628 Chapter 17
distance with a reference electrical input signal. The
most common standard is dB-SPL at 1 meter with a
1-watt input. Since a loudspeaker’s impedance varies
with frequency, and since a power amplifier is actually a
voltage-controlled voltage source, the 1 watt figure is
usually translated into an rms voltage (e.g., 2.83 Vrms
into 8 := 1 W). Also noteworthy is the fact that the
actual measurement will generally not be accurate if it
is actually carried out at a 1-meter distance, since it will
not be in the loudspeaker’s farfield. Instead, the testing
is done at a greater distance and the results normalized
to the 1-meter reference distance. The discussion that
follows is based on the premise that we are interested in
knowing the acoustic output characteristics of a loud-
speaker with known voltages applied to its input.
17.9.3 Network Transfer Function
The performance characterization of electrical circuits is
a well-developed realm and is employed as the basis for
much data presented in regard to loudspeaker behavior.
The impulse response, H(t), and LaPlace transfer func-
tion, L(H[t]) =F(s), of an electrical circuit are widely
used models for the linear portion of a circuit’s
behavior. A large body of practical mathematics has
been developed to aid in manipulating transfer functions
of circuits, and the subject is covered at the undergrad-
uate level in almost every engineering discipline.
A necessary item in applying these concepts to an
electrical network is a definition of which terminals
constitute the input and which will be considered the
output. Using these definitions, the performance of the
circuit may be modeled and/or measured and the
resulting data used to evaluate the suitability of the
circuit for an intended use.
17.9.4 Loudspeaker Transfer Function
Before developing these concepts further, we should
recognize the importance of the correlation between the
response behavior of a loudspeaker and its audible (i.e.,
subjectively evident) performance. The inevitable limi-
tations in the resolution of measured loudspeaker data
should ideally be determined with the capabilities and
limitations of human hearing in mind. Data that is too
highly resolved will reveal a number of details, or arti-
facts, that are not likely to be audibly significant, while
insufficiently resolved data will tend to smooth over
relatively serious imperfections that may easily be
heard. With respect to frequency resolution, constant
percentage octave (log frequency) resolution would
appear to correlate best with the capabilities of human
hearing. If measurements are taken so as to yield
1/6-octave resolution above 100 Hz, it is unlikely that
greater resolution would reveal additional features that
can be distinguished by human hearing.
The concept of frequency response (more appropri-
ately, amplitude response) is a direct consequence of the
transfer function model. This is the most familiar of the
many possible ways of graphically representing
portions of a transfer function. It is widely assumed that
a loudspeaker may be characterized by one frequency
response, usually measured at a point defined to be on
axis of the loudspeaker. This assumption is incomplete:
a loudspeaker has infinity of transfer functions (or,
interchangeably, impulse responses), one for each point
in 3D space. In the interest of compactness, we could
say equivalently that a loudspeaker has a transfer func-
tion with four independent variables instead of one:
where F(s) is sufficient for electrical circuits, for loud-
speakers the equivalent expression (in Cartesian coordi-
nates) will be F(s,x,y,z).
When we consider loudspeakers, the analogy with
electrical networks is incomplete due to the nature of the
device’s output. Whereas a two-port electrical network
has a single output, a loudspeaker radiates energy into
free space in all directions. If only one listener were
present, and if there were also no reflections in the
acoustic environment, then the loudspeaker’s response
at a single point—the listening location—would be
sufficient to characterize what that listener would hear.
If multiple listeners and reflections are present, it is no
longer sufficient to consider only the single transfer
function: we need much more information.
If we limit our consideration to the far field (i.e.,
distances many times greater than the largest dimension
of the loudspeaker), then the dependence of the transfer
function on distance will be reduced, in most practical
cases, to a characteristic delay of(17-10)where,
r is the distance from the source to the observation
point,
c the phase velocity of sound in air, plus a change in
acoustic pressure that is inversely proportional to
distance from the source due to the inverse square law.
Both of these quantities may be assumed to be
frequency-independent, although there are some
exceptions.W r
c=--