horizontal set of measurements. i.e., left but not right, up but not down. For
multi-way systems, a bit more insight is provided if both up and down off-axis
responses are shown, since asymmetry reveals lack of time alignment. The crossover
region of our example systems shows asymmetry at the crossover region, as expected
from a passive system.
Representations for lots of data
In order to represent all measured data for all frequencies some fairly complex plots are
required. Another isobaric display can be seen in Figure 9 where we have
measurement angle as a function of frequency. In this case, isobars are plotted in 1 dB
increments from 0 (blue) to 6 dB (black). The latter would represent the coverage
angle. The flat portion of the isobars in the 2-12.5 kHz region shows the constant
directivity nature of the horn in this system. Figure 10 is a more colorful and
contemporary image, but showing basically the same of information. In this case, a
single 15” subwoofer is used; the narrowing and lobing with increasing frequency are
Page 5. Joe Brusi, A game of numbers. Understanding Directivity Specifications
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angle in degrees
angle in degrees
shown (a bit like a “line array”, except that a correctly design isophasic waveguide
should not show any side lobes).
Figure 9a. Isobar display from 0 to -6 dB
in 1 dB intervals
Figure 9b. Isobar display from 0 to 10 dB in 1 dB intervals
Modeling
Electro-acoustic modeling software can be used to display directivity information quite
effectively through images that look like weather maps , and visually show how even
coverage is throughout the listening area.
In the past, these programs utilized fairly crude directivity balloons, typically with 10
degree resolution in octave bands. For the last 10 years the AES (Audio Engineering
Society) has been trying to produce a standard for high resolution directivity data to be
used by modeling software, but so far not much has been achieved. The CLF universal
format (with a maximum 5-degree angular and 1/3rd band resolution) has been adopted
by CATT, Ulysses and LARA software. Data resolution on EASE (the de-facto
standard), originally 10-degree and 1-octave band, became 5-degree and 1/3rd octave
on version 3, and version 4 introduced a format whereby an arbitrary resolution can be
used.
Modeling software is used by designers to ensure all areas are covered. It is often
required by customers when spec’ing a system and it also provides graphically
attractive presentation material. The most elaborate programs will be able to show fairly
realistic looking rooms in 3D, which, aside from the visual impact side of things, will
help customers understand the proposed concept, as they will be able to see the
location of the speakers within the room.
The projection of the loudspeaker’s directivity onto a coverage area can be seen for
EASE and CADP2 (a wonderful but defunct piece of software, made by JBL, that did
not make it to the XXI century) in Figures 11 and 12 respectively. A color scale can be
seen of the right hand side of each of the images.
Page 6. Joe Brusi, A game of numbers. Understanding Directivity Specifications
Figure 11. Loudspeaker directivity information as
shown EASE's area projection
Figure 12. Loudspeaker directivity information as